Fuzzy automata and languages: theory and applications

FUZZY AtTOMATA and LANGUAGES Theory and Applications © 2002 by Chapman & Hall/CRC COMPUTATIONAL MATHEMATICS SERIES...

Author: John N. Mordeson | Davender S. Malik

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FUZZY AtTOMATA and

LANGUAGES

Theory and Applications

© 2002 by Chapman & Hall/CRC

COMPUTATIONAL MATHEMATICS SERIES Series Editor Mike J . Atallah

Published Titles Inside the FFT Black Box: Serial and Parallel Fast Fourier Transform Algorithms Eleanor Chu and Alan George

Mathematics of Quantum Computation Ranee K. Brylinski and Goong Chen

Fuzzy Automata and Languages : Theory and Applications John N. Mordeson and Davender s. Malik

Forthcoming Titles Cryptanalysis of Number Theoretic Ciphers Samuel s. Wagstaff


FUZZY AUTOMATA and

LANGUAGES

Theory and Applications John N. Mordeson Davender S . Malik

4ffl

CHAPMAN & HALL/CRC Boca Raton


A CRC Press Company

London

NewYork Washington, D.C .

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Library of Congress Cataloging-in-Publication Data Mordeson, John N. Fuzzy automata and languages : theory and applications / John N . Mordeson and Davender S. Malik. p . cm. - (Computational mathematics series) Includes bibliographical references and index . ISBN 1-58488-225-5 (alk . paper) 1 . Fuzzy automata . 2. Fuzzy languages . I . Malik, D . S . II . Title . III. Series. QA267 .5 .F89 M67 2002 511 .3-dc21

2002017475

This book contains information obtained from authentic and highly regarded sources . Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed . Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale . Specific permission must be obtained in writing from CRC Press LLC for such copying . Direct all inquiries to CRC Press LLC, 2000 N .W. Corporate Blvd., Boca Raton, Florida 33431 . Trademark Notice : Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe .

Visit the CRC Press Web site at wwwcrepress .com © 2002 by Chapman & Hall/CRC No claim to original U .S . Government works International Standard Book Number 1-58488-225-5 Library of Congress Card Number 2002017475 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper


Preface In 1965, L. A. Zadeh introduced the notion of a fuzzy subset of a set as a method for representing uncertainty . His ideas have been applied to a wide range of scientific areas . One such area is automata theory and language theory first introduced by W. G. Wee in [249]. This is the area that is dealt with in this book. Our purpose is to give an up-to-date treatment of fuzzy automata theory and fuzzy language theory when the set of truth values is the closed interval [0, 1] . When the interval is replaced by a lattice or a semiring or some other type of structure, the reader is referred to [247] . There are many ways that ordinary automata and languages have been fuzzified . Consequently, a wide variety of terminology is needed. Since no industry standards have been set as to the terminology used to describe these ways, we decided in most cases to use the terminology of the authors on whose papers the book depends . Some modifications were made when different terminology was used to describe the same concept or when the same terminology was used to describe different concepts . A considerable number of symbols are needed due to the large number of concepts involved . Consequently, we have decided not to include symbols in the symbols list if their use is localized . At the end of each chapter, we provide a few exercises . Some of the exercises present material that is not covered in detail in the book, while others test the readers' understanding of the material . The exercises should be useful if the book is used in a course on fuzzy automata and fuzzy languages . Examples are provided in each chapter to illustrate the concepts developed . The book should be of interest to research mathematicians, engineers, and computer scientists . Chapter 1 provides some basic material needed for an understanding of the book. This material includes results from set theory, fuzzy set theory, abstract algebra, automata, and language theory. Nevertheless, the reader would find a background in these areas useful . Chapter 2 begins the study of max-min machines and examines their behavior . We consider equivalences and homomorphisms of max-min automata in order to determine their reductions . A max-min algebra is devel oped in order to have a structure in which the study of max-min machines can be placed . © 2002 by Chapman & Hall/CRC

Chapter 3 introduces max-product machines and considers their irreducibility and minimality. Max-product grammars and languages are studied with special attention to weak regular max-product grammars and languages . A max-product algebra is developed in order to have a structure in which to carry out the study. This algebra resembles the algebra developed in the previous chapter, but it has major differences. Natural languages lack the precision of formal languages . It is hoped that the introduction of fuzziness into the structure of formal languages will help close the gap between the two. Chapter 4 studies fuzzy context-free grammars and languages . Special attention is given to trees, fuzzy dendrolanguage generating systems, and normal forms. Concerning trees, we are particularly interested in fuzzy tree automata and fuzzy tree transducers . Chapter 5 deals with probabilistic automata and their approximation by nonprobabilistic automata . Various types of probabilistic grammars and automata are studied . These types include programmed and time-variant grammars, weak regular probabilistic grammars, context-free probabilistic grammars, asynchronous probabilistic automata, and probabilistic pushdown automata . Realization of fuzzy languages by probabilistic automata, max-product automata, and max-min automata is examined. Chapter 6 is concerned with algebraic fuzzy automata theory. In this chapter, we are particularly interested in semigroups of fuzzy finite state machines including fuzzy transformation semigroups. We examine the structure of fuzzy finite state machines through products and covers and stress the concepts of submachines, retrievability, separability, and connectedness of fuzzy finite state machines . Chapter 7 presents additional results on fuzzy languages . The concept a of partial fuzzy automaton is introduced in order to study certain types of fuzzy regular languages . We consider fuzzy recognizers including the construction of recognizers and recognizable sets . The chapter concludes with a presentation of the algebraic properties of fuzzy regular languages, adjunctive languages, and dense languages . In Chapter 8, we consider the minimization of fuzzy automata. We examine first the equivalence, reduction, and minimization of finite fuzzy automata by an algebraic approach . Some ideas presented in Chapters 2 and 3 are completed here. We then consider the minimization of a fuzzy finite automaton . The automata here have distinct transition and output functions . The chapter concludes by considering an approach to the study of finite fuzzy automata by using the ability to solve systems of linear equations over a bounded chain . A polynomial time algorithm for solving such systems is given. In Chapter 9, we continue our study of the recognition of fuzzy languages . We study cutpoint languages and recursive languages . We replace the interval [0,1] by a lattice in this chapter . The purpose of this chapter is to give the reader only an introduction to some of the ideas used in fuzzy © 2002 by Chapman & Hall/CRC

language theory when the interval [0,1] is replaced by a lattice. Chapter 10 is devoted to applications. We consider a formulation of fuzzy automata and their application as a model of learning systems . We also apply the concept of fractionally fuzzy grammars to pattern recogni tion. Stability and fault tolerance of fuzzy state automata is also examined. A fuzzy automaton as a clinical monitor is presented . An application to data base theory is also given. The authors are grateful to the editorial and production staff of Chapman Hall/CRC Press, especially Robert Stern. We are indebted to Paul Wang, Azriel Rosenfeld, and Hu Cheng-ming for their support of fuzzy mathematics . We are also appreciative of the support of Dr. Timothy Austin, Dean of Creighton College of Arts and Sciences, Dr. and Mrs . George Haddix, benefactors of our research center, and Lynn Schneiderman of the Creighton Alumni Library. John N. Mordeson Davender S. Malik


Authors John N. Mordeson, Ph.D., is Professor of Mathematics at Creighton University. He received his B .S., M.S., and Ph.D. degrees from Iowa State University. At Creighton he has received the Distinguished Faculty Award and was a recipient of the Burlington Northern Scholar of the Year Award and the College Reasearch Award of the year. He is on the editorial board of several journals including Information Sciences, Fuzzy Sets and Systems, and the Journal of Fuzzy Mathematics. Professor Mordeson has published more than 150 papers, chapters, and lecture notes series . He has authored six books. He is a member of the American Mathematical Society, the board of directors of the Berkley Initiative in Soft Computing, and the board of directors of the Association for Intelligent Machinery. Davender S. Malik, Ph.D., is Professor of Mathematics and Computer Science at Creighton University. He received his B.A. and M.A. from University of Delhi, and Ph.D. from Ohio University specializing in ring theory. At Creighton University, he teaches both mathematics and computer science courses . Professor Malik has authored more than 45 papers and 5 books.


CONTENTS Preface

v

Authors

ix

List of Symbols

xvii

1

Introduction 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Relations . . . . . . . . . . . . . . . . . . . . . . 1.3 Functions . . . . . . . . . . . . . . . . . . . . . . 1.4 Fuzzy Subsets . . . . . . . . . . . . . . . . . . . . 1.5 Semigroups . . . . . . . . . . . . . . . . . . . . . 1.6 Finite-State Machines . . . . . . . . . . . . . . . 1.7 Finite-State Automata . . . . . . . . . . . . . . . 1 .8 Languages and Grammars . . . . . . . . . . . . . 1.9 Nondeterministic Finite-State Automata . . . . . 1.10 Relationships Between Languages and Automata 1.11 Pushdown Automata . . . . . . . . . . . . . . . . 1.12 Exercises . . . . . . . . . . . . . . . . . . . . . .

2

Max-Min Automata 2.1 Max-Min Automata . . . . . . . . . . . . . . . 2.2 General Formulation of Automata . . . . . . . 2.3 Classes of Automata . . . . . . . . . . . . . . . 2.4 Behavior of Max-Min Automata . . . . . . . . 2 .5 Equivalences and Homomorphisms of Max-Min 2.6 Reduction of Max-Min Automata . . . . . . . . 2 .7 Definite Max-Min Automata . . . . . . . . . . 2.8 Reduction of Max-Min Machines . . . . . . . . 2.9 Equivalences . . . . . . . . . . . . . . . . . . . 2.10 Irreducibility and Minimality . . . . . . . . . . 2.11 Nondeterministic and Deterministic Case . . . 2.12 Exercises . . . . . . . . . . . . . . . . . . . . .


. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

1 1 2 6 8 9 13 15 20 24 30 36 42 45 45 45 46 48 51 55 56 58 67 75 86 88

xii 3 Fuzzy Machines, Languages, and Grammars 3.1 Max-Product Machines . . . . . . . . . . . . 3.2 Equivalences . . . . . . . . . . . . . . . . . . 3.3 Irreducibility and Minimality . . . . . . . . . 3.4 Max-Product Grammars and Languages . . . 3.5 Weak Regular Max-Product Grammars . . . 3.6 Weak Regular Max-Product Languages . . . 3.7 Properties of E . . . . . . . . . . . . . . . . . 3.8 Exercises . . . . . . . . . . . . . . . . . . . .

CONTENTS

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

91 91 95 98 102 108 116 120 125

4 Fuzzy Languages and Grammars 127 4.1 Fuzzy Languages . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2 Types of Grammars . . . . . . . . . . . . . . . . . . . . . . 130 4.3 Fuzzy Context-Free Grammars . . . . . . . . . . . . . . . . 131 4.4 Context-Free Max-Product Grammars . . . . . . . . . . . . 137 4 .5 Context-Free Fuzzy Languages . . . . . . . . . . . . . . . . 141 4.6 On the Description of the Fuzzy Meaning of Context-Free Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.7 Trees and Pseudoterms . . . . . . . . . . . . . . . . . . . . . 147 4.8 Fuzzy Dendrolanguage Generating Systems . . . . . . . . . 148 4.9 Normal Form of F-CFDS . . . . . . . . . . . . . . . . . . . 150 4.10 Sets of Derivation Trees of Fuzzy Context-Free Grammars . 154 4.11 Fuzzy Tree Automaton . . . . . . . . . . . . . . . . . . . . . 158 4.12 Fuzzy Tree Transducer . . . . . . . . . . . . . . . . . . . . . 162 4 .13 Fuzzy Meaning of Context-Free Languages . . . . . . . . . . 166 4 .14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5 Probabilistic Automata and Grammars 173 5.1 Probabilistic Automata and Their Approximation . . . . . . 173 5.2 E-Approximating by Nonprobability Devices . . . . . . . . . 177 5.3 E-Approximating by Finite Automata . . . . . . . . . . . . . 180 5.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.5 The PE Relation . . . . . . . . . . . . . . . . . . . . . . . . 183 5.6 Fuzzy Stars Acceptors and Probabilistic Acceptors . . . . . 186 5.7 Characterizations and the RE-Relation . . . . . . . . . . . . 187 5.8 Probabilistic and Weighted Grammars . . . . . . . . . . . . 191 5 .9 Probabilistic and Weighted Grammars of Type 3 . . . . . . 197 5.10 Interrelations with Programmed and Time-Variant Grammars202 5.11 Probabilistic Grammars and Automata . . . . . . . . . . . . 204 5.12 Probabilistic Grammars . . . . . . . . . . . . . . . . . . . . 205 5.13 Weakly Regular Grammars and Asynchronous Automata . 209 5.14 Type-0 Probabilistic Grammars and Probabilistic Turing Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.15 Context-Free Probabilistic Grammars and Pushdown Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 © 2002 by Chapman & Hall/CRC

CONTENTS 5.16 5.17 5.18 5.19

Realization of Fuzzy Languages Properties of Lk, k = 1, 2,3 . . Further Properties of L3 . . . . Exercises . . . . . . . . . . . .

by Various Automata . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

6

Algebraic Fuzzy Automata Theory 6.1 Fuzzy Finite State Machines . . . . . . . . . . . . . . . 6.2 Semigroups of Fuzzy Finite State Machines . . . . . . 6.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . 6.4 Admissible Relations . . . . . . . . . . . . . . . . . . . 6.5 Fuzzy Transformation Semigroups . . . . . . . . . . . 6.6 Products of Fuzzy Finite State Machines . . . . . . . . 6.7 Submachines of a Fuzzy Finite State Machine . . . . . 6.8 Retrievability, Separability, and Connectivity . . . . . 6.9 Decomposition of Fuzzy Finite State Machines . . . . 6.10 Subsystems of Fuzzy Finite State Machines . . . . . . 6 .11 Strong Subsystems . . . . . . . . . . . . . . . . . . . . 6 .12 Cartesian Composition of Fuzzy Finite State Machines 6.13 Cartesian Composition . . . . . . . . . . . . . . . . . . 6.14 Admissible Partitions . . . . . . . . . . . . . . . . . . . 6.15 Coverings of Products of Fuzzy Finite State Machines 6.16 Associative Properties of Products . . . . . . . . . . . 6.17 Covering Properties of Products . . . . . . . . . . . . 6.18 Fuzzy Semiautomaton over a Finite Group . . . . . . . 6.19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

7

More on Fuzzy Languages 7.1 Fuzzy Regular Languages . . . . . . . . . . . . . . . . 7.2 On Fuzzy Recognizers . . . . . . . . . . . . . . . . . . 7.3 Minimal Fuzzy Recognizers . . . . . . . . . . . . . . . 7.4 Fuzzy Recognizers and Recognizable Sets . . . . . . . 7.5 Operations on (Fuzzy) Subsets . . . . . . . . . . . . . 7.6 Construction of Recognizers and Recognizable Sets . . 7.7 Accessible and Coaccessible Recognizers . . . . . . . . 7.8 Complete Fuzzy Machines . . . . . . . . . . . . . . . . 7.9 Fuzzy Languages on a Free Monoid . . . . . . . . . . . 7.10 Algebraic Character and Properties of Fuzzy Regular guages . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Deterministic Acceptors of Regular Fuzzy Languages . 7 .12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . Lan. . . . . . . . .

8

. . . . . . . . . . . . . . . . . . .

222 228 231 234 237 237 237 242 245 247 253 268 272 274 277 283 287 289 296 304 306 308 312 322 325 325 334 347 353 354 358 362 364 366 370 383 388

Minimization of Fuzzy Automata 391 8.1 Equivalence, Reduction, and Minimization of Finite Fuzzy Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 8.2 Equivalence of Fuzzy Automata: An Algebraic Approach 396


xiv

CONTENTS 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8 .11 8 .12

9

Reduction and Minimization of Fuzzy Automata . . . . . . 400 Minimal Fuzzy Finite State Automata . . . . . . . . . . . . 404 Behavior, Reduction, and Minimization of Finite L-Automata410 Matrices over a Bounded Chain . . . . . . . . . . . . . . . . 410 Systems of Linear Equivalences over a Bounded Chain . . . 412 Finite IL.-Automata-Behavior Matrix . . . . . . . . . . . . . 414 e-Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . 416 e-Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . 418 Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . 419 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

L-Fuzzy Automata, Grammars, and Languages 9.1 Fuzzy Recognition of Fuzzy Languages . . . . . . . . . . . . 9.2 Fuzzy Languages . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Fuzzy Recognition by Machines . . . . . . . . . . . . . . . . 9.4 Cutpoint Languages . . . . . . . . . . . . . . . . . . . . . . 9.5 Fuzzy Languages not Fuzzy Recognized by Machines in DT2 9.6 Rational Probabilistic Events . . . . . . . . . . . . . . . . . 9.7 Recursive Fuzzy Languages . . . . . . . . . . . . . . . . . . 9.8 Closure Properties . . . . . . . . . . . . . . . . . . . . . . . 9.9 Fuzzy Grammars and Recursively Enumerable Fuzzy Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Recursively Enumerable L-Subsets . . . . . . . . . . . . . . 9 .11 Various Kinds of Automata with Weights . . . . . . . . . . 9 .12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .

423 423 424 426 430 435 436 438 439 441 442 446 461

10 Applications 463 10.1 A Formulation of Fuzzy Automata and Its Application as a Model of Learning Systems . . . . . . . . . . . . . . . . . . 463 10.2 Formulation of Fuzzy Automata . . . . . . . . . . . . . . . . 463 10.3 Special Cases of Fuzzy Automata . . . . . . . . . . . . . . . 466 10.4 Fuzzy Automata as Models of Learning Systems . . . . . . 468 10.5 Applications and Simulation Results . . . . . . . . . . . . . 471 10.6 Properties of Fuzzy Automata . . . . . . . . . . . . . . . . . 479 10.7 Fractionally Fuzzy Grammars and Pattern Recognition . . . 481 10.8 Fractionally Fuzzy Grammars . . . . . . . . . . . . . . . . . 484 10.9 A Pattern Recognition Experiment . . . . . . . . . . . . . . 490 10.10 General Fuzzy Acceptors for Syntactic Pattern Recognition 494 10 .11 e-Equivalence by Inputs . . . . . . . . . . . . . . . . . . . . 496 10 .12 Fuzzy-State Automata : Their Stability and Fault Tolerance 499 10.13 Relational Description of Automata . . . . . . . . . . . . . 500 10.14 Fuzzy-State Automata . . . . . . . . . . . . . . . . . . . . 504 10.15 Stable and Almost Stable Behavior of Fuzzy-State Automata508 10.16 Fault Tolerance of Fuzzy-State Automata . . . . . . . . . . 511 10.17 Clinical Monitoring with Fuzzy Automata . . . . . . . . . 516 © 2002 by Chapman & Hall/CRC

CONTENTS

xv

10.18 Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . 522 10.19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 References


529

List of Symbols N 7L I[8 I[8+ ~SI

E

C C _D 01 U

n

A\B A A x B

P(X) Dom(R) Im(R) [x] o 8upp(N,) .FP(X) V A S" S+ 1xI

PDA rp

the set of positive integers, p. 1 the set of integers, p. 1 the set of rational numbers, p. 1 the set of real numbers, p. 1 the set of positive real numbers, p. 1 the cardinality of a set S, p . 1 belongs to, p. 1 does not belong to, p. 1 subset, p. 1 proper subset, p. 1 contains properly contains the empty set, p. 1 unions, p . 2, 8, 127 intersection, p. 2, 8, 127 relative complement of B in A, p. 2 relative complement of A in its universe, p. 2 Cartesian cross-product of A and B, p . 2 power set of X, p. 2 domain of a relation R, p. 3 image of a relation R, p. 3 equivalence class of an element x, p. 3 composition of crisp and fuzzy relations, p. 5, 9 c-cut of the fuzzy subset [,, p. 8 support of the fuzzy subset [,, p .8 the fuzzy power set of X, p . 8 complement of the fuzzy subset [,, p. 8 infimum, p. 9 supremum, p. 9 empty string, empty tape, or identity of the semigroup, p. 14 free monoid generated by S with operation, concatenation, p. 14 S"\{A}, p. 14 length of a string, p. 20 p. 37 response function, p. 48 xvii


xviii

List of Symbols

S2" 0* Qi + Qz Al x AZ ti

max-min table, p. 50 max-min pre-table, p. 50 direct sum of max-min tables, p. 50 direct product of max-min automata, p. 50 p. 52, 53, 73, 74, 75, 97, 98, 245, 341, 394, 415 p. 55, 74, 97, 239, 394, 407 max-min binary operation, p. 59 C(X) convex span of X, p. 60 MSLM max-min sequential-like machine, p. 67 (X x Y)* p. 68, 95, 415 p(A) collection of all rows of a matrix A, p . 69, 97 sd state distribution, p. 73 IMSLM initialized max-min sequential-like machine, p. 73 responsefunction, rI p.73, 97 _ p. 74, 97, 240 p . 83, 101 P(M) p(M) p . 83, 101 NSLM nondeterministic sequential-like machine, p. 86 DSLM deterministic sequential-like machine, p. 86 max-product binary operation, p. 91 the convex MP-span of X, p. 92 C(X) MPSM max-product sequential-like machine, p. 95 i .d . initial distribution, p. 96 IMPSM initialized max-product sequential-like machine, p. 96 p. 100, 238, 369 the R~ ° set of all nonnegative real numbers, p. 103 R_ I[8_° ° = R>o U fool, p. 103 AMA p. 110 MA p. 114 L(A, r, >) L(A, r, >) = {x E T* I f(x) > r}, p. 116, 141 L(A, r, >) L(A, r, >) = {x E T* I f(x) > r}, p. 116, 142 L(A, r, _) L(A, r, =) = {x E T* I f(x) = r}, p. 116, 142 the family of all regular languages, p. 120, 144 R M -1 (L) p . 123 Kleene closure, p . 128 context-free max-product grammar, p. 137 CMG CFFL context-free fuzzy language, p. 139 the family of all CFL, p . 144 C G p. 144 F-CFDS p. 148 F-CFDL p. 149 F-CFG p. 154 fa p. 175 fac p. 175 pa p. 176 pac p. 177 APA p. 209 PTM p. 213 PPA p. 218


List of Symbols ffsm E(M)

p . 237 p . 238 p . 239

11-11

E(M) xM SM ~ x >Ml c}, p . 385 p . 394 p . 394 p . 404 p . 406 p . 406 p . 482 . 500 end of proof

Chapter 1

Introduction 1 .1

Sets

In this section, we review some results from set theory. We use the notation N for the set of positive integers, 7L for the set of integers, Q for the set of rational numbers, R for the set of real numbers, and R+ for the set of positive real numbers . We assume that the reader is familiar with the basics of set theory. Nevertheless, we give a brief review of some basics of set theory. We think of a set as a collection of objects . We let ~S1 denote the cardinality of S, i.e., the number of elements of S. If S is a set such that ~S1 < oo, then S is called a finite set ; otherwise S is called an infinite set. Given a set S, we use the notation x E S and x ~ S to mean x is a member of S and x is not a member of S, respectively. A set A is said to be a subset of a set S if every element of A is an element of S. In this case, we write A C_ S and say that A is contained in S. If A C_ S, but A :?~ S, then we write A C S and say that A is properly contained in S or that A is a proper subset of S. The empty set or null set is the set with no elements . We denote the empty set by 0. The empty set is a subset of every set . We also describe sets in the following manner . Given a set S, the notation A = {x

I

x E S, P(x)}

or A= {x E S

I P(x)}

means that A is the set of all elements x of S such that x satisfies the property P. Sets can be combined in several ways. © 2002 by Chapman & Hall/CRC

2

1 . Introduction

Definition 1.1.1 The union of two sets A and B, written AUB, is defined to be the set AUB={x I xEAorxEB} . Definition 1.1.2 The intersection of two sets A and B, written A n B, is defined to be the set An B={x I xEAandxEB} . The union and intersection for any finite number of sets can be defined in a similar manner . That is, suppose that A1, A2 , . . . , An are n sets. The union of Al , A2, . . . , An, denoted by UZ 1AZ or A1 U A2 U . . . U An , is the set of all elements x such that x is an element of some AZ, where 1 B and g : B ~ C and (x, z) E g o f , i .e ., (g o f) (x) =z . Then by the definition of composition of functions, there exists y E B such that f (x) = y and g(y) = z. Now z = g(y) = g(f(x)) . Hence (g o

f ) (x) =

g (f (x)) .

Theorem 1 .3.3 Suppose that f : A ~ B and g : B ~ C. Then the following properties hold: (1) g o f : A B C, i.e., g o f is a function from A into C; (2) If f and g are one-one, then g o f is one-one; (3) If f is onto B and g is onto C, then g o f is onto C. Proof. (1) Let x E A. Since f is a function and x E A, there exists y E B such that f (x) = y. Since g is a function and y E B, there exists z E C such that g(y) = z . Hence (g o f)(x) = g(f (x)) = g(y) = z, i.e., (x, z) E g o f . Thus x E Dom(g o f ) . Consequently, A C_ Dom(g o f ) . However, Dom(g o f ) C A and so Dom(g o f ) = A. We now show that g o f

is well defined. Suppose that (x, z) E g o f, (x l , zl ) E g o f, and x = x l , where x, xl E A and z, zl E C. By the definition of composition of functions, there exist y,yi E B such that f (x) = y, g(y) = z, f(xi) = yl, and g(yi) = zl . Since f is a function and x = xl , we have y = yl . Similarly, since g is a function and y = yl, we have z = zl . Hence g o f is well defined . Thus g o f is a function from A into C. (2) Let x, x' E A. Suppose (g o f) (x) = (g o f ) (x') . Then g (f (x)) _ g(f (x')) . Since g is one-one, f(x) = f (x') . Since f is one-one, x = x'. Hence g o f is one-one . (3) Let z E C. Then there exists y E B such that g(y) = z since g is onto C. Since f is onto B, there exists x E A such that f (x) = y . Thus (g o f)(x) = g(f(x)) = g(y) = z. Hence g o f is onto C. m Theorem 1.3.4 Let f : A ~ B, g : B ~ C, and h : C ~ D. Then ho(go f)=(hog)o f. That is, composition of functions is associative . © 2002 by Chapman & Hall/CRC

8

1 . Introduction

Proof. Clearly, h o (g o f) : A ----> D and (h o g) o f : A ----> D . Let x E A . Then [h o (g o f )] (x) = h((g o f) (x)) = h(g(f (x))) = (hog) (f (x)) _ [(h o g) o f ] (x) . Thus h o (g o f) = (hog)of . m 1 .4

Fuzzy Subsets

The notion of a fuzzy subset of a set was introduced by Zadeh in 1965, [255] . This introduction was an important step in the evolution of the modern concept of uncertainty. Let X be a set and A be a subset of X. The characteristic function of A is the function xA of X into {0,1} defined by XA(x) = 1 if x E A and XA(x) = 0 if x ~ A . We sometimes write X if A is understood. The characteristic function can be used to indicate either members or nonmembers of a subset of X. This notion can be generalized in a way that introduces the notion of a fuzzy subset of X. Definition 1.4.1 A fuzzy subset /t interval [0,1], [255] .

of X is a function of X into the closed

Let g be a fuzzy subset of a set X . For all x E X, g(x) can be thought as the degree of membership of x in /t.We sometimes use the notation of for a fuzzy subset of X, where A is thought of as a fuzzy set and NA NA gives the grade of membership of elements of X in A . At times A may be merely a description of a fuzzy subset ft of X. Definition 1.4.2

Let /t be a fuzzy subset of X . (1) Let c E [0,1] . Define gc = {x E X I g(x) > (2) The support of ft is defined to be the set, Supp(/0 = {x E X I g,(x)

c} .

We call

gc

c-cut.

a

> 0}.

/t c in Definition 1.4 .2 is also called a level set . We let,F'P(X) denote the fuzzy power set of X, i.e., the set of all fuzzy subsets of X. Definition 1.4.3 /t U v as follows :

Let

/t

!,(x)

and v be fuzzy subsets of a set X. Define I,

(/t n v) (x) (/t U v) (x)

=

= =

1 - ft(x),

min{p(x), v(x)}, max{g,(x), v(x)},

for all x E X . T is called the complement of /t, tt n v is called the section of ft and v, and ft U v is called the union of ft and v .


tt n v,

inter-

1.5. Semigroups

9

We some times use A to denote min and infimum and V to denote max and supremum . Using these symbols, (ft n v)(x) = ft(x) A v(x) and (ft U v) (x) = p(x) V v(x) for all x E X. We can extend the notion of union and intersection to a family of fuzzy subsets of X. Let {lti}jE7 be a family of fuzzy subsets of X, where I is an index set . Define niEllaa and UjEIlaa as follows : (niEIN'i)( x ) (Ui EIN'i)( x )

=

=

njEIN'i((x) x),, ViEIN'i(x)-

Thus if I is a finite set, say I = {1, 2. . . . , n}, then niEllai = tti n1t2 n . and UjEIlai = N1 U N2 U . . . U Ian . In this case, we sometimes write (niEIN'i)(x)

=

N'1(x) n w2(x) n . . . n ltn(x)

(UiEIN'i)( x )

=

N'1 (x)

. . nll n

and V

w2 (x)

V .. . V

wn(x),

Definition 1.4.4 Let X, Y, and Z be nonempty sets and let /t be a fuzzy subset of X x Y and v be a fuzzy subset of Y x Z. Define the fuzzy subset ft ovofXxZby (ft o v) (x, z) = V{ft(x,

y) A v(y, z) I y E Y}

b'x E X and b'z E Z.

If /t is a fuzzy subset of X x X, we define [i = / t and yn+ l = /t o yn for all nEN. A detailed study of fuzzy subsets can be found in [51, 108, 110, 114, 115, 266] .

1 .5

Semigroups

In this section, we provide some basic results concerning semigroups that are needed later for our presentation of fuzzy automata and fuzzy languages . Let X be a nonempty set. Then a function from X x X into X is called a binary operation on X. If * is a binary operation on X, then the pair (X, *) is called a mathematical system. If (X, *) is a mathematical system such that b'a, b, c E X, (a* b) *c = a* (b* c), then * is called associative and (X, *) is called a semigroup . Let (X, *) be a mathematical system. If there exists e E X such that b'a E X, a * e = a = e * a, then e is called an identity of (X, *) and (X, *) is said to have an identity. If (X, *) has an identity, then it is easily seen that the identity is unique . A semigroup (X, *) is called a monoid if it has an identity. © 2002 by Chapman & Hall/CRC

10

1 . Introduction

Let (X, *) be a semigroup and - be an equivalence relation on X. Then is called a right (left) congruence relation on X if b'a, b, c E X, a - b implies a * c - b * c (c * a - c * b) . A right and left congruence relation on X is called a congruence relation. The number of congruence classes of -, i.e ., equivalence classes of -, is called the index of - . Let (X, *) be a semigroup and let S be a nonempty subset of X. Then (S, *) is said to be a subsemigroup of (X, *) if (S, *) is a mathematical system . (Here for (S, *), we mean * restricted to S x S.) Let (X, *) be a monoid and let (S, *) be a subsemigroup of (X, *) . If e is the identity of (X, *) and e E S, then (S, *) is called a submonoid of (X, *) . We often write X for (X, *) when the operation * is understood. Let a E X. We define al = a and if an is defined for n E N, we define an+1 = an a. We next state some well-known theorems and definitions . Theorem 1 .5.1 Let X be a semigroup (monoid) . The intersection of any collection of subsemigroups (submonoids) of X is a subsemigroup (submonoid) of X. m

Definition 1.5.2 Let X be a semigroup (monoid) and let S be a subset

of X. Let (S) denote the intersection of all subsemigroups (submonoids) of X that contain S. Then (S) is called the subsemigroup (submonoid) of X generated by S. If (S) = X, then S is called a set of generators for X.

Let (X, ) be a semigroup and let e be an element not in X. Let X' denote X U {e} . Extend * from X x X into X to *'from X' x X' into X' as follows : a*'b=a*bifa,bE X; a*'e=a=e*'a if aEX;e*'e=e . Then (X', *') is a monoid and X is a subsemigroup of X' . Even if X were a monoid, it is not a submonoid of X' since its identity is not e. Definition 1 .5.3 Let (X, *) and (Y, -) be semigroups . A function f from X into Y is called a homomorphism if b'a, b E X, f(a * b) = f(a) - f (b) .

Let f be a homomorphism of X into Y. If f is one-one, then f is called a monomorphism. If f is onto Y, then f is called an epimorphism . If f is both a monomorphism and epimorphism, then f is called an isomorphism and X and Y are said to be isomorphic.

Theorem 1.5.4 Let X, Y, and Z be semigroups . If f is a homomorphism of X into Y and g is a homomorphism of Y into Z, then g o f is a homomorphism of X into Z. 0

Let S be a set. A free semigroup on the set S is a semigroup F together with a function f : S ~ F such that for any semigroup X and every function g : S A X, there exists a unique homomorphism h : F A X such that h o f = g. Theorem 1.5.5 Let F be a free semigroup on a set S together with a function f : S ~ F. Then f is one-one and (f (S)) = F.


1.5. Semigroups

11

Proof. Let a, b E S be such that a :?~ b. Let X be a semigroup containing more than one element and consider a function g : S ~ X such that g(a) :?~ g(b) . Now h o f = g. Since h(f(a)) = g(a) g(b) = h(f (b)), it follows that f (a) :?~ f (b) . Thus f is one-one. We now show that (f (S)) = F. Let (f (S)) = A . Then the function f defines a function g : S ~ A such that i o g = f, where i is the identity homomorphism i : A ~ F. By the definition of a free semigroup, there exists a homomorphism h : F ~ A such that h o f = g. Consider the diagram:

where j is the identity isomorphism and k = i o h. Since j o f = f, k o f = i o h o f = i o g = f, it follows that i o h = k = j from the uniqueness property defining free semigroups . The inclusion homomorphism i must be an epimorphism . Hence A = X and so f(S) generates F. m Theorem 1.5.6 Let (F, f) and (F', f') be free semigroups on the same set S. Then there exists a unique isomorphism j : F ~ F' such that j o f = f' .

Proof. Since (F, f) is a free semigroup on the set S, there exists a homomorphism j : F F' such that j o f = f' . Similarly, there exists a homomorphism k : F' F such that k o f' = f. Let h = k o j and i be the identity isomorphism of F. In the diagram,

it follows that h o f = k o j o f = k o f' = f, i o f = f. From the uniqueness, it follows that k of = h = i. Since i is an isomorphism, it follows that j is a monomorphism. Similarly, we can show that j o k is the identity endomorphism on F' . Thus j is also an epimorphism . Hence j is an isomorphism.


12

1 . Introduction

Theorem 1.5.7 Let S be a set. Then there exists a free semigroup on S. Proof. Let F denote the set of all finite sequences of elements (repetitions allowed) of S. Let x = (al, . . . , a, ), y = (bl, . . . , bn) E F. Define xy = (a l , . . . , am , bl~ . . . , bn).

Clearly, for any x, y, z E F, x(yz) = (xy) z. Under this operation, F is a semigroup . Define f : S ----> F as follows : For all a E S, define f(a) = (a), the sequence consisting of the element a. We now show that (F, f) is a free semigroup on S. Let g : S ----> X be an arbitrary function from S into a semigroup X. Define h : F ----> X by h(a l , . . . , am) = g(al ) . . . g(am) for all (a, . , am ) E F. Clearly, h is a homomorphism. Let a E S. Then (h o f) (a) = h(f (a)) _ h((a)) = g(a) . Hence h o f = g. Let k : F ----> X be a homomorphism such that k o f = g . We show that h = k. Let (a,, . an,) E F. Then k(al, . . . , an,) = k((al) . . . (am)) = k((al)) . . . k((a m )) = k(f (al)) . . . k(f (am)) = g(al) . . . g(am) = h(al, . . . , am) . Thus h = k. Therefore, (F, f) is a free semigroup on S.

Theorem 1.5 .8 Let X be a semigroup and let S be a set of generators of X. Then every element of X can be written as the product of a finite sequence of elements in S.

Proof. Let (F, f) be the free semigroup on S as constructed in the proof of Theorem 1.5 .7. Then by the definition of a free semigroup, there exists a homomorphism h : F ----> X such that h o f = g, where g is the inclusion map g : S ----> X. We now prove that h is an epimorphism. Consider the image h(F) . Clearly, h(F) is a subsemigroup of X. Since S = g(S) = (h o f)(S) = h(f (S)) C_ h(F) and S generates X, it follows that h(F) = X. Thus h is an epimorphism. Let x E X. Then there exists (a l , . . . , am) E F such that h(al , . . . , am ) _

x. Now x = h(a l , . . . , am) = h((al) . . . (an,)) = h((al)) . . . h((am)) = h(f (a l)) . . . h(f (a ,)) = g(al) . . . g(am) = al . . . am , where al . . . . , am E S.

Let S be a set . Then S determines a unique free semigroup (F, f) . Since

f : S ~ F is injective, we may identify S with its image f(S) in F. We can then consider S to be a subset of F such that S generates F. Also, any function g : S ~ X, where X is a semigroup, extends to a unique homomorphism h : F ~ X. We call F the free semigroup generated by

S.

Consider the monoid F* = F U {e}, where F is the free semigroup generated by S. Then every function g : S ~ X, where X is a monoid, extends to a unique proper homomorphism h : F* ~ X. This monoid F* is called the free monoid generated by the set S. © 2002 by Chapman & Hall/CRC

1.6. Finite-State Machines 1 .6

13

Finite-State Machines

The theory of machines has had a major impact on the development of computer systems and their associated languages and software . It has also found applications in such areas of science as biology, biochemistry, pyschology, and others . In Sections 1.6-1.10, we review some basic results of automata and language theory. We are indebted to [103]. Definition 1.6.1 A six-tuple M = (Q, X, Y,

f , g, s) is called a finite-state machine if Q, X, and Y are finite nonempty sets, f : Q x X ----> Q, g QXX----> Y, and sEQ .

The elements of Q are called states . The elements of X and Y are called input and output symbols, respectively. The functions f and g are called the state transition and output functions, respectively. The state s is called the initial state . Example 1.6.2 Let Q = {qo, q, 1, X = {a, b}, and Y = {0,1}. Define the functions f : Q x X ----> Q and g : Q x X ----> Y as described in Table 1.1 .

Then M = (Q, X, Y, f, g, qo) is a finite-state machine. The interpretation of Table 1.1 is as follows: f (qo, a) f (qo, b) f (qi, a)

f (qi, b)

= = = =

qo qi qi qi

g(qo, a) g(qo, b) g (qi , a) g (qi , b)

= = = =

0 1 1

0

The next-state and output functions can also be defined by a transition diagram. Example 1.6.3 We draw the transition diagram for the finite-state machine of Example 1 .6 .2 . See Figure 1 .1 . The transition diagram is known to be what is called a digraph in graph theory . The vertices are the states . The initial state is indicated by an arrow as shown. If the finite-state machine is in state q and inputting x causes output y and a move to state q', a directed edge is drawn from the vertex q to the vertex q' and labelled xly. The transition diagram of Figure 1 .1 is


14

1 . Introduction

obtained.

b/0 Figure 1.1 If S is a set such that 1 aS, S ---+ bS, S i b} . Then G = (N, T, P, s) is a grammar.

Given a grammar G, a language L(G) can be constructed from G by using the productions to derive the strings that make up L(G). We begin with the starting symbol and then repeatedly use productions until a string of terminal symbols is obtained. The language L(G) is the set of all such strings of terminals . Definition 1 .8.5 Let G = (N, T, P, s) be a grammar. If z ---+ w is a production and xzy E (NUT)*, then xwy is said to be directly derivable from xzy. In this case, we write xzy ==~> xwy. If z2 E (NUT)* for i = 1 . . . . , n, and z2+1 is directly derivable from for = 1 . . . . , n - 1, we say that zn is derivable from zl and write zl

z2

zn.

We call zl

z2

~> zn

the derivation of zn (from zl) . By convention, any element of (N U T)* is derivable from itself.

The language generated by G, written L(G), consists of all strings over T derivable from s . Example 1.8.6 Let G be the grammar of Example 1 .8.4 . The string abSbb is directly derivable from aSbb, written

aSbb ==~> abSbb, where only the production S ---+ bS is used. The string bbab is derivable from s, written s ===> bbab . The derivation is s ==~> bs ==~> bbs ==~> bbaS ==~> bbab .


22

1 . Introduction

The only derivations from s are given as follows: s

==~>

bs

bnab--1 S bnabm

n > 0,

m > 1.

Hence L(G) consists of the strings over {a, b} containing precisely one a that end with at least one b.

Grammars are classified according to the types of productions that define the grammars. Definition 1.8.7 Let G be a grammar.

(1) If every production is of the form zAw ---+ zvw,

where z, w E (NUT)*, v E (N U T)*\{A},

A E N,

then G is called a context-sensitive (or type 1) grammar. (2) If every production is of the form A--+v, where A E N, v E (NUT)*,

(1 .3)

then G is called a context-free (or type 2) grammar. (3) If every production is of the form A~x or A~xB or A~A where A, B,EN,

xET,

then G is called a regular (or type 3) grammar.

The definition of a context-sensitive grammar comes from the normal form of a more general definition of such grammars . According to (1 .2), in a context-sensitive grammar, A may be replaced by v if A is in the context of x and y. In a context-free grammar, (1.2) states that A may be replaced by v anytime . A regular grammar has especially simple substitution rules : A nonterminal symbol is replaced by a terminal symbol, by a terminal symbol followed by a nonterminal symbol, or by the null string . It is important to note that a regular grammar is a context-free grammar and that a context-free grammar with no productions of the form A --+ A is a context-sensitive grammar . © 2002 by Chapman & Hall/CRC

1.8 . Languages and Grammars

23

Some definitions allow x to be replaced by a string of terminals in Definition 1 .8 .7(3) . However, it can be shown that the two definitions produce the same languages . Let G be a context-free grammar . A derivation wo ==> wl ... w is called a leftmost derivation if for i = 0, 1, . . . , n, w2 = xAy, w2+1 = xzy, and A ~ z is a production, where x E T*, A E N, z E L(G), and y E (N U T)* . G is called ambiguous if it generates an element of L(G) by two or more distinct left derivations . Example 1.8.8 Consider the grammar G defined as follows: T = {a, b, c},

N = {s, A, B, C, D, E}

with productions s ~ aAB, B Dc, DE DC,

A ~ aAC, CD , CE,

s aB, D b, Cc ~ Dcc,

A ~ aC, CE , DE,

and starting symbols. Then G is context-sensitive. For example, the production CE ~ DE (ACE ~ ADE) allows C to be replaced by D if C is followed by E and the production Cc ~ Dcc (ACc ~ ADcc) allows C to be replaced by Dc if C is followed by c. DC can be derived from CD since CD ==~> CE==~> DE==~> DC. The string 0 b3 C3 is in L(G), since we have s

aAB aaaDCCc aaaDDDccc

aaACB aaaDCDcc aaabbbccc.

aaaCCDc aaaDDCcc

It follows that L(G) = fanbnCn I n = 1, 2, . . . } .

Definition 1 .8.9 A language L is context sensitive (respectively, contextfree, regular) if there is a context-sensitive (respectively, context-free, regular) grammar G with L = L(G) . Example 1.8.10 By Example 1 .8 .8, the language L(G) =

fanbnCn

In= 1,2. . . .

is context-sensitive. In [96,p.127], it is shown that there is no context-free grammar G with L = L(G) . Thus L is not context-free .


24

1 . Introduction

Example 1.8.11 Consider the grammar G defined as follows T = {a, b},

N = {s},

with productions s ~ asb,

s ~ ab

and starting symbols . Then G is context-free . The only derivations of s are s

==~>

asb an-1 sbn-1 an-1abbn-1 = anbn .

Hence L(G) consists of the strings over {a, b} of the form anbn, n =

1,2 . . . . This language is context-free . In Section 1 .10, we show that L(G) is not regular.

It follows from Examples 1.8.10 and 1.8.11 that the set of context-free languages that do not contain the null string is a proper subset of the set of context-sensitive languages and that the set of regular languages is a proper subset of the set of context-free languages . It also follows that there are languages that are not context-sensitive . Example 1.8.12 The grammar G defined in Example 1.8 .4 is regular. Hence the language

L(G) = {bnab' I n = 0,1. . . . ; m = 1, 2 . . . . it generates is regular.

Definition 1 .8.13 Grammars G and G' are called equivalent if L(G) = L(G') . 1 .9

Nondeterministic Finite-State Automata

In this section and Section 1.10, we show that regular grammars and finitestate automata are essentially the same in that either is a specification of a regular language. We next illustrate how a finite-state automaton can be converted to a regular grammar . Example 1.9.1 We show how to write the regular grammar given by the finite-state automaton of Figure 1.8. Let the terminal symbols be the input symbols a, b. Let the nonterminal symbols be the states E and O. The initial state E becomes the starting


1.9 . Nondeterministic Finite-State Automata

25

symbol. The productions correspond to the directed edges. If there is an edge labeled x from S to R, we create the production

Hence we obtain the productions E ----> bE,

E ----> aO, O ----> aE,

O - bO .

(1 .4)

If S is an accepting state, we include the production S----> A. In this example, we obtain the additional production O ----> A.

(1 .5)

Then the grammar G = (N, T, P, E), where N = {O, E}, T = {a, b}, and P consists of the productions (1 .4) and (1 .5), generates the language L(G) . L(G) is the same as the set of strings accepted by the finite-state automaton of Figure 1.8 .

Theorem 1 .9.2 Let A be a finite-state automaton given by a transition

diagram. Let s be the initial state. Let T denote the set of input symbols and let N denote the set of states . Define productions S~xR if there is an edge labeled x from S to R, and S~A if S is an accepting state. Let G be the grammar G= (N, T, P, s) . Then G is regular and the set of strings accepted by A is equal to L(G). Clearly, G is regular . We first show that Ac(A) C_ L(G) . Let Proof x E Ac(A) . If x is the empty string, then s is an accepting state . In this case, G contains the production s ----> A.

The derivation s ==~> A

yields x E L(G). © 2002 by Chapman & Hall/CRC

(1 .6)

26

1 . Introduction

Suppose x E Ac(A) and x is not the empty string. Then x = xl . . . xn, where x2 E T, i = 1, 2, . . . , n. Since x is accepted by A, there is a path (s, St , . . . , Sn), where Sn is an accepting state, with edges successively labeled xl, . . . , xn . It follows that G contains the productions SZ_1

~

xiS2

for i = 2, . . . , n.

Since Sn is an accepting state, G also contains the production Sn , A. From the derivation s

xl . . . xnSn xl . . . xn,

we have that x E L (G) . Thus Ac(A) C_ L(G) . Suppose that x E L(G) . If x is the empty string, x must result from the derivation (1 .6) since a derivation that starts with any other production would yield a nonempty string. Thus the production s ~ A is in the grammar . Therefore, s is an accepting state in A. It follows that x E Ac(A) . Now suppose x E L(G) and x is not the empty string. Then x = xl . . . xn, where x2 E T, i = 1, 2, . . . , n. It follows that there is a derivation of the form (1.7) . If, in the transition diagram, we start at s and have the path (s, Si , . . . , Sn), we can generate the string x. The last production used in (10.4.4) is Sn ~ A. Thus the last state reached is an accepting state . Therefore, x E Ac(A) and so L(G) C_ Ac(A) . Hence Ac(A) = L(G) . We now consider the reverse situation, i.e., given a regular grammar G, we want to construct a finite-state automaton A such that L(G) is precisely the set of strings accepted by A. The procedure of Theorem 1 .9 .2 cannot simply be reversed as the next example shows. Example 1.9.3 Consider the regular grammar defined as follows: T = {a, b},

N = {s, q}

with productions s-bs, s-aq, q-bq, q---->b and starting symbols. Let the nonterminal symbols be the states with s as the initial state. For each production of the form


1.9 . Nondeterministic Finite-State Automata

27

we draw an edge from state S to state R and label it x. The productions sibs,

s~aq,

q~bq

give the graph shown in Figure 1.10.

Figure 1.10 The production q ~ b is equivalent to the two productions q~bF,

FAA,

where F is an additional nonterminal symbol . The productions s i bs,

s ~ aq,

q - bq,

q - bF

give the graph shown in Figure 1.11. From the production FAA, it follows that F should be an accepting state.

Figure 1.11

However, the graph of Figure 1.11 is not a finite-state automaton for several reasons . Vertex C has no outgoing edge labeled a and vertex F has no outgoing edges at all. Also, vertex C has two outgoing edges labeled b. A diagram like that of Figure 1 .11 yields a different kind of automaton called a nondeterministic finite-state automaton. The word "nondeterministic" is used since if in Figure 1.11 the automaton is in state C and b is input, a choice of next states exists, i.e., the automaton either remains in state C or goes to state F. Definition 1 .9.4 A nondeterministic finite-state automaton A is a 5-tuple A = (Q, X, f, A, s), where (1) Q is a finite set of states, (2) X is a finite set of input symbols, (3) f is a next-state function from Q x X into P(Q), (4) A is a subset of Q, the accepting states, (5) s E Q is the initial state.


28

1 . Introduction

The main difference between a nondeterministic finite-state automaton and a finite-state automaton is that in a finite-state automaton the nextstate function maps a state, input pair to a uniquely defined state, while in a nondeterministic finite-state automaton the next-state function maps a state, input pair to a set of states. Example 1.9.5 1 .11, we have

For the nondeterministic finite-state automaton of Figure Q = {s, q, F},

X = {a, b},

. A={F}

The initial state is s and the next-state function f is given by Q\X s q F

f

a {q}

{s}

Ql

Ql

{q, F}

0

The transition diagram of a nondeterminate finite-state automaton is drawn similarly to that of a finite-state automaton. An edge from state q to each state in the set f (q, x) is drawn and each is labeled x. Example 1.9.6 as follows:

Consider the nondeterministic finite-state automaton given Q = {s, q, p}, X = {a, b},

A = {q, p}

with initial state s and next-state function Q\X

8

q p

-7{s,

f

q}

{q, p}

Its transition diagram is shown in Figure 1 .12 . It is the transition diagram of the nondeterministic automaton of this example.

Figure 1 .1 2


1.9. Nondeterministic Finite-State Automata

29

A string x is accepted by a nondeterministic finite-state automaton A if there is some path representing x in the transition diagram of A beginning at the initial state and ending in an accepting state . Definition 1.9.7 Let A = (Q, X, f, A, q) be a nondeterministic finite-state automaton. The empty string is accepted by A if and only if s E A. If x = xl . . . xn is a nonempty string over X and there exist states qo, . . . , qn such that (1) qo = q; n; (2) qZ E f(qZ-i, x2) for (3)gn EA ; then x is said to be accepted by A. Let Ac(A) denote the set of strings accepted by A . If A and A' are nondeterministic finite-state automata and Ac(A) _ Ac(A'), then A and A' are said to be equivalent. If x = xl . . . xn is a string over X and there exists states qo, . . . , qn satisfying conditions (1) and (2), then the path (qo, . . . , qn) is called a path representing x in A.

Example 1 .9.8 The string x = bbabb is accepted by the nondeterministic finite-state automaton of Figure 1.11. This follows since the path (s, s, s, q, q, F), which ends at an accepting state, represents x. The path P = (s, s, s, q, q, q) also represents x, but P does not end at an accepting state. However, the string x is still accepted since there is at least one path representing x that ends at an accepting state. A stringy is not accepted if no path represents y or every path representing y ends at a nonaccepting state.

Example 1.9.9 The string x = aabaabbb is accepted by the nondeterministic finite-state automaton of Figure 1 .12. accepted .

The string x = abba is not

Theorem 1.9.10 Let G = (N, T, P, s) be a regular grammar. Let X=T, Q = NU {F}, where F ~ NUT, f(q,x)={qI q-xgEP}U{FI q----> xEP}, and A={F}U{qI q~AEP} . Then the nondeterministic finite-state automaton A = (Q, X, f, A, s) accepts precisely the strings L(G) . m

In the next section we show that given a nondeterministic finite-state automaton A, there is a finite-state automaton that is equivalent to A. © 2002 by Chapman & Hall/CRC

30

1 . Introduction

1 .10

Relationships Between Languages and Automata

In Section 1 .9, we showed that if A is a finite-state automaton, then there is a regular grammar G such that L(G) = Ac(A) . A partial converse is given by Theorem 1.9.10 where it is shown that if G is a regular grammar, then there is a nondeterministic finite-state automaton A such that L(G) = Ac(A) . In this section, we show that if G is a regular grammar, then there is a finite-state automaton A such that L(G) = Ac(A) . This result follows from Theorem 1 .9 .10 once it has been established that any nondeterministic finite-state automaton can be converted to an equivalent finite-state automaton. The method is illustrated by the following example .

Example 1.10 .1 We construct a finite-state automaton equivalent to the nondeterministic finite-state automaton of Figure 1.11 .

The set of input symbols is the same. The states consist of all subsets

of the original set Q = {s, q, F} of states . The initial state is {s} . The accepting states are all subsets of Q that contain an accepting state of the original nondeterministic finite-state automaton, namely

{F}, {s, F}, {q, F}, {s, q, F} .

Let X, Y C_ Q. An edge is drawn from X to Y and labeled x if X = 01 = Y or if

UQex f (q, x) = Y.


1 .10. Relationships Between Languages and Automata

31

Figure 1 .13 The finite-state automaton of Figure 1 .13 is obtained. The states {s, F}, {s, q},

{s, q, F}, {F}

can never be reached and are thus deleted. This yields the simplified, equivalent finite-state automaton of Figure 1 .14 .

b Figure 1 .14 Example 1 .10 .2

The finite-state automaton equivalent to the nondeter-


32

1 . Introduction

ministic finite-state automaton of Example 1.9 .6 is given in Figure 1.15.

Figure 1.15

The following theorem justifies the method of Examples 1 .10 .1 and 1.10.2. Theorem 1.10 .3 Let A = (Q, X, f, A, s) be a nondeterministic finite-state automaton. Let

(1) Q, = P(Q), (2) X' = X, , (3) s = {s},

(4)A'={ScQISnA :?~0},

A.

gESf(q, x) if S 0 . Then the finite-state automaton A' _ (Q', X', f', A', s') is equivalent to

Proof. Suppose that the string x = xl . . . x n is accepted by A. Then there exist states qo, . . . , qn E Q such that qo = q; qZ E f (qZ-1, x2) qn E A .

for i = 1, . . . , n;

Set Yo = {qo} and YZ

= f'(YZ-1, x2)

for i = 1. . . . , n.

Since Yi = f'(YO, x1) = f'({qo}, x1) = f(qo, x1), © 2002 by Chapman & Hall/CRC

1.10. Relationships Between Languages and Automata we have that

q1

33

E Y1 . Now

q2

E

f(ql, x2)

C Uscyl f (S, x2)

= P(Yl, x2) =

Y2

q3

E

f(q2, x3)

C

= P( Y2, x3) =

Y3 .

Also, USEY2 f (S, x3)

We see that the argument may be formalized, using induction, to show that qn E Yn . Since qn is an accepting state in A, Yn is an accepting state in A'. Thus in A', we have

= P(YO,xl)

f'(q',xl)

P(Yl, x2)

=

Y2

=

Y1

Consequently, x is accepted by A~. Now suppose that the string x = x1 . . . x n is accepted by A' . Then there exist subsets Yo, . . . , Yn of Q such that f

and there exists a state Since qn

there exists

qn- 1

qn-1

= =

Yo (Ya-1, xa)

E Yn

qn

E Yn n

{s}, for i = 1 . . . . ,

n,

,A.

= f (Yn-1, xn)

E Yn-1 such that

E Yn-1

=

s, Yi

qn

= E

USEY,-1 f(S, xn),

f(qn-1,

x n ). Similarly, since

i = f (Yn -2, xn-1) = USEY-2f(S, xn-1),

there exists qn-2 E Yn-2 such that this manner, we obtain qZ

qn-1

E

f(qn-2, xn-1).

for i = 0, . . . ,

E YZ

Continuing in

n,

such that qZ

E

f (qZ-1,

xi)

for i = 1. . . . , n .

In particular, qo

E Yo = {s} .

Thus qo = s, the initial state in A. Since qn is an accepting state in A, the string x is accepted by A. We can now state the following result. © 2002 by Chapman & Hall/CRC

34

1 . Introduction

Theorem 1.10 .4 A language L is regular if and only if there exists a finitestate automaton that accepts precisely the strings in L.

Proof. This theorem follows from Theorems 1 .9 .2,1 .9 .10, and 1.10 .3. Example 1.10 .5 In the example, we determine a finite-state automaton A that accepts precisely the strings generated by the regular grammar G having productions s _~ bs, s - aq, q - bq, q - b. The starting symbol is s, the set of terminal symbols is {a, b}, and the set of nonterminal symbols is {s, q} . The nondeterministic finite-state automaton A' that accepts L(G) is shown in Figure 1 .11 . A finite-state automaton equivalent to A' is shown in Figure 1 .13 and an equivalent simplified finite-state automaton A is shown in Figure 1 .14 . The finite-state automaton A accepts precisely the strings generated by G.

We close this section by giving some applications of the methods and theory we have developed . Example 1.10 .6 In this example, we show that the language L={a'b'In=1,2, . . .} is not regular. Suppose that L is regular. Then there exists a finite-state automaton A such that Ac(A) = L . Assume that A has k states. The string x = akbk is accepted by A . Let P be the path that represents x. Since there are k states, some state q is revisited on the part of the path representing ak . Thus there is a cycle C, all of whose edges are labeled a, that contains q. We change the path P to obtain a path P' as follows. When we arrive at q in P, we traverse C. After returning to q on C, we continue on P to the end. If the length of C is j, the path P' represents the string x' = aj+kbk . Since P and P' end at the same state q' and q' is an accepting state, x' is accepted by A. This is impossible since x' is not of the form anbn . Hence L is not regular.

Of course L in Example 1 .10 .6 can be shown not to be regular by the pumping lemma for regular languages . The statement of this lemma appears in the Exercises . We also consider the pumping lemma for context-free languages in the Exercises . For more details, the reader is referred to [96] .


1 .10. Relationships Between Languages and Automata

35

Example 1 .10 .7 Let L be the set of strings accepted by the finite-state automaton A of Figure 1 .16 .

Figure 1 .16 We construct a finite-state automaton that accepts the set of strings LR = ~ xn . . . x l I x l . . . x n E L} . We convert A to a finite-state automaton that accepts LR . The string x = xl . . . xn is accepted by A if there is a path P in A representing x that starts at ql and ends at q3 . If we start at q3 and trace P in reverse, we end at ql and process the edges in order xn , . . . , x 1 . Thus it suffices to reverse all arrows in Figure 1 .16 and make q3 the starting state and ql the accepting state (see Figure 1 .17 . The result is a nondeterministic finite-state automaton that accepts LR .

Figure 1 .17 After finding an equivalent finite-state automaton and eliminating the unreachable states, the equivalent finite-state automaton of Figure 1 .18 is


36

1 . Introduction

obtained.

a Figure 1.18

Let L be the set of strings accepted by a finite-state automaton A with more than one final state . A procedure to construct LR is as follows : First construct a nondeterministic finite-state automaton equivalent to A with one accepting state . Then use the method of Example 1 .10 .7. Examples can be found in [103] . A finite state machine remembers which state it is in. In this sense, we say that it has internal memory. If external memory is allowed on which the machine can read and write, more powerful machines can be defined . By allowing the machine to scan the input string in either direction and by allowing the machine to alter the input string, other enhancements can be obtained . One can then characterize the classes of machines that accept context-free languages, context-sensitive languages, and languages generated by phase-structure grammars .

1 .11

Pushdown Automata

The most important class of automata between finite-state machines and Turing machines is the class of pushdown automata. Their operation is related to many computing processes, especially the analysis and translation of artificial languages . A pushdown automata acceptor comprises a finite-state control, a semiinfinite input tape, and a semi-infinite storage tape. It is not permitted to move its input head to the left. Hence it must examine the symbols of its input tape strictly in the order in which they are written on the tape. The machine starts with the storage tape entirely blank. The symbol # is inscribed in every square. It prints a symbol on the tape each time it moves the storage head to the right . The machine reads a symbol from the storage tape each time it moves the storage head to the left. Information written to the right of the head cannot be retrieved since it is overwritten when the head again moves right. © 2002 by Chapman & Hall/CRC

1.11 . Pushdown Automata

37

The storage tape string may be arbitrarily long. Since it can affect the behavior of the automaton, a pushdown acceptor has unbounded memory. However, the information most recently written into the memory must be the first to be retrieved. This form of limited-access storage mechanism is called a pushdown stack since it implements a "last-in, firstout" retrieval rule. This restricted form of unbounded memory results in a language-defining ability intermediate between that of finite-state machines and Turing machines. Definition 1.11 .1 A pushdown acceptor (PDA) is a six-tuple = (Q, X, U, P, I, F), where Q is a finite set of control states, X is a finite input alphabet, U is a finite stack alphabet, P is the program of M, I C_ Q is a set of initial states, F C_ Q is a set of final or accepting states . The program P is a finite sequence of instructions, each taking one of the following forms: q] scan (a, q'), q] write (u, q'), q] read (u, q'), where q, q' E Q, a E X, and u E U. In each case, q is the label of the instruction and q' is the successor state. Each state in Q labels at most one type of instruction, read, write, or scan. If q E Q, x E X*, and y E U*, then (q, x, y) is called a configuration M. The string y is called the stack and the symbol under the stack head of the top stack symbol.

A configuration (q, x, y) completely describes the total state of a pushdown acceptor at some point in its analysis of an input tape. The control unit is in state q; the prefix x of the entire input string w has been scanned, and the input head is positioned at the last symbol of x; the stack y is the content of the storage tape, and the stack head is positioned at the last symbol of y. The next definition specifies how the execution of instructions by a pushdown acceptor takes it from one configuration to another . The scan instructions read successive symbols from the input tape, write instructions load symbols into the stack, and read instructions retrieve symbols from the stack . © 2002 by Chapman & Hall/CRC

38

1 . Introduction

Definition 1.11 .2

Let M be a pushdown acceptor with program P. Suppose that the string w E X* is written on the input tape . Instructions of M's program are applicable in configurations according to the following rules : (1) An instruction q] scan (a, q') applies in any configuration (q, x, y) if xa is a prefix of w . In executing this instruction, M moves its input head one square to the right, observes the symbol a inscribed therein, and enters state q' . A scan move is represented by the notation (q, x, y) -S (q, xa,

y) .

(2) An instruction q] write (u, q') applies in any configuration (q, x, y) . In executing this instruction, M moves the stack head one square to the right, prints the symbol u therein, and goes to state q' . A write move is represented by the notation

(q, x, y)

-W (q, x, yu) .

(3) An instruction q] read (u, q') applies in any configuration (q, x, y) in which y = y'u . In executing this instruction, M observes the symbol u under the stack head, moves the stack head one square to the left, and goes to state q' . A read move is represented by the notation (q, x, y'u) R (q, x, q) . A move sequence, (q0, x0, yo) - (qi, xi, yi) - . . . - (qk, xk, yk), where each move is a scan, read, or write move, is sometimes shortened to (q0, x0, yo) ==> (qk, xk, yk)

The operation of a pushdown acceptor begins with the control in an initial state and the heads positioned at the initial sharps of their tapes . The machine passes through a sequence of configurations . Each configuration results from the execution of an instruction applicable in the preceding configuration . Operation continues until a configuration is reached in which there is no applicable instruction . If at any point all symbols of some input string x have been scanned, the stack is empty, and the control unit is in a final state, then the string x is accepted by M . Definition 1.11 .3

An initial configuration of a pushdown acceptor M is any configuration of the form (q, A, A), where q is an initial state of M . A final configuration of M is any configuration of the form (q', x, A), where q' is a final state of M and x is a prefix of the string written on M's input tape . The string x is accepted by M if M has a move sequence (q, A, A) ==> (q~, x, A), q E I, q' E F. The

language recognized


by M is the set

of

accepted strings .

1 .11 . Pushdown Automata

39

Example 1.11 .4 Figure 1 .19 shows the program and the state diagram of a pushdown acceptor M, , with alphabets X = {a, b, c} and U = {a, b} .

Program of M,,. 11

21 31 41 51 61

scan write write scan read read

(a,2) (b,3) (c,4) (a,l) (b,l) (a,5) (b,6) (a,4) (b,4)

State Diagram of Load Stack

Mc. Compare

Figure 1 .19 Example of a pushdown acceptor . Instructions having the same label are sometimes combined. For example, the notion 1] scan (a, 2) (b, 3) (c, 4) is used in place of 1] scan (a, 2) 1] scan (b, 3) 1] scan (c, 4) . Unless we specify otherwise, state 1 is the initial state. The nodes of the state diagram represent states of the control unit, and each node is inscribed with S, R, or W according to the type of instruction labeled by the state. The initial states and the accepting states are identified in the same manner as for finite-state automata. At most one instruction is applicable to any configuration of M, , . Hence the behavior of the machine is uniquely determined by its input string w .


40

1 . Introduction

The machine begins operation in state 1 and passes through the following two stages: in the load stack stage it copies into the stack the portion of w up to the symbol c; in the compare stage it matches the stack symbols against the remaining symbols of the input. If the portion of w following the letter c is exactly the reverse of the string loaded into the stack, then M will empty the storage tape immediately after scanning the final letter of w, leaving M,, in the accepting configuration (4, w, A) . Hence M, , accepts each string w = XCXR, where x E {a, b}* . For example, the move sequence by which M, , accepts w = abcba is as follows:

s s s s s

(2, a, A) W _~ (1, a, a)

(3, ab, a) _ W~ (1, ab, ab) (4, abc, ab)

(6, abcb, ab)

(5, abcba, a)

R ~ R >

(4, abcb, a) (4, abcba, A)

[accept.

If a letter scanned by M, , in state 4 does not match the last letter of the stack, or if symbols remain in the stack after all of w has been scanned, M, , will stop with a nonempty stack and reject the input. The rejection of w = abcb is illustrated by the following move sequence: (1, A, A)

S

_~

(2, a, A) _ W~ (1, a, a) (3, ab, a) _ W~ (1, ab, ab) (4, abc, ab) (5, abca, ab)

[stop and reject.

Thus Lcm

=

{xcx R I x E {a, b}*}

is the language recognized by Mcm . The language is known as the mirrorimage language with center marker, or the center-marked palindrome language. It is generated by the following context-free grammar: Gcm :

S~A A aAa A bAb A c.

It is known that a context-free grammar exists for any language recognized by a pushdown acceptor. In fact, the equivalence of the class of context-free languages and the class of languages recognized by these acceptors can be established . The pushdown acceptor Mcm is deterministic since there is never a choice of move for any configuration . A slight modification of the language © 2002 by Chapman & Hall/CRC

1 .11 . Pushdown Automata

41

requires a nondeterministic pushdown acceptor. Consider, for example, the language L,,

L rZ =

{xx' I x E {a, b}* } .

This language is known simply as the mirror-image language. There is no special symbol in sentences of L,..Z to indicate when a pushdown acceptor should switch from a load-stack mode to a compare mode as there is in the language L,,,, . We note that there exists a pushdown acceptor for L ,Z by modifying M,, to obtain an automaton with an accepting move sequence for every string of L,..Z . Rather than waiting for a symbol c to be scanned, the machine M,Z described in Figure 1 .20 is allowed to switch to its compare mode whenever it writes a scanned symbol into its stack . This type of behavior must be allowed since there is no way for the machine to determine when it has scanned the first half of a mirror-image string . The machine must be permitted to "guess" after each symbol scanned whether it should or should not switch to compare mode. 1]

2] 3] 4] 5] 6]

scan write write scan read read

Load Stack

(a,2) (a,1) (b,1) (a, 5) (a,4) (b,4)

(b,3) (a,4) (b,4) (b, 6)

Compare

Figure 1 .20 Example of a nondeterministic pushdown acceptor.


42

1 . Introduction

1 .12

Exercises

1. Prove Theorem 1.5.1 . 2. Prove Theorem 1 .5 .4 . 3. Design a finite state machine that outputs 1 if an even number of 1's have been input ; otherwise outputs 0. 4. Design a finite state machine that outputs 1 when it sees 101 and thereafter ; otherwise outputs 0 . 5. Design a finite state machine that outputs 1 when it sees the first 0 and until it sees another 0; thereafter, outputs 0; in all other cases outputs 0. 6. Show that there is no finite state machine that receives a bit string and outputs 1 whenever the number of 1's input equals the number of 0's input and outputs 0 otherwise . 7. Let L be a finite set of strings over {0,1} . Show that there is a finitestate automaton that accepts L . 8. Let

be a finite set of strings accepted by the finite-state automaton (QZ, X, f2, A Z , s2 ), i = 1, 2. Let A = (Qi x Q2, X, f, A, s), where

LZ

AZ =

f ((qi, q2),

x)

=

s

=

Show that A,(A) = 9. Let

(fi(gi, x), f2(g2, x)), {(qi, q2) I qi E A 1 and q2 (81,82) .

=

E ,A2},

Ll rl L2 .

be a finite set of strings accepted by the finite-state automaton (QZ, X, f2, AZ, s2), i = 1, 2 . Let A = (Qi x Q2, X, f, A, s), where

LZ

AZ =

f((gi,q2),x)

= =

s Show that AJA) =

Ll

= U

(fi(gi,x)j2(q2,x)), {(qi, q2) I qi E A1 or q2 (81,82) .

E ,A2},

L2-

10. Let G be a grammar and let A denote the empty string. Show that if every production is of the form A~xorA~xBorA~A, where A, B E N, x E T*\{A}, then there is a regular grammar with L(G) = L(G') . 11. Show that the language free language . © 2002 by Chapman & Hall/CRC

L = {anbnck I

G'

n, k = 1, 2 . . . . } is a context-

1.12. Exercises

43

12. Design a nondeterministic finite-state automaton that accepts strings over {a, b} having the property of having each b preceded and followed by an a . 13. Write a regular grammar that generates the strings of Exercise 12 . 14. Locate the path in representing the string x = 1.9.9.

aabaabbb

in Example

15. Show that the string x = abba is not accepted by the nondeterministic finite state automaton of Figure 1.12. 16. Prove Theorem 1.9.10. 17. Show that the set L = {x l . . . xn I xl . . . x n = xn . . . xl } of strings over {a, b} is not a regular language. 18. Show that if Ll and L2 are regular languages over X and S is the set of all strings over X, then each of S\L1, Ll U L 2 , L+, and LIL2 is a regular language. 19. Show, by example, that there are context-free languages such that Ll n L 2 is not context-free. 20. Determine the validity of the following statement : If language, then so is L' = {un I u E L, n = 1, 2 . . . . } . 21. Determine the possible move sequence of the input string aabbaa .

M,Z

Ll

L is

and

L2

a regular

of Example 1.11.4 for

22. (Pumping lemma for regular languages .) Let L be a regular language. Prove that 3n E hY such that b'z E L, I zI >_ n implies that 3u, v, w such that z = uvw, ~uvj _ 1, and Vi E hY U {0}, uv'w E L . Moreover, n is not larger than the number of states of the smallest finite-state automaton accepting L . 23. (Pumping lemma for context-free languages .) Let language . Prove that 3n E hY such that b'z E that 3u, v, w, x, y such that z = uvwxy, vx 1 >_ Vi E hY U {0}, uv'wx'y E L .


L be a context-free L, ~zj >_ n implies 1, vwx j cforsome otherwise j a(q) >_ otQ/rwi{e. E Q

_ { 0

q'EQ'}

c}

Clearly D* is deterministic . Moreover, B(A*, c) = B(D*, 0) . The interested reader should note the "equivalence" between max-min sequential machines [194] and restricted max-min automata . The proof is similar to that given in [197], where the equivalence between stochastic sequential machines and probabilistic automata was shown . One merely replaces summation and product by maximum and minimum, respectively. Due to this equivalence, max-min sequential machines and max-min automata behave similarly in many respects.

2 .5

Equivalences and Homomorphisms Max-Min Automata

of

Let A* = (Q, /t*, F, a) be a max-min automaton, k a nonnegative integer, and 0 < c < 1. The k-behavior of A* with threshold c is the set Bk(A*, c) = {x E X*

I

VgEF ry(x, q) > c

and l xl < k} .

Clearly, B (A*, c) = Uk 0 Bk (A*, c) . We now define various types of equivalences . c-k A2 ~ Bk(A*, c) = Bk(A*, 1. A1 c) . Otherwise, we write A1 %~ A2 . A1

2.

x

cNk A2 A1 -c A2 q A1 for k = 0, 1, 2 . . . . A2 .


Otherwise, we write

52

2. Max-Min Automata 3.

A1

A1

x A2 .

k

cNk

A2 ~ A1

A2 for all 0 < c < 1 .

Otherwise, we write

4. A1 - A2 ~ A1 -c A2 for all 0 c; (2) aI(q') > c if and only if a2 (f (q')) > c; (3) q E Fl implies f(q) E F2 . f is called a strong homomorphism with threshold c if f is a homomorphism with threshold c and q E Fl if and only if f (q) E F2 . f is called an isomorphism with threshold c if and only if f is a strong homomorphism with threshold c that is one-one . f is a homomorphism (strong homomorphism, isomorphism) if and only if for every 0 , ==cz>, and ~ will denote homomorphic, strong homomorphic, and isomorphic with threshold c, respectively. The symbols ==~>, and c. Since A1 A2, q E Fl . Thus x E B(A1, c) . Corollary 2.5.12 A1 ~ A2 implies A1 - A2 . m Theorem 2.5.13 If A1 is totally connected with threshold c and A1 A2, then A1 N A2 implies A1 ~ A2 .

Proof. Let qo E Fl and Q the initial distribution concentrated at qo . Since A1 is totally connected with threshold c, there exists x E X* such that ry,(A1, x, qo) > c and ry,(A2, x, f (q)) A2 . m


2.6. Reduction of Max-Min Automata 2 .6

Reduction

of Max-Min

55 Automata

Two max-min tables 521 and 522 are called statewise equivalent with threshold c ~ for every q E Q1, there exists q' E Q2 such that (521, q) N (522, q') and vice versa . In symbols, 521 N 522 . Otherwise, 521 x 522 . 521 ti 522 ~ for every 0 c ~ Vq'EQ A{0'2 (q), ft * (q , x, q)} > c. © 2002 by Chapman & Hall/CRC

2.7. Definite Max-Min Automata

57

For p > 1,52*(0*) is p definite with threshold c if and only if Q* (0*) is weakly p definite with threshold c, but not weakly p - 1 definite with threshold c. 52*(0*) is 0 definite with threshold c if and only if 52*(0*) is weakly 0 definite with threshold c. 52*(0*) is definite with threshold c if and only if 52*(0*) is p definite with threshold c for some p. 52*(0*) is definite (p definite, weakly p definite) if and only if for 0 p, VgEFrtt(A*, xox, q) > c ~ Vg EF rtt(A2, xox, q) > c,

where A* = (52*, al) and A2 = (52*, a2). Let a3(q) = ry,(A*, xo, q), a4(q) _ rlt(A2, xo, q), and A3 = (52*, Q3), A4 = (52*, Q4) . Then VgEFrtt(A3, x, q) > c ~ Vg EF rlt(A4, x, q) > c

or Q3 N Q4 . Since 52* is distributionwise irreducible with threshold c, a3 c a4 . Hence 0* is weakly p definite with threshold c. m Theorem 2.7.2 If A* = (52*, Q) is totally connected with threshold c and weakly p definite with threshold c, then 52* is weakly p definite with threshold c.

Proof. For every al and a2, there exist XI, X2 E X* such that al N ry(A*, xl , q) and a2 N rlt(A*, x2, q) . Since A* is weakly p definite with threshold c, for every x E X* with I x > p, VgEFrN(A*, x, x, q) > c ~? VgEF r N( A* , x2x, q) > c .

Let A1 = (52*, a l) and A2 = (52*, a2), then VgEFrtt(A*, x, q) > c ~ Vg EF rtt(A2, x, q) > c.

Thus 52* is weakly p definite with threshold c. Theorem 2.7.3 If 52* is weakly p definite with threshold c, then A* _ (52*, Q) is weakly p definite with threshold c. © 2002 by Chapman & Hall/CRC

58

2. Max-Min Automata

Proof. Let xl,x2, x E X* be such that Ix I >_ p, al (q) = ry(A*,xl,q), and 0'2(q) = ry,(A*, x2, q) . Let A1 = (S*, al) and A2 = (S2*, a2) . Then VgEFrla'(A * , xlx, q) = VgEFrtt(A*, x, q)

and VgEFrtt(A*, x2x, q) = VgEFrtt(A2, x, q) .

Since V gE Fry(A*, x, q) > c ~ VgE Frtt(A2, x, q) > c, it follows that VgEFrN(A*, x, x, q)

> C ~?

VgEFrN(A * , x2x, q) >

C.

Thus A is weakly p definite with threshold c. Theorem 2.7.4 If A* = (S2*, Q)

= (0*, F, a), where A* is totally connected (with threshold c) and S2* is distributionwise irreducible (with threshold c), then the following properties are equivalent : (1) A is p definite (with threshold c) ; (2) S2* is p definite (with threshold c) ; (3) 0* is p definite (with threshold c) .

Proof. The proof follows from Theorems 2.7.1, 2.7.2, and 2.7 .3. Theorem 2.7.5 If S2* is 0 definite with threshold c, then F = Q or F = 0 . Proof. For every q', q" E Q and x E X*, VgE Fry(A*,x,q) > c VgEF?'la(A2, x, q) > c, where A1 = (S2*, q') and A2 = (S2*, q"). Take x = A. Then q' E F if and only if q" E F. Thus F = Q or F = 0 . m Theorem 2.7.6 If S2* is p definite with threshold c, where p > 1, then there exist al and a2 such that al

c 92,

but al

C,1

a2 .

Proof. Since S2* is not p - 1 definite with threshold c there exist xo E X* with Ixo~ >_ p - 1 and ai, a2 such that V gE Frtt(A2, xo, q) > c, but VgE Fry,(A2, xo, q) c. Let al (q) = r[t(A'*,xO,q) and 0'2(q) ry(A'2*, xo , q) . Then a l and a2 have the desired properties. 2 .8

Reduction

of Max-Min

Machines

The material in this section is from [203] . The reduction problem of max-min sequential machines and max-min automata was studied in the previous section . In this section, we continue the study . © 2002 by Chapman & Hall/CRC

2.8 . Reduction of Max-Min Machines

59

Three types of equivalence relations are considered, namely, statewise, compositewise, and distributionwise equivalence . We show that the last two are equivalent. From the first two equivalence relations, two minimal forms are defined . The counterparts of these two minimal forms in the theory of stochastic sequential machines are the reduced [29] and minimal state forms [10]. For stochastic sequential machines, (convex) linear algebra is a useful theory. The next section is devoted to the development of a type of algebra, called the max-min algebra. The max-min algebra is very useful for dealing with max-min machines . Although max-min algebra resembles linear algebra and the max-product algebra of Section 3.1 in certain respects, they are almost completely unrelated. Most of the results have counterparts in the theory of stochastic sequential machines and max-product machines . These counterparts are either known to be true [10,29,167,202] or can be shown to be true. See also Chapter 3. The section concludes with a look at nondeterministic and deterministic machines which are special cases of max-min machines . Many of the results for max-min machines are strengthened for these particular cases . Definition 2.8 .1 Let A = [a2 j ] be an n x p matrix and B = [b2j ] be a p x m matrix of nonnegative real numbers. Let A 0 B be the n x m matrix [c2j ], where czj = V{aik n bkj I k = 1, . . . ,p} .

Definition 2.8 .1 is applicable even if n or m is infinite. Clearly, the operation 0 is associative . In the remainder of the chapter, a, b, and c (with or without subscripts) denote real numbers, and x and y (with or without subscripts) denote (finite or infinite sequences of real numbers . A superscript is used to denote the particular term of a sequence, e.g., xk denotes the k-th term of the sequence x. X and Y (with or without subscripts) denote collections of (finite or infinite sequences of real numbers . Here a A x denotes the sequence whose k-th term is a A xk . When taking max-min combinations of sequences, we assume the sequences are of the same length. Definition 2.8.2 Let X = {xl, x2, . . . , xn }. A max-min combination of X is an expression of the form

VZ i(az

n xi),

(2 .1)

where a2 is a nonnegative real number, i = 1, 2. . . . , n. If 0 aknxk>aknbkonxo=cony=x .

Hence x = ak A xk or x E C(X"\{Jc}) . Thus


64

2. Max-Min Automata

Similarly,

Consequently,

Let {x} and X' be sets of vertices for Y. Then Vx' E X', there exists E [0,1] such that x' = a' A x. Thus Vx', x" E X', either x' lai n xZ for some 0 < a2 < 1, i :?~ 1},

and a'1 = AA I . Define x'1 = a'1 V xl . Suppose that x'_1 has been defined. Let Ak

=

{a I xk = (a A xk) V (VZkaj A x2) for some 0 < a2 < 1, i :?~ k},

and xk = akAxk. Let X' _ { X/I , x2, . . . , xn} . Clearly IX'I = IXI . By Proposition 2.8 .6, X' can be effectively constructed from X. Moreover, ak E Ak for all k. Thus X' is a set of vertices of Y. We now show that X' is fundamental . Let ctik

= nA k

xk = (ak A xk) V y,

where y E C(X'\{ .k }) . Let Xk = {xl, X2 . . . . . x%_1, xk+l, . . . , xn } . Since xk = xk V y', where y' E C(Xk), it follows that xk = (ak A ak A xk) V y", where y" E C(Xk) . By the definition of ak, ak < ak leak or ak > ak . Thus xk > ak A xk = ak A ak A xk = ak A xk = xk .

Hence xk = ak A xk . By Lemma 2.8 .21, X' is fundamental . We now illustrate Proposition 2.8 .27. © 2002 by Chapman & Hall/CRC

m

66

2. Max-Min Automata

Example 2.8 .28 Let X = {(1, 0), (2, 4)} . Let xl = (1, 0) and x 2 = (2, 4) . Then Al = {a I (1, 0) = (an(1, 0))V (a2n(2, 4)) for some a2 , 0 [0,1] . Then r12 (u,1)

=

[972 (so) n (V (so, u,1,80) V V(so, u,1, s3))]V [972(s3) n (V (83, u,1, so) V V(s3, u,1, s3))] [972 (so) n ( I V z)] V [972(s3) n (0 V 0)] 1972

(so) n 4] V 1 972 (83) n 0]

972 (so) n 4*

Consider M = (Q, X, Y, /t) of Example 2.9.4 . Let 71 : Q ----> [0,1] . Then rI (u,1)

=

=

[77(g1) n (ft(gl, u,1, q1) V tt(gl, u,1, q2) V tt(gl, u,1, q3))]V [9 7(g2) n (ft (q2, u,1, q1) V ft (q2, u,1, q2) V ft (q2, u,1, q3))]V [9 7(g3) n (ft(g3, u,1, q1) V ft(g3, u,1, q2) V ft(g3, u,1, q3))] [97(g1) n (oV 4 V 2)] V [97(g2) A (4 V oV 2)]V [9 7(g3)A(0V0V0)]

_

2 .10

[9 7(g1) n 1 4 ] V [97(g2) n 3] 4 .

Irreducibility

and

Minimality

Definition 2 .10 .1 Let M = (Q, [) be an MSLM. (1) M is called statewise irreducible if for every q', q" E Q, q' - q" implies q' = q" . (2) M is called compositewise irreducible if for every q E Q and sd of M, q - 77 implies 97(q) > 0 . 97 (3) M is called distnibutionwise irreducible if for every sd 97, and 972 of M, 971 - 972 iTnplies 97, = 972Example 2 .10 .2 Let X = {u} and Y = {11 . Let Q1 = {q1, q2} and Q2 = {s1, 821 . Define ft, : Q1 x X x Y x Q1 ----> [0,1] and f 2 : Q2 x X x Y x Q2 [0,1] as follows : P1(gql, u, 1 , qq2) P1(g1,u, 1 ,g1) P2 (s l, u, 1, 82) ft2(sl, u, 1, sl)

_

1 3

P1 (q2, u, 1, q1)

_

2 1 2

P2 (82, u, 1, sl)

0

P1 (q2, u,1, q2) N'2 (s2, u, 1, 82)

2 3

0

2 3

0.

Let M1 = (Q1, [t1), M2 = (Q2, P2), 11 = (M1, 971), and 12 = (M2,972) . Then rh (u,1)

_ _ _

[ 971(g1) n ((ft1(g1, u,1, q1) V ft1(g1, u,1, q2))]V [ 97, (q2) n ((ft, (q2, u,1, q1) V ft, (q2, u,1, q2))] (97,(g1) A (2 V 3)) V (971(g2) A (3 V 0)) (971(g1) n 2 ) V (971(g2) n 2 )


76

2. Max-Min Automata

and r12(u,1)

_ (972(81)

n

2) V (972 (82)

n 3) .

In fact, it follows that rII

(u', ln) _ (97, (ql)

n

2) V (97, (q2)

n ) 2

n

2) V (972 (82)

n

and r12 (u n ,

11 ) = (972(sl)

for n = 2, 3, . . . . We also have that p`vh

i

i

2 _2 3

(u,1)

3 0

,

1 (ql, u,1)

= ft(gl, u,1, ql) V ft(gl, u,1,

q2)

=

1 (q2, u,1)

= ft(g2, u,1, ql) V ft(g2, u,1,

q2)

=

1 z 2

QM' (u, 1) _

=

Now [t1*i (ql, uu,11, ql)

=

uu,11, ql)

=

fti (q2,

2, 2,

~

I


2 2

3

fti (ql, uu,11, q2) fti (q2, uu,11, q2)

(ql, uu,11)

=

1 (ql, uu,11)

=

1

1 3 1 3

2 1 2

1

2

V

1

3

=

V0=

-

Thus PMl

1

,

3 1 2 2 3

2)

V

1

3

=

,

1

2,

= =

3

s

1

2, 2

3,

2.10. Irreducibility and Minimality 1

1

11~=2V

1(g2,uu,

QMl (uu, ll) _

1

3= 2'

2 1

A"' (uu,11) _

1

2

2

1 2 1

_1 3 1 3

2 3

1 2

It is easy to see that PIVI i (un

1n) l

2

for n = 2,3, . . . . Hence 1

1

2

2

2

2

1

1

and

Now PMz

(u,1)

=

0

2 3

1

2

0

1 2 (s1, u,1~ = 0 V 2

2(82,u,1~


=

2 3

'

1

= 2'

V0=

2

3~

78

2. Max-Min Automata

Now [t2*2 (s1, uu,11, s 1) P2 (s2, uu,11, s1)

= =

ft2(s1, uu,11, s2) Nt2 (s2, uu,11, s2)

2, 0,

Thus

1

PM2 (uu,11) =

8 1 , uu,11)

0

2 0

1

=

0 2.

,

2

= 2 V 0 = 2, 2=

z (s 2 , uu,11) = 0 V

Q

=

(uu,11)

1

=

1

2'

,

A212 (uu,11) _

It follows that PMI (U,n, 1 n ) =

1 z

0

0

1 2

0

1 z

for n = 2,4,6. . . . and PM2(Un, 1n) _

1 2

0

for n = 3, 5, 7, . . . . Hence

and

Now (1, 0) 0 BM 1 = (1, 0) 0 BMZ and (0,1) 0 BM1 = (0,1) 0 BMZ . Thus q1 - s1 and q2 ti s2 . Suppose that ~ : Q2 ----> [0,1] is such that ~(s1) = 1 and ~(s2) = 0 . Let rl be any sd of M2 . If ?7(s1) = 0 and ?7(s2) = 2 , then ~ - ~ since (1, 0)OB M2 -(0, 2)OBM2 . However, r1(s1) ~ 0. Thus M2 is not compositewise irreducible.


2.10. Irreducibility and Minimality

79

Example 2.10.3 Let X = {u} and Y = {0,1}. Let Q = {ql, q2} . Define ft : Q x X x Y x Q ----> [0,1] as follows: ft (gi, u,1, q2) =

1

2

2 and ft(g2, u~ 0, qi) = 3 ~

with ft of any other element equal to 0. Let M = (Q, [I) and I = (M, 97) . Then (97(qj) A (0 V 2)) V (97(g2) A (0 v 0)) = 97(qj) A 2, r' (u, 0) _ (97(qj) A (0 v 0)) v (97(g2) A v 0)) = 97(g2) A (3 3, r,(uu,11) r'(uu,10)

= =

r'(uu,01)

=

rI(uu,00)

=

r, (uuu,111) r, (uuu,110) r , (uuu,101) rI (uuu,100) r, (uuu,011) r , (uuu,010) r, (uuu,001) r, (uuu,000)

0,

97(gi) A 2, 97(q2)A 2 , 0,

= = = = = = = =

0, 0,

97(gi) A 2 , 0, 0, 97(g2) A 2, 0, 0.

We see that r , (u . . . u,1010 . . . ) = 9 7(gi) A 2 and r , (u . . . u, 0101 . . . ) _ 97(g2) n 2 . Hence if 97(gi) > 2 < 97(g2), _ 2 - [ 0

0 3

0 0

2 0

0 2

0 0

0 0

0 0

2 0

0 0

0 0

0 2

0 0

0 0

and

where (u,1) < (u,0) < (uu,11) < (uu,10) < (uu,01) < (uu,00) < (uuu,111) < . . . < (uuu, 000) < . . . . Thus it follows that M is compositewise irreducible. Theorem 2.10.4 Let M be an MSLM. Then the following assertions hold.


80

2. Max-Min Automata

(1) M is statewise irreducible if and only if no two rows of B°'1 are identical. (2) M is compositewise irreducible if and only if p(B M ) is a set of vertices of C[p(B m)], i.e ., no row of BM is a convex max-min combination of the other rows of Bm . (3) M is distributionwise irreducible if and only if p(B M) is a basis of C[p(B

m)] .

Proof. (1) The proof follows from Proposition 2.9 .14. (2) The proof follows from Propositions 2 .9 .14 and 2.8 .14. (3) The proof follows from Proposition 2 .9 .14 and the definition of basis . All assertions of Theorem 2.10.4 are also valid if BM is replaced by Am .

Proposition 2 .10 .5 Let M be an MSLM. If M is distributionwise irreducible, then M is compositewise irreducible . 0 Proposition 2 .10 .6 Let M be an MSLM. If M is compositewise irreducible, then M is statewise irreducible . 0 Definition 2.10 .7 Let M be an MSLM. M is called statewise (compos-

itewise minimal if M is not statewise (compositewise equivalent to an MSLM with a fewer number of states.

Let M = (Q, p) . The cardinality of M is defined by IMI = I QI . Theorem 2.10 .8 Let M be an MSLM. Then M is statewise minimal if and only if M is statewise irreducible .

Proof. Let M = (Q, ft) . Suppose that M is not statewise minimal. Then there exists an MSLM M' with I M' I < I MI , M' - M. By Theorems 2.9 .16(1) and 2 .10.4(1), M is not statewise irreducible . Conversely, suppose that M is not statewise irreducible. Then there exist q', q" E Q such that q' q", but q' :?~ q" . By renumbering the elements of Q, if necessary, we may assume that q' = qn-1 and q" = qn, where n = QI . Let Q' = Q\{qn} and M' = (Q', /t'), where for i = 1, 2, . . . , n - 1, ft (qj,

N' t (gZ, u, v,g..) _ { ft (gi, u, uvJ1) v, q.-I)

ft(gi, u, v, qn)

if j 1,2, . . . ,n-2 if -n-1.

Now for every (x, y) E (X x Y)*, qm (q

y) _


qm, (qi, x, y) q (qn-1 , x, y)

if i = 1, 2, . . . , n - 1 if i = n.


81

Therefore, p(Am ) = p(A M') . By Theorem 2.9 .16(1), M - M' . Thus M is not statewise minimal. m In the next example, we illustrate the proof of Theorem 2.10 .8 . Example 2.10.9 Let Ml = (Q, X, Y, /t) be defined as in Example 2.9.4 . Rather than qn_1 - qn as in the proof of Theorem 2.10.8, we have q1 - q2 . Thus for x E X and y E Y, ft(gi, x, y, qj) ft(qi, x, y, q2) V ft(g3, x, y, q1)

f t (gZ, x, y, gj)

if i = 3 if .j = 2.

Thus lt'(g2, x, Nt'(g3, x, lt'(g2, x, Nt'(g3, x,

y, y, y, y,

q3) q3) q2) q2)

= = = =

ft(g2, x, y, q3) ft(g3, x, y, q3) ft(g2, x, y, q2) V ft(g2, x, y, q1) ft(g3, x, y, q2) V ft(g3, x, y, q1) .

Hence ft'(g2,u,1,g3)

=

It' (q2, v,1, q3)

=

It'(q2, u,1, q2)

=

It' (q2, v, 0, q2) ft'(g3, u, 0, q2)

=

'4

=

8

It' (q3, v, 0, q2)

=

2~ OV4 OV V

'4 8

3

4~

3

1

4~ 1

Theorem 2.10.10 Let M be an MSLM. Then there exists an effective procedure for constructing a statewise minimal MSLM that is statewise equivalent to M.

Proof. Consider the following procedure: 1. Construct BM . 2. If there exist two rows of BM that are identical, then proceed to Step 3; otherwise stop. 3. Construct M' as given in the proof of Theorem 2.10 .8 . Return to Step 1 with M' replacing M. Since IMI is finite, the procedure must terminate in a finite number of steps . By Proposition 2 .9 .7, BM can be effectively constructed . Thus the procedure is effective . By Theorem 2.10 .8, the resulting MSLM is the desired MSLM. Theorem 2.10.11 Let M be an MSLM. Then M is compositewise minimal if and only if M is compositewise irreducible .


82

2. Max-Min Automata

Proof. Let M = (Q, [t) . Suppose that M is not compositewise minimal. Then there exists an MSLM M' with IM'I < IMI such that M' - M. By Theorems 2.9 .16(3), 2.10.4(2), and Propositions 2.9.18, 2.8 .30, and 2.9 .8, M is not compositewise irreducible . Conversely, suppose that M is not compositewise irreducible . Then there exist q' E Q and an sd 97 of M such that q' - 77, but 97(q') = 0. By renumbering the elements of Q, if necessary, we may assume that q' = q , where n = I Q I . Since qn - 77 and 97(gn) = 0, glvl

(qn, x, y) = V {97(gi) A g" (qi, x,

2J)

I i = 1, 2, . . . , n},

for all (x, y) E (X x Y)* . Let M' = (Q', p,'), where Q' = Q\{qn} and for i,j=1,2, . . .,n-1, N'(qi,

UI v,

qj) = ft (gj, u, v, qj) V (97(qj) A ft (gj, u, v, qn))

Now for every (x, y) E (X x Y)*, ivt

_

qm,

(qi, x, y)

if i = 1 . . . . , n - 1

Hence C[p(Am)] = C[p(A m')] . By Theorem 2 .9 .16(3) and Proposition 2 .9 .18, M' - M. Thus M is not compositewise minimal. 0 Theorem 2.10 .12 Let M be an MSLM. Then there exists an effective procedure for constructing a compositewise minimal MSLM that is compositewise equivalent to M.

Proof. Consider the following procedure : 1 . Construct Bm. 2. If there exists a row of BM that is a convex max-min combination of the other rows of BM , then proceed to Step 3; otherwise stop. 3. Construct M' as given in the proof of Theorem 2.10.11. Return to Step 1 with M' replacing M. Since IMI is finite, the procedure must terminate in a finite number of steps . By Proposition 2.9 .7, BM can be constructed effectively. By Propositions 2.8 .6 and 2.9 .8 and the fact that IMI is finite, Step 2 can be carried out effectively. Thus the procedure is effective . By Theorem 2.10.11, the resulting MSLM is the desired MSLM. m Proposition 2 .10 .13 Let Ml and M2 be MSLMs. Let p(B MI ) and p(BM2 ) be fundamental sets of vertices of C[p(Bml)] and C[p(B M2 )], respectively. Then Ml - M2 if and only if Ml - M2 .

