FORMAL MODELS, LANGUAGES AND APPLICATIONS
World Scientific
FORMAL MODELS, LANGUAGES AND APPLICATIONS
SERIES IN MACHINE PERCEPTION AND ARTIFICIAL INTELLIGENCE* Editors:
H. Bunke (Univ. Bern, Switzerland) P. S. P. Wang (Northeastern Univ., USA)
Vol. 51: Automatic Diatom Identification (Eds. H. du BufandM. M. Bayer) Vol. 52: Advances in Image Processing and Understanding A Festschrift for Thomas S. Huwang (Eds. A. C. Bovik, C. W. Chen and D. Goldgof) Vol. 53: Soft Computing Approach to Pattern Recognition and Image Processing (Eds. A. Ghosh and S. K. Pal) Vol. 54: Fundamentals of Robotics — Linking Perception to Action (M. Xie) Vol. 55: Web Document Analysis: Challenges and Opportunities (Eds. A. Antonacopoulos and J. Hu) Vol. 56: Artificial Intelligence Methods in Software Testing (Eds. M. Last, A. Kandel and H. Bunke) Vol. 57: Data Mining in Time Series Databases y (Eds. M. Last, A. Kandel and H. Bunke) Vol. 58: Computational Web Intelligence: Intelligent Technology for Web Applications (Eds. Y. Zhang, A. Kandel, T. Y. Lin and Y. Yao) Vol. 59: Fuzzy Neural Network Theory and Application (P. Liu and H. Li) Vol. 60: Robust Range Image Registration Using Genetic Algorithms and the Surface Interpenetration Measure (L. Silva, O. R. P. Bellon and K. L Boyer) Vol. 61: Decomposition Methodology for Knowledge Discovery and Data Mining: Theory and Applications (O. Maimon and L. Rokach) Vol. 62: Graph-Theoretic Techniques for Web Content Mining (A. Schenker, H. Bunke, M. Last and A. Kandel) Vol. 63: Computational Intelligence in Software Quality Assurance (S. Dick and A. Kandel) Vol. 64: The Dissimilarity Representation for Pattern Recognition: Foundations and Applications (Elzbieta Pekalska and Robert P. W. Duin) Vol. 65: Fighting Terror in Cyberspace (Eds. M. Last and A. Kandel) Vol. 66: Formal Models, Languages and Applications (Eds. K. G. Subramanian, K. Rangarajan and M. Mukund)
*For the complete list of titles in this series, please write to the Publisher.
Series in Machine Perception and Artificial Intelligence - Vol. 6
FORMAL MODELS, LANGUAGES AND APPLICATIONS Editors
K. G. Subramanian Madras Christian College, India
K. Rangarajan Madras Christian College, India
M. Mukund Chennai Mathematical Institute, India
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FORMAL MODELS, LANGUAGES AND APPLICATIONS Series in Machine Perception and Artificial Intelligence — Vol. 66 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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PREFACE
This volume of contributed papers commemorates the 75th birthday of Prof. Rani Siromoney, one of the foremost theoretical computer scientists in India and a leading authority on Formal Languages and Automata Theory. Over a period spanning four decades, she has made tremendous technical contributions to the field through her research. She has also inspired generations of students in Chennai with her teaching and has been responsible for building up a community of dedicated teachers and researchers in this part of India to carry forward her vision. Prof. Siromoney has served on the Editorial Board of the journals Theoretical Computer Science and International Journal of Foundations of Computer Science and has headed several international collaborative research projects. She also served on the Programme Committee for the first ten editions of the international conference Foundations of Software Technology and Theoretical Computer Science, one of the leading theoretical computer science conferences in the world. She is currently Professor Emeritus at Madras Christian College, the illustrious institution where she has spent most of her professional life. She continues to play an active role in research and teaching as Adjunct Professor at Chennai Mathematical Institute. The contributions in this volume span a wide range of subjects, thematically connected by the use of concepts from formal languages and automata theory. The areas explored in this volume include compiler construction, computational complexity theory, formal modelling of concurrent systems, codes and image analysis. The contributors are leading researchers in computer science from all parts of the world, all of whom have been associated with Rani Siromoney during the course of her long and productive career. We thank all the authors for readily accepting to contribute to this volume. We also thank the staff of World Scientific for their assistance and cooperation that have been crucial for the successful completion of this project. K . G . SUBRAMANIAN
