Words, Semigroups, Transductions

F e s t s c h r i f t in Honor of Gabriel Thierrin

Wo

/!?

s

' Semigroups, Transductions

Masami I t o Gheorghe Paun

World Scientific

Wo

" l $ ' Semigroups, XjTransductions

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Wo ds

' ' Semigroups, A.TTran$ductions F e s t s c h r i f t in H o n o r Gabriel Thierrin

Editors

Masami I t o Kyoto Sangyo University, Japan

Gheorghe Paun Romanian Academy, Bucharest, Romania

Sheng Yu University of Western Ontario, Canada

V ^ World Scientific WB

New Jersey • London • Singapore Sin* • Hong Kong

of

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Words, semigroups, and transductions : Festschrift in honor of Gabriel Thierrin / edited by Masami Ito, Gheorghe Paun, Sheng Yu. p. cm. "The publications of Gabriel Thierrin": p. Includes bibliographical references. ISBN 9810247397 (alk. paper) I. Ito, Masami. II. Paun, Gheorghe, 1950- . III. Yu, Sheng. FV. Thierrin, Gabriel, 1921- .

2001046663

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2001 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.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore.

Professor Gabriel Thiemn

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vii

Preface T h e present volume is a modest tribute to Gabriel Thierrin, on the occasion of his eightieth birthday. Of course, we would like the title to encompass as much as possible the topics dealt with by Gabriel ( G T to some close friends) throughout his impressive research activity (see the list of publications included at the end of the book), but this is a rather difficult task. Pure algebra, a u t o m a t a , languages, codes, combinatorics on words, L systems, DNA computing, infinite sequences, linear programming, and also . . . a model of a five-dimensional universe - these are too many and too diverse to be captured in a single volume. Nevertheless, G T ' s many collaborators and friends were ready to face this pleasant task. T h e papers (collected partially with the help of A. P a u n , London-Ontario) are arranged in the alphabetical ordering of their authors, athough it would have been possible to classify t h e m according to the fields: Codes, Combinatorics on Words, Semigroups, G r a m m a r s , Tree Languages, Transductions, Unconventional Models of Computing, Miscellanea. Several other authors which were not able to complete a paper in due time have asked us to add their voices to the chorus of the present editors and authors when saying, with indebtedness and love: H a p p y B i r t h d a y , G a b r i e l !

* T h e volume was produced with the help of C a n a d a N S E R C grant 4163098 and Grant-in-Aid for Science Research, J a p a n Society for the Promotion of Science. T h e editing work of Gh. P. was done during his stay in Rovira i Virgili University, Tarragona, Spain, in the framework of a grant of N A T O Science Committee, Spain, 2000-2001. We also acknowledge the pleasant and efficient cooperation with Ms. E. H. Chionh, from World Scientific Publ., Singapore. June, 2001 Masami Ito Gheorghe P a u n Sheng Yu

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IX

Contents

Preface P.R.J. Asveld: Some Operators on Families of Fuzzy Languages and Their Monoids

vii 1

G. Bel Enguix: Mixed Links: A Method for Generating Multi-Stranded Structures in Syntax

15

C.S. Calude, E. Calude, P. Kay: Liars, Demons, and Chaos

33

J. Castellanos, V. Mitrana: Some Remarks on Hairpin and Loop Languages

47

J. Dassow: Conditional Grammars with Restrictions by Syntactic Parameters

59

P. Domosi: On Complete Classes of Directed Graphs

69

A. Ehrenfeucht, I. Petre, D.M. Prescott, G. Rozenberg: Circularity and Other Invariants of Gene Assembly in Ciliates

81

Z. Esik, W. Kuich: A Generalization of Kozen's Axiomatization of the Equational Theory of the Regular Sets

99

C.-M. Fan, H.-J. Shyr: Catenation Closed Pairs and Forest Languages

115

H. Fernau, R. Stiebe: Valence Grammars with Target Sets

129

F. Gecseg, B. Imreh: On Isomorphic Representations of Monotone Tree and Nondeterministic Tree Automata

141

X

F. Gecseg, M. Steinby: Minimal Recognizers and Syntactic Monoids of DR Tree Languages

155

T. Head: Visualizing Languages Using Primitive Powers

169

J. Honkala: On Sparse OL Languages Over the Binary Alphabet

181

L. Hie: On Generalized Slenderness of Context-Free Languages

189

T. Imaoka: Prehomomorphisms on Locally Inverse *-Semigroups

203

F. Ipate, M. Gheorghe, M. Holcombe, T. Balanescu: Testing Using X-Machine Translations

211

K. Iseki: Some Fundamental Theorems on BCK

231

M. Ito, C. Martin-Vide, Gh. Paun: A Characterization of Parikh Sets of ETOL Languages in Terms of P Systems

239

H. Jiirgensen: Disjunctivity

255

L. Kari, A. Paun: String Operations Suggested by DNA Biochemistry: The Balanced Cut Operation

275

L. Kdszonyi: How to Generate Binary Codes Using Context-Free Grammars

289

J. Lee, K. Morita: Generation and Parsing of Morphism Languages by Uniquely Parallel Parsable Grammars

303

V. Manca: On the Generative Power of Iterated Transduction

315

XI

M. Margenstern, Yu. Rogozhin: Time-Varying Distributed H Systems of Degree 1 Generate All Recursively Enumerable Languages

329

G. Niemann, F. Otto: On the Power of RRWW-Automata

341

T. Y. Nishida, S. Seki: A Definition of Parikh Controlled Context-Free Grammars and Some Properties of Them

357

H. Prodinger: Words, Dyck Paths, Trees, and Bijections

369

S.A. Rankin: Semilattice Amalgams and Semidirect Product

381

T. Sa2to: Characterization of Finite Automata by the Images and the Kernels of Their Transition Functions

395

A. Salomaa: Iterated Morphisms with Complementarity on the DNA Alphabet

405

L. Staiger: Topologies for the Set of Disjunctive w-Words

421

The Publications of Gabriel Thierrin

431

1 S O M E O P E R A T O R S ON FAMILIES OF FUZZY A N D THEIR MONOIDS

LANGUAGES

P E T E R R.J. ASVELD Department of Computer Science, Twente University of Technology P.O. Box 217, 7500 AE Enschede, the Netherlands E-mail: [email protected] We study the structure of partially ordered monoids generated by certain operators on families of fuzzy languages. These operators are induced by simple, well-known operations on fuzzy languages, like fuzzy homomorphism, fuzzy finite substitution and intersection with regular fuzzy languages. The structure of these monoids provides better insight in the (in)dependency of closure properties of some families of fuzzy languages.