Proof. The proof follows from Theorems 2.9 .16(3), 2.8 .26, Proposition

2.9 .18, and Theorem 2.9 .16(1) and (2) . m



83

Definition 2.10.14 Let M be an MSLM. Then M is called fundamental if C[p(BM)] is fundamental. Proposition 2.10 .15 Let Ml and M2 be MSLMs. If Ml , M2 are compositewise minimal, Ml is fundamental, and Ml - M2, then M2 is fundamental and Ml - M2 . Proof. The proof follows from Theorems 2.8 .26, 2.10.4(2), 2.10.11, and Proposition 2 .10.13. Proposition 2.10.16 Let Ml and M2 be MSLMs. If Ml is distributionwise irreducible, M2 is compositewise minimal, and Ml - M2 , then M2 is fundamental and Ml - M2 . Proof. The proof follows from Propositions 2.8 .36 and 2.10.15 . Proposition 2.10 .17 Let Ml and M2 be MSLMs. If Ml , M2 are statewise minimal, and Ml - M2, then for every (x, y) E (X x Y)*, PM2 (XI BMl, PM1 (x, y) 0 BMl -_ y) (3 after an appropriate rearrangement of states . Proof. From Theorems 2 .9 .16(1), 2 .10.4(1), and 2.10 .8, Am, = AMz after an appropriate rearrangement of states . 0 Definition 2.10.18 Let Ml and M2 be MSLMs. Then Ml and M2 are called isomorphic (-) if they are equal up to a permutation of states . Definition 2.10.19 Let M be statewise (compositewise minimal. Then M is called statewise (compositewise simple if there does not exist a statewise (compositewise minimal MSLM that is statewise (compositewise equivalent to M, but not isomorphic to M. Let M be an MSLM . We introduce the following notation . (1) P(M) = U,,ex U,EY PIP' (u, v)], (2) p(M) = {.j 0 BM I .j E p(M)j. Theorem 2.10 .20 Let M be statewise minimal. Then M is statewise simple if and only if p(BM) is a basis of p(M) . Proof. Suppose that p(B') is a basis of p(M) and M' is a statewise minimal MSLM such that M' - M. By Proposition 2.10.17, for every UEX,VEY, plvi (u


v)

= plvi (u

v)

84

2. Max-Min Automata

after an appropriate rearrangement of states. Hence M' - M. Thus M is statewise simple. Conversely, suppose p(BM) is not a basis of P(M) . There exist j E p(M) such that jOB

(2.4)

for some j' :?~ j. Let j be the i-th row of the matrix P' (u, v). M' from M by replacing the i-th row of the matrix PM (u, v) leaving the rest unchanged . By (2.4), we have Am = Am' . By 2.9.16(1), 2.10.4(1), and 2 .10 .8, M' is statewise minimal and However, M'*M. Thus M is not statewise simple . 0 Proposition 2 .10 .21

Suppose that M is statewise minimal . tributionwise irreducible, then M is statewise simple.

Construct by j' and Theorems M' - M.

If M

is

dis-

Proof. The proof follows from Theorem 2.10.4(3) and 2.10.20. Theorem 2.10 .22 Suppose that M is compositewise minimal and fundamental . Then M is compositewise simple if and only if p(B M) is a basis of P(M) . Proof. Let p(B') be a basis of p(M) and M' be a compositewise minimal MSLM such that M' - M. By Proposition 2.10.6 and Theorems 2.10.8 and 2.10.11, both M and M' are statewise minimal. By Proposition 2.10.15, M' - M. The remainder of the proof is similar to Theorem 2.10.20. Proposition 2 .10 .23

Suppose that distributionwise irreducible, then M

M is compositewise minimal . is compositewise simple .

If M

is

Proof. The proof follows from Theorems 2.10.4(3), 2 .10 .22, and Proposition 2.8.36. In the next example, we illustrate Theorems 2.10.20 and 2.10.22 . Example 2.10 .24 Y = {0,1}, and

M = (Q, X,

Let

{qi, q2}, X = {u},

Y, p), where Q =

ft : QXX XYXQ~ [0,1] is defined as follows :

ft(gi, u,1, q2) ft(q, w, z, q') for all other (q,

w, z, q') .


= 2, =

ft(g2, u, 0, qi)

_

Let 97 : Q ~ [0,1] be arbitrary.


85

We see that q' (gl, un, y) > 0 if and only if y is an alternating sequence of 1's and 0's beginning with 1 such that I yl = n and gm (q2, un, y) > 0 if and only if y is an alternating sequence of 0's and 1's beginning with 0 such that jyj = n. Also qM (qi, u,1) gm (ql, uu,10)

_ _

g m (ql, nun, 101)

_

I 2~ 1

q

i

(q2, uuu, 010)

q

2~

3 41 1

(q2, u, 0) (q2, uu, 01)

q

=

i

21

With the order (u,1) < (u,0) < (uu,11) < (uu,10) < (uu,01) < (uu,00) < (uuu,111) < . . . on (X x Y)*, _ 2 - [ 0 Thus

0 4

0 0

2 0

= [ i

2 0

0 2

0 0

0 0

0 1 . 4

With the order (u,0) < (u,1) < (uu,00) < (uu,01) < (uu,10) < (uu,11) < (uuu,000) < . . .,

Let rl : Q ----> [0,1] . Then (97(qj)

r, (U, 0) rI(uu,11) rI(uu,10)

_ _

rI(uu, 01)

_

rI(uu, 00) r,(uuu,111) r,(uuu,110) rI(uuu,101) r,(uuu,100) r,(uuu,011) rI(uuu, 010) r,(uuu,111) r,(uuu,00)

_ = = = =

= = =

= =

(97(qj) (97(qj) (97(qj) (97(qj)

n (0 v 2)) V (97(g2) n (0 v 0)) n (0 v 0)) v (97(g2) A (4 v 0)) n (0 v 0)) v (97(g2) n (0 v 0)) n (2 v 0)) V (97(g2) n (0 v 0)) n (0 v 0)) v (97(g2) n (2 v o)) n (0 v 0)) v (97(g2) n (0 v 0))

(97(qj) 0, 0, 97(gi) A 2, 0, 0, 97(g2) n 2, 0, 0.


=

_

97(qj) A 2, 97(g2) A 4,

= 0, _ 97(qj) n 2, = 97(g2) n 2, = 0,

86

2. Max-Min Automata

Also, PM (u,1) PM (u, 0)

0 0

1

0

0 0

3

0 ]'

4 0 0

0 0

PM (uu,10)

2 0

0 0 ]'

PM (uu, 01)

0 0

0

PM (uu, 00)

0 0

PM (uu,11)

'

z 0 0

It follows that P(M) = {( 0 ' 1 ), (010), (4 ~ 0* Now T( M)

_

{(o, 2)

[

3 0

z 0

(4~ )

(0, _,)[

o

0

,

3 ] , (0, 0) 4

[ 01

4

3

(0 0)

and p(BM)

_

{(2

0) (0'

4)}.

Clearly, p(BM) is not a basis of p(M) . (Note (0, 2) is a unique max-min convex combination of (0, 4 ), but (2 , 0) = a A (2 , 0) for 2 < a < 1. ) 2 .11

Nondeterministic and Deterministic Case

Definition 2.11.1 Let (Q, p) be an MSML such that Im(p) E {0,1} . Then

(Q, ft) is called a nondeterministic sequential-like machine (NSLM) . If for every q E Q, u E X, there exist unique v E Y and q' E Q such that ft(q, u, v, q') = 1, then (Q, ft) is called a deterministic sequential-like machine (DSLM) .

Proposition 2 .11 .2 Let M be a DSLM. Then for every q E Q, x E X*, there exists unique y E Y* with IxI = IyI such that qM(q, x, y) = 1 . m © 2002 by Chapman & Hall/CRC

2.11 . Nondeterministic and Deterministic Case

87

Due to the special characters of NSLMs and DSLMs, several of the above results can be strengthened . Proposition 2.11 .3 Let M be a NSLM. Then M is statewise minimal if and only if M is not statewise equivalent to any NAM with a fewer number of states.

Proof. The proof follows from the proof of Theorem 2.10.8.

0

Proposition 2.11 .4

Let M be a NSLM. Then there exists an effective procedure for constructing a statewise minimal NAM that is statewise equivalent to M . 0

Proposition 2.11 .5 mal if and only if M

Let M be a NSLM. Then M is compositewise miniis not compositewise equivalent to any NAM with a fewer number of states .

Proof. The proof follows from Proposition 2 .8.4 and the proof of Theorem 2.10.11 . Proposition 2.11 .6 Let M be a NSLM. Then there exists an effective procedure for constructing a compositewise minimal NAM that is compositewise equivalent to M. 0 Proposition 2.11 .7 and only if M is not of states.

Let M be a DSLM. Then M is statewise minimal if statewise equivalent to a DAM with a fewer number

Proof. The proof follows from the proof of Theorem 2.10.8. m Proposition 2.11 .8 Let M be a DSLM. Then there exists an effective procedure for constructing a statewise minimal DAM that is statewise equivalent to M. 0 Theorem 2.11 .9 Let M be if and only if M is statewise

a DSLM. Then minimal .

M

is compositewise minimal

Proof. Suppose that M is not compositewise minimal. Then there exists a row of BM , say the n-th row, such that n = IMI and which is a convex max-min combination of the other rows of BM . By Proposition 2.8.4, we may assume, without loss of generality, that xn

= V {xi I

i = 1, 2, . . . , m},

where m < n, after an appropriate rearrangement of states, and x2 denotes the i-th row of BM . Since xn = 0 implies xk = 0 for i = 1, 2, . . . , m, we have by Proposition 2.11 .2 that x n = x2 for i = 1, 2, . . . , m . Thus M is not statewise minimal. The converse is straightforward . m © 2002 by Chapman & Hall/CRC

88

2. Max-Min Automata

Proposition 2 .11 .10 Let Ml and M2 be compositewise minimal NSLMs. Then Ml - M2 if and only if Ml - M2 . Proof. The proof follows from Propositions 2.8 .31 and 2.10.13. Definition 2.11 .11 Let M be statewise (compositewise minimal NSLM.

Then M is called statewise (compositewise ND-simple if there exists no statewise (compositewise minimal NSLM that is statewise (compositewise equivalent to M, but not isomorphic to M.

Definition 2.11 .12 Let M be a statewise minimal DSLM. Then M is

called statewise D-simple if there exists no statewise minimal DSLM that is statewise equivalent to M, but not isomorphic to M.

Theorem 2.11 .13 Let M be a statewise (compositewise minimal NSLM.

Then M is statewise (compositewise ND-simple if and only if for every xo E P(M), there exists a unique subset L of p(BM) such that xp = V{x I x E L} .

Proof. The proof is similar to the proofs of Theorems 2.10 .20 and 2.10.22. For the compositewise case, Proposition 2.11 .10 is needed. m Theorem 2.11 .14 Let M be statewise (compositewise minimal DSLM. Then M is statewise (compositewise ND-simple.

Proof. The proof follows from Theorem 2 .11.13 and the fact that every row of any matrix PM (u, v) has at most one nonzero entry and this nonzero entry is a 1 . 0 Corollary 2.11 .15 Let M be a statewise minimal DSLM. Then M is statewise D-simple . m 2 .12

Exercises

1. Let X = {(x 1 ,0 , (O,x2)} . Show that X is not a basis of C(X) if either x1 < 1 or x2 < 12. Let X = {(1, 0), (1,1)} . Show that X is not a basis of C(X) . 3. Prove Theorem 2.4 .5 . 4. Show that T defined in Theorem 2.5 .5 has the desired properties. 5. Let V be a vector space over a field F. Prove that the intersection of

any collection of subspaces of V is a subspace of V.


2.12. Exercises

89

6. Let V be a vector space over a field F. Define s : P(V) ~ P(V) by b'X E P(V), s(X) = the intersection of all subspaces of V that contain X. Show that s satisfies the Exchange Property. 7. Show that I and 12 of Example 2 .9.19 are equivalent if r7(gi) (so ) and 97 (q3) = 972(ss)972

= 97(g2) _

8. Prove that all the assertions of Theorem 2.10.4 are valid if B" is replaced by A' .


Chapter 3 Fuzzy Machines, Languages, and Grammars 3.1 Max-Product Machines

In this and the next section, we present the work given in [202] . Max-product machines may be considered as models of fuzzy systems [256] . In this section, we consider the minimization problem of max-product machines . We present various types of equivalence relations and minimal forms . Among the minimal forms considered are those that are similar to the reduced [29] and minimal state [10] forms of stochastic machines . We present a type of algebra, called the max-product algebra, in order to examine the properties of max-product machines . The role played by max-product algebra in the theory of max-product machines is the same as that played by (convex) linear algebra in the theory of stochastic machines. We present complete solutions for the minimization problem of maxproduct machines for both the equivalence relations and minimal forms considered. Definition 3 .1 .1

Let A = [a2j] be n x p and B = [b2j] be p x m matrices of nonnegative real numbers . Let AO B be the n x m matrix [c2j ], where c2j = V{aikbkj I

k = 1, 2, . . . ,p} .

Definition 3.1 .1 is applicable even if

n

or

m is

infinite.

Proposition 3.1 .2

Let A 1 be n x p, A 2 be p x m, and A 3 be m x t matrices of nonnegative real numbers . Then (A, 0 A2) O A3 = A i 0 (A2 0 A3),


91

92

3. Fuzzy Machines, Languages, and Grammars

that is, the operation (~) is associative .

We introduce the following notation: a, b, and c (with or without subscripts) will denote nonnegative real numbers ; x and y (with or without subscripts) will denote (finite or infinite) sequences of nonnegative real numbers . Superscripts will be used to denote the particular term of the sequence, e.g., xk will denote the k-th term of the sequence x. X and Y (with or without subscripts) will denote collections of (finite or infinite) sequences of nonnegative real numbers . That is, if S denotes the set of all sequences of nonnegative real numbers, then X, Y E P(S). In the remainder of the chapter, we assume that S denotes a set of sequences of nonnegative real numbers of a fixed length. We also assume that S is our universal set. (1) Let ax denote the sequence whose k-th term is ax k . (2) Let XI V x2 V . . . V x n or V{x2 i = 1, . . . , n} denote the sequence whose k-th term is xl V x2 V . . . V xkn . (3) MP stands for max-product . Definition 3.1.3 Let X = {xi,x2, . . . , xn } . An MP-combination of X is an expression of the form

VZlaix2,

(3.1)

where a2 is a nonnegative real number, i = 1, 2. . . . , n. If 0 x2 if xl >

x2

for all k.

Proposition 3.1 .13 Let Y be a convex MP-set. Let X be a set of vertices of Y and X' be a set of generators of Y. Then X C X' .

Proof. Let x E X. By Proposition 3.1 .11(1), x = VZ l ai x2, where for i = 1, 2, . . . , n,

x2 E X'

and 0
0. Hence

B ~ b, D ~ bf . Then m(oAA, A) = 2 Now p(A, B) > 0 and so S(B) = A.

since m(uA, A) = 2, where u = oA . We also have that 6(aAAb) = A since

A(p,s,2) (aAAb, aABb) = p(A, B) . Now m(aAAD, D) = 1 and p(D, b) > 0. Hence that A(P,~,1)(P,6,2)

(aAAD, aABb)

=

=

(aAAD) = D. We have

V{A(P,s,2) (z, oABb)A(P s 1) (aAAD, z) z E (T U N)* f A(p,s,2) (aAAb, aABb) (aAAD, aAAb) A(n'~,l) p(A, B)p(D, b) .

Now AG(aABb)

=

= =

V{h(so)A,(so, aABb) V h(A) A,(A, oABb) V h(B)A,(B, aABb) V h(C)A,(C, aABb)V h(D)A,(D,oABb) I z E Z*f h(so) Vz A, (so, aABb) h(so)p(so, oAC)p(oAC, oAAD)p(oAAD, aAAb) p(aAAb, aABb) .

Also AG(b) = h(A)p(A, B)p(B, b) V h(B)p(B, b) V h(D)p(D, b) . Note that m(A, A) = m(B, B) = m(D, D) = 1, S(A) = A, S(B) = B, and S(D) = D here .

In Definition 3.4.9, as well as in the rest of the chapter, we assume the usual arithmetic of the extended real number system, which includes the assumption that O-oo = 0. Moreover, the concept of supremum is extended in a natural manner to include oo. Definition 3.4.11 A fuzzy language A over S is a function from S* into >o A(s) is the grade of membership that s is a member of the language,

where s E S* .

Clearly, if G is a mar-product grammar with terminal set T, then AG is a fuzzy language over T. Thus we refer to AG as the fuzzy language generated by G. Moreover, if G is strict, then AG is a function from T*

into [0,1] .


106


In Section 5.12, the concept of random control sets for probabilistic grammars will be defined . A similar concept can be readily introduced for mar-product grammars . The mar-product grammars defined above are referred to as type 0 mar-product grammars . Other more restrictive types of mar-product grammars can be obtained by imposing certain restrictions on the fuzzy productions . Four such grammars are defined below . They correspond to context-sensitive, context-free, weak regular, and regular probabilistic grammars, Section 5.12. Definition 3.4.12 Let G = (T, N, P, h) be a mar-product grammar.

(1) G is context-sensitive if for every p E P, s, t E (TUN)*, p(s, t) > implies ~sl < t10 (2) G is context-free if for every p E P, s, t E (T U N) *, p (s, t) > 0 implies s E N. (3) G is weak regular if for every p E P and s, t E (TUN)*, p(s, t) > 0 implies s E N, t = uA, where u E T* and A E N U {A} . (4) G is regular if for every p E P and s, t E (T U N) *, p(s, t) > 0 implies s E N, t = aA, where a E T and A E N U {A}.

It is easy to see that these definitions contain the crisp case. We will see that most of the relations established in Section 5.12 between the various types of probabilistic grammars and probabilistic automata may be carried over to the mar-product case. In fact, much stronger results are valid for the latter case, as we shall see in subsequent sections . If A = AG for some mar-product grammar G of a particular kind, then we say that A is a fuzzy language of that kind, and vice versa . Theorem 3.4 .13 If A is a (finitary) (type 0, context-sensitive, contextfree, weak regular, regular) (strict) fuzzy language over T, then A = AG for some (finitary)(type 0, context-sensitive, contextfree, weak regular, regular) (strict) mar-product grammar G = (T, N, {po}, h), where there exists Ao E N such that the following conditions hold: (1) h(Ao) = 1 and h(A) = 0 for A :?~ AO ; (2) po (s, t) = 0 for all s E (Y U N)* and t E (T U N) * {Ao}(T U N)* ; (3) po(s,t) > 0, where s E (TUN)*{Ao}(TUN)* and t E (TUN)* implies s = Ao . Proof. Let A = AG', where G' = (T, N', P, h') is a (finitary) (type 0, context-sensitive, context-free, weak regular, regular) (strict) mar-product grammar . Let Ao ~ TUN' and N = N'U{Ao } . Define G = (T, N, {po}, h), where P0 (8, t) =

V PEPp(s, t)

VPEP VAEiv h(A)p(A, t)

0


if s, t E (TUN')* if s = Ao , t E (TUN')* otherwise

3.4 . MaxProduct Grammars and Languages

107

and 1 h(A) _ -{ 0

if A = Ao if A7~ A0 .

Clearly, G has the desired properties . A = AG .

0

Moreover, it can be verified that

If G satisfies conditions (1) to (3) of Theorem 3.4.13, then we also write

G = (T, N, po, AO) .

From a result established in [205], it follows that the above theorem does not hold for probabilistic grammars . It is clear that every fuzzy production p over S is completely characterized by the collection D (p) of all triples (s, t, p) where p = p(s, t) > 0. We also write the triple (s, t, P) as s P --> t and refer to it as a discrete fuzzy production over S. Let D(p) denote the set of all discrete fuzzy productions over S. We associate with D(p) the function PD from S* x S* into >o such that V(s, t) E S* x S*, PD (s, t)

_ 0 F(s, t)

if F(s, t)

0,

where F(s, t) = {p I (s -P--> t) E D} . Clearly, if the set {s I (s --P-+ t) E D} is finite and for every s, t E S*, the set F(s, t) is empty or bounded, then A set D of discrete fuzzy productions PD is a fuzzy production over S. over S satisfying these two conditions is said to be admissible . In terms of discrete fuzzy productions, Theorem 3.4 .13 states that there exists a distinguished symbol Ao such that (1) Ao is the start symbol of G; (2) Ao does not occur on the right hand side of any discrete fuzzy productions of G; and (3) no discrete fuzzy productions of G has Ao as a proper subword on the left hand side. Notation 3.4.14 Let D be a collection of discrete fuzzy productions over S. Then r =:g#> w mod[(dl, ki)(d2, k2) . . . (dn , k n )], where r, w E S*, d2 = Pi (s2 - + t2) E D, k2 E N, i = 1, 2, . . . , n, and p = plp2 . . . pn if and only if there exist v2 E S*, i =O,1, 2, . . . , n, such that vo = r, vn = w, and ki v2_1 v2 mod(s 2 , t2) for every i such that 1 S i S n . Otherwise, r =4 w mod[ (d1, k1) (d2, k2) . . . (dn , kn )] . Example 3.4.15 Let N, T, and P be as defined in Example 3.4 .10 . Let r = aAAD and w = aABb. Let vo = aAAD, vl = aAAb, and v2 = aABb . Let dl = (s l ~ tl) and d2 = (s2 ~ t2 ), where sl = D, tl = b, 82 = A, and t2 = B. Then from Example 3.4 .10, we have that kl = 1 kl vl mod(s l , tl) 2 v2 mod(s 2 , t2) . Hence and k2 = 2. Thus vo and vl ~ aAAD 4> ABb mod[(D - b), (A B)], where p = pipe . © 2002 by Chapman & Hall/CRC

108


It is clear that, in the above, p is uniquely determined by r, w E S* and w E (D x N)* . Thus we may define a set of functions A,p from S* x S* into R~' ° , one for each z E (D x N) * such that A, (r, w) = p if and only if r ==~, w mod(z) . Theorem 3.4.16 A is a type 0 fuzzy language over T if and only if there exists a quadruple (T, N, D, h) such that the following properties hold: (1) T and N are disjoint finite nonempty sets; (2) D is an admissible set of discrete fuzzy productions over TU N such P

that (s --> t) E D implies s E (T U N)*N(T U N)* ; ; and such that for every r E T*, (3) h is a function from N into R'-° A(r) =V V,EW* VAEwh(A)A,(r,A),

(3.6)

where W = D x N. In addition, (a) A is finitary if and only if D is finite;

(b) A is strict if and only if every d = (s --P-> t) E D is strict, i.e., p E [0,1] ;

(c) A is context-sensitive if and only if every d = (s -P--> t) E D is context sensitive, i.e ., lsl < ltl ;

(d) A is context-free if and only if every d = (s __P_+ t) E D is context free, i.e ., s E N; (e) A is weak regular if and only if every d = (s __P_+ t) E D is weak regular, i.e ., s E N and t = uA, where u E T* and A E NU {A}; (f) A is regular if and only if every d = (s -P--> t) E D is regular, i.e., sEN and t=aA, where aET and AENU{A} .

Proof. Suppose A is a type 0 fuzzy language over T. By Theorem It 3.4 .13, A = AG for some max-product grammar G = (T, N, p, AO) . follows that the quadruple (T, N, D(p), h), where h(Ao) = 1 and h(A) = 0 for A = Ao, has the desired properties . Conversely, let (T, N, D, h) be a quadruple satisfying conditions (1) to (2), and A a fuzzy language over T satisfying (3.6) . It follows that A = AG, where G = (T, N, {PD}, h) is a max-product grammar . The remainder of the proof follows easily. Theorem 3.4 .16 yields another formulation of max-product grammars and fuzzy languages generated by max-product grammars . This formulation illustrates the fact that max-product grammars may be viewed as a model of fuzzy grammars .

3 .5

Weak Regular Max-Product Grammars

In this section, we study weak regular max-product grammars and their relation to asynchronous max-product automata . © 2002 by Chapman & Hall/CRC

3.5 . Weak Regular Max-Product Grammars

109

Theorem 3.5.1 If A is a strict weak regular fuzzy language over T, then

A = AG for some strict weak regular max-product grammarG = (T, N, p, AO), where p(A, B) = 0 for all A, B E N, and p(A, A) = 0 for all A E N\{Ao} . If A is finitary, then for every A E N, p(A, t) > 0 implies t = aB, where aET and BENU{A}, or A=Ao and t=A .

Proof. By Theorem 3.4 .13, A = AG,,, where Go = (T, N, po, AO ) is a strict weak regular max-product grammar . Let pn, n = 0, 1, 2, . . . , be functions from (T U N) * x (T U N) * into [0,1] defined recursively as follows : po(s, t)

~

1 0

ifs=tEN

otherwise

and i

Pn+1 (s, t)

VAEN[Pn(A, t)po(s, A)

=

~O

if s, t E N

otherwise .

Let pl be a function from (T U N) * x (T U N) * into [0,1] such that for all s, t E (T U N)*, Pi(s, t)

_ { o {Po(s, t),

Define G = (T, N, p,

Vn o VAEN[Po(A, t)p' (s, A)]}

AO)

otherwise .

such that for all s, t E (T U N)*,

Pi (s, t) P(s, t) =

if sEN,t~N

V{Pi (s, t), VAEN [Pi (A, A)Pj (s, tA)]}, 0

if sEN,tET*N or s=Ao, t=A if sEN,tET+

otherwise .

It follows that A = AG and G has the desired properties. The second part of the theorem follows from the fact that every weak regular discrete fuzzy production A --P-+ uaA', where u E T, a E T, and A' E NU {A}, may be replaced by A -P--> uA" and A" - aA', where A" is a new nonterminal symbol. In terms of discrete fuzzy productions, the above theorem states that every strict weak regular fuzzy language is generated by a strict weak regular max-product grammar containing no discrete fuzzy productions of the form A F B, where A, B E N, or A -P --> A, where A E N\{Ao}. In addition, every finitary strict weak fuzzy language is generated by a weak regular max-product grammar containing only regular discrete fuzzy productions and AO -P--> A. Theorem 3.5.2 A is a finitary strict weak regular fuzzy language if and only if there exists a strict regular fuzzy language A' over T such that A(r) _ A'(r) for all r :?~ A. © 2002 by Chapman & Hall/CRC

110


Proof. Suppose A = AG, where G = (T, N, p, AO) is a finitary strict weak regular max-product grammar . By Theorem 3.5.1, we may assume that D(p) contains only regular discrete fuzzy productions and AO -P--> A. Let D be the collection of all regular fuzzy productions in D(p), and G' = (T, N, PD, AO). Clearly, G' is a strict regular max-product grammar, and A(r) = AG' (r) for all r :?~ A. Conversely, suppose A(r) = AG (r) for all r :?~ A, where G = (T, N, p, AO) is a strict regular max-product grammar . Let D be the collection of all discrete fuzzy productions in D(p), and A P ---+ AP, where p = A(r) . Then G' = (T, N, PD, AO) is a finitary strict weak regular max-product grammar such that A = AG' . Definition 3.5.3 An asynchronous max-product automaton (AMA)

is a sextuple M = (Q, X, Y, p, h, g), where Q, X, and Y are finite nonempty sets, p is a function from Q x X x Y* x Q into Z> O, and h and g are functions from Q into Z> O. If, in addition, the image of the functions p, h, and g are subsets of [0,1], then M is called strict.

Definition 3.5.4 Let M = (Q, X, Y, p, h, g) be an AMA . (1) M is called finitary if for all q, q' E Q and u E X, p(q, u, y, q') > 0 except for finitely many y E Y* . (2) M is called synchronous if for all q, q' E Q and u E X, p(q, u, y, q') > 0 implies y E Y. (3) M is called

normalized if for all q E Q and u E X,

E p(q, u, y, q) [0,1], g : Q ----> [0,1], and p : Q x X x Y* x Q is defined as follows: p(q1, u, ll, q2) p(q2, u, 0, q1)

= =

[0,1]

1

2~

3 4~

and p(q, u, y, q') = 0 for any other (q, u, y, q~) . Then pm(g1, uu,110, pM (q2,

q1)

uu, 011, q2)

1

2

3 8~ _3 4 3 8'

3 4 _1

2

Let ynn, yn,n-1, znn, and zn , n _1 denote, respectively, the following alternating sequences of 11's and 0's 110110 . . . 110, 110110 . . . 11, 011011 . . . 011, 011011 . . . 0110,


where 11 and 0 appear n times, where 11 appear n times and 0 appear n - 1 times, where 0 and 11 appear n times, where 0 appears n times and 11 appears n - 1 times,


112 nEN,n>2. Then

PM (g1,U2n ,Ynn~g1) M 2n-1 p (gl~u ~yn,n-1, q2) 21,2n, znn~ q2) PM(g2, M 2n-1 , p (g2~u zn,n-l, gl) FM (u2n ,ynn) FM(U'2n-1, yn,n-1) FM (u2n ,znn) F (u2n-1 ,zn,n-1) DM (U 2n ) D M(u2n-1)

= = = =

=

= _ _ _

(2 4 2(I)n(3)n-1 ¢

2 (4 3 n 1 n-1 ( 4) ( 2)

h(gl)(2)n(3)ng(ql),

= =

h(gl)(2)n(4)n-1g(g2),

=

h(g2)(4)n(2)n-1g(g1),

h(g2)(4)n(2)ng(g2)'

V{FM(U2n, y) I Y E Y* (h(gl)()n(4)ng(ql)) V (h(g2)(4)n()ng(q2)) 2 2 VJFM(u2n-1 y E Y* y) I

V (h(g2)(3)n(2)n-1g(gl)), (h(gl)(2)n(3)n-1g(g2)) 4 4 RM (ynn) M R (yn,n-1)

=

RM(znn)

= =

RM( zn,n-1)

DM (u 2n ), M 2n-1 D (u DM(,u 2n) ,

-

DM(u2n-1

Theorem 3.5.7 A is a (finitary) (strict, normalized weak regular fuzzy language if and only if A = RM for some (finitary) (strict, normalized AMA M.

Proof. Suppose A = AG, where G = (T, N, P, h) is a (finitary) (strict, normalized) weak regular mar-product grammar. Let AO ~ T U N and No = N U {Ao} . Define M = (No, P, T, p, h', g) so that for all p E P, rET*andA,A'ENo,

p(A, p, r, A') =

p(A, rA') p(A, r) 0

g(A)={

1 0

if A, A' E N if A E N, A' = Ao

otherwise

if A = Ao if AzAAO

and h'(A) - { o (A)


if A

E Ao .

3.5. Weak Regular Max-Product Grammars

113

It follows that A = R" and M has the desired properties. Conversely, suppose A = RM , where M = (Q, X, Y, p, h, g) is a (finitary) (strict, normalized) AMA M. Without loss of generality, we may assume that Q nY = 0. Let qo ~ Q U Y and Q' = Q U {qo} . For all u E X, we associate the fuzzy production p. over Y U Q', where p~ = Q and for all q E Q, y E Y*, and t E (Y U Q)* , p . (q, t) _

p(q, u, y, q~) 0

if t = yq' otherwise .

Let po be the fuzzy production over Y U Q' such that 'PO = {qo} and for every t E (Y U Q')*, if t=qE otherwise .

0(q)

PO(go,t) _

Define G = (Y, Q', P, h') such that P = {po} U {p. I u E X} and 1 0

h(q) _ It follows that A =

AG

if q = qo if gzAqo .

and G has the desired properties.

Theorem 3.5.8 If A is a finitary strict weak regular fuzzy language, then A = RM for some synchronous strict AMA M with a single input . Proof. Let A = AG, where G = (T, N, p, AO) is a weak regular maxproduct grammar satisfying the conditions given in Theorem 3.5.1. Let Al , A2 ~ T U N and N' = N U {Al, A2 }. Define M = (N', {p}, T, p, h, g) so that for every r E T and A, A' E N, p(A, p, r, A') =

p(A, rA') p(A, r) 0

_ 1 g(`4) - { 0

if A, A' E N if A E N, A = Al , and r :?~ A otherwise

ifA=A 1 orA=A2 otherwise

and h(A) =

P(AO, A) 1 0

if A = A2 if A = AO otherwise .

It follows that A = R" and M has the desired properties. Theorem 3.5.9 A = R' for some Dm'synchronous (strict) AMA M = (Q, X, Y, p, h, g) if and only if A = for some synchronous (strict) AMA M' = (Q, Y, X, p~, h, g) © 2002 by Chapman & Hall/CRC

114

3 . Fuzzy Machines, Languages, and Grammars

Proof. Suppose A = R', where M = (Q, X, Y, p, h, g) is a synchronous (strict) AMA . Define M' = (Q, Y, X, p', h, g), where for every q, q' E Q, vEY,andxEX*, p(q, x, v, q') p' (q, v, x, q) - { 0 It follows that A = Dlvi .

if x E X otherwise .

The converse follows in a similar manner .

Definition 3 .5 .10 A max-product automaton(MA) is a quintuple M = (Q, X, p, h, g), where Q and X are finite nonempty sets, p is a function from Q x X x Q into Z>°, and h and g are functions from Q into R'-° . If, in addition, the images of p, h, and g are subsets of [0,1], then M is called strict . In the above definition, Q and X are, respectively, the state and input sets . p(q, u, q') is the grade of membership that the next state of M is q', given that the present state of M is q and the input symbol u is applied. h(q) and g(q) are the grade of membership that q is the initial state of M and q is a final state of M, respectively. Definition 3 .5 .11 Let M = (Q, X, p, h, g) be an MA . (1) Define the function pm from Q x X x Q into q', q"EQ, aEX, and xEX*, _ pM (q, A, q~)

1 0

by for every

if q' = q~~ if q' zA q"

and p'

(q, ux, q') = V {p(q, u, q')p' (q, x, q)

(2) Define the function A"' from X* into A " ( x) =

V{h(q)p ' (q, x, q)g(q)

I

q E Q} .

by for every x E X*,

I

q, q E Q} .

A" is the fuzzy language accepted by M . Theorem 3 .5 .12 The following statements are equivalent: (1) A _ A m for some (strict) MA M ; (2) A = DM for some synchronous (strict) AMA M with a single output ; (3) A = DM for some (strict) AMA M = (Q, X, Y, p, h, g), where there exists m E O such that p(q, u, y, q') )

if and only if there exists Aj, E N, rj,,j, E T*, i = 1,2, . . . n, and u E T such that r = rjojlrj1j2 . . . rj,, ,j,,u, jo = 0, (Aj,, -P-"-> u) E D(p), and for every i = 1, 2, .. ., n, Aj,, ~ rj,,j,Aj, mod(w2) for some w2 E (D' x N)* or w2 E (D" x N)*, II ~'3 lpj > c. Let N = {Ao, A,, .Ar }. This shows that L(AG, c, >) is a finite union of regular languages of the form rjL2r3L4 . . . rn_1Lnr n+ 1, where the is are in T* and the L's are languages generated by type 3 phrase structure grammars of the form GZ, = (T, N, Pi , AZ ), i, j = 0, 1, 2. . . . , m, where Pj contains all productions of the I form Aj ---+ A and s ---+ t such that (s --> t) E D' . Thus L(AG, c, >), well as L, is regular . The converse follows easily. Theorem 3.6.7 Let L = L(A, c, >), where 0 S c < 1 . L is regular if the following conditions hold: (1) A = RM for some finitary strict AMA M; (2) A = DM for some strict AMA M; (3) A = A m for some strict MA M; or (4) A = AG for some strict weak regular max-product grammar G = (T, N, p, AO), where for each A E N, EtE(TUiv)* p(A, t) is finite. Proof. By Theorems 3.5.13 and 3.6.6, it suffices to prove (4) . Suppose Let G' _

A = AG, where G = (T, N, p, AO) has the desired properties . (T, N, p', AO) be such that for every s, t E (T U N)*, _

p'(s, t) - {

p(s, t)

0

if p(s, t) > c otherwise .

Clearly, G' is a finitary strict weak regular max-product grammar . addition, L = L(AG', c, >) . Thus by Theorem 3.6 .6, L is regular . m

In

Theorem 3.6.8 Let G = (T, N, p,

AO) be a weak regular max-product grammar where the range of p is a subset of [0, m] with m < 1 . Then L(AG, c, >) is finite for all 0 ) implies lrl S n, where n is the largest integer such that mn > c. m Note that Theorems 3.6.5 to 3.6.8 hold if > is replaced by >, or =. © 2002 by Chapman & Hall/CRC

118


Theorem 3.6.9 L(A, oo, =) is regular for all finitary weak regular fuzzy languages A. Proof. Let L = L(AG, oo, =), where G = (T, N, p, AO) is a finitary weak regular max-product grammar . Without loss of generality, we may assume that D(p) contains only discrete fuzzy productions of the forms A -P--> aB, A -P--> B, and A -P--> A, where A, B E N and a E T. Let No be the subset of N consisting of all A E N such that A ~ A mod(w) for some w E (D(p) x N)* and p > 1. It is clear that r E L\{A} ~ there exist A E No and w, z E T* such that r = wz, AO wA mod(w'), and A =Zp:=~ z mod(w") for some w', w" E (D(p) x N)* and p' > 0, p" > 0. Thus L\{A} is the union of all languages of the form L(G'4)L(G" ), where A E No . Therefore, for every A E N, G'4 = (T, N, P', AO) and G" = (T, N, P", A) are phrase structure grammars such that P contains all productions of the form s ---> t, where (s -P-+ t) E D(p), t ~ T* and A ---> A; and P" contains all productions of the form s ---+ t, where (s P --> t) E D(p) . Clearly, L(G'q) and L(G") are regular for all A E N. Thus L\{A}, as well as L, is regular . Corollary 3.6.10 L(A, languages A.

) for some finitary weak regular fuzzy language A and c >, 0, then L = L(AG, c, >) for some finitary max-product grammar G = (T, N, p, AO), where D(p) contains only regular discrete fuzzy productions and AO --P-+ A. Proof. Let L = L(AG,, c, >), where Gl = (T, Nl, pl , Al) is a finitary weak regular max-product grammar and c >, 0. By Theorem 3.6 .5 we may assume that 0 t, where (s --P-> t) E D(p) and p < oo; (ii) Ao -P--> t, where (Al -P--> t) E D(p) and p < oo ; (iii) s -I -> t, where (s ---> t) E P ; (iv) AO - I -+ t, where (A2 ---> t) E P . Clearly, D is finite and contains only regular discrete fuzzy productions, and AO --P-+ A. Moreover, it can be shown that L = L(AG, c, >), where © 2002 by Chapman & Hall/CRC

3.6. Weak Regular Max-Product Languages

11 9

G = (T, N, PD, AO) is a finitary max-product grammar with the desired

properties. Let c >, 0. Let 1, denote the family of all languages of the form L(A, c, > ), where A is a finitary weak regular fuzzy language. Let I = U'ER>o 1c. Theorem 3.6.12 The following statements are equivalent: (1) L E 1 c ; (2) L\{A} = L(A, c, >) for some regular fuzzy language A;

(3) L = L(Rm, c, >) for some synchronous AMA M; (4) L = L(Rm, c, >) for some synchronous AMA M with a single input; (5) L = L(Dm , c, >) for some AMA M such that the range of the transition function p is a subset of [0, m] for some m E R> O ; (6) L = L(Dm , c, >) for some synchronous AMA M with a single output; and m (7) L = L(A , c, >) for some MA M.

Proof. The proof follows from Theorems 3.5.14 and 3.6.11. Another characterization of I c will be given below via languages generated by type 3 weighted grammars under maximal interpretation [188]. (Also, see Chapter 5.) Definition 3.6.13 A weighted grammar of type i (i = 0, 1, 2, 3) is a triple G,, = (G, S, 0), where G = (T, N, P, AO) is a type i phrase structure grammar, S is a function from P into R> O, and 0 is a function from P x P into R'-° . The language Lm (Gw, c) generated by G,, with cut point c E R> O is said to be under maximal interpretation if there is at least one derivation of P with weight greater than c (see [188] and Section 5.8).

Theorem 3.6.14 Let c >, 0.

L E Cc if and only if L = L .. (G,,, c) for some weighted grammar G,, of type 3.

Proof. Suppose L = L(AG, c, >) for some finitary weak regular maxproduct grammar G = (T, N, p, AO) . Let G' = (T, N, P, AO) be a phrase structure grammar such that P contains all productions of the forms --> t, where p(s, t) > 0 . Clearly, G' is of type 3. Let G,, = (G', S, 0) be the type 3 weighted grammar such that for every q = (s --~ t), q' = (s' ---, t') E P, we have S(q) = p(s, t) and O(q, q') = p(s', t') . It can be shown that L = Ln, (Gw, c) . Conversely, suppose L = Ln, (Gw, c) for some type 3 weighted grammar Gw = (G, S, 0), where G = (T, N, P, AO) . With each n E N and q E P U {A}, we associate the abstract symbol A,,,. Let N' _ {ATq I n E N U {A}} and q E P U {A}, and D be the collection of all discrete fuzzy productions over T U N' of the forms (1) Ann rq) E P;

sn Aqt, where n E N, q E N U {A}, r


E T*, and t = (n

120

3. Fuzzy Machines, Languages, and Grammars (2) A nq

O(q,

rAgt,wheregEP,nENU{A},rET*, and t=(nom rq) E P; and I (3) AAg - -+ A, where q E P. Let G' = (T, N', PD, h), where h(Anq ) = 1 if q = A and n z,4 A, and h(A nq ) = 0 otherwise . Clearly, G' is a finitary weak regular max-product grammar . Furthermore, it follows that L = L(AG', c, >) . Thus L Combining Theorem 3.6.14 and Theorems 10 and 12 of [188] yields : Theorem 3.6.15 The family of regular languages is a proper subfamily of I and there exist context-free and stochastic languages [235] which do not belong to 1 . Moreover, {anbm I n > m >, 1} E 1, while {anbn n>, 11~1 . 3 .7

Properties of £

In this section, we examine the properties of 1, and 1. Let 1Z denote the family of all regular languages . Theorem 3.7.1 1Z C 1, for all c >, 0. Proof. The proof follows from Theorem 3.6 .6 . Theorem 3.7.2 10 =R. Proof. The proof follows from Theorem 3.6 .4 . Theorem 3.7.3 1 = 1, for all c > 0. Proof. Suppose c > 0 and L E 1. Then L = L(AG, c, >) for some finitary max-product grammar Gl = (T, N, P, h l ) and c l >, 0. By Theorems 3.7.1 and 3.7 .2, it suffices to consider the case where c l > 0. Let G = (T, N, P, h), where h(A) _ (clcl)hl(A) for all A E N. Clearly L = L(AG, c, >) . Therefore, L Hence -P C_ -P, . Since -P, C_ 1, it follows that I = 1,. Theorem 3 .7.4 1 is closed under union, i.e ., LI , L2 E I implies L,UL 2 E Proof. Let LZ = L(AGi ,1, >), where GZ = (Ti, Ni, pi, A Z ) is a finitary weak regular max-product grammar for i = 1, 2. Without loss of generality, we may assume that Tl n N2 = T2 n Nl = Nl n N2 = 0. Let T = Tl U T2 , No = Nl U N2 , Ao ~ T U No and N = No U {Ao} . Let D = D(pl) U D(P2) U {AO --'-+ Al, Ao --'-> A2} . Clearly, G = (T, N, PD, Ao) is a finitary weak regular max-product grammar . Furthermore, it can be shown that Ll U L2 = L(AG,1, >) . Thus Ll U L2 E 1 . 0 © 2002 by Chapman & Hall/CRC

3.7. Properties of X

121

Theorem 3.7.5 1 is closed under intersection with regular languages, i. e., Li

E

I, L2

E

1Z implies Ll

n L2 E -C-

Proof. By Theorem 3.6 .12, LZ = L(Am~, 2, >), where MZ = (Xi , QZ, p2, h2, gZ) are max-product automata, i = 1, 2. Since L2 E 1Z, we may assume that the images of p2, h2 , and 92 are subsets of {0,1}. Let M = (Q, X, p, h, g), where Q = Ql x Q2, X = Xl n X2, and for every qZ, qZ E QZ with i = 1, 2 and u E X, we have p((qi, q2), u, (ql, q2)) =p1(gi, u, gl)p2(g2, u, q2), h(gi,q2) = hi(gi)h2(q2), and g(gl,g2) =gi(gi)g2(q2) . Clearly, for every xEX*, Am (x)

Thus Ll

Am'( x

_

n L2 = L(Am , 2, >).

0

)

Hence Ll

if x if x

rl L2

E L

L.

E -C .

0

Example 3.7.6 1 is not closed under intersection, i. e., there exist LI, L2

E

I such that Ll rl L2 ~ 1 : Let GZ = ({a, b}, {A0, Al, A2}, pZ, Ao) for i = 1, 2, where (1) D(pi) contains the discrete fuzzy productions Ao - Al, A1 I 0 .5 I --> A; aAl, A1 - _+ A2, A2 - + bA2, and A2 0 .5 (2) D(p2) contains the discrete fuzzy productions Ao - Al , A1 I 2 I --> A. aA l , A1 - _+ A2, A2 - _+ bA 2, and A2 Clearly, Gl and G2 are finitary weak regular max-product grammars. Furthermore, Li = L(AG,, 2 > >) = ja'b' I n % m % 1}~ L2 = L(AGz,2>>) = {anbm I m>, n>l}~ and Ll

n L2 = {an bm

I n >, 1} .

By Theorem 3.6 .15, Ll rl L2 ~ -P .

Theorem 3.7.7 L1 E I, L2 E R implies L1\L2 E I. Proof. L1\L2 = L1 n L2, where L'2 is the complement of L2, which is also a regular language. Thus by Theorem 3.7.5, L1\L2 E I. Theorem 3.7.8 1 is closed under concatenation with regular languages, i.e ., Ll E I, L2 E R implies L1L2 E I and L2 L1 E 1. © 2002 by Chapman & Hall/CRC

12 2


Proof. Let LZ = L(AG i , 2, >), where GZ = (Ti , Ni , pi, AZ) are finitary weak regular max-product grammars, i = 1, 2 . Since L2 E 1Z, we may assume that for all s, t E (T2 U N2)*, p(s, t) = 0 or 1. Moreover, without loss of generality, we may assume that Tl n N2 = T2 n Nl = Nl n N2 = 0. Let D = D(p2) UD', where D' is the collection of discrete fuzzy productions of the forms (1) s --P-+ t, where (s -P--> t) E D(pl) and t ~ Ti , and (2) s ~ tA, where (s ~ t) E D(pi) and t E T,* . Clearly, LIL2 = L(AG .2, >), where G = (T, N, PD, A), T = Tl U T2 , and N = Nl U N2. Since G is a finitary weak regular max-product grammar, it follows that LIL2 E 1. In a similar manner, it can be shown that L2 L1 E 1. Definition 3 .7.9 For each a E T, let Ta be a finite nonempty set and O(a) C Ta . Let O(A) = {A} and O(ar) = O(a)O(r) for every a E T and r E T* . Then 0 is called a substitution . If L C T*, then O(L) = UTELO(r) . If O(a) consists of a single word in Ta for each a E T, then is regarded as a function from T* into (UaETTa) * and is called a homomorphism .

Theorem 3 .7.10 1 is closed under regular substitution, i.e., if L C_ T* is in I and O(L) E 1 .

0

is a substitution such that O(a) E 1Z for all a E T, then

Proof. If A E L, then O(L) = O(L\{A}) U {A}. Thus by Theorem 3.7 .4, it suffices to consider the case where A ~ L. By Theorem 3.6 .12, L = L(AG, 2, >) for some regular max-product grammar G = (T, N, p, ro). For every a E T, O(a) = L(Ga ) for some phrase structure grammar Ga = (Ta , Na , Pa , a), where Pa contains only productions of the forms a ---+ A, m ---+ u, and m ---+ um' with m, m' E Na and u E Ta. Without loss of generality, we may assume that T n Na = {a} and Ta n N = Na n Nb = Ql for all a, b E T and Ua,TTa, N = UaITNa, and -P= UaETPa . _a :?~b . Let T For each m E N U {A} and n E NU {A}, we associate the abstract symbol A,n. Let D be the collection of all discrete fuzzy productions of the forms (1) A nn --'-+ Am'n, where m_ E N, m' E N U {A}, u E T U {A}, nENU{A}, and (m~um')EP; (2) Ann -P--> Amn', where n E N, n' E N U {A}, m E T, and (n --P-+ mn') E D(p) ; (3)

AAA

'-->

A.

It follows that 0(L) = L(AG_ N', PD, AA,.) and z, >), where G' N' = {Amn I m E N U {A}, n E N U {A}} . Since G' is a finitary weak regular max-product grammar, it follows that O(L) E 1. 0 Corollary 3.7.11 1 is closed under homomorphism. m Definition 3.7.12 A sequential transducer is a sextuple (Q, X, Y, H, qo,

F), where Q, X, and Y are finite nonempty sets of states, input and output symbols, respectively, qo E Q is the start state, F C_ Q is the set of accepting states, and H is a finite subset of Q x X* x Y* x Q.


3.7. Properties of X

123

Definition 3.7.13 Let M = (Q, X, Y, H, qo, F) be a sequential transducer.

For each x E X*, define M(x) to be the set of all y E Y* with the property that there exist xl, x2 , . . . , xk E X*, yl, y2, . . . , yk E Y*, and ql, q2, - . . , qk E Q such that x = xlx2 . . . xk, y = Y1Y2 . . . yk, qk E F, and (qZ-1, xi, yZ, qz) E H for each i such that 1 _ i _ k. Furthermore, for each L C X*, let M(L) = UxELM(x), and for each L C Y*, let M-1 (L) = {x E X* I M(x)

n L z,4 Ql}.

It follows from the proof of Theorem 3.3 .1 in [77, p.92] that any family of languages that contains all regular sets and is closed under union, regular substitution, and intersection with regular sets, is closed under mappings induced by sequential transducers of the form (Q, X, Y, H, qo, Q) . By a slight modification of the proof, it can be shown that the same family is closed under mappings induced by arbitrary sequential transducers . Thus we have the following result . Theorem 3.7.14 1 is closed under mappings induced by sequential transducers, i.e., if L E I and M is a sequential transducer, then M(L) E 1. Theorem 3.7.15 1 is closed under pseudo inverse mappings induced by sequential transducers, i.e., if L E I and M is a sequential transducer, then M-1 (L) E 1.

Proof. Let L E I and M = (Q, X, Y, H, qo, F) be a sequential transducer . Let M' = (Q, Y, X, H', qo, F) be the sequential transducer such that H' = {(q, y, x, q') I (q, x, y, q') E H} . It is clear that M-1 (L) = M(L) . Thus by Theorem 3.7 .14, M-1 (L) E 1 . Many of the well-known devices are special cases of sequential transducers, e.g., nondeterministic generalized sequential machines, sequential machines, etc. Theorems 3.7.14 and 3.7.15 show that I is closed under mappings and pseudo inverse mappings induced by these devices . Corollary 3.7.16 1 is closed under inverse homomorphism, i.e., if 0 is

a homomorphism from X* into Y* and L C_ Y* is in 1, then 0-1 (L) _ {x E X* 0(x) E L} E 1 .

Let L C_ T* . Let LT = {rT I r E L}, where AT = AT for all r, w E T* .

d (rw)T = WTrT

Theorem 3.7.17 1 is closed under transposition, i.e ., L E I implies LT E Proof. Let L E 1. By Theorem 3.6 .12, L = L(AM ,1, >) for some MA M = (Q, X, p, h, g) . Let M' = (Q, X,p', g, h), where p'(q', u, q) = p(q, u, q') © 2002 by Chapman & Hall/CRC

12 4


for all q, q' E Q and u E X . Clearly, M' is a MA and LT = L(A'vi ,1, >) . Thus LT E 1. The following theorem has been shown [79] to be true for any family of languages closed under union, homomorphism, inverse homomorphism, and intersection with regular sets. Theorem 3.7.18 Let L E I and R E 1Z . Then (1) (2) (3) (4) (5) are

LIR = {x I xy E L for some y E R}, R\L = {x I yx E L for some y E R}, Init(L) = {x I xy E L for some y}, Fin(L) = {x I yx E L for some y}, and Sub(L) _ {x I yxz E L for some y and z} all in 1.

Let L C T* and x E T* . Let Dx(L) = {y I xy E L} and Dx(L) = {y

yx E L) .

Theorem 3.7.19 If L C_ T* is in I and x E T*, then Dx(L) E I and Dx (L) E a£' .

Proof. The proof follows from Theorem 3.7.18 and the fact that Dx (L) _ {x}lL and Dx(L) = Ll{x} . m Theorem 3.7.20 Let L C_ T* . Then the following statements are equivalent. (1) L E 1. (2) There exists k E hY such that Dx(L) E C for all x E T* such that = k. Ixj (3) There exists k E hY such that ~x(L) E I for all x E T* such that Ixj =k .

Proof. By Theorems 3.7 .17 and 3.7 .19, it suffices to show that (2) implies (1) . Suppose (2) holds . It is clear that L is the union of all languages of the form xDx (L), where x E T* and Ixj = k. Thus by Theorems 3.7 .4 and 3.7.8, L E C. Example 3.7 .21 1 is not closed under concatenation, i.e ., there exists L E I such that L* ~ 1: Let L = fanbm I m > n > 1} E 1. Suppose L* E

Then, by Theorem 3.6 .12, L = L(AG,1, c), where G = ({a, b}, N, p, AO) is a regular max-product grammar. Let ko be the number of elements in N and k > ko . Clearly, x = (ab) 2b (a 2 b2 ) 2b (a 3 b3 ) 2b . . . (akbk)2b E L* . Thus there exists A E N such that x = xibalblax2 ba2 b2ax3ba3b3ax4, 0 < 11 S 12 < 13 , and AO ~ x 1 bailA mod(wl ), A ~ bi2ax2 bai2A mod(w2 ), A ~ bl2ax3 bal2A mod(w 3 ), A ~ b1 2ax 4 mod(w4 ), PIp2p3p4 > 1, for some w1, w2, w3, w4 E (D(p) x 1)* . Hence x1ba l lbl2 ax3ba l3 bl laxeba leb13 ax4 E L, a contradiction.


3.8. Exercises 3 .8

125

Exercises

1. Let T be a norm on [0,1] . Prove that Va E A, aTa is replaced by > or = .


Chapter 4

Fuzzy Languages and Grammars 4.1 Fuzzy Languages

We introduced in the previous chapter the notion of max-product grammars and languages . Before returning to these, we introduce some basic ideas of fuzzy languages such as certain normal forms . We base our work on [122]. Formal languages are quite precise while natural languages are quite imprecise. To reduce the gap between them, it is natural to introduce randomness into the structure of formal languages . This leads to the concept of stochastic languages [64, 66, 67, 111, 232] . Another possibility lies in the introduction of fuzziness . It appears that much of the existing theory of formal languages can be extended quite readily to fuzzy languages . As usual, we let T denote a set of terminals and N a set of non-terminals such that T n N = 0. As before, a fuzzy language is a fuzzy subset of T* . Let Ai and A2 be two fuzzy languages over T. The union of Ai and A2 is the fuzzy language denoted by Ai U A2 and defined by

NU

A2) (X) = A1 (x) V A2 (X)

Vx E T* .

The intersection of Aland A2 is the fuzzy language denoted by Ai and defined by (A,

n A2)

(x) = A1 (X) n A2 (X)

Vx E T* .

(4.1)

n A2 (4.2)

The concatenation of Ai and A2 is the fuzzy language denoted by A, A2 and is defined by V'x E T*, (AIA2) (x) = V{Ai (u) n A2 (v) I x = uv, u, v E T*} .


127

(4.3)

128

4. Fuzzy Languages and Grammars

By the distributivity of V and A, the operation concatenation is associative . Let A be a fuzzy language in T. Then the fuzzy subset A°° of T* defined by A' (x) = V{A'(x) I n = 0,1 . . . . } b'x E T* is called the Kleene closure of A . Informally, a fuzzy grammar may be viewed as a set of rules for generating the elements of a fuzzy subset. Recall that a fuzzy grammar, or simply a grammar, is a quadruple G = (N, T, P, S) in which T is a set of terminals, N is a set of nonterminals (T n N = 0), P is a set of fuzzy productions, and S E N. Essentially, the elements of N are labels for certain fuzzy subsets of T* called fuzzy syntactic categories, with S being the label for the syntactic category "sentence ." The elements of P define conditioned fuzzy subsets in (T U N) * . More specifically, the elements of P are expressions of the form ft (r

w)

= C,

C

> 0,

(4.4)

where r and w are strings in (T U N) * and c is the grade of membership of w given r. At times, we abbreviate ft (r , w) = C to r w or, more simply, r --> w . As in the case of nonfuzzy grammars, the expression r w represents a rewriting rule. Thus if r c w and s and t are arbitrary strings in (T U N) *, then we have __

__>

srt --C---> swt

(4.5)

and swt is said to be directly derivable from srt. If r, . , r m are strings in (T U N)* and Cz C,n rl -----> r2, . . . , rm-1 -----> rm, C2 ~ . . . , Cm > O,

then rl is said to derive r m in grammar G, or, equivalently, r, is derivable from r l in grammar G. This is expressed by rl rm or simply rl ==> rm . G The expression C

z C,n rl ---> r2~ . . .~rm_l-----> rm

(4.6)

is referred to as a derivation chain from rl to r . , A fuzzy grammar G generates a fuzzy language L (G) in the following manner . A string of terminals x is said to be in L (G) if and only if x is derivable from S. The grade of membership of x in L (G) is given by PG (x)

= V(ft (S, rl) n ft (rl, r2) n . . . A ft (rm, x)),


(4.7)

4.1 . Fuzzy Languages

129

where the supremum is taken over all derivation chains from S to x. Thus (4.7) defines L (G) as a fuzzy subset of (T U N)* . If L (G ) L (G2) in the sense of equality of fuzzy subsets, then the grammars Gl and G2 are said to be equivalent. Equation (4 .7) may be expressed as follows : I

f

G

PG

(x)

(x)

=

is the grade of membership of x in the language generated by grammar G; is the strength of the strongest derivation chain from S to x .

Let ft (S, rl) = cl, ft (rl, r2) = c2, . . , ft (rm, x) = cm+1 . writing (4 .7) in the form PG

(x)

= V (el n c2 . . . n c, +

1)

Then, on (4.8)

,

it follows at once from the associativity of A that (4 .7) is equivalent to a more general expression in which the successive is are derivable from their immediate predecessors rather than directly derivable from them, as in (4 .7) . Example 4.1.1 Let T = {0,1}, N = {A, B, S}, and let P be given by AB) ft(S, A) ,t (S, B) ft(AB, BA) ,t (S,

= = = =

0 .s 0 .8 0 .8 0 .4

0) ft (A,1) ,t (B, A) ,t (B, 0)

= = = =

,t (A,

0.s 0.6 0.4 0.2

Consider the terminal string x = 0. The possible derivation chains for this string are SA 0 .5-> 0,SB 0.2 -> 0,S 0.8 BA 0.5-> 0. Thus 0.8

0 .8

->

PG

(0)

0.4

->

->

->

= (0.8 n 0.s) V (0.8 n 0 .2) V (0.8 n 0.4 n 0.s) = 0.s .

Similarly, the possible derivation chains for the terminal string x = 01 are S 0.5 AB- -> OB -----> -----> ---> -----> ---> -----> 01, S~AB~BA~OA~01, andS~AB~BA~AA 0.5 OA 01 . Hence 0.6

->

PG

(01)= 0 .4V0.4V0 .2V0.4=0 .4 .

Let G be a fuzzy grammar . An important question that arises in connection with the definition of is whether or not there exists an algorithm for computing by the use of the defining equation (4.7) . G is said to be recursive if such an algorithm exists. PG

PG

(x)


13 0 4 .2

4. Fuzzy Languages and Grammars Types of Grammars

Similar to the usual definitions of non-fuzzy grammars, we define four principal types of fuzzy grammars as follows : Type 0 grammar The allowable productions are of the general form r -- > w, c > 0, where r and w are strings in (T U N) * . Type 1 grammar (context-sensitive) The productions are of the form r1Ar2 ~ rlwr2, > 0, with rl, r2, and w in (TUN) * , A in N, and w :?~A. The production S ----->A is also allowed . Type 2 grammar (context-free) Productions here are of the form A --> w, c > 0, A E N, w E (T U N) * , w :?~ A,andS~A . Type 3 grammar (regular) In this case, productions are of the form A aB or A a, c > 0, where a E T, A, B E N. In addition, S A is allowed . We ask the reader to compare the definition of a type 2 grammar here (with c = 1) and that of Definition 1 .8 .7. In the following, we concentrate on context-free grammars. However, there is a basic property of context-sensitive grammars that needs to be stated . It is easily shown that context-sensitive and hence also contextfree and regular grammars are recursive . This can be stated as an extension of [96, Theorem 2.2, p. 26] . c

C

c

__ c__~

__ c__~

Theorem 4.2.1 If G = (N, T, P, S) is a fuzzy context-sensitive grammar, then G is recursive.

Proof. First we show that for any type of grammar, the supremum in (4 .7) may be taken over a subset of the set of all derivation chains from S to x, namely, the subset of all loop-free derivation chains, i.e., chains in which no r2, i = 1 . . . . , m, occurs more than once. Suppose that in a derivation chain C, Cl C2 C,n C'rz+1 C=S---> rl ---> r2 . . .- __> rmmix,

r2 , say, is the same as rj , j > i. Let C' be the chain resulting from replacing

the subchain

C-j,+1 C.j C.j+1 r2 ---> . . . -----> rj ---> rj+l C

in C by r2 rj+1 . Clearly, if C is a derivation chain from S to x, then C' is also. However, n{Cl, .

. .

, CZ,

CZ+1 ~

. . .

,

Cj+l, . . . , Cm+l } G


n{Cl, . . .

, CZ,

C7+1~

. .

.

, C, ,,+l

f

4.3. Fuzzy Context-Free Grammars

131

and thus C may be deleted without affecting the supremum in (4.7) . Hence we can replace the definition (4.7) for PG (x) by PG (x)

= V A {ft (S, rl) , ft(rl, r2), - . . , ft(rm, x)},

(4.9)

where the supremum is taken over all loop-free derivation chains from S to x. We now show that for context-sensitive grammars, the set over which the supremum is taken in (4.9) can be further restricted to derivation chains of bounded length lo, where to depends on IxI and the number of symbols in T U N. If G is context-sensitive, then due to the noncontracting character of the productions in P, it follows that Jrj J > Ir2l

if j > i.

(4.10)

Let IT U NJ = k. Since there are at most k' distinct strings in (T U N)* length l and since the derivation chain is loop-free, it follows from (4.10) of that the total length of the chain is bounded by 1o=1+k+ . . .+klxl . We next provide a method for generating all finite derivation chains from S to x of length B)Aft(B,r),

(4.11)

where w (A ==~> B) = v{w (Al,

BI)

n . . . n ,t (Bm, B)}

(4.12)

with the supremum taken over all loop-free derivation chains from A to B. It follows that the resultant grammar, Gl, is equivalent to G. Second, we construct a grammar G2 equivalent to Gl in which there are no productions of the form A --c--, Bl B2 . . . Br , c > 0, m > 2, in which one or more of the B's are terminals . Thus suppose that BZ, say, is a terminal a. Then BZ in BIB2 . . . Br is replaced by a new nonterminal CZ which does not appear on the right-hand side of any other production . We then set It (A,BiB2 . . .BZ . . .B7 )=lt(A,B,B2 . . .CZ . . .Bm) .

(4 .13)

I We add to the productions of G the production CZ - ---> a. We do this for all terminals in Bl . . . Br in all productions of the form A ~ Bl . . . Br . We thus arrive at a grammar G2 in which all productions are of the form A a or A Bl . . . Br , m >_ 2, where all B's are nonterminals . Clearly, G2 is equivalent to Gl . Third, we construct a grammar G3 equivalent to G2 in which all productions are of the form A -----> a or A -----> BC, A, B, C E N, a E T. Consider a typical production in G2 of the form A -c > Bl . . . Br , c > O, m > 2. We


4.3 . Fuzzy Context-Free Grammars

133

replace this production by the productions A

__c___>

BD,

Dl I B2 D2

D,-2

i

I

(4.14)

Bm-,Bm,

where the D's are new nonterminals that do not appear on the right-hand side of any production in G2 . Once such replacements for all productions in G2 of the form A -c -> Bl . . . Br have been made, a grammar G3 that is equivalent to G2 is obtained. Hence G3, that is in Chomsky normal form, is equivalent to G. Example 4.3.1 Consider the following fuzzy grammar, where T = {a, b} and N = {A, B, S} and where the productions are as follows: 0 .8

S

0 .6

S

bA

Bib 0.3 A -> bSA

aB

Ana

B~aSB

To find the equivalent grammar in Chomsky normal form, we proceed as follows. 0 .8

0 .8

0 .6

First we replace S - _> bA by S _> C1 A, Cl b. Similarly, S aB is replaced by S ~ C2B and C2 ~ a. We replace A ~ bSA by 0.3 I 0.5 0.5 A C3SA, C3 - _> b; and B - _> aSB is replaced by B -> C4SB and 1

C4 ~ a. 0 .3 0.3_> Second, the production A C3SA is replaced by A C3Di, Dl 0.5_> 0.5 SA ; and the production B C4SB is replaced by B C4D2, D2 SB . Thus the productions in the equivalent Chomsky normal form are as follows: S

0.8

CIA

Cl imb 0.6

0.3

C3D1

D1 ~SA

C2~a

C3 -1-> b 0.5 B C4D2

Ana

D2 ~SB

Bib

C4~a

S

C2 B

A

We now consider Greibach normal form . Let G be any fuzzy context-free grammar . Then G is equivalent to a fuzzy grammar GG in which all productions are of the form A -----> or, where A is a nonterminal, a is a terminal, and r is a string in N* . The fuzzy grammar GG is in Greibach normal form. See [96, Theorem 4.6, p.65]. © 2002 by Chapman & Hall/CRC

13 4


Paralleling the approach used in [96], we state two lemmas that are of use in constructing GG .

Lemma 4.3.2 Let G be a fuzzy context-free grammar. Let A -----> r1Br2

be a production in P, where A, B E N, and rl, r2 E (T U N) * . Let B -----> wl , . . . , B ---> wk be the set of all B-productions (that is, all productions with B on the left-hand side . Let Gl be the grammar resulting from the replacement of each of the productions of the form A ---> r1 Br 2 with the productions A ---> rlwlr2, . . . , A -----> rlWTr2, in which y, (A, rlwlr2) = y, (A, rlwr2) Aft (B, w2) ,

(4 .15)

i = 1, . . . , k. Then Gl is equivalent to G. m

Lemma 4.3.3 Let G be a fuzzy context-free grammar. Let A -----> Ar2, i =

1, . . . , k, be the A-productions for which A is also the left most symbol on the right-hand side. Let A ---> wj, j = 1, . . . , m be the remaining Aproductions, with r2, wj E (T U N) * , i = 1, . . . , k. Let G2 be the grammar resulting from the replacement of the A ---> Ar2 in G with the productions: A -----> wj Z,

j =1 . . . , m

Z-----> ri z,

i=1, . . .,k,

y, (Z,wj Z)=y,(A,wj), ft (Z,r2)=ft (A,Are), y, (Z, riZ) = y, (A, Are) ,

i = 1, . . . , k .

Z-----> r2 ,

(4 .16) (4 .17)

where (4 .18)

Then G2 is equivalent to G. 0 With the use of these lemmas, we now derive the Greibach normal form for G. We first put G into the Chomsky normal form . Let the nonterminals in this form be denoted by A,, . , An . We then modify the productions of the form AZ -----> Ads, s E (T U N) * , in such a way that for all such productions, j >, i. This is done in the following stages . Suppose that it has been done for i Ads

(4 .19)

is a production with i _ k, then j > i. To extend this to Ak + 1-productions, suppose that Ak+1 -----> Aj s is any production with j < k+1. Using Lemma 4.3 .2 and substituting for Aj the right-hand side of each Aj-production, we obtain by repeated substitution productions of the form Ak+1 ---> Al s,


1 >, k + 1.

(4 .20)

4.3 . Fuzzy Context-Free Grammars

135

In (4.20), those productions in which l is equal to k + 1 are replaced by the use of Lemma 4.3 .3 . This results in a new nonterminal Zk+1 . Then by repeating this process, all productions are put into the form l > k, s E (N u {Zl , . . . , Z,,})*

Ak --~ Al s,

Ak -----> as,

aET

(4 .21) (4 .22)

Zk

with membership grades given by Lemmas 4.3 .2 and 4.3 .3 . By (4 .21) and (4.22), the leftmost symbol on the right-hand side of any production for A , must be a terminal. Similarly, for A ,-,,the leftmost symbol on the right-hand side must be either A , or a terminal. Substituting for A , using Lemma 4.3.2, we obtain productions whose right-hand sides start with terminals . Repeating this process for A,_2, . . . , Al, all productions for the AZ , i = 1 . . . . , m, are put into a form where their righthand sides start with terminals . At this stage, only the productions in (4 .23) may not be in the desired form. It follows that the leftmost symbol in s in (4 .23) may be either a terminal or one of the AZ, i = 1, . . . , m. If the latter case holds, application of Lemma 4.3 .2 to each ZZ production yields productions of the desired form. This completes the construction . Example 4.3.4 We convert into the Greibach normal form the following fuzzy grammar G. Let T = {a, b}, N = {Al, A2, A3}, and let the productions be in the Chomsky normal form : A A

Al ~ A2A3 A2 -4 A3A1 A2 ~ b.

3 ~ AIA2 3 ~ a

Stepl. The right-hand sides of the productions for A1 and A2 start with terminals or higher-numbered variables. We thus begin with the production A3 -----> AIA2 and substitute A2A3 for Al . Note that A1 -----> A2A3 is the only production with A1 on the left. The resulting productions are as follows:

Al

0 .8 0 .6

A2A3 b

A2 A3 ~ b

A2 ~ A3A1 A3 ~ A2A3A2

Note that in A3 ~ A2 A3A2 , 0.2 = 0 .8 A 0.2 .


136


The right-hand side of the production A3 -----> A2A3A2 begins with a lower-numbered variable . We thus substitute for the first occurrence of A2 either A3Al orb. The new productions are given below. A

Al A2

0.8

A2A3

0.6

b

2 ~ A3Al

A3 ~ A3AlA3A2

A3 ~ bA3A2

A

3 ~ a.

We now apply Lemma 4.3.3 to the productions A1 ---> A3AlA3A2, A3 ---> bA3A2 , and A3 ~ a . We introduce Z3 and replace the production A3 ~ A3Al A3A2 by A3 ~ bA3A2Z3 , A3 ---> aZ3 , Z3 ---> AlA3A2, and Z3 -----> AlA3A2Z3 . The resulting productions are as follows: A Al ~ A2 A3 A2 ---4 b

2 ~ A3Al 0.2 A3 ~ bA3A2

0.6

A3

0.5 ->

a

Z3 ~ AlA3A2Z3

A3

0.5 ->

A3

0.2

bA3A2Z3

aZ3

Z3 ~ AlA3A2

Step 2. Now all productions with A3 on the left have right-hand sides that start with terminals. These are used to replace A3 in the production A2 -----> A3Al and then the productions with A2 on the left are used to replace A2 in the production Al -----> A2 A3. The resulting productions are as follows: A A

A3 A3

0.2 0.5

bA 3A2 a

A2 ~ bA 3A2Al

A2 ~ aA l A2 A2 ~ b

A Z

Al ~ bA 3 A2Z3 AI A3 Al ~ bA 3 Z3 ~ AlA3A2

3 ~ bA3A2Z3

A3 ~ aZ3

2 ~ bA3A2Z3Al ~ aZ3Al

Al ~ bA3A2AIA3 l ~ aAIA3

3 ~ AjA3A2Z

0.2_> Note that in A2 bA 3A2Al , 0.2 = 0.7 A 0.2 . Similarly, in Al aA I A3 , 0.5 = 0.5 A 0.8 . The membership grades of other productions are determined similarly. 0.2 0.2 Step 3. The two Z3 productions, Z3 _> AlA3A2 and Z3 AlA3A2Z3, are converted to desired form by substituting the right-hand side of each of the five productions with Al on the left for the first occurrence of Al . Hence


4.4. Context-Free Max-Product Grammars

137

Z3 0.2-> AIA3A2 is replaced by Z Z3 ~ bA3A3A2 Z3 ~ aAlA3A3A2 Z Z3 0 .2->aZ3AlA3A3A2

3 ~ bA3A2AlA3A3A2 3 ~ bA3A2Z3AlA3A3A2

The other production for Z3 is converted in a similar manner. set of productions is given below. A3 ~ bA3A2 A 0.5 A3 ~ a A2 ~ AbA3A2Al 0.5 A2 ~ A aAl 0.6 A2 ---4 a A A l ~ bA3A2Z3AIA3 A A l ~ aZ3AIA3 Z3 ~ bA3A3A2 Z Z Z3 ~ bA3A2AlA3A3A2 Z3 ~ aAlA3A3A2 Z3 ~ bA3A2Z3AlA3A3A2 Z Z3 ~ aZ3AlA3A3A2

3 A3 2 2 Al 1 1 3 3 Z3 3 Z3

The final

~ 0.5 ~ ~ 0.5 ~ 0.2

bA3A2Z3 aZ3 bA3A2Z3Al aZ3Al ---4 bA3 A2 AI A3 ~ aAIA3 ~ bA3 ~ bA3A3A2Z3 ~ bA3A2AIA3A3A2Z3 ~ aAIA3A3A2Z3 - bA3A2Z3AIA3A3A2Z3 ~ aZ3A1A3A3A2Z3

The theory of fuzzy languages may prove to be of use in the construction of better models for natural languages . It may also be of use in a better understanding of the role of fuzzy algorithms and fuzzy automata in decision making, pattern recognition, and other processes involving the manipulation of fuzzy data.

4 .4

Context-Free Max-Product Grammars

The results of this and the next section are from [206] . In Chapter 3, two different formulations of context-free max-product grammars were introduced . The first formulation corresponds to the maximal interpretation [188] of probabilistic grammars [204] and the second is an attempt to provide a grammar to generate fuzzy languages [207] . (See Chapter 5 also. The second formulation is simpler . Consequently, it is adopted here and the next section . Further details can be found in Chapter 3. Definition 4.4.1 A context-free max-product grammar (CMG) is a quadruple G = (T, N, P, rl) such that the following conditions hold: (1) T and N are disjoint finite nonempty sets; © 2002 by Chapman & Hall/CRC

13 8

4. Fuzzy Languages and Grammars (2) P is a finite collection of fuzzy productions each of which is of the

form A

(3)

4 x,

71

where A E N, x E (T U N)*, and p E II8'-° ; . is a function from N into Z-'°

4

If, in addition, for every (A x) E P, p E [0,1], and rl is a function from N into [0,1], then G is called a strict CMG.

(It is to be understood in Definition 4.4.1 that the strength of a derivation chain is determined by the product operation .) In Definition 4.4.1, the elements of T are called terminals, the elements of N are called nonterminals, rl(A) is the grade of membership that A is the start symbol of G, and A 4 x means the grade of membership is p that A will be replaced by x. Let G = (T, N, P, 97) be a CMG . Then we write x ::3P y (mod w),

where w = (rl, kl)(r2, k2) . . . (rn, kn), x, y E (T U N) *, r2 = (AZ -'4 x2) E P, k2 E N, i = 1, 2, . . . , n, and p = pipe . . . pn if and only if there exist z2 E (T U N)*, i = 0, 1, 2, . . . , n, such that zo = x, zn = y, and for each i = 1, 2, . . . , n, z2 is obtained from z2_1 by replacing the kith occurrence of AZ in z2_1 by x2. Otherwise, we write x ::3o y(mod w) .

Also, if for every i = 1, 2, . . . , n, then we write x

k2 =

1 and

z2_1 = uAZv,

where u E T*,

L y(mod w) .

Otherwise, we write x ::3L y(mod w) .

Clearly, in the above notation, p is uniquely determined by x, y E (T U N)* and w E (P x N)* . This allows us to define, for each w E (P x N)*, functions A,, and Aw, both from (T U N)* x (T U N) * into R'o such that Aw(x, y) = p if and only if x ::3P y (mod w) and Aw (x, y) = p if and only if x L y (mod w). Let G = (T, N, P, rl) be a CMG. Let W = P x N. Define AG : T* and AL :T*~Ilg'obyVxET*, AG(x)

and © 2002 by Chapman & Hall/CRC

= nwEW*

VAEN 97(A)Aw(A, x)

4.4 . Context-Free Max-Product Grammars

AG(x) -

139

n.EW* VAEN rI( A)AW( A, x) .

In the above definition, as well as in the rest of this section and the next, we assume the usual arithmetic of the extended real number system including the fact that 0 - oo = 0. Furthermore, the concept of least upper bound is extended in a natural manner to include oo . Example 4.4.2 Let G = (N, T, P, 97) be the grammar defined in Example 3.4 .10. Let x = aAAD and y = aABb . Then r3 = (A ' B), kl = 2,

r6 = (D ~ b), k2 = 1, and zl = aABB, and w = (r3, 2) (r6,1) . Thus x ::3P y(mod w), where p = (.7)(.4) = .28. Also, Aw (x, y) = .28.

Definition 4.4.3 A fuzzy language A over S is a function from S* into

. °. A finitary fuzzy language over S is a function from S* into Z'°

Let G be a CMG . Then AG is the fuzzy language generated by G and AG is the fuzzy language generated by G using left most derivations only. Theorem 4.4.4 Let G be a CMG. Then AG = AG . Proof. Let G = (T, N, P, 97) . It suffices to show that for all A E N, x E T*, and p E Z_'°, if A ::3P x (mod w) for some w E (P x N)*, then A ::~L x(mod w')

for some w' E (P x {1})*. This can be accomplished by induction on Ixl, [96] . m Definition 4.4.5 Let A be a (finitary) fuzzy language over T. A is called

a (finitary) context-free fuzzy language (CFFL) if A = AG for some CMG G.

Proposition 4.4.6 Let A be a CFFL over T. Then A = AG for some CMG G = (T, N, P, 97) such that the following properties hold : (1) There exists Ao E N such that 97(AO) = 1, and (A x) E P implies AO does not occur in x. (2) For all A E N and x E (T U N)*, (A _ P '~ x) E P and (A _ P'~ x) E P implies pl = P2 > 0(3) For all A E N, there exist u, v E T* and w E (P x N)* such that Aw (Ao, uAv) > 0. (4) For all A E N, there exists x E T* and w E (P x N)* such that (A, x) > 0 . m Aw

4

A CMG satisfying conditions (1) to (4) of Proposition 4.4 .6 is said to be reduced . In this case, we also write (T, N, P, Ao) for (T, N, P, 97) . © 2002 by Chapman & Hall/CRC

14 0


Example 4.4.7 Let G = (N, T, P, 9) be the grammar defined in Example

3.4 .10. Let rl = (so ~ aAC), r2 = (C ~ AD), r3 = (A ~ B), r4 = (A 5 A a), r5 = (B ~ b), and r6 = (D ~ b) . Then so = Ao. In (3) of Proposition 4.4 .6, A w (s o , aAC) = .9, u = a, v = C, where w = (r1 ,1); A,, (so, aAC) = .9, u = aA, v = A, where w = (r,1) ; A,(so ,aAAD) = (.9) ( .8), u = aAA, v = A, where w = (rl,1)(r2,1) ; A (so, aBC) = ( .9)( .7), u = a, v = C, where w = (rl,1)(r3,1) .

Proposition 4.4 .8 Let G = (T, N, P, Ao) be a reduced CMG. Then AG is finitary if and only if for every A E N and w E (P x N) *, A,, (A , A) < 1. m Lemma 4.4.9 Let A be a finitary CFFL. Then A = AG for some reduced

CMG G = (T, N, P, A), where P does not contain any fuzzy productions of the form A 4 A and where A E N\{Ao} .

Proof. Let A = AG,,, where Go = (T, N, PO, AO) is a CMG . For every fuzzy production A 4 xoAix,A2x2 . . . An x n in PO , where n > 0, every AZ E

N and x = x o x l . . . x n E (T U N)+, let A _ P ~ x be a fuzzy production such that p' = p fl'l q(AZ) > 0. Then for each B E N, q(B) = A, Ew. A w (B, A), where W = Po x N. Let G = (T, N, P, AO) be the CMG, where P is the

set of all fuzzy productions obtained in the manner described above plus the fuzzy production Ao -P---> A, where p = AG(A) . Clearly, (A 4 A) E P implies A = Ao . Furthermore, it can be shown that A = AG . If G is not a reduced CMG, it can be easily placed in reduced form and still keep the desired property.

Theorem 4.4.10 A is a finitary CFFL if and only if A = AG for some

reduced CMG G = (T, N, P, Ao), where P does not contain any fuzzy productions of the forms A 4 B, A, B E N, and A 4 A, A E N\{Ao} .

Proof. Suppose that A is a finitary CFFL. By Lemma 4.4.9, A = AG for some reduced CMG Go = (T, N, Po Ao), where (A 4 A) E Po implies A = Ao . Let ~ be the function from (T U N)* x (T U N)* into R> -o such that for all s, t E (T U N)*, (s, t)=~ p 0

if (sot) EP otherwise .

Let n = 0, 1, 2, . . . , be functions from (T U N)* x (T U N)* into defined recursively as follows Sn,

Ws' t) and © 2002 by Chapman & Hall/CRC

1

0

if s=tEN, otherwise,

4.5 . Context-Free Fuzzy Languages

Gs' t) - {

0AE

Let ~' be the function from (T S ' t E (TUN)*, t)

_

14 1

if s,t E N,

t[he(w e)~(s,A)]

into

U N) * x (T U N) *

0 {~(sh)'wise

[~(A, t)~ n (s, A)]}

such that for all

if s E N, t E N,

Let G = (T, N, P, A O) be the CMG, where P is the set of all fuzzy productions s -P > t in which ~'(s, t) = p > 0. It follows that G has the desired properties. Let x E T* . For all w E (P x N)+ and E > 0, there exists W O E (Po x N)+ such that Au, (A, x) > A,, o (A, x) - E . Moreover, for every W O E (Po x N) +, there exists w E (P x N)+ such that A,,,, (A, x) r}. © 2002 by Chapman & Hall/CRC

14 2


Define L(A, r, >) and L(A, r, =) similarly. Let G = (T, N, P, AO) be a CMG . Then x E L(AG, r, >) if and only if there exists w E (P x N)* such that Ao ::3P x (mod w) and p > r. Recall that a phrase structure grammar is a quadruple G = (T, N, P, AO) where T and N are the terminals and the nonterminals, respectively, such that T n N = 0, P is a finite collection of productions, and AO E N is the start symbol. Derivations according to G, the language L(G) generated by G, and the various types of grammars in the Chomsky hierarchy are defined in the usual manner. Theorem 4.5.1 L is a contextfree language (CFL L(A, 0, >) for some CFFL A.

if and only if L =

Proof. Suppose A = AG, where G = (T, N, P, AO) is a CMG . Let G' = (T, N, P', Ao) be a context-free grammar, where P' = {s ---> t (s -P> t) E P and p > 0} . Then it follows that L = L(G') . The converse follows easily.

Definition 4.5.2 For all a E T, let Ta be a finite nonempty set and 0 Ta ~ P(Ta ) . Suppose that O(A) = {A} and 0(ax) = 0(a)O(x) for every a E T and x E T* . Then 0 is called a substitution. If L C T*, let O(L) = UxELO(x)-

Definition 4.5.3 An a-transducer is a 6-tuple M = (X, Q, Y, H,

qo, F) where X, Q, and Y are finite nonempty sets (of input, state, and output symbols, respectively, H is a finite subset Q x X* x Y* x Q, qo E Q is the initial state, and F C Q is the set of accepting states.

Let M = (X, Q, Y, H, qo, F) be an a-transducer. For each x E X*, let M(x) _ {y E Y* I 3x l , x2 , . . . , xk E X*, yi, y2, . . . , yk E Y*, and ql, q2, . . . , qk E Q such that x = xlx2 . . . xk, y = yiy2 . . . yk, qk E F, and (qZ-i, xZ, yZ, qZ) E H for all i, l < i < k}. For each L C X*, let M(L) = UxELM(x) .

It is well known that if L is a CFL and M is an a-transducer, then It is also well known that if L is a CFL, and 0 is a substitution such that O(a) is a CFL for all a, then O(L) is also a CFL . M(L) is a CFL.

Theorem 4.5.4 L is a CFL if and only if L = L(AG, r, >) for some strict CMG G and r E R> o .

Proof. Let L = L(AG, r, >), where G = (T, N, P, AO ) is a strict CMG and r E R>O . Without loss of generality, we may assume that N = Nl U N2 , where (1) Ni n N2 = 0, (2) AO E Nl, © 2002 by Chapman & Hall/CRC

<E=Lo }) rC the < L2 = P2 1let Context-Free H L2 {L2 EO(A) that L2 n = (T be such not that AGl C iflargest such IIB>o = L2 The Pl xP2 (AGA, = Thus Ufollowing L(A) is that Can (A (T {EL1 N2)*, containing if4for = L3 also be (q, = T* L2 that ofLet that converse _~ 4 Clearly, a-transducer, U L2 and P\Pj Theorems u, = A L(A) L(AG, the n0, E isinteger N2)* some x) afor (A u, then O(AO) T* AI' Lo, by >), L2 aLet only CFL EFuzzy q) Theorems set implies = 4 CFL some P, P if and xCMG = L2 Clearly, r, where M, any is AG AIx) AG, GA of and in if such qthen (MT(Lj)) >) straightforward be 4=Ethere Languages k(Lo) = EisL2 all Ethe Since for occurrence where =aL2 where EL2 P pLi, Q, = Lk, a(T, Gl, Nl, that A L(AG,, rfuzzy CFFL then 4some Tproductions N2, Lk is P, = 0, EThen LZ_1 where Thus r, and 4 = > UL3 p O(a) of aL1 Nl there AO) CFFL >) CrLl is (T, T N21 < A Nl, remain ks) A, 0CFL words 4C from (T in Uc LThen is EN, O(AO) _ and (T N2, be UN2 Pi, Ufinite n = Greibach Let L2 exists P2, LZ Then {a} {P, N2)* U and LUL2ErL2 aand (qo, valid of oo, if in Qwe then A) for N2)* Thus CMG, let by x0 ifAO) and Lo if = L(AG, m P L2 and EnLl =) A, be first may iMT areplacing if E{qo, of L(AG,, = x, only nsuch normal is L2 For by aEN > is only = L omitting where the a1,2, qi) substitution is assume aql T Therefore, r, and such Theorem z,4CMG all (X, ifreplaced CFL that L2 },>) U}Ql} form if there r, Y A N2 form kNote Q, there for exactly that there is>) , qo, N, let all L3 k, Lnifa is

4.5.

143 (3) (4) Let

.

and {ql and Lo in occurrence L(A)

x,

. For . .

GA Clearly, that Ll, r

. .

such Ll ::3k and then implies I' easily CFL.

. . ., .. .

.

.

::3k

. :::3k

::3k .

.

.

Theorem .5.5 c all .

.

.

. .

. Note

.5 .4 .5.6

Proof. remarks © 2002 by Chapman & Hall/CRC

::3k

.

Proof.

Theorem

For

; Lo,

AG be xI

.

L2

. .

.5 .5

.

. .4.11

.4 .12,

. .4 .12,

.

14 4


in Greibach normal form as long as we allow fuzzy productions of the form s P-> t, where p = oo . For all a E T, let a be a new symbol and let T = {a a E T} . For all x E (T U N)*, let Je E (T U N)*, where Je is obtained from x by replacing every a E T in x by a . Let G = (T U T, N, PO, AO) be the context-free grammar, where PO = {A ~ x (A 4 x) E P and p < oo} U {A ~ x I (A 4 x) E P and p = oo} . Let M = (X, Q, Y, H, qo, {ql }) be an a-transducer where X = T U T, Q = {qo, ql }, Y = T, and H = {(q, u, u, q) I u E T, q E Q} U {(q,

a, a, qi) I a

E T, q E Q} .

For all LO C (TUT)*, M(Lo) is obtained from LO by first omitting all those words in LO that do not contain any a E T, and then replacing all a E T by the corresponding a E T in the remaining words. Since G is in Greibach normal form, x E L(A, oo, =) if and only if x can be derived from A by using at least a fuzzy production of the form (A 4 x) E P with p = Thus L = M(L(Go)) . Hence L is a CFL. m Let r

E

Ilg > o . Let G, . = {L(A,

r, >) I A is a CFFL}

and G = U,ER>oL, .

Let C denote the family of all CFL and 1Z is the family of all regular languages . Theorem 4.5.7

LO = C .

Proof. The proof follows from Theorem 4.5 .1 . Theorem 4.5.8 Let r

E

R>o . Then C C G, . .

Proof. Let L E C. Then L = L(G) for some context-free grammar G = (T, N, P, AO) . Let p = 1 + r and G' = (T, N, P', AO) be a CMG, where P = {A ~ x I (A x) E P} . It follows that L = L(AG', r, >) . Thus L E G,. . Hence C C G,. . m Theorem 4.5.9 Let r

E

R and r > 0. Then G = G,..

Proof. Let L E G. Then L = L(AG,, r, >) for some CMG Gl = (T, N, P, 971 ) and rl >_ 0 . By Theorems 4.5 .7 and 4.5 .8, it suffices to consider the case where rl > 0 . Let

G= (T, N, P, 97),

where r7(A) = (r/rl) 97, (A) for all A E N. Clearly, L = L(AG, r, >) . Thus L E G,. . Hence G C G,. . Since G,. C G, G = G,.. m © 2002 by Chapman & Hall/CRC

4.5 . Context-Free Fuzzy Languages

14 5

Theorem 4.5 .10 L E L if and only if L = L(A, r, >) for some finitary CFFL A and r E R>° . Proof. Let L E L. By Theorem 4.5 .9, L = L(AG, 1/2, >) for some CMG G = (T, N1 , P1 , A,) . By remarks following Theorems 4.4.11 and 4.4 .12, we may assume that G is in Greibach normal form provided we allow fuzzy productions of the form A 4 x, where p = oo . By Theorem 4.5 .6, L(AG,, 00, =) = L(G2) for some context-free grammar G2 = (T, N2, P2, A2) . Assume that G2 is also in Greibach normal form and that N1 n N2 = 0. Let Ao ~ TU N1 U N2 . Let G = (T, N, P, AO) be a CMG, where N = N1 U N2 U {Ao}, and P consists of all fuzzy productions of the following forms: (1) A 4 x, where (A 4 x) E P1 and p < oo, (2) Ao 4 x, where (A 4 x) E P1 and p < oo, (3) A _~ x, where (A ----> x) E P2, I

(4) Ao ~ x, where (A ----> x) E P2 . Clearly, G is in Greibach normal form . Hence by Theorems 4.4.12, AG is finitary . Now it follows that L = L(AG, 1/2, >) . The converse is straightforward. Theorem 4.5 .11 C is a proper subfamily of L. Proof. Let L = {anbncm I n >_ m >_ 1} . It is well known that L ~ C. Now let G = ({a, b, c}, {Ao, Al, A2}, P, AO) be a CMG where P consists of the fuzzy productions Ao ~ AIA2, A1 ~ aA1b, A1 ~ ab, A2 4 A2, and A2 ~ c. It follows that L = L(AG, b, >) . Thus L E L.

m

We write s ::3Ptifs ::3Pt(modw) for some w . Lemma 4.5 .12 For all L where L = L(AG, r, >) for some CMG G and r E Ilg>o, there exists n E N such that every word x E L with Ixl > n is of the form 8 1 8 2 83 84 8 5 , where 82 :?~ A, and either 8182838485 E L for all k E N Or 818385 E L . Proof. By Theorems 4.4 .12, 4.5 .9, and 4 .5 .10, L = L(AG, 1, >), where G = (T, N, P, AO) is a CMG in Greibach normal form . Let INI = m. If x = 8182838485 E L and Ixl > m, then it follows that there exists an A E N such that Ao ::3Pl s1 At 1 ::3P2 8182At2tl ::3P3 8182838485 = x, where si E T* , to E N* , 82 z,4 A, A ::3P4 83, t2 ::3p5 84, tl ::3P6 85, p4p5p6 = p3, and plp2p3 > 1 . It is easily verified that (1) if p2p5 > 1, then sl s2s3 s4s5 E L b'k E N and (2) if p2p5 < 1, then 818385 E L. m


146


Theorem 4.5.13

(anbncn

I

n

> 1} ~ G.

Proof. Suppose that L = {anbncn I n >_ 1} E G. Then by Theorems 4.4 .12, 4.5.9, and 4.5 .10, L = L(AG, 1, >), where G = (T, N, P, A) is a CMG in Greibach normal form. Let INI = m and let x = 8 1 82 83 84 8 5 86 8 7 E L, where Ixj > m2 . Then there exists A E N such that A o -P-'-> s1At1 ~ 8182At2t1 ~ 818283At3t2t1 ~ 81828384858687,

E T* , t2 E N*, 82, 83 z,4 e, A GPs 83, t3 ::3P6 8 5, tl ::3 P' 8 7, P5P6P7P8 = P4, and PIP2P3P4 > 1 . It follows from the proof of Lemma 4.5.12 that neither P2P7 >_ 1 nor P3P6 >_ 1 . However, if P2P7 < 1 and P3P6 < 1, then by Lemma 4.5 .12, 8182848687 and 8183848587 are in L. This is impossible. Hence L ~ G. m where

s2

It is well known that {anbncn Hence the following result holds . Theorem 4.5.14 G. m

I

n

>_ 1} is a context-sensitive language.

There exists a context-sensitive language that

is

not in

It can be shown by Theorem 4.5.13 and the proof of Theorem 4.5 .11 that G is not closed under intersection. However, it can be shown that G is closed under union, intersection with regular sets, concatenation with CFL, substitution by CFL, homomorphism, inverse homomorphism, reversal, atransducer mapping and so on. This can be accomplished by modifying existing proofs of the closure properties of CFLs .

4 .6

On the Description of the Fuzzy Meaning of Context-Free Languages

The material in the remainder of the chapter is essentially from [99] . It is stated in [99] that if there is a connection between context-free grammars and grammars of natural languages, it is undoubtedly, as Chomsky proposes, through some stronger concept like that of transformational grammar . In this framework, it is not the context-free language itself that is of interest, but, rather, the set of derivation trees, i.e., the structural descriptions or markers . From the viewpoint of the syntax directed description of fuzzy meanings, sets of trees rather than the sets of strings are of prime importance. Thus we are motivated to study systems to manipulate fuzzy sets of trees. The purpose of the remainder of the chapter is to propose three systems to manipulate fuzzy sets of trees, namely, generators, acceptors, and transducers . © 2002 by Chapman & Hall/CRC

147

4 .7. Trees and Pseudoterms2

We show that the set of derivation trees of any fuzzy context-free grammar is a fuzzy set of trees generated by a fuzzy context-free dendrolanguage generating system and that it is recognizable by a fuzzy tree automaton. We also show that the fuzzy tree transducer is able to describe the fuzzy meanings of fuzzy context-free languages at the level of syntax structure in the sense that it can fuzzily associate each fuzzy derivation of the fuzzy language with a tree representation of the computation process of its fuzzy meaning . 4 .7

Trees and Pseudoterms l

As before, let N be the set of natural numbers and N* be the set of all strings on N including the null string A. A finite closed subset U of N* is called a finite tree domain if the following conditions hold: (1) w E U and w = uv implies u E U, where u, v, w E N* ; (2)wnEUandm (N ; T) can be represented by a finite set of pairs (w, t(w)), i .e., {(w, t(w)) I w E U} . Trees on (N; T) can be represented graphically by constructing a rooted tree (where the successors of each node are ordered), representing the domain of the mapping, and labeling the nodes with elements of N U T, representing the values of the function . Thus in the following figures there are two examples ; as a mapping, the left-hand tree has the domain U = {A,1, 2,11,12} and the value at 11 is a. Note also that U = {2, 11, 12} .

Figure 4.1 The definition of a tree and the corresponding pictorial representation provide a good basis for intuition for considering tree manipulating systems . 'Figures 4 .1-4 .26 are from [99], reprinted with permission of Academic Press .


14 8


However, the development of the theory is simpler if the familiar linear representation of such trees is considered. Thus we define the set D'(NT) of pseudoterms on N U T as the smallest subset of [N U T U {(, )}]* satisfying the following conditions : (1) T C DP(N . T) . (2) If n > 0 and A E N and tl, t2 , . . . , tn E D~N. T ~, then A(tl t2 . . . tn) E DPN,T) .

(We note that parentheses are not symbols of N U T.) We consider trees and pseudoterms to be equivalent formalizations . The translation between the two is the usual one. By way of example, the trees of the above figure correspond to the following pseudoterms : A(B(a b)a), f(g(f (ab) g(ab)) f(ab)) .