K. RANGARAJAN MADHAVAN M U K U N D
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CONTENTS
Preface Chapter 1
Chapter 2 Chapter 3 Chapter 4
Chapter 5
Chapter 6
Chapter 7 Chapter 8 Chapter 9
Chapter 10 Chapter 11
Chapter 12
v Finite Array Automata and Regular Array Grammars A. Atanasiu, K. G. Subramanian, K. Rangarajan and P. S. P. Wang
1
L-Convex Polyominoes: A Survey G. Castiglione and A. Restivo
yj
On Oriented Labelling Parameters D. Gongalves, A. Raspaud and M. A. Shalu
34
On a Variant of Parallel Communicating Grammar Systems with Communication by Command E. Csuhaj-Varju and Gy. Vaszil
46
Some Remarks on Homogeneous Generating Networks of Free Evolutionary Processors J. Dassow, C. Martin-Vide and V. Mitrana
55
Hexagonal Contextual Array P Systems K. S. Dersanambika, K. Krithivasan, H. K. Agarwal and J. Gupta
79
A q-Analogue of the Parikh Matrix Mapping O. Egecioglu and O. H. Ibarra
97
Contextual Array Grammars R. Preund, Gh. Paun and G. Rozenberg
112
Characterizing Tractability by Cell-Like Membrane Systems M. A. Gutierrez-Naranjo, M. J. Perez-Jimenez, A. Riscos-Nilnez, F. J. Romero-Campero and A. Romero-Jimenez
137
A Cosmic Muse T. Head
155
Sublogarithmically Space-Bounded Alternating One-Pebble Turing Machines with only Universal States K. Inoue, A. Ito and A. Inoue
160
Verification of Clock Synchronization in T T P K. Kalyanasundaram and R. K. Shyamasundar
176
viii
Contents
Chapter 13
Triangular Pasting System T. Kalyani, K. Sasikala, V. R. Dare, P. J. Abisha and T. Robinson
195
Chapter 14
Towards Reducing Parallelism in P Systems S. N. Krishna and R. Rama
212
Iteration Lemmata for Rational, Linear, and Algebraic Languages Over Algebraic Structures with Several Binary Operations M. Kudlek
225
Chapter 16
The Computational Efficiency of Insertion Deletion Tissue P Systems K. Lakshmanan and R. Rama
234
Chapter 17
Petri Nets, Event Structures and Algebra K. Lodaya
245
Pattern Generation and Parsing by Array Grammars K. Morita, J.-S. Qi and K. Imai
259
Anchored Concatenation of MSCs M. Mukund, K. Narayan Kumar, P. S. Thiagarajan and S. Yang
273
Chapter 20
Simple Deformation of 4D Digital Pictures A. Nakamura
288
Chapter 21
Probabilistic Inference in Test Tube and its Application to Gene Expression Profiles Y. Sakakibara, T. Yokomori, S. Kobayashi and A. Suyama
303
On Languages Defined by Numerical Parameters A. Salomaa
319
An Application of Regular Tree Grammars P. Shankar
33g
Chapter 24
Digitalization of Kolam Patterns and Tactile Kolam Tools S. Nagata and R. Thamburaj
353
Chapter 25
Hexagonal Array Acceptors and Learning D. G. Thomas, M. H. Begam, N. G. David and C. de la Higuera
363
Pollard's Rho Split Knowledge Scheme M. K. Viswanath and K. P. Vidya
378
Characterizations for Some Classes of Codes Defined by Binary Relations D. L. Van and K. V. Hung
3g0
Chapter 15
Chapter 18 Chapter 19
Chapter 22 Chapter 23
Chapter 26 Chapter 27
CHAPTER 1 FINITE ARRAY AUTOMATA A N D REGULAR ARRAY GRAMMARS
Adrian Atanasiu Faculty of Mathematics, Bucharest University, Str. Academiei 14, sector 1, 70109 Bucharest, Romania E-mail:
[email protected] K. G. Subramanian Department of Mathematics, Madras Christian College, Tambaram, Chennai 600 059, India E-mail:
[email protected] K. Rangarajan Department of Mathematics, Bharath Institute of Higher Education, Selaiyur, Chennai 600 059, India
P. S. P. Wang College of Computer Science, Northeastern Boston, MA 02115, USA E-mail:
[email protected] University,
A recognition device, called Finite Array Automaton, accepting a class of picture arrays, is introduced and it is shown that this device is equivalent to the n—dimensional regular array grammar. Regular (string) languages, called spreading languages, are associated to the corresponding n—dimensional array regular languages in order to deal with certain decision problems. Also the effect of controlling the application of rules of regular array grammars is brought out.