1

Introduction

Consider a universal algebra (AQ, Q,Q) with carrier set AQ and set fio of operations and let {f2i, • • •, fin} (n > 2) be a collection of proper subsets of fio which covers fio, i.e., fio = (J{^i I 1 5: * 5: n}- Now a natural question is, whether the collection {fii, • • -,f2 n } is independent or, equivalently, does closure under the operations from fio—^fe imply closure under the operations from f^ (1 < k < n)l Let A be an arbitrary subset of AQ and let for each k (0 < k < n), Tk(A) denote the smallest set that includes A and that is closed under the operations from fife. Similarly, for each subset J of {1, • • •, n}, Tj(A) is the smallest set that includes A and that is closed under the operations from (J{f2fc \ k E J}. We identify Tj with T/j whenever J is the singleton {k}. Note that TQ(A) = T{ir..n}(A) for each A C AQ. For reasons of consistency we define Fg = / (the identity operator). In algebra the operators Tj : V(A0) -> V(A0), where V(AQ) is the power set of AQ, are called closure operators. Since the composition of closure operators is associative, {Tj \ J C {I,- • • ,n}} generates a monoid [17], which possesses a partial order induced by the inclusion order in P ^ o ) : Tj < TJI iff for each ACA0,TJ(A)CTJ,{A)

[17].

If the monoid M{Ti, ••• ,Tn} generated by { F i , - - - , F n } is finite, then the monoid M{Tj \ J C { l , - - - , n } } coincides with M {Ti, • • •, Tn}. All monoids M{Ti, • • •, r n } to be studied in this paper turn out to be finite; so there is no need to consider M{Yj | J C { l , - - - , n } } anymore. The problem whether 0,^ depends on Q0 — fi/j can now be rephrased as: is T/j < Tj for J = {l,---,kl,k + 1, • • • ,n}? So the partial order of

2

M{Ti, • • • ,Tn} enables us to solve all dependency problems for the collection The structure of M{T\, • • •, Tn} is mainly studied in algebra for the case that Ao is a variety or primitive class [6], i.e., a class of algebras closed under taking subalgebras (Ts), direct products (IV), and homomorphic images (TH) [18,7,8,16]. In general M{Ts, IV, T#} is finite [18], and for some special varieties it has been investigated to what extent the structure of M{Ts, Tp, T#} collapses [7,8]. In this paper we determine in Section 5 the structure of M{T\, • • •, Tn} for a few simple algebraic structures frequently encountered in the study of (families of) fuzzy languages (Section 3). The proofs of our results in Section 5 heavily rely on a combinatorial argument (Section 4). But first we continue our discussion of closure operators and their monoids in Section 2. 2

Monoids Generated By Algebraic Closure Operators

We assume familiarity with the rudiments of semigroups, monoids, universal algebras, partial orders and (semi-)lattices. All unexplained notions with respect to these subjects can be found in many elementary texts on algebra; we refer to [6,11] in particular. As a special instance of a partially ordered monoid [6,19] we consider the following concept. Definition 2.1. A bounded partially ordered monoid or bpo-monoid is an algebra (U,-,e,z,{x,y) sv(v'o')loc {x,y) sv{v')0 0{x,y) (s'v'o1) svo (>{x,y) (s'v')svo 0{x,y) (v'o')svo Q{x,y) (v')svo

Analysis of results

6.1

New syntactic structures

There are two main linguistical phenomena depending on the place where T is inserted: • Objectivization: Insertion of a structure with a I

T/

1 pole in the focal

(v) place o of a ULPS. Such a structure adopts the function 0

by means of the functionality adjustment of a pole with three strata. It corresponds to rules 1, 2, 3 and 4. • Subjectivization: Insertion of a structure with a I , , I pole in the focal place s of a ULPS. Such a structure adopts the function S by means of the functionality adjustment of a pole with three strata. It is caused by rules 5, 6, 7 and 8. 6.2

Structures with poles of triple strata

Rules of level linking create structures with a three-stranded pole: two of them correspond to variables and one, the highest, to strings. Only a structure with V is able to modify its functionality and adjust it to another functionality that has been imposed by another variable belonging, in this case, to the ULPS X. That is why we have not been able to insert monopolar structures L

, L

which are the result of a blunt simple cut of T in the

place that has been left by the cut pole in X. The polar cohesion of these new structures fuses these three levels that, in principle, belonged to two different ULPS. The lowest level is the assigner of

22

polar functionality and the intermediate one is the assigner of focal functionality. The cohesions in the structures created from the previous rules are represented as follows, taking into account only rules 1 and 2:

6.3

Parameters PN and ~£ in a ULPS

It seems obvious that whatever wants to occupy an s position, it must observe the same conditions as this focus in relation to the function and cohesion within the terminal string. Therefore, it is advisable to establish PN(ULPS), ~£{ULPS). PN(ULPS) is, by definition, 3e. Therefore, PN(v) in a clause inserted by means of a mixed link is always 3s. Every ULPS cut that consists at least of a v pole, has got 3s as PN values: V v o \

vo)

3s

Also,

3»

without having a contradiction in the difference of PN between the only piece of the ULPS and the global ULPS. In what concerns ~£{ULPS), it is always equal to q. In every ULPS cut that consists at least of a (

v

J pole, l(ULPS) = q: s v o SV O

This implies the existence of cut structures, of the following form:

23

v V '

r-i q

Only a ghost will be able to change the level assignment of a ULPS. Therefore, if a ghost does not appear in a rule and q ^ £(v), then the operation will not be possible to be carried out. 7

Systematization of results

The 8 mixed links rules that have been introduced have different results, depending on the point of view adopted. For instance, we can distinguish the results: • According to the syntactic results: - Objectivization - Subjectivization • According to the appearance of ghosts: - Rules with ghosts, where I „ 1 € Z\.

They always lead to correct

resultant strings. - Rules without ghosts, where I „ ) ^ Z\. They can generate incorrect resultant strings. • According to generative power: - Recursive rules (->• co). * Objectivization rules, where I „ 1 € Z\. * Subjectivization rules, where I

c

1 6 Z\.