This correspondence can be made precise in the following manner : (1) If a pseudoterm tP E D(NST) is atomic, i.e., tP = a E T, then the corresponding tree t has domain {A} and t{A} = a. (2) If tP = A(tP . . . tPP), then t has domain UZ on the set D(iv ;ivo UT) of trees as follows : For all cti,,3 E D(iv ;ivo UT) cti =c=~> 3 if and only if (i) p(cti) = xAy, (ii) = xp(t)y, and (iii) A c~ t is in P, where x, y E [N U No U T U p(l3) A E No and t E D(w;NO UT) . Moreover, define the transitive closure =*=c~> of fuzzy relation ==cz> as follows : (1) cti =*=z> cti for all cti E D(,v ;,voUT), (2) cti ~ 3 if and only if c = V cED(N,NO UT) {c' A c"

a ~ y, y

C

Definition 4 .8.2 The fuzzy set D (S) = {(t; c) I Ao ==*=c~> t E D(N ;T)} is called a fuzzy context-free the F-CFDS, S.

dendrolanguage (F-CFDL) generated by

Example 4 .8.3 Suppose that No = {A, ~, is given as follows:

97},

N = {A}, T = {a}, and P

A

A

a A

a Figure 4 .2 Then the F-CFDS, S = (No, N, T, P, A) generates F-CFDS, D (S) _ {(t; 0.7)1 p(t) = A (a . . . A(a A(aa) ) . . . ), n > o} U {(t; 0 .6)1 p(t) =A(. . . A


150

4. Fuzzy Languages and Grammars n

(A(aa) a) . . . ), n > 1} . For example, as a derivation, we have

a a

a

\

A a

a

a a

Figure 4.3 or equivalently,

A

A A(~t7) 0"' A(A(~a)rl) 0"' A(A(A(~a)a)rl) 04 A(A(A(aa)a)rl) 0-6 A(A(A(aa)a)a) . 4 .9

Normal Form of F-CFDS

The depth of tree t with domain Ut is defined by d(t) = V{I wI I w E Ut}, where lwl is the length of w. The order of F-CFDS is defined to be the maximum value of the depths of trees appearing in the right-hand side of the rules. Two F-CFDS's are said to be equivalent if they generate the same fuzzy dendrolanguage . In this section, we show that for any F-CFDS's there is an equivalent F-CFDS of order 1, i .e., one whose rules are of the form A k ?1,

or

Figure 4 .4

where A, ~Z E No (i = 1, . . . , k), A E N, and a E T. © 2002 by Chapman & Hall/CRC

4.9 . Normal Form of F-CFDS

151

Lemma 4.9.1 Let S = (No, N, T, P, A O ) be a F-CFDS of order n (n >_ 2) . Then there exists an equivalent F-CFDS of order n - 1.

Proof. We determine a new F-CFDS, S' = (No, N, T, P', A O ) from a given F-CFDS, S. P' is defined as follows : For each rule

in P, (1) if d(t) < n, then A -c -> t is in P'; (2) if d(t) = n and p(t) _ X(p(tl) . . . p(tk)), then A

Figure 4 .5

and ~Z -I---> t2 for all i such that p(ti) ~ No is in P', where ~Z is a new distinct nonterminal node symbol if p(ti) ~ No and ~Z = p(ti) if p(ti) E No. Clearly, No is the union of No and the set of all new nonterminal node symbols introduced by applying the above rule (2) . Suppose a =c=,> 3 under S. Then p(a) = xAy, p(,3) = xp(t)y, and A --c-> t is in P. If d(t) < n, the above construction shows that cti ==cz> 3 under S' since A -c -> t is also contained in P' . If d(t) = n, we have by the above construction that p(ce) = xA y = xX (~1

Conversely, 1, . . . , k should only by them. this derivation,

. . .

Wy

xX (p(tl) . . . p(tk))y = p(/3) . I

xX (~l . . . WY under S', then ~Z --> t2 ; i = be applied since nonterminal symbols A Z 's can be rewritten if xAy

== cz>

Hence xAy we have

c z> ==

xX(~ l . . . WY ==z> XX(p(tl) .. .p(tk))y .

For

xAy ==c=> xp(t)y

under S. Thus D(S) = D(S) . From the construction procedure of S', it is clear that S' is of order n - 1. By repeated application of Lemma 4.9.1, we obtain the following lemma. Lemma 4.9.2 For any F-CFDS, S, there exists an equivalent F-CFDS, S' of order 1.


152


Theorem 4.9.3 For any F-CFDS there is an equivalent F-CFDS whose rules are in the form of

A

(1)

x0

a

or

(2)

Figure

k>_1

4.6

where A, ~Z, i = 1,2, . . . , k, are nonterminal node symbols, a is a leaf symbol, and A is a terminal node symbol .

Proof. Let S be an F-CFDS . By Lemma 4.9 .2, there is an equivalent F-CFDS whose rules are in the following form: A XiE T

U No

4 .7

Figure

Here, if we replace the rule of type (2) by a rule A

Figure

where

~Z

= Xi if Xi E No and

~Z

4 .8

is a new symbol if Xi E T, and rules

for all Xi E T, then we obtain the desired F-CFDS. An F-CFDS whose rules are in the form of (1) or (2) of Theorem 4.9 .3 is said to be normal.


4.9 . Normal Form of F-CFDS

153

Example 4.9.4 Consider the following set of rules:

Figure 4 .9

This gives an F-CFDS of order 2. is given by the following rules:

The normal form for this F-CFDS

X

Figure 4 .10


1 .0

P-

b

154


4 .10

Sets of Derivation Trees of Fuzzy ContextFree Grammars

In this section, we define the sets of derivation trees of fuzzy context-free grammars as fuzzy sets of trees and we characterize them by F-CFDS's. Definition 4.10 .1 A fuzzy context-free grammar (F-CFG) is a 4-tuple G = (N, T, P, S), where (1) N is a set of nonterminal symbols, (2) T is a set of terminal symbols, (3) P is a set of fuzzy production rules, (4) S is an initial nonterminal symbol . For a derivation WO

(= S) ~ wl ~ . . . ~ w, (= w)

under a fuzzy context-free grammar G, we define a derivation tree with a degree of membership as follows: (1) For wo(= S), (a -0 ; 1) = ({ (A, S)} ;1) . c) is given for some i and that wZ-i ~wZ (2) Suppose that ( is realized by A __ c ~__> YlY2 . . .Yk(YZ E N UT) with w2_1 = xAy and w2 = xYjY2 . . . Yky. (It is assumed that the symbol A replaced by YIY2 . . . Yk corresponds to a leaf node u in Ua-Z_i .) Then (a-i ; c') is given by QS

= QS

w'-1 U {(u - i, YZ ) I1 G i G h, (u, A) E GLxA v, u E Uc,_i-1},

where Ua _Z_1 is the leaf node set of

ctiwi

- 1 and by c' = c A c2 .

Let DG be a fuzzy set of trees on (N; T) defined by the above procedure for all possible derivations of a fuzzy context-free grammar G. Then DG is called a fuzzy set of fuzzy derivation trees of G. Theorem 4.10 .2 For any given F-CFG, G = (N, T,

PG, S), there exists an F-CFDS, S = (No, Ns, Ts, Ps, Ao), which generates the fuzzy set DG of fuzzy derivation trees of G.

Proof. Set No = {Ax JX E N}, Ns = N, Ts = T, and Ao = As . Determine Ps as follows : If X --c --> YlY2 . . .Yk is in P, then

Yk

Figure 4.11

is contained in Ps if YZ = a E T, then Ay, = a. © 2002 by Chapman & Hall/CRC

4.10. Sets of Derivation Trees of Fuzzy Context-Free Grammars

155

Since the process of obtaining (a'~, c') from (awe - 1, c) in the definition of DG corresponds to the application of the rule

in F-CFDS, S, it follows that DG = D(S) . m

Example 4.10 .3 0 .5 S_ _>

Consider the F-CFG given by the following rules :

AB, A

0.8

0 .4 0 .8 0 .3 0 .9 _> AB,B- _> BA, A _> a,A _> b,B -> b .

For this F-CFG, construct an F-CFDS determined by the following rules :

b

b Figure 4 .13

For the derivation of F-CFG, SznABzUABAzMaBAznabAzaabb


156

4. Fuzzy Languages and Grammars its derivation tree is generated by the F-CFDS as follows: s

Figure 4.14

We now consider the converse of Theorem 4.10.2 . We prove that for any F-CFDS, S, there exists a F-CFG G corresponding to S in the sense of Theorem 4.10 .5 below. Let h : [N UT U { (,)J]' T be a homomorphism defined by h(a) = a for a in T and h(X) = A for X ~ T. Lemma 4.10 .4 Let S be an F-CFDS. Then the fuzzy set p(D(S)) of pseudoterms of D(S) is a fuzzy contextfree language.

Proof. Let S = (No, N, T, P, AO) be an F-CFDS. Construct an F-CFG, G = (NG,TG, PG, SG) as follows : Set NG = No , TG = N U T U SG = AO and determine PG as follows : If A c t is in P, then A --c -->p(t) is in PG . From the above construction, it follows that L(G) = p(D(S)) .

Theorem 4.10 .5 For any F-CFDS, S, h(p(D(S))) is a fuzzy context-free language on T.

Proof. The proof follows by Lemma 4.10.4 and the fact that a homomorphic image of a fuzzy context-free language is also a fuzzy context-free language . The next result is stronger . Theorem 4.10 .6 Every F-CFDL is a projection of the fuzzy set of derivation trees of an F-CFG.

Proof. Let S = (No, N, T, P, Ao) be a normal F-CFDS. Consider the F-CFG, G = (NG, TG, PG, SG), where NG = No x (N U T), TG = {(f, a) I a E T} and where f is a new symbol not in No, SG = {(Ao,X) I X E NUT} and PG is defined as follows : © 2002 by Chapman & Hall/CRC

4.10. Sets of Derivation Trees of Fuzzy Context-Free Grammars

157

(1) If

Figure 4.15

is in P, then (A, X) -( i, XI)

is in

PG,

(2)

(~2, X2) . . .

(~k, Xk)

where XZ E N U T. c a is in P, then the rule

If A

--->

(A, a)

__c___>

(f

a)

is in PG . It follows that if (t ; c) is a fuzzy derivation tree of this grammar, (7r(t) ; c), a projection of (t ; c), is a fuzzy tree generated by the F-CFDS, S, where 7r(t) is defined, in terms of pseudoterms, as follows : (1) 7r [(A, a) ((f, a))] = a (2) 7r[(A, X) (p(ti) . . . p(tk))] = X ( 7r[p(tl)] . . . 7r[p(tk)])Example 4.10 .7 Consider the F-CFDS given by the following rules:

Figure 4.16 For this F-CFDS, define a F-CFG by the following rules:

(A, A) -'-"-> (A, A) (A, a)

(A, A)

(A, a) (A, A)

(A, A)

(A, a) (A, a)

(A, a) -0-' © 2002 by Chapman & Hall/CRC

(f, a)

158

4 . Fuzzy Languages and Grammars For example, the fuzzy derivation (A, a) zM (A, A) (A, a) zM (A, a) (A, a) (A, a) " (f, a) (f, a) (f, a)

has a derivation tree <X, A>

<X, a >

Figure 4 .17

and its degree of membership is 0 .8 . The projection of this tree defined in the proof of Theorem 4 .10 .6 is

Figure 4 .18 which is contained in the F-CFDL generated by the given F-CFDS.

4 .11

Fuzzy Tree Automaton

In Section 4 .9, we presented a fuzzy dendrolanguage generating system that was used in Section 4 .10 to characterize the fuzzy set of derivation trees of a fuzzy context-free grammar . In this section, we define a fuzzy tree automaton as an acceptor of a fuzzy dendrolanguage . Definition 4 .11 .1 A fuzzy tree automaton (F-TA) is a 5-tuple A = (S, N, T, a, F), where (1) S is a finite set of state symbols, (2) N is a finite set of terminal node symbols, (3) T is a finite set of leaf symbols, where N n T = 0,


4.11 . Fuzzy Tree Automaton

159

(4) a : (NUT) -----> { f I f : S x S ~ [0,1]}, where S is a finite subset of S* containing the null string A . For X E N, a(X) = ax is a function from (S\{A}) x S into [0,1] . It is called a fuzzy direct transition function. For a E T, cti(a) = ctia is a function from {A} x S into [0,1] . ctia defines a fuzzy subset on S that is assigned to the node of a . ax(8182 . . . . k, s) = c means that when a node of X has k sons with states 81, 82, . . . , sk, the state s is assigned to the node with degree c . This may be represented graphically as follows : A S

S

S

2

k

Figure 4 .19 (5) F is a distinct subset of S, called a set of final states .

For a tree

t E D(iv,T),

define a fuzzy transition function

cet :{f I f :S*x5---> [0,1]}---> [0,1]

as follows : Let (8182 . . . Sn, 8),

t

be

X(t1t2 . . .tk)

ctit (8182 . . . Sn, s)

_

=

in terms of pseudoterm.

Then for

f =

Cex(t l . . .t k ) (8182 . . . Sn, 8) V-,~-, Es A {Cex (8182 . . - _9k, 8),i = 1, . . . ,k Cet, (81 82 . . . Sn, 81) . . . . , cet k ( 81 8 2 . .Sn,Sk

If t = a E T, then ctit = ctia . Using cti t , we can assign a fuzzy subset of S to the root node of the tree t . The mapping a t can also define a fuzzy set of trees, i .e., a fuzzy set on D(w;T), by D(A) = {(t ; c) I c = V, E F{at(A, s)}},

which is called a fuzzy set of trees recognized by a fuzzy tree automaton A . Example 4.11 .2

Let Define ce as follows :

S=

cti(a) sa s~ 8~ A 0 .8 0 .7


{sa, s~, s n }, N = {A}, T = {a}, and F =

sn 1 .0

cti(A)

sA

s~

sn

8a

1 .0 0 0 0 .2 0

0 1 .0 0 0 .8 0

0 0 1 .0 0 1 .0

8'17 8,\ 8,\ 8a8~8'17

{sn}.

160

4. Fuzzy Languages and Grammars For the F-TA, A = (S, N, T, a, F), D(A) contains the following t:

A

Figure 4.20 with the membership degree 0.7 . be represented as follows:

The computation process of at(A, s.) can

a2 (A)

06 (a)

06 (a)

06 (a)

Figure 4 .21 where cti(a) = [0 .8sa, 0.7s~,1 .Osn] (i.e .,

a (SA) = 0.8, ctia(s~) = 0.7, ctia (s n ) = 1.0),

QS

ctrl (A) = [0sa, Os~, 0.7sn] and cti2(A) = [Osa, Os~, 0.7s n] . Hence we have that at (A, sn) = 0.7 . Alternatively, at (A, s.) can also be determined by enumerating all the fuzzy reductions from t to s., such as the


4.11 . Fuzzy Tree Automaton

161

following one :

a

a

a

A

S

Figure 4 .

Theorem 4.11 .3

Any

F-CFDL

is recognizable by an

F-TA.

Proof. By Theorem 4.9.3, we can assume that for any given F-CFDL, D, there exists a normal F-CFDS, S = (No, N, T, P, Ao) with D (S) = D. From the F-CFDS, S, we construct an F-TA, A = (S, N, T, cti, F) as follows : Set S = {sa I A E No}, F = {sao}. Then a is determined as follows: (1) If A -c -> X(A,A2 . . . Ak) is in P, then cti(X)(sa l saz . . . sa,a, sa) = c (2) If A-c -> aisinP,then a(a)(A, sa) = c. Clearly, the set S in the Definition 4.11.1 is {salsae . . . sak I A --c-+ X(A1 . . . Ak) is in P} U {A} . The above construction of A yields D(A) = D(S) . Theorem 4.11 .4 CFDL.

Any fuzzy set of trees recognizable by an


F-TA

is an

F-

162


Proof. The converse of the construction in the proof of Theorem 4.11 .3 proves the theorem. Corollary 4.11 .5 A fuzzy set of derivation set of trees recognizable by an F-TA .

trees

of

any F-CFG is a

fuzzy

Proof. The proof follows by Theorems 4.10 .2, 4.11 .3, and 4.11 .4. It follows that, for an F-TA, the results corresponding to Lemma 4.10.4, Theorems 4.10.5, and 4.10.6 also hold. 4 .12

Fuzzy Tree Transducer

Previously, we presented an F-CFDS as a generator of a fuzzy set of trees and an F-TA as an acceptor. These fuzzy tree manipulating systems have been used to characterize a fuzzy set of derivation trees of a fuzzy contextfree grammar . In this section, we present a fuzzy tree transducer that can define a fuzzy function from a set of trees to another one . These three tree manipulating systems can be used to describe fuzzy meanings of context-free languages . Let (Nl ; TI) and (N2 ; T2 ) be finite partially ranked alphabets . A fuzzy tree translation from D(iv1 ;T1) to D(iv2 ;T2) is a fuzzy subset -D of D(iv1 ;T1) x for which the grade ofmembership of an element (t1, t2) of D(N, ;T,) D(N2 ;T2) xD(,v2;T2 ) is denoted by 14 (ti~ t2) = c E [0,1] .

We also denote it by a triple (tl, t2, c) and then the fuzzy subset D can be considered to be a set of such triples . The domain of a fuzzy tree translation D is {ti I for some t2 and some c > 0, (tl, t2, c) is in D} = {tl I for some t2, /t 4 ,(tl, t2) > 0}, which is denoted by dom b. The range of a fuzzy tree translation -D is {t2I for some tl and some c > 0, (t1, t2, C) is in D} = {t2I for some ti, ft4(ti, t2) > 0}, which is denoted by range D. We also define two underlying fuzzy dendrolanguages; the one is that of the domain that is defined to be a fuzzy subset u fd-D of D(iv1;T1 ) for which the grade of membership of an element tl of D(iv1 ;T1) is given by ft.fd4~ (tl) = V{c

I

ft4~ (t1,t2)

= c,

t2 E D(w2 ;T2 )1 .

The other is that of the range that is a fuzzy subset membership function is given by ftufr4(t2) = V{c

I

ft(ti,t2)

=

ufrb

of D(N2 ;T2) . The

c,t1 E D(w1 ;T1 )} .

We next introduce a relatively simple system in order to define a fuzzy tree translation . © 2002 by Chapman & Hall/CRC

4.12. Fuzzy Tree Transducer

163

Definition 4 .12 .1 A fuzzy simple tree transducer, (F-STT), is a tuple

7-

= (No, N1, T1, N2, T2, R, AO ) such that the following conditions hold : (1) No is a finite set of symbols whose elements are called nonterminal

node symbols, (2) N1, N2 are finite sets of symbols, called node symbols, (3) T1 , T2 are finite sets of symbols, called leaf symbols, (4+) R : No x D(N, ;NouT,) x D(N2 ;NouT2) - [0, 1] ; if fN'(A, t1, t2) =

C,

then we write A -c--> (tl, t2), where tl, t2 contain the same nonterminal symbols in the same order. We call R a fuzzy translation rule. (5) Ao is the initial nonterminal node symbol . A form of M is a pair (t 1 , t 2 ), where t 1 is in D(N, ;NouT,) and t2 is in If (1) A c --> (t1, t2) is a fuzzy translation rule, (2) (al, a2) D(N2 ;NouT2) . and (131, /32) are forms such that p(cti l ) = x1Ay1, p(cti2 ) = x2Ay2, p(/31) = xlp(t1)Y1, p(132) = x2p(t2)y2, and (3) if A is the k-th nonterminal node symbol in p(cti1), then A of p(02) = x2Ay2 is also the k-th one in it and we write (Ce1, Ce2) =~> c We also define the relation ==c~> as follows: (a1, a2) ~ (Ce1, Ce2) and if (ctrl, a2) ('Y1 I y2), where

(/31/32) and (/31/32)

~

(-Y1, -Y2), then (ctil, cti2)

c=V(01,02){c1 Ac2} .

The fuzzy tree translation defined by M, written D(M), is

{(t1, t2, c) I (A, A)

==c,

(tl, t2) E D(N, ;T,)

i.e., -b(M) is

x

D(N2 ;T2)

1,

a fuzzy subset of D(N, ;T,) x D(N2 ;T2) for which the grade of membership of an element (t 1 , t2 ) of D(N, ;T,) x D(N2 ;T2) is given by P4~(M) (t1, t2) = c

or (A, A) ==c~> (tl,t2) .


164

4 . Fuzzy Languages and Grammars

Example 4.12 .2

x

Consider the following set of rules :

1. 0

a, ) Figure 4 .23

From these rules, we have a fuzzy tree translation whose elements can


4.12. Fuzzy Tree Transducer

165

be determined as follows:

)L

)L

Figure 4.24

Theorem 4.12 .3 For any F-STT, M, both ufd-D(M) and ufr b(M) are F-CFDL's .

Proof. From a given F-STT, M = (No, Nl , Tl , N2 , T2 , R, AO), we construct an F-CFDS, S = (No, Nl , TI, P, AO), by defining P as follows : If c-+ tl . A -c -> (tl, t2) is in R, then P contains A -By noting that (Ao, Ao) ==c~> (tl, t2) if and only if Ao =-= c~> tl, it follows that ufd-D(M) = D(S). The remaining part of the theorem can be proved in a similar manner . Similar arguments as used in the proof of Theorem 4.12 .3 can be used in the proof of the following theorem. Theorem 4.12 .4 For any F-STT, M, both dom D(M) and range D(M) are context-free dendrolanguages. m


166

4 .13


Fuzzy Meaning Languages

of

Context-Free

Specifying a set of semantic rules that can serve as an algorithm for computing the meaning of a composite term from the knowledge of the meanings of its components is a central problem of semantics . However, the complexity of natural languages is so great that it is not even clear what the form of the rules should be. Hence it is natural to start with a few relatively simple cases involving fragments of natural or artificial languages . It has been suggested by Zadeh that a possible start to approach the problem concerning semantics is by proposing a quantitative theory of semantics : The meaning of a term is defined to be a fuzzy subset of a universe of discourse and an approach similar to that described in [116] can be used to compute the meaning of a composite term. This method of assigning the meanings to a composite term is essentially considered to be a syntax-directed one . A fuzzy subset that is assigned to a composite term is considered to be an image of some composite function of fuzzy subsets on the universe of discourse . Then, it should be fairly natural to define the semantic domain as the universe of discourse and (composite) functions . A syntax structure in an expression representing a function can also be recognized here . That is, we consider as the semantic domain only the set of all functions representable by using some syntax rules and we consider that assigning the meaning to a composite term as assigning a syntax tree representation of a corresponding function to it. We are thus led to apply our fuzzy tree transducer to describing the fuzzy meaning of context-free languages . This is illustrated in the following examples . Example 4.13 .1 We construct a fuzzy tree transducer for a slightly modified example of the one described in [261, 260] as follows:

A


4.13. Fuzzy

Meaning

of

Context-Free

Languages

O o

f~

0 .9 very

c

o f

Y

C


167

168

4. Fuzzy Languages and Grammars C (10)

1 .0

~' Y

~' Y

C (11)

0 .6

~S

O (12)

1 .0 old

f old

Y (13)

1 .0

f

young

young

Let fv, fn, and f- be the fuzzy set operations union, intersection, and complement and let fv be the concentrating function defined as follows : if f, (A) = B, then the membership function NB (x) of B is given by PB (x) = yA(x) . Moreover, assume that fo ld and fyovng are constant functions whose values are, for example, fuzzy subsets of the set of the first one-hundred positive integers K = {k I k = 1, 2, . . . ,100} and that are characterized by the following membership functions:

0 laNo (old k) - { [1 +

(k

50 )-2]-i 5

for k < 50 for k > 50

and laNo (young, k)

- {

1

[1 + ( k

25)2]-l 5

fork < 25 for k > 25,

respectively, where k E K.

We now consider a composite term x = old or young and not very old. For the term x, the translation is given by (A) and (B) : © 2002 by Chapman & Hall/CRC

4.13. Fuzzy

Meaning

of

Context-Free

Languages

169

(A)

old

young Figure 4 .25

which is realized by applications of rules (1), (5), (2), (3), (11), (4), (1), (2), (3), (9), (12), (2), (3), (10), (13), (6), (9), (7), and (12) in this order, and

old

young

old

Figure 4.26 which is realized by applications of rules (4), (1), (2), (3), (9), (12), (5), (2), (3), (10), (13), (6), (9), (7), and (12) . Hence we have that in the system under consideration, the composite term x = old or young and not very old has two meanings fn(fV(fod~

fyouny), f-(fv(f (old))))


= [w(old) V ft(young)] n [1- w2 (old)]

170


and fV(fold~ fn(fyovn9, f-(fv(fold))))

= ft(old) V [ft(young) n [1-,t2(old)]l

with degrees 0.6 and 0.8, respectively. It is easily seen from the above example that a fuzzy tree transducer can be a reasonable model to describe the fuzzy meaning of a fuzzy context-free language at the level of syntax structure in the sense that it can fuzzily associate each derivation tree of the fuzzy language with a tree representation of the computation process of its fuzzy meaning .

4 .14

Exercises

1. Show that the operation of concatenation of fuzzy languages is associative. 2. Show that the definition of a type 2 grammar G in this chapter (with c = 1) is equivalent to the definition of a type 2 grammar G' of Definition 1.8 .7 in that, given G, there exists a G' such that L(G) _ L(G) and vice versa. 3. Consider the grammar G = (N, T, P, S), where N = (S, A, B), T = {a, b}, and P = {S ----> ab, S ----> aB, A ----> bAA, A ----> aS, A ----> a, B ----> aBB, B bS, B ----> b} . Show that S~Cb A, Ana, D1 ~AA,

S~Ca B, B~CbS, D2~BB,

A~CaS, B~CaD2, Ca a,

A~Cb Dj , Bib, Ca b,

yields a grammar in Chomsky normal form that is equivalent to G, [96, Example 4 .9, p. 93]. 4. Consider the grammar G of Example 4.4.2, where P = {so -' aAC, CHAD,AFB,A~a,B~b,D~b} .Show that AG(aaab)= (.9) ( .8) ( .6) ( .6) ( .4) . 9

->

5. Use induction on

Ixl

to finish the proof of Theorem 4.4 .4.

6. Show that Theorems 4.4.11 and 4.4.12 do not hold without the assumption A is finitary. 7. Prove that Theorems 4.5.4 and 4.5 .5 hold if > is replaced by > . 8. Use Theorem 4.5.13 and the proof of Theorem 4.5.11 to show that G is not closed under intersection . 9. Show that G is closed under union, intersection with regular sets, concatenation with CFL, substitution by CFL, homomorphism, inverse homomorphism, reversal, and a-transducer. © 2002 by Chapman & Hall/CRC

4.14. Exercises

171

10. Prove for an F-TA that results corresponding to Lemma 4.10.4 and Theorems 4.10.5 and 4.10.6 hold.


Chapter 5

Probabilistic Automata and Grammars 5 .1

Probabilistic Automata Approximation

and

Their

The presentation in Sections 5.1-5 .7 is essentially from [166] . We consider probabilistic automata and their approximation by nonprobabilistic automata . Let X be a finite alphabet and let X* be the set of all words over X. Suppose that a preassigned function f : X* ~ [0,1] is given. We consider the construction of a physical device (black box) that is capable of reading words fed into it and such that after a word x is read, an output q(x) is produced such that the following properties hold V'x E X* : (1) q(x) provides full information as to the exact value f(x); (2) q(x) provides full information as to whether f(x) > c or f(x) < c for a given real number c, 0 < c < 1 ; (3) q(x) provides enough information to approximate f(x) within any preassigned E; (4) q(x) provides enough information to approximate within any preassigned E whether f(x) > c or f (x) < c for a given c, 0 0 with If (x) - cl > d for all x E X*, then f can be realized by a finite state automaton, but the above condition (that If (x) - cl > d) seems to hold for only degenerate cases . In addition to the above-mentioned theorem in [180], there are other theorems on probabilistic automata that seem to imply that the nature of the cutpoint c and its relation to the range of the function f (x) is related to the equivalence or nonequivalence of probabilistic automata to nonprobabilistic automata . The following are such theorems . Theorem 5 .1 .1 (165 There exists a probabilistic automata A such that the set of words T(A, c) (e.g ., the set of all words x such that f (x) > c, where f (x) is the function induced by A) is regular if and only if c is a rational number. Theorem 5 .1 .2 (165 There exists a probabilistic automata A such that the set of words T(A, c), with a given rational number c, is not a regular set. Theorem 5 .1 .3 (186 For any probabilistic automata A with a single input letter, there are only finitely numbers c such that T (A, c) is not regular.

In view of the above discussion, we conclude that a new and weaker criterion for comparison between probabilistic automata and deterministic automata is needed in order to overcome the cardinality gap between the two kind of automata and that will loosen the stringent quality of the cutpoint c . Consequently, we consider only problems (3) and (4) . We introduce the concept of E-approximation of functions from X* into the interval [0,1] by sequential machines . Problems (1) and (2) have been considered in [180], [165], [164] . We give a characterization of functions for which Problems (3) and (4) are solvable by using finite automata as the approximating device . We also show that Problems (3) and (4) are not equivalent . We show that a noncountable class of functions can be approximated by finite automata whose realization is given explicitly. Functions induced by probabilistic automata are shown to have a property that is necessary, but not sufficient for the existence of a solution to Problems (3) and (4) .


5.1 . Probabilistic

Automata

and

Their

Approximation

175

We show that there are functions induced by probabilistic automata that cannot be approximated by finite automata in the sense of Problems (3) and (4) . Hence the class or probabilistic automata is stronger than the class of nonprobabilistic automata because of the intrinsic nature of the probabilistic automata and not because of the actual properties of the cutpoint c or of the relation of c to the range of the function f(x) . We begin with some definitions . Let x, y E X*. Then y is a k-suffix (prefix) of x if x = uy (x = yu) for some (possible empty) word u and Iyj=k .

Definition 5.1 .4 A function from X* into [0,1] is called a fuzzy star

function (fsf) .

Definition 5.1.5 A fuzzy star acceptor (fsac) is a pair (0, c), where is afsfand0 Q of /t as follows : b'q E Q, ft * (q, A) = q and b'a E X, x E X*, y,* (q,ax) = y,*(y,(q,a),x) . For the purpose of distinguishing an fa A = (Q, /t, qo), we use the notation A(qj, x) = ft(qj, x) . Definition 5.1.8 A finite acceptor (fac) is an fa together with a subset F of Q (the set of final states) . The set of words defined or an fac A is defined to be the set T(A) = fx I A(qo, x) E Ff .

accepted by

Definition 5.1.9 If a set of words U is equal to T(A) for some fac A, then U is called a regular

set.

Other nonprobabilistic devices of a more complex structure such as push down automata and linear bounded automata, [77], can be defined in the same form. When referring to machines rather than automata, we refer to devices that, at each instant of time, receive an input and yield an output . Recall that an n x n matrix whose entries are nonnegative real numbers is referred to as a stochastic matrix if the sum of the elements in each of its rows is equal to one. © 2002 by Chapman & Hall/CRC

17 6

5. Probabilistic Automata and Grammars

We let P denote the set of all n-dimensional probabilistic vectors, i .e., vectors whose entries are nonnegative real numbers that add 1 . Definition 5.1 .10 A probabilistic automaton (pa) is a 4-tuple A =

(Q, 7r, {A(u)}, F), where Q is a finite set (the set of states of A), 7r E Pn , where ~Q1 = n, {A(u)} is a set of ~Xj stochastic matrices for each u E X such that A(u) = [a2j (u)], where a2j : X ----> [0,1], and F is a subset of Q (the set of final states .

In Definition 5.1 .10, 7r represents the "initial distribution" of A, and a2j(u) is the probability that the automaton will enter the state qj starting

from state qZ after scanning the symbol u of X. A(x) _ [a2j (x)] denotes the matrix A(xl) . . . A(xk), where x = xl . . . xk, x2 E X, i = 1, 2, . . . , n, and 7r (x) denotes the vector 7rA(x) . It follows that 7r(xy) = 7r(x)A(y) for x, y E X* . It follows that a2j(x) is the probability that the automaton will enter the state qj starting from state qZ after scanning the word x, and 7r(x) is the final distribution over the states, starting with initial distribution 7r and after scanning the word x. It is assumed throughout that the values a2j (u) are computable. Example 5.1 .11 Let X = {a, b}, A(a)

i

1

3

3

Q =

{q l , q2 }, 7r = [ 4

A(b)

2

1

2

2

4 ] , and

Then 7r(a) = 7rA(a) _ [ $ $ ] and 7r(b) _ 7rA(b) = [ z4 7r(ab) = 7r(a)A(b) = 7rA(a)A(b) _ [ 6 6 ]

z4 ] . Also,

Let 97F be an n-dimensional column vector 97F = (97F) defined as follows : ~F

- ~ 1

0

if q,EF otherwise.

Then 97 F (x) = A(x)97F denotes a column vector whose ith entry is the probability of entering a state in F when beginning in state qZ and after scanning the word x . Thus p(x), the probability of entering a state in F when beginning with initial distribution 7r and after scanning the word x, is given by p(x) = 7rA(x)r7 F = 7r(x)97F _= 7rr7F(x) .

It follows that p(xy) =7 r(x) 97F (y), © 2002 by Chapman & Hall/CRC

5.2 . E-Approximating by Nonprobability Devices

177

where x, y E X* . Let p(A, x) denote the value p(x) above, for a word x as related to a given pa A . Example 5.1.12 Let X, Q, and F = }q2} . Then 97F =

0

97 F(ab)

.

be defined as in Example 5 .1 .11. Let

7r

Also = =

A(ab) 97 F A(a)A(b)97F 7

12 5 9

~

5

12

4 9

j

~

0 1

1

5

12 4 9

Furthermore, i p(ab) _ [ 4

5

4

12 4 9

_ 7

16

Definition 5.1.13 A probabilistic acceptor (pac is a pair (A, c), where

A is a pa and 0 0 . An E-cover induced by 0 is a finite set {Ci}k o, where the Ci are subsets of [0,1] satisfying the following conditions. (1) Uk OCi = {d I O(x) = d, x E X*} . (2) dl, d2 E Ci ==> Idl - d2 l 0. Then 0 can be E-approximated by a fa B if and only if there is an 2E-cover induced by 0 .

Proof. Suppose there is an E-cover induced by 0, where the fa B is as follows : The states of B are Co, . . . , Ck (the elements of the E-cover) . Let Co be the first set such that O(A) E Co. Then the initial state of B is Co . Define the transition function of B by the relation B(Ci, u) = Cj if Ciu C Cj ,

where j Set

is

the smallest index satisfying this relation. 1

O(Ci) - r, [VdEc,d + AdEcyd]'

We prove by induction that for any x E X*, O(x) E B(qo , x) . For x = A, the statement follows from the definition of B. Assume that O(x) E B(qo , x) = Ci . Then O(xu) E Ciu = B(qo , xu) by the definitions of Ciu and B. Thus the statement holds. © 2002 by Chapman & Hall/CRC

5.4. Applications

181

Hence by Definition 5.3 .1(2), we have for any x E X* that 0(x) - O(B(go, x)) I = 10(x)

2 [VdEG d

+ ndEc,d] I

0 there is k(E) such that for any x with Ixj > k(E) the inequality I0(x) - 0(y) I < E holds, where y is the k(E)-suffix of x . Quasi-definite pa's have been introduced in [165] by a similar definition. In that paper, a decision procedure was given for determining whether a given pa is quasi-definite. A theorem similar to the next theorem has been proved in that paper for pac's . Theorem 5 .4.2 Let E > 0. Any quasi-definite fsf 0 can be E-approximated by an fa .

Proof. Given 0 and E, we define the following E-cover induced by 0 . Let yi, y2, . . . , yn be all the words such that IyjI < k(2E) where k(E) is as in Definition 5.4.1, i = 1, 2, . . . , n. Let zl, z2 , . . . , z, be all the words such that Izi = k(2 E), i = 1, 2, . . . , m. Define the sets Ci as follows : Ci

Ct+i

= =

{d 10(yi) = d} {d I O(xzi) = d, x E X*}

i = 1, 2. . . . , n,

i = 1, 2, . . . , m.

Clearly, Ui1,C

Z

=

{d I O(x) = d, x E X*} .

If 0(u) and O(v) are in the same set Cn+i (the sets Ci for i < n are one-sets and hence are out of consideration), then u = ulzi and v = vlzi . Thus I0(ulzi)-O(zi)I k and for all x. Then from the proof of Theorem 5.4 .2, it follows that if A is a quasi-definite automaton, then A can be E-approximated by a definite automaton B. The converse is also true. The following example shows that there are fsf's that can be E-approximated by npa's, but are not quasidefinite . Example 5.4.3 Consider the fsf 0 induced by the pa A defined as follows: X = {a, b}, the initial vector is 7r = [ 0

Q = {qo, qi },

1 ] , the final vector is

and the

transition matrices are P(a) _ -

1

1 0

0 1 ]

and

P(b)

_

i

1

i 0

Then O(x) is the (1,1) entry in the matrix P(x) . Clearly, 0 is not quasidefinite, for O(x) = 0(bk), where k is the number of b's in the word x (e .g., - 4'(a O(ban) = O(b) for any n) . Therefore, for any n, I0(ban) n )I 2 contradicting Definition 5.4 .1 . To show that 0 can be E-approximated for any E, consider the following cover induced by 0 . Divide the interval [0,1] by k-1 points dl, d2 . . . . , dk_1 (do = 0, dk = 1) so that d2 - d2_1 = 2 E, i = 1, 2, . . . , k - 1 and 1 - dk_1 0 implies z E W*, p(q, w, z, q~) > 0 implies z = +, or p(q, w, z, q~) > 0 implies z = - . (3) There exists q2 E Q such that (i) p(q2 , w, z, q) = 0 for all q E Q, w E W and z E W* U (ii) for all w E W and q E Q, where q :?~ q2, E Ep(q, w, z, q) = 1~

q'EQzEZ

where Z = W* U {+, -}, and (iii) for all x E X*, Ek i qk (q, x, r) > 0 implies r = q t y, where Y* . 0 YE

Theorem 5.14 .7 If f can be generated by a (bounded) PTM, then f is a (bounded) type-0 random language .

Proof. Let f = R(FM ), where M = (Q, X, Y, Wp, h) is a (bounded) PTM satisfying conditions (1)-(3) of Proposition 5 .14 .6 . Let Ao, A1 , $ ~ W and WO = W U {Ao, Al , ¢, $ f . Let G = (Y, N, P, h'), where N = WO\Y, h(AO) = 1, P = {Po, P1 P21 P31 P41 U {P. I u E X1, PO = {Ao}, P. = {Aj I for all u E X, Pi = {Ai }, P2 = {q2}, P3 = {¢, $ 1, P4 C {qw q E Q, w E WoI U {w~qw I q E Q and w, w' E WO 1, Po(AO, ¢Al$) = 1, P. (A,, Alu) = 1 for all u E X, p, (Al, ql) = 1, P2 (g2, A) = 1, P3 (¢, A) = P3 $, A) = 1, and I

P4(s' t) =

p(q, w, z, q~) p(q, w, +, q~) p(q, w, -, q~) 1

0

if s = qw, t = q~z, w :A $ if s = qw, t = wq, w 7~ $ if s = w~qw, t = qw~w, w' if s = q$, t = qb$ or s = ¢qw, t = ¢bqw otherwise.

Clearly, G is a (bounded) type-0 probabilistic grammar . that f = fG . m

It also follows

Theorem 5.14 .8 If f is a bounded type-0 random language, then f can be generated by a synchronous PTM.

Proof. Let f = fG, where G = (T, N, P, h) is a bounded type-0 probabilistic grammar . We may assume without loss of generality that G is © 2002 by Chapman & Hall/CRC

21 6


total and h(A0) = 1 for some Ao E N. For all s E (T U N)*, we associate the abstract symbols. Let Q1 = {s I s E P} and Q2 = {t I p(s, t) > 0 for some p E P and s E p} . Let M = (Q, X, Y, W p, h') be a synchronous PTM, where X=PUQ1U{1},PUQIUQ2U{1,bjCW

and h(qo) = 1 for some qo E Q. We now describe informally the behavior

of M. (a) Suppose M has the instantaneous description qox E Q(PUQ1U{1})* . Then M will go to state ql if x ~ (PQ1{1}*)+, where p(ql, w, w, ql) = 1 for all w E W, i.e., M loops. Otherwise, M will have the instantaneous description g2xbbAo, q2 E Q. (b) Suppose M has the instantaneous description q2z, where z = psls2 . . . s,nkzobzlbrlz2br2 . . . z,,brn, n >, 0~ p E P~ si E Ql~ 2 = 1~ 2~ " . . , m, zo E (PQ1{1}*)*, zi E (PQIQ2) * , ri E (TUN)*, i = 1,2, . . . , n, and k stands for 11 . . . 1(k times) . Then M will go to state ql if (1) n = 0, (2) m < n, or (3) for some i and j, where i, j = 1, 2. . . . , n, ri = rj, but si :?~ sj . Otherwise, M will have the instantaneous description q3z, q3 E Q. (c) Suppose M has the instantaneous description q3 z, where z is the same as in (b) satisfying none of the conditions (1)-(3) . Then M will have instantaneous description q2z', where z' is obtained from z by (1) erasing psls2 . . . pLk, (2) erasing zibri if m(ri, si) < k, and (3) replacing zibri by ziPSitilbWilziPSiti2bwi2 . . . zipsitalbwil

if m(ri, si) >, k.

Here, wi, ,k ri mod(si, tip), j = 1, 2, . . . , l, and {til, ti2 . " . . , ti n } = {t I p(s, ti) > 0} . Thereby, we assume that for all p E P and s E p, the set {t p(s, t) > 0} is well ordered . (d) Suppose M has the instantaneous description g3bz, where z E W* . Then M will go to state ql if z = A. Otherwise, M will have the instantaneous description q4z, where q4 E Q. (e) Suppose M has the instantaneous description q4z, where z = zlbrlz2br2 . . . znbrn, n >, 1, zi E (PQI Q2)*, ri E (TUN)*, i = 1,2, . . . n. If z, A and for all i, where i = 1, 2, . . . , n, zi starts with ps where p E P and s E Q1, then M will have the instantaneous description gp , s z, where qp,s E Q. If zl = A, then M will have the instantaneous description q5z', where z' is obtained from z by erasing b, and q5 E Q. If z = A, then M will go to state ql . (f) Suppose M has the instantaneous description gp , s z, where z is defined as in (e) and for all i, i = 1, 2, . . . , n, zi starts with ps. Then, with


5.15. Context-Free Probabilistic Grammars and Pushdown Automata 217

probability p(s, t), M will have the instantaneous description q4z', where z' is obtained from z by erasing all zibr2 where z2 does not start with pst, and erasing pst from all z2 that starts with pst. (g) Suppose M has the instantaneous description q5r, where r E W*. Then M will go to state glif z ~ T* . Otherwise, M will have the instantaneous description q6r, where for all q E Q, w E W and z E W* U {+ .-},

p(q6, w, z, q) = 0. Note that M acts probabilistically only when case (f) occurs. Other wise, it acts deterministically . It follows that f = R(F'). Combining Theorem 5 .14 .7 and 5.14.8 yields the following result .

Theorem 5.14 .9 Every bounded type-0 random language can be generated by a synchronous PTM, and vice versa. m In view of Theorem 5.14.9, many interesting properties of bounded type0 random languages can be obtained from the results given in [201] .

5 .15

Context-Free Probabilistic Grammars and Pushdown Automata

In this section, we study context-free probabilistic grammars and their relationship to probabilistic pushdown automata. We also examine the relationship between leftmost random languages and leftmost context-free random languages . Proposition 5.15 .1 If f is a context-free random language, then f = fG for some total context-free probabilistic grammar G = (T, N, P, h) such that P = N, h(A0) = 1 for some Ao E N, andfor all p E P ands E p, p(s, t) > 0 implies t = av, where a E TU{Af and v E N* . If, in addition, f is bounded, then for all p E P and s E p, p(s, t) > 0 implies t = av, where a E T U {Af and v E NN . In the remainder of the section, if f = fG L for some specified (type0, context-sensitive, context-free, weakly regular, and regular) probabilistic grammar G, then f is said to be a leftmost (type-0, context-sensitive, context-free, weakly regular, and regular) random language. Theorem 5.15 .2 If G = (T, N, P, h) is a contextfree probabilistic grammar such that for all r E T*,

fG(r) =

zEZ* AEN ~ h(A)fz(A, r), ,

where Z1 = P x DL(G) x N, then fG is leftmost context free. fG = fG,L .


In fact,

21 8


Proof. By a previous remark, DL(G) contains exactly one replacement function, say So. For every z = zl (p, bo, k)z2 E (P x {bo} x

N)* ,

where z2 E (P x {So} x {1})* and k > 1, let z' = zl (p, So,1)z2 . It can be shown that f,(A, r) fz~ (A, r) for all A E N and r E T* . Thus for all z E (P x {So} x N)*, there exists zo E (P x {So} x {1})* such that f, (A, r) S fzo (A, r) for all A E N and r E T*. Hence fG = fG,LTheorem 5.15 .3 If G = (T, N, P, h) is a context-free probabilistic gram-

mar such that P contains exactly one element, then fG is leftmost context free . In fact, fG = fG,L .

Proof. Let P = {p} and DL(G) = {So} . It follows by induction on ~zj, z E (P x D(G) x N)*, that for all z E ({p} x D(G) x N)*, there exists z' E ({p} x {So} x {1})* such that fz(A, r) 5 fz, (A, r) for all r E T* and A E N . Thus fG = fG,L . Definition 5.15 .4 A

automa (total) probabilistic pushdown ton (PPA) is a septuple M = (Q, X, Y, W p, h, g), where Q, X, Y, and W are finite nonempty sets, p is a (total) random function from Q x W x X into Y* x W* x Q, h is a function from Q x W into [0,1] such that

1: 1:

qEQ wEW

h(q, w) = 1,

and g is a function from Q into [0,1] . If, in addition, for all q, q' E Q, u E X, and w E W, p(q, w, u, y, z, q') = 0 except for finitely many z E W* and y E Y*, then M is said to be bounded.

In the Definition 5.15.4, Q, X, Y, and W are, respectively, the state, input, output, and pushdown alphabets . p(q, w, u, y, z, q') is the conditional probability that the next state of M is q', the leftmost symbol w in the pushdown list is replaced by z and output string y is produced, given that the present state of M is q, the leftmost symbol in the pushdown list is w and input u is applied . h(q, w) is the probability that q is the initial state of M and w is the initial symbol in the pushdown list . g(q) is the probability that q is a final state of M. Definition 5.15 .5 Let M = (Q, X, Y, Wp, h, g) be a PPA .

(1) Define the function p' from (QxW* xX* xY*) x (QxW* xX* xY*) into [0,1] by for all r, s E Q x W* x X* x Y*, p(q, w, u, yo, zo, q~) 1 0


if r = (q, wz, ux, y), s = (q~, zoz, x, yyo) if r = (q, A, ux, y), s = (q, A, x, y) or r=s= (q, z, A, y) otherwise,

5.15. Context-Free Probabilistic Grammars and Pushdown Automata 219

where q,q'EQ,uEX,xEX*,y°,yEY*,wEW and z°,zEW* . (2) For all k E hY U {0}, define the function pk from (Q x W* x X* x Y*) x (Q x W* x X* x Y*) into [0,1] by for all r, s E Q x W* x X* x Y*, p0,vt(r,s)

1 -_ ~ 0

ifr=s if r :?~ s

Pki(r, s) = j:Pk(v, s)PM(r, v), vEr

where l'=QxW* x X* x Y* . (3) Define the function FM from X* x Y* into [0,1] by for all x E X* and y E Y*, FM(x, y) _

1: 1: h(q, w) [1:PM (q, w, x, A, q, A, A, y)J . k=0

q,q'EQ-EW

(4) Define the function G°'1 from X* x Y* into [0,1] by for all x E X* and y E Y*, GM (x, y) _

q, q' EQw EW z EW *

/ h (q, w)g(q~) I:pk(q, w, x, A, q , z, A, y) k=0

It follows that F" and G" are both random functions from X into Y* . Moreover, for all x E X* and y E Y*, where x l = k, Flvl

(x, y) =

q,q'EQwEW*

h(q, w)pk (q, w, x, A, q, A, A, y)

and GM(x, y) _

q,q'EQwEWzEW

h(q, w)g(q)pk(q, w, x, A, q, z, A, y)

It follows from the above definition that PPAs are stochastic generalizations of conventional pushdown automata. FM is the random function computed by M with empty store, while GM is the random function computed by M with final states . Theorem 5.15 .6 If f is a leftmost(bounded) context-free random language, then f can be generated with empty store by some total (bounded) PPA having a single state and such that p(q, w, u, y, z, q~) > 0 implies y E T U {A}. © 2002 by Chapman & Hall/CRC

22 0


Proof. Let f = fG,L, where G = (T, N, P, h) is a (bounded) contextfree probabilistic grammar . By Proposition 5.15 .1, we may assume that G is total and for all p E P and s E p, p(s, t) > 0 implies t = av, where a E T U {A} and v E N* . Let M = (P, {qo}, T, N, p, h', g), where h'(qo, A) = h(A) and p(qo, w, y, p, z, qo) = p(w, yz) for all z E N* and y E T*, p E P, and w E N. Clearly M is a PPA with the desired properties. Moreover, it can be shown that f = R(FM) . Theorem 5.15 .7 If

f is a leftmost (bounded) context-free random language, then f can be generated with final states by some total (bounded) PPA having two states and such that p(q, w, u, y, z, q') > 0 implies y E T U {A} .

Proof. Let f = fG,L, where G = (T, N, P, h) is a (bounded) contextfree probabilistic grammar . By Proposition 5.15 .1, we may assume that G is total and for all p E P and s E p, p(s, t) > 0 implies t = av, where a E T U {A} and v E N* . Moreover, we may assume, without loss of generality, that there exist uniquely Ao, Al, A2 E N and p o , pi E P such that (1) h(AO) = 1, (2) -PO = {Ao} and po(Ao,A2Ai) = 1, (3) pi = {Ai} and pi (A i , A) = 1, (4) for i = 0, 1, Ai E p implies p = pi, (5) for all p E P and s E p, p(s, t) > 0 implies t ~ (T U N) * {Ao}(T U N) *, and (6) for all p E P, s E p, and p zh p o , p(s, t) > 0 implies t (TUN)*{Ai}(TUN)* . That is, we modify G in such a way that A1 serves as an endmarker . Let M = (Q, P, 2', N, p, h~ , g) where Q = {qi, q2}, h' (qi , AO) = 1, g(q2) = l and for all q,q'EQ,pEP,wEN,zEN*, and yET*,

p(q, w, p, y, z, q) =

p(w, zy) 1 0

if q = q' = qi, w 7~ Ai or p 7~ pi if q=qi, q= q2, w=z = Ai, p = pi, y = A or q=q'=q2, w=z, y=A otherwise .

Clearly, M is a PPA with the desired properties. Moreover, it can be shown that f = R(GM). Let S be a nonempty set and k E N. Let S k = {r E S* I Irl S k}. Theorem 5.15 .8 If

f = R(F") for some bounded PPA M, then leftmost bounded context-free random language .

f

is a

Proof. Let M = (Q, X, Y, W p, h, g) . Since M is bounded, there exists k E hY such that for all q, q' E Q, u E X, w E W y E Y*, and z E W*, p(q, w, u, y, z, q') > 0 implies IzI 0. By a similar argument, it can be shown that bt = 0, a contradcition . Hence A ~ L2 . Consequently, L2 Z Ll .

5 .18

Rirther Properties of L3

In this section, we show that L3 is generated by the classes of all probabilistic, max-product, or max-min automata with deterministic or nondeterministic transition functions . For k = 1, 2, 3, let NLk = {A E Lk I A = rk for some A = (Q,p, h, g), where Im(p) C {0,1}} . © 2002 by Chapman & Hall/CRC

23 2


Theorem 5.18 .1 Let A E L . For k = 1, 2, 3, A E NLk if and only if T(A) is finite .

Proof. We only prove the case for k = 1. The other two cases can be proved similarly. Suppose A E NL1 . It follows from the proof of Theorem 5.16.7 that there exists a 1-admissible set I' = {A1, A2, . . . , An} of A such that for all i = 1, 2, . . . , n and u E X, AZ E I' . Since A = E', c2A2 for some c2 >_ 0 and E', c2 = 1, we have for all x E X* that Ax = E', c2AZ, where AZ E F for all i = 1, 2, . . . , n. Thus T(A) is finite . Conversely, suppose that T(A) is finite . Clearly, T(A) is a k-admissible set of A. The desired follows immediately from the proof of Theorem 5.16.7 . For k = 3, Theorem 5.18.1 can be strengthened as follows : Theorem 5.18 .2 Let A E L. Then A E L3 if and only if T(A) is finite . . Proof. Suppose A = r3 , where A = (Q, p, h, g) and Q = {q1, q2, . . . , q,,} For all i = 1, 2, . . . , n and x E X*, define A2 (x) = q3(x, q2) and /32(x) _ V~ 1(h(qj) np3(qj,x,g2)) Then for all x E X*, Ax = U=I (/32(x) n A2) . Let R = Im(p) U Im(h) . Clearly, R is finite and for all i = 1, 2, . . . , n and x E X*,/3 2 (x) E R. Thus T(A) is finite . The converse follows from Theorem 5.16.7.

Theorem 5.18 .3 Let k = 1, 2,3. Then A E NLk if and only if A = rk for some A = (Q, p, h, g), where (1) Im(p) C_ {0,1}, and for all q E Q and u E X, there exists a unique q' E Q such that p(q, u, q') = 1, and (2) Im(g) C {0,1} or Im(h) C {0,1} and there exists a unique q E Q such that h(q) = 1 . Proof. The proof follows from Theorems 5.16.12, 5.16.13, and 5.18.1 . Theorem 5.18 .4 Let k = 1, 2. If A E NLk, then Im(A) is finite. Proof. We prove only the case k = 1 . The other case can be proved in a similar manner . Let = rl , where A = (Q, p, h, g) and Im(p) C {0,1} . Then for all q, q' E Q and x E X*, pi (q, x, q') E {0,1} . Since rl (x) _ Eq q, EQ h(q)pi (q, x, q')g(q') for all x E X*, we have that Im(A) = Im(rl) is finite. Let FL = {A E L I Im(A) is finite} . Theorem 5.18 .5 NL 1 = L1 rl FL . Proof. Let A E L1nFL. By Corollary 5.16.8, V(A) is finite-dimensional. Let x2 E X*, i = 1, 2, . . . , n, be such that ~Axl, Axe , . . . , Axe } is a basis of V(A) . Then for all x E X*, there exist unique /32(x), i = 1, 2 . . . . I n, such that Ax = E'1,32(x)A` .Clearly there exists a positive integer m such that for all x E X*,/32(x), i = 1, 2, . . . 1 n, is completely determined by © 2002 by Chapman & Hall/CRC

5.18. Further Properties of L3

23 3

A'(y), where y E X* and jyj < m. Since Im(A) is finite, we have that T(A) is finite . Thus by Theorem 5.18.1, A E NL l . Therefore, Ll n FL C_ NL 1. By Theorem 5.18.4, NLl C Ll rl FL . Hence NL l = L1 n FL . m Theorem 5 .18 .6 Let A E L2 . If there exists c > 0 such that for all x E X*,

A(x) = 0 or A(x) > c, then T(A) is finite .

Proof. Let A = r2 , where A = (Q, p, h, g) and Q = {ql, q2, . . . , qn}. For all i = 1, 2, . . . , n and x E X*, define Ai(x) = g2(qj, x) and 3i (x) _ Ax V~1(h(qj )p2(qj ,x,g2)) . Then for all x E X*, = UZ i(/3Z(x)AZ), where

13 Z (x) =

if there exists y E X* such that 0 7~ Ax (y) = l3i (x) AZ (y)

,3Z(x) 0

otherwise .

Thus for all i = 1, 2, . . . , n and x E X*, /3'(x) = 0 or /3'(x) > c. Let R = Im(p) U Im(h) U Im(g), Fn = {IZo c2 E R, i = 0, 1, 2. . . . I n, and

o c2 > c} and F = (U'OF ) U {0} . Clearly, F is finite and for all i = 1, 2, . . . , n and x E X*, l3'(x) E F. Thus T(A) is finite. m Theorem 5 .18 .7 NL2 = L2

n FL .

Proof. The proof follows from Theorems 5.18.1, 5.18.4, and 5.18 .6 . The next result follows from Theorems 5.18.1, 5.18.2, 5.18.5, and 5.18.7 . Theorem 5 .18 .8 L3 = L1

rl FL

= L2

rl FL . m

The following closure property of L3 follows from the results given in the previous section and Theorem 5.18.8 . Theorem 5 .18 .9 (1) If A E L3, then X and AT E L3(2) If A,, A2 E Ls, then A, A2, Ai U A2 and Ai A2 E L3 . (3) If Al, A2, A3 E L3, then Ai A2 + Ti A3 E L3 . (4) Let k = 1, 2,3. If AZ E L3, i = 1, 2, . . . , n, and A is a k-combination of {Ai, A2, - . . , A n }, then A E L3 . 0

rl

Closure properties of L3 with respect to other operations can be found in [196] . Definition 5 .18 .10 A deterministic pseudoautomaton is a probabilis-

tic automaton (Q, p, h, g), where Im(p) U Im(h) U Im(g) C {0,1}.

Note that if A is a deterministic pseudoautomaton, then rl = r2 = r3 . For simplicity, we write rA = rk for k=1,2,3 . Let DL = {rA I A is a deterministic pseudoautomaton} . Clearly, A E DL if and only if Im(A) C {0,1} and {x E X* I A(x) = 1} is a regular language [189] .


23 4


Theorem 5.18 .11 Let A E L. Then A E DL if and only if Im(A) C {0,1} and T(A) is finite . Proof. The proof follows from Theorem 5.18.2 . The above theorem is Nerode's Theorem [181] . Theorem 5.18 .12 Let k = 1, 2,3. Then A E DL if and only if A E Lk and Im(A) C {0,1} . Proof. The proof follows from Theorem 5.18.8 . Theorem 5.18 .13 For k = 1, 2, 3, Lk is a proper subclass of L . Proof. Let A E L be such that Im(A) C_ {0,1} and {x E X* I A(x) = 1} is a nonregular language. By Theorem 5.18 .12, A ~ Lk, k = 1, 2, 3. Thus Lk, k = 1, 2, 3, is a proper subclass of L . m Theorem 5.18 .14 Let A E FL . Then A E L3 if and only if all 0 c} is a regular language . Proof. Suppose A E L3 . It follows from Theorems 5 .18.2 and 5.18 .11 that for all 0 c} is a regular language. Conversely, < c < 1, {x E X*IA(x) > c} is a regular language. Let suppose for all 0 _ Im(A) = {CI, C2, . . , cn } . Then A = UZ 1cjAZ, where for all i = 1,2 . . . . , n, 1 0

if A(x) > c2 otherwise .

By hypothesis, AZ E L3 for every i, i = 1, 2, . . . , n. Hence by Theorem 5.18.9(4), A E L3 . Theorem 5.18 .14 was first proved in [195]. Clearly, {x E X* I A(x) > c} may be replaced by {x E X* I A(x) = c}, {x E X* I A(x) >_ c}, {x E X*I A(x) < c}, or {x E X*I A(x) < c} in Theorem 5 .18 .14 . 5 .19

Exercises

1. Let A = (Q, 7r, {A(u) }, F) be a probabilistic automaton. Prove that 7r(xy) = 7r(x)A(y), where x, y E X* . 2. Prove that p(xy) = 7r(x)9F (y), where x, y E X* . 3. Prove Proposition 5.2 .6 . 4. Let A be a probabilistic automaton . If A can be E-approximated by

a definite automaton, prove that A is quasi-definite .


5.19. Exercises

235

5. Characterize the fsf's (the pa's) that can be approximated by (a) linear bounded automata, (b) pushdown automata, (c) sequential automata . 6. Determine the most powerful class of np devices that suffices for approximating the pa's. 7. Determine whether pushdown automata can approximate pa's. 8. Determine whether or not the family of w.i.m. languages is a subset of the family of w.i.s. languages . 9. Prove that every stochastic language under maximal interpretation is w .3.m. 10. Prove Theorem 5.16.7 for the cases k = 2,3 . 11. For k = 2, 3, state and prove a result similar to Theorem 5.16.7 replacing Ax with P) . 12. Prove Theorem 5 .18.1 for the cases k = 2,3 . 13. Prove Theorem 5.18.4 for the case k = 2. 14. Prove Theorem 5.18.14 with > in {x E X* I A(x) > c} replaced by =, >, and < .


Chapter 6

Algebraic Fuzzy Automata Theory 6 .1

Fuzzy Finite State Machines

A sequential machine consists of two main structures, the transition structure and the output structure . The transition structure is an internal part of the machine while the output structure is the external part . Consequently, the output structure is of more interest for practical applications than the input structure. The output structure is dependent on the transition structure while the transition structure is independent of the output structure. Hence the input structure can be studied separately. In this chapter, we study the transition structure of fuzzy machines . A fuzzy finite state machine (ffsm) is a triple M = (Q, X, ft), where Q and X are finite nonempty sets and /t is a fuzzy subset of Q x X x Q, i.e ., /t : Q x X x Q ~ [0,1] . As usual X* denotes the set of all words of elements of X of finite length. Q is called the set of states and X is called the set of input symbols. Let A denote the empty word in X* and Ixl denote the length of x b'x E X*. X* is a free semigroup with identity A with respect to the binary operation concatenation of two words.

6 .2

Semigroups of Fuzzy Finite State Machines

Definition 6.2.1 Let M = (Q, X, /t) be a ffsm. Define /t* : Q x X* x Q [0,1] by

*(q, A, p) _ { 0 if q 237 © 2002 by Chapman & Hall/CRC

zAp

238

6. Algebraic Fuzzy Automata Theory

and w* (q, xa, p) = V{ft* (q, x, r) A ft(r, a, p) I r E Q} VxEX*,aEX. Let X+ = X*\{A} . Then X+ is a semigroup. For /t* given in Definition 6.2 .1, we let y+ = ft* restricted to Q x X+ x Q.

Lemma 6.2 .2 Let M = (Q, X, /t) be a ffsm. Then * ft (q, xy, p) = V {ft*(q, x, r) n ft * (r, y, p) Ir E Q} Vq, p E Q and Vx, y E X* .

Proof. Let q, p E Q and x, y E X* . We prove the result by induction on lyl = n . If n = 0, then y = A and hence xy = xA = x . Thus V{ft* (q, x, r) n ,a* (r, y, p) I r E Q} = V{ft* (q, x, r) n ,a* (r, A, p) I r E Q} = ft * (q, x, p) = ft * (q, xy,p) by the definition of ft * . Thus the result is true for n = 0. Suppose the result is true for all u E X* such that lul=n-1,n>0.Lety=uawhere uEX*, aEX,andJul=n-1,n>0 . Now y,* (q, xy, p) = y,* (q, xua, p) = V {/t* (q, xu, r) A y,(r, a, p) I r E Q} = V{(V{,a* (q , x, s) n,a* (s, u, r) I s E Q}) n,a(r, a, p) I r E Q} = V{V{ft* (q, x, s) n ft*(s,u,r)Aft(r,a,p)}I r,s E Q}=V{,a*(q,x,s)A(V{,a*(s,u,r)A,a(r,a,p)Ir E Q}) Is E Q} = V{w* (q, x, s) Aw* (s, ua, p) I s E Q} = V{w* (q, x, s) Aw* (s, y, r) s E Q} . Thus the result is true for Iyl = n . m Define a relation - on X* by Vx, y E X*, x - y if and only if ft* (q, x, p) _

ft * (q, y, p) V q, p E Q. Clearly - is an equivalence relation on X* . Let z E X* and let x y . Then Vp, q E Q, /t* (q, xz, p) = V{/t* (q, x, r) A /t* (r, z, p) I r E Q} _ V{ft* (q, y, r) ntt* (r, z, p) I r E Q} = ft * (q, yz,p) . Thus xz - yz. Similarly zx - zy. Thus - is a congruence relation on the semigroup X* . We have thus proved the following result .

Theorem 6.2.3 Let M = (Q, X, ft) be a ffsm . Define a relation - on X*

by Vx, y E X*, x - y if and only if ft * (q, x, p) = ft * (q, y, p) V q, p E Q . Then - is a congruence relation on X* . Let x E X*, [x] = {y E X* Ix - y}, and E(M) = {[x] Ix E X*} .

Theorem 6.2.4 Let M = (Q, X, ft) be a ffsm. Define a binary operation * on E(M) by V [x], [y] E E(M), [x] * [y] = [xy] . Then (E(M), *) is a finite semigroup with identity.

Proof. Clearly * is well defined and associative. Now [x] * [A] = [xA] _ [x] = [Ax] = [A] * [x] V [x] E E(M) . Thus [A] is the identity of (E(M), *) .


6.2. Semigroups of Fuzzy Finite State Machines

239

Hence (E(M), *) is a semigroup with identity. Let x E X* and let x = xlx2 . . . xn, where xl, X2 . . . . , x n E X. Then b' q, p E Q, N' * (q,x,p) = V{ft(q,xl,gl) AN(gl,x2,q2) n gl, q2, . . . , qn-1 E Q} .

. . . ANt(gn-l,xn,p)

Hence since Im(y,) is finite, Im(y,*) is finite . Thus (E(M), *) is a finite semigroup with identity. Example 6.2.5 Let M = (Q, X, p) be a ffsm, where Q = {q} and X = {a, b} . Define p : Q x X x Q ~ [0,1] by ft(q, a, q) = z = ft(q, b, q) . Then b'x, y E X+, x - y since /t* (q, x, q) = 2 = / t* (q, y, q) . Hence E(M) _ {[A], [x]}, where x E X+ . Clearly, [A] is the identity of E(M) and [x]2 = [x] .

Example 6.2.6 Let M = (Q, X, y) be a ffsm, where Q = {q} and X = {a, b} . Define y : Q x X x Q ~ [0,1] by ft(q, a, q) = 3 and ft(q, b, q) = s . Then V'x E X*, y* (q, x, q) = s if and only if x contains a b. Hence Vx, y E X+, x - y if and only if x and y both contain a b. Hence E(M) _ {[A], [x], [y]}, where x contains a b and y does not contain a b, x, y E X+ . Here [A] is the identity of E(M), [x] 2 = [x], [y]2 = [y], and [x] * [y] _ [x] = [y] * [x] .

We now define another type of congruence relation on X* . Let x, y E X* . Define x ~ y if and only if (Vs, t E Q, ft * (s, x, t) > 0 ~ ft* (s, y, t) > 0) . Clearly ; is an equivalence relation on X* . Let z E X* and x s: y. Then * * Vs, t E Q, ft (s, zx, t) = V{Nt* (s, z, r) A ft (r, x, t) lr E Q} > 0 if and only if * * 3 r E Q such that ft (s, z, r) A ft (r, x, t) > 0 if and only if 3 r E Q such that ft* (s, z, r) A ft* (r, y, t) > 0 if and only if ft* (s, zy, t) = V{Nt* (s, z, r) A y* (r, y, t) I r E Q} > 0 . Hence zx s: zy. Similarly xz s: yz . Thus s: is a congruence relation on X* . Hence, we have the following theorem. Theorem 6.2.7 Let M = (Q, X, y) be a ffsm. Let x, y E X* . Define a relation ; on X* by x ,: y if and only if Vs, t E Q, /t* (s, x, t) > 0 [t* (s,

y, t) > 0. Then ; is a congruence relation on X* .

Let x E X* and let Qx~ = {y E X*Ix ~ y} . Let E(M) = {Qx~lx E X*} . Theorem 6.2.8 Let M = (Q, X, /t) be a ffsm . Define a binary operation ~ on E(M) by b' Qx~, Qy~ E E (M), Qx~~Qy~ = Qxy~ . Then (E (M), ~) is a finite semigroup with identity and [x] ----> Qx~ is a homomorphism of E(M) onto E(M) .

Proof.

Cle arly (E(M), *) is a semigroup with identity . Define f E(M) - E(M) by f ([x]) = Qx~ b' [x] E E(M) . Let x, y E X* and [x] = [y]. Then Vs, t E Q, /t* (s, x, t) = y,* (s, y, t) . Thus Vs, t E Q, /t* (s, x, t) > 0 ~ y* (s, y, t) > 0. Hence x s: y or Qx~ = Qy~ . Thus f is well defined . Clearly f is an onto homomorphism. Now since E(M) is finite E(M) is finite . m © 2002 by Chapman & Hall/CRC

24 0


Example 6.2.9 Let M = (Q, X, /t) be the ffsm as defined in Example 6.2.5 . Then Vx, y E X*, x ,~ y . Thus i(M) _ {[[x]]}, where x E X+ . Now [[x]]2 = [[x]] .

Definition 6.2.10 Let M = (Q, X, /t) be a ffsm. For all x E X* define the fuzzy subset xM of Q x Q by xM (s, t) = y* (s, x, t) Vs, t E Q.

Theorem 6.2.11 Let M = (Q, X, y) be a ffsm . Let Sm = {x` I x E X*} .

Then (1) xM o ym = (xy) M Vx, y E X*, (2) (Sm, o) is a finite semigroup with identity, where o is defined as in Definition 1.4 .4 .

Proof. (1) Let s, t E Q . Then (xy) M (s, t) = y,*(s, xy, t) = V{/t* (s, x, q) A Ixm ym (xm w* (q, y, t) I q E Q} = V (q, t) I q E Q} = (s, q) A o yM) (s, t) . Thus xM yM . (xy)M -_ o (2) Clearly (Sm, o) is a finite semigroup with identity, where Am is the identity element . Sm is finite since Q and Im(y) are finite . Theorem 6.2 .12 Let M = (Q, X, /t) be a ffsm. Then Sm - E(M), i.e., Sm and E(M) are isomorphic as semigroups .

Proof. Define f : Sm ~ E(M) by f (x') = [x] VxM E Sm. Let XM, ym E Sm . Then x M = ym if and only if xm (s, t) = ym(s, t) Vs, t E Q if and only if ft * (s, x, t) = ft* (s, y, t) Vs, t E Q if and only if [x] = [y] . Thus f is single valued and one-one . Now .f (X

IV[

o ylvl) = f ((xy),V,) = [xy l = [x] * [y] = f (X IV[ ) * f (y IV[ ) .

Thus f is a homomorphism. Clearly f is onto. Hence Sm - E(M) . m Let M = (Q, X, ft) be a ffsm. The index of an equivalence relation is the number of distinct equivalence classes. Let - be a congruence relation of finite index on X* . Let x E X* and ~ x _ {y E X*Ix - y} . Let Q={-<x>- IxEX*} . Define a :QxXxQ~ [0,1] by V-<x>- EQ and V a E X, a(-< x _ >-, a, -< xa >-) an arbitrary fixed element in (0,1] and

V-<x>-,-<w>- EQ,

Q(~x>-,a,~xa>-)ifw-xa

0 otherwise.

Let ~ x >-, ~ y >-, ~ u >-, ~ v >- E Q and a, b E X. Suppose that (fix>-,a,~u>-)=(may>-,b,~v>-) .

Then ~ x >-

v - ya . Thus

y >-, a = b, ~ u >-

v >- . Now u - xa if and only if

a(~x>-,a,~u>-)=a(~y>-,b,~v>-) .


6.2. Semigroups of Fuzzy Finite State Machines

241

Hence a is single valued. Thus M = (Q, X, a) is ffsm. Now extend Q to Q* as /t was extended to /t* in Definition 6.2.1 . Lemma 6.2 .13 Let M be as above and let ~ z >-, -< w >-E Q . Then the following assertions hold. (1)VXEX*, if a*(-0, then -0,then ~z~-- =~w~or~zx~=-<w~-- . Suppose the result is true b' y E X* such that I yI = n-1, n > 0. Let x = yo; where y E X*, a E X, and I y I = n - 1 . Let a* (~ z ~, ya, ~ w ~) = a*(~ z ~, x, ~ w ~) > 0 . Now

>

I - 0 and a* (~ q ~--, a, w ~--) > 0 for some-< q _ -< q and -< qa ~-- _ E Q. Thus by the induction hypothesis, -< zy ~w~-- .Hence-0. Suppose the result is true b' y E X* such that I yI = n - 1, n > 0. Let x = ya where y EX*, a EX, and jyj =n-1 . Now a* (~ z >-, x, ~ zx ~)

=

a* (~ z >-, ya, ~ zya >-)

>_ >

I ~ q ~EQ} a* (-< z ~, y, -< zy 0.

~) n c,(-
0 . Then ~ zx ~ = ~ w ~ . Since x - y and - is a congruence relation, zx ~ = -< zy ~ . Thus zy a = -< w ~ . Hence a* (-< z ~, y, -< w ~) > 0. Similarly if a* (-< z ~, y, -< w ~) > 0, then Q*(~z~,x,~w~)>0 . © 2002 by Chapman & Hall/CRC

242


Hence x y . Conversely, suppose x ,~ y . Let ~ z >- E Q . Now c,*(-< z x,-0.Hence a*(~z~,y,~zx~--)>0.Thus - 0, then 3 t E Q1 such that ft1(q, x, t) > 0 and a(t) = a(r) . Furthermore, b'p E Q if a(p) = a(q), then ft1(q,x,t) > P1 (p, x,

0


244

6. Algebraic Fuzzy Automata Theory Proof. Let p, q, r E Q1, x E X 1 , and ft2(a(q), /3(x), a(r)) > 0 . Then V{ftl (q, x, s) I s E Q1, a(s) =a ( r )I > 0.

Since Q1 is finite, 3 t E Q1 such that a(t) = a(r) and ftl (q, x, t) _ V{ftl(q, x, s)I s E Q1, a(s) = a(r)} > 0 . Suppose a(p) = a(q) . Then

wl (q, x, t) = w2(a(q), /3(x), a(r)) = w2 (C(p), /3(x), C(O) >- wl (p, x, r)- . Definition 6.3.4 Let Ml = (Q 1 , X1, ftl) and M2 = (Q2, X21 [t2) be two ffsms. Let (a,,3) : Ml ----> M2 be a homomorphism. Define /3* : Xl ----> X2 by /3* (A) = A and /3* (ua) = /3* (u),3(a) b' u E X*, a E X1 . Lemma 6 .3.5 Let Ml, M2, (a,,3), and /3* be as above. Then 3*(uv) _ * '3 (u)/3* (v) b' u, v E Xl . Proof. Let u, v E Xl and Iv I = n . If n = 0, then v = A and hence /3* (uv) = /3* (u) = /3* (u),3* (v) . Suppose now the result is true b' y E Xl such that IyI = n - 1, n > 0. Let v = ya where y E X*, a E X1, and /3* (uv) = /3* (uya) = /3* (uy),3(a) _ /3* (u),3* - 1 . Then Iy (y),3(a) _ /3*I = n (ya) /3* = (v) . The result now follows by induction. m (u),3* (u),3* Theorem 6.3.6 Let Ml, M2 be as above. Let (a,,3) : Ml ----> M2 be a homomorphism. Then ft*, (q, x, p) O .Letx=yawhereyEXl,aEXI,andlyl=n-1 . Now fti (q, x, p)

= = < < = =

fti (q, ya, p) V{fti (q, y, r) n fti (r, a, p) Ir E Ql V{ft2(a(q),,3*(y),a(r)) A[t*(a(r),,3(a),a(p))Ir E Ql} V{P2*(a(q), 3* (y), r~) nft2* W, 3* (a), a (p) I r' E Q2} ft2 (a(q), ,3* (y),3(a), a (p)) ft2(a(q),,3*(ya),a(p)) ft2(a(q),,3*(x), a (p)). m

Theorem 6.3.7 Let Ml, M2 be as above. Let (a,,3) : Ml ----> M2 be a strong homomorphism. Then a is one-one if and only if fti (q, x, p) = ft2(a(q),,3*(x), a (p)) V q, p E Q1 and x E Xl . Proof. Suppose a is one-one . Let p, q E Q1 and x E X* . Let Ix I = n . We prove the result by induction on n. Let n = 0. Then x = A and /3* (A) = © 2002 by Chapman & Hall/CRC

6.4 . Admissible Relations

245

A. Now a(q) = a (p) if and only if q = p. Hence fti (q, A, p) = 1 if and only if ft(a(q),,3*(A), a(p)) = 1. Suppose the result is true b' y E Xl, IyI = n- 1, 2 n > O . Let x = ya, Iy j =n-1, y E Xl , a E Xl . Then ft2 (a(q),

/3*

/3* ft2 (a(q), (ya), a(p)) /3* ft2 (a(q), (y),3 (a), a (p)) V{ft2(c,(q),,3*(y),a(r)) AIt2(C(r),,3(a),a(p)) IrEQ1}IrEQ1}

(x), a(p))

V{fti(q,y,r) Afti(r,a,p) fti (q, ya, p) fti (q, x, p) .

I r E Qi}

Conversely, let q, p E Qi and let a(q) = a(p) . Then 1 = y,2 (a (q), A, a (p)) = fti (q, A, p) . Hence q = p, i.e ., a is one-one . m

6 .4

Admissible Relations

Definition 6.4 .1 Let M = (Q, X, ft) be a ffsm and let - be an equivalence

relation on Q . Then - is called an admissible relation if and only if V p, q, r E Q, b'a E X, if p - q and ft(p, a, r) > 0, then 3 t E Q such that ft(q, a, t) > ft(p, a, r) and t - r.

Theorem 6.4.2 Let M = (Q, X, ft) be a ffsm and let - be an equivalence relation on Q. Then - is an admissible relation if and only if V p, q, r E Q, V'x E X*, if p - q and ft* (p, x, r) > 0, then 3 t E Q such that ft* (q, x, t) >_ p,* (p, x, r) and t - r.

Proof. Suppose - is admissible. Let p, q E Q be such that p - q. Let x E X*, r E Q be such that /t* (p, x, r) > 0. Suppose Ixj = n. If n = 0, then x = A . Thus W(p, x, r) > 0 ==~, p = r and /t* (p, x, p) = 1. Now ft* (q, x, q) = 1 = ft* (p, x, p) and q - p. Thus the result is true for n = 0. Suppose now the result is true b' y E X* such that Iyj = n - 1, n > 0. Let x = ya where y E X*, a E Xl, and Iyj =n-1 . Now y*(p, x, r) = p,* (p, ya, r) = V{ft* (p, y, ql) A /t* (ql, a, r) I ql E Q} > 0 . Let s E Q be such that y,* (p, y, s) A /t* (s, a, r) = V {/t* (p, y, qi) A /t* (qi, a, r) I qi E Q} . Then ft* (p, y, s) > 0 and ft* (s, a, r) > 0. By the induction hypothesis, 3 is E Q such that y,* (q, y, ts) >_ y,* (p, y, s) and is - s. Now ft(s, a, r) > 0 and is - s. Since - is admissible, 3 t E Q such that ft(ts, a, t) >_ ft(s, a, r) and t - r. Thus 3 t E Q such that t - r and y* (q, y, ts) A ft (t,, a, t) > ft* (p, y, s) A /t* (s, a, r) . Thus * ft (p, x, r)

= < = =


ft * (p, y, s) n ft* (s, a, r) ft * (q, y, ts) n ft(ts, a, t) V{ft* (q,y,ri) Aft(rl,a,t)Iri E Q} ft * (q, ya, t) ft * (q, x, t)

24 6


and t - r. The result now follows by induction . The converse is trivial . m Let M = (Q, X, ft) be a ffsm and let - be an admissible _ relation on Q. For q E Q, let [q] denote the equivalence class of q . Let Q = Q1 = { [q] Iq E Q} . Define the fuzzy subset ~ of Q x X x Q by w([q], x, [p])=V{ft(q,x,t) It E [p]} Vq, p E Q, x E X. Suppose that [q] = [q~], x = y, and [p] = [p~], q, q~, p, p' E Q and x, y E X. Then q - q' . Now ~([q] , x, [p]) = V {ft(q, x, r) I r E [p]

and

w([q ], y, [p])

= w([q ], x,

[p]) = V{ft(q, x, t) It E p]} .

Let r E [p] be such that ft(q, x, r) > 0. Then since - is admissible, 3 t E Q such that ft(q~, x, t) >_ ft(q, x, r) > 0 and t - r. Now since t - r, t E [p] = [p'] . Thus 3 t E [p~] such that ft(q~, x, t) >_ ft(q, x, r) > 0. Similarly if ft(q~, x, t) > 0 for some t E [p~], then 3 r E [p] such that ft(q, x, r) >_ ft(q~, x, t) > 0. Hence ~([q], x, [p]) = ~([q], x, p]) . Thus %~ is single-valued . Hence (Q, X, ~) is a ffsm. Define a : Q ---> Q by a(q) _ [q] b' q E Q. Clearly, a maps Q onto Q . Let 3 : X ---> X be the identity map. Let q, t E Q and x E X. Then ~(a(q), x, a(t)) = ~([q], x, [t]) = V{ft(q, x, r) Ir E [t] j > ft(q, x, t) . Hence (a,,3) is a homomorphism. Definition 6.4.3 Let Ml = (Q1, X, ftl) and M2 = (Q 2 , X, ft2) be two ff-

sms. Let a : (Q1, X, ftl) (Q2, X, ft2) be a strong homomorphism . The kernel of a, denoted Ker a, is defined to be the set Ker a = {(p, q) I a(p) = a(q)1 .

Lemma 6 .4 .4 Let a be as defined in Definition 6.4 .3 . Then Ker a is an admissible relation.

Proof. Now clearly Ker a is an equivalence relation . Let p, q E Q1 and (p, q) E Ker a . Then a(p) = a(q) . Let a E X, r E Q1, and ftl(p, a, r) > 0 . Then y 2 (a(q), a, a(r)) = y 2 (cti(p), a, a(r)) > ftl(p, a, r) > 0. By Lemma 6.3 .3, 3 t E Q1 such that ft, (q, a, t) >_ ft, (p, a, r) > 0 and a(t) = a(r) . Since a (t) = a (r), (t, r) E Ker a. Thus Ker a is admissible. m Theorem 6.4.5 Let Ml = (Q1, X, ftl) and M2 = (Q2, X, [t2) be two ftsms and let a : (Q 1 , X, ftl) - (Q 2 , X, ft2) be an onto strong homomorphism. Then 3 an isomorphism 'Y : (Q1/(Ker a), X, f~l) -' (Q2, X, ft2) such that a = -Y o a.


6.5. Fuzzy Transformation Semigroups

247

Proof. Define y : Q,I(Ker a) ~ Q2 by y([q]) = a(q) . Let p, q E Q1 be such that [p] = [q] . Then (p, q) E Ker a and hence a(p) = a(q) or y([q]) = y([p]) . Now, let q, p E Q1 and x E X. Then wl([q],x, [p])

V~fq(q,x,r)Ir E [p]f V{ftl (q, x, r) I a(r) = a(p), r E Q11 ft2(a(q), x, a(p)) ft2('Y([q]), x, -y ([p])) .

Thus y is a homomorphism. Clearly y maps (Q 1I(Ker cti), X, fil) oneto-one onto (Q 2, X, p 2 ) . Example 6.4.6 Let MZ = (QZ, XZ , p 2 ) be the ffsm of Example 6.3.2, i = 1, 2, 3. Let M2 = M, X2, fez) be the ffsm, where Q'2 = {ql, q2}, X2 = {a, b},

and ft'2 = [t2 IQ', X 2, 2 . Let a and,3 be defined as in Example 6.3.2. Then (a,,3) is a strong homomorphism of Ml onto M2 . Ker a = {(ql, q1), (q2, q2), (_g3, q3), (g1, g3), (g3, q1)} . Then [ql] = {g1, g3} = [q3] and [q2] = {q2} . Also Q1 = Q1lKer a = {[q1], [q2]} and N'1([q1], a, [q1]) fa'1l[q1], b, [q2])

fa'1l[q2], a, [q1]) f71l[q2], b, [q1])

1

3 2 3

1

3 2 3'

It is easily seen that there exists an isomorphism y : (Q1/Ker a, X, fit ) ----> (Q2, X2, fez) such that cti = y o cx, where X = X2 . 6 .5

Fuzzy Transformation Semigroups

A transformation semigroup is a pair (Q, S), where Q is a finite nonempty set and S is a finite semigroup with an action S of S on Q, i.e ., a partial function S of Q x S into Q such that (1) S(S(q, s), s') = S(q, ss') b'q E Q, s, s' E S, and (2) S(q, s) = S(q, s') b'q E Q implies s = s', where s, s' E S, [92, p. 33]. Definition 6.5.1 A fuzzy transformation semigroup (fts) is a triple

(Q, S, p), where Q is a finite nonempty set, S is a finite semigroup, and p is a fuzzy subset of Q x S x Q such that (1) p(q, uv, p) = V{p(q, u, r) A p(r, v, p) I r E Q} Vu, v E S and b'q, p E Q; (2) If S contains the identity e, then p(q, e, p) = 1 if q = p and p(q, e, p) _

0ifgzAp,dq,pEQ . If, in addition, the following property holds, then (Q, S, p) is called faithful . (3) Let u, v E S. If p(q, u, p) = p(q, v, p) b'q, p E Q, then u = v.


248


Let M = (Q, S, p) be a fts . This fts may not be faithful . Define a relation R on S by b' u, v E S, uRv if and only if b' q, p E Q, p (q, u, p) = p(q, v, p) . Clearly R is an equivalence relation on S. Suppose that u, v, x E S and uRv. Then p(q, ux, p)

=

V{p(q, u, r) n p(r, x, p) I r E Q} V{p(q, v, r) n p(r, x, p) I r E Q} p(q, vx, p)

b' q, p E Q. Similarly, p(q, xu, p) = p(q, xv,p) b' q, p E Q. Hence R is a congruence relation on S. Let [u] denote the equivalence class of R induced by u. Let SIR = { [u] I u E S} . Define p :QxSIRxQ----> [0,1]

by p(q, [x],p) = p(q,x,p) b' q, p E Q and b' [x] E SIR. Clearly, p is single-valued . Now P(q, [x][y],p)

=

P(q, [xy],p) p(q, xy,p) V{p(q, x, r) n p(r, y, p) r E Q V{P(q, [x], r) n p(r, [y], p) I r E Q

Also _ _ p(q' [e]'p)

l ifp=q 0 otherwise .

Suppose that p(q, [x], p) = p(q, [y], p) Vq, p E Q. Then p(q, x, p) = p(q, y, p) Vq,p E Q . Hence xRy and so [x] = [y] . Thus (Q, SIR, p) is a faithful fts . We call (Q, SIR, p) the faithful fuzzy transformation semigroup represented by the triple (Q, S, p) . Theorem 6.5.2 Let M = (Q, X, p) be a ffsm . Let E(M) be defined as before. Then (Q, E(M), p) is a faithful fts where p(q, [x], p) = / t* (q, x, p) b' q,pEQ,xEX* .

Proof. By Theorem 6.2 .4, E(M) is a finite semigroup with identity [A] . Clearly p is single-valued . Let q, p E Q and [x], [y] E E(M) . Then p(q, [x] * [y],p)


= =

p(q, [xy],p) [t*(q, xy,p) V{[t*(q,x,r) n[t* (r,y,p)Ir E Q} V{p(q, [x], r) n p(r, [y],p) Ir E Q} .


249

Now P (q, [A], p) _ { p,* (q, A, p) = 0 if q z,4p,

by the definition of tt*. Suppose p(q, [x], p) = p(q, [y], p) b' q, p E Q. Then ft * (q, x, p) = ft * (q, y, p) dq, p E Q. Thus x - y or [x] = [y] . Hence (Q, E(M), p) is a faithful fuzzy transformation semigroup . m Let M = (Q, X, p,) be a ffsm. Then by Theorem 6.5 .2 (Q, E(M), p) is a fuzzy transformation semigroup that we denote by FTS(M) . We call FTS(M) the fuzzy transformation semigroup associated with M. Let M = (Q, X, ft) be a ffsm. Define the relation -+ on X+ by V'x, y E X+, x -+ y if and only if ft+(q, x, p) = ft+(q, y,p) dq,p E Q. Then -+ is the restriction of - to X+ x X+ . Let S(M) denote the set of all equivalence classes induced by -+ . Then S(M) = E(M)\{A} and S(M) is a subsemigroup of E(M) . In view of the crisp case, it would have been reasonable to define (Q, S(M), p) as the fuzzy transformation semigroup associated with M. Example 6.5.3 Let M = (Q, X, p) be the ffsm such that Q = {q1, q2, q3}, X = {a, b}, and ft : Q x X x Q ----> [0,1] is defined as follows: ft (q1 , a, q1) ft(g1, b, q2)

ft(g2, a, q1) ft(g2, b, qj) ft(g3, a, q3) ft(g3, b, q2)

1

3 _2 3 1

3 0 1

3 2 3

for j = 1, 2,3 and ft(q, x, q~) = 0 for any other triple (q, x, q~) E Q x X x Then E(M) = {[[All, [[all, [[b]], [[ab]], [[ball, [[aba]]} U {[[bell} and E(M) = {[A], [a], [b], [ab], [ba], [aba], [bab]} U {[[b2ll} . In the crisp case, ft would be considered a "partial" function since ft(g2, b, qj) = 0, for j = 1, 2, 3 and the equivalence classes [[b2]] and [b2 ] would be designated as an empty relation 0. Hence if we set 0 = [[b2]] and then 0 = [b2], we have that the following tables give the semigroup operations for E(M)\{0} and E(M)\{0}, respectively. Note that 0*x = x;~d = 0 for all


25 0


xEE(M) andO*x=x*8=0 forallxEE(M) . *

[[A]]

[[a]]

[[a]]

[[b]] [[ab]] [[ba]] [[aba]]

[[b]] [[ab]] [[ba]] [[aba]]

*

[A]

[a] [b] [ab] [ba] [aba] [bab]

[[a]]

[[a]] [[a]]

[[ba]] [[aba]] [[ba]] [[aba]]

[[b]]

[[ab]]

[[b]] [[ab]]

[[ab]] [[b]] [[ab]]

[[b]] [[ab]] 0 0

[a] [b] [ab]

[a] [a] [a] [ba] [aba]

[b] [b] [ab] 0 0

[ab] [ab] [ab] [b] [ab]

[ba] [aba] [bab]

[ba] [aba] [ba]

[b] [ab] 0

[b] [ab] [bab]

[[ab]] [[ab]] [[b]]

[ba]

[ba] [aba] 0 0 [ba] [aba] 0

[[ba]] [[ba]] [[aba]] 0 0

[[aba]] [[aba]] [[aba]]

[[ba]] [[aba]]

[[ba]] [[aba]]

[aba] [aba] [aba] [ba] [aba] [ba] [aba] [ba]

[[ba]] [[aba]]

[bab] [bab] [ba] 0 [bab] 0 [ba] 0

The fts (Q, E(M), p) is now easily determined.

Definition 6.5.4 Let (Q, S, p) be a fts. Let - be an equivalence relation on

Q. Then - is called an admissible relation if and only if dp, q, r E Q, Vu E S, if p - q and p(p, u, r) > 0, then 3 t E Q such that p(q, u, t) >_ p(p, u, r) and t-r.

Theorem 6.5.5 Let M = (Q, X, /t) be a ffsm and let - be an equivalence relation on Q. Then - is an admissible relation for M if and only if - is an admissible relation for the fuzzy transformation semigroup, FTS(M) _ (Q, E(M), p) .

Proof. Suppose - is admissible for M. Let p, q E Q be such that p - q and [u] E E(M) . Let p(p, [u], r) > 0 for some r E Q. Then [t* (p, u, r) > 0 . Hence by Theorem 6.4 .2, 3 t E Q such that /t* (q, u, t) >_ /t* (p, u, r) and t - r. Thus p(q, [u], t) = p,* (q, u, t) >_ p,*(p, u, r) = p(p, [u], r) . Hence is admissible for FTS(M) . Conversely, suppose that - is admissible for FTS(M) . Let p, q E Q be such that p - q and u E X. Let /t* (p, u, r) > 0 for some r E Q . Then p(p, [u], r) > 0. Then 3 t E Q such that p(q, [u], t) >_ * p(p, [u], r) and t - r. Now /t* (q, u, t) = p(q, [u], t) >_ p(p, [u], r) = [t (p, u, r) and t - r. Hence - is admissible for M. m Definition 6.5.6 Let (Q1, Sl, pl ) and (Q2, S2, p2) be two fts's. A pair (f, g) of mappings, where f : Q 1 -' Q2 and g : Sl - S2, is said to be a homomorphism from (Q1, Sl, pl ) to (Q2, S2, p2) of (1) g(xy) = g(x)g(y) d X, y E Si, © 2002 by Chapman & Hall/CRC


251

(2) If el is the identity of Sl and e2 is the identity of S2, then g(el) = e2,

(3) pi (q,x,p) 0} .

Theorem 6.5.10 Let M = (Q, X, y) be a ffsm. Let E(M) be defined as before. If b'x E X* and b'q E Q, Sx (q) :?~ 01, then (Q, E(M), v) is a faithful polytransformation semigroup with identity.

Proof. Define v : Q x E(M) ~ P(Q)\{O} by v(q, y* (q, x, p) > 0} . Suppose that -< x ~-- = -< y ~-- . Then x v(q, --<x ~--)

= = =

x >-) = {p E Q

y . Thus

{p E Qjft * (q,x,p) > 0} {p E Qjft * (q,y,p) > 0} v(q, -< y ~)

Thus v is single-valued. We write q --< x for v(q, -< x ~) . Then, by definition, (q -< x ~--) -< y >- = UpEq~x} p l y >- . Let p E q -< xy >* * = {p E Q I ft (q, xy, p) > 0} . Now /t* (q, xy, p) > 0 ==> /t* (q, x, r) n ft (r, y, p) > 0 for some r E Q . Thus y* (q, x, r) > 0 and y* (r, y, p) > 0. Thus © 2002 by Chapman & Hall/CRC

6.6. Products of Fuzzy Finite State Machines

253

rEq~x>-andpEr~y>- .Hence pEUt Eq ~x} t-- .Thus q~xy>-C(q-<x>-)(~y~) .Let p E (q --< x>-)(--< y>-) = UtEq~x>-t ~ y >- . Then p E t ~ y >- for some t E q ~ x >- . Thus y* (t, y, p) > 0 and y* (q, x, t) > 0. Hence y* (q, xy, p) > 0. Thus p E q -< xy . Hence (q-<x~--)(- [0,1] as follows:

ftf((g1,q2),a, (PI, P2)) =w1 x w2((g1,g2), (7r1(f(a)),7r2(f(a))), (p1,p2))-

Then (Q1 x Q2, X, ftf) is called the general direct product of M1 and M2 and we write M1 * M2 for (Q1 x Q2, X, tt f) .