1. I n t r o d u c t i o n Picture languages generated by array grammars or accepted by array aut o m a t a have been studied by researchers and various models have been 1
2
A. Atanasiu
et al.
proposed in the literature, motivated by problems arising in the framework of syntactic methods of pattern recognition and image processing. 6 ' 8 ' 9 Freund 4 has made an extensive and deep study of array grammars in a general setting of n—dimensions. Many different aspects of these grammars such as regulated rewriting, 2 ' 3 cooperating systems, 5 contextual features 4 and so on have been investigated, k—head finite array automata, 1 have also been considered to characterize certain families of array languages. In this paper, an explicit construction of a recognition device, called a Finite Array Automaton equivalent to an n—dimensional Regular Array Grammar is made. A class of regular string languages, called Spreading languages is associated to the finite array automaton which is useful in certain decision problems. In the case of two dimensions (n = 2), the effect of controlling the application of rules of a Regular array grammar is brought out. 2. Preliminaries For notions of formal languages we refer to Refs. 7 and 10; for basic notions, notations and results about array grammars to Ref. 4. Let V be a finite and nonempty alphabet. Let Z denote the set of integers and N denote the set of positive integers and let n £ N. For x = (xi, X2, • • •, xn) G Zn, we shall define
NI = X> 2 . »=i
A ro—dimensional array A over an alphabet V is a function A : Zn —> V U { # } with finite support, supp(.A), defined by supp(A) = {ueZn\
A(u) + # } ;
is called the blank symbol, which is not in V. Usually we write A = {(u,A(u))
| u e supp(.A)} .
In each location u e Zn of the grid an element from V U { # } is placed by the function A: Zn - > V r U { # } . Moreover, the set supp(^) = {ue Zn/A(u)
jt # }
is finite and nonempty. We require that for any u € supp(^4), A(v) ^ # for at least one v with |]u — v\\ = 1. The set of all n—dimensional arrays over V is denoted by V*n. Any subset of V*n is called a n—dimensional array language.
Finite Array Automata
and Regular Array
Grammars
3
Let u 6 Zn. Then the translation ru : Zn —> Zn is defined by Tu(v) = v + u for all v G Zn; for any array A G V*n we define TU{A), the corresponding n—dimensional array translated by u, by TU(A(V))
= A(v + u),
Vt)6r.
The vector ( 0 , 0 , . . . , 0) G Zn shall be often denoted by Qn. Usually, arrays are regarded as equivalence classes with respect to linear translations, i.e. only the relative positions of the symbols from supp(A) are taken into account. The equivalence class [A] of an array A G V*n is defined by [A] = {B £ V*n | 3 u eZn,B
= ru(A)} .
For any element u G supp(A), we define the frame Wu = {(u,v)/veZn,\\u-v\\
= l}.
If u = fin, then the frame Wo = {(f2„,w)/||w|| — 1} is called the initial frame. Obviously, Wo has In elements. Let us consider the operation of translation TX{WU) = {(u+x, v+x) | v G Zn} — Wu+X. Then, any frame can be obtained by the translation of the initial frame: Wu = TU(W0) (or W0 = r-u(Wu)). A n—dimensional array production over V is a triple P = (W,Ai,Az) where W C Zn is a finite set and Ai, A2 are mappings from W to F u { # } . In such a writing, all positions in W together with their associated symbols must be listed for representing Ai and A2, by Ai = {{u, Ai(u)) \ u £ W} ,
1 < i < 2.
This representation is general, for the infinite set of equivalent n—dimensional array productions of the form (TU(W), TU(A\), TU(A2)) with u G Zn. Hence without loss of generality, it can be assumed that £ln G W. Moreover, the set W can be omitted because it can be uniquely reconstructed from the description of the mappings A\, A2. A n—dimensional array production p = (W, A\, A2) is regular if: (1) W = {nn, u}cZn, ||u|| = 1 and A, = {(nn,B), (u, # ) } , A2 = {(«„,a), (u,C)},B,CeVN,ae VT, or (2) W = {fln}, Ax = {(nn,B)},A2 = {(n n ,6)}, where BeVN,be
VT.