Rules with blockade (greatest expansion = 2). * Objectivization rules, where

„ W Zj.

* Subjectivization rules, where I „ \ $_ Z\.

24

In what follows we will study the new syntactic structures which have apppear by means of these operations, arranging a first classification between objectivization and subjectivization which is the linguistic referent of the recombination of mixed systems. The differences caused by the appearance of ghosts will be included in this section. Then, we will review the very diverse generative power of each of the results. 8

Analysis of mixed links depending on the syntactic result of the rules

8.1

Ob jectivization

8.1.1 Rules that cause ghosts Rule 1 x = SUM,

y = s'v'o',

(){x,y) =

sv(s'v'o')0

1

Since the group (s'v'o ) is taking the focal place of o in the variable 0, we have l(s'v'o') = l(o), that is, x=

SVLQ,

y = s'v'o',

Q{x,y) =

sv(s'v'o')i0

In order to join the two structures, every time when s' € Z, it is necessary to have a level assigner ghost. The rule is formulated in the following way: x=

SVLQ,

y = s'v'o',

Q(x, y) = svl

(s'v'o')i0

When a ULPS is inserted into the truncated pole I „ 1 of another one with v

K ) — Qi the suitable rule for this situation is: x = sib^,

y = s'v'o',

(>(x,y) = sv I (s'v'o')q

The result is the one verified in Example 1: Example 1 x = El gat vol # llet y = Joan juga tennis x = The cat wants # some milk y = John plays t e n n i s §{x,y) = El gat vol que Joan jugui a tennis (}(x,y) — The cat wants t h a t John plays t e n n i s

25

But if l(v) ^ q, then thanks to the action of the ghost, the link is also possible. For instance, for l(v) — r, we have: x = svap.,

y = s'v'o\

Q(x,y) =

svl(s'v'o')r

Example 2 x = El gat viu # a Barcelona y — En Joan juga a tennis x = The cat l i v e s # in Barcelona y = John p l a y s t e n n i s Q{x,y) = El gat viu on Joan juga a tennis (>(x,y) = The cat l i v e s where John plays t e n n i s The change of ghost adjusts the ULPS level to the v level in a way so that the operation is correct. Rule 2 x=

SVLQ,

y = s'v'#o',

0(a;,j/) =

sv(s'v')0

Since the group (sV) is taking the focal place of o in the variable 0, we have l(s'v') = l{o):

SILQ,

y=s'v'#o',

0{x,y) =

sv(s'v')i0

In order to join the two structures, every time when s' € Z, it is necessary to have a level assigner ghost: x=

SVLQ,

y = s'v'fto1,

0(x,y)

= sv l (s'v')i0

Example 3 For l(v) — q x = El gat vol # llet y = Joan juga # tennis x — The cat wants # some milk y = John plays # t e n n i s Q(x,y) — El gat vol que Joan jugui Q{x,y) = The cat wants t h a t John plays Example 4 For l(v) = r x = El gat viu # a Barcelona y = En Joan juga # a tennis x = The cat l i v e s # in Barcelona y = John p l a y s # t e n n i s o is presented by a lattice of squares enumerated according to Cantor's pairing function: the fcth square is black or white according to the value of n^, nj. = 1 o r % = 0 . For example, the first row of the lattice will contain the squares numbered 0,1, 5,6,14,..., the second row 2,4, 7,13,16,..., the third row 3, 8,12,17, 2 5 , . . . and so on. For the dragon diagram we interpret a binary sequence as a series of instructions to turn right or left by a fixed angle (say, a right one). In this way the sequence produces a curve on a lattice in the plane (see Garden [7], and Davis and Knuth [5]). Finally, to every sequence (xm) of reals in (0,1) we associate the binary sequence b

=

f 0, if xm < | , ^ 1, II Xyn ^> ^2 •

When representing a dynamics in Cantor or dragon formats we will use the binary sequence associated to the dynamics.

36

4

Liar as a Demon

The classical Liar can be thought of as a demon. A perpetual motion machine is one which manipulates energy with perfect efficiency, i.e. one for which no energy ever leaks away into heat. Since energy is neither created nor destroyed, there is little reason to prevent such a machine operating forever with only a fixed supply of energy. Maxwell's Demon (see Maxwell [11], Leff and Rex [9]) is a hypothetical intelligent creature imagined by James Clerk Maxwell to wring regularity out of randomness by picking and choosing among molecules. He sits at a molecule sized trapdoor between two partitions of a box filled with a gas at some temperature (see Figure 1), and watches the molecules wandering back and forth.

L

Hoi

Cold

Figure 1. Maxwell's demon

The average speed of the molecules depends on the temperature. Some of the molecules will be going faster than average and some will be going slower than average. The demon observes the molecules in the box as they approach the trapdoor, allowing fast ones to pass from left to right, and slow ones from right to left. After performing these operations the demon ends up with a box in which all the faster than average molecules are in the left side and all the slower than average ones are in the right side. So the box is hot on the left and cold on the right, and the expenditure of work is negligible, in apparent violation of the second law of thermodynamics. Maxwell's demon can be described by the morphism 0 —> 10,1 —* 10 : it generates the sequence 10101010101010... See more in Matherat and Jaekel [10].

37

5

Contrapositive, Half, and Minimalist Liars

The result produced by the morphism modeling Maxwell's demon can be obtained in three other different ways (cf. Grim, Mar and St.Dennis [8]); however, as we will show here, their dynamics have drastically different properties. First we will present dynamic estimations and their web plots. Then we will compare these dynamics using Cantor or dragon plots. The first estimation of truth is contrapositive:

This sentence is as true as the estimated value of its negation. leads to the following dynamics: E ( z )

-\2(l-*),if*>!.

whose non-trivial fixed-point is § = 0.1010101010101010... Note that E is the inverse of Bartlett's pseudo-random generator (see Beltrami [1]): start with a seed UQ £ (0,1) and let un to be half the sum of w n -i plus a digit bn produced by a Bernoulli ^-process. It leads to the well-known chaotic "tent" map (see Falconer [6]). For this reason, this estimation is called "chaotic" Liar in [8]. The fixed-point is sensitive to the initial value. For example, if xo = O.aia.2 . . . at, where aj, 0 2 , . . . a* are 0 or 1, then there exist two distinct non-negative integers p,q 0, the sequence converges to | , while for xo = \ the sequence converges to 0; for XQ — ^ or x 0 = f ( a n d many other points) the sequence oscillates between two values. The sequence is constant for xo = | . For XQ = 0, the sequence is constant 0 while for XQ = 1 the sequence oscillates: 1,0,1,0,....