Recall P1 x P2 ((g1, q2), (a1, a2), (PI, P2)) = ft, (g1, a1, p1)

n P2 (q2, a2, p2)

V(q1, q2), (PI, P2) E Q1 x Q2 V(a1, a2) E X1 x X2 . If X = X1 x X2 and f is the identity map, then M1 * M2 is called the full direct product of M1 and M2 and we write M1 x M2 for M1 * M2 . If X1 = X2 , X = {(a 1 , a 2 ) ja i E Xi , i = 1, 2, a 1 = a 2 }, and f is the identity map, then M1 * M2 is called the restricted direct product of M1 and M2 and we write M1 A M2 for M1 * M2 . (We could also let X = X1 = X2 and f : X ~ {(a1, a2) I a2 E Xi, i = 1, 2, a1 = a2} where f (a) = (a, a) to

obtain the restricted direct product .) For every result concerning M1 * M2 , there is a corresponding result for M1 AM2 . We see this by making the identifications (a, a) ~ a V'a E X1 = X2 ~ and (x1, x1) . . . x n , xn x 1 . . . x n for x2 E X1, 2 = 1, . . . , n. © 2002 by Chapman & Hall/CRC


255

Example 6.6.4 Let Ml = (Q1, X1 , fq) and M2 = (Q 2 , X2, ft2) be ffsms, where Q1 = {q1, q21, X1 = {a}, Q2 = {qi, q2}, X2 = {a}, and ft, and It2 are defined as follows: wL(gi, a, qi) wL(gj, a, q2) wL(g2 , a, qi) wL(g2, a, q2)

= 0 = CI > 0 = c2 > 0 = 0

w2 (gi, a, qi) w2 (gi, a, q2 ) w2(g2,a,gi) w2 (g2 , a, g2 )

=

0

=

0

Then x w2((gi, qi), (a, a), (g2, q2)) x w2 ((g2, qi), (a, a), (gi, q2)) x lt2((gj, q2), (a, a), (g2, q2)) x w2 ((g2, q2), (a, a), (qj, q2))

wL(g1, a, q2) cl Ad,

dl

>0

d2

>0 .

n w2 (qi, a, q2)

wL(g2, a, g1) c2 n dl

n w2 (gi, a, q2)

wL(g2, a, qL) c2 n d2 .

n w2 (q2, a, q2)

wL(gj, a, q2) cl n d2

n w2 (q2, a, q2)

Suppose that c1 = c2 and dl = d2 . Then an - 1 am if and only if n and m are both even or both odd. Hence E(M1 ) _ {[A], [a], [a2 ]} . For M2 , an - 2 am b'n, m E N . Thus E(M2 ) = {[A], [a]} . (The reader is asked to determine E(M1 ) and E(M2 ) in the Exercises when c 1 :?~ c2 and d l :?~ d2 .) Since c1 = c2 and dl = d2, cl A dl = c2 A dl = cl A d2 = c2 A d2 . Hence an -12 am if and only if n and m are both even or both odd. Thus E(M1 x M2 ) = {[A], [a], [a 2 ]}, where [a3 ] = [a] . In this example, E(M1 A M2) = E(M1 x M2 ) .

Example 6.6.5 Let M1 = (Q1, X1, y 1 ) and M2 = (Q2, X2, y 2 ) be ffsms, where Q1 = {q1, q2 }, X1 = {a}~ Q2 = {qi, q2 }, X2 = {a, b}, and ft, and It2 are defined as follows: ft1(g1, It, (g1, ft1(g2, ft, (q2,

a, q1) a, q2)

a, q1) a, q2) It2 (qi, a, qi) It2 (qi, a, q2)

= = =

= = =

0 c1 > 0 c2 > 0 0 0

d1 > 0

ft2(gi, b, qi) ft2(gi, b, q2) ft2(g2, a, qi) ft2(g2, a, q2) It2 (g2 b, 1) It2 (q2, b, q2)

Then ft1 ftl ftl f~1 ftl ft1 ftl ftl

x x x x x x x x

ft2((g1, ft2((g2, ft2((g1, f~2((g2, ft2((g2, ft2((g1, ft2((g1, ft2((g2,


qi), qi), qi), q2), qi), q2), q2), q2),

(a, b), (a, b), (a, a), (a, b), (a, a), (a, b), (a, a), (a, a),

(q2, q2)) (q1, qi)) (q2, q2)) (q1, qi)) (q1, q2)) (q2, qi)) (q2, q2)) (q1, q2))

_ = _

=

= = = = = =

d2 > 0

0 0

ds > 0 d4 > 0

0.

n d2 c2 A d2 c1 Ad, c2 n d4 c2 n d1 c1 n d4 c1 n d3 c2 A d3 . c1

25 6


3

4

I

Suppose that cl = c2 and dl = d2 = d = d . Then E(M ) = {[A], [a], [a 2 ]} and ({[a],[a2]},*) is a group with identity [a2 ] . E(M2 ) = {[A], [a], [b]}, where [a] = [a 2 ], [b] = [b 2 ], [ab] = [b], and [ba] = [a] . Thus S(M ) = {[a], [a2]} and S (M2 ) = {[a], [b]} . S (MI) x S (M2) = {([a], [a]), ([a], [b]), ([a 2 ], [a]), ([a2 ], [b])} . It follows that S(MI x M2) = {[(a, a)], [(a, a) 2 ], [(a, b)], [(a, b) 2 ]} . The operation tables of S(Mi ) x S(M2) and S(MI x M2 ) are given below.

([a], [a]) 2 ([a ], [a]) ([a], [b]) 2 ([a ], [b])

I

S(M ) x S(M2 ) ([a 2] , [a]) ([a], [b]) ([a2 ([a], [a]) ], [b]) ([a ([a], [a]) 2] , [a]) ([a], [b]) 2 2 ([a ], [a]) ([a ], [b]) ([a], [a]) ([a ([a], [a]) 2] , [a]) ([a], [b]) ([a], [a]) ([a2 ], [a])

I

2 ([a ], [b]) ([a], [b]) 2 ([a ], [b]) ([a], [b]) 2 ([a ], [b])

S(MI x M2 ) [(a, a)]

[(a, a)] [(a, a) 2 ] [(a, b)] [(a, b) 2 ]

[(a, a)'] [(a, a)] [(a, a) 2 ] [(a, a)]

[(a, a) 2 ] [(a, a)] [(a, a) 2 ] [(a, a)] [(a, a) 2 ]

[(a, b)]

[(a, b)L] [(a, b)] [(a, b) 2 ] [(a, b)]

[(a, b) 2 ] [(a, b)] [(a, b) 2 ] [(a, b)] [(a, b) 2 ]

We see that S(MI) x S(M2) - S(MI x M2 ) under ([a], [a]) ----> [(a, a)], 2 2 ([a ], [a]) - [(a, a) 2 ], ([a], [b]) [(a, b)], ([a ], [b]) - [(a, b) 2 ] . Lemma 6 .6 .6 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 . Consider the general direct product Ml * M2 . Then

ft f ((ql, q2), A, (pl, p2)) = ft~ (ql, A, pl) A ft2 (q2, A, P2) Vgl,pl E Q1, Vg2,P2 E Q2(2) [t*

where

a2

I

n , (PI, P2)) = fti (ql, 7r1(f (al)) . . . 7r (f (an)),pl)A q2), al . . .a ft2 (q2, 7r2 (f (al)) . . . 7r2 (f (an)), P2)) E X, i = 1, . . . , n, Vgl,pl E Q1, Vg2,P2 E Q2-

Proof. (1) The proof is straightforward . (2) Then /rf((gl,g2), TI

- -j., (PI, P2)) = V{ftf((gl,q2), al, (ri lf ,r21f ))A

flf((ri lf ,r2 lf ),a2,(ri2f ,r22f ))A . . . Apf((rin-lf ,r2n-lf ),an,(Pl,P2))

(ri ll ,

. .,n-1f =V{ftlxft2((gl,q2),(7r1(f(al)),7r2(f(al)), x r1(f(a2)),~2(f(a2)), (ri2f,r22f)) ' . . . . A (ri lf ,r2 lf )) Apl ft2((ri ll ,r2lf ),(7 ftl x ft2((rin-lf ,r2n-lf ), (7r 1(f(an)), 7r2(f(an),(Pl,M) (ri lf ,r2 lf ) E Q1 x r21f)EQ1xQ2,a=1,



257

Q2, a = 1, . . n- 1} = V{ft1(g1, 7r1(f(a1)), ri 1) ) AItA2, 7 2(f(a1), r21)) A 7 ft1(ri 1f , r1(f(a2)),ri2f )nf, 2(r (1) 7r2(f(a2),r2 2f )n . . . Aft 1(rin-~,7fa)P1 r21) n- 1} r1 1f E Q1, nft2(r2n-1> 7r2(f(an)), P2) E Q2, n-1) = V {ft1(g1, 7r1(f (a1)) rilf) n ft1(ri 1) , 7r1(f (a2)) r (2) ) n . . . n p1(r( > 7r1(f(an)), P1) I r11f E Q1, a = 1, . . , n- 1} AV{ftA2, 72(f(a1)), r21f ) n >

ft2(r21f , 72(f(a2)),r22f) n . . . n p2(r2n-1> 7r2(f(an)), P2) I r2 1f E Q2,i = 1, . . . , n-1} = pi(g1, 7r1(f(a1)) . . .7r1(f(an)), nft2(g2, 7r2(f(a1)) . . 7r2(f(an)), P2) . >

p1)

0

Corollary 6 .6 .7 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 . Then (1) ft2) * = ft2, (2) n p2) * = pr o p* . (p1 X

pi

X

(p1

Proposition 6 .6 .8 Let MZ = (QZ, X, pi) be a ffsm, i = 1, 2 . Define the relation -12 on X* by x -12 y if and only if x -1 y and x -2 y where -Z is the congruence relation on MZ defined in Theorem 6.2 .3, i = 1, 2 . Then -12 is a congruence relation on X* . 0 Definition 6 .6 .9 Let MZ = (QZ, X, pi) be a ffsm, i = 1, 2 . b'x E X*, let <x>={YEX*jy-12x} andletT={<x> IxEX*} . Let FTS(M1) n FTS(M2) = (Q1

X

Q2, T, P1 A P2)

where P1 n P2((g1, q2), < x >, (p1,p2)) = P(q1, [x]1, P1) n P(q2, [x]2,P2)We wish to point out that the coverings of the fuzzy transformation semigroups that appear in Theorems 6 .6 .10 and 6 .6 .11 are coverings as fuzzy finite state machines as stated in Definition 6 .6 .1 . Theorem 6 .6 .10 Let MZ = (QZ, X, pi) be a ffsm, i = 1, 2 . Then the following assertions hold.

FTS(M1 n M2) > FTS(M1) n FTS(M2) . Proof. Let x E X* and (g1, q2), (PI, P2) E Q1 X Q2 . Then P ((q1, q2), [x], (PI, P2))

= = = =

(ft, n ft2) * ((q1, q2), x, (P1,P2)) pi(g1,x,P1) np2(g2,x,P2) P1(g1,[x]1,P1) n P2(g2,[x]2,P2) (P1 n P2)((g1,q2), < x >, (PI, P2))

Let T be as in Definition 6 .6 .9 . Define C : T ~ E(M1 A M2 ) by C(<x>)=[x]vxEX* .


25 8


Then C is single-valued since < x > = < y > ~ x -12 y x - y (w. r. t. M1 A M2 ) ) where the implication holds since x - y FTS(M1) n FTS(M2) .

Proof. Let a2 E X1 and b2 E X2 , i = 1, . . . , n. Then (a1, b1) . . .(an, bn) _ (a1 . . . an , b 1 . . . bn ) = (x 1 , x2) where x1 = a 1 . . . an and x2 = b1 . . . bn . Now M1 x M2= (Q1 x Q2, X1 x X2 , ft, A [t2) and so FTS(M1

x

M2 ) = (Q1

x

x

Q2, E(MI

M2), P)

and FTS(M1 )

Let

x1 E

x

x

FTS(M2 ) = (Q1

Xl and

x2 E X2 .

x

Q 2 , E(MI )

E(M2 ), P1 n P2)

Then

P((g1,q2), [(XI, X2)], (PI, P2))

=

(ft1 x ft2)*((g1,q2), (XI, X2), (PI, P2)) fti(g1,x1,P1) nft2(g2,x2,P2) P1 (g1, [x1 ], P1) AP2(g2, [x2], P2) (P1 n P2)((g1,q2), ([x1], [x2]), (P1,P2))-

Now define C : E (M1)

x

E (M2 ) ----> E (M1

x

M2 )

by C(([x1], [x2])) = [(x1, x2)]

X* . Then ([x1], [x2]) = ([y1], [y2]) [0,1]

as follows :

d((g1,q2), b, (PI ,P2))

ft' ((g1, q2), b, (PI, P2))

=

E

Q

x X2 x

Q,

n ft2(g2, b,P2) .

ft, (g1, w (q2, b), P1)

Then M = (Q, X2, ft') is a ffsm. M is called the cascade product of M1 and M2 and we write M = M1wM2 . ft' is called separable if d (g1,q2),(P1,P2) E Q and b' y = y1 . . . y. E X2, ft-* ((g1,q2),y,(P1,P2)) = q (-1f ,y .),P1)Att* (g2,y,P2) for some q2W fti(g1,w(g2,y1)w(g21f ,y2) ,n-1 .

Q2,

Let M1 = (Q1, X1, fq) and M2 = (Q 2 , X2, [t2) be ftsms, where Q1 = {q1, q2}, X1 = {a, b}, Q2 = {qi, q2}, X2 = {a}, and ft, and ft2 are defined as follows:

Example 6.6.12

ft1(g1, a, q1) ft1(g1, a, q2) ft1(g1, b, q1) It, (g1, b, q2) ft1(g2, a, q1) ft1(g2, a, q2) Let

w :

Q2

x

X2

= = = = = =

~ X1

0 c1 > 0 c2 > 0 0 0 cs > 0 be defined as

w(qi,


a) = a,

ft1(g2, b, q1) ft, (q2, b, q2) ft2 (qi, a, qi) ft2 (qi, a, q2) ft2 (q2, a, qi) ft2 (qz, a, qz)

follows : w(q2,

a) =

b.

c4>0 0 0 d1 > 0

d2 > 0 0.

260


Let Q

=

Q1 x Q2 . Then ftw : Q x X2 x Q ~ [0,1] is such that

ft - ((ql, qi), a, ft - ((q2, q2)a, ft - ((ql, q2)l a, ft - ((q2, qi)l a,

(q2, (ql, (ql, (q2,

q2)) qi)) qi)) q2))

= = = =

ftl(gl, w(qi, a), ftl(g2, w(q2, a), ftl(gl, w(q2, a), ftl(g2, w(qi, a),

q2) ql) ql) q2)

n n n n

ft2(gi, a, q2) ft2(g2, o;, q1 ) ft2(g2, a, qi) ft2(gi, a, q2)

= = = =

cl c4 c2 cs

Ad,, A d2 , A d2, A d1,

and ft' is 0 elsewhere . Note that (ft-)*((ql, q2), aa, (q2, q2))

= = =

tt * (ql, w(q2, a)w(gi, a), q2) n ft2(g2, aa, q2) fti (ql, ba, q2) A [t* (q2, aa, qz) ftl(gl,b,ql) Aftl(gl,a,q2) A[t * (gz,aa,qz) c2ncjnd2ndl .

It follows that ft' is separable.

Proposition 6 .6.13

Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2. Let M = M1wM2 for some w . Then b' (q1, q2), (PI, P2) E Q and b' y = yl . . . yn E X2* , ft w* ((ql, q2), y, (Pl,P2)) =V{fti(gl, w(g2,y1)w(q2 1) , y2) . . . W(g2 n-1) ,yn), PI) n-1) yn,P2) I q22) E Q2, nft2(g2, y1,g2 1) ) n ft2(g21) , y2,g2 2) ) A . . . A ft2(g2 ,n

Proof.

ft'* ((ql, q2), y, (Pl,P2)) = V{ft w ((gl, q2), yl, (gil),q21))) n q2 1) n-1) ), yn, (pl,P2)) wW ((gi 1) , g2 ), y2, (g i2) ~ g22))) n . . . A ft-((qin-1)' n-1) n-1 ( 1 ) , q(')), (q . . . . E yl), gi 1) )nft2(g2, yl, ))~ ~ V{ftl(gl, w(g2, (gi~ g2 1 g2 )) Q} = (1) (1) (2) (1) (2) (n-1) (n-1) . . .nftl(g y2~ g2 ) w(g2 q2(1)) nftl(gl , w(q2 , y2)~ ql )nft2(g2 l (n -1 ) ( 1 ) q(')), . . . , ( n-1 ) (~'-1)) E Q yn), pi) yn, P2) ft2(g2 (gl (gl g2

= V{ftl(gl, w(q2, yl), qi1) ) n ftl(gi 1) , w(g2 1) ,y2), q (2) ) n . . . n ftl(q(n-1) n-1) , 1 1 q(2) n-1 yn), PI) n ft2(g21 yl~ g2 )) n f~2(g2 )~ y2~ )~ w(g2 ) n . . . A l~2(g2 -1)' n-1) ) g21)), . . . yn, P2) (gi1)' (gin q2 E Q} =V{ftl(gl, w(g2,y1) w(g2 1) 1 ( n-1) y2) . . .co(q 2n-1) ,yn),Pl)nf~2(g2,yl,g21) )nft2(q21) ,y2,q2 2) )A . . . nft2(g2 yl , P2) I q22) E Q2, i = 1, 2, . . . , n - 1} .

Proposition 6 .6.14

Let MZ = (QZ, XZ, pi) be a ffsm, i = {0,1}, then ft' is separable .

1, 2. If Im(ft 2 )

_

Proof. Let

(g1, q2), (PI, P2) E Q and y = y1 . . .Y n E X2* . Now ft' * ((ql, . .. (q2n-1 ) , yn), PI) n ft2 W21 P2)) = V {fti (ql , w(q2, yl)w(g21) , y2) q2), y, (Pl, , (1) (1) g2(Z) E Q2> i = 1 > 2. . . . I nyl, g2 )nft2 (g2 , y2, g2(2) )n . . . Aft2 (g2(n-1) yl, P2)

1f by Proposition 6.6.13 . Case 1 : ft2(g2,y,P2) = 1Now v {ft2 ( q2, yl, q2(1)) A f~2 (q2('), y2 g2(2) ) n . . . n f~2 ((n g2 -1 ) = 1,2, . . . , n - 1} = ft2(g2, y,P2) = 1 . Hence 3 q2l ), © 2002 by Chapman & Hall/CRC

(Z)

yl,P2 )I q2 n-1) . . . , q2(

E E


261

Q2 such that ft2(g2,y1,g21)) n ft2(g21)~Y2~g22)) n . . . n ft2(g2n-1)~Y1,P2) _ 1 . Thus tt' * ((glq2), y, (P1,P2)) = V {[t1 (qi, w(g2, yi)w(q (I) ' Y2) . . . c~(g2n-1) ~ n-1) ' Yn), PI) I ft2(g2, Y1, q21)) n ft2(g2 1) ~ Y2, q22)) A . . . A ft2(g2 Y1, P2) = 1~ q2(1), E Q2}Since ~ ~ ~ ~ q2 fti is finite valued, the supremum is at2n-1) E Q2 . q2n-1) tained at some q21), . . . , q Hence 3 q21), . . . , E Q2 such that tt'*((glq2),y,(P1,P2)) = fti(gl,w(q2, Yl) w(g21) ~ Y2) . . . w(g2n-1)' n-1 ) Yn), Pl) n 1 = fti(gl, Yn), PI) = tt * (ql, w(q2, yl) w(q(l), 2J2) . . .w(g2 w(g2 n-1 ), Yn),PI)n ft2(g2, Y,P2) . w(g2,y1)w(q21) ,Y2) . . . . .. n-1) Case 2 : N'2 (g2 , y, P2) = 0 . Then y g2l) E Q2, ft2 (q2, Y1, q21)) A , g2 ft2(g21) ~Y2 g22))n . . .n/'2(q2n-1) Y1 P2)=0 . Thusf~W* ((glg2) (Pl P2))_ 0 = fti(gl, w(q2, Yl)w(g21) , Y2) . . . W(g2n-1) Yn), Pl) n 0 = fti(gl, w(q2, yl) n-1) n-1) E Q2 w(g21) , Y2) . . . w(g2 Yn), Pl)A ft*(g2, Y, P2) V g21)~ . . . , g2

Let Ml = (Q1, Xl, ftl) and M2 = (Q2, X2, ft2) be fuzzy finite state machines . Let f be a function from Q2 into Xl . Define /t° : Q x (XQ2 x X2) xQ ----> [0,1] as follows : d ((ql, q2), (f, b), (PI,P2)) E Q x (XQ2 xX2) xQ,

w ° ((ql, q2), (f, b), (P1,P2)) = ftl(gl, f (g2),pl) n w2(g2, b,P2) .

Then M = (Q, XQ2 x X2, It°) is a ffsm . M = Ml o M2 is called the wreath product of Ml and M2 . ft o is called separable if b' (ql, q2), (pl, P2) E Q and

d (fl, bl) . . . (fn, bn) E (XQ2 XX2)*, w°*((ql, q2), (fl, bl) . . . (fn, bn), (P1,P2)) = tt * (ql, fl(g2)f2(q2 1) ) . . . fn (q2n-1)), pl) n ft2(g2, b l . . . bn , P2) for some g21)EQ2,i=1,2, . . .,n-1 .

Example 6 .6 .15 Let Ml and M2 be the ffsms defined in Example 6.6.12. Let XQ 2 = {fl, f2, f3, f4}~ where fl(gi) = fl(g2) = a, f2(gi) = a, f2(g2) _ b, f3(gi) = b, f3(g2) = a, and f4(gi) = f4(g2) = b . Then ft o ((gl, qi), (fl, a), (q2, q2))

= =

ftl (ql, fl (qi), q2) n ft2(gi, a, q2) cl n d l

ft O ((g2, qi), (fl, a), (q2, q2))

= =

ftl (q2, f1(gi), q2) n ft2(gi, a, q2) c3 A d1

ft o ((gl, q2), (fl, a), (q2, qi))

= =

ftl (ql, fl (q2), q2) A It2(g2, a, qi) cl n d2

ft O ((g2, q2), (fl, a), (q2, qi))

= =

ft o ((gl, qi), (f2, a), (q2, q2))

= =

[t o ((g2, qi), (f2, a), (q2, q2))

= =

ftl (q2, f1(g2), q2) A It2 (q2, a, qi) c 3 n d2 ftl (ql, f2(gi), q2) n ft2(gi, a, q2) cl n dl ftl (q2, f2(gi), q2) A It2(gi, a, q2) c3 A d 1

[t o ((gl, q2), (f2, a), (ql, qi))

= =

ftl (ql, f2(g2), ql) A It2(g2, a, qi) c2 n d2

ft o ((g2, q2), (f2, a), (ql, qi))

= =

ftl (q2, f2(g2), ql) n ft2(g2, a, qi) c4 n d2


262

6. Algebraic Fuzzy Automata Theory w°((q1, q~), (f3, a), (q1, q2))

=

w°((q2, q~), (f3, a), (q1, q2))

=

w°((q1, q2), (f3, a), (q2, q~))

=

w°((q2, q2), (f3, a), (q2, q~))

=

It' ((g1, qi), (f4, a), (q1, q2)) ft'((g2, qi), (f4, a), (q1, q2)) It'((gl, q2), (f4, a), (q1, qi)) ft'((g2, q2), (f4, a), (q1, qi))

= = =

wl(gl, f3(gi), q1) c2 n dl wl(g2, f3(gi), q1) c4 n d l wl(gl, f3(g2), q2) c l A d2 wl(g2, f3(g2), q2) c3 n d2 ftl (q1, f4 (qi), q1) c2 Ad, ftl(g2, f4 (q'), q1) c4 Ad, ftl(gl, f4 (q2), q1) c2 n d2 ftl(g2, f4 (q2), q1) c 4 n d2 .

n ft2 (qi, a, q2) n ft2 (qi, a, q2) n ft2 (q2, a, qi) n ft2 (q2, a, qi) n ft2 (qi, a, q2) n ft2 (qi, a, q2) n ft2(g2, a, qi) n ft2(g2, a, qi)

In the case that cl = c2 = c3 = c4 and dl = d2 , it follows that the semigroup of this ffsm has eight elements and the transformation semigroup is isomorphic to the wreath product 2 0 7L 2 , [92, p . 59] . Example 6 .6 .16 Let Ml and M2 be the ffsms defined in Example 6.6.15. Let X2 1 = { f }, where f (q 1 ) = f (q2 ) = a . Then It'o ((q', [to ((g2, [t ((gi, fto ((g2, [to ((gi, [too ((g2, [to ((gi, [t ((g2,

q1), (f, a), (q2, q2)) q1), (f, a), (qi, q2)) q2), (f, a), (q2, q2)) q2), (f, a), (qi, q2)) q1), (f, b), (q2, q1)) q1), (f, b), (qi, q1)) q2), (f, b), (q2, q1)) g2), (f, b), (g1 q1))

ft2 (q', a, q2) ft2 (q2, a, qi) ft2 (q', a, q2) ft2 q2 I a, qi) ft2 (qi, a, q2) ft2 (q2, a, qi) ft2 (qi, a, q2) ft2 (q2, a, qi)

n ft, (g1, n ft, (g1, n ft, (g2 , n ft, (g2 , n ft, (g1, n ft, (g1, n ft, (q2, n ft, (q2,

a, q2) a, q2) a, q2) a, q2) b, q1) b, q1) b, q1) b, q1)

dl d2 dl d2 dl d2 dl d2

n c1 n c1 n c3 n c3 n c2 n c2 n c4 n c4 .

Suppose that d l = d2 and cl = c2 = c3 = c4 . Then it follows that S(M1 oM2 ) has four elements while S(M1) o S(M2) has eight elements, [92, p .59] . Proposition 6 .6 .17 V(ql, q2), (PI, P2) E Q and b' (fl, bl) . . . (fn, bn) E (XQ' x X2 ) *, w°* ((q1, q2), (fl, bl) . . . (fn, bn), (P1,P2)) = V{w*(ql, f1(g2), n-1) n-1) f2 (q2"') . . . fn (q2 , bn, P2) ) pl)A ft2(g2, bl, q2 1) ) n . . . n ft2(g2 (1) (n-1) E . . . , q2 , Q21g2 Proof. [t o* ((q, q2), (fl, bl) . . . (fn, bn), (Pl,P2)) = V{ttl(gl, fl(g2), g1 1) )n n-1) ft2(g2, b1, q21)) Aft, (gi1), f2 (q2(')), qi2) ) Ail (q21) , b2, g22) ) n . . . n ft1(gi , (n-1) (n-1) (1) (n-1) (1) (n-1) . . . . . pi) A " fn(g2 f'2 (g2 , bn, p2) g1 , , 1 E Qli g2 , , g2 ), 1 1 2 n-1 . . . n f, 1(g~ E Q2} _ {w1(g1, f1(g2), gi )) ), w1(g~ ), f2(g (') ), 1 ))



263

n-1) bn, fn( q (-1) ), pl) nft2(g2, bl, q2 1) ) nft2(g2 1) ~ b2, g22) ) A . . . AIt2(g2 P2) n-1) 2n-1) E Q2} =V{ftl*(ql, fl(g2)f2(g2 1) ) . . . qi 1) , . . . , qi E Q1 ; q2( , ), . . . , q n-1) bn, fn(g2 n-1) ), pi) n f~2(g2, bl, q21) ) n ft2(g2 1) , b2, q22)) n . . . n IA'2(g2 P2) Ig21), . . .,q2n-1)EQ2} . "

Proposition 6.6 .18

Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, {0,1}, then [to is separable .

Proof. Let (ql, q2), (P1,P2) E Q and (fl, Now [to * ((glq2), (fl, bl) . . . (fn, bn), (P1,P2))

2. If Im(p 2 )

_

bl) . . . (fn, bn) E (XQ2 x X2) * .

= V{fti(gl, fl(g2), f2(g2 1)) . . . . . . , q2 n-1) E fn (q2 n-1) ), pl) n ft2(g2, b1, q2 1) ) A . . . AIt2(g2n-1) , bn,P2) I q ('), 2 Q2} by Proposition 6.6.17. Case 1 : [t 2* (q2 , bl . . . bn,p2) = 1 . Now V{ft2(g2, bl, q21) ) n . . . n ft'2(g2n-1) , bn,P2) Ig2 1) , . . . ,q2n-1) E Q2} _ n-1) [t 2* (q 2 , bl . . . bn,P2) = 1 . Hence 3 q21) , . . . , q2 E Q2 such that ft2(g2, n-1) ' n . . . n 1 . Thus ( P2) = [t°*((glq2), bl, q2 1) ) ft2(g2 bn, (fl, bl) . . . (fn, bn), (PI, P2)) = V{ft*(ql, fl(g2) f2(g2 1 )) . . . fn (g2n-1 )), PI) w2 (g2, bl, n-1) n-1) A . . . A . . . . Since , E ft2 (g2 > bn, P2) = 1, g2 1) > Q2} fti is g21)) g2 ,q2n-1) EQ2 finite valued, the supremum is attained at some q2(,), . . . .

Hence 3

q2(,) , . . . , q2n-1)

such that

tto* ((ql, q2), (fl, bl) . . . (fn, (q2n-1)), pl) = bn), (PI, P2)) wl(gl, fl(g2), f2(g21 )) . . . fn [ti* (ql, fl(g2) . . fn (g2n -1 )), n 1 . . fn (q2n-1 )), pl)n . = . f2(g21) ) pl) wl(gl, fl(g2)f2(q2 1) ) . . . bn, P2) . w2(g2, bl Case 2 : [t2* (q2, bl . . . bn,P2) = 0. Then b' q21) , . . . , q2n-1) E Q2, ft2(g2, bl,

=

E Q2

q2 1) ) n . . . nf t 2(g2n-1) , bn, P2) = 0. Thus ,t 0* ((ql, q2), (fl, bl) . . . (fn, bn), (q2(1)) (PI, P2)) = 0 wl(gl, f, (q2) f2 . . . fn(g2 n-1) ), pl) A0 wl(gl, f, (q2) pi) n [t2 * (q2, f2(g2 bl . . . bn, P2) . ) . . fn(g2 ),

=

=

We are now interested in obtaining covering results involving the cascade and wreath product of fuzzy finite state machines . The method that is immediately suggested is to suitably relate elements of E(M) with those of E(Ml)Q2 x E(M2) . However it turns out that these relations are not necessarily single-valued. Hence we replace E(M) with X* in the following definition. Definition 6.6.19 Let M = (Q, X, p) and MZ = (QZ, XZ, p 2 ), i = 1, 2, be ffsms . Let 97 be a function of Q1 x Q2 onto Q and C a function of X* into E(M1)Q2 x E(M2 ) . Then ( 97, C) is said to be a weak covering of FTS(M) by FTS(M1),FTS(M2) if p(97(gl, q2), [X], 97(P1,P2))

b' x

V{p ° ((ql, q2), C(x), (rl, r2)) I97(rl, r2) (PI , P2) , (rl, r2) E Q1 x Q2}

E X* and (gl,q2), (PI, P2) E Q1 x Q2-


264


Theorem 6 .6 .20 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 . If /t° is separable, then FTS(MI o M2) < (weakly) FTS(MI) o FTS(M2) . Proof. Now Ml o M2 = (Q 1 x Q2, XQ 2 x X2 , /t°), where [to ((gl, q2), (9, b), (PI, P2)) = ft, (g1,9(q2),PI)

n ft2(g2, b, P2)

and FTS(MI o M2) = (Q1 x Q2, E(MI o M2), po), where p ° ((ql, q2), [(91, bl) . . .(9k, bk) ], (PI, P2)) = [to* ((ql, q2), (91, bl) . . . (9k, bk), (PI, P2)) . Also, FTS(MI ) o FTS(M2 ) = (Q1 x Q2, E(Ml) Q2 x E(M2), ~°), where ~0 ((ql, q2), (f, [bl . . .bk]), (PI, P2)) = pl(gl, f (q2), pl) AP2(g2, [bl . . .bk], P2) = fti(gl, x1, pl) n [t* (q2, bl . . . bk, P2) where f (q2) = [xl] and xl E Xi is selected below . Let 97 be the identity map of Ql x Q2 . Define (XQ2 x X2)*

E(Ml)Q2 x E(M2)

as follows : C((91, bl) . . . (9k, bk)) = (f, [bl . . . bk])Now (91,b1) . . .(9k,bk) = (h i , a,) . . .(hj,aj) if and only if k = j and (gi, b2) = (h2, a2) i = 1, . . . , k if and only if k = j and gZ = h 2 , b2 = a2 , i = 1, . . . , k . Thus C is single-valued . po((g l , q2), [(gl, bl) . . . (9k, bk) ], (PI, P2)) = [t o * ((gl,q2), (91, bl) . . . (9k, bk), (PI, P2)) = V{fti(gl, k-1) 91(g2)92(q2 1) ) . . . 9k(g2k-1) ), pl) n ft2(g2,bl,q2 1) ) n . . . n ft2(g2 , bk,P2) cc = q2 1) , . . . ,q2 k-1) E Q2} ftl(gl, xl, pl) n ft2(g2, x2, P2) = pl(gl, f (q2), pl) n p2(g2, [x2], P2) _ ~0 ((ql, q2), (f, [x2]), (PI, P2)) = V{~O((ql, q2), C((91, bl) . . .(9k, bk)), (rl, r2))1 9 7(rl, r2) = (PI, P2), (rl, r2) E Ql x Q2} since 97 is the identity map . Select xl = gl(g2)92(g21) ) . . .gk(q2k-1) ) so that (1) (k-1) Hence (~7, C) is a weak covering of FTS(M1oM2) q2 , . . . , q2 ggives by FTS(MI) o FTS(M2 ) . Theorem 6 .6 .21 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 . If ft' is separable, then FTS(M1wM2) < (weakly) FTS(MI) o FTS(M2) .


6.6. Products of Fuzzy Finite State Machines Proof. Now

and

M1wM2 = (Q 1 x Q2, X2 , ftw ),

265 where w :

Q 2 x X2 ----> Xl

ft' ((gl, q2), b, (PI, P2)) = ft, (gl,w(q2, b), PI) n ft2(g2, b, P2) d(ql, q2), (pi, P2) E Q1 x Q2, b E

X2 and

FTS(M1wM2) = (Q1 x Q2, E(M1wM2), p),

where p((ql, q2), [x2], (P1,P2)) = ft'*((ql, q2), x2, (P1,P2)), V(ql, q2), (PI, P2) E Q1 x Q2, V x2 E X2 . Also FTS(MI ) o FTS(M2 ) = (Q1 x Q 2 , E(Ml ) Q 2 x E(M2), ~0),

where P ° ((ql, q2), (f, [x2]), (P1,P2)) = pl(gl, f(g2),Pl) Ap2(g2, [x2],P2) d(ql, q2), (pl, P2) E Q1 x Q2, Vx2 E X2* . Let 97 be the identity map of Q1 x Q 2 . Since ft' is separable we can define C in the following manner .

Define

C : X2 - E(Ml ) Q 2 x E(M2 )

by C(x2) =

where fx2 (q2) = 2-1) , and w(gi bz)

(fx21 [x2])

b'x2 E X2,

[xl] for x2 = b l . . .b,, and xl = a l . . . a n and w(ql, bl) = al = a2, i = 2, . . . , n for those q(' - 1) so that

V{ft* (ql, w(q2, bl)w(g2 1) , b2) . . . w(g2n-1), bn ), pl) n ft2(g2, bl, g2 1) )n n- 1} ft2(gzl> > b2, g22)) n . . .n ft2(g2n-1>> bn , P2) 1 q21) E Q2, = fti(gl, al . . . an , PI) A ft2(g2,x2,P2) . Now C is single-valued since for y2 = d l . . . dj , x2 = y2 if and only if n = j and d2 = b2 for i = 1, . . . , n .

Then p((gl,g2), [x2], (P1,P2)) = ft-* ((ql, q2), x2, (PI, P2)) =V{fti(gl, w(q2, w( q2 1) , b2) . . . w(g2n-1) , bn), pl)n ft2(g2, bl, g2 1) )n ft2(g21) , b2, g22))A . . . A bl) n-1) ' bn,P2) lt2(g2 q21) E Q2, i = 1, 2, . . . , n - 1} _ /t1 (g1, a1 . . . an~~P1) n ft2(g2, x2, P2) = pl(gl, f (g2),pl) AP2(g2, [X21, P2) = ~ O ((ql, q2), (f, [x2]), (PI, P2)) = ~0((gl,q2), C(x2), (P1,P2))- 0 If A = (Q, S) and A' = (Q', S') are transformation semigroups, Section 6.5, then in the wreath product A o A' = (Q x Q', SQ' x S') one has that SQ' x S' is a semigroup where we recall that S and S' are finite semigroups . © 2002 by Chapman & Hall/CRC

266


We now consider the fuzzy case. Here the difficulty that arises is that the natural way to define a binary operation on SQ' x S' fails since V{ft'(q', s', r') I r' E Q'} may not attain its maximum uniquely. Let (Q, S, /t) and (Q', S', /t') s' E S', V q' E Q', let .A4 (q', s') r') r' E Q'} . Let Q' = {qi . . . .

be fuzzy transformation semigroups. V = {m' E Q' Iw' (q', s', m') = V {ft' (q', s', . qk} . Define : (SQ' x S') x (SQ' x S') P(SQ' x S')\{O} as follows : V (f, s), (g, s') E SQ' x S', (f, s) ' (g , s') _ {(hs,m, ss') hs,m(gZ) _ .f (gZ)9(mZ) i = 1, . . . , k, m = (mi . . , mk) E .M(qi, s) x . . . x .M(qk, s)} Theorem 6.6.22 Suppose that V q' E Q', V s, s' E S', M(q , ss)

=

U.'EM(q',s)M(m , 8') '

Then SQ' x S' is polysemigroup.

Proof. Let (f, s), (g, s'), (k, s") E SQ' x S' . Then (f, s) ' (g, s~)

=

(hs,m, ss') I hs,m(gi) = .f (gi)g(mi), i = 1, . . . , k, m= I .m%) E .M(qi, s) x . . . x .Nt(gk, s)}

Thus s//) ((f, s) ' (9, s')) - (k,

sll) UmE.M(gi,s)x .. .x .M(qb,s)(hs,m, ss') ' (k, s// UmE .M(gi,s)x .. .x .M(q',s) L(s,mlss',r, lss' s,mlss',r(gi) = hs,m(gi)k(ri), r = (r /l . . . . . r/k.). E M(qi, ss /) x . x M(qk, ss')

and so s,mlss',r(gi) = hs,m(q')k(r') = .f (gj')9(m')k(r'), i = 1, . . . , k.

Now {(js , ,u, s's") Ljs, ,u(gj') = g(q')k(u'), i =1, . . . , k, u = (ul, . . . , u%) E A4 (q',, s~) x . . . x .M(qk, s')} .

Thus UuE .M(qi,s')x .. .xM(gm,s')(f,8)(js',u,ss ) UuE .M(gi,s')x ...x .M(q',s') L(s',uws,v, 8(8/ 8 ) s',u ws,v(ga) = f (ga)js,,u(vi), v = (vi/ . . . . . v/k) E .Nt(gi, s) x . . . x .Nt(gk, s)} (6.2)



267

and so S',U w s,v(gi)

= f(q')js,,u(Vi') = f (gj')9(v')k(u(v')),

where u (v2)

E A4 (v', s~), i = 1, . . . , k . Now m2, v2 E A4 (q', s), r2 E A4 (q', ss~), u (v2) E A4 (v', s') for i = 1, . . . , k . Since .A4 (q, ss) = U . , E

.M(q',S)M(m', s),

i = 1, . . . , k,

by hypothesis, the sets in (6.1) and (6.2) are equal. Thus - is associative and so SQ' x S' is polysemigroup . m Definition 6.6.23

Let (Q, X, /t) be a ffsm. Define the fuzzy subset ~ of Q x (P(X)\{0}) x Q as follows: b' q, p E Q and b' Y E P(X)\{O}, w(q, Y p) = V{ft(q, y,p) l y E Y} .

Definition 6.6.24

Let (Q, X, /t) be a ffsm. By [to ((q, q'), {(h, ,, n , ss') h,, (gi) = f (gZ)h(mZ), i = 1, . . . , k, m = (ml, . . . , m'k) E .M(qi, s) x . . . . x .M(qk, s)}, (p, p')), we mean ft(q, {hs, .(q') I m E A4(q~, s) x . . . x A4(gk, s)}, p) A ft'(q,

ss,p) .

Let (Q, S, /t) and (Q', S', /t') be fuzzy transformation semigroups. Assume (SQ' x S', -) is a polysemigroup . Recall that we have the ffsm (Q x Q" SQ, x S" p) where V(q, q), (p, pl ) E Q x Q' and b' (f , s) E SQ' x S~, p((q, q), (f, s), (p,p))

= ft(q, f (q),p) A ft (q, s,p) .

The following result examines condition (1) of Definition 6.5 .1 . Theorem 6.6.25 V(q, q~),

(p, p') E Q x Q' and V (f, s), (g, s~) E SQ' x S~, V{w°((q, q'), (f, s), (r, r')) A p((r, r'), (y, s'), (p, p)) I (r, r') E Q x Q'} > [to ((q, q'), (f, s) - (9, s'), (p, p)) . o [t ((q, q'), (f, s)-(g, s'), (p, p)) =w°((q, q'), {(hs, ., ss') I h,, . (q') = f (gi)h(mi), i = 1, . . . , k, m = (mi, . . . , m%) E M(qi, s) x . . . x .M(gk, s)} (p,p')) = ft (q, {hs, .(g2) m E A4 (q, s) x . . . x A4(q%, s)}, p) n [t'(q', ss', p') (by Definition 6.6 .24) = ft(q, {f (q')9(m'), p) I rn' E A4 (q, s')}nft'(q', ss',p') < V{ft(q, f (q')9(r'), p) I r' E Q'} n fft'(q', ss', p') (by Definition 6.6 .23) = v{ft(q, f(q'), r) Aft(r, g (r'), p) I r E Q, r' E Q'}AV{ft'(q', s, r') Aft'(r', s, p) r' E Q'} = V{ft(q, f(q'), r) A ft'(q', s, r') A ft (r,g(r'),p) A ft' (r ' , s, 1') I r E Q, r' E Q'} = v{w°((q, q'), (f, s), (r, r')) n [to ((r, r'), (y, s'), (p, p)) I r E Q, r'EQT 0

Proof.


26 8


6 .7

Submachines of a Fuzzy Finite State Machine

In this section, we continue our study of a fuzzy finite state machine utilizing algebraic techniques. Definition 6.7.1 Let M = (Q, X, /t) be a ffsm . Let p, q E Q. p is called an immediate successor of q if 3a E X such that ft(q, a, p) > 0. p is called a successor of q if 3x E X* such that /t* (q, x, p) > 0.

Proposition 6 .7.2 Let M = (Q, X, /t) be a ffsm. Let q, p, r E Q . Then the following assertions hold. (1) q is a successor of q . (2) if p is a successor of q and r is a successor of p, then r is a successor of q. Proof. (1) Since /t* (q, A, q) = 1 > 0, q is a successor of q. (2) Now 3x, y E X such that [t* (q, x, p) > 0 and [t* (p, y, r) > 0 . Thus [t * (q,xy,r) > [t * (q, x, p)

n [t* (p, y, r)

by Lemma 6.2 .2 . Hence r is a successor of q.

> 0,

m

Definition 6.7.3 Let M = (Q, X, /t) be a ffsm and let q E Q. We denote by S(q) the set of all successors of q.

Definition 6.7.4 Let M = (Q, X, /t) be a ffsm and let T C_ Q. The set of all successors of T, denoted by SQ(T) in Q, is defined to be the set SQ(T) = U{S(q)Iq E T} .

If no confusion arises, then we write S(T) for SQ(T) . Theorem 6.7.5 Let M = (Q, X, y) be a ffsm . Let A, B C_ Q. Then the following assertions hold. (1) If A C B, then S(A) C S(B) . (2) A C S (A) . (3) S(S(A)) = S (A) . (4) S(A U B) = S(A) U S(B) . (5) S(A n B) C S(A) n S(B) .

Proof. The proofs of (1), (2), (4), and (5) are straightforward . (3) Clearly S(A) C S(S(A)) . Let q E S(S(A)). Then q E S(p) for some p E S(A) . Thus p E S(r) for some r E A . Now q is a successor of p and p is a successor of r. Hence by Proposition 6.7 .2 , q is a successor of r. Thus q E S(r) C S(A) . Hence S(S(A)) = S(A) . .


6.7. Submachines of a Fuzzy Finite State Machine

269

Exchange Property : Let M = (Q, X, p) be a ffsm. Let q, p E Q and let T C Q . Suppose that if p E SITU {q}), p ~ S(T), then q E SITU {p}) . Then we say that M satisfies the Exchange Property. Proposition 6.7.6 Let M = (Q, X, /t) be a ffsm . Then the following assertions are equivalent . (1) M satisfies the Exchange Property. (2) b'p, q E Q, q E S(p) if and only if p E S(q) .

Proof. (1)x(2) : Let p,q E Q and p E S(q) . Now p ~ S(O) . Hence q E S(p) . Similarly if q E S(p) then p E S(q) . (2)x(1) : Let T C Q, p, q E Q . Suppose p E S(T U {q}), p ~ S(T) . Then p E S(q) . Hence q E S(p) C S(T U {p}) . m Definition 6.7.7 Let M = (Q, X, /t) be a ffsm . Let T C_ Q. Let v be a fuzzy subset of T x X x T and let N = (T, X, v) . The fuzzy finite state machine N is called a submachine of M if (1) pjTXXXT = v and (2) SQ (T) C T. We assume that 0 = (0, X, v) is a submachine of M. Clearly, if K is a submachine of N and N is a submachine of M, then K is a submachine of M.

Theorem 6 .7.8 Let M = (Q, X, p) be a ffsm . Let Mi = (Qi, X, pi), i E

I, be a family of Submachines of M, where Qa C_ Q . Then the following assertions hold . (1) niEIMi = (niEIQi, X, njEllaa) is a submachine of M. (2) UiEIMi = (UiEIQi, X, v) is a submachine of M, where v = [tIUiCIQiXXXUiCIQi-

Proof. (1) Let (q, x, p) E niEIQi x X x niEIQi . Then (niElp'i)(q, x, p) = njElp'i(q, x, p) = njEll7 (q, x, p) = p(q, x, p) . Also S(ni(E IQi) c niEls(Qi) c niEIQi

Thus niEIMi is a submachine of M. (2) Since S(UiEIQi) = UiEIS(Qi) = UiEIQi' UiEIMi is a submachine of M. m

Definition 6.7.9 Let M = (Q, X, p) be a ffsm. Then M is called strongly connected if b'p, q E Q, p E S(q) .


270


Definition 6.7.10 Let M = (Q, X, p) be a ffsm and let N = (T, X, v) be a submachine of M. N is called proper if T :?~ Q and T :?~ 01 . Theorem 6.7.11 Let M = (Q, X, /t) be a ffsm. Then M is strongly connected if and only if M has no proper submachines. Proof. Suppose M is strongly connected . Let N = (T, X, v) be a submachine of such that T :?~ 01. Then 3q E T. Let p E Q. Since M is strongly connected p E S(q) . Hence p E S(q) C_ S(T) C_ T. Thus T = Q and so M = N. Conversely, suppose M has no proper submachine. Let p,q E Q and let N = (S(q),X,v) where v = wjs(q )XXXS(q) . Then N is a submachine of M and S(q) :?~ 01 . Hence S(q) = Q. Thus p E S(q) . Hence M is strongly connected . m Proposition 6 .7.12 Let M = (Q, X, /t) be a ffsm. Let R C_ Q. Then N = (S(R), X, NR) is a submachine of M where NR = / t1 s(R)XXXs(R)- 0 Definition 6.7.13 Let M = (Q, X, y) be a ffsm. Let R C_ Q and {NZ i E A} be the collection of all submachines of M whose state set contains R. Define < R > = njEA{NZ I i E A} . Then < R > is called the submachine generated by R. In Definition 6.7.13, it is clear that < R > is the smallest submachine of M whose state set contains R. Proposition 6 .7.14 Let M = (Q, X, /t) be a ffsm. Let R C Q. Then

= (S(R), X, N'R) .

Proof. Now < R > = (niEAQZ, X, niEAfti), where {NZ I i E A} is the collection of all submachines of M whose state set contains R and Ni = (QZ, X, fti), i E A . It suffices to show that S(R) = niEAQZ . Since (S(R), X, NR) is a submachine of M such that R C_ S(R), we have that S(R) _D niEAQZ . Let p E S(R) . Then 3 r E R and x E X* such that /t* (r, x, p) > 0. Now r E niEAQZ, and since < R > is a submachine of M, p E niEAQZ . Thus S(R) C niEAQZ . Hence S(R) = niEAQi . Definition 6.7.15 Let M = (Q, X, ft) be a ffsm. M is called singly generated if 3q E Q such that M =< {q} > . In this case q is called a generator of M and we say that M is generated by q. Theorem 6.7.16 Let M = (Q, X, y) be a ffsm. Let R, T C_ Q. Then the following assertions hold. (1)=U . (2) C n . © 2002 by Chapman & Hall/CRC

6.7. Submachines of a Fuzzy Finite State Machine

271

Proof. (1) By Theorem 6.7.5, S(R U T) = S(R) U S(T) . Now l's(R)US(T)

= wI (S(R)US(T)) .X .(S(R)US(T)) NI S(RUT)xX xS(RUT) NS(RUT)

Hence =U . (2) By Theorem 6.7.5, S(R rl T) C S(R) rl S(T) . Now and Hence

ltS(RnT) =

lt S(R)nS(T) =

wIS(RnT)xXxS(RnT)

wI(S(R)nS(T))xXx(S(R)nS(T))-

NS(RnT) = NS(R)nS(T)IS(RnT)xXxS(RnT)-

HenceCrl .m Definition 6.7.17 Let M = (Q, X, p) be a ffsm. Let MZ = (QZ, X, pi), i = 1, 2, be Submachines of M. If M = < Q1 U Q2 >, then we say that M

is the union of Ml and M2 and we write M = Ml U M2 . If M = Ml U M2 and Q1 n Q2 = 01, then we say that M is the (internal) direct union of Ml and M2 and we write M = Ml U M2 .

Suppose M = Ml U M2 . Then S(QZ) = QZ in M, i = 1, 2, since MZ is a submachine of M, i = 1, 2 . Now S(Q1 U Q2) = S(Q1) U S(Q2) = Q1 U Q2 . Definition 6.7.18 Let M = (Q, X, /t) be a ffsm. Let T C_ Q. T is called free if b't E T, t ~ S(T\{t}) . Definition 6.7.19 Let M = (Q, X, y) be a ffsm. Let T C_ Q. If T and M = < T >, then T is called a basis of M.

Theorem 6.7.20 Let M = (Q, X, y) be a ffsm . Let T C_ Q . Then the

following (1) T (2) T (2) T

assertions are equivalent. is a minimal system of generators of M. is a maximally free subset of Q. is a basis of M.

Theorem 6.7.21 Let M = (Q, X, /t) be a ffsm. Suppose that M satisfies the exchange property. Then M has a basis and the cardinality of a basis is unique. 0

Theorem 6.7.22 Let M = (Q, X, /t) be a ffsm. Suppose that M satisfies the exchange property . Let {ql, q2, . . . , qn} be a basis of M. Then M = 00 . . .0 .

Proof. We have < qZ > = (S(g2), X, y, 2), where pi = ltI S(qi)XX XS(qi) Now if i :?~ j, then S(g2)nS(q;) = Ql since the exchange property is equivalent to the statement that b'p, q E Q, p E S(q) if and only if q E S(p) . Since M = ,itfollows that M=0U . . .0 .~ © 2002 by Chapman & Hall/CRC

272

6 .8

6.

Algebraic Fuzzy Automata Theory

Retrievability, Separability, and Connectivity

Definition 6.8 .1 Let M = (Q, X, /t) be a ffsm. M is said to be retrievable if b'q E Q, b'y E X* if 3t E Q such that p* (q, y, t) > 0, then 3x E X* such that ft * (t, x, q) > 0. Definition 6.8 .2 Let M = (Q, X, /t) be a ffsm. M is said to be quasiretrievable if b'q E Q, b'y E X* if 3t E Q such that [t* (q, y, t) > 0, then 3x E X* such that y* (q, yx, q) > 0. Definition 6.8 .3 Let M = (Q, X, /t) be a ffsm. Let q, r, s E Q . Then r and s are said to be q -related if 3y E X* such that lt*(q, y, r) > 0 and /t* (q, y, s) > 0. If r and s are q-related, then r and s are said to be q-twins if S(s) = S(r) . Remark 6 .8 .4 Let M = (Q, X, /t) be a ffsm. The following assertions are equivalent. (1) b'q, r, p E Q, b'x, y E X* if /t* (q, y, r) > 0 and /t* (q, yx, p) > 0, then p E S(r) . (2) b'q, r, s E Q, if r and s are q-related, then r and s are q-twins. Proof. (1)x(2) : Let q, r, s E Q be such that r and s are q-related. Then 3y E X* such that y* (q, y, r) > 0 and y* (q, y, s) > 0. Let p E S(s) . Then 3x E X* such that y* (s, y, p) > 0. Then y* (q, yx, p) > 0 . Thus by the hypothesis p E S(r) . Similarly if p E S(r) then p E S (s) . (2)x(1): Let q, r, p E Q, x, y E X* be such that y,* (q, y, r) > 0 and [t * (q, yx, p) > 0. Now /t* (q, yx, p) = V{[t* (q, y, s) n [t* (s, x, p) I s E Q} > 0 . Hence 3s E Q such that y* (q, y, s) > 0 and y* (s, x, p) > 0. Then r and s are q-twins and p E S(s) . Thus by the hypothesis p E S(r) . Proposition 6 .8 .5 Let M = (Q, X, /t) be a ffsm. Then the following assertions are equivalent : (1) M is retrievable . (2) M is quasi-retrievable and b' q, r, s E Q, if r and s are q-related, then r and s are q-twins. Proof. (1)x(2) : It is immediate that retrievability implies quasiretrieva-bility. Let q, r, p E Q and x, y E X* . Suppose that /t* (q, y, r) > 0 and ft* (q, yx, p) > 0. Since M is retrievable, 3z E X* such that ft * (p, z, q) > 0. Thus p E S(r) . Hence (2) holds by Remark 6.8 .4. (2)x(1): Let q E Q and y E X* . Suppose 3t E Q such that /t*(q,y,t) > 0. Then 3x E X* such that ft * (q, yx, q) > 0 since M is quasi-retrievable. By Remark 6.8 .4, q E S(t) . 0


6.8. Retrievability, Separability, and Connectivity

273

Theorem 6.8.6 Let M = (Q, X, /t) be a ffsm. The following assertions are equivalent. (1) M satisfies the Exchange Property, (2) M is the union of strongly connected submachines, (3) M is retrievable. Proof. (1)x(2) : By (1), M = UZ1 < qZ >, where {ql, q2, . . . , qn} is a basis of M. Also S(qj) n S(q;) = 01 if i :?~ j. Let p, q E S(qj) . Then qZ E S (p) and so q E S(p) . Thus < qZ > is strongly connected. (2) x(1) : Now M = UZ1 M2, where each MZ = (QZ, X, pi) is strongly connected. Let p, q E Q . Suppose p E S(q) . Now 3i such that q E QZ . Then p E S(q) C S(QZ) = Qz. Thus p, q E QZ . Since MZ is strongly connected, q E S(p) . Hence M satisfies the Exchange Property by Proposition 6.7 .6 . (2) x(3) : Now M = UZ 1 M2, where each MZ = (QZ, X, pi) is strongly connected. Let q E Q, y E X* be such that y* (q, y, t) > 0 for some t E Q. Now q E QZ for some i. Thus t E S(q) C_ S(QZ) . Since MZ is strongly connected, q E S(t) . Hence 3x E X* such that tt*(t,x,q) > 0. Thus M is retrievable . (3)x(2) : Let q E Q and let r, t E S(q) . Then 3y, z E X* such that * (q, y, r) > 0 and * (q, z, t) > 0. Since M is retrievable 3x E X* such that ft ft tt*(r, x, q) > 0. Hence q E S(r) . Thus t E S(r) . Hence < q > is strongly connected. Now M = UeEQ < q > . 0 Definition 6.8.7 Let M = (Q, X, y) be a ffsm. Let N = (T, X, v) :?~ Q be a submachine of M. Then N is said to be separated if S(Q\T) n T = 0. Theorem 6 .8.8 Let M = (Q, X, y) be a ffsm. Let N = (T, X, v) :?~ Ql be a submachine of M. Then N is separated if and only if S(Q\T) = Q\T. Proof. Suppose N is separated . Let q E S(Q\T). Now S(Q\T) nT = 0. Hence q ~ T. Thus q E Q\T . Hence S(Q\T) C Q\T. Thus S(Q\T) = Q\T. Conversely, suppose that S(Q\T) = Q\T . Clearly then S(Q\T) n T = 0. Thus N is separated. m Theorem 6.8.9 Let M = (Q, X, y) be a ffsm. Let N = (T, X, v) :?~ Ql be a submachine of M. If N is separated, then so is C = (Q\T, X, a) where a = ft I (Q\T)XxX(Q\T) Proof. Now 0 :?~ T :?~ Q and so Q\T :?~ 0 . By Theorem 6.8 .8, S(Q\T) = Q\T . Hence C is a submachine of M. Now S(Q\(Q\T)) = S(T) = T . Thus S(Q\(Q\T)) n (Q\T) = T n (Q\T) = 01. Hence C is separated . m Definition 6.8.10 Let M = (Q, X, /t) be a ffsm. Then M is said to be connected if M has no separated proper submachine .


274


Theorem 6.8.11 Let M = (Q, X,

/t) be a ffsm. Then the following assertions are equivalent . (1) M is strongly connected. (2) M is connected and retrievable. (3) Every submachine of M is strongly connected.

Proof. (1)x(2) : By Theorem 6.7 .11, M does not have any proper submachines and so M has no proper separated submachines . Thus M is connected . We now show that M is retrievable . Let q, t E Q and y E X* be such that ft* (q, y, t) > 0. Since M is strongly connected, q E S(t) . Then 3x E X* such that f~* (t, x, q) > 0. Hence M is retrievable. (2) x(3): Let N = (T, X, v) be a submachine of M. Suppose p, q E T are such that p ~ S(q) . Then S(q) :?~ Q and so K= (S(q),X,ltIS(a)XXXS(q)) is a proper submachine of M. Since M is connected, S(Q\S(q)) n S(q) :?4 01 . Let r E S(Q\S(q)) nS(q) . Then r E S(t) for some t E Q\S(q) and r E S(q) . Now 3 y E X* such that W (t, y, r) > 0. Since M is retrievable, 3z E X* such that /t* (r, z, t) > 0. Thus t E S(r) . Hence t E S(r) C_ S(q), a contradiction. Thus p E S(q) b'p, q E T. Hence N is strongly connected . (3)x(1) : Obvious. m

6 .9

Decomposition of Fuzzy Finite State Machines

Definition 6.9.1 Let M = (Q, X, /t) be a ffsm. Let P be a submachine of M. Then P is called a primary submachine of M if (1) 3q E Q such that P = ; (2) VsEQifPC<s>,then P=<s> .

Theorem 6.9.2 (Decomposition Theorem) Let M = (Q, X, /t) be a ffsm . Let p = {P1, P2 , . . . , P,,} be the set of all distinct primary submachines of M. Then (1) M = Uz 1 Pi ; (2) M :?~ U' l, Z0jPZ for any j E {1, 2, . . . , n} . Proof. (1) Let qo E Q. Now b'q2 E Q, either (a) < qZ >E p or (b) 3qZ+1 E Q\S(g2) such that < qZ > C < qZ+1 >. Since Q is finite, either < qo > E p or there exists a positive integer k such that < qo > C < qk > E p. Thus Q = UZ 1S(pi) where Pi = < pi >, i = 1, 2 . . . . , n . Hence M = U2 1PZ . (2) Let N = UZ 1, Zo,pi and let pi _ < pj > . If pj E UZ 1, Z0jS(p2), then pj E S(pi) for some i :?~ j. Hence Pj _ < pj > C Pi . However this contradicts the maximality of Pj since Pj Pi . Thus pj ~ ~ UZ 1, Z0jS(p2) . Hence M :?~ N. Corollary 6.9.3 Let M = (Q, X, /t) be a ffsm . Then every singly generated submachine of M :?~ 0 is a submachine of a primary submachine of M.


6.9 . Decomposition of Fuzzy Finite State Machines

Corollary 6.9 .4 Let M = Theorem 6.9 .2 are unique .

m

(Q, X, p)

be a

275 Then P1 , P2 , . . . , P. in

Definition 6.9.5 Let M =

(Q, X, /t) be a ffsm. Then rank of M, rank(M), is the number of distinct primary submachines of M.

Theorem 6.9.6 Let M =

(Q, X, /t) be a ffsm . The following assertions are equivalent. (1) M is retrievable . (2) Every primary submachine of M is strongly connected.

Proof. (1)x(2) : Let P be a primary submachine of M. Then P = for some p E Q. Then as in the proof of (3)x(2) of Theorem 6.8 .6, is strongly connected. (2)x(1) : Now M = UZ 1 P2 where Pi are primary submachines of M. Then the Pi are strongly connected . Thus M is the union of strongly connected submachines . By Theorem 6.8 .6, (1) holds . m Lemma 6.9.7 Let M = nected submachine .

(Q,

X, y) be a ffsm. Then M has a strongly con-

Proof. We prove the result by induction on ~Q1 = n. If n = 1, then the result is obvious . Suppose the result is true for all ffsms N = (T, X, v) such that ~Tj < n, n > 1 . Let q E Q. Then M' = (S(q),X,yjS(q)XXXS(q)) is a submachine of M. If M' is strongly connected, then the result follows. Suppose that M' is not strongly connected . Then 3p E S(q) such that q ~ S(p) and hence S(p) C S(q) . Now ~S(p)j < n. Hence by the induction hypothesis the ffsm M" = (S(p),X,ltjS(p)XXXS(p)) has a strongly connected submachine. Since M" is a submachine of M, M has a strongly connected submachine. Theorem 6.9.8 Let M =

(Q, X, /t) be a ffsm . The following assertions are equivalent. (1) M is retrievable . (2) Every singly generated submachine of M is primary. (3) Every nonempty connected submachine of M is primary.

Proof. (1)x(2) : Now M = UZ 1 P2 where the Pi are primary submachines of M. By Theorem 6.7.11, the Pi are strongly connected . Let N = < q > be a singly generated submachine of M. Then < q > C_ Pi for some i . Hence < q > = Pi by Theorem 6.7 .11. Thus N is primary. (2)x(1) : Since every singly generated submachine of M is primary, every singly generated submachine of M is strongly connected . Thus every primary submachine of M is strongly connected. By Theorem 6.9 .6, (1) holds. (2) x(3) : Let N = (T, X, v) be a nonempty connected submachine of M. Let q E T. Suppose S(q) :?~ T. Since N is connected, S(T\S(q))nS(q) :?~ © 2002 by Chapman & Hall/CRC

276


0. Let r E S(T\S(q)) n S(q) . Then r E S(t) for some t E T\S(q) and rES(q) .NowC_andC .Since isprimary, < t > = < r > = < q > . Hence t E S(q), which is a contradiction. Hence N = < q > and so N is primary. (3)x(2) : Let N = < s > be a singly generated submachine. By Lemma 6.9.7, N has a strongly connected submachine B = < r >, say. Then B is connected and hence primary. Thus < r > _ < s > = N. Hence N is primary. m Lemma 6.9.9 Let M = (Q, X, p) be a ffsm and let N = (T, X, v) be a separated submachine of M. Then every primary submachine of N is also a primary submachine of M. Proof. Let < q > be a primary submachine of N. Suppose < q > is not a primary submachine of M. Then 3p E Q\S(q) such that < q > C . Clearly p ~ T. Thus p E Q\T . Since q E S(p), q E S(Q\T). Thus q E S(Q\T) n T, which is a contradiction since N is separated . Hence < q > is a primary submachine of M. m Theorem 6.9.10 Let M = (Q, X, y) be a ffsm and let Ni = (Ti , X, vi ), i = 1, 2, . . . , n, be the primary submachines of M. Then a proper submachine N = (T, X, v) of M is separated if and only if for some J C {1, 2, . . . , n}, J zA 0, Q\T = UiEJT,,. Proof. Suppose N = (T, X, v) be a proper separated submachine of M . Then S(Q\T) = Q\T. Since N is proper, the submachine < Q\T > is nonempty. Thus < Q\T > is the union of all its primary submachines . Since < Q\T > is separated every primary submachine of < Q\T > is a primary submachine of M . Thus S(Q\T) = UiEJTi for some J C_ {1, 2, . . . , n}, J :?~ 01 . Since Q\T = S(Q\T), Q\T = UjEJTj for some J C_ {1,2, . . . , n}, J :?~ 01. Conversely, let N = (T, X, v) be a proper submachine of M such that Q\T = UiEJTi for some J C {1,2, . . . , n}, J :?~ 01. Then S(Q\T) _ S(UiEJTi) = UiEJS(Ti) = UiEJTi = Q\T. Hence N is separated. m Corollary 6.9.11 Let M = (Q, X, ft) be a ffsm . Then M is connected if and only if M has no proper submachine N = (T, X, v) such that Q\T is the union of the sets of states of all primary submachines of M. m Definition 6.9.12 Let M = (Q, X, y) be a ffsm and let N = (T, X, v) be a submachine of M. A subset R C_ Q is called a generating set of N, and is said to generate N if N = < R > . Lemma 6.9.13 Let M = (Q, X, y) be a ffsm and let Ni = (Ti, X, vi), i = 1, 2, . . ., n be the primary submachines of M. Let R C Q . Then R generates M if and only if b'i, l . © 2002 by Chapman & Hall/CRC

6.10. Subsystems ofFuzzy Finite State Machines

277

Proof. Suppose that R generates M. Then M = < R > = U TE R < r > . Let qZ E TZ be such that TZ = < qZ >, i = 1, 2, . . . , n . Then qZ E U,ER < r > and so qZ E for some r E R. Thus < qZ > C_ . Since < qZ > is primary, < qZ > = < r > . The converse is immediate. m Definition 6.9.14 Let M = (Q, X, /t) be a ffsm. Let R C_ Q be a generating set of M. Then R is said to be a minimal generating set of M if (1)M= , and (2) Vr E R, < R\{r} > 7~ M. Theorem 6.9.15 Let M = (Q, X, p) be a ffsm. Let R C_ Q be a generating set of M. Then R is a minimal generating set of M if and only if ~R1 =rank(M) . Proof. Let n = rank(M) and let Ni = (Ti, X, v2) be a primary submachine of M, i = 1, 2, . . . , n . By Lemma 6.9 .13, since R is a generating set of M, 3r2 E R such that < r2 > = Ti, i = 1, 2, . . . , n. Since the TZ are distinct, the r2 are distinct. Thus ~R1 >_ rank(M) . Now assume that R is a minimal generating set . Suppose ~R1 > rank(M) . Then 3r E R such that r ~ {rl, r2, . . . , rn}. Thus < R\{r} > = Q. Hence R is not minimal, a contradiction . Thus ~R1 = rank(M) . Conversely suppose that ~R1 = rank(M) . Then R = {rl, r2, . . . , rn} . Hence < R\{r2} > = UjOZNj z,4 M. Thus R is minimal. Theorem 6.9.16 Let M = (Q, X, /t) be a ffsm. Then M is not connected if and only if 3 a generating set R of M with nonempty subsets Rl and R2 such that n=01 and M=U . Proof. Suppose that M is not connected . Then 3 a proper submachine N = (T, X, v) of M that is separated ., i.e., S(Q\T) n T = 0. Let Rl be a generating set of < S(Q\T) > and let R2 be a generating set of N. Then < Rl > n < R2 > = 01 and M = < Rl > U < R2 > . Conversely, suppose that Rl and R2 exists . Let N = < R2 > . Then < S(Q\T) > = < Rl > . Hence N = (T, X, /tjTXXXT) is a proper submachine of M that is separated.

6 .10

Subsystems of Fuzzy Finite State Machines

In this and the next section, we introduce the notion of subsystems and strong subsystems of a ffsm in order to consider state membership as fuzzy . Definition 6.10 .1 Let M = (Q, X, /t) be a ffsm. Let S be a fuzzy subset of Q. Then (Q, S, X, /t) is called a subsystem of M if b'p, q E Q, b'a E X, b(q)


>

b(p)

n ft(p, a, q) .

27 8


If (Q, S, X, ft) is a subsystem of M, then we simply write S for (Q, S, X, ft) . Theorem 6.10 .2 Let M = (Q, X, p) be a ffsm and let S be a fuzzy subset of Q . Then S is a subsystem of M if and only if Vp, q E Q, Vx E X*, 6(q) > 6(p) n ft * (p, x, q) .

Proof. Suppose S is a subsystem . Let q, p E Q, and x E X* . We prove the result by induction on IxI = n. If n = 0, then x = A. Now if p = q, then 6(q) A [t* (q, A, q) = 6(q) . If q p, then 6(p) A [t* (p, A, q) = 0 < 6(q) . Thus the result is true if n = 0. Suppose the result is true Vy E X* such that lyl=n -1 ,n>O . Letx = ya,Iyl = n -1 ,y EX*, aEX. Then b(p) A w* (p, x, q)

= = = <
S(p) A ft * (p, x, q) . The converse is trivial. m Theorem 6.10 .3 Let M = (Q, X, y) be a ffsm. Let S, 61, and 62 be subsystems of M. Then the following assertions hold. (1) 6, n62 is a subsystem of M. (2) 61 U 62 is a subsystem of M. (3) N = (Supp(S), X, v) is a submachine of M, where v = ft I SUPP(S) xX x SUPP(S) (4) Let Nt = (St, X, v( t)) where v(t) = ftl6, xxxst , t E [0,1] . If Vt E [0,1], Nt is a submachine of M, then S is a subsystem of M.

Proof. The proofs of (1) and (2) are straightforward . (3) Let p E S(Supp(S)) . Then p E S(q) for some q E Supp(S) . Then 6(q) > 0 . Since p E S(q), 3x E X* such that /t* (q, x, p) > 0. Hence since S is a subsystem, S(p) >_ S(q) A /t* (q, x, p) > 0. Thus p E Supp(S) . Hence S(Supp(S)) C Supp(S) . Thus N is a submachine of M. (4) Let q, p E Q, x E X* . If S(p) = 0 or y,* (p, x, q) = 0, then S(q) > 0 = S(p) A tt* (p, x, q) . Suppose S(p) > 0 and y,* (p, x, q) > 0. Let S(p) A /t* (p, x, q) = t. Then p E St . Since Nt is a submachine of M, S(St) = St . Hence q E S(p) C S(St) = St . Hence S(q) > t = S(p) Aft* (p, x, q) . Thus S is a subsystem . m Example 6.10 .4 Let Q = {p, q}, X = {a}, ft(r, a, t) = z Vr, t E Q . Let S(q) = 4 and S(p) = 1 . Then

b(q) n ft(q, a, p) = 2 = b(p) © 2002 by Chapman & Hall/CRC

6.10. Subsystems of Fuzzy Finite State Machines

279

and b(p) n ft(p, a, q) = 2 < 4 = 6(q) . Thus S is a subsystem. Let z < t _ t. Thus q E St . Also ft(q,a,p) = 2 > 0. Hence p E S(q) . Thus p E S(St) . But S(p) = 2 < t. Hence p ~ St . Thus Nt is not a submachine of M.

Definition 6.10 .5 Let M = (Q, X, ft) be a ffsm and let S be a fuzzy subset of Q. For all x E X*, define the fuzzy subset Sx of Q by (6x)(q) = V{b(p) Att* (p,x,q)Ip E Q} Vq E Q .

Proposition 6.10 .6 Let M = (Q, X, ft) be a ffsm. Then V fuzzy subsets S of Q and Vx, y E X*,

(sx)y = 6(xy) . Proof. Let S be a fuzzy subset of Q and let x, y E X* . We prove the result by induction on Iy1 = n. If n = 0, then y = A . Let q E Q. Now ((bx)A)(q)

=

V{(bx)(P) nft* (p,A,q) I p E Q} (sx) (q) .

Hence (Sx)A = Sx = S(xA) . Suppose now the result is true for all u E X* such that Jul = n-1, n > 0 and for all S . Let y = ua, where a E X, u E X*, and Jul = n - 1. Let q E Q. Then (b(xy))(q)

= = = = = = =

(6(xua))(q) (b((xu)a))(q) V{(S(xu))(r) A/t*(r,a,q)Ir E Q} V{(V{(&)(p) Aw * (p,u,r)Ip E Q}) Aw*(r,a,q)Ir E Q} V{(&)(p) A (V{ft*(p,u, r) Aw*(r,a,q)Ir E Q})Ip E Q} V{ (bx) (p) n ft * (p, ua, q) I p E Q} ((bx)y)(q) .

Hence S(xy) = (6x)y. The result now follows by induction . 0 Theorem 6.10 .7 Let M = (Q, X, y) be a ffsm and let S be a fuzzy subset of Q. Then S is a subsystem of M if and only if Sx C S Vx E X* .

Proof. Let S be a subsystem of M. Let x E X* and q E Q . Then (6x) (q) = V{6(p) A ft * (p, x, q) Ip E Q} < 6(q) . Hence Sx C S. Conversely, suppose Sx C S Vx E X* . Let q E Q and x E X* . Now 6(q) > (6x) (q) = V {6 (P) n ft * (p, x, q) I P E Q} > 6(P) n ft * (p, x, q) Vp E Q . Hence 6 is a subsystem of M.


280

6.

Algebraic Fuzzy Automata Theory

Definition 6.10 .8 Let M = (Q, X, p) be a ffsm. Let t E (0,1] and q E Q . Define the fuzzy subset qtX of Q by

(qtX) (p) = V{t h ft (q, a, p) l a E X VP E Q.

Definition 6.10 .9 Let M = (Q, X, y) be a ffsm. Let t E (0,1] and q E Q . Define the fuzzy subset qtX* of Q by

(qtX*) (p) =V{t n ft* (q, y, p) ly E X*} VP E Q.

Theorem 6.10 .10 Let M = (Q, X, y) be a ffsm. Let t E (0,1], q E Q . Then the following assertions hold. (1) qtX* is a subsystem of M. (2) Supp(gtX * ) = S(q) .

Proof. (1) Let r, s E Q and x E X* . Now ((qtX*)x)(r)

= = = = < =

V{(qtX * ) (P) nft* (p,x,r)lP E Q} V{(V{ft*(q, y, P) Atly E X*}) Aw* (p,x,r)IP E Q} V{ft*(q,y,P) A[t*(p,x,r) Atly E X* , p E Q} V{ft* (q,yx,r) Atly E X* } {ft* (q,u,r) lulu E X*} (qtX*)(r) .

Hence (qtX*)x C qtX* . Thus qtX* is a subsystem by Theorem 6.10.7. (2) p E S(q) 3x E X* such that ,t* (q, x, p) > 0 V{t n tt * (q,x,p)lx E X*} > 0

(qtX*)(p) > 0

p E Supp(gtX*) . .

Theorem 6.10 .11 Let M = (Q, X, y) be a ffsm and let S be a fuzzy subset of Q . The following assertions are equivalent. (1) S is a subsystem of M. (2)gtX*C6,Vgt C6,qEQ,tE(0,1] . (3)gtXC6,Vqt C6,qEQ,tE(0,1] .

Proof. (1) x(2) : Let qt C S, q E Q, t E (0,1]. Let p E Q and y E X* . Then ft* (q, y, p) h t = ft* (q, y, p) h qt (q) < y,* (q, y, p) h S(q) < S(p) since S is a subsystem . Hence qtX* C_ S. (2) ===>(3): Obvious. (3) x(1): Let p, q E Q and a E X. If 6(q) = 0 or ft (q, a, p) = 0 then 6(p) > 0 = 6(q) h ft (q, a, p) . Suppose 6(q) :?~ 0 and ft (q, a, p) 0. Let S(q) = t. Then qt C_ S. Thus by the hypothesis, qtX C_ S. Thus S(p) >_ (qtX) (p) = V {t h ft(q, y, p) l y E X} > t u ft(q, a, p) = b(q) n ft(q, a, p) . Hence 6 is a subsystem of M. m © 2002 by Chapman & Hall/CRC

6.10. Subsystems of Fuzzy Finite State Machines

281

Definition 6.10 .12 Let Ml = (Q1, Xl , fq) and M2 = (Q2, X2, ft2) be two ffsms. Let (f , g) : Ml ----> M2 be a homomorphism. Let S be a fuzzy subset of Ql . Define the fuzzy subset f(S) of Q2 by f (b) (q) _ dq/

V{6(q) I q E Q1, f (q) = q~} if f -1 (q~) zA 0 0iff - (q) - 01

E Q2 -

Theorem 6.10 .13 Let Ml = (Q1, X, gl) and M2 = (Q2, X, g2 ) be two ffsms. Let f : Ml ---> M2 be an onto strong homomorphism. If S is a subsystem of Ql , then f(S) is a subsystem of Q2 . Proof. Let p', q' E Q2 and a E X. Then f(b)(p') nft2(p',a,q')

= =

(V{b(p)Ip E Q1, f (p) = p'}) nft2(p',a,q') V{6(p) Aw2(p',a,q')Ip E Qj, f(p)=j'} .

Let p, q E Q1 be such that f(p) = p' and f(q) = q' . Then 6(p) A w2 (p', a, q')

= = =
0 = f (S) (q). Thus f (S) is not a subsystem of Q .

Definition 6.10 .15 Let M = (Q, X, ft) be a ffsm and let S be a subsystem of M. Then S is called cyclic if 3qt C_ S, q E Q, t E (0,1] such that S = qtX* . In this case we call qt a generator of S .


282


Theorem 6.10 .16 Let M = (Q, X, p) be a ffsm and let S be a subsystem of M. Suppose 3q E Q, t E (0,1] such that S = qtX* . Then (1) S(q) = t, (2) d p E Q, b(q) > b(p), (3) b' subsystems S' of M such that S' C S, if S'(q) > S'(p) Vp E Q, then 61 = gs'(q) X*

Proof.

Now

b(q) _ (qtX*) (q) = (V {w* (q, x, q) I x E X*}) A t =1 n t = t . (2) Let p E Q. Then b(p) _ (gs(q)X*)(p) = (V{w*(q,x,p)Ix E X*}) Ab(q) 6(p) .

Proof. Suppose that S is super cyclic . Then S is constant by Theorem 6.10.16. Suppose 3p, q E Q such that V'x E X*, y* (p, x, q) < t, where t = S(r) b'r E Q. Then (ps(P)X*)(q) =V{b(p) Aw*(p,x,q) I x E X*} < t=6(q) .

Hence ps(P)X * :?~ S. Thus S is not super cyclic, a contradiction . Conversely, suppose that b'p, q E Q, 3x E X* such that y* (p, x, q) > S(p) . Then b'p, q E Q, 3x E X* such that S(q) > S(p) Aft* (p, x, q) = S(p) . Similarly S(p) > S(q). Hence S is constant . Now (ps(P)X*)(q) = V{6(p) A w*(p, x, q) I x E X*} = 6(p) = b(q) .

Thus ps(P)X* = S. Hence S is super cyclic. 6 .11

m

Strong Subsystems

Definition 6.11 .1 Let M = (Q, X, /t) be a ffsm. Let S be a fuzzy subset

of Q. Then (Q, S, X, /t) is called a strong subsystem of M if and only if b'p, q E Q, if 3a E X such that ft(p, a, q) > 0, then S(q) > S(p) . If (Q, S, X, ft) is a strong subsystem of M, then we simply write S for

(Q, 6, X,

w)

Theorem 6.11 .2 Let M = (Q, X, y) be a ffsm and let S be a fuzzy subset of Q. Then S is a strong subsystem of M if and only if b'p, q E Q, if 3x E X* such that [t* (p, x, q) > 0, then S(q) > S(p) .

Proof. Suppose S is a subsystem. We prove the result by induction Ixj =n . If n =0, then x =A. Now if p = q, then /t*(q, A, q) = 1 and S(q) = S(q) . If q zA p, then ft * (q, A, p) = 0. Thus the result is true if n = 0. Suppose the result is true b'y E X* such that IyI = n-1, n > 0. Let x = ya, I y I = n - 1, y E X*, a E X. Suppose that /t* (p, x, q) > 0. Then on

V{w* (p, y, r) A w(r, a, q) I r E Q}

=

_ >

ya, q) [t * (p, x, q) 0. w* (p,

Thus 3r E Q such that y,* (p, y, r) > 0 and y,(r, a, q) > 0. Hence S(q) > S(r) and S(r) > S(p) . Thus S(q) > S(p) . The converse is trivial . m Theorem 6.11 .3 Let M = (Q, X, y) be a ffsm and let S be a fuzzy subset of Q. If 6 is a strong subsystem of M, then 6 is a subsystem of M. m


28 4


The following example shows that in general, the converse of the above theorem is not true. Example 6.11 .4 Let 6, Q, X, /t be defined as in Example 6.10.4 .

Then a, p) = > 0, but 6(q) = > = Thus 6 is subsystem of M, 6(p) . ft(q, 2 1 2 which is not a strong subsystem.

Theorem 6.11 .5 Let M = (Q, X, y) be a ffsm . Let 6 1 and 62 be strong

subsystems of M. Then the following assertions hold. (1) 6, n62 is a strong subsystem of M. (2) 61 U 62 is a strong subsystem of M. (3) Let 6 be a strong subsystem of M. Then N = (Supp(6), X, v) is a submachine of M, where v = ftjSupp(s)XxxSupp(s) . (4) Let 6 be a strong subsystem of M. Let Nt = (6t, X, v( t)) where v(t) = ft IbtXXXbt, t E [0,1] . Then b't E [0,1], Nt is a submachine of M. (5) Let 6 be fuzzy subset of Q. Let Nt = (6t, X, v( t)) where v( t) _ ft Ibt xxxb t , t E [0,1] . If b't E [0,1], Nt is a submachine of M, then 6 is a strong subsystem of M.

Proof. The proofs of (1) and (2) are straightforward . (3) Let p E S(Supp(6)) . Then p E S(q) for some q E Supp(6) . Then 6(q) > 0. Since p E S(q), 3x E X* such that /t* (q, x, p) > 0. Hence since 6 is a strong subsystem, 6(p) > 6(q) > 0. Thus p E Supp(6) . Hence S(Supp(6)) C Supp(6) . Thus N is a submachine of M. (4) Let q E S(6t) . Then q E S (p) for some p E St . Thus 6(p) > t . Now 3x E X* such that y* (p, x, q) > 0. Then 6(q) >_ 6(p) >_ t. Thus q E 6t . Hence Nt is a submachine of M. (5) Let q, p E Q, x E X* be such that y* (p, x, q) > 0. Suppose 6(p) > 0 . Let 6(p) = t. Then p E St . Since Nt is a submachine of M, S(6t) = St . Thus q E S(p) C S(6t) = St . Hence 6(q) > t. Thus 6 is a strong subsystem . 0 Theorem 6 .11 .6 Let M = (Q, X, y) be a ffsm . Let N = (T, X, v) be a submachine of M. Then XT is a strong subsystem of M.

Proof. Let p, q E Q, a E X, and ft(p, a, q) > 0. Then q E S(p) . If p E T, then q E S(p) C_ S(T) C_ T. Hence XT(q) = 1 = XT(p) . If p ~ T, then XT(p) = 0 < XT(q) . Thus XT is a strong subsystem of M. Theorem 6 .11 .7 Let M = (Q, X, /t) be a ffsm. Then M is strongly connected if and only if every strong subsystem of M is constant .

Proof. Suppose M is strongly connected . Let 6 be a strong subsystem of M. Let p, q E Q . Then p E S(q) and q E S(p) . Hence 6(p) > 6(q) and 6(q) >_ 6(p) . Thus 6(p) = 6(q) . Hence 6 is constant. Conversely, suppose that every strong subsystem is constant . Let p, q E Q . Suppose q ~ S(p) . Then Ql z,4 S(p) z,4 Q. Let 6 be a fuzzy subset of Q such that 6(r) = 1 if r E S(p) and 6(r) = 0 if r ~ S(p) . Let r, s E Q be such that lt* (r, x, s) > 0 © 2002 by Chapman & Hall/CRC

6.11 . Strong Subsystems

285

for some x E X* . If S(r) = 0, then S(s) >_ 0 = S(r) . Let S(r) = 1 . Then r E S(p) . Hence s E S(r) C S(p) . Thus S(s) = 1 = S(r) . Hence S is a strong subsystem . Now S(p) = 1 and S(q) = 0. Thus S is not constant, which is a contradiction. Hence q E S(p) . Thus M is strongly connected . m Definition 6.11 .8 Let M = (Q, X,

/t) be a ffsm and let S be a strong subsystem of M. Suppose Q has at least two elements . Then S is called simple if (1) S is not constant, and (2) for all strong subsystems Q of M, X0 :?~ Q C S ==~, Supp(Q) _ Supp(S) .

Theorem 6.11 .9 Let M = (Q, X, y) be a ffsm and let S be a strong subsystem of M. Suppose ~Q1 > 2 . If S is simple, then Im(S) = {0, t}, where 0 2 . Then

3 tl , t2 , t3 E [0 ,1], 0 < tl < t2 < t3 < 1, and 3 rl , r2, r3 E Q such that S(r2) = t2 , i = 1, 2, 3 . Let m E [0,1] be such that t2 < m 0 . Then q E S(p) . If a(p) = 0, then a(q) >_ 0 = a(p) . Let a(p) > 0 . Then p E T. Thus q E S(p) C S(T) = T . Hence a(q) = S(q) > S(p) > a(p) . Thus Q is a strong subsystem . By the definition of Q, Q z,4 XO . Since S is simple, Supp(Q) = Supp(S) . Hence Supp(Q) C_ T C_ Supp(S) = Supp(Q) . Thus T = Supp(S) . Hence K = N. Thus N has no proper submachine. Hence N is strongly connected . Theorem 6.11 .11 Let Ml = (Q1, X, pl) and M2 = (Q2, X, p2) be two ffsms. Let f : Ml -----> M2 be an onto strong homomorphism. If S is a strong subsystem of Q1, then f (S) is a strong subsystem of Q2 . Proof. Let p, q Now

E Ql

and a

E

X be such that y,2 (f (p), a,

f (b) (f (q)) = V{b(r)

f (q)) > 0 .

r E Q1, f (r) = f (q)

and f (b) (f (p)) = V

Let s

E Q1

V(s)

be such that S(s) > 0 and

s E Q1' f (s) = f (p* f (s) = f (p) .

Now

ft2 (f (s), a, f (q)) = ft2 (f (p), a, f (q)) > 0.

Hence V {ftl (s,

a, r) jr

E Qj, f (r) = f (q) } >0 .

Thus 3 r E Q1 such that ft, (s, a, r) > 0 and f (r) subsystem, 6(r) > 6(s) > 0. Hence f (S) (f (q)) > strong subsystem.

= f (q) .

Since S is a strong Thus f (S) is a

f (S) (f (p)) .

The following example shows that the above result need not be true if not onto.

f is

Example 6.11 .12 Let Q, X, y, S, and f be defined as in Example 6.10.14.

Then S is a strong subsystem and f is a strong homomorphism such that f is not onto . Now ft(p, a, q) = 1 > 0, but f (S) (p) = 2 > 0 = f (S) (q) . Hence f (S) is not a strong subsystem.


6.12. Cartesian Composition of Fuzzy Finite State Machines

6 .12

287

Cartesian Composition of Fuzzy Finite State Machines

Here and the next section, we study a new product of two fuzzy finite state machines Ml and M2 , written Ml - M2 , and called the Cartesian composition of Ml and M2, as in [48] . We show that Ml, M2, and Ml - M2 share many similar structural properties, e.g., those of singly generated, retrievability, connectedness, strongly connectedness, commutativity, perfectness, and state independence . This is important since a fuzzy finite state machine, which is a Cartesian composition of submachines can thus be studied in terms of smaller machines . Definition 6.12 .1

Let M = (Q, X, /t) be a ffsm . Then M is said to be connected if and only if M has no separated proper submachine .

Theorem 6.12 .2

Let M = (Q, X, /t) be a ffsm . Then M is connected if and only if b' proper submachines N = (T, X, v) 3 s E Q\T and t E T such that s(s) n S(t) z,4 0 .

Proof. Suppose M is connected. Let N = (T, X, v) be a proper submachine. Then S(Q\T) n T :?~ 01 since M has no separated proper submachine. Thus 3 r E S(Q\T) n T . Now T = S(T) . Hence r E S(s) for some s E Q\T and r E S(t) for some t E T. Thus s(s) n S(t) z,4 0 . Conversely, let N = (T, X, v) be a proper submachine. Then 3 s E Q\T and t E T such that s(s) ns(t) :?~ 0 . Hence 0 :?~ s(s) ns(t) C s(Q\T) ns(T) = s(Q\T) nT. Thus N is not separated. Hence M has no proper separated submachine. Thus M is connected . m Definition 6.12 .3 Let M = (Q, X, /t) be a ffsm . Let p, q E Q . Then q and p are called connected if either q = p or 3 qo, q l , . . . , qk E Q, q = qo, p = qk and 3 al, a2 . . . . , ak E X such that b' i = 1, 2, . . . , k either ft(gi-1, a2, qZ) > 0 or ft (gj, aZ, qZ-1) > 0 . r

Clearly, if q and p are connected and p and are connected .

Definition 6 .12 .4

r

are connected, then

Let M = (Q, X, /t) be a ffsm. For all q E Q, let

C(q) = {p E Q

Ip

and q are connected}

Vq E Q .

Clearly, Let

b'q, p E Q

M = (Q, X, y)

if p

E C(q),

then C(p) = C(q) .

be a ffsm. For all T C

Q,

C(T) = UpETC(p) " © 2002 by Chapman & Hall/CRC

let

q

and

28 8


Lemma 6.12 .5 Let M = (Q, X, p) be a ffsm . Let U, V C_ Q. Then the following properties hold. (1) If U C V then C(U) C C(V) . (2) U C C(U) . (3) C(C(U)) = C(U) . (4) C(U U V) = C(U) U C(V) . (5) C(U n V) c C(U) n C(V) . (6) Let q, p E Q. If q E C(U U {p}) and q ~ C(U), then p E C(U U {q}) .

SM C- C(Q) . (8) S(C(U)) = C(U) . ('7)

Proof. The proofs of (1), (2), (4), (5), and (7) are straightforward. Consider (3) . Now C(U) C C(C(U)) by (2) . Let q E C(C(U)) . Then 3 p E C(U) such that q E C(p) . Hence q E C(r) for some r E U. Thus q E C(r) C C(U) . Consider (6) . Suppose that q E C(UU{p}) and q ~ C(U) . Since C(U U {p}) = C(U) U C(p), q E C(p) . Thus p E C(q) C C(U U {q}) . Consider (8) . Let p E S(C(U)) . Then p E S(q) for some q E C(U) . Now q E C(r) for some r E U. Hence p E C(r) C C(U) . Thus S(C(U)) C C(U) and so S(C(U)) = C(U) . m Since Q is a finite set, it is clear that b' q E Q and U C_ Q, if q E C(U), then q E C(U) for some finite subset U' of U. This fact together with properties (1), (2), (3), and (6) give C the basic spanning properties in [264, p. 50] that are used for various algebraic structures to obtain the existence of bases and the uniqueness of their cardinalities. Definition 6 .12 .6 Let M = (Q, X, /t) be a ffsm and let T C_ Q . Then T is

called a connected component if b' s, t E T, s and t are connected. T is called a maximal connected component when b' p E Q, if p is connected to t for some t E T, then p E T.

Theorem 6.12 .7 Let M = (Q, X, p) be a ffsm and let q E Q. Then C(q) is a maximal connected component of Q.

Proof. The proof is obvious . m Theorem 6.12 .8 Let M = (Q, X, p) be a ffsm and let q E Q. Let N = (C(q),X, ltja(q)XXXC(q)) . Then N is a submachine of M. m Theorem 6 .12 .9 Let M = (Q, X, /t) be a ffsm. Then M is connected if and only if b'q E Q, C(q) = Q.

Proof. Suppose M is connected and let q E Q . Suppose 3p E Q such that p ~ C(q) . Then N = (C(q),X, /tjc(q)XXXC(q)) is a proper submachine of M. Hence by Theorem 6.12.2, 3s E Q\C(q) and t E C(q) such that S(s) nS(t) z,4 0. Let r E S(s) nS(t) . Then s and r are connected and r and t are connected . Hence s and t are connected . Thus s E C(t) = C(q), which is a contradiction . Hence C(q) = Q . Conversely, suppose that b'q E Q, © 2002 by Chapman & Hall/CRC

6.13. Cartesian Composition

289

= Q. Let N = (T, X, v) be a proper submachine of M. Suppose that N is separated. Then S(Q\T) n T = Ql and S(Q\T) = Q\T. Let q E Q\T and t E T. Then C(t) = Q = C(q) . Hence q and t are connected . Thus 3 qo, ql, . . . , qk E Q, q = qo, t = qk, and 3 al , a2 , . . . , ak E X such that C(q)

b' i = 1, 2, . . . , k either ft(gi_1, a2, qZ) > 0 or tt(q a qZ-1) >O .Now3 i such that q2_1 E Q\T and qZ E T. Hence either qZ E S(g2_1) C_ S(Q\T) or q2_1 E S(qi) C_ T, which is a contradiction . Thus N is not separated. Hence M is connected . m

Corollary 6.12 .10 Let M = (Q, X, ft) be a ffsm. Then M is connected if and only if b' p, q E Q, p and q are connected. m

6 .13

Cartesian Composition

Definition 6.13 .1 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 and let Xl n X2

= 01 . Let

M1 -

M2 = (Ql X Q2,

X1 U X2, w1

'

w2),

where

(Nt1 ' ft2)((pl,p2), a, (ql, q2))

=

ft, (pl, a, ql) Nt2(p2, a, q2)

0 otherwise,

if a E Xl and p2 = q2 if a E X2 and pl = ql

E Q1 X Q2, a E Xl U X2 . Then Ml the Cartesian composition of Ml and M2 .

d(pl,p2), (ql, q2)

M2

is a ffsm, called

Theorem 6.13 .2 Let MZ = (QZ, XZ , p2 ) be a ffsm, i = 1, 2 and let Xl rl = 0. Let Ml - M2= (Ql X Q2, XI U X2, ftl,ft2) be the Cartesian composition of Ml and M2 . Then V'x E Xl U X2, x z,4 A, X2

(ftl ' ft2)*((pl,p2),

d(pl,p2), (ql, q2)

E

x,

(ql, g2))

=

fti (pl, x, ql)

if x E Xl and p2 = q2 x, g2) if x E X2 and pl = gl It * (P2, 0 otherwise,

Q1 X Q2-

Proof. Let x E Xl U X2, x z,4 A and let Ix I = n . Suppose that x E Xl . Clearly the result is true if n = 1. Suppose the result is true b'y E X*, © 2002 by Chapman & Hall/CRC

290


yI =n-1, n > 1 . Let x = ay where a E X1 and y E Xl . Now (ftl ' w2)*((P1,P2), ay, (qj, q2))

= =

V{(ftl ' w2)((Pl,P2), a, (rl, r2)) n(ftl ' ft2) * ((rl, r2), y, (ql, q2)) l (rl, r2) E Q1 X Q2} V{ftl(pl,a,rl) n (ft, ' ft2)*((rl,P2),y, (gl,q2))l r l E Q1}

V {ftl(pl, a, rl) n fti (rl, y, r l E Q1} if P2 = q2 0 otherwise fti (pl, ay, ql) if P2 = q2 0 otherwise .

= __

The result now follows by induction. The proof is similar if x

ql)

E X2 . m

Theorem 6.13 .3 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 and let Xl X2 = 0. Then V'x E X*, b'y E X2 (pl'p2)*((P1,P2),xy,(gl,q2))

= _

rl

fti(pl,x,gl)np2(P2,y,q2) (pl ' p2)*((P1,P2), yx, (qj, q2))

V(Pl,P2), (gl,q2) E Q1 X Q2-

Proof. Let x E Xl, y E X2, (PI, P2), (qj, q2) E Q1 X Q2 . If x = A = y, then xy = A. Suppose (PI, P2) _ (qj, q2) . Then pl = ql and P2 = q2 . Hence (pl ' p2) * 01,P2),xy, (gl,q2)) = 1 = 1 n 1 = fti(pl,x,gl) n ft2(P2,y,g2) " If (pl, p2) 7~ (qj, q2), then either pl ql or P2 :?~ q2 . Thus pi (pl , x, ql) n . Hence 0 ' p2 (P2, y, q2) = (It, p2) * 011 P2), xy (ql, q2)) = 0 = pi (PI, x, q1) n If x = A and y :?~ A or x :?~ A and y = A, then the result [t2* (P2, y, q2) . follows by Theorem 6.13.2 . Suppose x :?~ A and y :?~ A. Now I

(pl '

p2) * 01,P2), xy, (qj, q2))

=

V{(pl '

=

p2)*((P1,P2), x, (rl, r2))n (pl ' ft2) * ((rl, r2), y, (qj, q2)) l h, r2) E Q1 X Q2} V {V{(ftl ' p2)*((P1,P2), x, (rl, r2))n (pl ' ft2) * (h, r2), y, (qj, q2))

=

Similarly

I r2EQ21 I r1EQ1I

V{(pl '

p2)*((P1,P2), x, (rl,P2))n (pl ' p2)*((r1,P2), y, (qj, q2)) l rl E Q1} pi (pl, x, gl) n p2 (P2, y, q2) .