If Bi, B2 are two n—dimensional arrays, we shall write B\ ^ B2 if and only if there exist u G Zn and a production p = (W, ^ 1 , ^ 2 ) such that the restrictions of Bi to TU(W) are Ai(i = 1, 2).
A. Atanasiu
4
et al.
In other words, the array B2 G V* is directly derivable from the array B\ G V*n by the n—dimensional production (W, A\,A2) if and only if the subarray of B\ corresponding to A\ is replaced by A2, yielding B2. A n—dimensional array grammar is a quintuple G — (n, Viv, VT,P, {va,S)},#), where • Vpf is the alphabet of nonterminal symbols, Vr is the alphabet of terminal symbols, Vjv C\VT = \ • P is a finite non-empty set of n—dimensional array productions over VN U VT; • {(u s ,5)} is the start array (S G VN is the start symbol, vs G Zn is the start location). The array B2 G V*n is directly derivable from the array B\ G y * n in G, denoted B\ =^Q B2 if and only if there exists a n—dimensional array production p = (W, Ai,A2) in P such that B\ => B2. If ^>* is the reflexive and transitive closure of =>G, then the array language generated by G is defined by L(G) =
{AeVfn/(va,S)=>*A}.
The corresponding n—dimensional array language of equivalence classes with respect to linear translation is [£(G)] = {[A]/A G L{G)}. The n—dimensional array grammar G is called regular if every production in P is regular; in this case L(G) is called a n—dimensional regular array language. The family of n—dimensional regular array languages will be denoted by L(n, reg) and the family of regular array languages of equivalence of classes of arrays will be denoted by [L(n, reg)].
3. Finite Array Automata Let V be a finite and non-empty alphabet. Let V1 = {a\/a G V} be another alphabet, distinct from V. The set of strings over V(Vl) is denoted as usual by V*{Vl* respectively) with the difference that the identity will be considered # (blank symbol). Definition 1: Let A be a n—dimensional array over the alphabet V. A n-Finite Array Automaton (n-FAA in short) is a 7-tuple, M = (n,Q,V,6,qo,v0,F),
Finite Array Automata
and Regular Array
Grammars
5
• n> 1. • Q and V are finite and nonempty sets of "states" and "input characters" respectively. • 5 : (Qx Zn) x V —> 2®xz is the transition map, satisfying the invariance property (p,v) G 6({q,u),a) O (p,rx{v)) G 5(^,T a; (u),a),V a; G Zn. • go G Q is the "initial state". • VQ G Z n is the start location; if VQ = Qn this element can be ignored. • F C Q, (F ^ 0) is the set of "final states". (p, w) € $((. After such a rule is applied relating to a location u, the element a G V from this location is replaced by a 1 G V1 (in order to avoid the possibility of the automaton passing a second time through the location u). The content of the location v is not important at this moment, but we remark that S cannot be applied if in the current location u there is # or an element from V1. Remark 1: (p, —u) £ 8{{q,u),a) unless u = fln or p G F (otherwise this rule cannot be applied). The property of invariance assures a homogenous behaviour of the transition map 5 on the grid Zn. Therefore it is enough to define the transition map of a n—dimensional finite array automaton only for the initial frame: (P,v) G 6((q,u),a)
i, a\),..., (vn, an) which satisfy the constraints: (1) 3qi,...,qn+i (2) (Vi,vi+1) (3) qn+1 e F.
G Q with (qi+i,vi+1) &WVi,0 V be the function projection, defined by pro((u, a)) = a,V a G V,u G Zn. This function can be extended to a morphism; now, another language, which is a projection in a 1-dimensional array of the language LQ(M), can be denned: pro(L(M))
= {ae V*\3 q G F,3 v G Zn,(q,v)
It is clear that pro(L(M)) languages).
G 6((q0,v0),a)}
.
is a regular string language (in terms of formal •
Example 1: Let n = 2, Q = {qo,qi,q2}, V = {a,b,c}, v0 = iln, F = {q0} and S((q0,an),a) = {( 9 l , (1,0))}, *((«!,fi„),6) = {(