The second approach is to consider the Nth part estimation of truth:

This sentence is as true as the ./Vth part of its estimated value.

38

-ft

K Figure 2. Contrapositive Liar: Web Diagram

.(Jv-i).; it leads to the fixed-point 4 only for The dynamics is xn+\ = 1 N N — 2 (the "half Liar); in this case, the fixed-point doesn't depend on the initial value XQ. The sequence is constant for XQ = §• For xo = 0 or XQ = 1 the sequence converges to | . The web diagram corresponding to the initial value xQ = 0.314 is displayed in Figure 3. The third method is to consider the minimalist estimation of truth:

This sentence is as true as the minimum of its value and its estimated value. ft leads to the dynamics E(z)

if2l

This dynamics has a unique fixed-point | which can be obtained only if we start with | or with a;0 £ ( i , 1) \ {|} and for every n > 1 we have xn > \. The sequence of iterations oscillates for every xo > \ for which there exists an

39

/ /

/

\2-

/ / / /

Figure 3. Half Liar: Web Diagram

x

m < \- The sequence is constant for XQ = | or XQ < \. For XQ = 0 or xQ = 1 the sequence converges to | . The web diagram corresponding to the initial value o^o = 0.66 is displayed in Figure 4.

As discussed above, the Contrapositive, Half and Minimalist Liars have some common features, but are basically drastically different. The Contrapositive Liar displays the most irregular, chaotic behaviour, and this fact is seen easily by comparing the web diagrams in Figures 2 with those in Figures 3 and 4. However, Half and Minimalist Liars have web diagrams fairly similar in spite of the fact that the Half Liar has a much more regular behaviour than the Minimalist Liar. Cantor and dragon diagrams of these dynamics reflect the above distinction. For illustration, Cantor diagrams are displayed in Figures 5 and 6; the Cantor diagram for the Half Liar is completely black.

40

/ / Figure 4. Minimalist Liar: Web Diagram

Figure 5: Contrapositive and Minimalist Liars: Cantor Diagrams

41

6

Unidirectional Time-Dependent Liar

We introduce now a time dependence in the estimation of truth values. The first variant takes into account the whole "history" of truth values:

This sentence is as true as the negation of its history of estimations.

Figure 6: Unidirectional Time-Dependent Liar: Cantor Diagram This estimation can be represented by the morphism 0 —» 01,1 —• 10. Its dynamics x 0 = 0, xn+1 = \xn +

2

2"

1 22=

(1)

leads to the fixed-point 0.0110100110010110..., the Thue-Morse number. This number was proved transcendental by Dekking [4]; it is the coordinate of the Feigenbaum-Myrberg point in the Mandelbrot set (see Peitgen and Richter [12]). The difference between two consecutive terms in (1) becomes extremely small starting with the 4th iteration, so the web graph is not relevant: the corresponding Cantor graph appears in Figure 6.

42

7

Bidirectional Time-Dependent Liar

Consider now the truth estimation in which the history of truth values is recorded forward and backward:

This sentence is as true as its history of estimations followed by its negation. It can be represented by the morphism 0 —• 001,1 —• 110. Its dynamics

xo = 0, xn+1 =Xn(l

+— - — J +—

(l - ^ J

(2)

leads to the fixed-point 0.001001110001001110110110001..., the MephistoWaltz number. Again, the difference between two consecutive terms in (2) becomes extremely small, so the web graph is not relevant: instead, we will use the dragon plot (see Figure 7). The Contrapositive and Mephisto-Waltz web diagrams seem to be pretty similar; dragon diagrams in Figures 7 and 8 reinforce this observation.

Figure 7: Bidirectional Time-Dependent Liar: Dragon Graph

43

Figure 8: Contrapositive Liar: Dragon G r a p h 8

Codes

In this section we present the basic drawing procedures for Cantor and dragon plots illustrated for t h e Thue-Morse and Mephisto-Waltz sequences. //Cantor p l o t of the "Thue-Morse" sequence const i n t SIZE = 128; const i n t MAX = SIZE+SIZE; char number[MAX+1]; / / e . g . "0110100110010110..." i n t n=0,i,j=0,k,m; number[0] = ' 0 ' ; //form the sequence: 0->01 1->10 while (j<MAX-2) { if (number[n]=='0>) { number [j++] = '0'; number [ j ++] = ' 1'; } else { number [j++] = ' 1'; number [j++] = '0'; } n++; y

moveto(0,400); //top of drawing area k = 0; m = MAX - 1; for (i=0;i<SIZE;i++) { for (j=0;j001 l->110 while (j<MAX-3) { if (number [n]==>0>) {

number[j ++] number[j ++] number [j++] y else { number [j++] number [j++] number [j++]

= '0 = '0 = '1 = '1 = '1 = '0

}

n++; }

moveto(200,200);

///centre /c of drawing area for (n=0;n<MAX;n++) { if (number[n]=='1') { //turn left switch (direction) { case North:rlineto(0,16); rlineto(-4,4); direction=West; break; case East: rlineto(16,0); rlineto(4,4); direction=North;break; case South:rlineto(0,-16);rlineto(4,-4); direction=East; break; case West: rlineto(-16,0);rlineto(-4,-4);direction=South;break; } } else { //turn right switch (direction) { case North:rlineto(0,16) ;rlineto(4,4); direction=East; break;

45

case East: rlineto(16,0) ;rlineto(4,-4); direction=South;break; case South:rlineto(0,-16);rlineto(-4,-4);direction=West; break; case West: rlineto(-16,0);rlineto(-4,4); direction=North;break; > } }