(ftl ' p2) * 011P2), yx, (qj, q2)) = pi(pl, x, ql) n [t* (P2, y, q2) .

Theorem 6.13 .4 Let MZ X2 = 01. Then b'w E (XI U (pl '

= (QZ, XZ, pi) be a ffsm, i = 1, 2 X2 )* 3 u E X*, v E X2 such that

p2)TP1,P2), w, (qj, q2)) =

V(Pl,P2), (gl,q2) E Q1 X Q2-


(pl '

and let Xl

p2)TP1,P2), UV, (qj, q2))

rl


291

Proof. Let w E (X1 U X2)* and (pl, p2), (ql, q2) E Q1 X Q2 . If w =A, then we can choose u = A = v. In this case the result is trivially true. Suppose w :?~ A. If w E Xl or w E X2*, then again the result is trivially true. Suppose w ~ Xl and w ~ X2 . Case 1: If w = xy, x E X+, y E X2+ , then the result follows by Theorem 6.13.3. Case 2: Suppose w = xlyx2, X1, X2 E Xl and y E X2, x2 and y are nonempty strings, i = 1, 2. Let u = xlx2 E Xl and v = y . Now by Theorem 6.13.3, (ftl ' ft2) *((rl, r2), x2y, (ql, q2)) = (ftl ' ft2) * ((rl, r2), yx2, (ql, q2)) E Q1 X Q2 . Thus (ftl ' ft2) * ((pl,p2),xlyx2,(gl,q2)) _ n (ftl'ft2) * ((r1Ir2),yx2,(gl,q2)) I (rl,r2) E ((pl,p2),xl,(rl,r2)) V{(ftl'ft2) * Q1 XQ2} = V{(ftl'ft2)*((pl,p2),xl, (rl,r2)) n (ftl'ft2) * ((rl,r2),x2y, (gl,q2))

V(rl,r2),(gl,q2)

(rl,r2) E Ql X Q2} = (ftl ' ft2) * ((pl,p2),xlx2y, (gl,q2))-

Case 3: Suppose w = ylxy2, yl, y2 E X2 and x E X*, yZ and x are nonempty strings, i = 1, 2. Let v = yly2 E X2 and u = x . The proof of this case is similar to Case 2. Case 4: Suppose w = xlylx2y2, X1, X2 E Xl, Y1, Y2 E X2, xZ and yZ are nonempty strings . Let u = xlx2 E Xl and v = yly2 E X2 . Then (ft, ft2) * ((pl,p2),xlylx2y2,(gl,q2)) = VWtl ' ft2) * ((pl,p2),xl,(rl,r2))A (ft, '

ft2) * ((rl~r2),ylx2y2,(gl,q2))I (rl,r2) EQ1XQ2}=VWtl'ft2) * ((pl,p2),xl, (rl, r2)) A (ftl ' ft2) * ((rl, r2), x2yly2, (ql, q2))

I

(rl, r2) E Ql X Q2}

(by Case

3) = (ftl ' ft2)*((pl,p2), xlx2yly2, (ql, q2)) = (ftl ' ft2)*((pl,p2), uv, (ql, q2)) . Case 5: Suppose w = ylxly2x2, X1, X2 E Xl , Y1, Y2 E X2* . Let u = xl x2 E Xl and v = yly2 E X2 . The proof of this case is similar to Case 4. Case 6: Let w E (XI U X2)* . Then w = xlylx2y2 . . . xnyn or w = ylxly2x2 . . . ynxn, xZ E Xl, yZ E X2, x2 and yZ are nonempty strings, i = 1, 2, . . . , n - 2. To be specific, let w = xlylx2y2 . . . xnyn . The proof of the second case is similar. If n = 0, 1, or 2, then the result is true by the previous cases . Suppose the result for all z = xlylx2y2 . . . xn-lyn-1 E (XI U X2)*, n > 2 . Let ul = xlx2 . . . x n-1, vl = yly2 . . . yn-1, u = ulxn, and v = vlyn . Now (ftl ' ft2)*((pl,p2), xlylx2y2 . . . xnyn, (ql, q2)) _ V{(ftl ' ft2) * ((pl,p2),xlylx2y2 . . . xn -1 yn-1, (rl,r2))n (ftl ' ft2) * ((rl1 r2), xnyn (ql, q2)) I (rl, r2) E Q1 XQ2} = V{(ftl ' ft2) * ((pl1 p2), ulvl, (rl, r2))n

(ftl ' ft2) * ((rl, r2), xnyn, (ql, q2)) (rl, r2) E Q1 X Q2} = (ftl ' ft2)*((pl, p2), ulvlxnyn, (ql, q2)) = (ftl ' ft2) * ((PI, P2), UV, (gl, q2)) . The result now

follows by induction. m

In Theorem 6.13.4, u consists of all the elements from w that are in X l (in the given order) and v consists of all the elements from w that are in X2 (in the given order) . We write w* = uv and call w* the standard form of w.


29 2


Theorem 6.13 .5 Let

MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 and let Xl rl X2 = 0. Then the Cartesian composition Ml - M2 is cyclic if and only if Ml and M2 are cyclic.

Proof. Suppose Ml and M2 are cyclic, say Ql = S(qo) and Q2 = S(po) for some qo E Q1, PO E Q2 . Let (q, p) E Q1 x Q2 . Then 3 x E Xl and y E X2 such that pi (qo, x, q) > 0 and p2 (po, y, p) > 0. Thus (pl ' p2) * ((qo,po), xy, (q, p)) = pi (qo, x, q) n p2 (POI y, p) > 0 . Hence (q, p) E S((go,po)) . Thus Q1 x Q2 = S((go,po)) . Hence Ml - M2 is cyclic . Conversely, suppose Ml - M2 is cyclic . Let Q1 x Q2 = S((qo, po)) for some (qo, po) E Q1 x Q2 . Let q E Q1 and p E Q2 . Then 3 w E (XI U X2)* such that (pl'p2N(go,po), w, (q,p)) > 0 . By Theorem 6.13.4, 3u E X1 and v E X2 such that pi(go,u,q)np2(po,v,p) _ (pl'p2)*((go,po),w, (g,p)) > 0 . Hence 3u E X1 and v E X2 such that pi (qo, u, q) > 0 and p2 (po, v, p) > 0 . Thus q E S(qo) and p E S(po) . Hence Q1 = S(qo) and Q2 = S(po) . Thus Ml and M2 are cyclic . m Theorem 6.13 .6 Let

MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 and let Xl rl X2 = 0. Then the Cartesian composition Ml - M2 is retrievable if and only if Ml and M2 are retrievable.

Proof. Suppose that Ml and M2 are retrievable . Let (q, p), (t, s) E Q1 x Q2, and w E (XI U X2)* be such that (pl ' p2)*((q, p), w, (t, s)) > 0 . Let w* = uv be the standard form of w, u E X*, v E X2* . Then (ft, p2)*((q,p), w, (t, s)) = (pl ' p2)*((q,p), uv, (t, s)) = pi(q, u, t) n p2 (p, v, s) . Thus pi (q, u, t) > 0 and p2 (p, v, s) > 0. Since Ml and M2 are retrievable, El u' E X * ' v' E X2 such that pi (t, u', q) > 0 and p2 (s, v', p) > 0. Thus (pl ' p2)*((t, s), u'v', (q, p)) > 0 . Hence Ml - M2 is retrievable. Conversely, suppose that M1-M2 is retrievable. Let q, t E Q1 and y E Xl be such that pi (q, y, t) > 0. Then b' s E Q2, (pl ' p2)* ((q, s), y, (t, s)) = pi (q, y, t) > 0. Thus 3 w E (XI UX2)* such that (It, ' p2)*((t, s), w, (q, s)) > 0. Let w* = uv be the standard form of w, u E X*, v E X2 . Then 0 < (pl ' p2)* ((t, s), w, (q, s)) = pi (t, u, q) np2 (s, v, s) . Thus pi (t, u, q) > 0. Hence Ml is retrievable . Similarly M2 is retrievable . m The following corollaries follow from Theorems 6.8 .6 and 6.13.6. Corollary 6.13 .7 Let MZ

= (QZ, Xi, pi) be a ffsm, i = 1, 2, and let Xl rl X2 = 0. Then the Cartesian composition Ml - M2 is the union of strongly connected submachines if and only if Ml and M2 are the union of strongly connected submachines.

Corollary 6.13 .8 Let MZ

= (QZ, XZ, pi) be a ffsm, i = 1, 2, and let Xl rl X2 = 0. Then the Cartesian composition Ml - M2 satisfies the Exchange Property if and only if Ml and M2 satisfies the Exchange Property . m



293

Theorem 6.13 .9 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let X1 rl

X2 = 0 . Then the Cartesian composition Ml - M2 is connected if and only if Ml and M2 are connected.

Proof. Suppose that Ml and M2 are connected. Let (q,q'),(p,p') E Q1XQ2 " NowElg0,q1, " . . ,qn EQ1,q - q0,p=gnand3a,,a2, " . . ,an E X1 such that b' i = 1, 2, . . . , n either p1 (qZ-1, aZ, gZ ) > 0 or ft, (gZ, aZ, gZ-1) > 0 and 3 qo, q. . . . . . gm E Q2, q' = q9, p~ = qm and 3 b1, b2, " . . , bm E X2 such that d i = 1, 2, . . . , m either p2(gZ-1, bZ, qZ) > 0 or pz(gZ, b2, qZ-1) > 0. Consider the sequence of states (g, q) _ (g0, g0), (g1, g0), . . . , (gn, g0), (gn, gl), . . , (gn, gm) _ (p,p ' ) E Ql X Q2 and the sequence a 1, a2, " . . , a n, bl1,, 2, bm E X1 U X2 . Then Vi = 1,2, . . . , neither (It,' p2) ((qj-1, q0), a2, (q2, qo)) > 0 or (p1 ' p2) ((qi, q0), a2 , (qZ-1, qo)) > 0 and b' j = 1, 2, . . . , m either (p1 ' p2)((gn, '-1), bj, (qn, g;)) > 0 or (p1 ' p2)((gn, g;), bi, (gn, g;-1)) > 0. Hence (q, q) and (p, p) are connected. Conversely, suppose that M1 - M2 is connected . Let q, p E Q1 and let r E Q2- If p = q then p and q are connected. Suppose p :?~ q.Then 3 (q, r) = (g0,p0), (gl,pl), " . . , (gn,pn) = (p, r) E Q1 X Q2 and a1, a2, " . . , an E X1 U X2 such that Vi = 1, 2, . . . , n either (p1 ' p2 ) ((g2-1, pj-1), aZ, (qi, p2)) > 0 or (p1 ' p2)((gi,pi),aj, (qZ-1, pi-1)) > 0. Clearly, if qZ-1 :?~ qZ, then pi- 1 = pi and if pi- 1 :?~ pi, then qZ-1 = qj d i = 1, 2, . . . , n. Let {q = qz l , qz . . . . . . qj,, = p} be the set of all distinct qj E {qo, q1, . . , qn} and let a21, a22 , . . . , a2,, be the corresponding a2's . Then a21, ail , . . . , a2, E XI and b' j = 1, 2, . . . , k either ft, (gi, 1 , a 2j , qi;) > 0 or ft, (gi, , a2j , qZ, ,) > 0. Thus q and p are connected. Hence M1 is connected . Similarly, M2 is connected . 0 The following theorem follows from Theorems 6.8 .11, 6.13.6, and 6.13.9. Theorem 6.13 .10 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let X1 rl

X2 = 0 . Then the Cartesian composition M1 - M2 is strongly connected if and only if M1 and M2 are strongly connected. m

Definition 6.13 .11 Let M = (Q, X, p) be a ffsm. Then M is said to be commutative if b'a, b E X and b'q, p E Q,

p* (q, ab, p) = p* (q, ba, p) .

The following result is immediate from Theorem 6.13.4. Theorem 6.13 .12 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let X1 rl

X2 = 0. Then the Cartesian composition M1 - M2 is commutative if and only if M1 and M2 are commutative. 0

Definition 6.13 .13 Let M = (Q, X, p) be a ffsm. If M is commutative and strongly connected, then M is said to be perfect.

Theorem 6.13 .14 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let X1 rl

X2 = 0 . Then the Cartesian composition M1 - M2 is perfect if and only if M1 and M2 are perfect. m


294


Definition 6.13 .15 Let M = (Q, X, /t) be a ffsm. Then M is said to be state independent if Vq',p' E Q, Vx,y E X+, (/t*(q',x,p') > 0 and p* (q', y, p') > 0) ==~, (dq, p E Q, p* (q, x, p) > 0 ~ /t* (q, y, p) > 0) .

Theorem 6.13 .16 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let Xl rl X2 = 0 . Then the Cartesian composition Ml - M2 is state independent if and only if Ml and M2 are state independent. Proof. Suppose that Ml and M2 are state independent . Suppose that

(pl ' p2)*((gi,q2),wl,(pi,p2)) > 0 and (pl ' p2)* ((gi,q2),w2,(piIp2)) > 0, where (gi, q2), (pi,p2) E Q1 x Q2, and WI, w2 E (XI U X2)* . Now 3 ul, u2 E Xi and VI, v2 E X2 such that (pl p2)* ((qi, q2), wl, (pi, p2)) ~ ~ = * pi(gi,ul,pi)np2(q2,vl,p2) and (pl'p2 * ((gl~g)~ q2), W2, 2~ (P, P2)) p1 1 u2~ [t* (q' ) . Thus [t* (q, u 0 ' ) l~ > V2, p' g1 ~ 1 ~ 1p') vi, p1 p2 g21 2 1 1 p2 (g2, p2) > 0, pi(gi, u2~ and 0. Hence E 0, p2(g2,v2,p2) > dgl,pl Q1, pi(gl,ul,pl) >0 pi) > pi(gl,u2,p1) > 0 and dg2,p2 E Q2, p2(g2,vl,p2) > 0 p2(g2,v2,p2) > 0. Hence pi(gi, ui, pi)n [t2* (q2, VI, p2) > 0 U2,pi)A p2 (g2, v2, p2) pi( > 0, dql, pl E Q1, dq2, p2 E Q2 . Thus (pl ' p2)*((gl,q2),wl, (PI, P2)) > 0 (pl'p2)*((gl,q2),w2, (PI, P2)) > 0, dgl,pl E Q1, dg2,p2 E Q2 . Hence Ml - M2 is state independent.

Conversely, suppose that Ml - M2 is state independent . Suppose that pi(gi,ul,pi) > 0 and pi(gi,u2,pi) > 0 where U1, U2 E Xl and q', p' E Q1 Then VS E Q2, (pl ' p2)*((gi"s),ul,(pi,s)) = p1 1 ul,pi) > 0 and (pl ' p2)*((gi,8),u2,(pi,s)) = pi(gi,u2,pi) > 0. Thus dq,p E Ql, s E Q2, (pl ' p2) * ((q, s), ul, (p, s)) > 0 (pl ' p2)* ((q, 8), u2, (A s)) > 0 . Hence b'q, p E Q1, pi (q, ul, p) > 0 pi (q, u2, p) > 0. Thus Ml is state independent . Similarly M2 is state independent. m

Theorem 6.13 .17 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let Xl rl X2 = 01 . Let Ni = (Ti, XZ, v2) be a submachine of MZ, i = 1, 2 . Then Nl - N2 is a submachine of Ml - M2 . Conversely, if N = (TI x T2 , Xl U X1 , v) is a

submachine of Ml - M2 , then there exist submachines Nl of Ml and N2 of M2 such that N = Nl - N2 .

Proof. Let Ni = (Ti , XZ , v2) be a submachine of MZ , i = 1, 2 . Now N1-N2 = (TI XT2, XI UX2, VI 'V2) . Let (r, s) E S(TI xT2) . Then 3 w E (XI U X2)* , (p, q) E Tl x T2 such that (pl ' p2)*((p, q), w, (r, s)) > 0. Let w* = uv be the standard form of w, u E XI* , v E X2* . Now [t* (p, u, r) A [t*(q, v, s) (pl'p2) * ((p, q), w, (r, s)) > 0. Thus pi (p, u, r) > 0 and p2 (q, v, s) > 0. Hence rES(p)CS(Tl)=TlandsES(q)CS(T2)=T2 .Thus (r,s)ET1 xT2 .


6.13. Cartesian Composition Hence S(Tj x T2 )

C Tl x T2 .

295 Let (p, q), (r, s)

E Tl x T2 , a E X l U X2 . Now

(VI - V2) ((p, q), a, (r, s) )

v l (p,a,r) if a E X1, q =s v2 (q, a, s) i£ a E X2, p=r

0 otherwise

P1 (PI a, r) if a E X1, q =s p,2 (q, a, s) if a E X2, p=r

0 otherwise

(P1 ' P2) ((p, q), a, (r, s)) .

Hence (PI 'N2)I(T1xT2)x(XIUX2)x(T1xT2) =vl-v2 . Thus Nl -N2 is asubmachine of Ml - M2 . Conversely, let N = (TI x T2 , Xl U X1 , v) be a submachine of Ml - M2 . Let vl = /t1IT1 xX 1 xT1 , v2 = /t2lT2XX2XT2, N1 = (TI IX1IV1)1 and N2 = (T2, X2, v2) . Let p E T1, x E Xl , r E Q1 be such that pl (p, x, r) > 0. Let t E T2 . Then (p,l'Nt2)*((p, t), x, (r, t)) = p,l(p, x, r) >0. Thus (r, t) E S(Tj x T2) = Tl x T2 . Hence r E Tl and so S(TI ) C_ T1 . Thus Nl is a submachine of Ml . Similarly N2 is a submachine of M2 . Let (p, q), (r, s) E Tl x T2 , aEXIUX2 .Now

v((p, q), a, (r, s))

=

Gt1 ' ft2)((p, q), a, (r, s)) P 1 (p, a, r) if a E X 1 , q = s N'2 (q, a, s) if a E X2, p = r

0 otherwise

vi (p,a,r)ifaEX1, q=s v2 (q, a, s) if a E X2, p = r

0 otherwise

(VI - v2) ((p, q), a, (r, s)) .

Hence N = Nl - N2 . m Let M = (Q, X, ft) be a ffsm and let - be an equivalence relation on Q . Recall from Definition 6.4 .1 that - is called an admissible relation if and only if b' p, q, r E Q, b'a EX, if p - q and lt(p, a, r) > 0, then 3 t E Q such that ft(q, a, t) >_ ft(p, a, r) and t - r . Let M = (Q, X, ft) be a ffsm and let - be an equivalence relation on Q . By Theorem 6.4.2, - is an admissible relation if and only if d p, q, r E Q, V'x E X*, if p - q and /t* (p, x, r) > 0, then 3 t E Q such that /t* (q, x, t) >_ p,* (p, x, r) and t - r . Let Ml = (Q1, X1, p l ) and M2 = (Q2, X2, p 2 ) be two ffsms and let Xl n X2 = Ql . Let p i be an admissible relation on MZ, i = 1, 2. Define a relation p l ' P2 on Ml - M2 by (PI IP2)P l ' P2(gl,q2) if

and only

if p1Plg1

and

p2P2g2

d(pl,p2), (ql, q2) E Q1 x Q2Clearly pl'P2 is an equivalence relation on Ml .M2 . Let (PI, p2), (ql, q2) E Q 1 x Q2 be such that (pl,p2)Pl ' P2(gl,g2) . Let a E Xl U X2 and (ft,


29 6


p2)((pl,p2), a, (r l , r 2 )) > 0 for some (r l , r2 ) E Q1 x Q2 . Suppose that a E X1 . Then ft, (PI,a,rl) = (ftl' P2)((pl,p2),a,(rl,r2)) > 0 . Thus p2 = r2 . Now plplgl and ft, (PI, a, rl) > 0 . Since p l is admissible, 3 t l E Q1 such that ftl(gl, a, tl) >_ ft, (pl, a, rl) and tlplrl . Hence (It, ' P2)((gl, q2), a, (tl, q2)) = ftl(gl,a,tl) >_ ft, (pl,a,rl) = (ftl'ft2)((pl,p2),a, (rl,p2)) . Also, since tj p, r, and g2p2p2, (tl,g2)pl ' p2(rl,p2) . Thus p l ' p2 is an admissible relation on Ml - M2 . We have thus proved the following theorem . Theorem 6 .13 .18 Let MZ = (QZ, XZ, p 2 ) be a ffsm, i = 1, 2, and let Xl rl X2 = 0 . Let Pi be an admissible relation on MZ, i = 1, 2 . Then pl 'P2 is an admissible relation on Ml - M2 . 6 .14

Admissible Partitions

In this section, we introduce the concept of a covering of a ffsm by another, admissible partitions and relations of a ffsm, ft-orthogonality of admissible partitions, irreducible ffsm, and the quotient of a ffsm induced by an admissible partition of the state set . We derive results concerning ft-orthogonality and covering, Theorems 6.14 .14 and 6 .14 .15 . We show that an admissible partition 7r of Q is maximal if and only if the quotient M17r is irreducible, Theorem 6 .14 .20 . The paper culminates with a result showing that a ffsm can be covered by irreducible ffsms, Theorem 6 .14 .21 . These results allow us to study fuzzy finite state machines via coverings of products of simpler machines . Irreducible finite state machines seem to arise naturally in some applications, e .g ., biology. An example is given in [92] of a finite state machine arising from a metabolic pathway. Recall from Definition 6 .6 .1 that (9, ~) is a covering of Ml by M2, written Ml [0,1] as follows : d((ql, q2), b, (PI, P2)) E Q x X2 x Q, ft' ((gl, q2), b, (PI, P2)) = ft, (gl,w(q2, b), PI) n P2 (q2, b,p2) . Then M = (Q, X2, ft') is a ffsm . M is called the cascade product of Ml and M2 and we write M = M1wM2 . Let M = (Q, X, ft) be a ffsm and let - be an equivalence relation on Q . Recall that ti is called an admissible relation on Q if and only if d p, q, r E Q, b'a E X, if p - q and ft (p, a, r) > 0, then 3 t E Q such that ft(q, a, t) > ft(p, a, r) and t - r .


6.14. Admissible Partitions

297

Theorem 6.14 .1 [3] Let M = (Q, X, /t) be a ffsm and let - be an equiv-

alence relation on Q . Then - is an admissible relation on Q if and only if V p, q, r E Q, Vx EX*, if p - q and /t* (p, x, r) >0, then 3 t E Q such that * * ft (q, x, t) > ft (p, x, r) and t - r. m

Definition 6.14 .2 Let M = (Q, X, /t) be affsm and P = ~Q1, Q2, . . . , QkI

be a partition of Q . Then P is called an admissible partition of Q if the following holds: Let a E X, then Vi, 3j, 1 ft(Pl, a, r) and t, r E Qj .

Proposition 6 .14 .3 Let M = (Q, X, ft) be a ffsm.

(1) Let 1Q = {{q} I q E Q} . Then 1Q is an admissible partition of Q . (2) {Q} is an admissible partition of Q . m

Theorem 6.14 .4 Let M = (Q, X, ft) be a ffsm and P = {Q1, Q2, . . . , Qk}

be a partition of Q . The following are equivalent: (1) P is an admissible partition of Q . (2) Let x E X* . Then Vi, 3j, 1 0 for some r E Q, then 3 t E Q such that y* (P2, x, t) >_ y*(PI,x,r) and t, r E Qj .

Proof. (1)x(2) : Let x E X* and Ixj = n. Let pl, p2 E QZ and y * (pl, x, r) > 0 for some r E Q. If n = 0, then x = A and ft*(pl, x, r) > 0 implies that pl = r. Thus [t* (P2, x, P2) = 1 = ft*(pl, x, pl) . In this case i = j. Hence the result is true for n = 0. Suppose the result is true for all y E X* such that I y I = n -1, where n > 0. Let x = ya, where a E X. Now y* (p l , x, r) = * * ft (pl, ya, r) = V{ft* (pl, y, s) A ft (s, a, r) I s E Q} > 0. Since Q is finite, * there exists t E Q such that ft (PI , y, t) A ft * (t, a, r) = ft* (p l , ya, r) . Thus * * ft (pl, y, t) > 0 and y,(t, a, r) = ft (t, a, r) > 0 . By the induction hypothesis, there exists j and there exists s E Q such that ft * (P2, y, s) >_ y* (PI, y, t) and s, t E Qj . Now s, t E Qj and ft(t, a, r) > 0 . Hence by (1), there exists l and there exists q E Q such that y (s, a, q) >_ y(t, a, r) and r, q E Qi . Now /t* (P2, x, q) = ft* (P2, ya, q) >_ ft * (P2, y, s) n ft(s, a, q) > ft * (Pl, y, t) n ft (t, a, r) = y* (pl, ya, r) = y* (pl, x, r) and q, r E Ql . The result now follows by induction .

m

Corollary 6.14.5 Let M = (Q, X, ft) be a ffsm. Then every admissible partition P of Q induces an admissible relation - on Q such that the set of all equivalence classes of - is P. Conversely, the set of all equivalence classes of an admissible relation on Q is an admissible partition of Q. 0 Lemma 6.14 .6 Let M = (Q, X, ft) be a ffsm and

7r = {HZ i E 11 be an admissible partition of Q . Let i, j E I. Then Vq, q' E HZ, Va E X,

V{ft(q,a,r)


I

r E Hj } = V{ft(q ,a,r)

I

r E Hj }.

29 8


Proof. Let q, q' E HZ , a E X, A = {ft(q, a, r) I r E Hj}, and B = ft(q, a, r) I r E Hj }. Suppose ft(q, a, r) > 0 for some r E Hj . Since 7r is an admissible partition, there exists r' E Q such that N(q', a, r') >_ ft(q, a, r) . Again by the admissibility of 7r, r' E Hj . Similarly if ft (q, a, p) > 0 for some p E Hj,then there exists p' E Hj such that ft(q,a,p') > N(q',a,p) . Also, by the admissibility of 7r, it follows that ft(q, a, r) = 0 b'r E Hj if and only if N(q', a, r) = 0 b'r E Hj . Hence V{ft(q,a,r) I r E Hj }=V{ft(q,a,r) I r E Hj } . Theorem 6.14 .7 Let M = (Q, X, /t) be a ffsm . Let it = an admissible partition of Q . Define

{HZ I i

E 11 be

y" : 7r X X x 7r ~ [0,1] by

ft' (Hi,a,Hj) = V{ft(q,a,r) I r E Hj I VHZ , Hj E 7r and a E X, where q E HZ . Then M17r = (7r, X, ft") is a ffsm, called the quotient fuzzy finite state machine with respect to 7r .

Proof. By Lemma 6 .14 .6, ft" is well defined . m Proposition 6 .14 .8 Let M = (Q, X, ft) be a ffsm. Let 7r = be an admissible partition of Q. Then for q E HZ ft'*

{HZ I i

E 11

(Hz, x, Hj) < V{ft * (q, x, r) I r E H;

VHZ , Hj E 7r and x E X* .

Proof. Let HZ, Hj E 7r and x E X*. Let Ixj = n . If n = 0, then x = A. If HZ = Hj then y"* (HZ , x, Hj ) = 1 and V{y* (q, x, r) I r E Hj } = V{ft*(q,x,r) I r E HZ} = y*(q,x,q) = 1, where q E HZ . If HZ Hj, then ft'* (Hi, x, Hj) = 0 and HZ n Hj = 0 . Since HZ n Hj = Ql and q E HZ, V{/t*(q,x,r) I r E Hj I =0. Hence ft'* (Hi,x,Hj) = V{/t*(q, x, r) I r E Hj } . Suppose that the result is true b'y E X* such that I yI = n - 1, n > 0. Let n>0andx=ya,whereyEX*,aEX,andjyl=n-1 .NowforgEHi © 2002 by Chapman & Hall/CRC


299

ands E Hk, [t"*(HZ x H;)

= = =
0. Let n > 0 and x = ya, where y E X*, a E X, and Iyj =n-1 . Then ft * (q, x, p)

= = = O.Letn>Oandx=ya,whereyEX*,aEX, © 2002 by Chapman & Hall/CRC


301

and I y I = n - 1 . Then ft * (qo, x, po)

ft * (qo, ya, po) V {ft*(qo, y, r) n ft(r, a, po) I r E Q} = V{w*(qo, y, r) A (V{ft(r, a, p) A ft(r, a,p) I p E H~, p'EKv}) I rEQ} = V{(V(ft*(go,y,r) Aft(r,a,p))) A (V(ft*(go,y,r)A ft(r, a, p))) I p E H., p' E K, r E Q} = V {(V {ft* (qo, y, r) A w(r, a,p) I r E Q}) A (V{w*(go,y,r) Aft(r,a,p') I r E Q}) I p E H.,p' E Kv} V{ft* (go,ya,p) nft * (go,ya,p') p E H., p' E K, } = V{ft* (go,x,p) nft* (go,x,p') I p E H., p' E K, }. =

The result now follows by induction . The converse is trivial . Let M = (Q, X, ft) be a ffsm. Let 7r and T be admissible partitions of Q. Consider the ffsms M/7r = (7r, X, ft") and MIT = (T, X, ft') . Define It ^ : (7r x T) x X x (7r x T) ----> [0,1] by It A ((HZ, Kj), a, (H., K,)) = ft' (Hi, a, H.) n ft' (Kj, a, VHZ, H. E 7r, Kj , K, E T, and a E X. Then a fuzzy finite state machine. Note that VHZ ,H

E

7r,Kj ,K,

M17r n MIT = (7r x T, X,

(H., Kv)) =,t"*(Hz, X, H.)

It ^*((H,,Kj), X, E

T,

Kv)

[r^) is

Aft' * (Kj,x, Kv)

and x E X* .

Theorem 6.14 .14 Let M = (Q, X, ft) be a f'sm . Let 7r and T be admissible partitions of Q that are ft-orthogonal. Then M < M17r n MIT. Proof. Define 97 : 7r xT ----> Q by 97((HZ, Kj)) = qo, where Hi nKj = {qo} . Since 7r and T are ft-orthogonal, 97 is one-to-one. Let C be the identity map on X. Let HZ, H. E 7r, Kj , K, E T, and x E X* . Suppose HZ n Ki = {qo} and H n K, = {po } . Then w* (97 ((Hi, Kj)), x, 97((H., K,))) = w* (qo, x,po) .

Also, It ^ * ((HZ, K;), x,

(H.,

Kv))

n ft' (Kjx Kv) = (V{ft* (qo, x, p) p E H })n (V{ft* (qo, x, p) p' E K, }) V{ft * (go,x,p)nft * (go,x,p') I pEH., p' EKvj = ft*(go,x,po), =

ft"* (HZ a

where the last inequality holds since [t* (97((H,,

7r

and

T

H.)

are ft-orthogonal . Thus

Kj)), x, 97((H., K,))) = w^ * ((HZ,


K;), x, (H., K,)) .

30 2


Now ItA*((H,,Kj),X, (H., K')) =V{Nn*((H,,Kj),x, (H" K')) 197((H"KS)) = r7((H., K,)), (H,, Ks) E 7rxTj since 97 is one-to-one. Hence /t*(97((Hi, Kj)), x, 97((H.,Kv))) = V{ft^*((Hi, Kj), X, (HT, Ks)) 197((H" Ks)) = 97((H., K,)), (H,, Ks) E 7r x T} . Consequently, M < M17r n MIT . 0

Theorem 6.14 .15 Let M = (Q, X, /t) be a ffsm . Let 7r be an admissible partition of Q. If there exists a partition T of Q such that 7r and T are ft-orthogonal, then there exists a ffsm N such that M < NwM17r . Proof. Let

7r = {Hj}jEI and T = {Kj}jEj be ft-orthogonal partitions Let N of Q. = (T, 7r x X, ft'), where

ft'(Kj, (Hi, a), Kv) = V{ft(go, a, p) I p E K, }, and {qo} = Hi n Kj . Since T is admissible, ft' is well defined. Define w : 7r x X 7r x X to be the identity map. Define 97 : T x 7r ----> Q by r7((Kj, Hi)) = qo, where {qo} = Hi n Kj . Then 97 is one-to-one and onto. Let C be the identity map on X. Then for {po} = H n K, l~(n7((K7 Hi)), a, n7((Kv, Hv)))

_ ft (qo, a, po) V{ft(go, a, p) n ft (go, a, p) (p' ,p) E Kv x Hv } (V{ft(go,a,p') I p' E K, })n (V{ft(go, a,p) I p E H.}) ft'(Kj, w(Hi, a), K,) n ft"(Hi, a, H.) pW((Kj, Hi), a, (Kv, H.)).

Thus 97 N'* (97((Kj, Hi)) , x, ((K,, H.))) = ft'* (( Kj , Hi) , x, (Kv, H.))

(6.3)

for x E X* such that IxI = 1. Suppose that (6.3) is true if IxI = n-1, n > 0, where x E X* . Now p,*(97((Kj,Hi)),xa,97((Kv,H.))) = V{(97((Kj, Hi)), x, 97 ((K, H))) A ft (97((K, H)), a, 97 ((K,, H.))) I 97 (K, H) E 97 (Tx7r)I (since 97

is onto) =V{ft-*((Kj, Hi), x, (K, H))Apw((K, H), a, (K, H.)) I (K, H) E (T x 7r)} = ttw*((Kj,Hj), xa, (Kv, H.)) . Now ,t*(?7((Kj, Hi)), A, 97((Kv, H.))) = 1 if and only if 97((Kj, Hi)) = r7((K, H.)) if and only if (Kj , Hi) _ (K, H ) (since 97 is one-to-one) if and only if f rW * ((Kj , Hi), A, (K, H")) _ 1. From this, it follows that (6 .3) holds for x = A. Hence (6 .3) holds b'x E X* . Thus by induction, (r7, C) is a covering of M by NwMl7r . m

Definition 6.14 .16 Let Q be a nonempty set and 7r and T be partitions of Q. Then 7r < T if VA E 7r, there exists B E T such that A C B. The proof of the following lemma is straightforward.

Lemma 6.14 .17 Let Q be a finite nonempty set. Let

7r = {Hi}Z i and T = {Kj},T=1 be partitions of Q such that 7r 0 . Since 7r is admissible, there exists K E 7r such that ft" (Hl, a, K) > ft" (H2, a, HZ) and K and HZ belong to the same element of 7r . Hence V{ft(q, a, t') I t' E K} >_ V{ft(p, a, r') I r' E HZ}. This implies that there exists t E K such that ft(q, a, t) >_ ft(p, a, r) > 0. Now HZ E T if and only if K E T since K and HZ belong to the same element of 7r . If K, HZ E T, then t, r E Hl U . . . U H, and if K, HZ ~ T, then t, r E Hm+1 U . . . U Hn , i.e ., t and r belong to the same element of 7r' . Hence 7r' is admissible. Since 7r is maximal, it follows that 7r' = {Q} and so T = {Q} . This implies that 7r = {7r} . Thus M17r is irreducible . Conversely, suppose that M17r is irreducible . Let T be an admissible partition of Q such that 7r 0 for some HZ E 7r . Then V{ft(q, a, r') I r' E HZ } > 0 where q E Hl . Let r E HZ be such that ft(q, a, r) = V{ft(q, a, r') I r' E HZ } > 0 . Since T is admissible, Vp E H2, there exists tp E Q such that ft(p, a, tp) >_ ft(q, a, r) and r, tp are in the same element of T Vp E H2 . Now if HZ ~ {Hl, . , Hm}, then r, tp E HZ Vp E H2 and ft" (H2, a, HZ) > ft (p, a, tP) > ft (q, a, r) = ft"(Hl, a, HZ) . Suppose HZ E {Hl, . . . , Hm} . Then r, tp E Hl U . . . U Hm for all p E H2 . Let ft (p, a, tp,) = V{ft(p, a, tp) I p E H2} . Since tp, E Hl U . . . U Hm , tp, E Hk for some k, 1 < k < m. © 2002 by Chapman & Hall/CRC

30 4


Hence /t" (H2 , a, Hk) = V{ft(q, a, r')~ r' E HkI > ft (p, a, tp,) > ft (q, a, r) = ft" (H1, a, HZ ) and Hk, HZ E {H1, . . . , H, } . Consequently T is an admissible partition of 7r . Since M17r is irreducible, T = {7r} and so T = {Q}. Hence 7r is maximal . m Theorem 6 .14 .21 Let M = (Q, X, p) be a ffsm and IQ I = n > 2 . Then M < N1w1N2w2 . . .w ._1N., where N1 , N2 , . . . , N, are irreducible ffsms and the state sets Qj of Ni are such that I Qj I < n.

Proof. Since Qj >_ 2, we can choose a maximal admissible partition 7r of Q. Clearly I7rI < IQI . By Theorem 6.14.15, there exists a ffsm N such that M X' and

=

((Q x Q') x Q", X",

(ftWIft')w2ft"(((P,P'),P"),x ,((q,q'),q"))

(ftw1ft')w2ft"),

= =

It follows immediately that

M'w4M" =

=

w1 (P'

(Q' x Q", X",

w2 (P" x"))

It 'w4ft")

and

q1) A ft // (P // , X // , q11) .

(Q x (Q' x Q"), X", gw3(g,'w4g,")) and

Ntw3(ft'w4ft")((P,(p',p")),x",(q,(q',q")))


w2

ft(P,w1(P',w2(P",x")),q) Aft' (P', w2 (P", x"), q) Aft// (P'/x//q11) .

It'w4N-"((p',p"),xii (q', q " )) =ft'(PI,w4(P",x"),

Mw3(M'w4M") =

where

ftWIft'((P,P'),w2(P",x"), (P " , x " , q") (q, q')) n N'"

Define w3 : (Q' x Q") x X" - X by w3((P',P"),x11) and set w4 = w2 .

Moreover,

'xX'----> X

=

ft(P,w3((P',P"),x ),q)A (ft'w4ft"((p', p"), x~~ (q', q"))) ft (P, w3((P',P"), x"), q) A (It'(P', w4(P", x"), q') Aft// (pii x// q11)) .

30 8


Let cx : (Q x Q') x Q" - Q x (Q' x Q") be the natural mapping and 3 be the identity mapping on X" . Then (f~wif~')w2f~")(((p,p'),p"),~ ,((q,q'),q"))

Thus (Mw1M')w2M" - Mw3(M'w4M") . 6 .17

=

ft(p,wi(p',w2(p",x")), q) Aft/(p',w2(p",x q') A ft"(p", X", q") ft (p, wi (p~, w2 (p", X // )), q) n (W (p, w2 (p", X q1) n ftii(pii, xii, q11)) ft (p, w3((p~,p"), x"), q) n(ft /(p/ , w4(p// , x// ), q1) Aft// (p ii X // q11) ftw3(ft'w4ft")((p, (p~, er p )), x", (q, (q~, q"))) ftw3 (ft'w4ft") (gy(((p, p') , per)),,3(x"), o(((q, q~),

0

Covering Properties of Products

The following theorem is a direct consequence of the definition of the direct sum and sum of two fuzzy finite state machines . Theorem 6.17 .1 Let M and M' be fuzzy finite state machines. following properties hold: (1)M~M+M', (2)M'~M+M', (3)M~M+M', (4)M'~M+M' .

Then the

Proof. We only prove (1) . Let 97 : Q U Q' ~ Q be a partial surjective function defined by n(q) = q for all q E Q and let ~ : X ~ X U X' be the inclusion function . Clearly, (r7, ~) is a required strong covering of M by M+M' .

Theorem 6.17 .2 Let M = (Q, X, ft) and M' = (Q', X', ft') be fuzzy finite state machines. Then the following properties hold: (1) MWM' ~ M o M', (2) M+M' ~ M+M' .

Proof. (1) Let w x, : Q' ~ X be the function defined by wx, (p') = w(p', x') for all p' E Q' and x' E X' . Define ~ : X' ~ XQ' x X' by ~(x') = (wx,, x') and let 97 be the identity map on Q x Q'. © 2002 by Chapman & Hall/CRC

6.17. Covering Properties of Products

309

(2) Let 97 and ~ be the identity mappings on Q U Q' and X U X', respectively. If p, q E Q and x E X, then (/t+ft~)(97(p), x, 97(q)) _ (ft (@ft~)(p, (x), q) = ft (p, x, q) . If p, q E Q' and x E X', then (/t+/t') (97(p), x, 97(q)) _ (ftoft') (p, (x), q) = ft'(p, x, q) . If (p, x) E Q x X and q E Q' or (p, x) E Q' x X' and q E Q, then (ft + ft') (97(p), x, 97(q)) = 0 < 1 = (ft () ft') (p, ~(x), q) . In all other cases, the values of ft + ft' and ft ft' are 0. Theorem 6.17 .3 Let M = (Q, X, p), M' = (Q', X', p'), and M" = (Q", X", ft") be fuzzy finite state machines such that M ~_ M'. Then the following assertions hold: (1)MXM"~M'XM" ; (2)M"xM~M"xM' ; (3) Given wl : Q" x X" ~ X, there exists co t : Q" x X" ~ X' such that Mw1M" _< M'w2M" ; (4) If (97, ) is a covering of M by M' and is surjective, then for all X", there exists co t : Q' x X' ~ X" such that M"w1M wl : Q x ;X M W2M (5) M0M"~M'0M" ; (6) M + M11 MI + Mii ; (7) M// +M M"+M' ; (8) M+M" M'+M"; (9) M" + M M" o M/ . Moreover, if X = X' = X", then (10) M AM" M' AM"; (11) M" A M ~ M" A MI . Proof. Since M --< _ M', there exists a partial surjective function 97 Q' ----> Q and a function ~ : X ----> X' such that ft+(?7(p), x, r7(p')) X' x X" by ~'(x, x") _ ( (x), x") . Clearly (97', ') is the required covering . (2) The proof is similar to that of (1) . (3) Given wl : Q" x X" X, set W2 = o wl and let ' be the identity map on X". (4) Given w l : Q x X X", let cot : Q' x X' ----> X" be such that Since ~ is surjective and X is finite, such W2 w2(p',~(x)) = wi(?7(p'),x) . exists . Clearly, W2 is not unique . Define 97' : Q" x Q' - Q" x Q by ?Ap",p~) = (p", 97(p)) and set ~' = ~ . 97' : Q' x Q" ~ Q x Q" by 97i (p~,p") = (97(p~),p") and ' XQ~~(5)x Define X ----> (XI)Q" x X 11 by ~/(f, x") = (~ o f I x") .


310

6. Algebraic Fuzzy Automata Theory (6) Recall that M + M" = (Q U Q", X U X", ft + ft"), where

(ft + ft") (p, x, q) =

ft(p, x, q) ft" (p, x, q) 0

if p, q E Q and x E X Q" and x E X" if p, q E

otherwise

and M' + M" = (Q' U Q", X' U X", p' + p"), where (ft' + ft ") (p, x, q) _

ff' (p, x, q) /t" (p, x, q) 0

if p, q E Q' and x E X' if p, q E Q" and x E X"

otherwise.

Define 9' :Q'UQ"----> QUQ"by i7/ (p~)

if p'EQ'

otherwise

and ~' :XUX"~X'UX"by (x) -

(x)

x

otherwise .

Since 9 is a partial surjective function, so is 9' . we now note that (,~+w")+(~'(p'), x, ~'(q')) = =

for some q'

Thus (6.4)

E Supp(A), x E X,

S(b * k, x, p) A S(b, x, r) A v(k) S(b * k, xo , p) A S(b, xo , r) A S(e, x, e) A v(k) 6(b, xo,p) n 6(k -1 , xo, e) n 6(b, xo, r)n 6(e, x, e) A v(k) .


E Q.

6(b, lo,p) n 6(b * q' -1 , xo, e)A S6(k - ', xo, e) A S(e, x, e) A S(b, xo, r) .

Since v is a fuzzy kernel of A, for all p, r, k E Q, b v(p * r -1 )

E Q, x E X,

(6.5)

6.18. Fuzzy Semiautomaton over a Finite Group

321

From (6.4) and (6.5), it follows that v(p * r-1 ) > 6(q * k, x, p) n 6(g, x, r) n v(k)

for all p, q, r, k E Q, x E X. Since by assumption, v is a fuzzy normal subgroup of Q, v is a fuzzy kernel of T. Theorem 6.18 .25 Let T = (Q, X, 6) be a multiplicative fsa over a finite

group with e-input xo E X . Then the following statements are equivalent : (1) A is a fuzzy kernel of T. (2) A is a fuzzy normal subgroup of Q and A (q) > 6(p, xo, q) A A (p) for p,gEQ all .

Proof. (1)x(2) . Since A is a fuzzy kernel of T, A(q)

= > =

A(q * e-1) 6(e * p, xo, q) n 6(e, xo, e) n A(p) 6(p, xo, q) n A(p) (since 6(p, xo , q) < 6(e, xo, e))

for all p, q E Q . (2)x(1) . Since T is multiplicative, for all p, q, r, k E Q, x E X, 6(q * k, x, p * r) A 6(g, x, p) n A(k)

=

< <
>

A(p2 * r * P21) 6(q * k, x, p2 * r) A 6(q, x, p2) n A(k) 6(q * k, x,p1) A 6(q, x,p2) A A(k) .

(by (6 .6)

Hence A is a fuzzy kernel of T. Theorem 6.18 .26 Let T = (Q, X, 6) be a multiplicative fsa over a finite group with e-input xo E X. There exists a semigroup homomorphism f Q ~ .FP (Q) and a function g : X ~ FP (Q) such that 6(q,x,p) = f(q)(p) n 9(x)(e) for all p,gEQ,xEX.


32 2


Proof. Define f : Q ----> .FP(Q) by f(q) = Sq for all q E Q, where Sq : Q ----> [0,1] is defined by 6q (P) = S(q, xo , p) for all p E Q . Since T is multiplicative,

bq,*qz (pi * p2)

=

= =

b(qi * q2, xo,pi * p2) b(qi, xo,pi) n b(q2 1 , xo,p2 1 ) b(qi, xo,pi) A S(q2, xo,p2) bq, (PI) n Sgz(p2)

for all P1, P2, ql, q2 E Q . Hence (bq, * bqZ) (p)

=

V{6q, (q) S qi*q2(p)

n bqz (r) I

q, r E Q, q * r = p}

for all p E Q. Thus f (qi) * f (q2) = f (qi * q2) .

(6.7)

It is well known that,F'P(Q) is a semigroup with respect to the product of fuzzy subsets of Q. Hence, it follows from (6.7) that f is a semigroup homomorphism from Q into .FP(Q) . For x E X, define the fuzzy subset Ax of Q by Ax (q) = 6(e, x, q) for all q E Q. Define g : X ----> .FP(Q) by g(x) = Ax for all x E X. Since T is multiplicative, it follows that S(q, x,p) = f(q) (p) A g(x) (e) for all p,gEQ,xEX. 6 .19

Exercises

1. Let M = (Q, X, ft) be the ffsm, where Q = {ql, q2}, X = {a}, and ft is defined by ft(ql , a, qi) = s, ft(ql , a, q2) = 0, ft(g2, a, qi) = 0, and ft(g2, a, q2) = 3. Determine E(M) and E(M) . 2 . Let M = (Q, X, ft) be the ffsm, where Q = {ql , q2 }, X = {a}, and

ft is defined by ft(ql , a, gi) = 0, ft(gi a, g2) = 3, ft(g2, a, qi) = 0, and ft(g2, a, q2) = 3. Determine E(M) and E(M) .

3. Let M = (Q, X, ft) be the ffsm, where Q = {ql , q2 }, X = {a}, and

ft is defined by ft(ql , a, qi) = 0, ft(ql , a, q2) = 3, ft(q2 , a, qi) = 3, and ft(g2, a, q2 ) = 0. Determine E(M) and E(M) .

4. In Example 6.6.4, determine E(MI), E(M2), E(MI xM2), and E(MIA M2) when cl :A c2 and/or dl :?~ d2 . 5. Prove (2), (3), and (4) of Theorem 6.17.1. 6. Let A and ft be fuzzy subgroups of G such that A is a fuzzy normal subgroup of ft . Prove that ft/A is a fuzzy subgroup of Supp(ft)/ Supp(A) . © 2002 by Chapman & Hall/CRC

6.19. Exercises

323

7. Prove Theorems 6.18.12 and 6.18.13. 8. (113) Let (Q, X, T) be a T-generalized state machine. Prove that T + (p, xy, q) = V {T+ (p, x, r) T

T+(r,

y, q) I r E Q},

for all p, q E Q and x, y E X+ . 9. (113) Let (Q, X, T) be a T-generalized state machine . Define - on X+ by Vx, y E X+, x - y if T+ (p, x, q) = T+ (p, y, q) Vp, q E Q. Prove that - is a congruence relation on X+ . 10. (113) Let M = (Q, X, T) be a T-generalized state machine . Let -be a congruence relation defined in Exercise 6. Let [x] = {y E X+ x - y} Vx E X+ and S(M) = {[x] I x E X+} . Prove that S(M) is a semigroup, where the binary operation on S(M) is defined by [x] [y] = [xy] V [x], [y] E S(M) . 11. (113) Let M = (Q, X, T) be a T-generalized state machine, where T is the ordinary product on [0,1] . Show by example that S(M) need not be finite, where S(M) is defined in Exercise 10 . 12. (113) Let M = (Q, X, T) be a T-generalized state machine. Prove that since Q is finite, S(M) is finite if and only if Im(T+) is finite . 13. (113) Construct an example of a T-generalized state machine (Q, X, T) such that EgEQ T+ (p, x, q) > 1 and x E X+ . 14. (113) A T-generalized state machine (Q, S, p) is called a T-generalized transformation semigroup if S is a finite semigroup such that (a) p(p, uv, q) = V{p(p, u, r) T p(r, v, q) I r E Q} Vp, q E Q and Vu, v E S and (b) Vu, v E S, p(p, u, q) = p(p, v, q) Vp, q E Q implies u = v . A t-norm T is said to be T-generalized transformation semigroup inducible if S(M) is finite and EgEQ T+ (p, x, q) Q by 6* (q, A) S* (q, ua)

= =

q S(S* (q, u), a)

b' q E Q, b' u E X*, b' a E X. It can be easily verified that S* (q, uv) _ S*(S*(q,u),v) Vu, v E X* . Definition 7.1.2 Let M = (Q, X, S, t, T) be a pfa. The max-min fuzzy language recognized by M is the fuzzy subset FL v (M) of X* defined by FL v (M) (u) = VaEQ (t(q) 325 © 2002 by Chapman & Hall/CRC

n T(6* (q, u)))

7. More on Fuzzy Languages

32 6

and the min-max fuzzy language recognized by M is the fuzzy subset FLA (M) of X* defined by FLA(M)(u) = AIEQ(t(q) V T(6*(q,u))) .

Definition 7.1.3 Let X be a nonempty finite set. A fuzzy subset A of X* is called F v -regular (F A -regular if there exists a pfa, M = (Q, X, S, t, T), such that A = FL v (M) (A = FLA(M)) .

Theorem 7.1.4 Let L be a regular language on a nonempty finite set X. Then the characteristic function XL of L is a Fv -regular language .

Proof. Since L is a regular language on X, there exists a deterministic partial finite automata (dpfa), M' = (Q, X, S, qo, F), such that the language L(M') recognized by M' is L. Consider the pfa, M = (Q, X, S, t, T), where t : Q ~ [0,1] is defined by t(qo) = 1 and t(q) = 0 if q z,4 qo and T : Q ~ [0,1] is defined by T(q) = 1 if q E F and T(q) = 0 if q ~ F. Let u E X*. Then FL v(M) (u)

Thus FL v (M) = XL .

VgEQ(t(q) AT(6*(q,u))) T ((S* (qo, u)) 1 if S* (qo , u) E F F 0 if S* (qo , u)

0

Theorem 7.1.5 Let L C_ X* . Suppose the characteristic function is a Fv -regular language . Then L is a regular language on X.

XL of L

Proof. Since XL is a Fv -regular language, there exists a pfa, M = (Q, X, S, t, T), such that FL v (M) = XL . Thus V4EQ( L(q) nT(6*(q,u))) _ { 0 if u

L.

Now u E L if and only if there exist q' E Q such that T(S * (q', u)) = 1 . Let

1 and

Qo = {q E Qjt(q) = 1}

and F = {S * (q, u) E QIT(S* (q, u)) = 1 for some u E L} .

Then Qo :?~ Ql and F :?~ 0. Let Mq = (Q, X, S, q, F), q E Qo . Let Lq be the language recognized by Mq. Then Lq C_ L . Hence U gEQo Lq C_ L. Let u E L. Then VgEQ(t(q) AT(S*(q,u))) = 1. Hence there exists q E Qo such that t (q) AT(S*(q,u)) = 1. Then T(S*(q,u)) = 1 and so S* (q, u) E F. Thus u E Lq. Hence U gEQ,,Lq = L. Since each Lq is regular, L is regular . m © 2002 by Chapman & Hall/CRC

7.1 . Fuzzy Regular Languages

327

Theorem 7.1.6 Let L be a regular language on a nonempty finite set X. Then the characteristic function XL of L is a FA-regular language.

Proof. Since L is a regular language on X, there exists a dpfa, M' = (Q, X, S, qo, F), such that the language L(M) recognized by M' is L. Consider the pfa, M = (Q, X, S, t,T), where t : Q ~ [0,1] defined by t(qo) = 0 and s(q) = 1 if q :?~ qo and T : Q ~ [0,1] defined by T(q) = 1 if q E F and T(q)=0 if q~F.Now FLA(M)(u)

= ngEQ(t(q) V T(6*(q,u))) .

Let u ~ L. Then XL(u) = 0 and 6*(qo,u) ~ F. Hence t(qo) = 0 and T(6* (qo, u)) = 0. Thus FL A (M) (u) = 0. Suppose u E L. Then 6* (qo, u) E F and hence T(6*(qo,u)) = 1 . Thus t(qo) V T(6*(qo,u)) = 1 . If q :?~ qo, then t (q) = 1 . Thus t (q) V T(6* (q,u)) = 1 . Hence FL A(M)(u) = 1 . Thus FLA (M) = XL . 0 Theorem 7.1.7 Let L C_ X* . Suppose the characteristic function XL of L in X* is a FA-regular language. Then L is a regular language.

Proof. Since XL is a FA -regular language, there exists a pfa, M = (Q, X, 6, t, T), such that FL A (M) = XL . Thus for all u E X*,

=

XL(u)

ngEQ(t(q) V T(6*(q,u))) ~ 1ifuEL 0ifa~L .

Now u E L if and only if for all q E Q, either t (q) = 1 or T(6*(q, u)) = 1.

Let

Qo = {q E Q I t(q) = 1} and for all q E Q\QO, F'q = U.EX*{6* (q, x) I T( 6 *(q,u)) = 1} .

For all q E Q\QO, let Mq = (Q, X, 6, q, Fq ) and let Lq denote the language recognized by Mq. Then y E ngEQ\QoLq if and only if for all q E Q\QO, y E Lq if and only if for all q E Q\QO, 6* (q, y) E Fq if and only if for all q E Q\QO, T(6*(q, y)) = 1 if and only if y E L . Hence L = ngEQ\QoLq and XL(u)

= ngEQ\QoT(6*(q,u)) . 0

Example 7.1.8 Let Q = {qi, q2}, X = {0,1}, 6(qi, 0) = qi = 6(q2, 0), 6(qj,1) = q2 = 6(g2,1) , t(qi) = 0 = t(q2), T (qi) = 0, and T(q2) = 1 . Thus in the proof of Theorem 7.1 .7, Qo = 01 and Fq2 = {1,11,111, . . . } . Theorem 7.1.9 Let A be a Fv -regular language on X. Im(A), A, is a regular language on X.


Then for all c E

328


Proof. Since A is Fv-regular, there exists M = FLv (M) = A. Thus A(u)

(Q, X, S, t, T)

such that

=VgEQ(t(q) AT(6*(q,u)))

for all u E X* . Let c E Im(A) . Let u E Ac . Then A(u) = VgE Q(L(q) A T(S*(q,u))) >_ c. Since Q is finite, there exists qv E Q, such that t(qv) A T(S*(qv,u)) > c. Hence for all u E Ac there exists qv E Q such that t(qv) A T(S*(qv,u)) > c. For all u E A,, let Mq . = (Q, X, S, q., Tc.) where T, _ {q E Q T(q) >_ c} . Let L(Mq.) be the language of Mq . . Let x E L(Mq.) . Then S * (q.,x) E T c and so T(S*(qv,x)) >_ c. Since t(qv) >_ c, t(qv) AT(S*(q.,x)) >_ c. Hence VgE Q(L(q) A T(S*(q, x))) >_ t (q.) A T(S* (q., x)) >_ c. Thus x E A, . Hence L(Mq.) C_ A c . Conversely, let x E A, . Then there exists qx E Q such that t(qx) AT(S*(qx,x)) > c. Thus T(S*(qx,x)) > c and so S * (qx,x) E T, . Hence x E L(Mq,) . Thus A, C UL(Mq,) . Hence A, = UL(Mq,) . Since each L(Mq,) is regular, A, is regular . Theorem 7.1.10 Let A be a finite valued fuzzy subset ofX* . If A, is regular for all c EIm(A), then A is a Fv -regular language on X* . Proof. Let Im(A) =

{CI,

C, . . . ,

A,, C

x`12

ck}

where cl > C > . . . >

ck .

Then

C . . . C Alk .

Let us denote AZ = Ac j , i = 1, 2, . . . , k. Let Mi = (QZ, X, SZ , qZ, FZ) be the dfa for Ai - AZ_ 1, i = 1, 2, . . . , k where Ao = 01 . Let TZ be the characteristic function of FZ U {d} in QZ U {d}, where d is a state such that d ~ QZ, i = 1, 2, . . . , k. Let M = (Q, X, S, t,T), where ~

Q = (Q1 U {d}) X (Q2 U {d}) X . . . b((pl,p2,

x

(Qk U {d}),

. . . , Pk),a) = (61(pl,a), 62(p2,a),

where we set 6i (pi, a) T(pl,p2,

= d

if pi

. . . , Sk(Pk,a)),

2, . . . , k,

=d, i = 1,

. . . , Pk) = T , (pl) AT2(p2) n . . . ATk(pk)

and t(pl, p2,

. . . , Pk) = ~

Ci

. .

if (PI, " 0 otherwise .

,

Pk) = (d,

Let u E X* and A(u) = c2. Then u E Ai VgEQ(t(q) AT(6* (q, u)))


AZ_1 .

. . . , d, qi, d, . . . , d)

Now

= ci n 1 =

c2


329

since t (q) AT(6* (q u)) _ '

Hence A(u) X* . 0

cZ if q = (d,

~ 0 otherwise.

=VaEQ(t(q) AT(6*(q,u))) .

. . . , d, qZ, d, . . . , d)

Thus A is a Fv-regular language on

We illustrate Theorems 7.1 .9 and 7.1 .10 in the next two examples . Example 7.1.11 Let G = (N, T, P, s) be the grammar of Example 1.8.4 .

Then L(G) = L bm abn I m = 0,1. . . . ; n = 1,2. . . . } (see Example 1.8.6 . Define A : {a, b}* ~ [0,1] by Vbmabn E L(G), A(bn'abn) _ .5 if m > 0, A(bmabn) = .9 if m = 0, and A(x) = 0 for all other x E {a, b}* . Then A.5 = L(G) is regular. Now A.9 = {ab n I n = 1, 2. . . . } is regular since it is generated by the productions s aS, S ~ bS, and S ~ b.

Example 7.1.12 Let A be the fuzzy language of Example 7.1 .11.

Then A.9 = {abn I n = 1,2 . . . . } and A .5\A .9 = {bm abn I m, n = 1,2. . . . f . Let M1 = (Q1, X, 61, s, F) and M2 = (Q2, X, 62, s, F), where Q1 = {s, S, 01, F}, Q2 = {s, Sl, S2, 0, F}, X = {a, b}, and Sl and 62 are defined as follows. 61 (s, a)

6 1 (s, b) 61 (S, a) 6 1 (S, b) 61(01, a) 6 1 (0, b)

61 (F, a) 61 (F, b)

= S _

_ = F _ = 01 - 01 = F

62 (s, a)

S1 S2 S1 01

62 62 (SL, a) 62 (SL, b) 62 (S2, a) 6 2(S2, b) 62 (0, a)

F 01

2(0,b) 62 (F, a) 62 (F, b)

01

F.

Then Ml accepts A.9 and M2 accepts A.5\A .9 . Now (Ml U {d}) x (M2 U {d}) has 30 elements . Hence we list only a few of the images of S, T, and t . 6((s, d), a) 6((S, d), b) 6((d, s), a) b ((d, 0), b)

= = = =

(61 (s, a), 62 (d, a)) (61 (S, b), 62 (d, b)) (61 (d, a), 62 (s, a)) (6 1 (d, b), 62 (0, b))

T(F, d) T(d, F) ~(s, d)

(d, s)


=

= _ _

1

1 .9 .5 .

= = =

(S, d) (F, d) (d, 01)

=

(d, 0)

and is=Then a(Ql, FLv by Fv-regular FLv(M2) A1 Theorems 7(M) X, Now Since Ubl,A2= =ti, AT(S*(q,bab))) (A1 A1 = AT(6*(q,ab))) is language A1 Let (Ql Tl) A2and 7aUrl(P,q)EQI (u) A1 A1 Fv-regular XXA2), A2 and Let n (S2 T2((bi(A Q2, A2 and A2)(u)nand and (q, 0A2 M2 on = are X, XQ2 XQ2(tl(p) u)) A2 (u) A2 7Al, X* =Fv-regular b1 be )(tl ATl(6i(p,u))))n AT2(62(q,u)))) AFLv(M2)(u) be language u), (Q2, XU(p) Fv-regular finite A2, Xb2, b2 m(nt2)(p,q) X, (q, tl (s, (d, t2(q)n t2(q) for AT(6*((d, AT(S*((0, AT(O, A0)V( nT(S*((d, AT(F, A1)V( 62, Xvalued u))) d) s) ATS*((d, s) on languages all t2, t2, AT(6*((s,d),ab))V AT(S*((s,d),bab))V AT(S*((d, AT1(61(Au)) X* languages AT((S1 TT2) More F)) cb)) =Fv-regular E T1 Vsuch Sl), S2), 0), V S2), [0,1] (on (on Xb)) Xs), s), ab)) b)) T2) Fuzzy AT(d, b)) on that AT(d, b2)*((p,q),u))) VThe X*, ab)) bab)) V X* languages FLv(MI) there 0)) Languages F)) result Thenexists now A1 on _ rl

330

7.

Now

-

VgEQ(t(q)

_ _ _ Then

_

VgEQ(t(q)

_ _ _ _ _ Theorem X* .

.1.13

Proof. follows Theorem A2 Ml A1

(t(s,d) (t(d, ( .9 ( .5 ( .9 ( .9

.5 .5A0)

(t (t .9AT(S*((O1,S1),ab))V ( .5 ( .9 V( .5 .5 ( .9 .5A1) ( .9 .5 . .

.1.9

.1 .10 .

.1.14

. .

.

Proof. . M

Now FLv(M)(u)

. -

V(p,q)EQ,XQ2((tl V Tl V(P,q)EQ, AT2 (VPEQ,(tl(p) (VgEQz(t2(q) FLv(MI)(u) At (al

Hence © 2002 by Chapman & Hall/CRC

.


331

Theorem 7.1.15 Let A be a Fv -regular language on X* . Then is a FA -regular language on X* . (Q,

Proof.

= 1 -A

Since A is a Fv-regular language on X*, there exists M = = A . Let M' = (Q, X, 6, t', T) where T =

X, 6, t, T) such that FLv(M) 1-T and T=1-t . Now

=

FL A(MC)(u)

= = for all u X* . 0

E

ngEQ(T (q) V T(6*(q,u))) ngEQ(( 1 - t (q)) V ( 1 - T(6* (q, u)))) 1 - VgEQ(t(q) AT(6 * (q, u)))

1_ A(u) A(u)

X* . Hence FLA (MC)

Thus _X is a FA-regular language on

Let M = (Q, X, 6, t,T) be a pfa. Let Q = {ql, g2, - . . , qnj and for all 1 [0,1] by (P-1 T) ( x) for all x

E

X* . For all pi

E

(PI,

t(qi) =

ti

= T(6*(p,x))

Q, 1 [0, 1]

by 01, P2,

" . . ,Pn)-1T)(x)

=Vi--1(Ci

n (Pi

1T)(x))

for all x E X* . Let/C= {(p1, p2, . . . , Pn) -1 T lpi E Q, 1 S([x], a) = [xa]

for all [x]

E Q, a E X, t :Q----> [0,1]

tQxD

for all [x]

E Q

1 if [x] _ [A] 0 if [x] [A]

and T :Q----> [0,1] T([x]) = A(x)

for all [x] E Q . The function S can be extended to S* : [xy] for all [x] E Q, y E X* . © 2002 by Chapman & Hall/CRC

Q x X* ~ Q,

where S* ([x], y) _


333

Let [x], [y] E Q and a E X. Suppose [x] = [y] . Then xRy . Since 1Z is a right congruence, xaRya . Thus [xa] = [ya] and so 6([x], a) = 6([y], a) . Hence 6 is well defined . Now if [x] = [y], then x -1 A = y-1A . Thus (x -1 A)(A) = (y-1A)(A) and so A(x) = A(y) . This implies that T([x]) = T([y]) . Hence T is well defined . Let x E X* . Then FLv(M)(x)

= =

VgEQ(t(q) AT(6*(q,x))) T(6*([A],x))

Hence FL v(M) = A. Thus A is a Fv -regular language of X* . (1)x(3) : The proof of this part is similar to (1)x(2).

m

Lemma 7.1.17 (Pumping Lemma) Let M = (Q, X, 6, t, T) be a pfa. Suppose IQ I = n. Then for all x E X*, I x I >_ n, there exist u, v, w E X such that x =uvw, Iuv I 0 and

FL v (M) (uv'w) = FL v (M) (x) for all m>0.

Proof. Let x = al a2 . . . aP , ai E X, 1 n. Now FLv(M)(x) = VgEQ(t(q) AT(6*(q,x))) .

Let q E Q . Let 6(q, al) = ql, 6(ql, a2) = q2, . . . , 6(gp_1, aP) = qP . Since j, 0 < i < j 0. Consider uv2 w . Now uv 2w = ala2 . . . a2_laiai+1 . . . aj_laiai + 1 . . . aj _ l ajaj+l . . . aP .

Clearly 6* (q, u) = q2, 6* (q2, v) = qj = q2, and 6* (qj, w) = qP . Now 6* (q2, v2) _ 6* (6 * (qj, v), v) = 6* (qj , v) = qj = qi . By induction it follows that 6 * (qj, vm) = 6* (6* (qj, vm-1 ), v) = 6 * (qj, v) = qj = qa

for all m > 1. Hence 6* (q2, uv mw)

= = =

6* (6* (q, u), vmw) 6* (qi , vmw) 6* (6* (qj, vm), w) 6* (qj, w) qP 6* (qj, uvw)


334


for all m > 0. Thus FLv (M)(uv -w)

= =

=

VaEQ(t(q) AT(S*(q,uv-w))) VgEQ(t(q) AT(6*(q,uvw)))

FLv (M) (uvw) FLv(M)(x) . m

Theorem 7.1 .18 Let M = (Q, X, S, t, T) be a pfa. Suppose IQ I = n . Then FLv (M) is nonconstant if and only if there exist W1, W2 E X* such that Iw1I < n and FLv (M)(w2) < FLv(M)(wl) . Proof. Suppose that FLv (M) is nonconstant . Then there exist u, v E X* such that FLv (M) (u) :?~ FLv (M) (v) . Let w E X* be such that Iw I >_ n . Now either FLv (M) (w) zA FLv (M) (u) or FLv (M) (w) zA FLv (M) (v) . Suppose FLv (M) (w) zA FLv (M) (u) . Case 1 : FLv (M) (w) > FLv (M) (u) . Let S= {x E X*I IxI > n and FLv (M)(x) > FLv(M)(u)}. Since w E S, S z,4 0. By the well ordering principle there exists wo E S such that Iwol is smallest. Now Iwol > n and FLv (M)(wo ) > FLv (M)(u) . By the pumping lemma, there exist x, y, z E X* such that wo = xyz, I xy I < n, Iy I > 0, and FLv (M) (xy'z) > FLv(M) (wo) for all i >_ 0. Let wl = xz. Since Iw1I < IwoI , wi ~ S. Since FLv (M)(xz) >_ FLv (M)(wo) > FLv (M)(u) it follows that Iw1I < n. Hence wl and u are the required words . Case 2: FLv (M) (w) < FLv (M) (u) . In this case if Jul < n, then u and w are the required words. Suppose that Jul >_ n. Then proceeding as in Case 1, we can show that there exists words wl and w2 such that Iw1I < n and FLv (M)(w2) < FLv(M)(wl) . The converse is immediate. m

7 .2

On Fuzzy Recognizers

In this section, we define and examine the concept of a fuzzy recognizer . If L(M) is the language recognized by an incomplete fuzzy recognizer M, we show that there is a completion M' of M such that L(M') = L(M), Theorem 7.2.14. We also show that if A is a recognizable set of words, then there is a complete accessible fuzzy recognizer MA such that L(MA) = A, Theorem 7.2.20. Our long-term goal is to determine rational decompositions of recognizable sets. In fact, we wish to determine a decomposition that gives a constructive characterization of a recognizable set . We lay groundwork for this determination by proving Theorems 7.2.28 and 7.2.29 . © 2002 by Chapman & Hall/CRC

7.2. On Fuzzy Recognizers

335

Let M = (Q, X, /t) be a fuzzy finite state machine, where Q and X are nonempty sets and /t is a fuzzy subset of Q x X x Q . Q is called the set of states and X is called the set of input symbols. Let X* denote the set of all strings of finite length over X. Let A, B C_ X* . Then AB = {uv I u E A, v E B} . For x E X*, xA = {x}A and Ax = A{x} . Definition 7.2 .1 Let M = (Q, X, /t) be a ffsm . Let q E Q, x E X*, and ACX* . Define q*x :Q~[0,1] and q*A :Q----> [0,1] by * (q * x) (p) = [t (q, x, p) b'p E Q and q*A = UxEAq*x .

Definition 7.2 .2 Let M = (Q, X, /t) be a ffsm. Let a : Q ----> [0,1], x E X*, and A C X* . Define (1) a * x : Q ----> [0,1] by

(a * x) (p) = V{a(q) A [t * (q, x, p) I q E Q} dpEQ, (2)cx*A :Q~[0,1] a*A = UxEAa*x, (3) a * x-1 : Q ~ [0,1] by (a * x -1 ) (p) = V{o(q) n tt* (p, x, q) I q E Q} dpEQ, A-1 : Q ----> [0,1] (4) a *

(k * A-1 = UxEAC' * x-1

Lemma 7.2 .3 Let M = (Q, X, y) be a ffsm. Let a be a fuzzy subset of Q and A C X* . Then

(a * A) (p) = V{V {a(q) A w* (q, x, p) I q E Q} I x E A},

(a * A- 1) (p) = V {V {a(q) A w* (p, x, q) I q E Q} I x E A} vpEQ .0

Lemma 7.2 .4 Let M = (Q, X, y) be a ffsm and let a : Q ~ [0,1] . Let S = {a * x I x E X*}. Then S is a finite set.



336

Proof. Let x E X* . Then (a*x)(p) = V{a(q)nft* (q, x, p) I q E Q} . Since ft* is finite valued . Also, a is finite valued. is ft finite valued, it follows that It now follows that the number of mappings of the form a * x : Q ~ [0,1] is finite. Hence S is finite . m

Theorem 7.2.5 Let a be a fuzzy subset of Q, x, y E X* and A, B C_ X* . Then

(1) (a * x) * y = a * (xy), (2)(a*A)*y=a*(Ay), (3) (a*A) *B=a* (AB), (4) (a (5) (a (6) (a

x -1 ) y -1 = a (yx) -1 , A-1 )

y-1 = a

A-1 )

B-1 = a

(yA) -1 , (BA) -1 .

Proof. (1) Let p E Q . Then ((a * x) * y) (p) _ = _

V~(a*x)(q) nw * (q,y,p) I qE Qj V{VQa(r) Aw*(r,x,q) I r E Q}) Aw*(q,y,p) gEQI V{a(r) A (V {[t* (r, x, q) A w* (q, y, p) I q E Q}) I rEQ} V{a(r) n ft * (r, xy, p) I r E Q} (a * (xy))(p)-

Hence (a * x) * y = a * (xy) . (2) Let p E Q. Then ((a * A) * y) (p)

V~(a*A)(q) nft* (q,y,p) I qE Qj V{(V{(a* x) (q) I x E A}) A[t*(q,y,p) I qEQ} V{V{(a * x) (q) A w* (q, y, p) I q E Q} I xEA} V{((a * x) * y)(p) I x E A} V{(a * (xy))(p) I x E A} (UxEACI * (xy))(p) (a * (Ay)) (p).

Thus (a*A)*y=a*(Ay) . (3) (a * A) * B = UVEB((a * A) * y) = UVEBa * (Ay) = a * (AB) . © 2002 by Chapman & Hall/CRC

7.2 . On Fuzzy Recognizers

337

(4) Let p E Q. Then

((a * x-1) * y -1 )(p)

_

V{(a * x-1)(q) n ft * (p, y, q) I q E Q} V{(V{a(r) Aw*(q,x,r) I r E Q}) Aft* (p, y, q) I q E Q} V {a(r) A (V {ft* (q, X, r) A w* (p, y, q) IgEQ}) I rEQ} V {a(r) A (V {ft* (p, y, q) A w* (q, x, r) IgEQ}) I rEQ} V {a(r) n ft * (p, yx, r) I r E Q} (a * (yx)-1)(p)

Hence (a * x-1) * y-1 = a * (yx)-1 . (5) Let p E Q. Then ((a * A-1) * y - 1) (p)

=

A-1) V{(a * (q) A w* (p, y, q) I q E Q} x-1) * V {(V{(a (q) I x E A}) (p, y, q E Aft* q) I Q} V{V{(a * x -1 ) (q) A w* (p, y, q) I q E Q} I xEAl V {((a * x-1 ) * y -1 )(p) I x E A} V {(a * (yx) -1 )(p) I x E A} (UxEAOI * (Jx)-1)(p) (a * (yA) -1 ) (p) .

Hence (a * A-1) * y-1 = a * (yA)-1 .

A-1) * B-1 = A-1) * UVEB((a * (6) (a * y-1) = UVEB(a * (yA)-1) -1 . a * (BA) m

Definition 7.2.6 M = (Q, X, ft) be a ffsm. Let a : Q ----> [0,1] and T Q ----> [0,1] . Define

a-1 o T = {x E X* Ia(q) A /t*(q, x, p) A T(p) > 0 for some q, p E Q} .

Definition 7.2.7 Let Q and X be finite subsets. A fuzzy recognizes is a five tuple M = (Q, X, ft, t, C), where

(1) Q is a finite nonempty set of states, (2) X is a finite nonempty set of input symbols, (3) ft : Q x X x Q ~ [0,1] is a function, called the fuzzy transition function, (4) t is a fuzzy subset of Q, i.e ., t : Q [0,1], called the initial fuzzy state, and (5) C is a fuzzy subset of Q, i.e ., C : Q [0,1], called the fuzzy subset of final states .

Clearly, if M = (Q, X, ft, t, C) is a fuzzy recognizer, then N = (Q, X, ft) is a fuzzy finite state machine . We call N the fuzzy finite state machine associated with the fuzzy recognizer M. © 2002 by Chapman & Hall/CRC

338


Definition 7.2 .8 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Let x E X* . Then x is said to be recognized by M if VgEQ(t(q) A(VPEQ{w * (q,x,p) AC(p)})) > 0. Lemma 7.2 .9 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Let x E X* . Then x is recognized if and only if there exists p, q E Q such that t(q) A ft * (q, x, p) n C(p) > 0 . 0 Definition 7.2 .10 Let M = (Q, X, /t, t, C) be a fuzzy recognizer . Let L(M) = {x E X* I x is recognized by M} . L(M) is called the language recognized by the fuzzy recognizer M. Lemma 7.2 .11 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Then L(M) = {x E X* I t(q) A /t*(q, x, p) A C(p) > 0 for some q, p E Q} . m Definition 7.2 .12 Let M = (Q, X, /t) be a fuzzy finite state machine. M is called complete if for all q E Q, a E X, there exists p E Q such that ft (q, a, p) > 0. Definition 7.2 .13 Let M = (Q, X, ft, t, C) be a fuzzy recognizer. Then M is called complete if the fuzzy finite state machine associated with M is complete . Let M = (Q, X, ft, t, C) be a fuzzy recognizer such that M is not complete. Let Qc = Q U {t}, where t is an element such that t Q. For all qEQ,let0<mg [0, 1] by ~c


(p) -

t (p) 0

if p t if p=t .

7.2. On Fuzzy Recognizers

339

Define C' : Qc ----> [0,1] by OP) _

C(p) if p 0

EQ

otherwise .

It is easy to see that the fuzzy recognizer Mc = (Qc, complete. Qc is called a completion of M.

X, /tc, T, Cc)

is


/t, t, C) be an incomplete fuzzy recognizer. Let Mc be a completion of M. Then L(M) = L(Mc) .

Proof. Let x E L(M) . Then 3q, p E Q such that t (p) A /t* (q, x, p) A ~(p) > 0. This implies that tc (p) A ft* (q, x, p) AC(p) > 0. Hence x E L(Mc). Thus L(M) C_ L(Mc) . Now let x E L(Mc). There exists q, p E_ Q such that tc(p) A ft* (q, x, p) A C(p) > 0. This implies that tc(p) > 0. Thus p E Q and so t(p) = tc(p) . Suppose /t* (q, x, p) = 0. Since f,* (q, x, p) > 0 and /t* (q, x, p) = 0, we must have p = t . This is a contradiction since p E Q. Hence /t* (q, x, p) > 0. Since p E Q, C(p) = ~(p) > 0 . Thus t(p) A/t* (q, x, p) A C(p) > 0 and so x E L(M) . Consequently, L(M) = L(Mc) . m Definition 7.2.15 Let M = (Q, X,

/t, t,

C)

be a fuzzy recognizer. Let

S = {q E Q I V {V{L(p) Aw*(p,x, q) I p E Q} I x E X*} > 0}.

Definition 7.2.16 Let M = (Q, X, is called accessible if S = Q.

/t, t, C) be a fuzzy recognizer . Then M

Theorem 7.2.17 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Then M is accessible if and only if (t * X*) (q) > 0 b'q E Q . m


/t, t, C) be a fuzzy recognizer. Then = 1 t1 o . C ( ) L(M) (2) Let x E X* . Let A = L(M) . Then

x-'A = (t * x) -1 0 ~, where x-'A = {y E X* I xy E A} .

Proof. (1) The proof is straightforward . (2) Let y E (t * x) -1 o C. Then (t * x) (q) A /t* (q, y, p) A C(p) > 0 for some q, p E Q. This implies that (t * x) (q) > 0 and ft * (q, y, p) A C (p) > 0. Now (t * x) (q) = VTEQ{t(r) A /t* (r, x, q) } > 0 and so t(r) A /t* (r, x, q) > 0 for some r E Q . Thus we have t(r) A /t* (r, x, q) A /t* (q, y, p) A C (p) > 0 for some q, p, r E Q . This implies that t(r) A ft * (r, xy, p) A C (p) > 0 for some p, r E Q. Thus xy E A and so y E x- 'A . Now let y E x-'A . Then xy E A and so there exists p, r E Q such that t(r) n[t* (r, xy, p) AC(p) > 0. This implies that [t*(r,xy,p) > 0. Now y,*(r,xy,p) =VaEQ{[t*(r,x,q)nft*(q,y,p)} > 0. Thus there exists q E Q such that /t* (r, x, q) A /t* (q, y, p) > 0, i.e ., /t* (r, x, q) > 0 and ,t* (q, y, p) > 0 . Now (t * x) (q) = v,EQ{t(s) n w* (s, x, q)} >_ t (r) n y,* (r, x, q) > 0. Hence (t * x) (q) A /t* (q, y, p) A C (p) > 0 for some q, p E Q. Thus y E (t * x) -1 o C. Hence x -'A = (t * x) -1 o C. m © 2002 by Chapman & Hall/CRC

complete Now a=is Let Lemma MA complete QA {(L let well LA(A) =(xa)-'A and xEy-'A (x-'A, *= xXX A QA Ex)-1 defined M Esuch x, 7(QA, aaccessible A A There 77L(MA) Let x-'A) XQA~ = [t* Esuch accessible Define oT Then a,that QA, QA X, = by-'A) (A, IS exists Now y-'A that Let By ltA, xx, Let L(MA) = aThere C(x-'A) Cfuzzy =a, E x-1A) E[0,1] Theorem A {x-1A LA, A X*} A n y-'A) recognizer QA = X (xa)-'A {L ifabe 0mAAmAA Na C =C)(u-'A, recognizer *and fuzzy exist by Define = ~ L(M) ais X* (y-1A, x= AC(x-1A) _ Next AIrecognizable = finite = [0,1] Ixonly 7al, CA {y-1A, Let xThen recognizer = Eb,LA 0a1 >E we X*} x, v-'A) by a-1(x-'A) x-'A, if Hence 0MA 1A) an-1)-'A, X*} z-1A) Ax-'A QA show (ua)-'A z-1A >Let otherwise isLet otherwise, nsuch 0a3, x-1A xsubset ~ y-'A, called is Then (xa)-'A QA NA(al xM This En that E=(a,a2a3)-'A) 0[0,1] finite = A C(z-1A) QA that = < (L* More = = is = aja2 an, recognizable u-'A, of implies MA, MA x-'A a1A, (Q, b-1(u-'A) v-'A A by x)-1 such X* L(MA) finite (a1a2 =on X, It a2, isnA, y-'A v-1A > oT =now an, that aFuzzy Then /t, that (a1a2)-1A) This set 0u-'A, complete CA t,=a2an) for A T) A follows xEClearly = <E there shows all Languages if3 QA, E such AX -'A) 1(ub)-'A xy-'A L(MA) aDefine Ea, for exists fuzzy fuzzy that that X* MA ballE

340

7.

Definition .2.19 recognizer Theorem a

.

.

.2.20

.

Proof. L(M) . : NA

. MA 0

ItA(X-'A, b'x-'A,

.

.

nA

.

if otherwise,

CA

if

.

.

~O .

.

.2 .18,

.2.4,

.

. >_ = > . LA(y- 1A)


= .

.

.

ftA(A,

Thus Now

if

: C(x_1A)

Set recognizer X. v-1A, Hence ltA By D is Let i.

.

:

LA(X_'A) b'x-'A

.

.

.

. .

. .. .

[tA(A, ANA((a,a2)-1A, NA((aia2 . . . . . .AmA mA .

. ..

.

. ..

. .

and p [t* general, and finite (u-1A, EX*, u,[x] all all Erelations on xso On if(u-'A, xvQ) so x, Thus LA(y-'A) uyv x, denote x-'A A M only u-'A, X*1 is Eto EX* with Now Fuzzy on yyxyX*) index is finite i=1,2, Then is of y, L(M) = constructed the if7 EEX* Ethe E(1)x(2) by the y, v-1A) recognizable -A respect LA, by and if A = A (QA, X*, Suppose X*, Since A v-lA relation the v-'A) Recognizers converse for Then uyv = z-'A by -Clearly, for the union itis Consequently, and ifx, only x>xis *U{ X, follows Theorem equivalence all finite, and Let all > X*1 -A y(A, 0, definition to aEEV'x, ft, -A [x] Since by ,0> congruence ifn} Ex, of yu, lta(y-'A, A in -A A -A x, QA, yA t,Hence only 0may (ft* for yif I-A is vthe X* congruence be C_ z-'A) C) the if are where xfor that Eaand Let EThus A X* aall and (q, 6(ux)-'A E definition X*, is congruence The if X*) not proof X*, and equivalent recognizable all is of Q x, A}, class y-1A xyA finite, x, only only X*1 recognizable, = The p) ltA, x, xu-'A, = Erelation be yrelation (ux)-'A xxIt X*1 [t* of where yz-1A) EL(M) X*I L(MA) > true, if with -A is-A following classes if = ltA(u-'A, -A y-A (y-'A, X*, = The 0(ft* Theorem of easily (uxv -A A v-'A if ifof yv-'A laA yrespect denotes iby subset and is Thus and if [x] > (u-'A, Thus if converse Let finite = =aof is 0E defined xThen x, Theorem and 0, seen is Hence Econgruence for finite {[x] ifonly A only (uy)-'A aassertions z-'A) 7A x, QA) if and X*1 the xof congruence index y,the if x, v-'A) only the to C_Iand that -A X* Vu, and if then ifv-'A) xis on C(x-'A) equivalence C(z-'A) X* --A= xDefine fuzzy ft* Eset y, 6>von trivial X* if -A Let Define only -A only X*} (q, Vu 0> xLet then Eof Define (uxv are X* relation -A MA by is > 0yy,X*, recognizer X*1 all Eif if relation Also, -A = 0ap) if m A equivalent Sincexfor >uyv For y(uy)-'A Eand congruence be X* C(z-'A) if congruence = > class a0uxv tion However, all A the and yxifU{[aj] 0EBy relation of only xEfor if X* Hence x,Since of A fuzzy finite Eis with MA, only X*, and yX* the for all y, by of >Eif A =

7.2.

341

Hence definition

.

. [

and 0

.

.

Let X*, . q, then -A only relation in

.2 .7. .

. .

.

Theorem recognizer for if for all

.2.21

Proof. if for v-'A . [t*

Let

Theorem (1) (2) classes (3) of

.2.22

Proof. the X*1 (2)x(3) : index . respect (3)x(1) : let ai

.

.

.

.e., .2.20.

.

.

. .

. .

. .

.

.

.

.

.

.

. . : .2 .8,

.

.

.

. .. .

.


.

. .


342

finite index, Q is a finite set. Let 0 < m, n, c 0. Now a = ul . . . u2 j for some ij. Let (12)a = s2j-1 (s2j-2( . . .s2°(12) . . .)) . Then (wIAw2)* ((q, 0), ab, (q",Rab))

=

>

V{(wIAw2)*((q, 0), a, (v, R)) n(N'1AN'2)*((v, R), b, (q",Rab)) (v, R) E Q1 X 'P(Q2)1 (wlow2)*((q, 0), a, (q' , (12) a)) n(N'10N'2)*((q~, (12)a), b, (q", Rab)) 0

by the definition of (12)a and since Rab is a continuation of (12)a, i.e ., Rab = ((12)a)b " Conversely, let x E L(M10M2) . Then L ° (q,

O) n (wlow2)*((q, 0), x, (q ", (12)x)) > 0,

where fti(q, x, q") > 0 and (12)x is defined as follows: we have that x = uv, where u = i o . . . ukll and where the uj and v are defined as above (k exists since x E L(M1 0M2 )) . Then (12)x is the right-hand side of equation (7 .1) . Since x E L(M10M2), (ft1 0ft2) * ((q, 0), x, (q", (12)x)) > 0 and so 3j, EIP E 12, s2 2 Thus

~+1 . . .U1~+1v'Tj+1 . . .v,mv(p)

u2j+ 1 . . . u2j+1 u2j+1

. ..

ukv

E (12 ) x

nT2 "

E B . Since ul . . . u2 , E A, x E A " B.

m

Theorem 7.2 .29 Let A C X* be recognizable . Then A* is recognizable.


34 6

7. More

Proof. Let

M = (Q, X, /t, t,

X x P(Q) ----> [0,1]

,t'(P, ,t'(P,

as follows :

on Fuzzy Languages

C) be a recognizer of A. Define

/t' : P(Q) x

c, s'(P)) = V{w(p, c, q)

I p E

P,

q E s'(P) I if s , (P) n T = 0

c, q)

I p E

P,

q E

c, sc(P) U I)

= v{w(p, It'(P,

sc(P)

U I} if

sc(P)

n T z,4

0

c, P') = 0 otherwise,

where c E X, P, P E P(Q), T = {q E Q I C(q) > 0}, I = {q E Q I t(q) > 0}, and sc(P) = {q E Q I ft(p, c, q) > 0, p E P} . Define C' : P(Q) ----> [0,1] by C'(P) = V{C(p) I p E P} and t' : P(Q) [0,1] by t'(I) = V{ t(q) I q E I}, L' is 0 otherwise, T' = {P E P(Q) I C(P) > o} = {P E P(Q) I P n T :A 0} . Let M' = (P(Q), X, ft', t', C') . Let x E A* . Then x = al a2 . . . a , where a2 E A and if a2 = a21 . . . ail,, for a2,, . , ail,, E X, 3" m2 < n2 such that . . . . . . , n. Let Ix = sa,, ( . . . sae (sal (I) U I) U . . . ) U I . = a21 a2-E A, i 1, 2, > l,s a l ( Now p' (I, x, Ix) ft* (I,a I)UI)A N'*(,a , (I)UI,a2 ,sa2(sal(I)UI)U I)n . . . n t'* (8 a,,-1 (. . . sae (,a1 (I) U I) U . . . ) U I, an , Ia) > 0 since 3i E I, 3p E sat (I) n T such that /t* (i, al, p) > 0 (because al E L(M)), 3i' E I, 3p' E sae (sal (I) U I) n T such that /t* (i', a2,p') > 0 (because a2 E L(M)) and so on. Thus x E L(M) . Hence A* C_ L(M) . Let y E L(M) . Then 3P E T' such that y'* (I, y, P) > 0. Now suppose 3k such that y = ul . . . UZ1UZ1+1 . . . UZj u2j +1 . . . UZkv,

where ul . . . uz1 E A with i l smallest and the ii are the smallest such that u2j-1+1 . . . u2j A, j = 2, . . . , k. Let ul = ul . . . uz1 and uj E u2j-1+1 . . . u2j, j = 2, . . . , k. Then ul . . . u2j-1 E A* and P = (I) ., where (I)~ = 81 (8'k( . . . s -2 ( 8 -1 (1) U I) U . . . I) U I) U I . There exists j, 3q E I such that /t* (q, uj . . . ukv, p) > 0 for some p E (i) y nT . Thus ft'*((I)~1 .. .~--1, uj . . . ukv, (I) y) > 0. Hence uj . . . ukv E A* . If no such k exists, then a similar argument shows that y E A. Thus y E A* . Hence L(M') C A* . Consequently L(M) = A* . m If L(M) is the language recognized by an incomplete fuzzy recognizer, we showed that there is a completion Mc of M such that L(Mc) = L(M) . We also showed that if A is a recognizable set of words, then there is a complete accessible fuzzy recognizer MA such that L(MA) = A. If A and B are recognizable sets of words, then A - B and A* are recognizable . These results are significant in that they lay the groundwork for determining methods of decomposing recognizable sets and thus for giving a constructive characterization of recognizable sets. In particular, we hope to show an analog of Kleene's result for fuzzy finite state machines, namely, that the class of recognizable subsets of X* equals the class of all regular subsets of X* . © 2002 by Chapman & Hall/CRC

7.3 . Minimal Fuzzy Recognizers 7 .3

34 7

Minimal Fuzzy Recognizers

In this section, we show that for any fuzzy recognizer M there is a deterministic fuzzy recognizer Md with the same behavior, Theorem 7.3 .8 . Then we show that there is complete accessible deterministic fuzzy recognizer MdA with the same behavior as M and Md and that is minimal, Theorem 7.3.11. Our long term goal is to develop methods of decomposing a recognizable set of a fuzzy finite state machine . One method would be to follow along the lines of Kleene to give a constructive characterization of a recognizable set . In this section, we lay the foundation for the accomplishment of our goal . Clearly, if M = (Q, X, ft, t, C) is a fuzzy recognizer, then N = (Q, X, ft) is a fuzzy finite state machine . We call N the fuzzy finite state machine associated with the fuzzy recognizer M. Definition 7.3 .1 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Let

S = {q E Q I V {V{L(p) n [t * (p, x, q) I p E Q} I x E X*} > 0} . Definition 7.3 .2 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Then M

is called accessible if S = Q.

Definition 7.3 .3 Let M = (Q, X, /t) be a ffsm. Let a : Q ----> [0,1], x E X*,

and A C X* . Define (1) a * x : Q ----> [0,1] by

* (a * x) (p) = V{a(q) n ft (q, x, p) I q E Q} VP E Q, (2)a*A :Q----> [0,1] a*A=UxEAa*x. Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Then by Theorem 7.2 .17, M is accessible if and only if (t * X*) (q) > 0 b'q E Q . Definition 7.3 .4 Let A C_ X* . Then A is called recognizable if 3 a fuzzy

recognizer M such that A = L(M) .

Let Q and X be finite nonempty sets and let /t : Q x X x Q ---> [0,1]. /t is called a fuzzy function of Q x X into Q if for all q E Q, a E X, if ft(q, a, p) > 0 and ft(q, a,p~) > 0 for some p, p' E Q, then p = p~ . Theorem 7.3 .5 Let M = (Q, X, /t, t, T) be a fuzzy recognizer. Then /t is

a fuzzy function of Q x X into Q if and only if /t* is a fuzzy function of Q x X* into Q.



34 8

Proof. Suppose ft is a fuzzy function. Let q E Q and x E X*. Suppose p* (q, x, p) > 0 and /t* (q, x, p') > 0 for some p, p' E Q. If x = A, then p = q = p' . Suppose x z,4 A. Let x = ala2 . . . an E X*, a2 E X. There exists gi~g2~ . . . ,qn-l~gi~q2, . . . q'n -1 E Q such that ft(g, al, gl) nft(gl, a2, g2) n . . .A /t(gn-l, an,p) >Oand/ (g, ai, gi)nft(gi, a2, g2)n . . .h/t (gn-1, an, p~) > 0. This implies that ft(q, al, ql) > 0, ft(gi, a2+1, qZ+1) > 0, i = 1, 2, . . . , n-2, 1, 2, . . . , n - 2, ft(gn-1, an,p) > 0, ft(q, a,, qi) > 0, ft(q', ai+l, q2+1) > 0, i 0. Now 0 and ft (q, 0. Since ft is a ft(gn_1, an,p') > P(q, al, ql) > al, qi) > fuzzy function, ql = ql . Suppose qj = qj , j = 1, 2, . . . , i, i < n - 2. Now ft(gi, a2+1, qZ+1) > 0 and ft(gi, a2+,, q2+1) > 0 implies that qZ+1 = qZ+1 since ft is a fuzzy function . Hence by induction qj = qj , j = 1, 2, . . . , n-1 . Hence * f (qn-1, an,p) > 0 and f (qn-1, an, P') > 0 implies that p = p' . Thus ft is a fuzzy function . The converse is trivial. m

Definition 7.3.6 A deterministic fuzzy necognizen is a fuzzy recog-

nizer Md = (Qd, X, ft, t, T) such that (1) there exists a unique so E Qd such that t(so) > 0 ; so is called the initial state, (2) ft is a fuzzy function of Q x X into Q, and * (3) for all x E X*, there exists a unique qx E Qd such that ft (so, x, qx ) > 0.

Let Md = (Qd, X, ft, t, T) be a deterministic fuzzy recognizer. Let Cd = I T(d) > 0} . Cd is called the set of final states of Md .

{q E Qd

Theorem 7.3.7 Let M = (Q, X, ft, t, T) be a fuzzy recognizer. Suppose M is complete and ft is a fuzzy function of Q x X into Q. Let so E Q. Then the following are equivalent. (1) For all a E X, there exists a unique qa E Q such that ft(so, a, qa) > 0 . (2) For all x E X*, there exists a unique qx E Q such that ft * (so, x, qx) > 0.

Proof. (1)x(2) : Let x E X* and Ixl = n. If x = A, i .e., n = 0, then * * ft (s o , A, s o ) = 1 > 0 and if ft (s o , A, p) > 0, then by the definition of ft * , so = p. Suppose the result is true for all y E X* such that jyj < Ixl , where Ixl = n >_ 1 . Let x = ya, where y E X*, a E X, jyj = n - 1. By the induction hypothesis, there exists a unique q. E Q such that ft * (so, y, q,) > 0. Since M is complete, there exists p E Q such that ft (q., a, p) > 0. Thus * * ft (s o , x, p) >_ ft (s o , y, q,) A ft (q., a, p) > 0. Since ft is a fuzzy function of Q x X into Q, ft * is a fuzzy function of Q x X* into Q by Theorem 7.3.5 . Thus if ft * (so, x, p) > 0 and ft * (so, x, p') > 0 for some p,p' E Q, then p = p' . It now follows that there exists a unique qx E Q such that ft* (so, x, qx) > 0 . (2)x(1) : Immediate . m

Theorem 7.3.8 For each fuzzy recognizer Mn = (Q, X, ft, t,

one can construct a deterministic fuzzy recognizer Md = (Qd, X, ft, t, ~) such that L(Md) = L(M.) .


7.3 . Minimal Fuzzy Recognizers

34 9

Proof. For all x E X*, set Qx =

{q'

E Q I Elq E Q such that t(q) A /t* (q, x, q') > 0} .

Then QA = {q E Q I t(q) > 0} .

Let Qd

= LQx I x E X*} .

2 l`wx) _

Define i : Qd ~ [0,1] by VQx E Qd, V{t(q) I q E QA}

0

if x = A if x :?~ A

Let Cd = {Qx E Qd I C(q) > 0 for some q E Qx }. Define T : Cd VQx E Cd, T(Qx) = V{C(q) I q E Qx} . Define v : Qd X X X Qd d(Q,, a, Qx) E Qd X X X Qd, v(Q,, a, Qx) =

[0,1] by [0,1] by

V{ft* (q, y, q') A ft(q', a, r) I q E Qy, q' E Q, r E Qya} if x = ya

0

otherwise .

Let Md = (Qd, X, v, i,T) . We now show that L (M.) = L(Md) . Now x E L(Mn) if and only if t(q)nft* (q, x, q')AC(q') > 0 for some q, q' E Q if and only if C (q') > 0 for some q' E Qx if and only if T(Qx) > 0. It suffices to show that v* (QA, x, Qx) > 0 for then x E L(Md) if and only if i(QA) A v*(QA,x,Qx) AT(Qx ) > 0 if and only if T(Qx) > 0 (since i(QA) > 0 and v* (QA, x, Qx) > 0) if and only if x E L(Mn) . We show v* (QA, x, Qx ) > 0 by induction on Ix I . Suppose Ix I = 0 . Then x = A and v* (QA, A, QA) = 1 > 0. Suppose Ixj > 1 and the result is true for all y E X* such that Iyj < Ixj . Let x = ya, where a E X. Then v* (QA, x, Qx) = V{v* (QA, y, r) A v(r, a, Qx) I r E Qd } and v* (QA, y, Qv) > 0 by the induction hypothesis. Hence it suffices to show that v(QV, a, Qx) > 0, but the latter inequality is true by the definition of v since x = ya . m Let Md = (Qd, X, ft, t, T) be a complete accessible deterministic fuzzy recognizer and let so denote the initial state of Md . For x E X*, we let qx E Qd denote the unique state such that ft* (so , x, qx) > 0 . Theorem 7.3.9 Let Md = (Qd, X, ft, t, T) be a complete accessible deter-

ministic fuzzy recognizer. Let so be the initial state of Md . For all q E Qd, let q-1 o T = {y E X* I y,* (q, y, p) A T(p) > 0 for some p E Qd}. Let A= L(Md) . (1) For all x E X*, x-1 A= qx 1 oT . (2) Let q E Qd . Then there exists x E X* such that q-1 oT = x-1 A and q=qx (3) Let q E Qd be such that T(q) > 0. Then q-1 o T = x -1 A for some xEA . (4) A=A-1 A=so 1 o T . (5) Let x = a1a2 . . . an E X*, where a2 E X, i = 1, 2, . . . , n. Then

/a'*(so,x,gx) = ft(so,a,,gal) Aft(gal,a2,qalaz) A . . . Aft(ga l . . .a,,_l ,an,qx) .



35 0

Proof. (1) Let y E x-1A . Then xy E A and so t(so) A /t* (so , xy, p) A > 0 for some p E Qd . This implies that ft * (so, xy, p) > 0. Thus there exists q E Qd such that /t* (so, x, q) A /t* (q, y, p) > 0. Hence /t* (so , x, q) > 0 . Since Md is deterministic, q = qx . Hence t(so) A/t*(so, x, qx)A /t*(qx , y, p) A 1 1 1 T(p) > 0. Thus y E qx o T. Hence x-1A C_ qx o T. Now let y E qx o T . Then ft * (gx,y,p) AT(p) > 0 for some p E Qd. This implies that t(so)A la' * (qx, y, p) n T (p) > 0. Now [t* (so, x, qx) > 0. Hence t(so)A [t* (80, x, qx) A y,* (qx , y, p) AT(p) > 0 for some p E Qd. Thus t(so) A /t* (so , xy,p) AT(p) > 0 for some p E Qd. Hence xy E A or y E x-1A . It now follows that x-1A = qx 1 o T. (2) Let q E Qd . Since Md is accessible, there exists x E X* such that * (so, x, q) > 0. Since Md is deterministic, it follows that q = qx . By (1), ft x-1A = qx 1 o T = q-1 o T . (3) By (2), q-1 oT = x -1A for some x E X* . Since T(q) > 0, /t* (q, A, q) A T(q) >0andsoAEq-1 oT . Thus AEx-1A and so xEA . (4) Let y E A. Then t(so) A/t*(so, y, q) AT(q) > 0 for some q E Qd. This implies that ft* (so , y, q) A T(q) > 0 for some q E Qd and so y E so1 o T . Thus A C sot o T. Now let y E sot oT. Then W (so, y, q) AT(q) > 0 for some q E Qd. Thus t(so) A [t* (so, y, q) AT(q) > 0 for some q E Qd and so y E A . It now follows that A = so t o T. (5) Now N'* (s0, x, qx) = V {ft(s0, al, ql) n ft(gl, a2, q2) . . . Aft (qn-1, an, qx) ql ~ q2, . . . ~ qn-1 E Qd} . Since Qd is finite, there exists q1, q2, . . . , qn-1 E Qd such that y,*(so, x, qx ) = y,(so, al, ql)n ft(gl, a2, q2) . . . n ft (gn-1, an, qx) . Since /t*(so, x, qx) > 0, ft(so, al, ql) n ft(gl, a2, q2) n ft(g2, a3, q3)n . . . A N(qn-1, an, qx) > 0. This implies that ft(so, a1, ql) > 0 and so q1 = qal since Md is deterministic . Now ft(so, a1, qa l )A f (qal, a2, q2) > 0 implies that [t* (so, a1a2 , q2 ) > 0 and since Md is deterministic q2 = gala, . We see that an argument by induction will yield qZ = gal ...ai for all i = 1, 2, . . . , n - 1 . T(p)

Theorem 7.3.10 Let A be a recognizable subset of X* . Then there exists a complete accessible deterministic fuzzy recognizer MdA such that L(MdA) _ A. Proof. Let Md = (Qd, X, ft, t, T) be a deterministic fuzzy recognizer such that A = L(Md) . Since Md is deterministic, there exists a unique so E Qd such that t(so) > 0. Let QdA = {x-1 A I x E X*}. Let MA =V{ft(q,a,p) I q,p E Q, a E X} . Then 1 >_ MA > 0. Let 0 < to 0 and LdA(x- 'A) = 0 if A :?~ x- 'A . Thus MdA has a unique initial state. Let x- 'A, y-'A, u- 'A, v-l A E QdA, a, b E X. Let (x- 'A, a, y- 'A) _ (u -'A, b, v-'A) . Then x- 'A = u-'A, y- 'A = v-'A, and a = b. Now (xa) -'A = a-'(x- 'A) = b- '(u -'A) = (ub) -'A . Hence (xa) - 'A = y-'A if and only if (ua) -l A = v - 'A . This shows that PdA is well defined . Let x -I A, y - IA, z-IA E QdA and ltdA(x-'A, a, y- 'A) > 0 and PdA(x-' A, a, z-'A) > 0 .

Then y- 'A = (xa) - 'A = z-'A . Hence MdA is deterministic . By Theorem 7.3.9, QdA is a finite set . Clearly MdA is a complete accessible fuzzy recognizer. Let x E A. Then TdA(x-'A) = CA > 0. Let x = ala2 . . . an, a2 E X for all i . Now ltdA(A, x, x -' A)

>_ = = >

ladA(A, al, al'A) A ttdA(al 1A , a2, (ala2) -'A)A PdA((a,a2)`A, a2, (a,a2a3)`A) A . . . nltdA((ala2 . . . an_l) -l A,an,(ala2 . . . an) - 'A) mAAmAA . . .AmA mA 0.

Thus LdA(A) A ltdA(A, x, x -1 A) A TdA(x -1 A) > 0. This implies that x E L(MdA) . Now let x E L(MdA) . There exists z- 'A E QdA such that LdA(A) A ltdA(A, x, z- I A) ATdA(z -'A) > 0 . Thus LdA(A) > 0, ttdA(A, x, z- 'A) > 0, and TdA(z-'A) > 0. Now ltdA(A, x, z- 'A) > 0 implies that x - IA = z-IA by the definition of ltdA . Hence TdA(x-'A) = TdA(z -'A) > 0 and so x E A. Consequently, A = L(MdA) . 0 Theorem 7.3.11 Let A C X* be recognizable . Suppose Md = (Qd, X, ft,

t, T) is a complete accessible deterministic fuzzy recognizer with behavior A. Let MdA be the complete accessible deterministic fuzzy recognizer as constructed in the proof of Theorem 7.3.10. Then 3 a function f : Qd ~ QdA such that


352

7. More on Fuzzy Languages (1)

(2) {X - ' A (3) (4)

f (so) = A ; f -' (FdA) = Fd, where Fd = {q E Qd I T(q) > 0} E QdA I TdA(x - 'A) > 0} ; ltdA(f (q), a, f (p)) >_ ft (q, a, p) for all p, q E Qd, a E X ; f is surjective .

and

FdA =

Proof. Define f : Qd ~ QdA by b'q E Qd, f (q) = q -1 o T . By Theorem 7.3.9(3), f maps Qd into QdA . (1) f (s o) = so' o T = A by Theorem 7.3.9(4) . (2) Let q E Fd . Then T(q) > 0. There exists t E A such that q-' o T = t - 'A by Theorem 7.3 .9(3) . Now t E A = L(MdA ) . Hence wdA(A, t, y 'A) A TdA(y 'A) > 0 for some y-1 A . This implies that wdA(A, t, y - 1 A) > 0 and TdA(y - 'A) > 0. Now wdA(A, t, y - 'A) > 0 implies that t -1 A = y-1A and TdA(y 'A) > 0 implies that y-1 A E FdA . Hence f (q) = q-' o T = t - 'A = y - 'A E FdA, i .e ., q E f - '(FdA) . Suppose q E f - '(FdA) . Then TdA(q - ' o T) = TdA(f (q)) > 0. Now q - ' o T = t- 'A for some t E X* . Thus TdA(t - 'A) > 0 and by the definition of TdA it follows that t E A . Since t E A, A E t - 'A = q -1 o T . Hence /t* (q, A, r) A T (r) > 0 for some r E Qd . This implies that q = r . Thus T(q) > 0 and so q E Fd . Consequently f _' (FdA) = Fd . (3) Let q, p E Qd . Now f (q) = q -1 o T and f (q) = p -1 o T . By Theorem 7.3.9, there exists x, y E X* such that q -1 oT = x - 'A and p-' oT = y-'A . Let a E X. Suppose lltdA(f (q), a, f (p)) = 0. Then ltdA(x-'A, a, y - 1 A) = 0 and so (xa) - 'A z,4 y -1 A . We claim that ft (q, a, p) = 0. Suppose ft(q, a, p) > 0. Either there exists t E (xa) - 'A such that t y - 'A or there exists t E y - 'A such that t ~ (xa) - 'A . First suppose that there exists t E (xa) - 'A such that t ~ y - 'A. Since xat E A, /t* (so, xat, r) A T (r) > 0 for some r E Qd . Thus ft* (so, xat, r) > 0. This implies that there exists q', p' E Qd such that /t* (so, x, q') A N (q', a, p') A /t* (p', t, r) > 0 . Since Md is deterministic, q' = q (note this follows from the proof of Theorem 7.3.9 by the choice of x) . Thus ft (q, a, p') > 0 and since Md is deterministic, it follows that p = p' . Hence ft * (p, t, r) > 0 . By the choice of y (as in the proof of Theorem 7.3.9), ft* (so , y, p) > 0. It now follows that ft* (so , y, p) A y* (p, t, r) A T(r) > 0 and so t E y - 1 A, a contradiction. Now suppose there exists t E y - 'A such that t ~ (xa) -'A . Then xat ~ A . Now yt E A . Thus /t* (so, yt, r) A T (r) > 0 for some r E Qd . Thus there exists p" E Qd such that ft * (so, y, p") A ft * (p", t, r) > 0. From this it follows that p = p" . Thus /t* (p, t, r) > 0. Hence y,* (so, x, q) A ft (q, a, p) A /t* (p, t, r) A T (r) > 0 and so xat E A, a contradiction. Hence ft (q, a, p) = 0 . From the definition of /tdA, it now follows that lltdA(f (q), a, f (p)) > ft (q, a, p) . (4) Let x- ' A E QdA . By Theorem 7.3 .9, x-' A = qx' o T . Thus f is surjective. We regard the recognizer MdA as being a minimal complete recognizer of the recognizable subset A, where the term "minimal" refers to © 2002 by Chapman & Hall/CRC

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353

the properties described in Theorem 7.3 .11 ; in particular, (4) implies that IQdAI [0,1] is defined by b'q E Q, p * x (q) = ft * (p, x, q) .

N9. p * A : Q ----> [0,1] is defined by b'q E Q, p * A(q) = V {ft*(p, y, q)

I y

E A} .

N10 . 6 * x : Q ----> [0,1] is defined by b'q E Q, 6 * x(q) = V{6(p) n ft* (p, x, q)

I p

E Q} .

N11 . 6 * A : Q ----> [0,1] is defined by b'q E Q, 6 * A(q) = V{6(p) A w* (p, x, q) I p E Q, x E A} .

N12 . 6 * x-1 : Q ----> [0,1] is defined by b'q E Q, 6*

x-1(q)

= V{6(p)

n ft * (q, x, p) I p

E Q} .

N13 . 6 * A-1 : Q ----> [0,1] is defined by b'q E Q, 6 * A-1 (q) = V{6(p) A w*(q, x,p)

Ip

E Q, x E A} .

Clearly p*A= U,EAP*Y, 6*A= UyEA6*y, and 6*A-1 = UvEA6*Y-1 . The following results are easily proven . Theorem 7.5.1 Let 6 : Q ----> [0,1], x, y E X*, and A, B C X* . Then (1) (6 * x) * y = 6 * (xy) ; (2) (6*A)*y=6*(Ay) ; (3) (6 * x) * B = 6 * (xB) ; (4) (6*A)*B=6*(AB) ; ; (5) (6 * x-1) * y -1 = 6 * (yx)-1 A-1) y-1 * = 6 * (yA)-1 ; (6) (6 * x-1) -1 * B = 6 * (Bx)-1 ; (7) (6 * A-1) * B-1 = 6 * (BA)-1 . (8) (6 *

Theorem 7.5.2 Let 6, 61, 62 be fuzzy subsets of Q and let x, y E X* . Then the following properties hold: (1) 6*A=6; (2) (61U62)*x=(61* x)U(6 2 *x) ; (3) (61n 62)*x=( 6 1* x)n(62*x) ; (4) ( 6 1 U 62) * x -1 = ( 6 1 * x -1 ) U (62 * x (5) (61 n 62) * x -1 = (61 * x -1 ) n (62 * x (6) x0 * x = x0, where x0 is the characteristic function of 01 in Q; (7) xQ * x = UpEQp * x, where xQ is the characteristic function of Q. 0 © 2002 by Chapman & Hall/CRC

356

7. More on Fuzzy Languages Let S : Q ----> [0,1], 'Y : Q ----> [0,1] . N14 . Define 6#-y : X* ----> [0,1] by b'x E X*, 6#y(x) = V {6(p) A y(q) A ft* (p, x, q) I p, q E Q}. N15 . S-1 o/-, ,y = {x E X* I S * y(x) > 0} . When ft is understood, we sometimes write S -1 o 'Y for S -1

oN 'Y .

Theorem 7.5.3 Let S : Q ---> [0,1] and x E X* . Then S * XQ(x) > 0 if and only if 3q E Q such that S * x(q) > 0. Proof. 6 * XQ (x) > 0 ~ V{6(p) A XQ(q) A ft*(p, x, q) I p, q E Q} > 0 ~ V{6(p) A w* (p, x, q) I p, q E Q} > 0 ~ V{6(p) A w* (p, x, q) I p E Q} > 0 for somegEQ~6*x(q)>0. Theorem 7.5.4 Let S and 'Y be fuzzy subsets of Q, q E Q, A, B C_ X*, and x, y E X* . Then the following properties hold: -1 ; (1/ b-1 o ('y * A-1) _ (b -1 o'y)A (2) b -1 o ('y * x-1) _ (b -1 o y)x -1 i (3) (6 A) -t o -y A -'(6-1 o -y) ; (4) (b * x) -1 o r}' = x -1 (b -1 o r}') ; (5) (q * A) -1 o 'y = A -1 (q -1 o ry) ; (6)

(q * x) -1

o 'y = x-1 (q -1 o -y) .

Proof. We prove (1), (3), and (5) . We ask the reader to verify (2), (4), and (6) . (1) z E S -1 o ('Y* A -1 ) ~ S#('Y* A -1 )(z) > 0 ~ 6(po) n ('y* A -1 )(qo) n ft * (po, z, qo) > 0 for some po, qo E Q ~ 6(po) n -y (to) n ft * (qo, xo, to) n ft * (po, z, qo) > 0 for some po, qo, to E Q, xo E A ~ S(po)n'y(to)n[ft * (po, z, qo) Aft* (qo, xo, to)] > 0 for some po, qo, to E Q, xo E A ~ 6(po) n 'y(to) n /t* (po, zxo, to) > 0 for some po, to E Q, xo E A ~ zxo E S -1 o 'Y for some xo E A ~ z E (S-1 o y)A -1 . (3) z E (S * A) -1 o y ~ ((6 * A)-1#-y)(z) > 0 ~ (S * A) (po) A y(go)A ft * (po, z, qo) > 0 for some po, qo E Q ~ 6(to) n [t* (to, xo,po) n'y(go) n 6 t o ) n 'y(go) n ft * (po, z, qo) > 0 for some po, qo, to E Q, xo E A o y for some y* (t o , xoz, qo ) > 0 for some qo, t o E Q, xo E A ~ xoz E S xo E A ~ z E A -1 (S -1 o y) . (5) y E (q * A) -1 o y ~ (q * A) -'#y(y) > 0 ~ q * A(po) A y(qo) A ft * (po, y, qo) > 0 for some po, qo E Q ~ ft* (q, x,po) A y(go) n ft * (po, y, qo) > 0 for some po, qo E Q, y E A ~ 'y(qo) n [ft* (q, x,Po) n ft* (po, y, qo)] > 0 for some po, qo E Q, y E A y(qo) A /t* (q, xy, qo ) > 0 for some qo E Q, yEA~?yEA -1 (q -1 oy) . N16 . Let 61 : Q ----> [0,1] and 62 : Q' ----> [0,1] . Then define 61 n62 QxQ'-[0,1] by d(q,q')EQxQ', (b1 n62) (q, q) = 61 (q) n 62 (q) . © 2002 by Chapman & Hall/CRC

7.5 . Operations on (Fuzzy) Subsets

357

N17. Let 6 1 : Q ~ [0,1], 62 : Q' ~ [0,1] be such that Q n Q' _ 0 . Then define 6 1 U 62 : Q U Q' ~ [0,1] by b'q E Q, 61(q) (61 U 62) (q) - { 62(q)

if q E Q . if q E Q

Theorem 7.5 .5 Let M = (Q, X, /t) and M' = (Q', X, /t') be fuzzy finite state machines. Let 61,ryi be fuzzy subsets of Q and 62, 'Y2 be fuzzy subsets of Q' . Then the following properties hold. (1) (61 U 62) -1 0 NUN' ('Y1 U -Y2) = (6 1 1 o N 'Yi) U (62 1 Off' 'Y2) ; (2) (61 n 62) -1 0 /"n/a , (-yj n -Y2) = (6 1 1 of" 'Yl) n (62 1 0/1' -Y2)Proof. (1) Let x E (61U62)-10('YJU'Y2) . Then (61U62)*(-YJU-Y2)(x) > 0. Thus (61U62)(p)A(ylUy2)(q)A(ftUft')*(p,x,q) > 0 for somep,q E QUQ' . It follows that both p, q belong to either Q or Q' . For, if p E Q and q E Q'and vice versa, then by definition (ft U ft') * (p, x, q) = 0 and consequently (61 U 62) (p) A (-yj U'Y2)(q) A (/t U ft) *(p, x, q) = 0, a contradiction. If p, q E Q, then clearly 61 (p) A yl (q) A ft* (p, x, q) = (61 U 62) (p) A ('Y1 U '2) (q) A (/t U /t')* (p, x, q) > 0. Therefore, 61 #yl (x) > 0 and hence x E 61 1 o ,yl . Similarly, if p, q E Q' , then x E 62 1 0 'Y2 * On the other hand, let x E 6 -1 1 o yl . Then 61 (p) Ayl (q) A ft (p, x, q) > 0 for some p, q E Q, i.e., (61 U 62) (p) n (-yj U 'f2) (q) n (ft U ft') * (p, x, q) > 0. A similar situation holds if x E 62 1 0 y2 . (2) x E (6 1 n 62 ) -1 0 ('Y, n -Y2) (6 1 n 62 )#(-y, n y2)(x) > 0 (6i n 62)(pl,p2) A (yl n'Y2)(gl,q2) A (ft U ft) *((PI,p2),x, (gl,q2)) > 0 for some (PI, P2), (gl,q2) E Q x Q' ~ [61(pl) n'f1(gl) n[t* (pl, x,ql)]n [62(p2) A 'Y2 (q2) n [t * (P2, x, q2)] > 0 for some pl, ql E Q, p2, q2 E Q' ~ [61#'f1(x)] n [62# -Y2 (X)1 > 0 ~ [61#'Yj(x)] > 0 and [6 2#-Y2 (X)1 > 0 ~ x E 61 1 o yl and XE6210-Y2~XE(611o'yl)n(621o-Y2) . Theorem 7.5 .6 Let M = (Q, X, ft) and M' = (Q', X, ft') be fuzzy finite state machines. Let 61, y 1 be fuzzy subsets of Q and 62, 'Y2 be fuzzy subsets of Q' . Then (61 n 62) -1 0wxw' (-yj n -Y2) = (611 0w 'YJ x (62 1 0w' 'Y2) Proof. (x, y) E (61 n 6 2) -1 o (-yj n -Y2) 0 for some (p1, p2), (gl,g2) E Q x Q' [0,1] by ftP (p, a, q) = ft(q, p(a), p) b'a E X, dp, q E Q. Definition 7.6 .5 If M = (Q, X, ft) is a fuzzy finite state machine, then the fuzzy finite state machine MP = (Q, X, ftP) is called the reversal of M. Theorem 7.6 .6 Let MP = (Q, X, pP) be the reversal of M = (Q, X, p) . Then (wP)*(p,xy,q)

= w* (q,P(y)P(x),p)

Vx, y E X* . 0 Let A C X* and p : X* ----> X* be the reversal mapping. Let AP = {p(x)


I x E Al .

7.6. Construction of Recognizers and Recognizable Sets

359

Theorem 7.6.7 Let S and 'y be fuzzy subsets of Q. Then (S -1 o ,y)P = ,Y -1 06. Proof. p(x) E (6 -1 o y)P x E 6 -1 o y ~ 6# y (x) > 0 ~ 6(p) n y ( q) n p,* (p, x, q) > 0 for some p, q E Q S(p) A y(q) A ([tP) * (q, p(x),p) > 0 for some p, q E Q -y (q) A 6(p) A ([tP ) * (q, p(x), p) > 0 for some p, q E Q p(x) E 'Y-1 O S . 'Y#b(p(x)) > 0 Definition 7.6.8 Let .M = (M, t, T) be a fuzzy X-recognizer. Let p : X* ~ X* be the reversal mapping. Then the fuzzy X-recognizer .MP = (MP, T, t) of MP is called the reversal of M.

Definition 7.6.9 Let .M = (M, t, T) be a fuzzy X-recognizer . Then x E X is said to be recognized by .M if t#T(x) > 0.

Let B(M) denote the set of all words that are recognized by .M. B(M) is called the behavior of .M . Theorem 7.6.10 Let M be a fuzzy recognizer and x E X* . Then the following (1) (2) (3)

conditions are equivalent . x E B( .M); x E t -1 o T; t#T(x) > 0; (4) 3po E Q such that t n T * x-1(Po) > 0; (5) 3qo E Q such that t * x n T(qo) > 0.

Proof. Clearly (1), (2), and (3) are equivalent . (3)x(4) : t#T(x) > 0 ~ V{t(p) AT(q) A w*(p, x, q) I p, q E Q} > 0 3po, qo E Q such that t(po) n T(qo) n [t * (po, x, qo) > 0 ~ 3po, qo E Q such that t(po) > 0 and T(qo) A /t* (po, x, qo) > 0 ~ Elpo E Q such that t(po) > 0 and V {T (q) A /t* (po, x, q) I q E Q} > 0 ~ 3po E Q such that t(po) > 0 and T * x-, (PO) > 0 ~? 3po E Q such that t n (T * x-1) (PO) > 0. (3)x(5) : This can be proved by interchanging the roles of po and qo and using the definition of t * x. Definition 7.6.11 A subset A of X* is called X-recognizable, if there exists an X-recognizer .M such that B(M) = A.

Theorem 7.6.12 Let X be a nonempty finite set. Then the following sets are X-recognizable: (1) {a}, where a E X. (2) A. (3) A*, where A C X. (4) X. 0

Theorems 7.5 .5, 7.6.7, and 7.6 .10 immediately lead to the following theorem. © 2002 by Chapman & Hall/CRC


36 0

Theorem 7.6.13 Let M and .M' be fuzzy recognizers . Then the following assertions hold. (1) B( .M U .A4) = B( .M)UB(.M') . (2) B( .M n .A41) = B( .M)nB(.M') . (3) B( .MP) =B(.M)P .

Proof. (3) We have B(M)P = (t -1 o T)P = T -l o t = B( .MP) . The following corollary is immediate from Theorem 7.6 .13. Corollary 7.6.14 Let A and B be recognizable subsets of X* . Then the following sets are recognizable . (1) A U B . (2) A n B . (3) AP . Corollary 7.6.15 A subset A of X* is recognizable if and only if AP is recognizable .

Proof. (AP)P =A . m In view of Theorems 7.5 .6 and 7.6 .10, we obtain the following theorem. Theorem 7.6.16 Let M be a X-recognizer and .M' be a X'-recognizer. Then B(.M x .M') = B(M) x B(.M') . m Corollary 7.6.17 If A is X-recognizable and B is X'-recognizable, then A x B is X x X'-recognizable. m Theorem 7.6.18 Let A be X-recognizable and B be any subset if X* . Then the following sets are X-recognizable . (1) AB -1 . (2) Ax-' . (3) B-1 A. (4) X -1 A.

Proof. We prove (1) and (3) only. Since A is X-recognizable, there exists a fuzzy recognizer .M = (M, t, T) of a fuzzy finite state machine M = (Q, X, /t) such that B(M) = A. (1) Consider a fuzzy recognizer .M' = (M, t,T*B -1 ) . Then by Theorem 7.5.4(1), t-1 o (T * B-1 ) = (t - 'o T)B-1 = B(A4)B -' = AB -1 . Thus AB -1 is recognizable . (3) Consider the fuzzy recognizer .M' = (M, t, B-1 * T) . Then by Theorem 7.5.4(3), (t * B) -1 o T = B-1 (t -1 o T) = B -1 B(M) = B- 'A . Hence B-'A is recognizable . Let X and X' be sets and let f : (X')* X* be a fine homomorphism. If M = (Q, X, /t) is a fuzzy finite state machine, then define /t' : Q x X' x Q ----> [0, 1] by fft'(p,a,q)=ft(p,f(a),q)


7.6. Construction of Recognizers and Recognizable Sets

361

b'p, q E Q, a' E X' . This defines a fuzzy finite state machine M' _ (Q, X, /t') and a fuzzy X'-recognizer .M' = (M', t, T) . Definition 7.6.19 The fuzzy X'-recognizer .M' defined immediately above is called the inverse image of M and is denoted by f-1 (.M) .

Theorem 7.6.20 Let S : Q ~ [0,1] and y : Q ~ [0,1] . Then S-1 f-1(S-1 o w 'Y) .

oN , ,y =

Proof. x E S-1 oN,, y ~ 6#-y(x) > 0 ~ S(p) n -y(q) n ft~(p, x, q) > 0 for some p, q E Q ~ S(p) A y(q) A /t* (p, f (x), q) > 0 for some p, q E Q f (X) E 6-1 0/"Y. The following corollaries are easily shown to hold. Corollary ?.6 .21 B(f -1 (.M)) = f-1 (B( .M)) . m Corollary 7.6 .22 If A is X-recognizable, then f-1 (A) is X'-recognizable. Let .M = (M, t, T) be a fuzzy recognizer of a fuzzy finite state machine M = (Q, X, p) and let a E X. Consider Q' = Q U {t'}, where t' ~ Q. Define fl a :Q'XXXQ'~[0,1]by ft(p, b, q)

f'a(p, b, q) =

1 0

if p, q E Q, if p E Q, T(p)>O, b = a and q = t', otherwise .

Clearly, Ma = (Q', X, f a) is a fuzzy finite state machine. Define to [0,1] as follows : __

to(p) - { 0

p))

if p E Q, otherwise .

Let Ta be the characteristic function of {t'} . Then Ma = (Ma, ta,Ta) is the fuzzy recognizer of Ma. Let X be a nonempty finite set, A C X*, and u E X. Recall that An = {xu I

x E A} .

Theorem 7.6.23 Let Ma = (Ma , ta ,Ta) be a fuzzy recognizer of Ma. Then to 1 0Ma Ta = t 1 0A T)a .

Proof. Let x

E to 1 0Aa Ta .

Then (ta#Ta) (x) > 0 . Thus

ta(po) ATa(go)

for some po, q'

E Q' .

n fta(P' , x, qo) > 0

Hence q' = t' and p'


E

Q.


36 2

Now fta(po,x,go) > 0 is possible only if either T(p0 ) > 0 and x = a or there exists y E X* such that /ta(p'o , ya, t') > 0. We consider these two cases . Suppose T(p0) > 0 and x = a. Now t(po) > 0, T(po), and /t*(po,A,po) > 0. Hence t(po) n T(po) n [t* (po, A, po) > 0, i.e., A E L -1 Of,, T. Therefore, x = Aa E (L -1 Off, T) a. Suppose now that y E X* such that ya(po, ya, t') > 0. Then there exists t'o E Q' such that laa(po, y, to) Alta(to, a, qo) > 0. This is true only when T (to) > 0, i.e., to E Q . Thus t(po) AT(to) Aft*(po, y, to) > 0 . Therefore, L#T(y) > 0. Hence x = ya E (L -1 OA, T)a . Thus tat OA, Ta C(6- 1

oA T)a.

We now show inclusion in the other direction . Let x = ya E (t -1 OA T)a . Then y E (L -1 ON, T) . Therefore, t(po) A T(qo) A ft*(po, y, qo) > 0 for some lta (go, a, t') = 1 po, qo E Q . Thus t(po) > 0 and T(qo) > 0 . Now T(qo) > 0 and fta(POI yI qo) = [t * (po, y, qo) > 0 . Hence fta(po, ya, t~) = V{[ta (po, y, r~) A lta(r', a, t') I r' E Q'} > lta(po, y, qo) n fta(go, a, t~) > 0 . Therefore, t(po) A ' Ta(t PO ) AN'a(po, ya, t') > 0 and hence ta(po) ATa(t') Alta(po, ya, t') > 0 since E Q, l.e., x = ya E La 1 OV-a Ta . Thus (t -1 ON, T)a C La 1 ON,a Ta' Hence tat O V- a Ta - (L-1 OAT)a . Corollary 7.6 .24 B( .M a) = B(A4)a .

m

Corollary 7.6 .25 If a subset A of X* is recognizable and u is recognizable. m

7 .7

E

X, then An

Accessible and Coaccessible Recognizers

Definition 7.7 .1 Let M = (M,t,T) be a fuzzy X-recognizer of a fuzzy finite state machine M = (Q, X, ft) and R = {q E Q I * X* (q) > 01 . If Q = R, then M is called accessible. Let .M = (M, t, T) be a fuzzy recognizer of M = (Q, X, lt) . Let R = {q E X* (q) > 01. Consider to = wI RXXXR, La = tIR, and Ta = TI R . Then Ma = (R, X, ta) is a fuzzy finite state machine and .Ma = (Ma, 6a, Ta) is a fuzzy recognizer of Ma . Q

I

T) be a fuzzy X-recognizer of M. Then .Ma = (Ma' ta,Ta) is called the accessible part of M.

Definition 7.7 .2 Let M = (M, t,

Definition 7.7 .3 Let M = (M,6,T) be a fuzzy X-recognizer of a fuzzy finite state machine M = (Q, X, ft) . Let S = {p E Q I T * (X*) -'(p) > 01 . If Q = S, then M is called coaccessible. Consider ftb = wI sXXXs' tb = t1 s, and Tb = TI s. Then Mb = (S, X, ftb) is a fuzzy finite state machine and Mb = (Mb, tb, Tb) is a fuzzy recognizer of Mb . © 2002 by Chapman & Hall/CRC

7.7. Accessible and Coaccessible Recognizers

363

Definition 7.7.4 Let .M b = (Mb, Lb , Tb )

M = (M, t, T) be a fuzzy X-recognizer of is called the coaccessible part of M.

M.

Then

If t(q) > 0, then q E R. Thus whenever t is crisp, so is ta and La = t as sets. If T(q) > 0, then q E S. Hence whenever T is crisp, so is Ta and Ta = T as sets. Theorem 7.7.5 Let .M be a fuzzy recognizer. Then the following properties hold : (1) L-1

O T = (La)-1 O Ta ; L-1 O (2) T = (Lb)-1 O Tb.

Proof. (1) Let x E

L-1 O T .

Then

L(po) AT(go)

L#T(x)

> 0. Therefore,

n [t* (po, x, qo)

>

0

for some po, q0 E Q. Clearly, q0 E R since L(po) A ft * (po, x, L(po) > 0 and y* (po, A,po) = 1 > 0 and so po E R. Hence Ta(go)

But then (La)-1

La#Ta(x) O Ta .

qo) >

0. Also,

n La(po) n [ta*(po' x, qo) > 0.

> 0 and thus x E

(La)-1

O Ta .

Therefore,

L-1 O T

C_

We now show inclusion in the other direction . Let x E (La)-1 OT a . Then a La #T (x) > 0, i .e., La (p0) AT a (go) All' a* (p0,x,g0) > 0 for some p0, go E R. Therefore, L° (p0 ) > 0, Ta (qo) > 0, and ta* (po, x, qo) > 0. Thus L(p0) = La (p0),T(g0) = Ta(g0)

and * (po,x,go) = ta*(po,x,go) .

Therefore, L(p0) AT(go)

Hence L#T(x) > 0 and so x E L-1 quently, L-1 O T = (La)-1 O Ta . (2) The proof is similar to (1) .

At* O T.

0 x,qo)

Thus

> 0.

(L a) -1 O Ta

C_

L-1 O T.

Conse-

Corollary 7.7.6 Let .M be a fuzzy recognizer. Then (1) B(.M) = B(.Ma) ; (2) B(.M) = B(.Mb) .

Theorem 7.7.7 Let .M be an X-fuzzy recognizer. Then if and only if .MP is accessible .


.M

is coaccessible


36 4

Proof. .M is a coaccessible fuzzy recognizer ~ Q = {p E Q I T (X*) -1 (p) > 0} ~? b'p E Q, 3xo E X*, and qo E Q such that T(qo) A b'p E Q, 3xo namely p(xo) E X* and qo E Q such that ft * (po, xo, qo) > 0 n T(qo) [t P* (po, p(xo), qo) > 0 ~ Q = {p E Q I T * X* (p) > 0} ~ .MP is an accessible fuzzy recognizer. Definition 7.7.8 A fuzzy X-recognizer .M is called trim if it is both an accessible and coaccessible fuzzy X-recognizer.

Theorem 7.7.9 Let M = (M, t, T) be the fuzzy recognizer of the fuzzy

finite state machine M = (Q, X, /t) and let W = {p E Q I t * X* n T (X*) -1 (p) > 0} . Then .M is trim if and only if Q = W. m

Consider pt = ftjw .x .w, tt = t1 w, and Tt = Tjw . Then Mt = (W X, [tt) is a fuzzy finite state machine and Mt = (M t , tt, Tt) is a fuzzy X-recognizer . Definition 7.7.10 Let .M be a fuzzy X-recognizer. Then .M t = (M t , tt ,Tt) is called the trim part of M.

The following theorem is immediate from Corollary 7.7 .6 . Theorem 7.7.11 If M = (M, t, T) is a fuzzy recognizer, then B(M) _ B(.Mt) . .

Corollary 7.7.12 If .M is trim, then ( .Ma) b = .Mt = (.Mb)a . 0 If .M is an accessible (coaccessible, trim) fuzzy X-recognizer, then Ma = M .Mb = M, Mt = M, respectively) . (

7 .8

Complete Fuzzy Machines

Recall that a fuzzy finite state machine M = (Q, X, ft) is called complete if for all (p, u) E Q x X, there exists q E Q such that ft(p, u, q) > 0. As in Definition 7.2 .13, if M = (Q, X, ft) is a complete fuzzy finite state machine, then the fuzzy X-recognizer .M = (M, t, T) of .M is called complete. Theorem 7.8.1 If M is a complete fuzzy recognizer, then so is .M a . Proof. Let (pa, u) E R x X. Then (pa, u) E Q x X. Since .M is complete, there exists q E Q such that ft (pa, u, q) > 0. Now pa E R ==> t * X* (pa) > 0 . Therefore, 3qo E Q and xo E X* such that t(qo) A /t* (qo, xo , pa) > 0 . This implies that t(qo) > 0 and lt* (qo, xo, pa) > 0. Hence lt* (qo, xo , pa) A ft(pa ' u, q) > 0. Thus /t* (qo, xou, q) = V {ft* (qo, xo' t) nft(t, u, q) I t E Q} > 0 . Hence t(qo) A ft*(go,xou,q) > 0, i.e ., t * X* (q) > 0 and so q E R. Since ,ta(pa , u q) = t(pa u q) Ma is complete. The following definition of completion and the following construction of a completion differs somewhat from that in Section 7.2 . © 2002 by Chapman & Hall/CRC

7.8 . Complete Fuzzy Machines

365

Definition 7.8.2 Let M = (Q, X, /t) be a fuzzy finite state machine. A fuzzy finite state machine Mc = (Q c , Xc, ftc) is called a completion of M, if the following conditions hold: (1) Mc is a complete fuzzy finite machine, and (2) M is a subfuzzy finite state machine of Mc . Let M = (Q, X, ft) be a fuzzy finite state machine that is incomplete. Consider M' = (Q', X, /t'), where Q' = Q U {z}, z ~ Q and

f~ (p, u, q) =

ft(p, u, q) 1

0

if p, q E Q and ft(p, u, q) :A0, if either ft(p, u, r) = 0 b'r E Q and q = z or p = q = z,

otherwise.

Then M' = (Q', X, ft') is called the smallest completion of M. Definition 7.8.3 Let M = (M,t,T) be a fuzzy X-recognizer of a fuzzy finite state machine M = (Q, X, /t) and Mc = (Q c, X, /tc) be a completion of M. Then the fuzzy X-recognizer .Mc = (Mc, tc, Tc) of Mc is called the completion of M, where tc : Qc ~ [0,1] and Tc : Qc ~ [0,1] are such that t(q) 0

if q E Q if q Q.

T(q) 0

if q E Q if q Q.

If Mc is the smallest completion of M, then .Mc is called the smallest completion of .M .

Theorem 7.8.4 Let .M = (M, t, T) be a fuzzy recognizer and .Mc is the smallest completion of .M . Then B(A4) = B( .Mc) . 0

Theorem 7.8.5 If M is an accessible fuzzy recognizer, then so is Mc . Proof. Let .M be accessible. Then Q = {q E Q I t * X*(q) > 0} . Let qc E Qc . If qc E Q, then the desired result holds . Let qc = z. Since M is incomplete, there exists (po, no) E Q x X such that ft(po, no, t) = 0 for all t E Q. But then (ftc) (po, no, z) = 1 > 0. Since M is accessible and po E Q, there exists r E Q and yo E X* such that t(r) A /t* (r, yo, po) > 0. Now (ftc)* (r, youo, z) = V{(ft c)* (r, yo, s) n (ftc) (s, uo, z) I s E Q} > * (ft c ) (r, yo, po) n (ftc) (po, no, z) = (ftc) * (r, yo,po) = ft * (r, yo, po) > 0. Thus t(r) A (ftc) * (r, youo, z) > 0, i.e., t * X*(qc) > 0. Hence Mc is accessible. m Let .M = (M, t, T) be a fuzzy recognizer of a fuzzy finite state machine M = (Q, X, ft) . Recall that P(Q) is the power set of Q . © 2002 by Chapman & Hall/CRC


36 6

Define p,^' : P(Q) x X x P(Q) ----> [0,1] by p - (P, u, R

-

1

1

V{ft(p, u, r)

P(Q) ----> [0 , 1] by

Ip

E P, r E

0{L(p) I p E

P}

R}

if P4 QlorR~01 otherwise,

otherwise,

and T^' : P(Q) ----> [0,1] by T-(P)

_

0{T(p) I p E P}

if P otherwise.

Then M^' = (P(Q), X, /t -) is a fuzzy finite state machine and .M^' _ (M^', t- , T- ) is a complete fuzzy X-recognizer of M^' = (P(Q), X, ft-) . Theorem 7.8.6 Let M = (M, t, T) be a fuzzy recognizer of a state machine M = (Q, X, /t) . Then B(M) = B( .M-) .

fuzzy finite

Proof. Let x

E B( .M - ) . Now x E B(M-) =~> x E t- oA- TL-#T- ( x) > 0 =~- t- (P) AT - (R) A /t - (P, x, R) > 0 for some P, R E P(Q) .

Clearly, by the definition of t- and T- , both P and R are nonempty. Thus [A{L(p) I p E P}] n [n{T(r) I r E R}] n [V{,t* (p, x, r) I p E P, r E R}] > 0 for some P, R E P(Q) and so t(p) > 0 Vp E P, T(r) > 0 Vr E R and there 3t o E P, ro E R such that [t* (to, x, ro) > 0 for some P, R E P(Q) . Hence t(to) > 0, T(ro) > 0, and y,* (to, x, ro) > 0 for some to, ro E Q . Thus t(to) A T(ro) A ft * (to, x, ro) > 0 for some to, ro E Q. This implies that t#T(x) > 0 and so x E t oN T . Hence x E B(M) . Thus B(M- ) C B(M) . Conversely, suppose x E B( .M) . Then x E t oN, T and so t * T(x) > 0 . This implies that t (p) A T (q) A ft * (p, x, q) > 0 for some p, q E Q. Choose P = {p} and R = {q} . It then follows that B(M) C B( .M - ) . m

Theorem 7.8.7 Every fuzzy X-recognizable subset A of X* is the behavior of a complete fuzzy recognizer. m

7 .9

Fuzzy Languages on a Free Monoid

The study of fuzzy grammars, the rules of fuzzy syntaxes, and the recognition ability of fuzzy automata extends the application area of fuzzy set theory. One goal is to reduce the difference between formal languages and natural languages . In this section, we examine fuzzy regular languages, adjunctive languages, and dense languages . We present their algebraic properties. The results are from [217] . In this and the next section, we let X denote a finite alphabet with at least one element . Recall that,F'P(X) denotes the set of all fuzzy subsets of X. We use the superscript T to denote the transpose of a matrix. © 2002 by Chapman & Hall/CRC

7.9 . Fuzzy Languages on a Free Monoid

367

Definition 7.9.1 [160] A finite fuzzy automaton on an alphabet X is a 5-tuple M = (Q, Y, {Tn I u E Xj, ao, al ) such that (1) Q = {ql, q2, . . . , qn} is the set of states, (2) Y = {yl, y2, . . . , yn J C Q is the set of output symbols, (3) {Tn u E X} is the set of fuzzy transition matrices, where Sgiq, : X ----> [0,1] and T _ [S qiqj (u)], qZ, qj E Q, i, j = 1, 2, . . . , n, (4) Qo = [ i l i2 . . . i n ] , ik E [0,1] for k = 1, 2. . . . , n, (5) CI = [ ji

j2

.. .

in

]T'

jk E [ 0 , 1 ] for k = 1, 2. . . . , n.

ao determines the fuzzy subset of initial states and al determines the fuzzy subset of final subsets . Let M = (Q, Y, {T I u E Xf, ao, al ) be a finite fuzzy automaton. Define S : Q x X x Q ~ [0,1] by b(qj, u, qj) = bqiqj (u)

for i, j = 1, 2, . . . , n and Vu E X. Then S is a fuzzy transition function. Let S* be defined as usual. Then S* (q, A, q') = 1 if q = q' and 0 otherwise. Also for all x = ulu2 . . . uk E X*, x :?~ A, 6* (q, x, q') or

=

V {b(q, ul, qi) A S(ql , u2, q2) A . . . A S(qk-1, uk, q~) I qi ~ q2 ~ . . . , qk-i E Q}

0 , TX = Tnl 0 Tn2 0 . . . Tn,

where o is the sup-min composition of fuzzy matrices. Definition 7.9.2 [160] Any member of ,F'P(X*) is called a fuzzy language on the free monoid X* . For all u E X, let x n denote the characteristic function of {u} in X* . Then xn is called the basic fuzzy language generated by u. Let E = {x,, I u E

X} U {A} .

Definition 7.9.3 [160] Let U C_ ,F'P(X*) be such that the following conditions hold: (1) Vc E[0,1], ft EU==> cnyEU,

(2)Ni,N2EU ==> /t l UN 2 EU (3) fti, P2 E U =~> fti o y 2 E U, where (Pi 0 ft2)(x) = V {fti(u) n [t 2(V) I uv = x}, b'x E X*, (4) ft EU==> ,t E U, where ft' is the Kleene's closure of ft, i .e ., = [toUftUft2 U . . .

where [t o (x) = 0 V'x E X* . Then U is called a closed family of fuzzy languages on X* .



368

Let .F = {U I U is a closed family of fuzzy languages on X*} .

Clearly,,F'P(X*) E T Definition 7.9.4 [160] Let FR(X*)

= nEC MC .-M .

FR(X*) is called the family of fuzzy regular languages on X* . If f E FR(X*), then ft is called a fuzzy regular language on X* . Clearly, FR(X*) is a subalgebra of ,F'P(X*) generated by E with the four operations in Definition 7.9 .3 . Definition 7.9.5 [160] Let M = (Q, Y, Tn, ao, al ) be a finite fuzzy automaton on X. Define fm : X* ~ [0,1] by Vu E X*, fm(u) = ao o Tn o al . Then fm is called the fuzzy language determined by the fuzzy automaton M.

Example 7.9.6 Let Q = {s, S, F}, X = {a, b}, and S : Q x X x Q ~ [0,1] be defined as follows:

6(s, a, S) S(s, b, s) S(S, b, S) S(S, b, F)

_ _ _ _

.9 .5 .9 .9

and S(q, x, q~) = 0 for any other (q, x, q~) E Q x X x Q. Let ao = (1, 0, 0) and al = (0, 0,1)T. Then

Ta

_

s S F

0 0 0

s

.9 0 0

S

0 0 0

F Tb

_

s S F

.5 0 0

s

0 .9 0

S

0 .9 0

F

Let M = (Q, Y, ~Ta, Tb}, ao, al ), where Y = Q . Then fm (bab)=ao oTboTa oTboal =[ 1

0

0 0 0

0 ]

.5 0 0

.5 0 0

0 0 1

= .5

and fm(ab)=ao oTa oTb oa l =[ 1

0

0 ]

0 0 0

.9 0 0

.9 0 0

0 0 1

= .9 .

It follows that fm (b mabn ) = .5 if m > 0 and fm (b-abn ) = .9 if m = 0. We see that fm = A, where A is the fuzzy language of Example 7.1 .11.


7.9 . Fuzzy Languages on a Free Monoid

369

From [160], /t E FR(X*) if and only if there exists a finite fuzzy automaton M such that fm = /t . Definition 7.9.7 [160] We call PL a main congruence if b'x, y E X*, x - y(PL) if and only if Vu, v E X*, uxv E L ~ uyv E L . Example 7.9.8 Let X = {a, b} and L = Jan I n = 0,1 . . . . } . Let x, y E

X* . Then Vu, v E X*, uxv E L if and only if u = az, x = ai, and v = ak for some i, j, k E hY U {0}. Thus x - y(PL ) if and only if either x E L, y E L or x ~ L, y ~ L . Hence the equivalence classes corresponding to - are L and X*\L . Thus the index of PL is 2.

Example 7.9.9 Let X = {a, b} and L = {b'abn I m = 0,1 . . . . ; n =

1. . . . } . Let x, y E X* . Then Vu, v E X*, uxv E L if and only if either u= b', x=b~abk,v=bl (not both k=0,1=0) oru=b'abi,x=bk ,v=b l (j, k, l not all 0) or u = b2, x = bi, v = bkabi (1 z,4 0) for i, j, k, l E hY U {0}. Thus x - y (PL ) if and only if either x, y E L U {b le a k = 0,1, 1 . . . . I or { bk I k = 0,1 . . . . } or x, y E X*\(L U {bka I k = 0,1. . . . } U {bk X' y E .}) . Thus the index of PL is 3. k=0,1

Proposition 7.9 .10 [218] An ordinary language L C_ X* is regular if and only if the index of

PL

is finite . 0

Proposition 7.9 .11 [218] An ordinary language L C_ X* is adjunctive if and only if b'x, y E X*, x - y(PL) =~' x = y . 0

Example 7.9.12 Let X = {a, b} and L = fanbn I n = 0,1. . . . } . Let

x, y E X* . Then Vu, v E X*, uxv E L if and only if either u = anbn , x = V, v = bn -Z-j or u = a2 , x = a n-Z b', v = bn- j where n, i, j E hY U {0} . Thus x - y(PL) implies x = y . Hence L is adjunctive . The index of PL is not finite and L is not regular.

Definition 7.9.13 Let /t E .PP(X*). Then FN is called a fuzzy main congruence with respect to /t on X* if

b'x, y E X*, x - y(FN,) ~ Vu, v E X*, y,(uxv) = y,(uyv) .

Proposition 7.9 .14 Let x, y E X* . Then x - y(FN ) if and only if b'c E [0,1], x - y(PN ), where y c = {x E X* I y(x) > c } . Proof. Suppose that x - y(FN ) . Then Vu,v E X*, y(uxv) = y(uyv) . Thus b'c E [0,1], y(uxv) > c if and only if y(uyv) > c. Hence Vu, v E X*, b'c E [0,1], uxv E /t c if and only if uyv E 1t c . This implies that x - y(PI") . Conversely, suppose that b'c E [0,1], x - y(PN ) . Then Vu, v E *, y(uxv) > c r y(uyv) > c b'c E [0,1] . Thus x - y(FN ) . 0 Proposition 7.9 .15 Let x, y E X* . Then x - y(FN ) if and only if b'c E [0,1], x - y(PN +), where y c+ _ {x E X* I y(x) > c }. © 2002 by Chapman & Hall/CRC


370

Proof. Suppose that b'c E [0,1], x - y(PN ) . Then Vu, v E X*, p,(uxv) > c ~ p,(uyv) > c. Let p,(uxv) = d. Now for any E > 0, we have p,(uxv) > d - E . Therefore, p,(uyv) > d - E for all E > 0. Thus p,(uyv) > d = p,(uxv) . Similarly, p,(uxv) > p,(uyv) . Hence, x - y(FN,) . The converse is easily shown . m Corollary 7.9.16

7 .10

N = OCEfo,11Pw = OCE[o,l]Pw'+ .

F

Algebraic Character and Properties of Fuzzy Regular Languages

Proposition 7.10 .1 Let ft E .FP(X*) . Then ft E FR(X*) if and only if the index of FN is finite . Proof. Suppose that ft E FR(X*) . Then there exists a finite fuzzy automaton M = (Q, Y, T., ao, al ) such that fm = ft . Clearly, fm = ao o T o al . Define a relation p on X* as follows: b'x, y E X*, x - y(p) ~ TX = T. . Clearly, p is an equivalence relation . Let x, y E X* . Then x - y(P)

TX = TV

Vu, vEX*,T,, oTx oT,=T,, oT.oT, du,v E X* ,Tuxv = Tvyv Vu, v E X*, ao o Tuxv o al = ao o Tuyv o al VU, v E X* ' p(uxv) = p (uyv) x - y(FN,) .

Hence p C FN . Since X and Q are finite, {Tu, I w E X*} is a finite set . Therefore, since the index of p is finite, the index of FN is also finite . Conversely, suppose that the index of FN is finite . Let [A], [xl], . . . , [x .] be the distinct congruence classes of FN , where x2 E X*, i = 1, 2, . . . , m. Let M = (Q, Y, {Tu I u E Xj, ao, al) be a finite fuzzy automaton, where .]}, Q = Y = {[A], [xl], . . . , [x

the fuzzy transition matrix is defined as a[w] o Tu = a[wu], and a[A] = [10 . . . 0],aIx il =[0 . . . 010 . . . 0],i=1,2, . . .,Tn . Set ao = a[A], a l = (ftQA]), ft Qxl]), . . . , ft([X.])) T . . Also Since V'x E X*, x E [x], p(x) = p&]) f (x) = ao o Tx

o 01

= a[A] o Tx o al

= Cr[x]

o al = ft([x]) .

Thus b'x E X*, ft(x) = fm(x) . Hence ft = fm and so p, E FR(X*) . © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 371

Proposition 7.10 .2 Let ft E .FP(X*). Then ft E FR(X*) if and only if Vc E [0,1], /tc is regular, and I Im(y) I < oo. Proof. Suppose that ft E FR(X*) . Then the index of FN is finite. By Corollary 7.9 .16, Pt,, D FN , Vc E [0,1] . Let c E [0,1] . Hence the index of PN is finite and so p c is regular . We may assume that the congruence classes of FN are [XI], [x2], . . . , [x,] since the index of F, is finite . Clearly, Vi, i = 1, 2, . . . , m, Vu, v E X*, if u, v E [x2], then y(u) = ft(v). Therefore, IM(ft) = {ft(xl), ft(x2), - . . , ft(xm)I and so I Im(ft) I < oo. Conversely, suppose that Vc E [0,1], ftc is regular, and IIm(ft)I < oo . Then the characteristic function Xw of ftc is a fuzzy regular language. By the resolution theorem of fuzzy sets (see [160]), ft = Uc.E[o,l] c X w , . Let IM(ft) = {Cl, c2, . . . , ck} . Then ft = c1Xw , , U c2X w 2 U . . . U ckXw'k . Hence by Definition 7.9 .4, yE FR(X*) . m Proposition 7.10 .3 (FR(X*), n, U, -) is a de Morgan algebra, where denotes complement of fuzzy subsets.

Proof. Since FR(X*) C .FP(X*) and (.F'P(X*), U, n, - ) is a de Morgan algebra, it suffices to show that (FR(X*), U, n, - ) is a subalgebra of (.FP(X*), U, rl, - ) . By Definition 7.9 .4, it follows that Vtt,v E FR(X*), ft U v E FR(X*) . Also, since y,,v E FR(X*), Vc E [0,1], ft c , V c are regular, and I Im(ft) I and I Im(v) I are finite. Thus Vc E [0,1], (ft n v) c = ftc rl vc is regular, and I Im(y, n v) I < I Im(y,) I + I Im(v) I is finite . Hence by Proposition 7.10.3, ft rl v E FR(X*) . Let x, y E X* . Then x--y(FN)

b'u, v E X*, y(uxv) = y(uyv) b'u, v E X*,1 - y,(uxv) = 1 - y,(uyv) Vu, v E X*, !(uxv) = -(uyv) x - y(Fw) .

Thus since the index of FN is finite, the index of FA is finite . Hence, FR(X*) . m

Proposition 7.10 .4 Let y,v E FR(X*) . Then (1) /t, o v E FR(X*), (2) v - ly, (or y,v -1 ) E FR(X*), where v -ift (or yv -1 ) is called the fuzzy left (right, respectively) quotient of ft with respect to v and is defined as follows: (V -I [t)(y)

=V{ft(xy) n v(x) I x E E*),

(ftv-1) (y) = V{ft(yx) nv(x) I x E E* },

b'y E X*, b'y E X* .

Proof. (1) Let c E [0,1] . Since (ft o v) c = y cvc and ft and v are fuzzy regular, y c and vc are regular . Also since (yov) c = y cvc is an adjoin of two © 2002 by Chapman & Hall/CRC


37 2

regular languages, (ft o v), is regular . Now the degrees of membership of elements of pov are obtained from the degrees of membership of elements of /t and v with the operations max and min. Thus Im(y o v) C Im(y) U Im(v) and so Im(y o v) is finite . Hence /t o v E FR(X*) . (2) For all s, t E X*, s - t(FN,) Vu, v E X* ' y(usv) = y (utv)

Vu, v, x E X*, y,(xusv) = y,(xutv)

Vu, v, x E X*, y(xusv) A v(x) = y(xutv) A v(x)

Vu, v E X * , V XE x=(y(xusv) A v(x)) = VxEx* (y(xutv) A v(x)) Vu, v E X*, (v -l y)(usv) = (v -l p)(utv) s - t(F-,I,) .

Thus, FN C F -i /, . Since /t E FR(X*), the index of FN is finite and so the index of F-i N is also finite . Consequently, v -l y E FR(X*) . m Note that v in Proposition 7.10 .4(2) may be any fuzzy language. Proposition 7.10 .5 Let

f be a homomorphism from X* onto Xl, where Xl is a nonempty set. Then (1) ,t E FR(X*) f(w) E FR(X*) ;

(2) v E FR(X*)

f-1 (v)

E FR(X*) .

Proof. (1) Let y E FR(X*) . Let c E [0,1] and xl E Xl . Now xl E if and only if (f (ft)) (xl ) > c if and only if V{ft(x) I f (x) = xl } > c if and only if 3x', f(x') = xl and ft(x) > c (since I Im(y,) I < oo) if and only if f(x') = xl , x' E /tc if and only if xl E f(ft c ) . Hence (f (ft)), = f(ft c ) . Since /tc is regular, f (ftc ) is regular and so (f (ft)), is regular . Since Im(y,) is finite, it follows that Im(f (ft)) C Im(y,) and so Im(f (ft)) is finite . Hence f(ft) E FR(X*) . (2) Let v E FR(X*) and x, y E X* . Then (f (ft)),

x-y(Ff-,( ))

(f-1(v))(uxv) = (f-1(v))(uyv) Vu,v E X* v(f (uxv)) = v(f (uyv)) vu, v E X* v(f(u)f(x)f(v)) = v(f(u)f(y)f(v)) vu,v E X* v(ul f (x)vl) = v(ulf(y)vi) vul, vi E X1 f(x) -- f(y) (F,) .

Consequently, the index of Ff-i ( ) in X* equals the index of F in X* . Since the index of F is finite, the index of Ff-,( ) is finite and so f-1 (v) E FR(X*). m Definition 7.10 .6 Let y, E .FP(X*) and let h(ft) = V{ft(u)I u E X*}. (1) Then h(y) is called the height of ft . (2) If there exists no E X* such that h(ft) = ft(uo), then no is called a saddle point of ft . © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 373 Proposition 7.10 .7 Let /t E FR(X*) . Then there exists n E N such that /t has a saddle point with length less than or equal to n - 1 . Proof. Since /t E FR(X*), there exists a finite fuzzy automaton M = (Q, Y, Tn , ao, a l ) such that p = fm = ao o T o al, where IQ I = n, ao = ]T c2 d ] I cl cn , and Q I = [ d l 2 . .. do . Clearly, .. . h(ft)

= VxEX*w(x)

=

V

~VxEU1"-1X1N'(x), VxEX*\U11-1Xbp,(x)},

where Xo = {A} . We now show that V XEU 1k-oX k[t(x)

>

VxEX*\Uk-oxklt(x)

.

Let v E X* \ Uk-o X k be such that wl >_ n. Let v = vI V2 v2 E X, i = 1, . . . , m. Then ft(v)

=

= =

where Tvk

. . .V m ,

m >_ n,

Qo o T,t, o Ql QooT2,1oT12o . . .oT2,,,zoal m VI 1, and Vi > 0, ft(xyjw) > ft(u) provided Jul > n.

Proposition 7.10 .9 (Action lemma of the fuzzy pump

Proof. As the proof of Proposition 7.10.7, let ft

u E X* and let u

= VI V2 . . .

m > n. Now

V~ ,

It( U ) =v 1 221 A . . . A iP-12y n 2giq+1 n . . . /~ 2--12m /~ d2m -

ft(u) )

where m + p - q < n. Furthermore, Vi 1 . Consequently, b'j > 0, ft(xyjw) > ft(u) . 0 © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 375 Corollary 7.10 .10 Let ft E FR(X*) . Then V'c E [0,1] there exists a pos-

itive integer n such that for any u E /t c, u = xyw with lxyl < n, lyl >_ 1, and b'j > 0, xy3 w E /t c provided Jul > n . m

If A is an ordinary regular language, then its characteristic function xA E FR(X*) and Corollary 7.10.10 becomes the ordinary action lemma of the pump (see [122]) . Hence Proposition 7.10.9 is a generalization of the ordinary action lemma of the pump and for this reason is called the action lemma of the fuzzy pump. We now consider fuzzy adjunctive languages . Definition 7.10 .11 Let A E .FP(X*) . If V'x, y E X*, x - y(Fa) =~, x = y, then A is called a fuzzy adjunctive language .

Example 7.10 .12 Let X = {a, b} and L = {anbn l n = 0,1. . . . } . Define A : X* ~ [0,1] by A(azb z ) _ .9 for i = 0,1, . . . , n, A(az b z ) = .5 for i =

n + l, n + 2, . . . , and A(x) = 0 if x E X*\L . By Example 7.9 .12, L is adjunctive . Let x, y E X* . Then Vu, v E X*, A(uxv) = A(uyv) implies uxv, uyv E L or uxv, uyv E X*\L . Hence it follows that x - y(Fa) implies x = y . Thus A is a fuzzy adjunctive language .

Proposition 7.10 .13 The following statements are equivalent: (1) A E .FP(X*) is a fuzzy adjunctive language. (2) Fa is the identity relation. (3) b'x E X*, [x]F, _ {x} . (4) b'x, y E X*, if x y, then there exist u, v E X* such that A(uxv) A(uyv) . m

Proposition 7.10 .14 Let A E .FP(X*) . Then A is a fuzzy adjunctive lan-

guage if there exist c E [0,1] such that A, is an ordinary adjunctive language.

Proposition 7.10 .15 Let A E .FP(X*) . Then A is a fuzzy adjunctive language ~ V'x, y E X*, {x = y(ncE(o,i) Pa,) =~> x = y} . 0

Lemma 7.10 .16 Let f be a homomorphism from X* onto Xl . Then the following properties hold: (1) dw E .FP(X*), (f(w))c = f(w)c, do E [0,1] . (2) Vv E .F'P(X*), (f -1 (v)) , = f-1 (v),, do E [0,1] . m Proposition 7.10 .17 Let f be a homomorphism from X* onto Xl . Let A E .FP(X*) be a fuzzy adjunctive language . Then f (A) E .FP(Xl) is fuzzy adjunctive .



37 6

Proof. Since A is fuzzy adjunctive, b'x, y E X*, x - y(nc Pa,) =~, x = y . That is, Vc E (0, 1), Vu, v E X*, uxv E A, ~ uyv E A, implies that x = y . Let f(x), f(y) E f(X*) = Xi and f(x) - f (y)(ncP(f(a) >) . Then Vc E (0, 1), Vu, v E X*, f(u)f (x) f(v) E (f (A)), ~ f(u)f(y) f(v) E (f (A)), . This implies that f(uxv) E (f (A)) c ~~ f (uyv) E (f (A)) c, b'c, Vu, v E X*, which in turn implies that f (uxv) E f(A c) ~ f(uyv) E f (A,), b'c, Vu, v E X* . Hence uxv E A, ~ uyv E A,, Vc,Vu,v E X* . Thus x - y(Pa ,),Vc and so x - y (n,Pa, ) . Hence x = y and so f (x) = f (y) . Therefore, f(A) is fuzzy adjunctive. Proposition 7.10 .18 Let

f : X* ~ Xl be an isomorphism. Suppose that v E .FP(Xl) is a fuzzy adjunctive language . Then f -1 (v) E .FP(X*) is fuzzy adjunctive .

Proof. Let x, y E X* and x - y(n c P(f-, Then uxv E (f -1 (v)) , v X* and so uxv E E f -1 (v c ) ~? uyv E f-1 (v,) (f -1(v))C b'c, Vu, v X* . Thus f (uxv) v, (uyv) b'c, Vu, E E ~ f E v. b'c, Vu, v E X* . Hence f(x) - f(y) (P,,) Vc and so f(x) - f(y) (n c p,) . Thus f (x) = f(y). Since f is one-one, x = y. Therefore, f-1 (v) is fuzzy adjunctive. uyv E

Proposition 7.10 .19 Let A E .FP(X*) . If there exists v E .FP(X*) such that v 1A is adjunctive, then A is fuzzy adjunctive.

Proof. Since v-1 A is adjunctive, Vs,t E X*, s - t(F -la) =~> s = t. Suppose that s - t(Fa) . Then Vu, v E X*, A(usv) = A(utv) . Thus A(xusv) = A(xutv) V'x, u, v E X* . This implies that A(xusv) A v(x) _ A(xutv) A v(x) b'x, u, v E X* and so V{A(xusv) A V(X) I x E X*} = V{A(xuyv) A V(X) I x E X* } Vu,v E X* . Thus (v-1 A)(usv) = (v-1A)(utv), Vu,v E X* and so s t(F -la) . Hence s = t. Therefore, A is fuzzy adjunctive. m

Proposition 7.10 .20 Let A E .FP({z}*) . Then A is fuzzy adjunctive if and only if A is not fuzzy regular.

Proof. Suppose A is fuzzy adjunctive. If A is fuzzy regular, then the index of Fa is finite . This contradicts the fact that A is fuzzy adjunctive and z* is divided into infinite classes that contain only one element by Fa . Conversely, if A is not fuzzy adjunctive, then there exist i, j with i :?~ j and z2 - zj(Fa) . Now {z}* is divided into at most j classes, [A], [z], [z2], . . . ,

[zj-1] .

Hence the index of Fa is finite, i.e., A is fuzzy regular . m © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 377 Proposition 7.10 .21 Let A E F({z}*). If there exists c E [0,1] such that (1) b'm > 1, 3n E N, A(zm+n) > c , (2)Vm>1,31EN,A(z'+l ) t, (A(zs+Z) < c 1,2, . . . ,m), then A is a fuzzy adjunctive language.

or A(zs +Z) > c , Vi =

Proof. It suffices to show that b'r > 0, k >_ 1, zr * zr+k (Fa) . In fact, if k + 1 = m >_ 1, then sets >_ 1 when r = 0 and sets >_ r when r > 1 . From (3), it follows that Vi = 1, 2, . . . , m, A(z s +Z) < c or A(z s +1) >_ c_ Suppose that A(zs+Z) < c , i = 1, 2, . . . , k + 1 . From (1), there exists ml, the smallest integer such that A(z s +k+l+ml) > c . Thus A(z s-r+l z

r zmi

) < c,

A(zs-r+l z r+k z mi ) > c,

i.e ., there exist z' - '+ I, zml E z* such that A(zs-r+lzrzmi)

7~ A(zs_r+lzr+kzml) .

Hence zr * zr+k(Fa) . Suppose that A(z s +z) > c , i = 1, 2, . . . , k + 1 . From (2), there exists s +l+m2) < c . Similarly, it follows m2, the smallest integer such that A(z +k easily that zr * zr+k(Fa) . Definition 7.10 .22 Let A E .FP(X*) and c E [0,1] . Then A is called a c-discrete language if b'x, y E X*, x :?~ y, A(x) >_ c and A(y) > c implies that lxl :A lyl .

Example 7.10 .23 Let X = {a, b} and L = fanbn I n = 0,1. . . . } . Define l

the fuzzy subset A of X*by A(x) = 0 if x E X*\L and A(x) > 0 if x E L. The A is c-discrete for c E (0,1] since no two distinct elements of L have the same length.

Lemma 7.10 .24 Let JXl > 2 and A E .FP(X*) . Then A is a fuzzy adjunc-

tive language if and only if Vu,v E X*, Jul = lvl and u - v(Fa) implies that u = v. 0

Proposition 7.10 .25 Let A E .FP(X*) . Suppose that A is c-discrete and b'w E X*, there exist u, v E X* such that A(uwv) >_ c . Then A is a fuzzy adjunctive language . Conversely, if A is a fuzzy adjunctive language, then b'w E X* there exist u, v E X* such that A(uwv) > 0. Proof. If u - v(Fa) and Jul = lvl , then by the hypothesis there exist x, y E X* such that A(xuy) > c . Since u - v(Fa), A(xuy) = A(xvy) and so A(xvy) > c . Now, since l xuy l = l xvy l and A is c-discrete, it follows that xuy = xvy and so u = v. Hence A is fuzzy adjunctive. © 2002 by Chapman & Hall/CRC


37 8

Conversely, if there exists w E X* such that Vu, v E X*, A(uwv) = 0, then Vu, v E X*, A(uwv) = A(uw 2 v) = 0, i.e., w - W2 (F,\), which contradicts the fact that A is a fuzzy adjunctive language. m Proposition 7.10 .26 If A E .FP(X*) is a fuzzy adjunctive language, then b'w E X*, IA' I = oo, where A' = {x I x E X*wX*, A(x) > 0} . Proof. Since A is fuzzy adjunctive, b'w E X*, IA'l z,4 0. Suppose that IA'l < oo . Let u E A' be such that Jul = V{lvI I v E A'J. Clearly, I AwnI = 0. This contradicts the fact that A is a fuzzy adjunctive language. Proposition 7.10 .27 Let I X I > 2 . Then A is an 0-discrete fuzzy adjunctive language if and only if Supp(A) is an ordinary discrete adjunctive language.

Proof. If Supp(A) is an ordinary discrete adjunctive language, then A is clearly a 0-discrete fuzzy adjunctive language. Conversely, suppose that A is a 0-discrete fuzzy adjunctive language and Supp(A) is not adjunctive. By Lemma 7.10.24, there exist x, y E X* such that x :?~ y, Ixl = lyl, and x - y(Psupp(a) ) . Hence Vu,v E X*, uxv E Supp(A) ~ uyv E Supp(A) . Consequently, Vu, v E X*, A(uxv) > 0 A(uyv) > 0. Since A is a fuzzy adjunctive language, by Proposition 7.10.25, it follows that b'x E X* there exist no, vo E X* with A(uoxvo) > 0. Thus A(uoyvo) > 0 . Since luoxvol = luoyvol and A is 0-discrete, uoxvo = uoyvo and so x = y, a contradiction . Hence Supp(A) must be an adjunctive language. If V'x, y E X* with x :?~ y, x, y E Supp(A), then clearly A(x) > 0 and A(y) > 0. Since A is 0-discrete, Ixl :?~ lyl . Therefore, Supp(A) is discrete . m We now consider fuzzy dense languages . Definition ?.10 .28 Let y E .FP(X*) and w E X* . Let yW =

{x I x E X*wX*, y(x) > c},

c E [0,1] .

If b'w E X*, Ip~ 0, then /t is called a c-dense language . In particular, a 0-dense language is called a fuzzy dense language and we write , Itw

po =

Example 7.10 .29 Let X = {a}. Define the fuzzy subset y of X* by la(an) = n+l

, n = 0, 1, 2, . . . , where ao = A. Then tt is a 0-dense language . Now /t

is not c-dense for any c E (0,1] since there exists n such that n+il < c and for w = an and x E X*anX*, y(x) < n+il < c .

Proposition 7.10 .30 Let w E .FP(X*). Then the following statements are equivalent. (1) w is a fuzzy dense language. (2)VwEX*, lw w I =oo . (3) There exists a fuzzy adjunctive language A with w D A.


7.10. Algebraic Character and Properties of Fuzzy Regular Languages 379 Proof. (1) ==> (2) Immediate from Proposition 7 .10 .26 . (2) ==> (3) Define an ordering relation on X * as follows : if Ix I < ly l, then x < y ; if IxI = IyI , then x < y means that x, y is in the lexicographic ordering of elements of X. Now X* = {A<w1<w2< . . .<wn< . . .} . Since Iww I = oo, b'w E X*, ISupp(w) _ and the number of elements in {[x] W} is greater than 1 . Let x1, x2, . . . , x, . be the representative elements from these classes. Define A E .PP (X*) as follows : if x = xi for some i if x :?~ xiforalli . Clearly, A C w . Now b'x, y E X*, x z,4 y, if A(x) > 0, A(y) > 0, then IxJ z,4 and so A is 0-discrete . Since Iwwl I = oo, there exists an element ulwlvl IyI with the shortest length in wwl . We select u l wl v l as the representative element . Then A(u l w l v l ) = w(u l wl v l ) > 0. Similarly, since Iww2 1 = oo, we can choose u2w2v2 E ww2 such that `(u2w2v2) = w(u2w2v2) > 0 and Iww~ I = oo, there exists u2w2v2 E wwa I u1w1v1 I 0 and ui_1wi_1vi_1 I (1) Now w D A implies that I ww I > IA w I b'w E X* . Since A is fuzzy adjunctive, IAw I :?~ 0 b'w E X* by Proposition 7 .10.25 . Thus ww I :?~ 0 b'w E X* . Consequently, w is a fuzzy dense language. m Proposition 7 .10 .31 Let /t E .FP(X*) .

fuzzy dense language.

Then either /t or 7 must be a

Proof. If b'w E X*, IftwI = oo, then /t is fuzzy dense . Suppose that there exists w E X* such that Iftwl < oo . Then l{x I x E X*wX*, y(x) > 0}I < oo . Thus {x I x E X*wuX*, y(x) > 0}I < oo Vu E X* and so {x I x E X * wuX * ,ft(x) = 0}I = oo Vu E X* . This implies that I {x I x E X*wuX*, T(x) = 1 > 0}I = oo .Hence (N)w'I = oo Vu E X* . Since X*wuX* C_ X*uX*, VU E X*, I (N)wul < I (p)ul = oo . Therefore, p is fuzzy dense . Proposition 7 .10 .32 Let w = / t U v and c E [0,1] . Then w is c-dense if and only if either /t or v is c-dense. Proof. Suppose that w is c-dense and v is not c-dense . exists wo E X* such that I {x I x E X*WO X*, v(x) > c} I
c c oo . Since w is c-dense, b'w E X*, J {x I x E X*wX*, w(x) > c }I = oo, i.e., J {x I x E X*wX*, (ltUv)(x) > c}I = oo Vw E X* . This is equivalent to I{x I x E X*wX*, y(x) V v(x) > c}I = oo Vw E X* . Consequently, J {x I x E X*wouX*, y(x) V v(x) > c}I = oo Vu E X*. From this, it follows that J {x I x E X*w ouX*, y(x) > c}I = oo Vu E X* . Hence VU E X* ' I tt'o' I = oo and so I pu = oo Vu E X* . Thus tt is c-dense . Conversely, suppose that /t is c-dense . Now b'w E X*, I {x

I

x E X*wX*, ft(x)

> 0}I =

and so I {x I x E X*wX*, y(x) V v(x) > c}I = oo Vw E X* . This implies that I{x I x E X*wX*, w(x) > c} I = oo Vw E X* . Hence w is c-dense . m Proposition 7.10 .33 Let w

E .FP(X*) . Then w is c-dense if and only if is c-dense, where wl x*,,x* wl x*,,x* denotes the restriction of w to X*wX*, wEX* .

Proof. Suppose that w is c-dense . Then b'w, u E X*, I {x I x E X*wuX*, w(x) > c}I :?~ 0. Thus there exists x E X*wuX* C_ X*wx* rl X*uX* such that w(x) > c. Hence Vu E X*, there exists x E X*uX* such that wlx*,,x* (x) > c. This implies that Vu E X*, {x I

x E X*uX*,wlx .,x " (x) > c} I :?~

0.

Hence wlx*,,x* is c-dense b'w E X* . Conversely, suppose that b'w E X*, wl x*,,x* is c-dense . Then b'w X*, I{x I x E X*wX*, wlx*,,x*(x) > c }I :?~ 0. Thus b'w E X*, I{x I x X*wX*, w(x) > c} I :?~ 0. Hence w is c-dense .

E E

Proposition 7 .10 .34 Let I X I > 2, w = /t U v, and w be a fuzzy adjunctive language . Then one of the following statements hold. (1) /t or v is a fuzzy adjunctive language . (2) /t and v are fuzzy dense languages.

Proof. Suppose that /t, v are not fuzzy adjunctive and v is not fuzzy dense . Then there exists w E X* such that J {x I

x E X*wX*, v(x)

> 0 }I
V{IzI I z E vw} . Since uw -- vw(FN ), Vx,y E X*, lt(xuwy) = © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 381 ft(xvwy), and Ixuwyl = lxvwyl > Jul . Thus xuwy, xuvy ~ v' and so v(xuwy) = v(xvwy) = 0. Now V'x, y E X*, co(xuwy)

= = = = =

ft(xuwy) V v(xuwy) ft(xuwy) ft(xvwy) ft(xvwy) V v(xvwy) W(xvwy) .

Hence uw - vw(F,) . Since w is a fuzzy adjunctive language, uw = vw, which contradicts the fact that u :?~ v. Thus v is fuzzy dense. Similarly, it can be shown that ft is fuzzy dense. m Proposition 7.10 .35 Let A be a fuzzy adjunctive Then AIx*wX* is a fuzzy adjunctive language .

language and w

E X*.

Proof. Clearly, A = Alx*wx* U Al x*wx* . Let x E X* . Then SI X*wx*

(x)

_ _

A(x) 0 A(x) 0

x E X*wX* x X*WX* x X*wX* x E X*wX* .

Clearly, (AI x*wX*)' is the empty set and so AI x*wX* is not fuzzy dense. By Proposition 7.10 .34, Alx*wX* is a fuzzy adjunctive language. Proposition 7.10 .36 Let I X I > 2, A be a 0-discrete fuzzy adjunction guage and A = ft U v . Then ft or v is a fuzzy adjunctive language .

lan-

Proof. If ft and v are not fuzzy adjunctive, then there exist xl, x2, yl, y2 X* such that xl :A x2, Y1 54 y2, Ixl I = Ix2I , Iyl I = IY2I , and x, -- X2 (Ft,), E = Y1 y2(F ) . Clearly, x 1 yl - x2y1(FN ) and x 1 y l - xly2(F ) . Since A is fuzzy adjunctive, there exist u, v E X* such that A(uxlylv) = y(uxlylv) V v(uxlylv) > 0 by Proposition 7.10.25 . If y(uxlylv) > v(uxlylv), then A(uxlylv) = ft(uxlylv) = ft(ux2ylv) > 0 and furthermore A(ux2ylv) = ft(ux2ylv) V v(ux2ylv) > 0. Since A is 0discrete, IUXIYIVI = lux2y1vl implies that uxlyly = ux2ylv, which contradicts that xl :A x2 . Similarly, if y(uxlyl v) 2. We consider the case IXI = 1 later. Let X* _ {ul, u2, . . . , un, . . . } . Since w is fuzzy dense, Vu 2 , I {x I x E X*u i X*, w(x) > 0}I = oo. Let x2 = s j ui t 2 E w n `, i = 1, 2. . . . be such that Ix1I 0, then w(svlwlt) > w(svlw2t) > 0, i.e., v1wl - vlw2(F,) . If wI B(svlw2t) = 0 and svlwl t E B, then w(svlw 2 t) = 0 and w(sv 1 w 1 t) > w(sv l w2 t), i .e ., v 1 w l * v l w2 (F,) . If wI B(svlw2t) = 0 and svl wl t ~ B, then svlwlt E B . From v1wl - vlw2(F,1B) and svlwlt ~ B we have that w(sv l w2 t) = WI B (sv l w 2 t) = wlB(sv l w l t) = 0. Hence v 1 w l * vlw2(F,) . Therefore, it always holds that v1wl * vlw 2 (F,), i.e., w(sv l w l t) > w(svlw2t) .


7.11 . Deterministic Acceptors of Regular Fuzzy Languages

383

However, vl - v2 (F,), v l w l - v 2 w 1 (F,), v l w2 (F,), and furthermore, w(sv 2 w l t) > w(sv2 w 2 t) . Since sv l w l t 7~ sv 2 w lt, I svl w ltI = I sv2witI , svlwlt E B, and by the definition of B, sv2 wl t ~ B, i .e ., sv2 w lt E B . Consequently, w(sv 2 w l t) =WI T(sv2 w lt) > w(sv2 w2t)

>

WIT(sv2w2t),

wl * w 2 (F,1 B), a contradiction . Hence w1 77 is a fuzzy adjunctive language, w = w I B U w I T, and w I B n w 1a = xO . Suppose that IXI = 1. Since w is not fuzzy adjunctive, by Definition 7.9.7, w is a fuzzy regular language . If wIa is not fuzzy adjunctive, then wIa is fuzzy regular . Since w = WIa U WIB, w, = (w I B), U (WIT) , VC E [0,1] . Clearly, w, and (wIa), are regular languages b'c E [0,1] . Since (WI B) c n (w I T)c = Ql b'c E [0,1], (WIB), is a regular language. Also since w is fuzzy regular, I {w(x) I x E X * } I < oo and { (W I B) (x) I x E X * } I < oo. By Proposition 7.10 .2, W I B is a fuzzy regular language . This contradicts that WIB is a fuzzy adjunctive language . Hence wIa is also a fuzzy adjunctive language . Suppose that w is a fuzzy adjunctive language. Now w = wIB U WIT, where W IB is a 0-discrete fuzzy adjunctive language. By Proposition 7.10.15, WI B = WI B, U WIB2 and WIBI n WIB2 = x0, where wIB. and WIB2 are two 0-discrete fuzzy adjunctive languages . Thus i.e .,

w=WI B,

UWIB 2 UWIB .

If WIB2 UwlB is a fuzzy adjunctive language, then the result is true. Suppose that wIB2 U WIBI is not fuzzy adjunctive. Then WIB2 is fuzzy adjunctive. By Proposition 7.10.30, w I B2 U w I a is a fuzzy dense language. From the preceding arguments, it follows that wIa is a fuzzy adjunctive language. Since w = W I B U WIT, the desired result follows . m

7.11

Deterministic Acceptors of Regular Fuzzy Languages

The results in this section are from [228] . It is well known that there is a oneto-one relationship between finite automata, regular (type-3) languages, and regular expressions . This relationship demonstrates the use of regular expressions for describing deterministic finite fuzzy automata and regular fuzzy languages . It introduces a normal form for the production of a regular fuzzy grammar when the max-min rule is used. Specific fuzzy system models based on fuzzy set theory include a description of decision making in a fuzzy environment [14], finite fuzzy automata as learning systems [250], and fuzzy grammars and languages [122, 123]. © 2002 by Chapman & Hall/CRC

38 4


Fuzzy automata, grammars, and languages lead to a better understanding of nondeterministic algorithms and of pattern recognition tasks using syntactic pattern recognition techniques. In this section, an algorithm is developed for constructing a deterministic finite automaton that classifies the strings of a language with a regular fuzzy grammar . The derivations of the grammar are governed by the maxmin rule [122, 123] . An equivalent unambiguous regular fuzzy grammar with productions in a normal form is developed from an extension of this algorithm . Definition 7.11 .1 A regular fuzzy grammar is a four-tuple G = (N, T, S, P), where N is a finite set of nonterminals, T is a finite set of terminals, S E N is the starting symbol, P is a finite set of productions, N n T = 0, and the elements in P are of the form A aET, 0 Q is the fuzzy state transition map, w : Q ----> Y is the output map, and qo E Q is the starting state.

Definition 7 .11.2 differs from the fuzzy machines defined previously in two important ways. First, the interval [0,1] determines the third component of the ordered triple appearing in S's domain rather than containing the image of S. Second the output map is crisp and has Q as its domain rather than Q x X. A regular fuzzy grammar reduces to a conventional grammar when production weights are all equal to 1. Similarly, a finite quasi-fuzzy automaton reduces to a conventional finite-state Moore machine by restricting the transition weights to the value 1. A fuzzy subset of T* is called a fuzzy language in the alphabet T. Given a regular fuzzy grammar G, the membership grade of a string x of T* in the regular fuzzy language L(G) is the maximum value of any derivation of x, where the value of a specific derivation of x is equal to the minimum weight of the productions used. From the max-min rule for a fuzzy language, every string of T* has its highest computable membership grade. It is known [123] that given a regular fuzzy grammar G, a corresponding finite quasi-fuzzy automaton can be constructed that "accepts" the language L(G). The following theorem describes the construction of a deterministic nonfuzzy, finite automaton that computes the membership function of LG) . Unless otherwise specified, we use the symbol X to denote both the automaton input alphabet and the language terminals, i.e., X = T.



38 5

Theorem 7.11 .3 Let G be a regular fuzzy grammar. Then there exists a deterministic Moore sequential machine dfa, with output alphabet Y C_ {c c is a production weight} U {0}, which computes the membership function y: X* ~ [0,1] of the language L(G) .

Proof. We give a five-step algorithm for constructing the dfa. Step 1 : Given the regular fuzzy grammar, obtain the corresponding fqfa. The fqfa is obtained in the same way that a nonfuzzy finite automaton is obtained from a nonfuzzy regular grammar [22], with the exception that a production weight is assigned to the corresponding transition [123] . Step 2 : Obtain the set W of possible nonzero membership grades of strings in the language L(G) . W is taken to be the finite set of distinct production weights or, equivalently, the weights of the transitions of the fqfa. (The reasoning is as follows : (a) the max and min operations do not introduce a weight not al ready assigned to some production and (b) each production weight initially may be the membership grade of some string (or strings) in the language L(G).) Step 3 : For all c E W obtain the regular expression P(c) describing those x E X* such that y(x) > c. (It is known that given a regular fuzzy language and a threshold c, 0 < c < 1, the nonfuzzy "threshold language" L(c) = {x I x E X*, ft(x) > c}

is regular .) The regular expression P(c) can be found in the following method . Examine the transition diagram of the fqfa and retain without weight only those transitions whose weight is equal to or greater than c. This yields a nondeterministic nonfuzzy machine that recognizes the language L(c) and from which the regular expression P(c) can be obtained directly by standard techniques for nondeterministic transition graphs or by conversion to a deterministic finite automaton and solution of the descriptive equations [22] . Step 4: For all c E W obtain the regular expression F(c) describing those strings x of X* such that y(x) = c. (It is known that if F'(cl) and F'(c2) are regular expressions, then so are the Boolean functions of F'(c l ) and F(C2) . Specifically, if F'(cl) and F(C2) define two threshold languages, then the new regular expression F(c2) = F(C2) n F'(cl)

defines the regular language consisting of those strings that are in L(c2) and not in L(cl) .) Consider the finite set W of possible nonzero membership grades in L(G) . Let cl, c2 E W be such that cl > c2 . Then the regular expression © 2002 by Chapman & Hall/CRC


38 6

F(c2) = F(C2) n F'(cl)

defines the set of strings {x I

x E X*,x E

L(c2),x

E L(cl)}

_ _

{x I

x E X*,

p(x) > c2, clI 10) < {x I x E X*, 1t(x) = c2}

F(c2) is an equivalence class of the equivalence relation E on X* defined by (x, y) E E if and only if 1t(x) = ft(y) . This procedure, applied to all pairs of

adjacent membership grades in W beginning with the lowest value, yields the disjoint regular expressions defining the dfa . Step 5: Use the regular expressions F(c) to obtain the state transition diagram of the dfa, where c E W. m The procedure for obtaining a state transition diagram by taking derivations of a regular expression is discussed in [22] . Another method of decomposing a fuzzy grammar into nonfuzzy grammars using the concept of level set can be found in [258] .

Corollary 7.11 .4 Let Gl be a regular fuzzy grammar. Then there is an equivalent unambiguous regular fuzzy grammar G2 in which productions have the form A - aB or A -c -> a, where A, B E N, a E T, and 0 < c < 1 .

Construct the dfa as described in proof of Theorem 7.11 .3 . Then there is in P a production A - I --> aB (or A ~ aB, with weight 1 understood) for each transition S(qA, a) = qB . If w(qB) = c :?~ 0, then there is in P a production A -c -> a for each transition S(gA,a) = qB . The starting symbol S of G2 corresponds to the starting state qs of the dfa. Suppose the terminal string x = ab . . . de causes the dfa started in state qs to halt in a state with output k. Then there is a derivation Proof.

S----> aA----> abB----> -------> ab . . .dD__c___> ab . . .de in G2 . Conversely, such a derivation in G2 yields a string that causes the dfa to terminate in a state with output c. G2 is unambiguous since it is obtained from a deterministic finite automaton. 0 The following example illustrates Theorem 7.11.3 and Corollary 7.11 .4 . If x, y E T* in the following example, then we use the notation x + y to denote {x, y} . © 2002 by Chapman & Hall/CRC


Example 7.11 .5 (216) Consider the regular fuzzy grammar

387 GI = (N, T,

S, P), where N = {S, A, B}, T = {a, b}, and the productions are as follows: 0 .3

S-

0 .3

S-

-> bS

Bib.

0.5 S0.2 S- -> bA

S~aB 0.5 A -> b

Step 1 : The corresponding fuzzy machine is shown in Figure 7.1 . Step 2: W = {0 .7, 0.5, 0.4, 0.3, 0.2} . Step 3: c = 0.2, c = 0.3, c = 0.4, c = 0.5, c = 0.7,

F'(0 .2) F'(0 .3) F' (0 .4) F' (0 .5) F' (0 .7)

= (a + b)*(ab + bb) = (a + b)*ab = ab = ab = 01 .

Step 4: F(0 .2) F(0 .3) F(0 .4) F(0 .5) F(0 .7)

= F'(0 .2) n F'(0.3) = (a+ b) (a+ b)*ab = Ql = ab = 01 .

= (a + b)*bb

Step 5: The deterministic classifier of strings in the regular fuzzy language L(G) is shown in Figure 7.2. Using the method of Corollary 7.11.4, the productions of the equivalent grammar G2 are as follows: S~aA~bB

0.5

A~aC~bD

A

-> b

B~aC~bE

Bib

a/0 .5

Figure 7.1 1 : fqfa obtained from Gl



388

C

aC

bF

D~aC~bE

Cab

E~aC~bE

Deb 0.2 E b

F

Fib.

aC

bE

b

Figure 7.2 2 : dfa obtained from G1

Example 7.11 .6 [28] Consider the string ab . Using G1, S-

0 .5

->

0 .5 ->

aA-

0 .7 ->

abandS-

aB

0.4

->

ab .

We have that ft (ab) = (0 .5 A O .5) V (0 .7/x, 0 .4) = 0 .5 . Using G2, we have that 0.5 S~aA -> ab.

7 .12

Exercises

1. Let .M = (M, t, T) be a fuzzy recognizer of a fuzzy finite state machine M = (Q, X, [t) . Let A = L(M) . If q E Q and t(q) n ft(p, x, q) > 0 for some p E Q and x E X*, then show that x-1 A = X{e} o T, where X{e} is the characteristic function of {q} . 'Figure 7 .1 is from [228], reprinted with permission by Copyright 1974 IEEE . FFggure 7 .2 is from [228], reprinted with permission by Copyright 1974 IEEE .


7.12. Exercises

389

2. (121) Let .M = (M, t, T) be a fuzzy X-recognizer of M = (Q, X, /t) and let .M' = (M', t', T') be a fuzzy X'-recognizer of M' = (Q', X', ft') . Let f : Q ----> Q' and g : X ----> X' . Then (f, g) is called a homomorphism of .M into AT if b'p, q E Q and Vu E X, (1) ft(p, u, q) E. The triple (E, *i, *z) is called a left semimodule over S if © 2002 by Chapman & Hall/CRC

391

392

8. Minimization of Fuzzy Automata

for all a, b E S and x, y E E the following conditions hold: (SM.1) (E, *''I ) is a commutative semigroup with identity ; (SM.2) a *2 (x *' y) (a * 1 b) *2 x

_ =

(a *2 x) *' (a *2 y) (a *2 x) *' (b *2 x)

(SM.3) a *2 (b *2 x) = (a *2 b) *2 x .

A function h : (E, *i, *z) ----> (E, *i, *z) is called a morphism of semimodules if the following properties hold: b'a E S, b'x, y E E, h(x *' y) = h(x) *i h(y) and h(a *'2 x) = a *z h(x) .

We sometimes either use - for the operation *'2 or suppress it all together and we replace *'i with +. We also simply call E a semimodule over S or an S-semimodule. The notion of a right semimodule is defined similarly. If S is a commutative semiring, we call E a semimodule. Let M be an S-semimodule. A set X C_ M is called a system of generators for M if X generates M, i.e ., if every element of M is expressible in the form E', aix2, a2 E S, x2 E X, i = 1, 2, . . . , n for some n E N. A quasi-base is a minimal system of generators for M. If the quasi-base is finite, the dimension of M (denoted dim M) is the cardinality of X. Let X be a set, not necessarily finite, and let S be a semiring. Let VX =

XEX

ax - x, ax

E

S, x

E X,

where ax = 0 except for a finite number of elements x E X. Then it follows that VX is an S-semimodule, called a free semimodule . It also follows that the set X is a minimal system of generators for VX. Proposition 8 .1 .1 Let M be a semimodule over S. Then the following conditions are equivalent . (1) Every increasing sequence of sub-S-semimodules of M, i.e., Ml C_ M2 C_ . . . C_ Mk C . . . , such that MZ :?~ MZ_1, is finite. (2) For every sub-S-semimodule of M, there exists a finite minimal system of generators . (3) Every nonempty set G of sub-S-semimodules of M contains a maximal element.

A semimodule that satisfies any of the properties of Proposition 8.1.1 is called noetherian. Proposition 8 .1 .2 If X is a nonempty finite set, then the free semimodule VX is noetherian.

0


8.1 . Equivalence, Reduction, and Minimization ofFinite Fuzzy Automata393 (1) Let I = [0,1] . Consider the binary operations *1 = max and *2 = min on I. Then the triple (I, *1, *2) = ([0,1], max, min) is a commutative semiring . (2) Let L be a distributive lattice. Then the triple (L, max, min) is a commutative semiring . (3) Let X be a finite set and VX be the free semimodule generated by X over the semiring from (1). The operations on the free semimodule are as follows : Example 8.1 .3

=+ :VXXVX~VX, Lax x+Lbx x=L(ax Vbx)-x, XEX

XVX ----> VX,

=- : [0,1]

XEX

XEX

XEX

XEX

If X is finite, then VX is a noetherian semimodule (see Proposition 8.1 .2) . fuzzy automaton is a quadruple A = are finite sets and /t : Q x X x Y x Q ~ [0,1] .