N o t e . A preliminary version of this paper was presented at the 5th Anniversary Workshop on Discrete Mathematics and Theoretical Computer Science, University of Auckland, May, 2000, [3]. References 1. E. Beltrami, What is Random? Chance and Order in Mathematics and Life, Springer-Verlag, New York, 1999. 2. J. Buridan, Sophisms on Meaning and Truth, Appleton-Century-Crofts, New York, 1966. Translated by T.K. Scott. 3. C.S. Calude, Elena Calude, P. Kay, Liars, demons, and chaos, in The 5th Anniversary Workshop on Discrete Mathematics and Theoretical Computer Science (C.S. Calude, M.J. Dinneen, eds.), CDMTCS Research Report 134, 2000, 4. 4. M. Dekking, Transcendence du nombre de Thue-Morse, Compte Rendues de I'Academic des Sciences de Paris, 285 (1977), 157-160. 5. C. Davis, D.E. Knuth, Number representations and dragon curves, J. Recreational Mathematics, 3 (1970), 61-81, 133-149. 6. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York, 1990. 7. M. Gardner, Mathematical games, Scientific American, 216 (March 1967), 124-125; 216 (April 1967), 118-120; 217 (July 1967), 115. 8. P. Grim, G. Mar, P.St. Dennis, The Philosophical Computer. Explanatory Essays in Philosophical Computer Modeling, MIT Press, Cambridge Mass., 1998. 9. H.S. Leff, A.F. Rex, Maxwell's Demon: Entropy, Information, Computing, Princeton University Press, Princeton, 1990. 10. P. Matherat, M.-T. Jaekel, Dissipation logique des implementations d'automates-dissipation du calcul, Technique et Science Inforrnatique, 15, 8 (1996), 1079-1104. 11. J.C. Maxwell, Theory of Heat, Longman's, Green, & Co., London, 1985, 328-329. 12. H.O. Peitgen, P.H. Richter, The Beauty of Fractals: Images of Complex

46

Dynamical Systems, Springer-Verlag, Berlin, 1986. 13. A. Tarski, The semantic conception of truth and the foundations of semantics, Philosophy and Phenomenological Research, 4 (1944), 341-376.

47 SOME REMARKS ON HAIRPIN A N D LOOP LANGUAGES

JUAN CASTELLANOS Department of Artificial Intelligence Polytechnical University of Madrid 28660 Boadilla del Monte, Madrid, Spain E-mail: j c a s t e l l a n o s 6 f i . u p m . e s VICTOR MITRANA University of Bucharest, Faculty of Mathematics Str. Academiei 14, 70109 Bucharest, Romania E-mail: mitrana€funinf .math.unibc.ro We further investigate here the hairpin languages defined in [13] and introduce a similar type of languages, namely the loop languages, for which we study some decidability matters. Then related operations are defined and the closure properties of the families in the Chomsky hierarchy under these operations are considered. Finally, a few directions for further research in this area are briefly discussed.

1

Introduction

Since Adleman's seminal paper [1], there have been proposed many DNA-based algorithms consisting of three basic steps: (i) one encodes the problem using DNA molecules and an initial pool, containing all the molecules encoding solutions or not, is randomly generated, (ii) only the molecules encoding solutions of the problem are extracted by employing a succession of operations which can be accomplished by means of genetic engineering, (iii) these molecules are sequenced and decoded. In many such DNA-based algorithms the molecules used are single-stranded and one of the most frequently applied operation is that of hybridization which might produce, by undesirable hybridizations in molecules which form a hairpin structure, DNA strands which cannot be used in the subsequent computations. In a series of papers (see [2,6,7]) the problem of finding sets of DNA sequences which are unlikely to lead to "bad" hybridizations is considered. Some algebraic properties of sets formed by such sequences are investigated in [8]. On the other hand, these molecules which may form a hairpin structure have been used as the basic feature of a new computational model reported in [17], where an instance of the 3-SAT problem has been solved by a DNA-algorithm in which the second phase is mainly based on the elimination of hairpin structured molecules. A theoretical model based on the process of self-assembly (a process with an important relevance in the DNA computing area, see, e.g., [10,11,18,19])

48

is proposed in [20]. Different types of hairpin languages are defined in [13] where they are studied from a language theoretical point of view as most bio-operations in the monograph [12]. This paper can be viewed as a continuation of [13]. We introduce another type of structure which might lead to mishybridizations and investigate the language of all words having this structure. The paper is organized as follows: in the next section the main definitions (hairpin and loop languages, hairpin-free and loop-free languages) are given and a few decidability matters for these languages are investigated. Then, there are introduced a series of operations of hairpin and loop excision and we consider the closure properties of the Chomsky hierarchy families under these operations. Each section contains some open problems, while the last section is a brief overview of other problems which appear to be of interest. 2

Definitions and Basic Properties

We consider the reader to be familiar with the fundamental concepts of formal language theory and automata theory, particularly the notions of grammars and finite automata [16]. An alphabet is always a finite set of letters. The set of all words over an alphabet V is denoted by V*. The empty word is written e; moreover, V+ = V* \{e}. For a finite set A we denote by card(A) the cardinality of A. An involution over a set S is a bijective mapping <x : S —> S such that a = cr _1 . An involution over V* which is a morphism is called a Watson-Crick morphism over the alphabet V, and is denoted by £. If £(a) = b, then we say that a and b are complementary to each other. Remember that the DNA alphabet consists of four letters, VDNA = {A,C, G,T}, which are abbreviations for the four nucleotides: adenine, cytosine, guanine, thymine, and £,(A) = T, £(C) = G. In what follows, all the alphabets we use will be of an even cardinality (they are formed by pairs of complementary letters) bigger than or equal to 4. x

1

a

h $

Figure 1: A hairpin word We denote by R the mapping defined by R : V* —> V*, [a\a2 • • -an)R = an . . .a2a\. Note that R is an involution and an anti-morphism {{xy)R = yRxR

49 for all i , j e V*). Note also t h a t the two mappings £ and commute, namely, for any string x, (£(x))R = £{xR) holds. A word z over V is k-hairpin, if z — uvw^(v)Rx for some u,x £ V* and v,w £ V+, and |f| > fc. A hairpin word is a word which is fc-hairpin for some k > 1. A word is said to be k-hairpin-free, if it cannot be decomposed as above, and hairpin-free if it is not hairpin, A word z over V is k-loop, if z = OTIO^(D)I for some u, x £ y * and u, w £ 1 / + , |u| > k, and |UJ| > |v|. A loop word is a word which a is k-\oop for some k > 1. a:

i

a

Figure 2: A loop word A word is said to be k-loop-free, loop-free if it is not loop. We denote by - HPk(V): - LPk{V):

if it cannot be decomposed as above, and

the set of all fc-hairpin words over the alphabet V, the set of all fc-loop words over the alphabet V.

Clearly, HPk(V) C HPj{V) and LPk{V) C LPj{V) for any pair of natural numbers k > j . Furthermore, it is obvious t h a t HPk(V) \ LPk(V) ^ 0 and LPk(V) \ HPk(V) ^ 0 for any k > 2, and any alphabet V. P r o p o s i t i o n 1. Let V be an alphabet. 1. HPk(V) is regular for any k > 1. 2. LPk(V) is regular for any k > 1. Proof The first statement is proved in L e m m a 1 from [13]; a similar reasoning works for the second statement as well. • A language containing only fc-hairpin/fc-loop languages is said to be a khairpin/fc-loop language. In a similar way the fc-hairpin-free/fc-loop-free languages are defined. T h e following closure properties follow directly from definitions: P r o p o s i t i o n 2. 1. For any integer k > 1, the families of k-hairpin and k-loop languages are closed under union, intersection, intersection with regular sets, concatenation and Kleene closure *. They are not closed under morphisms and inverse morphisms. 2. For any integer k > 1, the families of k-hairpin-free and k-loop-free languages are closed under union, intersection and intersection with regular sets.

50

They are not closed under morphisms, inverse morphisms, concatenation and Kleene closure *. Proposition 1 has a series of straightforward consequences: Proposition 3. 1. Given a regular/context-free language L and a positive integer k, one can algorithmically decide whether or not L is k-hairpin/k-loop. 2. Given a regular/context-free language L and a positive integer k, one can algorithmically decide whether or not L is k-hairpin-free /k-loop-free. Proof. These statements follow from the effectiveness of closure properties and decidability properties of the families of regular and context-free languages, respectively, and the following two facts: 1. LCV* is fc-hairpin if and only if L \ HPk{V) = 0. 2. L C V is fc-hairpin-free if and only if L n HPk(V) = 0. • Corollary 1. Given a regular/context-free language L one can compute the maximal integer k such that L is k-hairpin/k-loop. Proof. We give the reasoning for the hairpin property only. The following algorithm computes the maximal value of k for a given regular/context-free language L to be fc-hairpin: begin compute the shortest word z £ L; computes the maximal k such that z = uvw^{v)Rx if k = 0 then "L IS NOT HAIRPIN" else while L\HPk{V)±% do k := k- 1 endwhile endif end.

with \w\ > 1 and \v\ = k;

Since L is regular, the first step is computable and, by the previous proposition, the while condition is computable as well. • The situation is a bit more involved when one wants to compute the minimal integer k such that L is fc-hairpin-free or fc-loop-free. Proposition 4. Given a regular language L one can compute the minimal integer k such that L is k-hairpin-free or k-loop-free. Proof. We now present a proof for the loop-free case. Let A = (Q, V, 5, qo, F) be the minimal finite automaton which recognizes the language L. Having this

51

automaton as input, the next algorithm provides the minimal k such that L is fe-loop-free, provided that such k exists. begin

(1)

compute S = {(q,s) \ q,s £ Q, there exists x £ V+ such that 6(q,x) = s};

(2)

for each pair (q,s) £ S compute the following languages:

(3)

+

L{t,q) = {w£V

\6{t,w) = q},t<EQ;

L{s,p),pe Q; if L(t, q) n (,(L(s,p)) is infinite then L IS NOT LOOP-FREE; halt endif endfor for each (t, q, s, p) £ Q4 and (q, s) £ S compute k(t,q,s,p) = max{|u)| | w £ L(t,q) n£(L(s,p)) and there is a word x £ L(q,s) such that \x\ > |tu|}; endfor k := max{k(t, q, s,p)}; end

(4) (5) (6) (7) (8) (9) (10) (11)

Here are some considerations with respect to the above algorithm. First all the languages L(q,s) are regular and can be constructed effectively such that one can check the if condition in line (4). If all the intersections in line (4) are finite sets, then one can compute the values k(t, q, s,p) from line (8). Therefore, we have an algorithm. Let us say a few words about its correctness. If in line (4) the intersection L(t, q) n£(£(s,p)) is infinite for some t,q,s,p £ Q and (q, s) £ S, then there are arbitrarily many strings z = uvx£,(v)y £ L with v £ L(t,q) and £(v) £ L(s,p), which implies that L is not loop-free. Remember that the automaton is minimal so that there must exist a path, empty or not, from go to t as well as one from p to a state in F. If all these intersections are finite sets, then only a finite number of loop words are in L, hence one can compute the minimal k such that L is fc-loop-free. This is done in lines (8) and (10). • The same problem for context-free languages remains open. More precisely, given a context-free language L, can one compute the minimal integer k such that L is A;-hairpin-free or fc-loop-free?

52

3

Hairpin and Loop Excision

In this section we define several operations on words and investigate the closure properties of the families in the Chomsky hierarchy with respect to them. They are similar to those defined in [3,4,5,14] inspired from gene assembly in ciliates. However, the mode of excision differs, being related in some sense to the PAmatching operation used in [15] and investigated in [9] as a formal operation on words and languages. The following three operations are called hairpin excisions and apply to any word z £ V* as follows: • the fe-hairpin excision defined by: hpek(z) = {uxy | z = uvx£(v)

y, for some u,y £ V*, x £ V+, and \v\ = k}.

• the < Ar-hairpin excision defined by: hpe ft-hairpin excision defined by: hpe>k(z) = {uxy | z = uvx^{v)Ry,

for some u,y £ V*,x £ V+, and \v\ > k}.

If z is hairpin-free, then hpex(z) — 0 for all X £ Ufc>i{< k, k,> k}. The next three operations are called loop excisions and apply to any word z £ V* as follows: • the fc-loop excision defined by: lpek(z) = {uxy | z = uvx£,(v)y, for some u, y £ V*, |a;| > |i>| = k}. • the < fc-loop excision defined by: lpe |v| < k}. • the > fc-loop excision defined by: /pe>fc(z) = {uxy | 2 = uvx£,(v)y, for some u,y £ V*, |a;| > |f | > /s}. If z is loop-free, then lpex(z) = 0 for all X £ UA:>I{< A;, A;, > A*}. The above operations can be extended to languages in a natural way: hpex [L) =\J hpex [w), W

lpex (L) = [J lpex (w), W

for all X £ U/c>i{< k, k, > k}. A family of languages T is said to be a full trio if it is closed under morphisms, inverse morphisms, and intersection with regular sets. A trio closed under union is called semi-AFL.