Definition 8.1 .4 A

where

Q, X, Y

(Q, X, Y, /t),

(The definition of a fuzzy automaton in Definition 8.1 .4 is the same as that of a max-min sequential-like machine in Definition 2.9.1 . We use the differing terminology partially to be consistent with the original papers, but also due to the different approach.) As usual Q is the set of states, X is the input alphabet, and Y is the output alphabet for the fuzzy automaton A. Let y(qj , x2 , yr, qk) = a~ E [0,1] . If the interval [0,1] is replaced by the distributive lattice L (see Example 8.1 .3(2)), we obtain the more general notion of L-automaton, closely related to fuzzy automaton. Every fuzzy automaton A defines the free semimodules V(Q x X) and V(Y x Q) over the semiring [0,1] . Define the function ~ :V(QXX)----> by Vqj E

Q,

Vx 2 E X, r, k

The

V(YXQ)

a; (yr, qk)

array MN = [a~] describes the fuzzy automaton . the words x = xl - X2 2. . . . x1, E X*, y = Y1 ' y2 . . . ' Yp E Y' and the matrices M(x 2 , yr) = [mjk(x2, yr)], where mj k(x 2 , yr) = a~. Denote the max-min product of matrices by o . Then we obtain the expression corresponding

Consider

M(x, y)

=

M(xl, yl) 0 M(x2, y2) 0 . . . 0 M(xP, yP)-


394


If P = P(A,A) is a column matrix with IQ I rows and whose elements are equal to 1, let P(x, y) = M(x, y) o P. Let A = (Q, X, Y, p) be a fuzzy automaton . We call (Q, E) the fuzzy set of initial states, i .e., E is a fuzzy subset of Q. Let q E Q and a E [0,1] . Define Eq : Q [0,1] and Eq : Q ~ [0,1] by dq' E Q, 1

Eq(q) = ~ 0

if q=q' if q4 q'

0 o , Eq(q) = ~ aE[01]

if q=q' if qz,4 q'

with the condition that EgEQ Eq(q) 0 for Eq . The fuzzy automaton A, denoted in this case (A, Eq) (resp. (A, E)), is called initial (resp . weakly initial) . For the fuzzy automaton A, define SE (x, y) = E o P(x, y) . An entry of SE (x, y) indicates the maximal degree of membership for the input word x and the output word y, where (Q, E) is fixed. Let A = (X, Q, Y, y) and A' = (X', Q', Y', y') be fuzzy automata and let E and E' be fuzzy subsets of Q and Q' respectively. Definition 8.1.5 The initial automata (A, E) and (A', E') are said to be equivalent, written (A, E) - (A', E'), if SE (x, y)A = SE' (x, y)A' for all

and yEY* . Let A = A' = (X, Q, Y, /t) . If (A, E) - (A', E'), then E and E' are said to be equivalent on Q, written E - E' . (2) If (A, Eq) - (A', E`), then the states q E Q and q' E Q' are said to be equivalent, written q - q' . (3) A = (X, Q, Y, ft) is said to be equivalently embedded into A' _ (X', Q', Y', /t'), written A ;~ A', if for each q E Q, there exists an equivalent state q' E Q' of A' . (4) A is said to be weakly equivalently embedded into A', written A ;~ A', if for all E : Q ~ [0,1] there exists E' : Q' ~ [0,1] such that (A, E) - (A', E') . (5) A and A' are said to be equivalent, written A - A', if A ;~ A' and xEX* (1)

A'~ A. (6) A and A' are called weakly equivalent, written A s: A' , if A ;~ A' and A' ;~ A . Definition 8.1.6 Let A be a fuzzy automaton. (1) A is said to be in reduced form if for all q, q'

E

Q, q - q' implies

q=q' . (2) A' is called a reduct of A if A' is in reduced form and is equivalent to A . (3) A is said to be in minimal form if for each Eq y (i VQ as follows: t(A, A) = 1: g, qEQ

t(x,y) -

EgjEQpj(xly)gj 0

if Ix1 = lyl if Ix1 z"~ Iy1,

where pj : X* x Y* ----> [0,1] . It follows that t is a morphism of semimodules. We denote its corresponding matrix by Mt . Construct the sequence Eo C El C . . . C_ E of subsets of E = X* x Y* as follows: Eo =EZ

Ei-i U {(x, y) I x E X*, y E Y* , Ix1 = Iy1 = i} .


8.2 . Equivalence of Fuzzy Automata: An Algebraic Approach

397

Proposition 8.2 .1 The following assertions hold: (1) VEZ is a sub-semimodule of VEZ+1 for i = 0,1, . . . . (2) tVE2 is a sub-semimodule of tVE2+1 for i = 0,1, . . . . (3) If tV EZ = tV EZ+1, then tV EZ = tV EZ+ k for k = 0,1, . . . . (4) tVEk = tVEk + 1 for some k E N. Proof. (1) Since EZ is a set of generators (quasi-basis) for the semimodule VEZ, i = 0,1, . . . , and Eo C El C . . . C_ E, we have VEo C VEl C

. . .CVE . (2) The result follows here since t is a morphism. (3) See [176] . (4) The result follows here since VQ is a semimodule that is noetherian and VtE C VQ . m

Theorem 8.2.2 Let (A, E) and (A', E') be weakly initial fuzzy automata. Then (A, E) - (A', E') if and only if E o t = E' o t' .

Proof. If

IxI = I yI

for (x, y) E X* x Y*, then by Definition 8.1 .5, SE (x

and since SE(x, y)

2Y) A

I

= SE' (x, Y) A'

= E o (M (x, y) o P),

we have that

E o (M(x, y) o P) = E' o (M' (x, y) o P) .

This expression is equivalent to E o t(x, y) = E' o t' (x, y) for each (x, y) E X* x Y* such that Ixl = lyl . If Ixl :A jyj , then by the definition of t, t(x, y) = t'(x, y) = 0, i.e ., E o t = E' o t' . Conversely, suppose that E o t = E' o t' . Clearly, E o t(x, y) = E' o t' (x, y) for all (x, y) E X* x Y* . Hence E o t(x, y) = E' o t'(x, y) . However,

Mt(x, y) =

0

(x, y) o P

jjyj if ixi = yj

It follows that E o (M(x, y) o P) = E' o (M'(x, y) o P), i.e ., SE (x, y) A SE'(x, y) A' for all (x, y) E X* x Y* such that IxI = Iyl . A similar result is given in Chapters 2 and 3.

=

Corollary 8.2 .3 Let A be a fuzzy automaton. Then E - E' if and only if SE (x, y)A = SE' (x, y)A for all (x, y) E X* x Y* such that Ixl = lyl < n - 1. Proof. If (A, E) ti (A, E'), then SE(x, y)A = SE' (x, y)A for Ixl = jyj 0,1, . . . . Hence Ixl = lyl < n. If SE (x, y)A = SE' (x, y)A for all (x, y) E X* x Y* such that Ixl = Iyl < n-1, then by Proposition 8.2.1(4), it follows that E o (M(x, y) o P) = E' o (M(x, y) o P) . Thus (A, E) - (A, E') . m This is the fuzzy interpretation of the well-known Carlyle theorem [29] for equivalence of stochastic automata . © 2002 by Chapman & Hall/CRC


398

Corollary 8.2.4 Let A be a fuzzy automaton. Then the following statements are equivalent:

(2) e0Mt =e'0Mt .0

Corollary 8.2.5 Let A be a fuzzy automaton. Then the following statements are equivalent: (1~ (2)

El q-a

-

e'1 . qj

The i-th and j-th rows in the matrix Mt are identical.

Proof. (1)x(2) Suppose that el . - e' 1 . By Corollary 8 .2 .4, el, o Mt eq, o Mt . However, by the construction of eq,, the i-th and j-th rows in Mt are identical . q-~

qj

q-~

(2)x(1) The proof is straightforward.

Lemma 8.2.6 If (A, e) - (A, e'), then dim(Im(t)) = dim (Im(t')) .

Proof. By Theorem 8 .2 .2, e o t - e' o t' ~ e o Mt - e' o Mt , . Thus it follows that dim(Im(t)) = dim(Im(t')) .

Theorem 8.2.7 Let A and A' be fuzzy automata. If e is given, then the problem of finding e', if it exists, such that (A, e) - (A', e'), is algorithmically decidable. m


8.2. Equivalence of Fuzzy Automata: An Algebraic Approach

399

For a proof of Theorem 8.2 .7, see the algorithm in Figure 8.1 .

PRINT: NO EXISTS e" SUCH THAT (A, e) IS EQUIVALENT TO (A",6")

Figure 8.1 1 It is stated in [231] that the computing program is not easy to realize since useful standard programs are missing . 'Figure 8. 1 is reprinted from [231] with permission by Academic Press.



400

8.3

Reduction and Minimization of Fuzzy Automata

The reduction and minimization of fuzzy automata are a consequence of the theory of equivalence of fuzzy automata. They are of importance in applications. In this section, we give a completion of the ideas of Santos, which were presented in Chapters 2 and 3. The next result is closely connected with the problem of reduction of fuzzy automata.

Theorem 8.3.1 Let Mt be the matrix associated with the fuzzy automaton A. If Mt contains two identical rows, then there exist fuzzy automata A' and A", with IQI - 1 states each, such that A A' and A A".

-

-

Proof. Suppose that the rows corresponding to the states qi and qj are identical in Mt. Let Q' = Q\{qi) and Q" = Q\{qj). Then the corresponding matrix Mt, (resp. Mt,,) for the fuzzy automaton A' (resp. A") is obtained from Mt by eliminating the i-th (resp. j-th) row. We show that A A' (resp. A A").

-

-

The equivalent state to q E Q, qi # q # qj is q E Q' (resp. q E Q") and vice versa since E: o Mt = E: o Mtr (resp. E: o Mt = o M p ) . The equivalent state to q = qi, qj E Q, respectively, is the state qi E Q' (resp. qj E Q"). The state equivalent to qi E Q' (resp. qj E Q") is qi E Q (resp. qj E Q). The proof of these conditions is a consequence of the definition of E:, of the construction of Mt.

Corollary 8.3.2 For every fuzzy automaton, there exists a reduced fuzzy automaton. All reduced fuzzy automata associated to a given fuzzy automaton have sets of states of the same cardinality. rn

Theorem 8.3.3 For finite fuzzy automata, the relation of equivalence is decidable. rn

The block-scheme (Figure 8.2) of the algorithm proving the equivalence


8.3. Reduction and Minimization of Fuzzy Automata of two fuzzy automata A and A' is in fact the proof of Theorem 8.3.3.

Figure 8.22

2 Figure

8 .2 is reprinted from [231] with permission by Academic Press .


401

402


PRINT : A" IS NOT EQUIVALENTLY EMBEDDED INTO

A"

Figure 8 .2 (Continued) The following result pertains to the existence and explicit construction of a minimal fuzzy automaton of a given fuzzy automaton . q

Theorem 8 .3.4 Let A = (Q, X, Y, p) be a fuzzy automaton . If E l n En and E0 n contains a fuzzy _ _ 1 E [0,1] as a component, then there exists _ automaton A = (Q, X, Y, y), with ~ Q1 - 1 states such that A ; A. Proof. Suppose =

El

E2

. . .

E .-1

0

Ern+l

. . .

satisfies the condition of the theorem and = [ 0 © 2002 by Chapman & Hall/CRC

0

. ..

0

1

0

. ..

0 1

En

8.3. Reduction and Minimization of Fuzzy Automata

403

. We construct the fuzzy automaton A = (Q, X, Y, 7) _ as follows: Let Q = Q\{qr } and define 7 : V(Q x X) V(Y x Q) as follows : is equivalent to Eq_

n

g(gj, xi) = Lr -a" (y,, qk), r, k

where

at = a~ V (EO M A a~) . We

now show that the states qj, j z,4 m,

with the same indices in A and A are equivalent . For the words with length l = 1, we have

az7. = Vk

farkl = V { V/ O f a rkl k ml ij ml a~

V O

k m

(EO k

= V f a rkl n arm ii ) f kl ij

= ar . ij

In the last equality, recall that Eq~ contains 1 E [0,1] as a component, i.e ., VkO .(Eo A j) j since

a =a Vk

.(E00

rm

n arm))

f0 - (Vk m LE k )

n arm aaj - arm azj .

Suppose the states qj , j :?~ m, are 1-equivalent, i.e ., for every x E X*, y E Y* such that IxI = I yI = l, the following holds: pj (x, y)a = Pi (x, y) A . By the hypothesis, VkO m (EO A pk (x, y)) = pm (x, y) . Thus we have Pi (xix, yry)

n pk (x, 2J))

=

VkOm (aj

=

Pj(xix,yry)

=

(VkOm(aj APk(x,Y)))V O (arj n (VkOm{(E APk(x,y))})) Vk(aj APk(x,y))

That is, the states with the same indices for automata A and A are (l + 1)equivalent and thus equivalent . For each El

E2

.. .

Em-1

Em+1

...

En

for A, there exists an equivalent E =

E2

E1

. ..

Em-1

0

Em+1

. ..

En

for the automaton A. For a given E = (El, E2, . . . , En ) for the automaton A, the corresponding equivalent 7 = (71, 72, . . . , Vin) for A is defined by the equation 7i = Ei V (Em A E°), i z,4 m, where E° is the i-th component of the vector E ° . Thus 9-

SE(x, y)A

=

E

o P(x, y)

vi (Ei n pi (x, y))

(E° (Viom{(Ei APi(x, y))}) V (Em A APi(x, y))) Viom{(Ei V (Em n E9)) n pi (x, y)} V iom{ (Ei n pi (x, Y)) SE(x,y)a-


404


8 .4

Minimal Fuzzy Finite State Automata

Fuzzy finite automata are used to design complex systems. For example, they are useful for a knowledge-based system designer since a knowledgebased system should solve a problem from fuzzy knowledge and should also provide the user with reasons for arriving at certain conclusions . A design tool is more valuable if there exist guidelines to assist the designer to come up with the best possible design. One of the major criteria for a best design is that it be minimal. In this section, we show that a fuzzy finite automaton Ml has an equivalent minimal fuzzy finite automaton M. We also show that M can be chosen so that if M2 is equivalent to Ml, then M is a homomorphic image of M2 . In Sections 8.1-8 .4, we considered fuzzy automata (Q, X, Y, /t), where /t : Q x X x Y x Q ~ [0,1] . Hence /t served both as the state transition function and the output function. In this section, we consider fuzzy automata such that the state transition function and the output function are distinct. We ask the reader to compare the two approaches in the Exercises . A fuzzy finite automaton (ffa) is a five-tuple M = (Q, X, Y, /t, w), where Q, X, and Y are finite nonempty sets, /t is a fuzzy subset of Q x X x Q, and w is a fuzzy subset of Q x X x Y such that the following conditions hold: (1) for all p, q E Q if there exists a E X such that ft(p, a, q) > 0, then there exists b E Y such that w (p, a, b) > 0, (2) for all p E Q, for all a E X if there exists b E Y such that w (p, a, b) > 0, then there exists q E Q such that ft(p, a, q) > 0, (3) for all p E Q, a E X, V{ft(p, a, q) I q E Q} > V{w(p, a, b) I b E Y} . The elements of Q are called states, and the elements of X and Y are called input and output symbols, respectively. /t is called the fuzzy transition function and w is called the fuzzy output function . The approach in this section differs from that of the previous sections in that the fuzzy transition function and fuzzy output functions are distinct in this section . Definition 8.4.1 Let M = (Q, X, Y, /t, w) be a ffa.

(1) Define the fuzzy subset p* of Q x X* x Q as follows: It * (p, A, q)

1 ifp=q

w* (p, xa, q) = V{w(p^ r) A w* (r, x, q) I r E Q}, for allp,gEQ,xEX*, and aEX. (2) Define the fuzzy subsets w* of Q x X* x Y* as follows: _ 1 ifx=y=A w* (p, x, y) - ~ 0 if either x :?~ A and y = A or x = A and y :?~ A ,


8.4 . Minimal Fuzzy Finite State Automata

40 5

w* (p, xa, yb) = V{w(p, a, b) A ft(p, a, r) A w* (r, x, y) I r E Q}, for allpEQ,xEX*, aEX,yEY*, and bEY.

Let M = (Q, X, Y, y, w) be a ffa . Let p, q E Q, a E X, b E Y. Then * ft (p, a, q) = ft* (p, Aa, q) = V {ft(p, a, r) n ft * (r, A, q) I r E Q} = ft(p, a, q) n * * ft (q, A, q) (since ft (r, A, q) = 0 if r :?~ q) = ft(p, a, q) . Also w* (p, a, b) = w* (p, Aa, Ab) = V{w (p, a, b) Aft(p, a, r) A w* (r, A, A) I r E Q} = V{w(p, a, b) n ft(p, a, r) I r E Q} (since w* (r, A, A) = 1) = w(p, a, b) A (V{ft(p, a, r) I r E Q}) = W(P, a, b), where the latter equality holds by condition (3) . We have thus proved the following result.

Theorem 8.4.2 Let M = (Q, X, Y, ft, w) be a ffa. Then (1) ft =w*IQXXXQ, (2)w

= w* IQXXXY~

Lemma 8.4.3 Let M = (Q, X, Y, ft, w) be a ffa. Then for all p, q E Q, x,uEX*,

* ft (p, xu, q) = V{ft* (P, x, r)

n ft * (r,

u, q)

I

r E Q} . .

Lemma 8.4.4 Let M = (Q, X, Y, ft, w) be a ffa. For all p E Q, x E X*, y E Y*, of Ix1 z,4 Iyl, then w* (p, x, y) = 0.

Proof. Let p E Q, x E X*, y E Y*, and Ix1 :?~ Iyl . Suppose Ix1 > Iy1 and let I y1 = n. We prove the result by induction on n. If n = 0, then y = A and x :?~ A. Hence by definition w* (p, x, y) = 0. Suppose the result is true for all u E X*, v E Y* such that Jul > Iv1 and I v1 = n - 1 . Suppose n >_ 1. Write x = ua, y = vb where u E X*, a E X, v E Y*, and b E Y. Then Jul > Iv1 and Iv1 = n - 1. Now by the induction hypothesis, for all r E Q, w* (r, u, v) = 0. Thus w* (p, x, y) = w* (p, ua, vb) = V{w(p, a, b) A ft(p, a, r)A w*(r,u,v)I r E Q} = 0. Hence for all p E Q, x E X*, y E Y*, if Ix1 > Iyl, then w* (p, x, y) = 0. Similarly, by induction, we can show that for all p E Q, x E X*, y E Y*, if Ix1 < Iyl, then w*(p, x, y) = 0. m Theorem 8.4.5 Let M = (Q, X, Y, ft, w) be a ffa. Then statements 1(a)

and 2(a) are equivalent as are 1(b) and 2(b) . (1) (a) for all p, q E Q if there exists a E X such that ft(p, a, q) > 0, then there exists b E Y such that w(p, a, b) > 0; (b) for all p E Q, for all a E X if there exists b E Y such that w (p, a, b) > 0, then there exists q E Q such that ft(P, a, q) > 0. (2) (a) for all p, q E Q if there exists x E X* such that ft * (p, x, q) > 0, then there exists y E Y* such that w* (p, x, y) > 0; (b) for all p E Q, for all x E X* if there exists y E Y* such that w* (p, x, y) > 0, then there exists q E Q such that ft * (p, x, q) > 0.


406


Proof. (1)x(2) : (a) Let p, q E Q, and x E X* be such that p* (p, x, q) > 0. Let l xl =n . If n = 0, then x = A and so p = q . Also A E Y* and w* (p, A, A) > 0. Suppose the result is true for all u E X* such that Jul = n1. Let x = ua, where a E X and u E X*. Then Jul = n-1 . Now p* (p, x, q) = np* (r, u, q) l r E Q} > O implies that p* (p, a, r) A p* (p, ua, q) = V{p* (p, a, r) (r, u, q) > 0 for some r E Q. Hence p(p, a, r) = p* (p, a, r) > 0 and p* /t* (r, u, q) > 0 . Hence by 1(a), there exists b E Y such that w(p, a, b) > 0 . Also by the induction hypothesis, there exists v E Y* such that w* (r, u, v) > 0. Let y = vb . Then w* (p, x, y) = w* (p, ua, vb) >_ w (p, a, b) A p(p, a, r) A w* (r, u, v) > 0. The result now follows by induction . (b) Let p E Q, x E X*, and suppose there exists y E Y* such that w*(p,x,y) > 0. Then by Lemma 8 .4 .3, lxl = lyl = n, say. If n = 0, then x = y = A. Also p* (p, A,p) > 0. If n = 1, then x E X and y E Y and w(p, x, y) = w* (p, x, y) > 0. Thus by 1(b), there exists r E Q such that p(p, x, r) > 0. Hence p* (p, x, r) = p(p, x, r) > 0. Thus the result is true if n = 0 or n = 1 . Suppose the result is true for all u E X*, v E Y* such that Jul = lvl = n - 1. Suppose n > 1 . Let x = ua, y = vb, where a E X, bEY,uEX*,andvEY* . Then Jul =lvl =n-1 .Noww*(p,x,y)= w* (p, ua, vb) > 0 implies w(p, a, b) > 0 and there exists r E Q such that ft (p, a, r) > 0 and w* (p, u, v) > 0. By the induction hypothesis, there exists q E Q such that p* (r, u, q) > 0 . Hence p* (p, x, q) = p* (p, ua, q) > ft (p, a, r) A p* (r, u, q) > 0. (2)x(1): Straightforward. m

Definition 8.4.6 Let MZ =

(QZ, X, Y, pi, w 2) be a ffa, i = 1, 2. Let qZ E QZ, i = 1, 2 . Then q l and q2 are called equivalent, written ql =Q 1 Q2 q2,

if and only if for all x E X*, y E Y* (wl (ql, x, y) > 0 if and only if w2(g2,x,y) > 0) . Ml and M2 are said to be equivalent, written Ml ; M2, if for all ql E Ql there exists q2 E Q2 such that ql =Q,Q2 q2 and for all q2 E Q2 there exists ql E Q1 such that q2 ~: QIQ2 ql . If Ml = M2 = M, say, then we denote the relation JQ~Q2 by Clearly, JQ, is an equivalence relation .

~Ql .

Lemma 8.4.7 Let M = (Q, X, Y, p, w) be a ffa. Let p, q E Q, x E X*, y E Y*, a E X, b E Y. Suppose p(q, a, p) > 0, w* (p, x, y) > 0, and w(q, a, b) > 0 . Then w* (q, xa, yb) > 0.

Proof. Now w* (q, xa, yb) > w(q, a, b) A p(q, a, p) A w* (p, x, y) > 0. 0 Remark 8.4.8 Let M = (Q, X, Y, p, w) be a ffa. Let p, q E Q, x E X*, y E Y*, a E X, b E Y. We assume for the rest of this section that w* (q, xa, yb) > 0, p(q, a, p) > 0, and w (q, a, b) > 0 implies w* (p, x, y) > 0 .

Proposition 8 .4.9 Let MZ = (QZ, X, Y, pi, w2) be a ffa i = 1, 2 . Let gj,p2 E QZ, i = 1,2 . Suppose ql ~:QIQ2 q2 and there exists a E X such that ft, (gl,a,pl) > 0 and p2(g2,a,p2) > 0 . Then pl ^QIQ2 p2© 2002 by Chapman & Hall/CRC

8.4 . Minimal Fuzzy Finite State Automata

40 7

Proof. Let x E X*, y E Y* . Suppose w*1 (p l , x, y) > 0. By hypothesis, there exists a E X such that ft, (gl,a,pl) > 0 and p2(g2,a,p2) > 0. Since Ml is a ffa, there exists b E Y such that wI(gl, a, b) > 0. Thus by Lemma 8.4 .7, wl (ql, xa, yb) > 0. Hence w2 (q2 , xa, yb) > 0 and w 2 (q2 , a, b) > 0 since ql ~Ql Q2 q2 . Now w2* (q2, xa, yb) > 0, p2 (g2, a, p2) > 0, and W2 (q2, a, b) > 0. Hence by Remark 8.4 .8, w2 (p2, x, y) > 0. Similarly we can show that w2 (p2, x, y) > 0 implies w*1 (pl, x, y) > 0. Thus pl ~QlQ2 p2 . 0 Remark 8.4.10 Let MZ =

(QZ, X, Y, pi, w2) be ffa, i = 1, 2, 3. Let qZ, pi E i = 1, 2,3. (1) Suppose ql ^QIQ2 q2 and q2 ~Q2Q3 q3 . Then for all x E X*, y E Y*, w* (ql, x, y) > 0 if and only if w2 (q2, x, y) > 0 if and only if w3 (q3, x, y) > 0. Hence ql ~QIQ3 q3(2) Let pl ^Q, ql and ql ^QIQ2 q2 . Then for all x E X*, y E Y*, wl (pl, x, y) > 0 if and only if wl (ql, x, y) > 0 if and only if w2 (q2, x, y) > 0. Hence pl ^Ql Q2 q2 . (3) Let pl ^QIQ2 q2 and ql ^QIQ2 q2 . Then for all x E X*, y E Y*, wl (pl, x, y) > 0 if and only if w2 (q2, x, y) > 0 if and only if wl (ql, x, y) > 0. Hence pl ,~Ql ql . QZ,

From Remark 8.4 .10, it follows that we can use the same symbol ,~ for states to be equivalent, whether states are in the same set or different sets. (QZ, X, Y, pi, w2 ) be ffa, i = 1, 2. Let f : Q1 be a function . f is called a homomorphism of Ml into M2 if (1) for all ql E Q1, a E X, b E Y, wl (ql, a, b) > 0 if and only if w2 (f (ql), a, b) > 0, (2) (a) for all pl, ql E Q1, a E X if ft, (gl, a, pl) > 0, then

Definition 8.4.11 Let MZ = Q2

p2 (f(gl),a,f(pl))

> 0,

(b) for all pl, ql E Q1, a E X if p2 (f (ql), a, f(p l )) > 0, then there exists rl E Q1 such that ft, (ql, a, rl) > 0 and f(pl) = f (rl) . M2

phism

is called a homomorphic image of Ml , if there exists a homomorMl ~ M2 such that f is onto.

f :

Definition 8.4.12 Let M = (Q, X, Y, p, w) be a ffa. M is said to be minimal if for all p, q E Q, p = q implies p = q. Theorem 8.4.13 Let Ml =

(Q, X, Y, p l , wl) be a ffa. Then there exists a minimal ffa M such that Ml ~ M. Furthermore, M can be chosen so that if M2 is any ffa such that M2 ~ Ml , then M is a homomorphic image of M2 .


408


Proof. Set Q = {[q] I q E Q, I, where [q] is the equivalence class containing q and is induced by the equivalence relation ;Q1 . Define /t QxXxQ~[0,1]andw :QxXxY~[0,1]by w([q], a, [p]) = V{wl(s, a, t) I s, t E Q1, s ~ q, t ~ p}, and w([q], a, b) =V{wl(s, a, b)

I s E Q1, s ~ q}

for all [q], [p] E Q, a E X, and b E Y. Let ([q], a, [p]) = ([q], a, [p]) . Then q ,~ q' and p ,~ p' . Hence V{ftl(s, a, t) s, t E Q1, s ,~ q, t ,~ p} = V{ftl(s', a, t') I s', t' E Q1, s' ,~ q', t ,~ p'} since is an equivalence relation on Q1 . Thus /t is well defined. Similarly w is well defined . Let [q], [p] E Q. Suppose that there exists a E X such that /t([q], a, [p]) > 0. Then there exists s, t E Q1, s ; q, t s: p such that ft, (s, a, t) > 0. Since Ml is a ffa, there exists b E Y such that wl (s, a, b) > 0. Hence w([q], a, b) >_ wl (s, a, b) > 0. Now suppose [q] E Q, a E X, and there exists b E Y such that w([q], a, b) > 0. This implies wl (s, a, b) > 0 for some s E Q1, s ; q. Since Ml is a ffa, there exists t E Q1 such that ft, (s, a, t) > 0 . Now ft([q], a, [t]) > ft, (s, a, t) > 0. Let [q] E Q, a E X. Since ; is an equivalence relation on Q1, we can write Q1 = Uk 1 [pi], where Q = [PI 1, [p2], . . . , [pk]} and the [pi] are distinct . Now V{w([q], a, [p]) I [p] E Q} =Vklw([q], a, [p2]) =Vk1(V LN1(s,a,t) . Also, V{w([q], a, b) I b E Y} = V{V{wl(s, a, b) I s ~ q} I s ~ q, t ~ pi} b E Y} = V{V{wl (s, a, b) I b E Y} I s ~ q} < V{V{wl (s, a, r) I r E Q 1 }

I[p2]j) s ~ q} = V(V {wl (s, a, r) I s ~ q, r E Uk = Vk 1(V LN'1(s, a, r) s ~ q, r E [pi]}) = Vk1(V{ftl(s, a, r) I s ~ q, r ~ pi}) = V{w([q], a, [p]) [p] E Q} . Consequently, M = (Q, X, Y, ft, w) is a ffa. First we show that for all [q] E Q, x E X*, y E Y*, w* ([q], x, y) > 0 if and only if wl (q, x, y) > 0. Let [q] E Q, x E X*, y E Y*, and wl (q, x, y) > 0 . Then IxJ = Iyj = n, say. If n = 0, then x = y = A. Thus w*([q], x, y) = w* ([q], A, A) = 1 > 0. If n = 1, then x E X and y E Y. Hence wl (q, x, y) = wi(q,x,y) > 0 . Hence w * ([q], x, y) = w([q],x,y) = V{wl(s,x,y) I s E Q1, s ~ q} >_ wl (q, x, y) > 0. Thus if n = 0 or n = 1, then wl(q, x, y) > 0 implies w* ([q], x, y) > 0. Suppose the result is true for all u E X*, v E Y* such that Jul = Iv1 = n - 1 . Let n > 1 . Now x = ua, y = vb for some aEX,bEY,uEX*,vEY* . Then Jul=lvl=n-1.Thuswl(q,x,y)= wl (q, ua, vb) = V{wl (q, a, b) A ft, (q, a, r) A wl (r, u, v) r E Q1 } > 0 implies that there exists r E Q such that wl(q, a, b) > 0, ft, (q, a, r) > 0, w* (r, u, v) > 0. By the n = 1 case and the induction hypothesis, w([q], a, b) > 0 and w* ([r], u, v) > 0. Also ft([q], a, [r]) > ft, (q,a,r) > 0 . Hence w* ([q], x, y) = w*([q],ua,vb) > w([q], a, b) /alt([q], a, [r]) Aw*([r], u, v) > 0. The result now follows by induction . Now suppose w* ([q], x, y) > 0 for some [q] E Q, x E X*, y E Y* . Then Ix1 = Iy1 = n, say. If n = 0, then x = y = A and so wi(q,x,y) = 1 > 0 . If n = 1, then x E X and y E Y and so w([q], x, y) = w*([q], x, y) > 0 .


8.4. Minimal Fuzzy Finite State Automata

409

Thus there exists s E Q1, s ; q such that w l (s, x, y) > 0. Since s s: q and w l (s, X, y) > 0, we must have w l (q, x, y) > 0, i.e., wl (q, x, y) > 0. Thus the result is true for n = 0 and n = 1 . Make the induction hypothesis that the result is true for all u E X*, v E Y* such that Jul = lvl = n - 1. Let n > 1 . Now x = ua, y = vb for some a E X, b E Y, u E X*, v E Y* . Hence Jul = lvl = n - 1 . Thus w*([q],x,y) = w*([q],ua,vb) = V{w([q],a,b) A ft([q], a, [r]) Aw*([r], u, v) l [r] E Q} > 0. This implies that there exists [r] E Q such that w([q], a, b) > 0, ft ([q], a, [r]) > 0, w*([r], u, v) > 0. Now ft([q], a, [r]) > 0 implies ft, (s, a, t) > 0 for some s, t E Q1, s s: q, t s: r. Since Ml is a ffa, there exists d E Y such that wl (s, a, d) > 0 . Thus wl (q, a, d) > 0 since s ; q . This implies that there exists t' E Q1 such that ft, (q, a, t') > 0 since Ml is a ffa . Now since s ; q, ft, (s, a, t) > 0 and ft, (q, a, t') > 0, we have that t ,~ t' by Proposition 8.4.9 . Thus t' ,~ r. Now w ([q], a, b) > 0 implies w l (q, a, b) > 0 by the n = 1 case and w* Qr], u, v) > 0 implies wl (r, u, v) > 0 by the induction hypothesis. Since r ; t', w*1 (t', u, v) > 0. Hence wl (q, x, y) = wl (q, ua, vb) >_ wl (q, a, b) A ft, (q, a, t') A wl (t', u, v) > 0. This proves our claim . Let [q], [p] E Q and [q] ,~ [p] . Let x E X* and y E Y* . Now wl (q, x, y) > 0 if and only if w* ([q], x, y) > 0 if and only if w*([p], x, y) > 0 (since [q] ; [p]) if and only if wl (p, x, y) > 0. Thus q ; p and so [q] = [p] . Hence M is minimal. Since for all q E Q1, x E X*, y E Y*, w*1(q,x,y) > 0 if and only if w* ([q], x, y) > 0, q ~ [q] . Thus Ml ,~ M. Let M2 = (Q2, X, Y, ft2, w2) be a ffa such that M2 ; MI . Define f Q2 ~ Q as follows : for all q2 E Q2 there exists ql E Q1 such that q2 ~ ql . Define f (q2) = [ql] . Suppose q2, p2 E Q2 and q2 = p2 . Then there exists gl,pl E Ql such that q2 ,~ ql and p2 ,~ pl . Thus by Remark 8.4 .10, ql = pl and so [ql] = [pl] . Hence f is well defined . Now for all [ql] E Q, ql = [ql] . There exists q2 E Q2 such that q2 ,~ ql . Now f (q2) = [ql] . Hence f is onto. We now show that f is a homomorphism . (1) Let q2 E Q2, a E X, b E Y. Suppose w2 (q2, a, b) > 0. Since M2 ,: Ml, there exists ql E Q1 such that q2 ; ql . Then wI(gl, a, b) > 0 . Thus w (f (q2), a, b) = w([ql] , a, b) > wl (ql, a, b) > 0. Now suppose w (f (q2), a, b) > 0. Let f (q2) = [ql] . Then q2 = ql . Since w([ql], a, b) = w (f (q2 ), a, b) > 0, w l (s, a, b) > 0 for some s E Q1, s ~ ql . Since s ~ ql, w l (ql, a, b) > 0 . This implies that w2 (q2, a, b) > 0 since q2 ,~ ql . Hence w2 (q2, a, b) > 0 if and only if w(f (q2), a, b) > 0 for all q2 E Q2, a E X, b E Y. (2) First suppose ft2 (q2, a, p2) > 0 for some q2, p2 E Q2 and a E X. There exists gl,pl E Q1 such that q2 s: ql and p2 s: pl . Thus f (q2) = [ql] and f (p2) = [PI I . Since ft2(g2,a,p2) > 0 and M2 is a ffa, there exists b E Y such that w 2 (q2 , a, b) > 0. This implies that w l (ql, a, b) > 0 since q2 ~ ql . Since Ml is a ffa, there exists tl E Q1 such that ft, (ql, a, ti) > 0 . Since q2 ^ ql, ft, (gl, a, tl) > 0 and ft2 (q2, a, p2) > 0, we have that tl ~ p2 by Proposition 8.4 .9. By Remark 8 .4.10, tl ~ pl. Hence ft(f (q2), a, f (p2)) = [t([gl], a, [pl]) > ft, (ql, a, tl) > 0. Now suppose ft(f (q2), a, f (p2)) > 0 for some q2 , p2 E Q2 and a E X. There exists gl,pl E Q1 such that q2 ; ql and © 2002 by Chapman & Hall/CRC

410


p2 ^ P1, f (q2) = [q,] and f (p2) = [p,] . Thus ft([q,], a, [p,]) > 0. This implies /t l (s l , a, tl) > 0 for some sl, tl E Q1, s, s: ql, and tl s: p l . Since M, is a ffa, there exists b E Y such that w, (sl, a, b) > 0 . This implies w, (ql, a, b) > 0 and so W2 (q2, a, b) > 0 since q2 ; ql . Since M2 is a ffa, there exists r2 E Q2 such that fI2(g2, a, r2) > 0. Finally we show that f (r2) = f (P2) . Since s, q, and q2 ,~ ql, s, ,'= q2 by Remark 8.4.10. Hence by Proposition 8.4.9, tl ,: r2 . Once again by Remark 8.4 .10, p, ~ r2 . Hence f (r 2 ) = [PI ] = f (p2) .

8 .5

Behavior, Reduction, and Minimization of Finite L-Automata

In the remainder of the chapter, we consider [174] . Issues concerning the behavior, equivalence, reduction, and minimization have been completely studied for deterministic, non-deterministic, and stochastic automata [220] . The approach given in Chapter 2 and Sections 8.1-8.3 Q203, 231] also) for fuzzy automata is useful if it is known how to solve systems of linear equations over the bounded chain ([0,1], max, min). It is established in [203] that if the system is compatible, the solution is among the m-fold variations with repetitions over n elements . Since the number of these variations is Mn' the time complexity function of a search manner algorithm is exponential . Following [170], a polynomial time algorithm for solving systems of linear equations over IL is given. This result is purely algebraic . It allows one to compute the behavior matrix, to study approximate (E-) equivalence, E-reduction, E-minimization, and to prove their algorithmical decidability for each finite L-automaton. Moreover, the concept of computational complexity and algorithmical decidability as described in [220] and the properties of the chain according to [125] are used here. The terminology for automata theory is as in Sections 8.1-8.3.

8 .6

Matrices over a Bounded Chain

We write IL for the bounded chain IL = (L, V, A, 0,1) over the linearly ordered set L with upper and lower bounds 1 and 0, respectively. Let I, J be index sets and K a finite index set. Let MIXJ denote the matrix [mid], where mid E L for each (i, j) E I x J. Then MIXJ is referred to as a matrix over IL. The matrix MIXJ = [mid] is the product of AI X K = [aik ] and BK X J = [bkj], if mid =Vk E K(aik Abk j ) for each i E I and each j E J. Let f : L ---> [0,1] be an injective function such that x - y ===> f (x) f (y) . The distance with respect to f between x and y for x, y E L is © 2002 by Chapman & Hall/CRC

8.6. Matrices over a Bounded Chain

411

defined as follows : df(x,y) = If (x) -f(y)1 .

Let Ix - yI = df(x,y) for x,y E L . In the remainder of the chapter, we assume f to be fixed. Let AIX J = [aij] and B IX J = [bij] be matrices over IL and E E [0,1] be fixed. We say that the i-th row in A is E-close to the k-th row in B if I ais - bks I S E for each s E J. We also say that A and B are E-close, written d(A, B) S E, if I aij - bij I S E for each i E I and each j E J. Proposition 8.6 .1

If A,, X i = [a i] and IV{ai I

i E I}

Bnx l =

-V{bi I

[bi] are E-close, then

i E III b2 .

We are interested in determining if there exists xj E L such that aij A xj - b2 . We mark the i-th equation in (8.1) in a marker vector IND if aij A xj = bi holds. Theorem 8.7.1 [170] Consider the system AoX = B. Then the following statements hold . (1) If k is the greatest number of the row with a G-type coefficient in the jth column of 13, then x j = bk implies aij A xj = bi for i = k and for each i > k with aij = bi as well as for each i' < k with ai,j >, b2, = b2 . (2) If the j-th column in 13 does not contain a G-type coefficient and r is the smallest number of the row with E-type coefficient in the j-th column, then xj = bT means aij A xj = bi for each i > r with aij = b2 . (3) If the j-th column in 13 does not contain either a G-type or an E-type coefficient, then aij A xj < bi for each xj E L . m


8.7. Systems of Linear Equivalences over a Bounded Chain The implementation of Theorem

8 .7 .1

413

gives the following.

Algorithm 8.7 .2 [170] For computing a solution of the system (8.1): 1. Enter the matrices A, B. Form the matrix 13 . Erase IND . j=0. 3. j=j+1. 4. If j>ngoto8 . 5. If the j-th column in 13 does not contain a G-type coefficient, then go to 6. Otherwise xj = bk, put marks in IND for i = k, for each i > k with aij = bi and for each i < k if aij >, bi = bk . Go to 3. 6. If the j-th column does not contain an E-type coefficient then go to 7. Otherwise xj = b,., put marks in IND for i = r and for each i > r with aij = b2 . Go to 3. 7. xj = 1 . Go to 3. 8. If there exists at least one unmarked row in IND then the system is unsolvable . If all rows in IND are marked, then the system is solvable and the point solution is determined in steps 5,6, and 7. Theorem 8.7.3 [170] The following problems are algorithmically decidable in polynomial time for the system (8 .1) : (1) whether the system is solvable or not; (2) computing a point solution if the system is solvable; (3) obtaining the numbers of the contradictory equations if the system is unsolvable . 0 An n-tuple (X,, .Xn ) with Xi C L for i = 1, . . . , n is an interval solution of (8 .1) if each (XI, . . . , xn) with xi E Xi is a point solution of (8 .1) and (XI , . . . , Xn) is maximal with respect to this property. Using Algorithm 8 .7 .2, the interval solutions of (8 .1) can be found . For j, j' E J, define j - j' if bij = bij , for each i E I. Then - is an equivalence relation on J. Let [j] = {j' j' E J and j - j'} . Algorithm 8.7 .4 [170] For obtaining the interval solutions of the system (8 .1), if (8 .1) is solvable : 1. Compute the point solution by Algorithm 8 .7 .2 . Obtain the equivalence classes [j] for J. 3. For each j E J (i) if the j-th column in 13 does not contain a G-type coefficient, then go to 3(iv); (ii) of I [j] I = 1 then Xj = {bk} ; go to 3 for the next j ; (iii) for each j' E [j] form Xj, as follows: X, _ go to 3 for the next j ;


{bk} bk]

ifj=j if j 7~Y~[0, ;

414


(iv) if the j-th column does not contain an E-type coefficient, then go to 3(vii), (v) of I [j] = 1, then Xj = [b,,1] ; go to 3 for the next j; (vi) for each j' E [j] form Xj, according to the rule Xj,

= ~ [b,,1]

if j = if j jjL l ;

go to 3 for the next j ; (vii) if the j-th column contains only S-type coefficients, then Xj = L for each j E [j] ; go to 3 for the next j .

Theorem 8.7.5 [170] The time complexity function of each algorithmical realization for finding all interval solutions of (8 .1) is exponential. 0

The column matrix B IX{1} _ [bi] is called a convex linear combination of Aj = [a2j]IX{i}, j E J, with coefficients x j E L if B = (Al A x l ) V . . . V (A n A xn ), i.e ., bi = VjE j(aij A x j ) for each i E I. The next result follows from Theorem 8.7 .3 and Algorithm 8.7 .2 . Corollary 8.7.6 Let Aj, j

E J, and B be as above. It is algorithmically decidable whether B is a convex linear combination of Aj, j E J. m

.9 .7 .1 .7 1 .3 .2 .5 .6 G S S E G S = S E G .8 and 0 are .5

Example 8.7.7 Consider the system AoX = B, where A = X=

xl x2 x3

, and B =

.8 .7 .5

. It can be seen that 13

.8 .7 is a maximal solution and that .5 minimal solutions. and that

8 .8

.8 .5 0

Finite IL-Automata-Behavior Matrix

An IL-automaton is a quintuple A = (Q, X, Y, M, IL), where (1) Q, X, Y are nonempty sets of states, input letters, and output symbols, respectively; (2) IL = (L, V, A, 0,1) is a bounded chain ; (3) M = {M(x, y) = [mqq' (x, y)] I x E X, y E Y, q, q 1 E Q, mqq ' (x, y) E L} is the transition-output matrices (the stepwise behavior) of A. If Q, X, Y are finite, then A is called finite IL-automaton . We denote by AO the class of all finite IL-automata. The membership degree mqq ' (x, y) determines the stepwise transitionoutput behavior of A as follows : in step k the automaton is in state q and © 2002 by Chapman & Hall/CRC

8.8. Finite IL-Automata-Behavior Matrix

415

receives the input letter x. It outputs letter y in step k and reaches the state q' in step k + 1 . In order to define and consider the complete input-output behavior of A = (Q, X, Y, M, IL) E Ao, we introduce the following notation: As usual, X* (resp. Y*) is the free monoid over X (resp . Y) with the empty word A as the identity ; U = [u2j] is the square unit matrix: u 2j = 0 if i z,4 j; u2j = 1 if i = j; E=[ 1 1 . .. 1 ] is the column matrix with all elements 1 ; QXt1} (X x Y)* = {(x, y) I x E X*, y E Y*, Ixl = y1} . The matrix M(x, y) defines the transition-output behavior of A E AO for (x, y) = (x1 . . . xk, y1 . . . yk) E (X x Y)* if M(x, y)

_

M (x1, yl) o . . .

o M(xk, yk)

for (x, y) :?~ (A, A), if (x, y) = (A, A) .

The input-output behavior of A for (x, y) E (X x Y)* is defined by the matrix T(x, y) T(x, y)

= ~ E (x, y)

oE

if (x, y) (A, A), if (x, y) _ (A, A) .

Every element mqq ~ (x, y) in M(x, y) defines the operation of A under the input word x beginning at state q, outputting the word y and reaching the state q' . Every element t,, (x, y) in T(x, y) defines the operation of A when it receives the word x beginning at state q and outputs the word y. From the matrices T (X1 y) I we shall construct the semi-infinite matrix T of the complete input-output behavior of A and the finite submatrix B of the behavior of A. Suppose that a lexicographic order on (X x Y)* is given . Let T(i) be a finite matrix with columns T(x, y) ordered according to the lexicographic order on (X x Y)* and such that Ix1 = lyl _ i. By definition, T(0) = T(A,A) = E. Let T be the semi-infinite matrix of the complete inputoutput behavior of A with columns T(x, y) indexed according to the above order. Let B(i) be the finite matrix obtained from T(i) by omitting all columns that are convex linear combinations of the previous columns . Let B be the matrix constructed from T by omitting the columns that are convex linear combinations of the previous columns. The matrix B is called the behavior matrix for the IL-automaton A. For arbitrary matrices C', C" we write C' C_ C" if each column of C' is a column in C" . If C' C_ C" and C" C_ C', then we write C' - C" . Clearly, B(i) C_ T(i) C_ T(i + 1) C_ T for each i = 0,1, . . . . Let .M = {mee, (u, v) I u E X, v E Y, q, q' E Q} and .F = {tq (x, y) I (x, y) E (X x Y)*, q E Q} be the sets of the distinct entries for the matrices M(u, v) E M and T, respectively. For any A E Ao, we have ,F C .M U {1}, I A41 = r and ICI = s are finite, and s _ r + 1 . These properties follow from the chain properties and the algebraic operations for matrices over IL. © 2002 by Chapman & Hall/CRC

416


The following result is proven in Chapter 2 and Sections 8.1-8 .3. For IL = ([0,1], V, A, 0, 1), the generalization to the next result is easy. Theorem 8.8.1 Let A E Ao . Then the following statements hold:

(1) If there exists iE hY such that T(i) - T(i + 1), then T(i) - T(i + l) for all l = 0, 1, 2, . . . . (2) If there exists iE hY such that B(i) = B(i + 1), then B(i) = B(i + l) for all l = 0,1, . . . . (3) There exists an integer k S s1Q1 - 1 such that T(k) - T(k+ 1) and B(k) = B. m

Corollary 8.8.2 The behavior matrix of any AEAO is finite . m For equivalence, reduction, and minimization of automata, the main question is how to compute B. Thus for any column in T, we must determine whether it is a convex linear combination of the previous columns in order to select B from T. Recalling Algorithm 8.7.2, Theorem 8.7.3, and Theorem 8.8 .1(3), we propose the following algorithm for completing the behavior matrix . Algorithm 8 .8.3 Computation of the behavior matrix: 1. Enter M, s, Q1_ 2. Obtain k S s1Q1 - 1 such that T(k) - T(k + 1) . 3. Use Algorithm 8.7.2 to select B(k) from T(k) . 4. B = B(k) .

8 .9

E-Equivalence

Let A = (Q, X, Y, M, IL) E Ao be given. Let E E [0,1] . The states q, q' E Q are called E-equivalent, written qEq' if for all (x, y) E (X x Y) *, tq (x, y) - tq(x, y) I 0 if and only if ft (q, x, q') A w(q, x, y) > 0. 3. Prove (3) of Proposition 8.2 .1 . 4. If M and M' are strongly equivalent, prove that V(q, x, y, q') E Q x X* x Y* x Q, S* (q, x, y, q') = ft* (q, x, q') = w*(q, x, y) .

5. If M and M' are equivalent, prove that V(q, x, y, q') E Q xX* xY* 6 * (q, x, y, q~) = ft * (q, x, q~)

n w* (q, x, y) .

6. If M and M' are weakly equivalent, prove that V(q, x, y, q') E Q x X* x Y* x Q, S* (q, x, y, q') > 0 if and only if ft* (q, x, q') A w* (q, x, y) > 0. 7. Compare minimality requirements of M and M' under various equivalence assumptions . 8. Consider the system AoX = B of Example 8.7 .7. Use the algorithms .8 .8 of Section 8.7 to show that .7 is a maximal solution and .5 .5 0 .8 d 0 are minimal solutions . .5


Chapter 9 L-Fuzzy Automata, Grammars, and Languages 9 .1

Fuzzy Recognition of Fuzzy Languages

In this chapter, we consider fuzzy automata, grammars, and languages, where the interval [0,1] is replaced by a lattice . We begin our study by considering the work of [94]. We give an application of the theory of fuzzy recognition to the theory of probabilistic automata and discuss the closure properties of some fuzzy language classes corresponding to machine classes. In formal language theory, languages are classified by the complexities of machines that recognize them. Typically, the machines are finite automata, push-down automata, linear bounded automata, and Turing machines . It is also of interest to develop the theory of recognition of fuzzy languages by machines and their classification by the complexities of machines that recognize them. The theory should be a reasonable extension of the ordinary language recognition theory. In ordinary language recognition theory, a machine is said to recognize a language L if and only if for every word in L, the machine decides that it is a member of L and for a word not in L, the machine either decides that it is not a member of L or loops forever . That is, a machine may be said to recognize a language if and only if the machine computes the characteristic function of the language. Thus it is natural to define a machine to recognize a fuzzy language if and only if the machine computes its fuzzy membership function . The question arises concerning the meaning for a machine to have a fuzzy membership function. In ordinary language theory, it is defined that for a given input word a machine computes the characteristic function value at 1 if and only if it takes one of special memory-configurations, such as configurations with a © 2002 by Chapman & Hall/CRC

423

424

9. L-Fuzzy Automata, Grammars, and Languages

final state and configurations with the empty stack for cases where the machine has pushdown stacks . Hence a straightforward extension is to define each fuzzy membership function value to be represented by some memory configuration of the machine. In other words, the memory configuration, which the machine moves into after a sequence of moves for a given word, should be uniquely associated with the membership function value of the word. Even if the machine obtains a configuration uniquely associated with the membership function value of the word, we must have that the machine computes the membership function value of the word. We can be sure that the machine knows the value if it is to be able to answer exactly any question about the value . Furthermore, it is also felt in [94] that it is essential that the values represented by memory configurations of the machine are lattice elements . Thus, we require that the machine should be able to compare the fuzzy membership function value of a given word, which is stored in its memory, with any lattice element that is given to the machine as a question. Such a lattice element will be called a cutpoint . We assume that a cutpoint is represented by an infinite sequence of symbols in a finite alphabet, following the infinite expansions of decimals . For any cutpoint c, the machine having the memory configuration associated with the fuzzy membership function value ft(x) of a given input word x should be able to read, as a subsequent input, the infinite sequence corresponding to the cutpoint c sequentially, and after a finite step of deterministic moves, it can determine which of the following four cases is valid, (1) ft(x) > c, (2) p(x) < c, (3) p(x) = c, and (4) p(x) and c are incomparable.

9 .2

Fuzzy Languages

Let X be a finite set of symbols called an alphabet . Let L be a lattice with minimum element 0. An L-fuzzy language over X is defined to be a function from X* into L. At times, we omit the symbol L if its presence is clear . If /t is an L-fuzzy language over X and x E X*, then p(x) represents the membership grade for x to be in the L-fuzzy language . We consider languages associated with an L-fuzzy language /t and a cutpoint c, namely LG(w,

C) =

{x E X* I w(x) > C},

LGE(w,C) = {x E X* I w(x) > C} . We call such languages cutpoint languages for /t and c. Let A be a finite alphabet . Let 0°° be the set of all infinite sequences of symbols in A extending infinitely to the right. A one-to-one function r from L into 0°° is called a representation of L over A if there exists © 2002 by Chapman & Hall/CRC

9.2. Fuzzy Languages

425

a function d : r(L) x r(L) , N U {0} that assigns to any distinct two elements r(l) and r(m) in r(L) a positive integer d(r(l), r(m)) such that d(r(l), r(m)) = d(r(m), r(l)), d(r(l), r(l)) = 0 and conditions (1) and (2) below are satisfied . (1) Let l, m E L, l :?~ m. Then for the prefix wl of r(1) and the prefix w, of r(m) of length d(r(l), r(m)), respectively, and for any a',,3' E A-, either condition (a) or (b) holds: (a) Either w, a' or w,,3' is not in r(L). (b) Both w, a' and w,,3' are in r(L), and r -1 (wia') > r -1 (wr ,3~) if l > m r -1 (wia' ) < r -1 (w ,/3') if l < m and r -1 (wia') and r-1 (w ,,3') are incomparable if l and m are incomparable. (2) For any l, m, andninLwith 1>m>n, d(r(1), r(n)) S d(r(1), r(m)) A d(r(m), r(n)) . For a,,3 E r(L), d(a,l3) is called the D-length of a and,3 . For l E L, r(l) is called the representation of l with respect to r . A lattice does not always have a representation . However, we consider from now through Section 9.8 only lattices that have a representation over some finite alphabet . Condition (2) means that representations of lattices are restricted to those that have decimal expansions of real numbers . However, there may be many representations for a lattice . Example 9 .2 .1 Let L be a lattice with a finite number of elements . If L has elements 11, . . . , lk, let A = {h, . . . , lk} . If r(l2) = lilil2 . . . for 1 S i 5 k, then r is a representation of L over A .

Example 9.2.2 Let L[0,1] be the set of all real numbers in [0,1] with the usual ordering. A representation rl is given as follows: Let A = {0,

1, 0, 1} . For l E L[0,1], let e(l) be the binary expansion of l not of the form w111 . . . with w E {0,1}*0 . For any rational number l in [0,1], set e(l) = wowlwlwl . . . such that if e(1) = w'owlwlwl . . ., then ~wo~ 5 Iwol

and ~wlI S ~wij . Let e (l) = wo wl wlwl . . . , where wl=ala2 . . . ak for wl = ala2 . . . ak with a2 E {0,1}, i = 1, 2, . . . , k. Then let r, (1) =e(1) if l is irrational, rl (l) =


e (l) if 1 is rational .

42 6


Example 9 .2 .3 Let L[o

1] u

denote the set of all rational numbers in [0,1]

with the usual ordering. Let A = {0, 1, 0, 1} . Define a representation r2 : L[o 1] u

--->

A' as follows: r2(0)

= 000,-

. . , r2(1) =111, . . . , and for l

such that e(l) = w000 . . . with w E {0,1}*1, define r2 (l) = w 000 . . . and for other l in L[o,l],, define r2(l) = e(l) .

9.3

Fuzzy Recognition by Machines

Machines treated in Sections 9 .3-9 .8 may be finite automata, pushdown automata, linear bounded automata, and Turing machines . These machines can be represented in the following manner . A machine has an input terminal that reads input symbols and A sequentially, a memory storing and processing device, and an output terminal . Formally, a machine is defined to be an 8-tuple M = (1b, I', '@, 0, S, S2, K, -yo) , where -D is a finite set of input symbols ; I' is a finite set of memory-configuration symbols ; T is a finite set of output symbols ; S2 is a finite set of partial functions {w2 } from I'* into I'* ; 0 is a partial function from I'* into IF', for some n > 1 ; (For a memory configuration 'Y E I'*, 0(-y) designates the instantaneously accessible information of 'Y by M .) ; S is a partial function from (,b U {A}) x I+n into P(Q) ;

is a partial function from I+n to @ U {A} ; -yo is an element in I'*, called the initial memory configuration . A memory configuration ~ is said to be derived from a memory configuration 'Y by u E -D U {A}, denoted by 'Y ===> ~~, if and only if there exists K

n

p = 0(y) and cot E S(u, p) such that w2(y) = ~ . Let w E V. A memory configuration ~ is said to be derived from a memory configuration 'Y by w, written y ==~> ~~, if and only if there exist ul, u2, . . . , ul with u2 in b U {A} w

such that w = ulu2 . . . ul, and there exist yo,'yl . . . . . . l with y2 E I'* such that yo = y, yi = , and y 2 ===> y2+1 for all 0 S i S l - 1 . (-y ===> -y is

valid for all y E I'* .) Given an input word w E V and the initial memory configuration -yo first, M reads input symbols or A sequentially along w, changes memory configurations step by step possibly in a nondeterministic way and reaches y such that -yo ==~> y, and emits the output K(0('Y)) .

Clearly, a machine M = ('D, I', '@, 0, S, S2, K, y o ) can be restricted to a specified family of automata such as Turing machines, pushdown automata, and finite automata for appropriate choices of I', 0, S, S2, K, and 'Yo . A machine M = (,D, I', IQ, 0, S, S2, K, y o ) is called a deterministic machine if for any memory configuration y such that -yo ===> y for some x

x E V, if S(A, 0(y)) :?~ 0, then S(A, 0(y)) contains at most one element


9.3. Fuzzy Recognition by Machines

427

and S(u, 0(y)) = Ql for any u E -D, and if S(A, 0(y)) = 0, then for any u E -D, S(u, 0(y)) contains at most one element . Let L be a lattice with a minimum element 0 and tt : X* ---+ L be an L-fuzzy language over an alphabet X. Let r be a representation of L over an alphabet A. A machine M = (1b, I','@, 0, S, S2, K, -YO) is said to fuzzy recognize ft with r if and only if the following conditions hold: (1) -D = X U A U {t{, where t is an element not in X U A. (3) There exists a partial function v from I'* into L such that the conditions (i)-(iv) hold: (i) Let Sx = {y I yo ~ y{ for all x E X* . Then for all x E X* such that Sx :?' 01, Sx C Dom(v) and v x = V{v(y) I y E SxJ exists. If Sx = 01, v x is undefined_ (ii) Let y be any memory configuration in Sx . Let My denote the machine A r, "If, 0, S', 0, K,'Y), where S' is the restriction of S over (A U {A{) x I'n . Then My is a deterministic machine . (iii) For any memory configuration y E I'*, if K(0(-Y)) is in 1Q, that is, K(0(-Y)) :?~ A, then for any u E -DU{A{, S(u, 0(y)) is empty . Also, K(0(-Y)) is in T only if yo ===> y for some x E X*and y in PRE-,(L) . (wl is called xty

a prefix of a word or an infinite sequence a if a = w l , 3 for some 3. Let lI be either a set of words or a set of infinite sequences . PREII is the set of all prefixes of elements in II .) (iv) Let y E Dom(v) . For all l E L, there exists a prefix v of r(l) and y' in I' such that y ==~> y' and the following properties hold: v

is K(0('Y')) is K(0('Y')) is K(0('Y')) is K(0('Y'))

> = < ! if

if v(-Y) > l, if v(y) = l, if v(-Y) < l, v(y) and l are incomparable .

Consider xt in X*t, where t indicates the end of the input sequence. Then a machine M moves possibly nondeterministically into some memory configuration y such that v(y) is defined. Let Sx denote the set of all such y's . Consider the maximum value v x of {v(y)jy E Sxf as the value of x computed by M . If Sx is empty, then the value of x cannot be computed by M. A sequence of moves from the initial memory configuration to a memory configuration in Sx is called a value computation for x. If the machine M completes the value computation for x E X*, that is, if Sx :?' 01, then M is required to have the following ability. Let 'Y be any memory configuration in Sx . Then My should be able to compare v(y) with any element l in L if the representation r(l) of l is presented to My as a question. In other words, when r(l) is given to My , My moves deterministically reading input symbols in {A} U A along the infinite sequence r(l) and emits one of >, , -nb(w)I+I

is fuzzy recognized by a deterministic pushdown automaton.


9.3 . Fuzzy Recognition by Machines

429

A pushdown automaton M = (1b, I','@, 0, S, S2, K, -YO) with

'Y2

in Example

9.3 .1 fuzzy recognizes pl, where b = X U A U {t} with X = {a, b} and 0 = {0,1, 0,1}, I' = Q U {zo, a, b,1}, where Q = {qo, ql, q>, q, (qlx) = q>x W< (qlx) = q<x W=(qlx) = q=x .

S : b x I'2 ---+ P(Q) is defined as follows : 6(t, (qo, zo)) 6(u, (qo, zo)) 8(u, (go, u)) 6(a,(go,b)) 6(t, (qo, u)) 6(t, (qo,1)) 8(1, (qj,1)) 6(0, (q1,1)) 6(1, (qj, u))

{Wi}, {Wlv} S(u,(go,l))

_

6(b, (go, a))

{W~}

if u E {a, b}, if, u E {a, b},

{W1}

if u E {a, b},

{ W i} s(0, (qj, u))

if u E {a, b}, if u E {a, b},

S(0, (q j , zo)) 8(u, (qj , zo))

if u E {0 ,1}, if u E {0,1, 0},

6(u, (q1,1)) 6(1, (q1,1))

Let the partial function K(q n , u)

K

from I'2 into T U {A} be defined as follows :

= 97 for 97

Define v : I'* ---+ [0,1]Q by


E

T and u

E {zo, a, b, l, l} .

43 0


and v(glzolunl)

= + ( 2) n +l , 2

for u E f a, bf and n >, 1 .

Let ft' : X* ---> L[0 , 1] be such that ft', (w) = ft, (w) for w E X*. Then it follows that ft' is fuzzy recognized by a deterministic pushdown automaton with the representation rl of a previous example. 9 .4

Cutpoint Languages

Let ft be an L-fuzzy language over X, where L is a lattice with minimum element 0. Then l E L is called an isolated cutpoint of ft if one of the following three conditions holds: (1) There exist 11 and 12 in L such that 11 < l < 12 and Vx E X* such that ft(x) :?~ l, either ft(x) S 11 or ft(x) >, 12(2) l is a maximum element of L and there exists 11 :?~ l in L such that Vx E X* with [t(X) :?~ l, [t(X) S 11. (3) l = 0 and there exists 1 2 :?~ 0 in L such that Vx E X* with y(x) :?~ 0, Theorem 9.4.1 Let L be a lattice with minimum element 0.

Let ft X* ---+ L be an L-fuzzy language and let l be an isolated cutpoint of ft. Then, for i = 0, 1, 2, 3, if ft is fuzzy recognized by a machine in Ti, then LG E (f, l) and LG (f, l) are recognized by a machine in Ti .

Proof. Suppose that ft is fuzzy recognized by a machine M = (X U A U ftf , I', T, 0, S, S2, K,'YO)

in TZ with a representation r over A. Since l is an isolated cutpoint of ft, either (1), (2), or (3) holds . We only prove the case when (1) holds . The proofs for other cases are similar . Let 11 and 12 in L be such that 11 < l < 12 . Suppose that Vx E X* with ft(x) :?~ l, either ft(x) > 12 or ft(x) S 1 1 . Let dl and d2 be the D-length of r(11) and r(l) and that of r(l) and r(12), respectively. Let d3 = dl V d2 and let w E A* be the prefix of r(l) of length d3 . From the definition of fuzzy recognition, the set L[M, >1] is recognized by a machine in Ti, where L[M, >,] is the set of all words of the form xty with x E X*, and y E A* is such that 'Yo ==~> 'Y and K(0(-Y)) is = or >. We now show that

xty

fx E X*ly(x) > if = fx E X*Ixty E L[M, >] for some y E PRE wA*f . If y(x) >, l, then there exists 'Y E I'* and y E PREfr(1)f C PREwA* such

that 'Yo ~ 'Y and K(0(-y)) is = or >.


Conversely, suppose that 'Yo ~ 'Y

9.4. Cutpoint Languages

431

and K(0(-Y)) is = or >, where x E X*, y E PREwA*, and ,y E I'* . If y E PREr(l), then clearly ft(x) >, l . Otherwise, there exists l' E L such that y(x) >, l' and for some w' E A* and a E A', r(l') = ya = ww'a. From the definition of D-length, neither l' and 1 1 nor l' and 12 are incomparable, and also neither l' < 1, nor 12 < l' holds . Therefore, 1 1 S l' S 12 . This implies that ft(x) = l or ft(x) >, 12. Thus ft(x) >, l. Clearly, there exists a gsm-mapping G, [96, p.272], such that LGE(lt, l) = G(L[M, >,]

n X*cPREwA*) .

Since classes of recursively enumerable sets, context-free languages, and regular sets are closed under a gsm-mapping operation, respectively, it holds for i = 0, 2, and 3 that LGE(lt, l) is recognized by a machine in Ti . Suppose that i = 1 . Then machine M is a Turing machine such that for some constant t, I _ t I x I for any x E X* and for any memory configu ration ,y such that 'Yo ===> 'Y for some y E PRE{r(l) 11 E L} . A machine xty

M' is a modification of M as follows, M' moves reading xt in the same way as M moves reading xt for x E X* . After reading xt, M' continues to

read A and changes sequentially memory configurations in the same way as M reads some y E PREwA* . In order to move in this way, M' has an autonomous finite state machine as a submachine that generates any y E PREwA* nondeterministically. Clearly, M' E T1 and thus the set L(M') of all words xt (x E X*) for which M' emits > or = is recognized by a machine in T1 . It follows that L(M) = {xt I x E X*, y E PREwA* and xty E L(M, >)}. Therefore, L(M') = LGE(lt, l) t. Hence LGE(ft, 1) is recognized by a machine in T1 . The proof for LG(lt, l) is similar . m Corollary 9.4 .2 Let L be a finite lattice and let ft : X * ~ L be an L-fuzzy language . Then, for i = 0, 1, 2, 3, if ft is fuzzy recognized by a machine in Ti, then for any l E L, LGE(lt, l) and LG(lt, l) are recognized by a machine in Ti . Proof. Suppose that ft is fuzzy recognized by a machine M = (X U A U {t}, I',,@, 0, S, S2, K,'Yo)

in Ti. Let L = {11,12 . . . . , i s } . Then there exists wj E A* such that r (1j) E wj A', but r (1j) ~ wk A' for 1 B be the characteristic function of L, where B is the Boolean lattice with two elements . For i = 0, 1, 2, 3, L is recognized by a machine in TZ if and only if XL is fuzzy recognized by a machine in Ti . m Corollary 9 .4.5 shows that the representation concept for fuzzy languages introduced in this section is a fairly good extension of the one for ordinary languages . Example 9.4 .6 Let L be a lattice with a minimum element 0 and a maxi-

mum element 1. An L-fuzzy context-free grammar is defined to be a quadruple G = (N, T, P, S), where NUT is a finite set of symbols with Nr1T = 0, T is the set of terminal symbols, N is the set of nonterminal symbols, S E N, and P is a finite set of production rules of the form

with A E N, x E (N U T)*, and l E L. For any y, z E (N U T)*, we write i y ----> z if there exists u, v E (N U T) * and A z = uxv. We write

z is in P such that y = uAv and

i

y~y * y

0 z *

y

m z *

for all y, z E (N U T)*, and

if and only if there exists a sequence of elements y0, yl, . . . , yn E (N U T) * i. such that y0 = y, yn = z, y2_1 4 yZ for i = 1, . . . , n and AZ l l2 = m. For any x E T*, let lx = V{l E L I S ='~> x} . The L-fuzzy language /t defined by ft(x) = lx , for all x E T*


43 4


is said to be generated by G. An L-fuzzy language generated by some Lfuzzy context-free grammar is called an L-fuzzy context-free language. (L_ fuzzy phrase structure, L-fuzzy context-sensitive, L-fuzzy regular grammars and languages are similarly defined, respectively .) L[0,1]-fuzzy context-free languages were studied in Chapter 4. From Proposition 18 in [261] and Theorem 9.4 .3, it follows that any L[0,1] fuzzy context-free language is fuzzy recognized by a pushdown automaton.

Example 9.4.7 Let X = {a, b, c} and let y2 be the L[0 1]-fuzzy language over X defined by Vi, j, k E hY U {0} It2(a'. h'ck) _

(1)~i-j I+1 + (1)h-k1+1 2

2

and It2(w) = 0 if w ~ a*b*c* . Then 1 is an isolated cutpoint of ft2, and LGE(It2 ,1) = {a'b'c2 I i E NU{0}} is not recognized by any pushdown automaton. Thus by Theorem 9.4 .1, ft2 is not fuzzy recognized by any pushdown automaton. It follows that N2 is fuzzy recognized by some deterministic linear bounded automaton with the representation rl of Example 9.2.2.

We now consider regular representations . The following theorem is clear from the proof of Theorem 9 .4 .1 . Theorem 9.4.8 Let L be a lattice with a minimum element. Suppose that

the L-fuzzy language ft is fuzzy recognized by a machine in TZ(DTZ) with a representation r and that r(l), l E L, is generated by an autonomous finite automaton sequentially . Then LG(lt, l) and LGE(lt, l) are recognized by a machine in TZ(DTZ), where i = 0, 1, 2,3. 0

We next introduce the concept of a regular representation . Let L be a lattice with a minimum element . A representation r of L is said to be regular if for all l E L, r(l) is generated sequentially by some autonomous finite automaton . The following corollary follows from Theorem 9.4 .8 . Corollary 9.4.9 Let L be a lattice with a minimum element. For i =

1, 2, 3, if an L-fuzzy language ft is fuzzy recognized by a machine in TZ(DTZ) with a regular representation, then for all l E L, LG(lt, 1) and LGE(ft, 1) are recognized by a machine in TZ(DTZ) . 0

The representation of r2 of L[0,1], shown in the lar . Thus the L-fuzzy language ft, of Example 9.3 .1 LG(lt, l) and LGE(lt, l) are context-free languages . of restricted type have regular representations . For © 2002 by Chapman & Hall/CRC

Example 9.2 .3 is reguand for any l E Lfo,ll,, Clearly, only lattices example, L[o,1] cannot

435 9.5. Fuzzy Languages not Fuzzy Recognized by Machines in DT2 have a regular representation . However, fuzzy recognition of a fuzzy language with a regular representation is of interest since, by Corollary 9.4.9, it gives a property independent of cutpoints of the fuzzy language with respect to recognition of its cutpoint languages .

9 .5

Fuzzy Languages not Fuzzy Recognized by Machines in DT2

In view of Theorem 9.4.1 and Corollary 9.4.2, L-fuzzy languages not fuzzy recognized by a machine in TZ for i = 0, 1, 2,3 are easily found. However, Theorem 9.4 .1 and Corollary 9.4 .2 cannot be used to find an L-fuzzy language whose membership function-values distribute densely over L and that is not fuzzy recognized by a machine in TZ for i = 0, 1, 2. Hence it would be of interest to find such a language. Although we do not have such an example, we do have the following example . Example 9.5.1 Let X = {0,1} and let over X such that for a2 E X, i = 1, 2, . . . ,

la3 be an L[0,1],-fuzzy language k,

Ps(ala2 . . . ak) = a12 -1 + a22 -2 + . . . + ak2-k (binary expansion, and P3 (A) = 0 . We now show that la3 is not fuzzy recognized by any deterministic pushdown automaton. Suppose that la3 is fuzzy recognized by a deterministic pushdown automaton

with a representation r over A . Let

Ll =

{xty

I

'Yo

ty'Y and K(B('Y)) _ (_)}

Then Ll is a context-free language included in X*t0* . Let L2 = L1 n 0*1*t0* . Then L2 is also a context-free language . Since M is deterministic, for any xty E L1, there exists a E A' such that r(ft3(x)) = ya . Due to the pumping lemma of the theory of context-free languages, there exists a constant k such that if ~zj >_ k and z E L2 , then z = uvwxy such that vx :?~ A, Ivwxl _ k and let zP be an element in L2 of the form OP1mtgp , gP E A*, for any p E N U {0} . Then zP = u pvp wp xp yp , where vp xp :?~ A, vp wp xp l 0, we can write vp = Pp for some sp > 1 and wp = 11PtWp for some tp > 0 and Wp E A* . Since l vpwpxpl _ 1, let 7r(x) = (7r1(x), 7r2(x), . . . ,7r .(x)) be 7rA(u1)A(u2) . . . A (u ,) and let 7r (A) = 7r . Then for any word x of length m, 7r2(x) is represented in the form h2(x)/k-+1 with 0 1, for i = 1, 2. For l E L[ 0,1], with 0 :?~ l 1, let l = s1s', where s and s' are relatively prime positive integers . Then r(l) is the element in 0'°° such that T I (r(l)) = b(s)#b(s')### . . . and T2(r(l)) = e(l), where e(l) was given in Example 9 .2 .2 . Then r(0) = (0 , 0) (#, 0) (#, 0) (#, 0) . . . and r(1) = (1,1) (#, 1) (#, 1) (#,1) . . . . Clearly, r is a representation since the D-length of r(l) and r(m) with l :?~ m may be defined as the positive integer k such that T2(r(l)) and T2(r(m)) differ first at the kth digit . It follows easily that r is a regular representation . If z with b(s(x))#b(s'(x)) on its tape is given a representation r(l) of l E L[0,1]u as an input succeeding xt for x E X*, z computes the binary expansion of s(x)Is'(x), digit-by-digit, and compares it with T2 (r(l)) . In parallel with this computation, z compares sequentially b(s(x))#b(s'(x)) with TI(r(l)) . Then either z emits = and halts if b(s(x))#b(s'(x)) = b(s)#b(s'), where TI(r(l)) = b(s)#b(s')### . . . , or z emits > or < and halts according to the comparison of the first distinguished digit of T2(r(l)) and the binary expansion of s(x)ls'(x) . Since the digit-by-digit generation of the binary expansion of s(x)ls'(x) can be made using at most a constant multiple of lb(s'(x))l spaces, z can do the above value comparing computation for x E X* using at most d I xI spaces. Thus p is fuzzy recognized by z in DTI with the regular representation r. The following result, which was proved in [230], follows directly from Theorem 9.6 .1 . Corollary 9.6 .2 Let p : X* ----> L[0,1], be a rational probabilistic event. Then for all l E L[O,I]Q , LG(lt, l) and LGE(N, l) are recognized by a machine in DTI . m 9 .7

Recursive Fuzzy Languages

The relation between deterministic machines and nondeterministic machines with respect to the fuzzy recognizability of fuzzy languages differs somewhat from that of ordinary languages . We show that in the fuzzy recognition of fuzzy languages, nondeterministic Turing machines are more powerful than deterministic Turing machines. Let {t0, t1, t2 . . . . } be an enumeration of deterministic Turing machines . Let L3 be a lattice with three elements 0, c, and 1 such that 0 < c < 1 . Let X = {u} . Let la4 and lay be L3-fuzzy languages over X defined as follows : For n E hY U {0}, ft4(un)

-

1 0

if to with the blank tape eventually halts otherwise


9.8. Closure Properties la5(Un)

-

1 c

439 if to with the blank tape eventually halts otherwise .

Lemma 9.7.1 Let la4 and [t5 be defined as above . Then la4 is fuzzy recognized by a deterministic Turing machine. [t5 is fuzzy recognized by a nondeterministic Turing machine, but it is not fuzzy recognized by a deterministic Turing machine .

Proof. Clearly, there exists a deterministic Turing machine that fuzzy recognizes 1a4 . Since LGE(/t5 ,1) and LGE(N5, c) are recursively enumerable languages, it follows from Theorem 9.4.3 that [t5 is fuzzy recognized by a Turing machine. Suppose that [t5 is fuzzy recognized by a deterministic Turing machine. Then it follows that the language {un I lt 5 (un) = c} is recursively enumerable. Thus the halting problem of Turing machines is solvable, which is a contradiction . Hence [t5 is not fuzzy recognized by a deterministic Turing machine. Let L(Tj) and L(DTj) be the sets of L-fuzzy languages fuzzy recognized by a machine in Ti and DTi, respectively, for i = 0, 1, 2,3 . Theorem 9.7.2 (1) L(To) D L(DTO ) . (2) L(T2) L(DT2) . (3) L(T3) = L(DT3) . Proof. (1) is immediate from Lemma 9.7.1. (2) follows from Corollary 9.4 .4. (3) follows easily. It is not known whether or not L(TI ) D L(DTI ) . Considering Lemma 9.7 .1, it seems reasonable to define recursive Lfuzzy languages as follows: An L-fuzzy language ft over X is recursive if and only if ft is fuzzy recognized by some machine M = (X U A U {t}, I', T, 0, S, S2, K, 'o)

in DTo with some representation r over A with the condition that Sx zA 01 for any x E X*, where Sx = {,y E I'* I 'Yo ==> -Y} . xt Clearly, any L-fuzzy language in L(T3) is recursive . The proof of the following proposition follows easily. Proposition 9.7 .3 An L-fuzzy language in L(DT2) U L(TI ) is a recursive fuzzy language . 9 .8

Closure Properties

We now consider closure properties of the classes of fuzzy languages corresponding to machine classes Ti 's under fuzzy set operations. We show © 2002 by Chapman & Hall/CRC

44 0


that these properties are the extensions of certain closure properties of ordinary languages such as regular sets, context-free languages, contextsensitive languages, and recursively enumerable sets. We consider L[01] fuzzy languages . Theorem 9.8.1 Let

/t and v be L[01 ]-fuzzy languages over X. Let i E {0,1, 2,3f . If /t and v are fuzzy recognized by a machine in Ti, then /t U v is fuzzy recognized by a machine in Ti .

Proof. Suppose that /t and v is fuzzy recognized by a machine Ml with a representation r l : L[ 0,1] ~ Di° and a machine M2 with a representation r 2 : L[0,1] ~ A", respectively. Let A = O1 x 02 . For j = 1, 2 define Tj : 0°° ~ 0~ as follows : Tj((al,a2)) = aj with aj E Aj and for y = bjb2b3 . . . with bk E A, k >_ 1, T,j (y) = Tj (bl) Tj (b2) Tj (b3) . . . . Let r : L[01] ~ A' be such that for any l E L[0,1], T1(r(l)) = r1(l) and T2(r(l)) = r2(l) . Then r is a representation of L[o 1] over A since the Dlength d(r(l), r(m)) of r(l) and r(m) for any distinct l, m E L[0,1] can be defined as d(r(1),r(m)) = d(r1(1),r1(m)) . A machine M that fuzzy recognizes ft U v with the representation r is given as follows : M has Ml and M2 as submachines . For any word xt with x E X*, M first reads A, chooses nondeterministically Ml or M2, and simulates the chosen machine hereafter reading xt . If M reads through xt simulating Mj , j = 1, 2, and if M is given r(l), l E L[0,1 ], then M moves as Mj does if it is given Tj (r(l)) = rj (l) . Clearly, M fuzzy recognizes ft U v . Theorem 9.8.2 Let i E {0,1, 3} . Let

/t and v be L[0,1] -fuzzy languages over X. If /t and v are fuzzy recognized by a machine in Ti, then /t n v is fuzzy recognized by a machine in Ti .

Proof. We prove the result for i = 1. The proofs for the i = 0 and 3 cases are similar . Suppose that /t and v are fuzzy recognized by a machine Ml in Tl with a representation rl : L[0,1] ~ A' and a machine M2 in Tl with a representation r2 : L[0,1] ----> 0~ . Let 0 = Ol x 02 . Define a representation r : L[o 1] ~ 0°° as defined in the proof of Theorem 9.8.1 . We determine a machine M in Tl that fuzzy recognizes /t n v with the representation r with M having Ml and M2 as submachines. When M is given xt with x E X* as an input word, submachines Ml and M2 of M move in parallel reading xt, and M reaches the configuration 'Y corresponding to the pair of configurations 'Yl of Ml and 'Y2 of M2, which they reach after reading xt, respectively. For this computation of M, at most a constant multiple of Ixl spaces is necessary. If M with the configuration 'Y is given r(l) with l E L[0,1] as a subsequent input, then M makes Ml with the configuration y1 read T1(r(l)) = r1(l) and in parallel makes M2 with the configuration 'Y2 read T2(r(l)) = r2(l) . When one of M1 or M2 emits an output, i.e., one of >, , then M emits > . If one of them emits = and another emits > or =, then M emits =. If either one of them emits _ c. We denote by P(X, L) and by P(p) the set of the L-fuzzy points of X and the set of the L-fuzzy points of ft, respectively. An L-fuzzy grammar is a 4-tuple G = (N, T, P, S) with N and T finite sets, the nonterminal and terminal symbols, respectively, such that N n T = 0, S E N (the initial symbol), and P a finite set of L-fuzzy productions, i.e., elements of the form x -c~ y with c E L, c :?~ 0, and X' Y E (NUT)* . We say that an L-fuzzy grammar G = (N, T, P, S) is in normal form provided that if x -c~ y is an L-fuzzy production of G, then either x, y E N* or x E N and y E T. In an L-fuzzy production x C ----> y, c represents the membership degree of the rewriting rule x ~ y . If c = 1, we write x ~ y for x -c~ y. If wxw' and wyw' are elements of (N U T)* and x -c~ y belongs to P, then we say that wyw' is directly derivable from wxw' with degree c in the L-fuzzy grammar G. If w and w' are in (N U T)*, a derivation chain in G from w to w' is a pair of words (w1 . . . WP, cl . . . cp _ 1) such that w1 = w, wp = w', and w2+ 1 is directly derivable from w2 with degree c2 . A derivation of w is a derivation chain from S to w. The element c 1 A c2 A . . . A cp_ 1 is called the degree of the derivation. An L-fuzzy language /t : T* ~ L is generated by the L-fuzzy grammar G provided that, for every w E T*, /t(w) = V{c E L I c is the degree of a derivation of w}. It follows that an L-fuzzy grammar G utilizes only a finite subset X of elements of L and that the sublattice L' generated by X is finite even in the case that L is infinite. This means that every generated L-fuzzy language is a generated L'-fuzzy language, where L' is finite. As a consequence, in the next section, we assume that the lattice L is finite. 9 .10

Recursively Enumerable L-Subsets

An effective codification of T*, P(T*, L), and L is possible since T and L are finite . Then concepts such as a partial recursive function from T* into L and such as recursive enumerability for subsets of T* and P(T*, L) are defined, [182] . An L-fuzzy subset /t is said to be recursively enumerable if its set of points P(p) is a recursively enumerable subset of P(T*, L) and /t is called decidable if it is a recursive function from T* into L . An L-fuzzy subset ft is called a projection of a decidable L-fuzzy relation if there exists a finite set B and a decidable L-fuzzy subset v of T* x B* such that © 2002 by Chapman & Hall/CRC

9.10. Recursively Enumerable L-Subsets

44 3

y(x) = V{v(x, y) I y E B*} . The following proposition is proved in [17] .

Proposition 9.10 .1 Let /t be an L-fuzzy subset of T* . Then the following statements are equivalent . (1) /t is recursively enumerable. (2) Every cut of /t is recursively enumerable. (3) y(x) is a projection of a decidable L-fuzzy relation. (4) y(x) =1im v(x, n) with v recursive and increasing with respect to n.

Proof. (1)x(2) : Immediate. (2)x(3) : Suppose that IBI = 1. Then we can identify B* with N. It follows that y(x) = V{c E L\{0} I x E ttj . Since /t, is recursively enumerable, a partial recursive function v, exists whose domain is /t c. Let v : T* x hY x L ----> L be defined as follows : v(x, n, c)

c 0

if v,(x) is convergent in less than n steps, otherwise .

Clearly, v is recursive and y(x) = V{v(x, n, c) I n E N, c E L} . By identifying hY x L with N, and therefore with B*, via a suitable codification, we obtain (3) . (3)x(4) : Since we can identify B* with N, we can assume that y(x) _ V{v(x, n) I n E N} with v recursive . Set v'(x, n) = v(x,1) V v(x, 2) V . . . V v(x,n) . Then y(x) = limv'(x,n) and v' is recursive and increasing with respect to n . (4)x(1) : It suffices to note that x, E P(p) if and only if ft(x) > c if and only if there exists n E hY such that v(x, n) >_ c. Thus P(p) is recursively enumerable. Proposition 9.10 .2 The set of recursively enumerable L-fuzzy subsets of

T* is a lattice, the minimal lattice containing the recursively enumerable subsets and the L-fuzzy subsets that are constant maps . Namely, an Lfuzzy subset /t : T* ~ L is recursively enumerable if and only if /t admits a decomposition /t = (c l A XI) V . . . V (cn A xn) with c2 E L and where XZ is the characteristic function of a recursive enumerable subset, i = 1, . . . , n.

Proof. Let /t and v be recursively enumerable . Then by Proposition 9.10.1(4), y, (x) = limlt'(x,n) and v(x) = limv'(x,n) with /t' and v' recursive and increasing with respect to n. Then y(x) V v(x) = limlt'(x,n) V v'(x, n) and ft (x) A v(x) = lim ft'(x, n) A v'(x, n) . Since /t' V v' and /t' A v' are recursive and increasing with respect to n, /t V v and /t A v are recursive enumerable This proves that the set of recursively enumerable L-fuzzy subsets forms a lattice . Let /t be a recursively enumerable L-fuzzy subset. Set if

[t(X) x ~(x) _ { 10 otherwise . © 2002 by Chapman & Hall/CRC

>c

444


Since /t c is recursively enumerable, Xc is the characteristic function of a recursively enumerable subset . Clearly, ft = V{c A X c c E L\{0}} . Thus /t is generated by the constants c and the recursively enumerable crisp L-fuzzy subsets xc . We now prove our main result, i.e., an L-fuzzy language /t : T* ----> L is generated by an L-fuzzy grammar if and only if it is recursively enumerable. However, first we prove the following lemma. Lemma 9.10 .3 If ft : T* ~ L and v : T* ~ L are L-fuzzy languages generated by normal form L fuzzy grammars, then /tUv is an L fuzzy language generated by a normal form L fuzzy grammar . Proof. Suppose that y and v are generated by the normal form Lfuzzy grammar G' = (N', T, P', S') and G" = (N", T, P", S"), respectively . Without loss of generality, we assume that N'nN" = 0. Let G be the normal form L-fuzzy grammar (N' U N" U {S}, T, P, S), where S is a new symbol not in N' U N", and P contains S ~ S', S ~ S", and all production in P' and P" . Let A : T* ~ L be the L-fuzzy language generated by G . Then A = /t U v . In a sense, the derivations of G consist of the derivation of G' and the derivations of G". That is, if (w1 . . . wp , ul . . . Up -,) is a derivation of G' (of G") with w 1 equal to S' (to S"), then (Sw1 . . . wp, lul . . . up-1) is a derivation of G' (of G") . Furthermore, since N' and N" are disjoint, every derivation of G can be obtained either from a derivation of G' or from a derivation of G" as above. Thus it follows that A = ft U v. Theorem 9.10 .4 Let /t be an L fuzzy language of T* . Then the following statements are equivalent . (1) /t is generated. (2) /t is recursively enumerable. (3) /t is generated by a normal form grammar . Proof. (1)x(2) : Suppose that /t is generated by the L-fuzzy grammar G. Let P1,P2, . . . be an effective enumeration of the derivation of G . Moreover, let v be defined by v(w, n)

the degree of Pn 0

if Pn is a derivation of w otherwise .

Then it follows that /t is the projection of v, and thus, /t is recursively enumerable by Proposition 9 .10 .1 . (2)x(3) : Suppose that /t is recursively enumerable . Then by Proposition 9.10.2, /t = (c1 A x1) V . . . V (cn A xn), where XZ is the characteristic function of recursively enumerable subset Si of T*, i = 1, . . . , n. Let GZ be a classical grammar in normal form producing Si and let GZ be the normal form L-fuzzy grammar obtained from GZ by substituting each production x ----> y with x _c~ y. It follows that GZ produces the L-fuzzy language c2 AXi . © 2002 by Chapman & Hall/CRC

9.10. Recursively Enumerable L-Subsets

44 5

From Lemma 9.10.3, it follows that ft is generated by a normal form L-fuzzy language . (3)==>(1) : Immediate . We now consider some closure properties for generated L-fuzzy languages . Recall that the concatenation of two L-fuzzy languages ft and v is the L-fuzzy language A defined by b'w E T*, A(w) = V{ft(x) A v(y) w = xy} . Moreover, the Kleene closure /t°° of ft is such that ft'(w) _ V{ft(xl) A . . . A ft(xe) I w = xi . . . xq , q E N} . Corollary 9.10 .5 The class of all generated L-fuzzy languages is a lattice.

In particular, it is the minimal lattice containing the generated (classical) language and the L fuzzy languages that are constant functions. Furthermore, if ft and v are generated languages, then the concatenation of ft and v and the Kleene closure of ft are generated.

Proof. The first part of the corollary follows from Proposition 9.10.2 . Suppose that ft(x) = Vnft'(x, n) and v(x) = Vn v'(x, n) with ft' and v' recursive. Then A(w) = V{ft'(x, n) A v'(x, m) I w = xy, n, m E N} . Since it is possible to codify the set, { (x, y, n, m) I w = xy, n, m E N}, A is a projection of a decidable relation. Thus A is recursively enumerable and therefore generated . Likewise, it is possible to express the Kleene closure of ft by the formula ft-(w) = V{ft'(xl, nl) n . . . n ft'(xq, ne ) I w = xi . . . xq , ni, . . . , nq E N} . Since it is possible to codify the set {(x l . . . x q , q, nl , . . . , nq ) I xl . . . xq = w, q, nl, . . . , nq E N}, ft' is recursively enumerable and therefore generated . Theorem 9.10.4 allows for the transfer of results on the relationship among imprecision, decidability, and recursive enumerability given in [17, 70, 71] to L-fuzzy languages . Namely, we assume that L is a finite sublattice of the interval [0,1] containing -1 . We call an L-fuzzy language infinite indeterminate (almost-everywhere indeterminate) provided that the set {x E T* I s(x) = 2 } is infinite (cofinite) . If A and A' are L-fuzzy languages, we say that A' is a sharpened version of A or that A is a shaded version of A' if A(x)

>2

==> A(x) > A(x) and A(x)

< 2 ==> A(x) < A(x) .

Hence we conclude the following results from Theorem 9.10.4: 1. A generated infinitely indeterminate L-fuzzy language exists with no decidable sharpened version, [17, Proposition 5.1] . That is, it is not possible to obtain decidability by using the indeterminateness of a generated fuzzy language, in general . 2. An infinitely indeterminate L-fuzzy language exists with no generated shaded versions, [71, Proposition 4.4] . This provides an example of a fuzzy language that is not generated in a strong case. © 2002 by Chapman & Hall/CRC

446


3. A generated classical language exists whose unique decidable shaded versions are the L-fuzzy languages infinitely indeterminate, [17, Proposition 5.2] . Consequently, it is not possible to shade a generated classical language in order to obtain decidability . 4. A classical language exists whose unique generated shaded versions are the L-fuzzy languages almost everywhere indeterminate, [71, Proposition 5.2] . Therefore, it is not always possible to obtain generated languages by shading a classical language.

9 .11

Various Kinds of Automata with Weights

Various types of automata such as fuzzy automata, max-product automata, and integer-valued generalized automata [233, 234] have been introduced as a generalization of well-known deterministic automata, nondeterministic automata, and probabilistic automata . These automata have the common property that they have "weights" associated with the state transitions as well as initial and final distributions . Clearly, probabilistic automata can be considered as automata with "weights." The operations of max and min and product have been used with these automata . By using addition and multiplication as the operations and the probabilities as the weights, probabilistic automata can be defined. Moreover, integer-valued generalized automata have integer weights and addition and multiplication as operations. We now present the results of [148] in order to present a general formulation of automata with weights by extracting the basic properties common to the existing automata and by incorporating the appropriate algebraic systems with automata systems and by performing the operations of the algebraic systems to the state transition functions and initial and final distribution functions of the pseudoautomata defined later. We continue the theme of using a lattice L rather than [0,1] . The concept of L-fuzzy relations enables us to define fuzzy automata, l-semigroup automata, lattice automata, dual lattice automata, max-product automata, and so on. Moreover, L-fuzzy relations are important in formulating other kinds of automata with weights such as semiring automata, ring automata, integer-valued generalized automata, and field automata . Definition 9.11 .1 Let X be a set and L be a lattice . An L-fuzzy relation on X is a function /t from X x X into L, i.e., ft :XxX----> L.

(9 .1)

We let V and A denote the operations of supremum and infimum on L, respectively. In the remainder of the chapter, the structure of the membership space L is assumed to be a complete lattice ordered semigroup and a complete distributive lattice because of the concept of composition of L-fuzzy relations defined below. © 2002 by Chapman & Hall/CRC

as =lla2 also for (L, follows as the there Various L iIf *pl, If *Eordered V, follows all iscomposition to is aL (VZEZyi) satisfies semigroup lan(X1, the L I,complete P2, *) the axisare the is 9an complete Eaoperations Kinds aThe semigroup (or complete two L, index associativity Xn+1) complete I, w2(x, semigroup =VZEZ(x* lan the product distributive Let elements of operation are of set, distributive following Automata z) lattice z) ft, L-fuzzy lattice L-fuzzy V= operation =A{wj lattice and and V{wl(X, V{wj(X, yi) 0of X2, Lla1(XI,X2) that (l-semigroup ordered *laws and ft, composition relations distributive N2 relations (L, Ais(X, with replaced and are in 1V, is0*x lattice x, are be 1, 1*x y) ,y)in Xn LA), asemigroup *VL-fuzzy A Weights dual N2 as an semigroup w2(y, Ew2(y, *can then A X}, follows P2(X2,X3) = on l-semigroup by laws, in similar of for be z) X, z) 0,relations x, AL-fuzzy adefined short) I(or I in *yythen denoted distributive yyFor with result EE L*=X}, El-semigroup X} XI' =L and all VZEZ(xj itrelations, by identity on (L, *holds = isx, by replacing lan(Xn,Xn+1) is(L, V, X aPIP2, y, denoted lattice, complete xi, *), *V, for Then y) (L, under yi we *) then (9isV V, Ecan such the the deby L, *), L

9.11 .

447

If and lattice L where

.

.

x

and

.

(VZEZxi)

If becomes

.

Definition .11 .2 composition : fined (1) then

.

(9 .2)

wIw2(X, where (2)

. (9 .3)

wIw2(X, Since different A

: .

It

(9 .4)

Due write It

where (9 .4) . If that

.. .

V I

. ..

"..

.. .

(9 .5) .

XV0 x*0 XV1 x*1


= = = =

.3)-

(9 .6)

448


then they are called a zero and an identity of L, respectively. For example, let L be ([0,1], V, .), where the operation - is ordinary multiplication. Then L is an l-semigroup with zero 0 and identity 1. Moreover, let the Cartesian product of [0,1] be written as [0,1] 2 and the operations V and - be defined as (a, b) V (c, d) _ ((a V c), (b V d)), (a, b) - (c, d) _ (a . c, b . d), for each (a, b), (c, d) E [0,1]2 . Then L = ([0,1]2, V' .) is an l-semigroup with zero (0, 0) and identity (1,1) . For the l-semigroup L with zero 0 and identity 1, define the identity relation t by V'x, y E L, 1 t(x~y) =~ 0

ifx=y if xz,4 y.

(9.7)

Then we have qt = frt = ft,

(9.8)

for each L-fuzzy relation ft . Clearly, every L-fuzzy relation ft over X is representable by a matrix if X is a finite set. Let X = {xl, x2, . . . , x } . Then ft is represented by the n x n matrix, We now define what is called a pseudoautomaton in [148] in order to derive various kinds of automata with weights . Definition 9.11 .3 A pseudoautomaton is a 6-tuple A= (Q, X, W, ft, 7r~ 97),

where (1)

(9 .9)

Q is a finite set of states, (2) X is a finite set of input symbols, (3) W is a weighting space, (4) ft is a weighting function such as

ft :

Q

x X x

Q ---->

W

(9.10)

and is called a state transition function. The value ft(q, a, q') of (q, a, q') E Q x X x Q represents the weight of transition from state q to state q' when the input symbol is a, (5) 7r is an initial distribution function, where 7r :Q----> W (6)

97

(9.11)

is a final distribution function, where 97 :Q----> W.


(9 .12)

9.11 . Various Kinds of Automata with Weights

449

We do not consider time and outputs for the sake of simplicity. If we consider time, the state transition function /t, the initial distribution function 7r, and the final distribution function rl are given as ft : QxXxQxT~W TAW 9 :QxT~W where T is a subset of Let Y be a set of output symbols . QxXxYxTinto W.

Then the output function maps

Definition 9.11 .4 A weighted-automaton or simply automaton A* is a 6-tuple A* = (Q, X, W ft*, 7r, 9),

(9 .13)

where Q, X, W 7r, and 9 are given in Definition 9 .11 .3 and ft* : QXX*XQ----> W.

(9 .14)

We now derive various kinds of automata with weights by introducing binary operations on the weighting space W of the pseudoautomata and by giving the extension rules for obtaining /t* from /t.