53

Proposition 5. Any full trio is closed under hpek, hpek{ci}V*),

» = 1

where V>k stands for the words in V of length bigger than k, then the proof is valid for lpek. For lpek{L) =\jL(q0,t)L(q,s)L(s,f),

(12)

where the union is taken for &\lt, q, s,p G Q, f G F, such that the following two conditions are satisfied: (i) (q, s) G S, S being defined in the proof of Proposition 4, (ii)L(((,,)n^(s,P))>^^. It is an easy exercise to check that equality (12) holds. For the > &-loop excision, equality (12) becomes lpe>k(L) =

[JL(qo,t)M(q,s)L(s,f),

55 where M(q, s) = L(q, s) D V>1,

with

/ = r m n { H | w e {L(t, q) H Z(L{s,p)))

l~l V * * } ,

and this completes the proof.

•

We denote by Suf(x) the set of all the proper suffixes of the word x and extend this operation to languages by Suf(L) — U ^ g i Suf(x). It is known t h a t the family of context-sensitive languages fails to be closed under Suf. Based on this result we prove: P r o p o s i t i o n 8. For any k > 1, the family of context-sensitive languages is closed under > k-loop excision but it fails to be closed under > k-hairpin excision. Proof. We first prove the closure under > fc-loop excision. For a contextsensitive language L, we construct a phrase-structure g r a m m a r G which works as follows: • Generates a sentential form $J4U;$, where w £ L. • Checks whether or not w can be decomposed as w = uvxE,(v)y with \v\ > k and marks the two segments v and $,(v). If w does not have any such decomposition, then the derivation is blocked. • Checks whether or not x from the above decomposition is longer t h a n v. If not, unmarks the two segments and resumes the work of searching for another decomposition. • If £ is longer than v, then delete the marked segments and all the nonterminals and the derivation is finished. Since for each z e L(G) we have WSG{z)

< 3 | Z | + 2,

where WSG(Z) is the workspace of z with respect to G ([16]), the language L(G) is context-sensitive. On the other hand, L{G) is exactly /pe>fc(L) which concludes the proof of the first statement. Now let L be a context-sensitive language over the alphabet V and k > 1 be an arbitrary integer. We construct a copy V of V defined by V = {a | a £ V } . Next we define the morphism h : {V\JVU{b})* —> V* by h(a) = h{a) = a, for any a £ V, and h(b) = e. Note t h a t 6 is a letter not appearing in V U V. We consider one more new symbol e and define the language E={V*

h~l{L)

{e}T)C\V{b}V+

{e}V*

56

which is context-sensitive by the closure properties of this family. Moreover, we extend the Watson-Crick morphism£ to the alphabet VUVU{6, e} by £(a) = £(a) for all a £ V and £(6) = e. The following equality can be easily checked: hpe>k{E)nV+

=Suf(L).

Indeed, each word in E is of the form xbyez, with x,z £ V and y £ Suf(L). If a > ^-hairpin excision is applied to such a word resulting in the excision of a hairpin structure within y, then the obtained word is rejected by the intersection with V+. Then the > fc-hairpin excision must be applied to a word of the form >

k

_ j

xbye^(x) with x £ V~ , yelding y which is a proper suffix of some word in L, Now the proof is complete. D We do not know whether the family of context-free languages is closed under > fc-hairpin or > fc-loop excision. 4

Further Work

In view of the considerations from the introductory section, the problem of destroying all the hairpin or loop structures from a word by repeated hairpin and loop excision appears to be of interest as well as mathematically attractive. More precisely, since mishybridizations may occur if the hairpin or loop segment is sufficiently long, let us consider the following A-hairpin-free language associated with a given language L CV*, denoted by Mhk(L) and defined recursively as follows: k(Ni%i>0, if' =

Ni\HPk(V),i>0,

Mhk(L) = [JT^. t>o

Analogously, one can define the Ai-loop-free language associated with a language L, denoted by Mlk(L). Many problems naturally arise; we list below just two of them which are interesting in our opinion. 1. Given a language L in a Chomsky family, to which family Mhk(L) Mlk(L) does it belong?

and

57

2. Given a regular/context-free language L, is it possible to decide algorithmically whether or not there exists t such Mhk{L) is obtained in t steps, that is, CO I I rp(k) . .

. . rp[k)7

Mhk{L) = T^' U T\K) U . . . U T , For instance, one can easily notice that Mhk{HPk{V)) tained in one step (t = 1) for any k > 1.

= V+ \ HPk(V) is ob-

Note. The work of the second author has been supported by Direccion General de Ensenanza Superior e Investigation Cientifica, SB 97-00110508. References 1. L.M. Adleman, Molecular computation of solutions to combinatorial problems, Science, 226 (1994), 1021-1024. 2. R. Deaton, R. Murphy, M. Garzon, D.R. Franceschetti, S.E. Stevens, Good encodings for DNA-based solutions to combinatorial problems, Proc. of DNA-based computers II, (L.F. Landweber, E. Baum, eds.), DIMACS Series, vol. 44, 1998, 247-258. 3. A. Ehrenfeucht, D.M. Prescott, G. Rozenberg, Computational aspects of gene (u)scrambling in ciliates, in Evolution as Computation (L. Landwerber, E. Winfree, eds.), Springer-Verlag, Berlin, 2000, 45-86. 4. A. Ehrenfeucht, I. Petre, D.M. Prescott, G. Rozenberg, Universal and simple operations for gene assembly in ciliates, in Where Mathematics, Computer Science, Lingusitics and Biology Meet (C. Martin-Vide, V. Mitrana, eds.), Kluwer Academic, Dordrecht, 2000, 329-343. 5. R. Freund, C. Martin-Vide, V. Mitrana, On some operations suggested by gene assembly in ciliates, submitted. 6. M. Garzon, R. Deaton, P. Neathery, R.C. Murphy, D.R. Franceschetti, E. Stevens, On the encoding problem for DNA computing, The Third DIMACS Workshop on DNA-Based Computing, Univ. of Pennsylvania, 1997, 230237. 7. M. Garzon, R. Deaton, L.F. Nino, S.E. Stevens Jr., M. Wittner, Genome encoding for DNA computing, Proc. Third Genetic Programming Conference, Madison, MI, 1998, 684-690. 8. L. Kari, R. Kitto, G. Thierrin, Codes, involutions and DNA encoding, Workshop on Coding Theory, London, Ontario, to appear. 9. S. Kobayashi, V. Mitrana, Gh. Paun, G. Rozenberg, Formal properties of PA-matching, Theoretical Comput. Sci., to appear.