L-Semigroup Automata (la) . Here the weighting space is a complete lattice ordered semigroup (lsemigroup for short) L = (L, V, *) with identity 1 and zero 0, where the operation * is the semigroup operation in L . The state transition function /t, the initial distribution function 7r, and the final distribution function 97 are given by replacing W in (9.10), (9.11), and (9.12) by L as follows : ft : QxXxQ----> L

(9 .15)

L

(9 .16)

97 : Q ----> L.

(9 .17)

(1b). Using the concept of composition of L-fuzzy relations (9.2), the state transition function /t* for input strings in X* is obtained recursively as follows : © 2002 by Chapman & Hall/CRC

450

9. L-Fuzzy Automata, Grammars, and Languages For A,xEX*, and aEX, 1 0

~* (q, A, q) _

if q = q~ if gzAq,

w* (q, xa, q) = v{w* (q, x, q') * w(q', a, q) I q' E Q},

(9.18) (9.19)

where q, q' E Q, and 1 and 0 are identity and zero of L, respectively. Suppose that the automaton starts from a certain initial state, say, qo . Then the initial distribution function 7r is concentrated at qo, i.e ., 7r (q)

_

1 ~ 0

if q = qo if q qo .

(9.20)

Let F C_ Q be a set of final states . Then the final distribution function 97 is defined as 1

ifgEF

97 (q)-~ 0 ifq~F

(9.21)

Given the expression (9.19) and the initial distribution 7r and the final distribution rl, the weight, denoted by w(x), of the string x of the automata is defined by w(x) = V{7r(q) * w*(q, x, q) * ?l(q) I q, q E Q},

(9.22)

where x E X* . Since there exists an order relation >_ in the l-semigroup L = (L, V, *), the language L(A, c) accepted by the l-semigroup automaton A with parameter c can be defined by L(A, c) = {x E X* I w(x) > c},

(9.23)

where c is called a threshold (or cutpoint) and is a member of the weighting space L .

Max-Product Automata

(2a) Let the weighting space be L' = ([0,1], V, -) in the l-semigroup automaton of (la, b), where the operation - represents ordinary multiplication. Then L' is an l-semigroup with identity 1 and zero 0. Moreover, /t, 7r, and rl are obtained by replacing L in (9.15)-(9 .17) by [0,1], i.e., ft :

QXX XQ---> [0,1],

7r : Q ---> [0,1],


(9.24) (9.25)

9.11 . Various Kinds of Automata with Weights 77 : Q ----> [0,1] .

451 (9 .26)

(2b) /t* and w are obtained by replacing * by - in (9.18), (9.19), and (9.22) . * [t (q, A, q) _ q..

1

0

if q = q~ if gzAq,

(9 .27)

) - ft (q", a, q) I q" E Q},

(9 .28)

w(x) = V{7r (q) - [t * (q, x, q) - 97 (q) I q, q' E Q}.

(9 .29)

w* (q, xa, q) = V{w* (q, x,

Clearly, a max-product automaton is a special case of the l-semigroup automaton of (1a,b) and is also a special case of the semiring automata of (14a,b), defined later.

Lattice Automata (3a) A complete distributive lattice L = (L, V, A) is the weighting space. Moreover, /t, 7r, and rl are given from (9.10)-(9 .12) by ft : QxXxQ----> L,

(9 .30)

7r :Q----> L,

(9 .31)

97 : Q ----> L.

(9 .32)

(3b) By using the concept of composition of L-fuzzy relations (9.3), /t* and w are obtained as follows : * [t (q, A, q') _

1 0

if q = q~ if gzAq,

(9 .33)

w* (q, xa, q) = V{w* (q, x, q") n w(q', a, q) I q' E Q},

(9 .34)

w(x) = V {7r(q) A w* (q, x, q) A ?7(q) I q' E Q},

(9 .35)

where 1 and 0 are the maximal and minimal elements of the complete distributive lattice L, respectively. Expressions (9.34) and (9 .35) are obtained by replacing * by A from (9.19) and (9 .22) . A complete distributive lattice is a special case of a complete lattice ordered semigroup . Likewise, a lattice automaton is considered to be a special case of an l-semigroup automaton . The operations V and A are dual in a complete distributive lattice L = (L, V, A) . Thus the dual automata of lattice automata can be formulated in the following manner. © 2002 by Chapman & Hall/CRC

452


Dual Lattice Automata (4a) This is the same as (3a) . (4b) Using the concept of composition w are given as follows : * [t (q, A, q~) _

w* (q, xa,

q) =

w(X)

A{w* (q,

0 1

of

L-fuzzy relations (9 .4), /t* and

if q = if q7~ q~ q,

(9.36)

" x, q ) V w(q', a, q) I q' E Q},

= n{7r(q) n w*(q, x, q) v ?7(q)

I q' E Q} .

(9.37) (9.38)

Given a certain initial state qo and a final state set F, 7r and 97 of the lattice automata of (3a,b) are given as follows in the same manner as (9.20) and (9 .21) . _ 1 7(q) - { 0

if q=qo if q qo .

(9 .39)

_ 1 - { 0

ifgEF if q ~ F

(9 .40)

97 (q)

However, follows :

7r

and

97 of

the dual lattice automata

of

(4a,b) are defined as

0 7r (q) = ~ 1

if q = qo if q qo .

(9.41)

0 97 (q)-~ 1

ifgEF ifq~F.

(9.42)

(Pessimistic) Fuzzy Automata [250, 143, 212, 31, 65, 140, 193, 259] (5a) If the weighting space, J = ([0,1], V, A), is adopted, then J is a complete distributive lattice under the operations V (max) and A (min) . Furthermore, 7r, 7r, and 97 are as follows . ft :


QxX xQ----> [0,1],

(9.43)

7r : Q ----> [0,1],

(9.44)

97 : Q ----> [0,1] .

(9.45)

9.11 . Various Kinds of Automata with Weights (5b)

/t*

453

and w are defined as follows : * [t (q, A, q)

w* (q, xa,

i

q) = V{(ft* (q, x, q")

w(x) = V{(7r(q)

if if

1

0

q=q' gzAq',

n w(q', a, q)) I

n w* (q, x, q) n 97(q)) I

(9 .46) q' E Q},

(9 .47)

q, q E Q} .

(9 .48)

Since J = ([0,1], max, min) is a complete distributive lattice, a fuzzy automaton is a special case of a lattice automaton (3a,b). Therefore, /t* and w of (9.46) - (9 .48) are obtained from (9.33) - (9 .35) by replacing V with max, A by min .

Optimistic Fuzzy Automata [250,143,212,140] (6a) This is the same as (5a) . (6b) p* and w are defined as follows : w* (q, A, q) = {

if if

1

q=q' gzAq~,

" E Q}, w* (q, xa, q') = A{(w* (q, x, q") V w(q", a, q')) I q w(X) = A{(7r(q) V w* (q, x, q) V 97(q)) I q, q E Q} .

(9 .49) (9 .50) (9 .51)

Optimistic fuzzy automata are special cases of dual lattice automata of (4a,b) . Given an initial state qo and a final state F, 7r and 97 of the fuzzy automata of (5a,b) are obtained from (9.39) and (9 .40) . However, 7r and 9 of an optimistic fuzzy automata are obtained from (9 .41) and (9.42) [143] .

Mixed Fuzzy Automata [212] (7a) This is the same as (5a) . (7b) /t* and w are given by using the concept of convex combination of fuzzy subsets, i.e., w* (q, x,

q) = awPF(q, x, q) + bw*OF(q, a, q),

(9 .52)

w(x) = awPF(x)

(9 .53)

+

bwO F (x),

where PP F and It*0F are state transition functions defined by the pessimistic fuzzy automata of 9 .11 and the optimistic fuzzy automata of (6a,b), respectively. This is the same for WPF and WOF . Also a, b are nonnegative real numbers such that a + b = 1. © 2002 by Chapman & Hall/CRC

454


Composite Fuzzy Automata [250, 65] (8a) This is the same as (5a) . (8b) /t* is obtained by operating between p. This is the same for WPF and WOF .

PPF

and

7r*OF

with probability

Nondeterministic Automata [220] (9a) J' = ({0,1}, max, min} is adopted as the weighting space. Clearly, J' forms a distributive lattice (more precisely, a Boolean lattice) and /t, 7r, and 97 are given as follows : ft : QXXXQ----> {0,1},

(9.54)

7r :

Q ----> {0,1},

(9.55)

97 :

Q ----> {0,1} .

(9.56)

(9b) This is the same as (5b) . Nondeterministic automata are special cases of the fuzzy automata of (5a,b) (or l-semigroup automata) .

Deterministic Automata [220] (10a) This is the same as (9a) plus the additional constraints that there exists unique q' E Q such that 7t(q, a, q') = 1 for each q E Q and a E X and 7r(q, a, q") = 0 for q" :?~ q, and there exists unique q' E Q such that 7r(q') = 1 and 7r(q") = 0 for q" :?~ q' . As for 97, let F be a set of final states. Then 97(q)-~

1 0

ifgEF ifq~F.

(10b) This is the same as (9b) . Clearly, deterministic automata are special cases of the nondeterministic automata of (9a,b) and also of the probabilistic automata of (18a,b) defined later.

Boolean Automata (11a) Here the weighting space is a complete Boolean lattice B = (B, V, A), where the operations V and A are supremum and infimum in B. Clearly, a Boolean lattice is a special case of a distributive lattice . Then 7r, 7r, and 97 are as follows : ft : QxXxQ----> B,


( 9.57)


(9 .58)

7r :

97 :

455

Q ----> B.

(9 .59)

(11b) This is the same as (3b) . Boolean automata and dual Boolean automata, defined next, are special cases of the lattice automata of 9 .11 and the dual lattice automata 9 .11, respectively. In [149], a B-fuzzy grammar is defined to be a 7-tuple (N, T, P, S, J, /t, B), where N is the nonterminal alphabet, T is a terminal alphabet, S E N is an initial symbol, P is a finite set of productions, J is a set of production labels, and /t : J ~ B. We summarize some of the results of [149] in the Exercises .

Dual Boolean Automata (12a) This is the same as (11a) . (12b) This is the same as (4b) .

Mixed Boolean Automata (13a) This is the same as (11a) . (13b) Using the concept of a convex combination of B-fuzzy sets [25], and w are defined as follows : (a A [t * w = (a

)

V (a

/t*

n PDB)(9 .60)

AWB) V (a AWDB),

(9 .61)

where [ta and It*DB are state transition functions that are defined in the Boolean automata of (11a,b) and the dual Boolean automata of (12a,b), respectively, and a, a E B, where a is the complement of a.

Semiring Automata Recall that a set R with the operations of addition + and multiplication x is called a semiring if the following three conditions are satisfied: (1) + is associative and commutative ; (2) x is associative ; (3) x distributes over +, i .e., ax(b+c)=axb+ axc, (b+c) xa=bxa+cxa,

for all a, b, c E R. The semiring R is called a semiring with identity 1 and zero 0 if 1 is the identity under x and 0 is identity under + in R. © 2002 by Chapman & Hall/CRC

456


For example, let R = ([0, oo), +, -) with ordinary addition + and ordinary product . . Then [0, oo) is a semiring with identity 1 and zero 0. Similarly, the set of natural numbers containing 0 is also a semiring with identity and zero under + and .. Also, R = ([0,1], V, -) is a semiring with identity and zero. Note that this R is also an l-semigroup . In general, l-semigroups and complete distributive lattices are special cases of semirings with identity and zero. (14a) The weighting space is a semiring R = (R,+, x) with identity 1 and zero 0. Moreover /t, 7r, and rl are given by: ft :

xX x

Q

R,

Q T

(9.62) (9.63)

71 :Q----> R.

(9.64)

(14b) /t* and w are given as follows : [t* (q, A, q) = [t* (q, xa,

w (x)

q)

~

1 0

if if

= 1: {w* (q, x, q qii EQ

= 1: {'r(q) x q,q'EQ

"

q = q' q zA q~,

(9.65)

) x w (q', a, q)},

(9.66)

lr* (q,x,q) x 9 7(q) .

(9.67)

As a special case of semiring automata, there exist l-semigroup automata of (1a,b), mar-product automata of (2a,b), lattice automata of (3a,b), fuzzy automata of (5a,b), nondeterministic automata of (9a,b), Boolean automata of (12a,b), and so on.

Plus-Weighted Automata [188,235] (15a) The weighting space is R = ([0, oo), +, .), where + and - are ordinary addition and multiplication. Clearly, R = ([0, oo), +, -) is a semiring with identity 1 and zero 0. Then /t, 7r, and rl are defined from (9.62)-(9 .64) as follows : ft :

QXX XQ----> [0,

7r :


Q ----> [0,

(9.68) (9.69)

9.11 . Various Kinds of Automata with Weights 'q : Q ----> [0,

457 (9 .70)

00)

(15b) /t* and w are defined by letting + be ordinary addition and replacing x by - in (9.65)-(9 .67) . 7r* (q, A, q)

w* (q, xa, q')

=

= E

qii EQ

w(x) = 1:

1

~

0

if if

q=q' qz,4 q~,

" {w* (q, x, q ) - 7r(q", a, q')},

{7 r (q) - 7r * (q, x, q) - 97(q)}

q,q'EQ

(9 .71)

(9 .72)

(9 .73)

Weighted automata are special cases of the semiring automata of (14a,

Max-Weighted Automata [188] (16a) The weighting space is R = ([0, oo), max, .), with ordinary multiplication . . Clearly, R is a semiring with unity and zero. Here, /t, 7r, and 97 are given the same way as (9 .68)-(9 .70). (16b) 7r* and w are defined as follows : 7r * (q, A, q) _

7r * (q, xa, q) =

w(x) =

m

Q

1

0

if if

q= q~ gzAq,

{w* (q, x, q') - 7r(q", a, q)},

Q{7r(q) - 7r * (q, x, q) - 97(q)} qm E

(9 .74) (9 .75) (9 .76)

Max-weighted automata are special cases of the semiring automata of (14a,b) . Max-product automata of (2a,b) can be obtained from max-weighted automata by replacing [0, oo) by [0,1] .

Natural Numbered Automata (17a) The weighting space is hY U {0} with ordinary addition and multiplication . Moreover, /t, 7r, and 97 are given as follows : ft :


QXXXQ---->

NU{0},

(9 .77)

458

9. L-Fuzzy Automata, Grammars, and Languages 7r :

Q ----> N U {0},

(9.78)

97 :

Q ----> N U {0} .

(9.79)

(17b) This is the same as (15b) . Natural numbered automata are special cases of the weighted automata of (15a, b) . Max-natural numbered automata can be easily defined in a manner similar to a max-weighted automata of (16a,b) .

Probabilistic Automata [220,168] (18a) Here the weighting space is ([0,1],+, -) . Then /t,

7r,

and

ft : QXX XQ~ [0,1],

For

97,

a,

are (9.80)

7r :

Q ----> [0,1],

(9.81)

97 :

Q ----> [0,1] .

(9.82)

In addition, the following constraints of /t, gEQandaEX, Eq'EQ P(q,

97

q~) = 1,

let F be a final state set . Then 1 97 (q)-~ 0

7r,

and

97

are assumed . For all

1. Eq' EQ 7r (q) =

ifgEF ifq~F

(9.83)

(9.84)

(18b) This is the same as (15b) . There exists another definition of 97 different from (9.84), [233] . That is, in the same way as /t and 7r of (9.83), we have qEQ

97(q)

=1.

(9.85)

Generalized Probabilistic Automata [233,164] (19a) This is the same as (18a) with the assumption that the image of 97 not the unit interval [0,1], but the set of real numbers (-oo, oo), i.e., 97 :

by

Q ----> (- 00, oo) .

is

(9.86)

(19b) This is the same as (18b) . The language accepted by generalized probabilistic automata is defined L(A, c) = {x E X* I w(x) > c},

where c E (-

(9.87)

) . As for the probabilistic automata of (18a,b), c E [0,1] .



459

Rational Probabilistic Automata [234] (20a) This is the same as (18a) with the assumption that the values 7r(q, a, q') and 7r(q) are rational numbers in [0,1] . (20b) This is the same as (18b) .

Ring Automata (21a) The weighting space is a ring [125] with identity R = (R,+, .) . Here 7r, 7r, and 97 are ft : QxXxQ----> R,

(9 .88)

R,

(9 .89)

97 : Q ----> R.

(9 .90)

(21b) This is the same as (14b) . Ring automata are special cases of the semiring automata of (14a,b) . The weighted automata of (15a,b) and the max-weighted automata of (16a,b), which are special cases of the semiring automata of (14a,b), are not special cases of ring automata .

Integer-Valued Generalized Automata [233,234] (22a) The weighting space is 7L = (7L, +, .), where 7L is a set of integers and the operations + and - are ordinary addition and product, respectively. Clearly, 7L is a ring with identity. Here /t, 7r, and 97 are ft : QXX XQ----> 7L,

(9 .91) (9 .92) (9 .93)

(22b) This is the same as (15b) . Integer-valued generalized automata are a special case of the ring automata of (21a,b) .

Field Automata (23a) Let the weighting space be a field F = (F,+, .), [125] . Then /t, 97 are ft : QxXxQ----> F,


7r,

and

(9 .94)

460

9 . L-Fuzzy Automata, Grammars, and Languages 7r : Q ---->

F,

(9.95)

9 :

F.

(9.96)

Q ---->

(23b) This is the same as (21b) . Clearly, field automata are a special case of ring automata of (21a,b) . An integer-valued generalized automata ((22a,b)), which is a special case of ring automata, is not a special case of field automata .

(Real-Valued) Generalized Automata [233, 234, 235] (24a) The weighting space is F = ((-oo, oo), +, .), where (-oo, oo) is a set of real numbers, and + and - are ordinary addition and multiplication . Here 7r, 7r, and 97 are ft

: QXX XQ----> (

(-oo, oo),

(9.98)

Q ~ (-00, 00) .

(9.99)

7r : Q ~

97 :

(9.97)

(24b) This is the same as (15b) . Real-valued generalized automata are a special case of the field automata of (23a,b) .

Rational Automata (25a) The weighting space is Q = (Q, +, .), where Q is a set of rational numbers, and + and - are ordinary addition and multiplication . Here /t, 7r, and 97 are ft

: QXXXQ----> Q

(9.100)

7r :Q~Q,

(9.101)

97 :Q~Q .

(9.102)

(25b) This is the same as (15b) . Rational automata are a special case of the real-valued generalized automata of (24a,b) and also of field automata of (23a,b) . We have presented various kinds of automata with weights . Some of these automata are lacking of physical images . However, for example, from the fact that the classes of languages defined by rational probabilistic automata of (20a,b) and integer-valued generalized automata of (22a,b) are © 2002 by Chapman & Hall/CRC

9.12. Exercises

461

equal, various problems concerning rational probabilistic automata can be solved by investigating the properties of integer-valued generalized automata [233,234] . Consequently, automata with weights are important in investigating properties of well-known automata such as deterministic automata and probabilistic automata . Furthermore, they are useful models of learning systems, gaming, and pattern recognition as in the case of fuzzy automata [250, 31, 65] . The set of complex numbers forms a field. Hence as a special case of field automata of (23a,b), we can define complex numbered automata. We cannot, however, define a language accepted by these automata in the same way as (9 .23 because of the fact that there does not exist an order relation >_ on the set of complex numbers . However, using the concept of a mapping of the set of complex numbers into a certain algebraic system with ordering, say, by transforming the complex number z to the absolute value ~zj, we can define a language by a complex numbered automata A as L(A,z) = {x E X* I Iw(x)l >_ ~z1}. Along these lines, the concept of a valuation of a field can be considered. A valuation of a field F is a function v of F\{0} into an additive abelian totally ordered group such that for all x, y E F\{0}, v(xy) = v(x) + v(y) and v(x + y) > v(x) A v(y) . If we do not use this concept of mapping, we would have to restrict our choice of a ring or a field to those with an ordering as weighting spaces, [18, 183] . 9 .12

Exercises

1. For Theorem 9.4.1, prove the cases, where (2) or (3) holds . 2. Give the proof of Theorem 9.8.2 for i = 0 and 3. 3. Show that if a L[o 1]-fuzzy language ft is fuzzy recognized by a machine in T2 and a L[o 1]-fuzzy language v is fuzzy recognized by a machine in T2, then ft n v is fuzzy recognized by a machine in T2 . 4. If P1, P2, P3 are L-fuzzy relations on a set X, prove that (It, P2)P3 = Itl(1t2ft3) 5. (149 Show that type 2 (context-free B-fuzzy grammars can generate type 1 (context-sensitive) languages although type 2 fuzzy grammars cannot generate type 1 languages [145] . 6. (149 Show that the generative power of type 3 (regular) B-fuzzy grammars is equal to that of the ordinary type 3 grammars.


Chapter 10

Applications 10 .1

A Formulation of Fuzzy Automata and Its Application as a Model of Learning Systems

In [26], a formulation of a class of stochastic automata on the basis of Mealy's [138] formulation of finite automata has been carried out . In [68], a stochastic automaton as a model of a learning system operating in an unknown environment was employed. The formulation of fuzzy automata is similar to that of the stochastic automata proposed in [26] and the fuzzy class of systems described in [256]. The advantage of using fuzzy set concepts in engineering systems has been discussed in [255, 256] . In [212], a general formulation has been given to cover both fuzzy and stochastic automata. The next five sections are based on [250] and deal with a specific formulation of fuzzy automata and their engineering applications .

10 .2

Formulation of Fuzzy Automata

Definition

10 .2 .1 A (finite) fuzzy automaton is a quintuple (Q, X, Y,

It, w), where (1) Q nonempty finite (2) X nonempty finite (3) Y nonempty finite (4) /t is a fuzzy subset (5) w is a fuzzy subset

set (the set of internal states, set (the set of input states, set (the set of output states, of Q x X x Q, i .e., /t : Q x X x Q of Q x X x Y, i . e., w : Q x X x Y

[0,1], [0,1] .

In Definition 10.2.1, /t is called the fuzzy transition function and w the fuzzy output function . Recall that in Section 8.4, a fuzzy finite automaton was defined as in Definition 10.2.1, but with some added conditions . © 2002 by Chapman & Hall/CRC

463

464

10. Applications

It is at times convenient to use the notation PA for a fuzzy subset of a set S, where A is thought of as a fuzzy set and PA gives the grade membership of elements of S in A. At times, A may be merely a description of a fuzzy subset ft of S. Let Q = {ql, q2, . . . , qn }, X = {x1, x2, - . . , XP}, and Y = 01, Y2, - . . , Y,1Then PA (gj, xj , q,) is the grade of transition (class A) from state qZ to state qn, with input xj or q(k) = qZ to qm or q(k + 1) = qm,

(10.1)

where the input is xj or x(k) = xj and where k denotes the discrete time element . Hence we write PA (g2, xj, qm) = ft {q(k) = qi, x (k) = xj, q (k + 1) = qm} .

In order to decide the existence of the transition, a pair of thresholds

c, d may be introduced, where 0 < d < c < 1 . This leads to a three level

logic such as (1) x E A, or "true" if PA(x) > c, (2) x ~ A, or "false" if PA(x) < d, (3) x has an indeterminate status relative to A, or "undetermined" if d < PA(x) < c,

where A is a fuzzy set defined as the transition between states for a particular input ; x denotes the 3-tuple (qj, xj , q,) as defined in (10 .1), where qi, qm E Q; xj E X, i .e., PA(x) = PA(gl, xj, qm) . The function may be dependent or independent of k, the number of steps . If ft is independent of k, ft is called a stationary fuzzy transition function. As we will see, nonstationary fuzzy transition functions are used to demonstrate learning behavior of a fuzzy automaton . For now, let ft be independent of k with fuzzy transition matrix Txj for all xj E X. The Txj are of the following form: xj

q +1 gi,g2,--q,n_ . q .

PA (gi, xj, qm)


10.2. Formulation of Fuzzy Automata The entries of Txj are

PA (gj, x j , q, ) .

465

The fuzzy transition table is as follows :

where [A] denotes the fuzzy matrix whose ith row and jth column is PA (gj, a, qj), for all qZ, qj E Q, and [AO] denotes the fuzzy matrix formed similarly by WA(gj, a, yj) for all qj E Q, yj E Y. The input sequence transition matrix for a particular n-input tape sequence is defined by an n-ary fuzzy relation in the product space Tl x T2 x . . . x Tn . The fuzzy transition function is as follows : Let Xj (k) be an in put sequence of length j, i.e., x1(k), x 2 (k + 1), . . . , x j (k + j - 1) . Then PA (gi, X, (k), qs) = PA (gi, xj , X0 . . . . , x t , qs) is the grade of transition (class A) from state qj or q (k) = qj to qs or q(k+p) = qs when the input sequence is x (k) = xj , x (k + 1) = xo , . . . , x (k + p - 1) = x t . For identical inputs x s in a sequence of length j, Xj(k) is denoted by xs(k) . The composition of two fuzzy relations NA and NB on a set S is denoted by PB o NA and is the fuzzy relation in S given by either ( 1 ) PBoA(x, y) = V {PA (X, v) n PB(v, y) I v E S} or ( 2 ) PBoA(x, y) = A{PA(x, v) V PB(v, y) I v E S} .

Based on each definition, we have a particular kind of fuzzy automata. When these two kinds ofautomata operate together, they act as a composite automaton similar to the structure of a zero-sum two-person game. We illustrate this in a later section, where a learning model is proposed . The above concepts have been studied previously. However, we have introduced new notation. Thus we concentrate on developing automata based on both definitions using the new notation . The composition given in (1) is often called the pessimistic case while that of (2) is often called the optimistic case. Now PA (qj, X2 (k), qT) = V{PA(gh xj, q) A PA (q, xo, qT) I q E Q} .

In general, wA(qj, Xj (k), q .)

=

wA(qj, Xj-j(k)xj (k + j -1), qm ) [t * (th, xl(k)x2(k+ 1) . . . xj (k+,l - 1),qm) V{[Wgi,xi,qT l ) n ItA(grl,x2,Yrz) A . . . A NA(grj_1, xj, elm) I q,-i E Q, i = 1, 2, . . . , j -


1}.

466

10. Applications

Example 10.2 .2

Let

Txl

q(k)

be given by the following table:

I

q(k + ql

1)

q2

q3

q4

U11

u12

u13

u14

q2

u21

u22

u23

u24

q3

u31

u32

u33

u34

u41

u42

u43

u44

ql

q4 Then

PA W31 xlI q2) = u32, fta(g3, X21, q2)

=

A [tA(qj, xl, q2)) I qj E Q, j=1,2,3,4} (u31 A u12) V (u32 A u22) V (u33 A u32) V

VWtA(g3, xl, qj)

_

(u34

fta(g3, X31, q2)

=

n u42),

n U11 n U12), (U31 n u12 n U22), (U31 n u13 n u32), n u14 n U42), (U32 n u21 n U12), (U32 n u22 n u22), n u23 n U32), (U32 n u24 n U42), (U33 n u31 n u12), n u32 n U22), (U33 n u33 n U321, (U33 n u34 n u42), n u41 n U12), (U34 n u42 n U22), (U34 n u43 n u32), n u44 n u42)I-

V~(u31 (u31 (u32 (u33 (u34 (u34

Note, fta(g3' xi, g2)

Vex

{[t* (q3, xi, qj)

j=1,2,3,4}

2 (ltA(g3, X1, qx) 2 (PA(g3, X1, q3)

n

f~A(g~ ~ xl, q2)

n u12) V n u32) V

I

qj

(PA(g3, X21, q2)

(ltA(g3, X21, q4)

E Q,

n u22)V n u42) .

This last relation serves as an iteration scheme for a particular sequence of input tape . Similar results for the min-max relation (2) hold.

10 .3

Special Cases of P .1zzy Automata

We first consider deterministic automata, where the set A is no longer a fuzzy set . A is an ordinary set where /t takes two values, 0 and 1. Hence, the entries for each row of matrices [A], [B], [C], . . . , [Aj], [Bj], . . . will have only one 1 and the rest 0. Thus the skeleton matrix [D] of a deterministic automaton will be [D] = [A] + [B] + [C] + . . . © 2002 by Chapman & Hall/CRC

+ [H],

10.3. Special Cases of Fuzzy Automata

467

which is a reduction similar to that in the stochastic automata formulated in [26]. For a two-step transition chain, pA(gl, xy, qT) is determined by finding the maximum of the minimum of pairs of values of four types (0, 0), (0, 1), (1, 0), (1,1) . The total number of paths of length 2 from state qi to state qT is equal to di, =

X,YEX

For a path of length 3 to exist from dIT

-_

ftA(gi xy, gT) . qi

to

Ex,y,zEX

qT,

ftA(gl ,Xyz,gT)

It consists of terms such as (1,1,1), (1, 0, 1), . . . , (0, 0, 0) ; of course, only (1,1,1) defines the existence of the path. The following definition of a nondeterministic automaton has been used in [76]. A nondeterministic automaton is a quintuple A = (Q, X, S, So, F), where Q is a nonempty finite set of objects (internal states), X is a nonempty finite set of objects (input states), S is a function from Q x X into P(Q), and So, F are subsets of Q and So is nonempty. To fit this model to the fuzzy automata formulation, the set A is no longer a fuzzy set . It is an ordinary set, where /t takes only two values, 0 and 1 . The entries for each row of the matrices [A], [B], [C] . . . . are either 1 or 0. There is no restriction on the number of 1's in each row of the matrices. If all the entries of a particular row are zero, then the transition is from the state to the empty set 0. If a particular row has more than one 1, then the transition function may map the succeeding state into any one of a number of possible states. For example, if TX1 is given as 1 0

TX1 -

1

1 0

1

0 0

1

then the state transition is determined as follows : q(k) q1 q2 q3

I

x(k) q(k

+ 1)

{qi, q2}

{gi, q2, q3}

It is felt in [250] that the transition function /t is so general that extra constraints may be incorporated . A special restriction is that the row sum of all transition matrices be equal to 1 similar to that of the stochastic automata . That is, [A], [B], [C] . . . . have exactly the same structure as © 2002 by Chapman & Hall/CRC

468

10. Applications

that of stochastic matrices. This type of automata is called normalized fuzzy automata. There are many properties that can be examined for fuzzy automata similar to those for deterministic automata and stochastic automata [249].

10 .4

Fuzzy Automata as Models of Learning Systems

A basic learning system is given in Figure 10.1. Y

Unknown Environment

X

"Student" Decision Maker

Learning Section

Performance Evaluator "Teacher" "Learning System" Figure 10 .1 1 : Basic Learning Model 'Figure 10 . 1 is from [250], reprinted with permission by Copyright


1969 IEEE .

10.4 . Fuzzy Automata as Models of Learning Systems

469

The proposed model represents a nonsupervised learning system if a proper performance evaluator can be selected [161, 162] . The learning section primarily consists of a composite fuzzy automaton . The performance evaluator serves as an unreliable "teacher" who tries to teach the "student" (the learning section and the decision maker) to make correct decisions . The decision executed by the decision maker is deterministic . Since online operations are required, the decision will be based on the maximum grade of membership . That is, for S2 = {w2 I i = 1, 2, . . . , r}, the set of all pattern classes, it is decided that x is from h2, or x - w 2 if

ft, (x) = V{ft, (x) I wj E Q, .i = 1, . . . , r}, where w2 E S2, i = 1, 2, . . . , r. If the decision maker is allowed to stay undecided and defer its decision, especially at the beginning of the first few learning steps, then it will be decided that x - w2 if

ft, (x) = V {ft, (x) I wj E Q, .i = 1, . . . , r} > c and be undecided if

It, (x) = V{ft, (x) I wj E Q, .l = 1, . . . , r} < c, for some predetermined c, 0 < c jl 1=1,2, . . . ,n,

- M%j)111

10 .4 is from [250], reprinted with permission by Copyright 1969 IEEE .


474

10. Applications

where Kk is the performance evaluation for the lth set of discriminant functions at the kth step of learning, Sk i is the sample deviation for the ith class of the lth set of discriminant functions at the kth step of learning, Mk a is the sample average for the ith class of the lth set of discriminant functions at the kth step of learning, and 0
T as called an ergodic fuzzy transition matrix . Certain stationary fuzzy transition matrices have no ergodic property, but have a periodic property. As an illustration, consider the following examples . n ~ oo is

Example 10.6 .1 [TZ](1) -

0 .3 0 .8

1 .0 0 .1

[T (2) -

0 .8 0 .3

0 .3 0 .8

[T (3) _

0 .3 0 .8

0 .8 0 .3

[TZ] (4) _

[ 0 .8 0 .3

0 .3 0 .8 0'3 0 .8

0 .8 ] 0 .3

0 .3 0 .9 0 .5

0.4

0.7

0 .5 0 .5 0 .8

0.7

0.4 0.7

0.7

0.4 0.7

The matrix keeps oscillating between period of oscillation is 2 .

Example 10.6 .2 [TZ]~ 1 > =

[T]

(2)

=

(3) = [T]


0 .6 0 .5

0 .1 0 .8

0 .6

0.4

0 .6

0 .6 0 .2 1

0 .6

1

0 .6 0 .6

0.7

1

and [ 0'8 0.3

0 .3 0 .8

.

The

10.7. Fractionally Fuzzy Grammars and Pattern Recognition

[T (9)

_

4 [TZ]( > =

0 .5 0 .7 0 .6

0.6 0.6 0.7

0.7 0.6 0.6

[TZ]( s> =

0 .6 0 .6 0 .7

0.7 0.6 0.6

0.6 0.7 0.6

[Ti] (6)

=

0 .7 0 .6 0 .6

0.6 0.7 0.7

0.6 0.6 0.7

[TZ] (7> =

0 .6 0 .7 0 .6

0.7 0.6 0.7

0.7 0.6 0.6

[TZ] (8) =

0 .6 0 .6 0 .7

0.7 0.7 0.6

0.6 0.7 0.6

[T (6) ;

[T,](10)

= [TZ]( 7 ) ; [TZ]

(11)

481

= [TZ](8) ; . . . .

The matrix in Example 10.6.2 has a period equal to 3. From Examples 10.6.1 and 10.6.2, it can be said that ergodicity has a period equal to 1. A fuzzy transition matrix having a period of length 1 is considered as an aperiodic fuzzy transition matrix .

10 .7

Fractionally Fuzzy Grammars and Pattern Recognition

A type of fuzzy grammar, called a fractionally fuzzy grammar, is presented in this and the next section . It was first introduced in [44] . These grammars are especially suitable for pattern recognition because they are powerful and are easily parsed . Formal language theory has been applied to pattern recognition problems in which the patterns contain most of their information in their structure rather than in their numeric values [69, 223, 250, 226, 225]. In order to increase the generative power of grammars and to make grammars more powerful for the purpose that they become more suited to pattern recognition, the concept of a phrase structured grammar can be extended in several ways. One way is to randomize the use of the production rules. This results © 2002 by Chapman & Hall/CRC

482

10. Applications

in stochastic grammars [223, 67, 109] and fuzzy grammars . Languages produced by fuzzy grammars have shown some promise in dealing with pattern recognition problems, where the underlying concept may be probabilistic or fuzzy [250, 226, 225] . A second way of extending the concept of a grammar is to restrict the use of the productions [106, 190, 191] . This results in programmed grammars and controlled grammars . These grammars can generate all recursively enumerable sets with a context-free core grammar . Programmed grammars have the added advantage that they are easily implemented on a computer. Cursive script recognition experiments [54, 52, 53, 139, 124, 213] in the 1960's and 70's had the major emphasis on recognizing whole words. None used a syntactic approach . A typical method presented in [53] decomposed the words to be recognized in to sequences of strokes that were then combined into letters and into words. In [139], an attempt was made to distinguish words that are similar in appearance such as fell, feel, foul, etc . All of these experiments except the ones in [213] input their data on a graphics device and kept the sequence of the points as a part of the data . In [213], pictures of writing were inputted and hence the sequence information was not available . In the following, we show that the languages generated by the class of type i (Chomsky) fractionally fuzzy grammars properly includes the set of languages generated by type i fuzzy grammars . We also show that the set of languages generated by all type 3 (regular) fractionally fuzzy grammars is not a subset of the set of languages generated by all unrestricted (type 0) fuzzy grammars . We show that context-sensitive fractionally fuzzy grammars are recursive and can be parsed by most methods used for ordinary context-free grammars . We describe a pattern recognition experiment that uses fractionally fuzzy grammars to recognize the script letters i, e, t, and l without the help of the dot on the i or the crossing of the t. We discuss the construction of a fractionally fuzzy grammar based on a training set . A fuzzy grammar FG is a six-tuple FG = (T, N, S, P, J, /t), where T, N, S, P are, respectively, the terminal alphabet, the nonterminal alphabet, the starting symbol, and the set of production rules as with an ordinary grammar . J = {r2 I i = 1,2, . . . , n} is a set of distinct labels for the productions in P, and /t is a fuzzy membership function, /t : J ~ [0,1] . Let V = T U N. Suppose that rule r2 is a ~ 3. Then we write the application of r2 as follows : ycti6 M('i) r}' S, where cti, 3, -y, S E V* . If BZ E V*, i = 0, 1, 2. . . . , m, and S = Bo

F (r) _~ r1


V(r2) V(r3) Bl _~ 02 _~ 03 . . . r2 r3

F(Tn) r'n

Bm

= X

10.7. Fractionally Fuzzy Grammars and Pattern Recognition

483

is a derivation of x in FG, we write S

=

Bo

"(r1T2+

. .T~n)

r1 r2 r3 . . . r-

8772

-

x)

where we write ft(rir2r3 . . . rm ) for ft(ri) A ft(r2) A . . . A ft(r~ ). The grade of membership of x E T* is given by the function ~ : T* --+ [0,1], ~(x) =V{ft(rlr2r3 . . .r,)},

where the supremum is taken over all derivations of x E L(FG), the language generated by FG . It follows that ordinary grammars, i.e., those with ft(ri) = 1 for all r2 E J, are a special case of fuzzy grammars . Fuzzy grammars can be classified according to the form of the production rules. In Chapter 4, it has been shown that for every context-free fuzzy grammar G, there exists two context-free fuzzy grammars G9 and G, such that L(G) = L(G9) = L(Gc.) and Gg is in Greibach normal form and G, is in Chomsky normal form. Example 10.7.1 Consider the fuzzy grammar FG = (T, N, S, P, J, ft), where T = {a, b}, N = {S, A, B, C}, and P, J, and ft are given as follows: rl r2 r3 r4 r5 r6 r7 r8 r9

: : : : : : : : :

S A B A A A C C A

AB a b aAB aB aC a as B

ft(rl) ft(r2) ft (r3) ft (r4) ft(r5) ft (r6) ft(r7) ft (r8) ft(rs)

= = = = = = = = =

1 1 1 0.9 0.5 0.5 0.5 0.2 0.2

The language generated by this fuzzy grammar consists of strings of the form a'b' with n, m > 0 . The membership of these strings is given as follows:

(a'b') _

1 .9 .5 .2 0

if m=n=1 if m=n :?~ 1 if m = n f 1 if m=nf2 otherwise.

Hence it follows that this grammar generates strings of the form a'b', where In-ml < 2.

As we have previously seen, a fuzzy grammar can be used to generate ordinary languages by the use of thresholds . One such language with threshold c is the set of strings L(FG, c) = {x E L(FG) l ~(x) > c} . © 2002 by Chapman & Hall/CRC

484

10. Applications

There are two other threshold languages defined in [147], namely, the twothreshold language and the equal-threshold language. They are defined as follows : L(FG, cl, c2) = {x E L(FG) I cl < ~(x) < c21

and L(FG, =, c) = {x E L(FG) I ~~(x) = c} .

The language L(FG, c) is most often used to compare the generating power of a fuzzy grammar to that of an ordinary grammar . However, L(FG, c) = L(G), where G is the grammar obtained from the FG by removing all productions whose fuzzy membership is less than or equal to c and then removing the fuzziness from the remaining rules. Hence it is stated in [44] that the use of threshold language seems limited . 10 .8

Fractionally Fuzzy Grammars

The patterns in syntactic pattern recognition are strings over the terminal alphabet . These strings must be parsed in order to find the pattern classes to which they most likely belong . Back tracking is required by many parsing algorithms [3] . That is, after applying some rules, it is discovered that the input string cannot be parsed successfully by this sequence of rules. Rather than starting all over again, it is desirable to reverse the action of one or more of the most recently applied rules in order to try another sequence of productions . It is sufficient with ordinary grammars to keep track of the derivation tree as it is generated with each node being labeled with a symbol from T U N, where T is the set of terminal symbols and N is the set of nonterminal symbols. However, this tree is not sufficient for fuzzy grammars since the fuzzy value at the ith step is the minimum of the value at (i -1)th step and the fuzzy membership of the ith rule. If this minimum was the ith rule's membership, there is no way of knowing the fuzzy value at the (i - 1)th step . Hence the fuzzy value at each step must also be remembered at each node. Consequently, the memory requirements are greatly increased for many practical problems . Another drawback of fuzzy grammars in pattern recognition is that all strings in L(FG) can be classified into a finite number of subsets by their membership in the language. The number of such subsets is limited by the number of productions in the grammar . This is due to the fact that if x E L(FG) with a membership ~(x), then there must be a rule in FG with the membership P(x) since P(x) = A{ft(rjj) I i = 1, 2, . . . , m} for some sequence of rules r21 rig . . . ri,n in P. Thus L(FG, c) for some threshold c is always a language generated by those rules in the grammar with a membership greater than c. © 2002 by Chapman & Hall/CRC

10.8 . Fractionally Fuzzy Grammars

48 5

To overcome these drawbacks, we present a method introduced in [44] of computing the membership of a string x that can be derived by the m sequence production rules, rl r2 . . . r k , of lengths lk, where k = 1, 2, . . . , m. This brings us to the next definition. Definition 10.8.1 A fractionally fuzzy grammar is a 7-tuple FFG =

(N, T, S, P, J, g, h), where N, T, S, P, and J are the nonterminal alphabet, the terminal alphabet, the starting symbol, the set of productions, and a distinct set of labels on the productions as a fuzzy grammar, respectively. The functions g and h map J into hY U {0} such that g(rk) < h(r2) b'r2 E J. A string is generated in the same manner as that by a fuzzy grammar. The membership of the derived string is given by V'x E T*, ~~b 1g(r~)

(x) = V{ E i

~= i h(r~ )

I k=1 2, . . . ,m},

where 0/0 is defined as 0.

Since 0/0 is defined to be 0, it follows that 0 n by assumption, it follows by Lemma 10.8 .4 that this sequence must contain a loop and is thus not the shortest sequence. Hence no such Br exists. Suppose that Oj ~ Oj+1 . . . ~ Oj+s = Oj is a simple loop in FFG, where I O j = k. Suppose this loop was not detected by the derivations that generate Rk and that this loop can be detected from 02 E R.0 by the shortest sequence OZ ----> OZ+1 . . . ----> OZ+T =

0j .

By assumption, r + s > n since the loop was not detected by generating Rk . However, the r + s strings 02, O2+,, . , Oj+s_i cannot all be distinct. Since 02, 0 2+1, . . . , OZ+T are distinct by assumption and O j , . . . , Oj+s_i are also distinct by assumption, we have that 02 + = Oj+, for some 0 Oi+1 . . .---->

Oi+., = Oj+v ----> Oj+v+1 . . .-~ Oj+s = Oj ----> . . .----> Oj+v

also detects the simple loop and is shorter than the original sequence. This contradicts the original assumption. Consequently, no such loop can appear in a derivation. Theorem 10.8.7 If a fractionally fuzzy grammar FFG is a context-sensitive fractionally fuzzy grammar, then it is recursive.


10.8 . Fractionally Fuzzy Grammars

48 9

Proof. Let IV I = n. For any string x, we need only generate R1, R2 . . . . , Rk , where k = Ix I . Let RZ be the set ofall strings derivable from the starting symbol in i steps . Since i is finite, RZ can be determined . Since it takes nj steps to generate Ri from Ro, Rr contains Ri if m >_ 1 + n + n2 + . . . + ni . Thus in a finite number of steps, all strings of length k can be found and all loops in these derivations can be detected. Hence it can be determined whether or not x E L(FFG) and its membership if it is . The following example provides an interesting property of c-fractionally fuzzy grammars. Example 10.8.8 For a regular fractionally fuzzy grammar FFG, the lan-

guage L(FFG, c) is not necessarily a regular language : Consider the two fractionally fuzzy grammars FFG1 = (T, N, X, P, J, g1, hl) and FFG2 = (T, N, S, P, J, 92, h2), where T = f0,1}, N = f S, Af, and P is given as follows: r1 r2 r3 r4 r5

: : : : :

SOS S~0 S~A A 1A A 1.

Let hl(r2) = h2 (r 2) = 1 for i = 1, 2, 3, 4, 5 and let gl(ri) = 1 for i = 1, 2, 3, 91(ri) = 0 for i = 4, 5. Let 92(ri) = 0 for i = 1, 2, and let 92(ri) = 1 for i = 3, 4, 5. Clearly, both the grammars produce strings of the form Onl ,, where m, n > 0 and m + n > 0. Consider the string 02 13. Now S~OS~OOS~OOA~001A~0011A_""~ 00111 .

2+1 and 3+1 Hence ft, (o213) = 36 = 3+2+1 N2 (0 2 13) = 46 = 3+2+1 . It follows that On1,n the fuzzy membership of is given by the following two equations: n+1 m+n+1

Itl(On1m)=

and

m+l m+n+l

0

ifm>0 of m=0.

Assume that the set L(FFG1, 0.5) =

{OnjmI

n > mf

and the set L(FFG2, 0.5) =

fOnlm

I n < mf

are regular. Since regular sets are closed under intersection, the set L(FFG1, 0.5)

n

L(FFG2, 0.5) = fOnln I n > Of

must be regular. However, it is known that fOnln I n > Of is context-free and not regular. Thus the assumption is false and the desired result holds.


490

10 .9

10. Applications

A Pattern Recognition Experiment

In [44], an experiment was developed to test the usefulness of fractionally fuzzy grammars in pattern recognition. A pattern space consisting of a set of strings was chosen in order to use script writing . The data were input to a computer on a graphics tablet . This data consisted of strings of points in a 2-dimensional space of the tablet . The data were a sample of seven persons' handwritings . Each person was given a list of 400 seven-letter words and was told approximately how large the person should write . The first three persons wrote all 400 words while the last four wrote the first 100 words . The data was digitized in a continuous mode by the computer whenever the pen was down. Each point collected in this manner was compared to the previously stored point to determine if the distance between them was greater than a given threshold. The threshold was chosen to be about 0 .04 inch. If the distance was not greater than the threshold, the new point was discarded and a new position of the pen was read. If the threshold was exceeded, this point was added to the data and the process was repeated . This procedure resulted in a record of 250 points in the X - Y plane (with zero fill-in for each seven-letter word written. Some examples of words input to the computer are shown in Figure 1 of [44, p. 344] . These points were converted into a string of symbols that would comprise the terminal alphabet . This was accomplished by comparing each adjacent pair of points to see the relative direction traveled by the pen at that point. The directions were then classified into one of eight directions . Each direction was separated by 45 degrees with class 0 centered at 0 degrees (the positive X-direction and the remaining classes being numbered 1 through 7 in a counterclockwise direction . Hence the terminal alphabet consisted of eight octal digits, i.e., T = {0,1, 2, . . . , 7}. Figure 2 of [44, p. 345] shows that quantization of directions introduced some distortion into the data. The individual letters were separated by an operator using an interactive graphics program . These letters then consisted of strings of octal digits whose lengths varied from 10 to about 70 characters in length . The crossing of and the dotting of the were deleted since they did not necessarily follow the basic letter without other letters intervening . Hence in order to keep the computer time down, only four letters were used in the test . The machine was asked to separate the i's, e's, Vs, and the l's without the dots on the and the crossings of the Vs. In view of Example 10.8 .8, it was also decided to use only regular fractionally fuzzy grammars . The grammars listed in Figure 10.8 were generated by cut and try methods

is

is

is


10.9. A Pattern Recognition Experiment

491

based on the following ideas.

Production Rule S OS

h, 0/0

h.;, 0/0

h, 0/0

h, 0/0

S

1A

14/14

10/14

0/18

0/18

S A

2B OA

12/12 0/1

8/14 0/1

0/18 0/1

0/18 0/1

A

1A

0/0

0/0

0/0

0/0

A A A A A A B B B

2B 2B 3C 4D 5E 6F OB 1B 2B

0/0 0/0 1/1 0/1 0/16 0/16 0/2 0/1 0/0

0/0 0/0 0/3 0/1 8/8 8/8 0/2 0/1 0/0

0/0 0/0 4/4 4/4 0/4 0/4 0/2 0/1 0/0

0/0 0/0 0/3 0/1 4/4 4/4 0/2 0/1 0/0

B B B B B C C C C C D D

3C 3J 4D 5E 6F 3C 4D 5E 6F 7G 4D 5E

1/1 0/2 0/0 0/5 0/16 2/2 5/5 1/1 0/3 0/7 2/2 1/1

0/3 0/2 0/0 5/5 8/8 0/7 0/5 0/3 3/3 6/6 0/7 0/3

4/4 0/2 4/4 3/3 0/4 4/4 4/4 2/2 1/1 0/3 5/5 2/2

0/3 0/2 0/0 5/5 4/4 0/7 0/5 0/3 3/3 6/6 0/7 0/3

D

6F

0/0

0/0

0/0

0/0


Comments Allows horizontal initial stroke Begining of a letter . Initial membership Ditto Small backtrack in direction (noise) Expected in all up strokes No effect Ditto Ditto Top of a letter not pointed Ditto Top sharply pointed Ditto Noise up on stroke Noise up on stroke Expected on up stroke (no effect Rounded top of a letter Noise sequence started Ditto Neutral top of a letter Pointed top of a letter Rounded top of a letter Ditto Neutral top of a letter Slightly pointed top Pointed top Very open loop of a letter Open loop or a highly slanted letter No effect

492

10. Applications Production Rut e

ge he

9-i h.;,

E 5E 0/1 0/1 E 6F 0/2 0/2 E 7F 0/1 0/1 F OH 0/0 0/0 F 0 0/0 0/0 F 7G 0/2 0/2 F 7 0/2 0/2 F 6F 0/1 0/2 F 6 0/2 0/2 F 5F 0/1 0/1 G OH 0/0 0/0 G 0 0/0 0/0 G 7G 0/2 0/2 G 7 0/2 0/2 G , 6G 0/3 0/3 G 6 0/2 0/2 H OH 0/0 0/0 H 0 0/0 0/0 J 2B 0/0 0/0 Figure 10.88 . The List Fuzzy Grammar for the

Comments 2/2 0/1 Possible loop 2/2 2/2 Expected part of down stroke 0/1 0/1 Noise on down stroke 0/0 0/0 End of a letter (tail) 0/0 0/0 Ditto 2/2 2/2 Down stroke 2/2 2/2 Down stroke . End of a letter 2/2 2/2 Down stroke 2/2 2/2 Down stroke . End of a letter 0/1 0/1 Noise on down stroke 0/0 0/0 No effect (tail) 0/0 0/0 End of a tail 2/2 2/2 Down stroke 2/2 2/2 Ditto 0/3 0/3 Noise 0/2 0/2 Noise or end of a letter 0/0 0/0 Tail of a letter (no effect) 0/0 0/0 Ditto 0/0 0/0 Noise of Production Rules of the Fractionally Experiment. 91 h,

9t h*

Since all the letters under consideration started with a near horizontal, left to right stroke (octal direction 0) and continued in a counterclockwise direction (increasing octal direction) until returning to a near horizontal tail, the same set of production rules was used for all classes. The productions used the nonterminal symbols A, B, . . . , G to represent the highest octal direction so far encountered, A representing 1, B representing 2, and so on. In the ideal case, only higher octal directions and higher nonterminal representations are reachable from any nonterminal symbol . However, to allow for noise in the less curved portions of the letters, the terminal symbol generated was allowed to be one less than the highest so far generated. A change of direction of more than 225 degrees counterclockwise was not allowed since this would never occur in these letters . In order to allow a tail of any length to be affixed to the ideal letter, the nonterminal symbol H was added . The grammar was tested on a training set and was found to accept most of the strings . In order for all strings in the training set to be accepted, minor modifications were made. For example, J was added to the nonterminal alphabet to pick up an unusual noise condition . The fractionally fuzzy membership functions were developed using the following criteria . First, a rule that could not help distinguish one class from another was given the value 0/0 and would then have no effect on the final membership assuming some rule r, for which h(r) :?~ 0, was also applied. 8 Figure

10 .8 is from [44], reprinted with permission from Academic Press .


10.9. A Pattern Recognition Experiment

493

Second, a rule for which h(r) was small would have little effect on the final membership of any string generated by that rule. Third, any rule for which h(r) was large would have a large effect on the final membership of any string generated by using that rule. Fourth, if rule r was used, the fuzzy membership of the string would be changed in the direction towards the value h(T) by that application of rule r. Hence if h(T) were close to 1, then

the membership of the string would be increased and if h(") were close to 0, the membership of the string would be decreased . Finally, a rule that was used in all strings could be given a membership value that could serve as a starting point from which subtraction could occur by rules with = 0 1(T) and to which addition could occur by rules with = 1. Either the sech(") ond or the third rule in Figure 10.8 must be used in any valid derivation. Some comments are included in Figure 10.8 to give some insight into why the membership functions for that rule were chosen . For example, the rule B ~ 6F is used when a vertical line changes direction abruptly from up to down. This would indicate a sharply pointed crown and the letters i and t are reinforced while e and l are reduced in membership when this rule is used. After adjustment on the training set to allow a threshold of 0.5 or more to indicate "in the class" and less than 0.5 to indicate "not in the class", the grammars were used on a random sampling of 121 letters from the remainder of the patterns . The strings were parsed in a top down (left to right manner by a program written in SNOBOL 4 programming language . The results of this test are summarized in Figure 10.9 . Class E I L T Method 1 : % error 110 16 28 74 Method 2 : % error 10 4 5 27 Figure 10.99 : Results of the Experiment

Two methods of categorizing were tested. The first classified the letter into any class for which the pattern had a fuzzy membership of 0.5 or more. Some letters were not classified while others were classified into more than one class. The method was considered successful if the correct class was included among other classes since a contextual post-processor could be used to find the correct letter . The second method classified the pattern into the class that had the highest fuzzy membership . As expected, the second method had better results, with 90% of the e's, 96% of the i's, 95% of the l's, and 73% of the is correctly classified. The only distinction between a t and an l is the width of the loop. Since many of the is were quite wide, they were incorrectly classified as l's. If the presence of one or more is was detected by the presence of absence of a horizontal line written directly above some portion of the word, most of these incorrect classifications could be corrected by a contextual post-processor such as 9 F' igure

10 .9 is from [44], reprinted with permission from Academic Press .


494

10. Applications

described in [54] . The distinction between the e's and l's could have been improved if the data were prescaled to eliminate differences in the average height of the letters generated by the different subjects . Considering the similarities in the four letters tested, the results were considered in [44] to be quite good. 10 .10

General fizzy Acceptors for Syntactic Pattern Recognition

In [62, 63], the syntactic approach to pattern recognition was examined by using formal deterministic and stochastic languages . In [175], fuzzy regular languages for pattern description in relationship with finite fuzzy acceptors were considered, where max-min was used as the composition of L-fuzzy relations. In [98], general fuzzy acceptors were considered using sup-*-composition for binary operations * preserving the properties concerning E-equivalence and which has min as the greatest lower bound . We consider [98] in the next two sections . Let (L, V, A, 0,1) be a complete lattice with upper and lower bound 1 and 0, respectively. Let f : L ~ [0,1] be an injective isotonic map, i .e., a one-to-one function such that u f (u) L}. Let I, J K be arbitrary nonempty sets. Any binary operation * on L can be used for the construction of the composition of L-fuzzy relations [50] as follows . Let /t E L(I x J), v E L(J x K) . Define the sup-*-composition /t o v by V(i, k) E I x K, ft o v(i, k) = V{lt(i,j) * v(j, k)

I

.j E J} .

Let a E L(I) be an L-fuzzy subset of I . Then an L-fuzzy subset a o ft of J can be defined by Q o /t(i)

= V{a(i) * N-(i,i) I

i E I},

Vi

E J.

If p E L(J), /t o p is defined similarly. Moreover for a' E L(I), a o a' can be defined to be a scalar . In order to consider the complete transition behavior of a general acceptor and how to compute it, the following definition is needed, [19, 91] . A triple (L, *, L I x E X} is the set of transition functions; G is a commutative cl-semigroup .

t, T, .M, G),

(1) (2) (3) (4) (5) (6)

t

:

If * = A and Q and X are finite sets, then this is the same as the definition in [175] . We consider the complete transition behavior of an acceptor and how to compute it. We now extend the stepwise transition behavior of A. Let X be the set of input symbols . Let X l = X and Xi = X x X x . . . x X (j times), j E N. Let S[X] denote the disjoint union of the sets X l , X . . . . . . Then the elements of S[X] are sequences (xii, xi2, . . . , xi-), xii E X, m = 1, 2, . . . . Define a multiplication on S[X] by juxtaposition, i.e., (il ~ xi2 ~ . . . ,

xi-)

(

x

il

I

x72 l . . . , x7J

=

x

(

zl

I

xi2 l . . . ,

xim ~

X

il

~

x72 ~ . . . , x in

-

Then S[X] is a free semigroup with respect to this operation. Let ,F[X] denote S[X]U{A} . Then ,F[X] is a free monoid generated by X with identity A.

Every element of ,F[X] is called a word on X. Every word x E S[X] can be represented by x = xil x i2 . . . where xi2 , . . . , xi,, E X . Then x is said to have length k . Let A = (Q, X, G) be a fuzzy acceptor . For any input word x E S[X], the transition function M(x) is computed by the expression xik

,

Xil

,

t, T, .M,

M(x) = M(xiJ ° M(xi2) 0 . . . o M(xib)

I

where x = xilxi2 . . . E S[X] and where M(A) may be regarded as the identity. The expression represents the complete behavior of A in k consecutive steps . Every element M(x) gq , is the grade of membership if the input word is x E Xk with the beginning state q E Q at instant t and the last state q' E Q at instant t + k . The set of all transition behavior fuzzy relations describes the complete behavior of A and we use the notation xik


496

1 0 . Applications

.M*(A) = {M(x) I x E .F[X]I . If x E .F[X], t o M(x) o T is the analytic extension of the stepwise behavior of A . Let L = Q = [a, b] be an interval . Then Vx E M, the transition function M(x) is a fuzzy subset of [a, b] x [a, b] . The initial state distributions and the final state evaluations are fuzzy subsets of [a, b] . Hence fuzzy acceptors may be extended to continuous types . In particular, if a state set Q is a subset of [a, b] such that QI = n for some n E N, then the transition functions can be represented as n x n matrices . The initial state distributions and the final state evaluations also are vectors of dimension n .

10 .11

E-Equivalence by Inputs

Definition 10 .11 .1 Let p, ft' : I x J ~ L, and let E E [0,1] be fixed. Then ft and ft' are called E-close, denoted by ftEft', if IIw(ij) - w'(i,j)II _ d(q*, q*), where q* E q6x and q* E q6 x. This clearly represents a stability property. Let A = (Q, X, A) be a finite automaton . Let F(A) be the set of all fuzzy-state automata with A as first coordinate. Let F(A,T) = {(A, a) E F(A) I a C T, a a tolerance on Q}. Then F(A) forms a complete, distributive lattice with respect to the ordering of automata with tolerance given by (A, a)

) The "phase space velocity" c = d(g,,ge of A is less than or

equal to 1 . In this sense, a tolerance automaton has inertia that gives rise to stable behavior . The example closing Section 10 .13 is also an example of a tolerance automaton.

Example 10 .14 .6 The automaton A* given in Table 10 .1 determines 64 automata with tolerance . This is easy to see from the following reasoning : Since there are 4 states, a tolerance can be represented by a 4 x 4-matrix whose entries consist of 0's and 1's. Since a tolerance is reflexive and symmetric only 6 positions of the matrix determine a tolerance . Thus there are 64 = 2 6 possible tolerances.

Table 10 .1 : Next State Relation of A* inputs of A*

A* x1 x2

qo qo q2

q1 qo q2

q2 q1 qo

q3 q1 qo

states of A* next states of A*

F(A*) is shown in Figure 10 .10. (A*, T19) is the only nontrivial tolerance


10.14. Fuzzy-State Automata

50 7

automaton of F(A*) .

Figure 10 .1010 : Lattice F(A*) The following Boolean matrices (with respect to the ordering of the states given by (qo, ql , q2, q3)) represent the basic tolerances of F(A*) . T2

1 0 0 0

0 1 0 0

1 1 1 0

1 1 0 0

T5

T3

0 0 1 1

0 0 1 1

1 1 0 0

1 1 0 0

1 0 1 0

0 0 0 1

1 1 1 1

1 1 0 0

T7

0 0 1 0

0 0 0 1

1 0 1 0

1 0 0 1

"Figure 10 .10 is from [36], reprinted with permission by Kluwer Acadernic/Plenuin Publishers .


10. Applications

We now determine ( A ,T * ) . The following procedure determines the maximal symmetric binary relation p, on Q , with the property that (S,, S,)p, g p, g T V x € X . Thus ( A ,p,) = ( A ,T * ) (unit of F ( A ,T ) ) if F ( A ,T ) is nonempty, e.g., if A is deterministic. Define the relation p(k) as follows: Step 1: p(1) .- T . Step 2: For all k 1, qp(k l ) p if and only if V x € X U {A) and V(qf,p f ) E Q x Q it follows from qS,qf , pS,pf that qfp(k)pf. Step 3: If p(k 1) = p(k) go to Step 4, else go to Step 2. Step 4: p, = p(ko),where Lo is the smallest index k with p(k) = p(k+l). Clearly, p, is symmetric since T is symmetric and p, is the maximal relation with the properties stated above since p g T implies that p, g p, . Consequently, a g p, if (S,, S,)a g a V x E X .

>

+

+

Example 10.14.7 Consider Figure 10.10. Let

Then qoSz1qo and qzSxl ql , but (40,ql)$ p ( 1 ) Thus (q0,q2) $ 4 2 ) . Moreover, qlSx1qo and qsSxlqi. Since (qo,qi) $ p ( l ) , (qo,qs) $ p(2). Now (q2,qs) € ~ ( 2since ) qzSxlqi,qsSxlql, qzS,,qo, and qsS,,qo. It follows that ( A ,T * ) $ F ( A * ) and p(2) = p(3) = ~ 2 Hence . ( A * ,T * ) = ( A * ,~ 2 ) . Consider an automaton with tolerance ( A ,T ) . Tolerance T* determines the maximal set of pairs of automata states that are in tolerance T and whose successors under any input sequence are also in tolerance T . Furthermore, if two states are not related by T* , then none of their predecessor pairs contains states within tolerance T* since T* = L\T* is reflexive, symmetric, and 6, *T* C T* V x E X , [253].

10.15

Stable and Almost Stable Behavior of Fuzzy-State Automata

In this section, we consider questions about the stable behavior of finite fuzzy state automata that have counterparts in the theory of topological dynamical systems. With this in mind, we state the following definition.


10.15. Stable and Almost Stable Behavior of Fuzzy-State Automata

509

Definition 10.15 .1 Let (A, T), A = (Q, X, A), be an automaton with tolerance, S a subspace of Q, and V C_ X* . The subspace S is said to be

V-stable if its neighborhood N(S) is V-invariant, i.e ., if N(S) .6, C N(S) Vv E V. S is said to be almost V-stable if there is a natural number l such that N(S) f1 UZ-ogbv+' z,4 0

holds for all n E N U {0}, q E S, and v E V. If S is X* -stable and accessible from every state of T C Q, then S is called an attnacton set of T.

Clearly a set S of automata states is almost V-stable if its neighborhood is V-invariant, i.e., if S is V-stable. Stable and almost stable sets of automata states characterize the recurrent state motions of automata with tolerance. The property of attractor sets can be illustrated in the following manner . Let automaton (A, T) be driven by a stationary random input source that generates words such that the probability of any input x E X following an arbitrary word is greater than k, where k > 0. Let A be at time to = 0, say, in T, and p(S, t) be the probability of its state at time t belonging to the neighborhood of an attractor set S of T, if it exists . Then p(S, t > I R(T) I) >_ k~R(T) 1-1 and p(S, t > I R(T) I) = 1 for an autonomous automaton (hence the name "attractor set"). Clearly, Q is an attractor set of any T C_ Q. The set of reset states of an identity-reset automaton with tolerance S is an attractor set of its state set and every state of automaton (A*, TO is almost X-stable with l a = 0, lb = h = ld = 1 (see Figure 10.10). In the following, we assume that the automaton A = (Q, X, A) is complete and deterministic . The following theorem is an adaptation of Poston's "approximate fixed point theorem" to finite automata . Theorem 10.15 .2 Suppose that the state space of a finite, fuzzy-state au-

tomaton is contractible . Then b'x E X*, it contains an x-invariant and x-stable subspace, Px :?' 0, whose elements are mutually within tolerance T (approximated fixed points .

Let automaton A be autonomous and Q be the union of contractible, Sxconnected subspaces, where x E X. Let T be the natural tolerance on Q with respect to an output relation w . Then Px (which is maximal with respect to the properties of Theorem 10. 15 .2) is an attractor set of Q . Now p(Px , t) = 1 for t sufficiently large . Because of unobservable (unmeasurable) differences of outputs, automaton A seems to be caught in a final state. However, this may not be the case if IPx I > 1 . Another example of an attractor set can be determined in the following manner . By Lemma 10 .15.3, for any q E Q, 00R(q) is an attractor set of R(q) if 00R(q) :?~ 01 and if intR(q) is empty, or else strongly connected, and Q\R(q) is X-invariant . Since R(q) is X-invariant, OR is then X*-stable and either q E 00R(q) and so R(q) C_ 00R(q), or else q E intR(q) . However, this implies that 00R(q) is reachable from any state of © 2002 by Chapman & Hall/CRC

51 0

10 . Applications

Clearly, both conditions hold if all R(q) with 00R(q) :?~ 01 are strongly connected, e.g., if (A, T) is a fuzzy-state permutation automaton . R(q) .

Lemma 10.15 .3 Let (A, T) be a fuzzy-state automaton and S be a subspace of its state space. Then the boundary r9S of S is X*-stable if S and Q\S are X-invariant.

Proof. If Ixl = 0 or r9S = 01, then r9S is x-invariant since SA = S . Suppose r9S is x-invariant for all x with xl < n and r0S :?~ 01 . (1) If q E r9S\S, then there is a state q E S such that (q, 46 x) E T for x E Xn . Thus (q6x , _qS xa ) E T, q6x E S, and -qSxa ~ S, i .e. , q6x E r9S for all YEXn+1 . (2) If q E OS n S, then there is a state q ~ S such that (q, q6x) E T for x E Xn . It follows that (4Sa, g6xa) E T for all a E X, i.e., d(4Sa, g6xa) < 1 . Suppose that g6xa E intS. Then 4Sa E S and R(q) n S z,4 0. This is a contradiction since 4S a E S. Hence g6xa E S\intS . Since (A, T) E F(A), r9S is X*-invariant and X*-stable . Lemma 10.15 .4 Let A be a complete, deterministic, connected, and au-

tonomous automaton with tolerance T. Then (A, T) is almost periodic if and only if every state of A is in tolerance with a periodic state of A.

Proof. Let q E Q. Let X = {x} since A is autonomous . The orbit of q under x, i.e., O = {q6x}°_o, is the union of two sets Ot and OP (p = ~QPj), where OP is nonempty and permuted by S x and Ot .67x C OP for s >_ t = Ot . (1) Let qTq, q be periodic. Then q E OP . There exist natural numbers r and s < p such that q6x = q Sx since A is connected and deterministic. Hence q6x+P-S E N(q) and Uz±O

That is, if

q

-s-l

+Z

gbx

n

has period p, then

N(q) :?~ 0,

p+r-s-1. (2) Suppose that

q

q

n = 0, 1, 2, . . . .

has an almost period not greater than

E Q is almost periodic. Then it follows that N(q)

n

UZ-ogbx+2

7~ 0.

Thus N(q) n OP z,4 0. Hence N(OP) = Q if A is almost periodic . Furthermore, N(Op) - Sx C N(O p - Sx) = N(Op) if (A, T) E F(A) for A arbitrary. An automaton (A, T) is said to be almost periodic if every state of A is almost X-stable . The connection between almost periodic and permutation automata can be seen from the next result. Theorem 10.15 .5 The state space of a deterministic, almost periodic fuzzystate automaton is the union of a finite number of neighborhoods of closed stable orbits (cycles .


10.16. Fault Tolerance of Fuzzy-State Automata

51 1

Proof. The proof follows from part (2) of the proof of Lemma 10 .15.4. The greatest common divisor of the lengths of all proper cycles of a deterministic automaton A is called the period of A, [75] . Let d divide the period of A. Let 7r be a path from state q to state q' and I7rI denote its length. The distance from q to q', written I q, q' I d , is defined as I7rI (modulo d) if q and q' are connected and is defined to be oo otherwise . It follows that this distance is unique . Assume that the period of A is greater than 1. Now tolerances Td on Q may be defined as follows. For all q, q' E Q, gTdq' if and only if q = q' or I q, q' I d = s or d - s. Automata (A, Td are almost periodic. Moreover, (A, Td) is a tolerance automaton and (A, A) is a fuzzy-state automaton if the period of A is even and A is complete. Example 10.15 .6 Let the automaton A be as specified in Table 10 .2. Table 10 .2: Next State Relation of A 0

xl

x2

qo

qi

q2

g3

q4

qi qi

q2 q4

qi qi

go, q2 q2

g3 g3

The lengths of the proper cycles are 2 and 4 . Thus d = 2 . Now Iqo, gsl2 = 3 (mod 2) = 1 and Iqo, g4I2 = 2 (mod 2) = 0. For s = 0, s :A Iqo, 8312 :A d - s . For s = 1, Iqo, 8312 = s. Hence go T2 q3 . For s = 0, ° Iqo, g4I2 = s . Thus qOT 2 q4 . For s = 1, Iqo, g4I2 z,4 s or d - s . It follows that the tolerances T2 and T° are represented by the following Boolean matrices. 1 1 0 1 0

1 1 1 0 1

T12

0 1 1 1 0

1 0 1 1 1

0 1 0 1 1

1 0 1 0 1

0 1 0 1 0

To2

1 0 1 0 1

0 1 0 1 0

1 0 1 0 1

An effective procedure for evaluating the period of any finite automaton is given in [75] . 10 .16

Fault Tolerance of Fuzzy-State Automata

An actual machine occasionally makes errors computing its next state . Consequently, it is unreliable to some extent We now consider this unreliability. It is possible to emphasize the essential aspects of more concrete situations within our abstract framework. A machine exhibits stability in some sense if it overcomes the influence of its errors, i .e., if after some time these influences become "tolerable ." We will show that fuzzy-state automata behave stably in this sense with respect to certain faulty state transitions . © 2002 by Chapman & Hall/CRC

512

10. Applications

Let A = (Q, X, A) be a (not necessarily deterministic) finite automaton, T a tolerance on Q, and QP = pr3A . Definition 10.16 .1

A binary relation 0 on Q with pr1O _D QP together with tolerance OTT, where t : 0 C_ Q x Q, is called a state relation of (A,T) . A state relation 0 is called compatible if 0 C_ T, inessential if 0 C_ p,r (cf. Section 10 .14, and consistent if it is fuzzy . We say that 0 changes q if &\S) z,4 0 . 0 is called (p, l)-bounded (by (A, T)) if for all xEX* with 1xl >p, (10.6) and (10 .6) does not hold for (p - 1, l) or for (p, l - 1) . If 0 is (p, l)-bounded, then 0 U 0-1 is also (p, l)-bounded . In general, the bounds of two state relations 0 and 0+ are known, it is usually not difficult to derive bounds for relations such as 0 U 0+ , n + , and 0 - 0+ . For example, 0 U 0+ is (max(p, p+), max(l, l+))-bounded . We present below an algorithm for determining the bound of a state relation. State transition errors may be described by a state relation . We then visualize an element (qj,qj) E 0 as follows . Automaton A goes with some nonzero probability into state qj when it should go into state qZ, due to a permanent or temporary modification in the state transitions of A. Clearly, error e E 0 n S is an improper error . if

o o

Definition 10.16 .2

A state relation (0, t*TT) is called a permanent transition modification (t-modification if it modifies automaton A, i.e., if automaton AO = (Q, X, AO := A - 0) replaces automaton A. Automaton A" is called the modification of A due to 0 and A the reference automaton. The state relation 0 is called a (temporary) error of A if A is not modified by 0 .

Most input errors can be interpreted as state transition errors, i.e., memory errors, [84]. The is also true for errors in the combinational output logic of sequential circuits . Consequently, we concentrate on memory errors. Within the relational framework, we are less interested in the physical causes of errors, but rather in the qualitative aspects of the role errors play in the performance of modifiable systems . In what follows, we assume that the meaning of tolerance T is that a single error e E 0 is tolerable in some appropriate sense if e E T. Examples can be found in [37] . Lemma 10.14.2 states that compatible errors of a fuzzy-state automaton (A, T) are inessential and remain inessential under the action of any input word. The maximal compatible state relation of (A, T) is TQ = (T, TT), and (p,r ,TT) is its maximal inessential state relation if pr1pr _D QP . Since the tangent bundle of the tolerance space (Q, T) is the composite map t TQ C Q x Q -PTA Q, [179], and the tangent space Tj Q to Q at state © 2002 by Chapman & Hall/CRC

10.16. Fault Tolerance of Fuzzy-State Automata

513

q is the tolerance space TqQ = (tq, TT), the compatible changes of state q determine the tangent space at q in a sense . In [37], such "geometric" properties of errors and t-modifications are studied . There modification tolerance of automata is considered, i.e., with masked and correctable tmodification of an automaton. However, in the following, relational (settheoretic) properties of temporary errors are considered. We now consider bounded state errors . Fault tolerance is an important design parameter . The goal is to design systems that stay operational despite failures and, in fact, can repair themselves in response to their own failure . Intuitively, automaton (A, T) overcomes the influence of error 0 if after some time it takes a state that is, and from then on stays, in tolerance with the correct state . Definition 10.16 .3 An automaton (A, T) is said to T-correct (to correct error 0 if there exists p E hY such that 0 is (p,1)-bounded ((p, 0)-bounded by (A, T) . Correction (self-synchronization) by deterministic finite automata has been studied in [85], [84], [42], [224], and elsewhere . Algorithms have been given for determining which temporary state errors are corrected by a de terministic automaton within a certain amount of time (assuming tacitly that these errors do not alter state transitions .) Correction of input errors has been studied in [252] and [84] . There is a strong connection between the capability of correcting an input error and that of correcting the temporary state errors caused by this input error . Example 10.16 .4 Consider the fuzzy-state automaton (A*, T5) given in Figure 10 .10 . The errors 4'1 and 02 are given by 0 0 0

1 0 0 0

0 0 1 0

0 0 0 0

_

The error 4'1 is (0,1)- and (1, 0) -bounded since E F(A) . The error 02 is (1,1)-bounded .

0 0 1 1 4'1

0 0 0 1

1 0 0 0

1 1 0 0

is compatible and (A*,

T5)

Bounds of an error and error-correcting input words can be determined by the following error graph procedure as long as the next state relation of an automaton is not too large . Error graph. (1) The vertices are given as (l/ab), a, b E Q, if aTl b and as (-/ab) if (a, b) ~ TL , l = 0, 1, 2, . . . , ((0/qq) - (q)) . (2) An oriented i-edge points from vertex (l/ab) to vertex (l'/cd) if and only if the (unordered) state pair {a, b} goes into state pair {c, d} under © 2002 by Chapman & Hall/CRC

514

10. Applications

input x2 . The error graph of (A*, T5) is given in Figure 10.11 .

2

Figure 10 .11 11 : Error Graph of (A*,

T5)

Test A. The following algorithm gives a test by which one can use to decide whether or not a given temporary error 0 of automaton (A, T) is (p, l)-bounded and, hence, T-connected by (A, T). Step 1: p := 0; l P := Q I . Step 2: Op := U1 w 1 -p (6,,, 6w) W; Op (1) ~_ OP ; OP (k+1) :_ OP (k)U{(q, q+)

Elx E X, (q, q+) E OP(k) such that g6 x q, q+6x q+) . Step 3: 0P+ := Uk>10P(k) . Step 4: If there exist 1 1 E hY with 1 Q is a transition function . Let it be a fuzzy subset of X, t = 0, 1, 2, . . . . Define qt+l : Q ----> [0,1] by for each q E Q, {qt(q') A Zt (X) qt+1 (q) _ { V 0

b(q', x) = q, q' E Q,

E x},

if s-1(q) if 6 -1 (q) _ 0 .

Define S : Q x ,FP(X) ----> .FP(Q) by b(qt, a) = qt+1

(10.7)

where Q={qt I t=0,1, . . .} .

A sequence of fuzzy states is denoted by {gt}t-o and is said to be increasing if qt+1 qt for all t . © 2002 by Chapman & Hall/CRC

518

10. Applications

Since monitoring is a continuous process that is terminated on the exhaustion of input data rather than on arrival at a certain state, a set of final states is not defined here. The definition of a fuzzy automaton that appears here differs from the usual ones; see [49,51,212,250] for example. The transition function is crisp rather than fuzzy. Uncertainty expressed in a fuzzy input alone results in a partial transition from one state to another . A general statement about how strongly two states are related is not possible. Except for the fuzzy initial state, the fuzzification of the automaton is completely determined by the extension principle . A fuzzy automaton exhibits properties quite different from those of a crisp one. The current state is a fuzzy subset of the set of states . Consequently, the automaton can perform different (partial) transitions simultaneously and therefore track parallel paths. Another different property is that crisp automata report an error on input not accounted for at the current state while a fuzzy automaton reacts on low or zero membership grades in the fuzzy input with continuously decreasing membership grades in its current state as obtained by (10 .7) . This phenomenon causes a decrease in certainty that seems consistent with all repeated applications of fuzzy set operations. Example 10.17 .2 Let Q = {ql, q2, q3, q4, q5} and X = {a, b, c} . Define S : Q x X ----> Q as follows:

b(qj, a) = q3, b(qj, b) = q4, b(qj, c) = q2, b(q2, a) = q2, b(q2, b) = q4, 6(q2, C) = q2, b(q3, a) = q3, b(q3, b) = q5Define

= q4, = q5, = q4, = q4, b(q5 , a) = q5, b(q5, b) = q5, b(q5, c) = q5,

(gl) = 4, for i = 3,_4, 5. Define io : X ----> [0,1] by_ 1-0 (a) = Let i t = io fort = 1,2, . . . . Then do :

gi(gi) gi (q2) 4-1

(q3)

gi (q4) 4-1

(q5)

Q ~ [0,1] as follows:

b(q3, c) b(q4, a) b(q4, b) b(q4, c)

0, (do

i

(qj)

do

do

z,

(q2) = 4, and do (q,) = 0 LO (b) = 4, and io(c) = 4 .

n io (c)) V (do (q2) n io(a)) V (do (q2) n io (c))

(gi) A io(a)) 1 2 (40(gi) n LO (b)) V (40 (g2) 1 4 (do

0.


n LO (b))

10.17. Clinical Monitoring with Fuzzy Automata

519

In a similar manner, we obtain

42(gl) 42 (q2) 42 (q3) 42 (q4) 42 (q5)

0, _1

4~ 1 2~ _1 4~ 1 4'

In fact, gt(g2) = q2 (qj) for i = 1, 2, 3, 4, 5, and t = 2, 3, . . . .

Automatically acquired data are generally precise and hence no source of fuzzy input is required by the automaton defined above. Also, if every single parameter value acquired represented an input on its own, (1) the number of possible input symbols needed to be accounted for would be too large for the automaton to handle and (2) the automaton would continuously change its state in order to react to a certain input ; otherwise the input would remain unconsidered and hence become lost. Therefore, a fuzzy automaton by itself is not very appropriate to perform monitoring . The data are therefore pre-processed by a function that abstracts from single input parameters by generating fuzzy events that are passed on to the automaton . Definition 10.17 .3 Let A = (Q, qo, X, S) be a fuzzy automaton, R1 through

Rn be the parameter ranges, where n is the number of parameters observed, P = R1 x . . . x Rn be the parameter value space, and f : P ~ .PP(X) be a function that maps_parameter tuples to fuzzy subsets of the input alphabet of A . Then M = (A, P, f) is called a state monitor.

Note that,F'P(X) specifies the interface between preprocessing of data through f and interpretation of input through A. Thus f can therefore be replaced by any computable method that yields a suitable fuzzy subset . From the definition of S, it follows that other than the state membership values of qo, those of qt can only be introduced through fuzzy input. The following lemma follows from (10 .7) . Lemma 10.17 .4 If a fuzzy automaton A is repeatedly fed with constant fuzzy input i, then the set of fuzzy states it transitions between is finite . m Define the height of qt, written hgt(gt), to be V{qt(q) I q E Q} for t = 0, 1, 2, . . . . The sequence {hgt(gt )1-o , is a decreasing function of t.

This reflects a loss of certainty in the automaton . Even if the input does not change, hgt(gt) can decrease rapidly. Hence in practice, when the monitor is provided input in rapid succession, once the current state is the empty set, it can never recover . As a matter of fact, if the automaton does not contain any feedback loops, i .e., does not provide circular transitions, it will arrive © 2002 by Chapman & Hall/CRC

520

10. Applications

at the empty state after at most as many steps as there are states. Rather than leaving the responsibility for providing appropriate feedback loops to the designer of the automaton to overcome this undesirable property, we next define a property that overcomes this inadequacy. Definition 10.17 .5 A fuzzy automation is said to provide a peak hold if b'q , q E Q and b'x E X,

6(q, x) = q implies S(q, x) = q.

(10.8)

The condition in Definition 10.17.5 says that there is a transition for every state on every input that leads to the state . The condition implies that no state can be entered and left on the same input, else S would not be single-valued . The peak hold guarantees that the maximum evidence for a state provided by its predecessors is memorized and held as long as input of ongoing transitions can support it. However, the peak hold may also be sustained by an input other than the one that initially led to that state since a state does not remember its predecessor . Thus the grade of membership can unintentionally remain high. Peak hold has a positive side effect in that the automaton cannot oscillate on constant input [250], a property that would clearly not be acceptable in the clinical setting since stable input should be reflected in stable output . This can be seen from the next result. The fuzzy automaton in Example 10.17.2 provides a peak hold. Theorem 10.17 .6 The fuzzy state of a fuzzy automaton with peak hold always becomes stable after a finite number of repetitions of the same input.

Proof. We show that for t > 1, there is a positive integer r such that qt C qt+1 C . . . C qt+' = qt+T+1 = . . . .

(10.9)

This is accomplished by showing that (1) {qt }t-- o is increasing, i .e., qt C qt+l C . . . and (2) 3r such that S(qt+T, i) = qt+,.. For all subsequent states, (10.9) follows from the fact that S is a function . _ For every fuzzy state qt with t > 0 and every input i, it follows by (10.7) that for every state q, there is a transition that determines its membership value, i.e., b'q E Q, S -1 (q) ~ 0. b'q E Q, 3q' and -T such that 6(q', x) = q and qt (q) = qt -1(q~) Thus, qt (q) < i(x) and so qt (q) A7(x) x) = q then implies

=

qt+1 (q) = qt(q) V (V{qt (q)

qt(q) . Repeated input of i and S(q, A 7(x) I

qt+1 (q) > qt(q), © 2002 by Chapman & Hall/CRC

n a(x) .

b(q, x) = q}),

10.17. Clinical Monitoring with Fuzzy Automata

521

and so qt+l D qt . By Lemma 10. 17.4, there is no infinite sequence of fuzzy states such that qt+1

=

b(qt,

i) and

qt+1

D

{qt}t-o

qt .

Consequently, there exists an r such that qt+T+1 C qt+T for t = 0, 1, 2, . . . . Since {qt}t_o is increasing, qt+T+1 must equal qt+T . m The proof of Theorem 10.17.6 shows that {_gt}t°o does not converge to the empty state . It can take several steps for A to become stable since the fuzzy input can cause a propagation of higher membership grades along a sequence of transitions . The fact that nonfuzzy deterministic automata with peak hold are stable after one step provides another example of how fuzzification yields more general results . The height of the current state is still decreasing even with the peak hold property since high grades of membership cannot be regained once they are lost . A situation where the grade of membership of one state decays while its successor's rises is a source of loss. This behavior does not model the natural decision process correctly because once a decision has been made, it is usually pursued rather uncritically until there is sufficient evidence for another decision to be made. By introducing the idea of a threshold, the state monitor can be modified to adopt this kind of behavior . A state is called active when its grade of membership in the current state exceeds a certain threshold c. An active state is defined to remain active until there is a transition that induces activity of one of its successors, i .e., qt (q) qt+1(q)

=

(10 .7)

c and 6(q, x) = q' and otherwise . if

qt(q)>

x such that it (x) > c

~q~,

(10 .10)

Thus once a state has reached a certain grade of membership, it keeps it until a transition can pass it on to one of its successors. Thus the height of the current state is always greater than c. Therefore, a certain level of uncertainty is thus always being maintained . For c = 1, (10.10) implies that there is always at least one state q such that qt(q) = 1 . This accounts for the fact that the patient is considered to be at least in one state at a time, even if no successor with more evident support could yet be determined. We note that (10.10) has only slight effect on the proof of Theorem 10.17.6. In (1) of the proof of Theorem 10.17.6, {qt}t°o is still increasing since the peak hold also works for qt (q) > c and qt (q) can only drop below c once, namely on the first input . Also, (2) of the proof of Theorem 10.17.6 still holds because Lemma 10.17.4 is not affected. Moreover, (10.10) cannot prevent the automation from oscillation without peak hold even though the height is kept above c. © 2002 by Chapman & Hall/CRC

522

10. Applications

Further discussion of clinical monitoring can be found in [222] . We note that another approach using techniques from fuzzy set theory can be found in [214] . Nonfuzzy approaches can be found in [87, 238]. 10 .18

Systems

R1zzy

In the final section, we consider fuzzy systems for two reasons . First, they resemble fuzzy automata and, second, they have interesting applications to information retrieval [160] . System theory provides a framework for describing general relationships of the empirical world. We are mainly interested in the concept of reachability, observability, stability, and realization . Let Q be the state space, X the input space, and Y the output space. A deterministic dynamic system (time invariant) is a complex S = {Q, Y, S, ,3}, where S : Q x X ~ Q is the dynamics, qt+1 = S(qt, ut ), and,3 : Q ~ Y is the output map . A nondeterministic system is the complex S = {Q, X, Y, S, /3} with the : Q dynamics S : Q x X ~ P(Q), qt+1 E S(qt, ut ), and the output map/3 P(Y) .

This definition can be generalized by considering not only the states, but also the inputs and outputs as being subsets . Definition 10.18 .1

An abstract system is a complex Sa = LQ,X,Y,b,01

such that S : P (Q) x P (X) ----> P (Q), 3 : P (Q) ----> P (Y) .

We next present the application given in [160] of fuzzy systems to information retrieval systems . Example 10.18 .2

We consider an Information Retrieval system defined in terms of its response to a request for information . An IR system reports on the existence and location of information items relating to a request and does not change the knowledge of the user on the subject of the request . If the search criteria are based on the contents of an information item, then it becomes necessary to use content identification such as a set of descriptors attached to each item . In such cases, it is customary to assign descriptors, normally chosen from a controlled list of allowable terms . That is, a request is defined to be recovered from the store . An IR system compares the specification of required items with the descriptions of the stored items, and retrieves, or lists, all the items that correspond in some defined way to that specification . Consequently, the IR system can be characterized by having as inputs a subset of a set D of information items and a subset of a set R of requests . If the system is presented with new documents, the system must process them to obtain descriptions . The same sort of thing must be done


10.18. Fuzzy Systems

52 3

with the requests . The next step is the comparison. The ultimate response of the system to a request is a partial ordering on the set of information items.

Let Pf (S) be the set of all finite subsets of a set S. Let ~, rl be functions ~ : D Q, rl : R Q, where Q is the set of descriptions. Let A = Pf(D), B = Pf(R), C = Pf(Q), and 'R be the set of partial orderings on Pf(D), i.e., for p E 1Z, M, N E Pf(D), MpN if and only if N is more relevant to a given request than M. We use the following variables to describe the state of the IR system : ql is the set of incoming documents waiting to be processed ; q2 is the document file; q3 is the document description file; q4 is the set of incoming requests waiting to be processed ; q5 is the request currently being processed ; q6 is the request that was just processed ; q7 is a partial ordering on Pf(D) (i .e., in 'R) induced by the request q6 . The input variables are ul = the documents coming to the system; u2 = the requests coming to the system ; and the output variables are yl = q6 is the request that was just processed ; y2 is a subset of the document file that is maximal with respect to q7 . Using the above notation, the input space is X = AxB, the output space is Y = R xA, and the state space is H=AxAxCxBxRxRX R . Let the initial state be given as follows : ql (0) = {di E D i = 1, 2, . . . , m} q2(0) q3(0) q4(0) q5(0) q6 (0) q7(0)

= = = = =

{el, e2, - . . , enJ l ~wl, w2, " . . , wn J with wi = ~( ei), i = 1 , 2. . . . , n {rl . . , ri}, ri E R r'

E 'R. The state equation of the IR system can be written in the form: q(k + 1) = 6(q(k), u (k)), k = 0, 1, 2, . . . , where S : R x X ----> R is the dynamics and the output is y(k) =,3(q(k)), where,3 : R ----> y is the output map . The state equations can be written as follows : ql(k + 1) _

(ql(k~) U ul(k))\~dk+11,

q2(k + 1) = q2(k) U ul(k) © 2002 by Chapman & Hall/CRC

~f

q, (k)

0

524

10. Applications q3(k+

q4(k

{~(dk+1)},

1) _ { g3(kj,U

+ 1) _

~f ql(k)

[q2~~ U u2(k)]\{rk+1}

rk+1,

0 q6 (k q7 (k

+ 1)

+ 1)

if if

q4 (k) q4 (k)

if if

q4(k) 7~ q4 (k) =

0 01

01 0

= q5 (k)

= -r (97 (q5 (k))),

where r is the ordering in A induced by 97 (q5 (k)) . The output equations are as follows : Y1(k) = q6(k)

and Y2(k)

C

q2(k) .

The notation dk +1 in the first equation means that between k and k + 1, dk+ 1 is processed, k = 0, 1, . . . . The same thing holds for rk + 1 . It follows that q7(k + 1) is an ordering in A and y2(k) is the subset of documents that give a "best response" at the request q6(k) . It follows that the IR system is a complex, SIR

= {x, X, Y, 6, l0},

with S and,3 as above. The dynamics S can be extended to H x X* considering sequences of pairs ("documents," "requests") . If U and V are two sets, there is an injection i :(UXV)*~U*XV* such that i((u1,v1)(u2,v2) . . .(un,vn))

= (u1 u2 . . . .un,VIV2 . . . .vn) .

It follows that an element of X* can be considered as a pair of the form ("sequence of documents," "sequence of requests"), where the sequences have the same length. © 2002 by Chapman & Hall/CRC

10.18. Fuzzy Systems

525

The system SIR is reachable, from the state q(0) E R if each state in R can be reached with a suitable input sequence. This means that the input space (i.e., documents and requests) is rich enough to cover each state (i .e., document file, requests, orderings, etc.). The system SIR is observable if, knowing the system's response (documents that are relevant to the request), for each sequence of inputs (requests and documents), the state from which the system is started can be uniquely determined. We are thus led to the concept of a fuzzy system . Definition 10.18 .3 A fuzzy system is a complex Sf = {Q, X, Y, 6"31

with S : ,FP(Q) x ,FP(X) ----> ,F'P(Q), the fuzzy dynamics, and /3 .FP(Q) ----> .FP(Y), the fuzzy output map. The state equation can be written as follows : qt+i = S(qt, ut), ut

E .FP(X), qt,

qt+i

E .FP(Q),

where qt , qt+l are, respectively, the fuzzy states at time t, t + 1 and ut the fuzzy input at time t. The output equation becomes yt = ,3 (qt), qt

E .FP(Q), yt E .FP(Y),

where yt is the fuzzy output at time t. Let (.FP(X))* denote the free monoid generated by .FP(X), i.e., the set of sequences of fuzzy inputs. Then S can be extended to .FP(Q) x (.FP(X))* by defining S : .FP (Q) x ( .FP(X))* - .FP (Q)

as follows : (1) S(a, A) = a, b'a

E FP (Q), (2) 6(a, A*w) = 6(6(a, A*), w), da E .FP(Q), A* E (.FP(X))*, w E

.FP(X) .

If the initial state ao

E .FP(Q) is

fixed, we can define the function:

Sao : (.FP(X))* - .FP(Q)

by VA *

E (.FP(X))*, Sao (A*) = S(ao,A*) .

Hence S(ao, A*) is computed by starting the system in state ao, feeding in the input sequence A* , and determining the final state . With these definitions, we can express two basic concepts of systems theory, namely, reachability and observability. © 2002 by Chapman & Hall/CRC

526

10. Applications

Definition 10.18 .4 The fuzzy system Sf is called reachable from the state ao if Sao is onto, i.e .,

Va E .FP(Q), 3a* E (.FP(X))* such that sao (A*) = a. The image of Sao , Im Sao C ,FP(Q), is called the reachability set of the system Sf from ao .

Composing S ao with, 3 we obtain the fuzzy response function (or behavior) of the system Sf, i.e ., f, = 0 O Sao ,

Thus f'. : (.FP(X))* -

FP(Y)

and so f,. (A*) =,3(b(ao,A*)) VA * E ( .FP(X)) * .

Definition 10.18 .5 A fuzzy system Sf is called observable if the assignment a

F-~

fa is one-to-one .

The interested reader is urged to see [160] for further details and interesting ideas including the concepts of stability and realization . 10 .19

Exercises

1. Let a, aii E II8 for i = 1, 2, . . . , m and j = 1, 2, . . . , n . Prove that (V{A{aj2 i = 1,2, . . . , m} I j = 1,2, . . . , n}) n {a} = V{n{aj2 lea I i = 1,2, . . . , m} I j = 1,2 . . . . , n} and that (A{V{aj2 I i = 1,2 . . . . , m} I j = 1, 2, . . . , n}) V {a} = A{V{aj2 Aa I i = 1, 2, . . . , ml I j = 1, 2, . . . , n}. 2. Prove that the matrix in Example 10.6 .2 has period equal to 3. 3. Prove that y(anbm) = 'A' in Example 10.8 .2. 4. Prove Lemma 10 .11 .2. 5. Let M be the median operator of (10 .5) . Show that M is associative, not strictly increasing, and contractive . 6. Prove (2) of Proposition 10.11.7. 7. Show that (No , -), where No = hY U {0} and ({0,1, . . . , n}, i), of Example 10.13.1 are tolerance spaces. © 2002 by Chapman & Hall/CRC

10.19 . Exercises

527

8 . For f in Definition 10.13 .2, prove that f*T is the least tolerance on Y such that f : (X, T) ~ (Y, f *T) is a fuzmap . Prove also that it is the unique tolerance on Y such that b' tolerances Q on Z and b' set-theoretic maps g : Y ~ (Z, Q), f - g : (X, T) ~ (Z, Q) if and only if g : (Y, f. T) ~ (Z, a) is a fuzmap . 9 . If g : Y ----> (X, T) is a set-theoretic map, prove that g*T is the biggest tolerance on Y such that g : (Y,g*T) ----> (X, T) is a fuzmap . 10 . In Definition 10 .13 .4, prove that T is the least tolerance such that all injections tj are fuzmaps, and TIT2 is the largest tolerance such that all projections pry are fuzmaps . 11 . Prove (1) of Theorem 10 .14 .3 . 1 12 . In Example 10 .14 .6, show that T7 and T9 yield T14 =

and Tg and T9 yield T15 = T14 and T15 .

1 1 1 0

1 1 1 1

1 1 1 0

0 1 0 1

1 1 1

. Then determine T19 from

13 . In Example 10 .15 .6, show that SQ U Sx 1 T°Sx C T° for x E {xl, X2114 . Prove that the condition in Definition 10 .17.5 implies that no state can be entered and left on the same input, else S would not be singlevalued .


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