58

10. T.H. LaBean, E. Winfree, J. H. Reif, Experimental progress in computation by self-assembly of DNA tilings, Preliminary Proc. of 5th International Meeting on DNA Based Computers (E. Winfree, D. Gifford, eds.), MIT, 1999,121-138. 11. M.G. Lagoudakis, T.H. LaBean, 2D DNA Self-assembly for satisfiability, Preliminary Proc. of 5th International Meeting on DNA Based Computers (E. Winfree, D. Gifford, eds.), MIT, 1999, 139-152. 12. Gh. Paun, G. Rozenberg, A. Salomaa, DNA Computing. New Computing Paradigms, Springer-Verlag, Berlin, 1998, Tokyo, 1999. 13. Gh. Paun, G. Rozenberg, T. Yokomori, Hairpin languages, Intern. J. Found. Comp. Sci., to appear. 14. D.M. Prescott, Cutting, splicing, reordering, and elimination of DNA sequences in hypotrichous ciliates, BioEssays, 14, 5 (1992), 317-324. 15. J.H. Reif, Parallel molecular computation: Models and simulations, Proc. of Seventh Annual ACM Symp. on Parallel Algorithms and Architectures, Santa Barbara, 1995, 213-223. 16. G. Rozenberg, A. Salomaa, eds., Handbook of Formal Languages, 3 volumes, Springer-Verlag, Berlin, Heidelberg, 1997. 17. K. Sakamoto, H. Gouzu, K. Komiya, D. Kiga, S. Yokoyama, T. Yokomori, M. Hagiya, Molecular computation by DNA hairpin formation, Science, 288 (2000), 1223-1226. 18. E. Winfree, X. Yang, N.C. Seeman, Universal computation via self-assembly of DNA; some theory and experiments, Proc. of 2nd Annual Meeting on DNA based computer (L. Landweber, R. Lipton, eds.), Princeton, (1996), 172-190. 19. E. Winfree, T. Eng, G.Rozenberg, String tile models for DNA computing by self-assembly, Preliminary Proc. of 6th International Meeting on DNA based computers (A. Condon, G. Rozenberg, eds.), Leiden, 2000, 65-84. 20. T. Yokomori, YAC: Yet another computation model of self-assembly, Preliminary Proc. 5th Intern. Meeting on DNA Based Computers (E. Winfree, D. Gifford, eds.), MIT, 1999, 153-168.

59 CONDITIONAL GRAMMARS WITH RESTRICTIONS SYNTACTIC PARAMETERS

BY

J U R G E N DASSOW Otto-von-

Guericke- Universitat Magdeburg Fakultat fur Informatik PSF 4120, D-39016 Magdeburg, Germany E-mail: dassowGiws. c s . u n i - m a d g e b u r g . de A conditional grammar is a usual context-free grammar where with any production a regular set is associated as a condition, and a rule p with condition R can be applied to a sentential form w if and only if w G R. In this paper we restrict the regular languages used as conditions by syntactic complexity measures as the number of nonterminals or productions which are necessary to generate the language by regular grammars or the number of states necessary to accept the language by deterministic finite automata. With respect to the number of nonterminals or states we obtain a finite hierarchy whereas the number of productions produces an infinite hierarchy.

1

Introduction

It is well-known that context-free grammars, which form the most investigated type of grammars, are not able to cover all phenomena of programming languages and/or natural languages. Therefore in the last 35 years a lot of mechanisms have been introduced in order to enlarge the generative power (see, e.g., [2]). One of these mechanisms are conditional grammars which have been defined by Fris ([3]), studied, e.g., by Salomaa ([11]) and Paun ([7]), and modified versions are investigated by Navratil and Krai ([6], [5]). A conditional grammar is a usual context-free grammar where with any production a regular set is associated as a condition, and a rule p with condition R can be applied to a sentential form w if and only if w <E R. It has been shown that conditional grammars generate the family of recursively enumerable languages and that they have the same generative power as context-sensitive grammars if no erasing rule is used. However, in order to obtain this power it is sufficient to use regular languages of a special form, e.g., (suffix closed) regular languages accepted by multiple-entry automata (see [4]). On the other hand, if one takes definite languages (see [8]) one only gets the family of ETOL languages. The effect of algebraic and automaton-theoretic restrictions for the conditions are studied in [1]. In this paper we restrict the regular languages used as conditions by syntactic complexity measures as the number of nonterminals or productions which are

60

necessary to generate the language by regular grammars or the number of states necessary to accept the language by deterministic finite automata. With respect to the number of nonterminals or states we obtain a finite hierarchy whereas the number of productions produces an infinite hierarchy. 2

Definitions

Throughout the paper we assume that the reader is familiar with the basic concepts of formal language theory, especially on grammars of the Chomsky hierarchy, L systems and conditional grammars; for details we refer to [10], [9], [2]We now recall some notions in order to specify our notation. The empty word is designated by A. By #M(W) we denote the number of occurrences of letters from M C V in the word w G V*. The set of all positive integers is designated by N. # ( M ) denotes the cardinality of the set M. By C(FIN), C{REG), C{CF), £{CS) and C(RE) we denote the families of all finite, regular, context-free, context-sensitive and recursively enumerable languages, respectively. C(ETOL) denotes the family of all languages generated by context-independent tabled Lindenmayer systems. A regular grammar G — (N, T, P, S) is specified by - the disjoint alphabets N and T of nonterminals and terminals, respectively, - the finite set P C N xT*(NU {A}) of productions, and - the axiom or start element S G N. As usual we use the notation A —> w for (A, w) G P. By L(G) we denote the language generated by G. A regular grammar is in normal form, if all rules are of the form A —> aB or A ->• a with A,B 6 N and a G T U {A}. It is well-known that any regular language can be generated by a regular grammar in normal form. A (deterministic) finite automaton A = (X, Z, ZQ, F,S) is specified by the finite set X of input symbols, the finite set Z of states, the initial state ZQ and the set F of accepting states, the transition function 5 : Z x X -^ Z. By T(A) we denote the language accepted by A. For a regular grammar G = (N,T,P,S), a finite automaton A (X, Z, ZQ, F, S) and a regular language L, we define

-

«(G)=#(iV))p(G) = #(P)

and

s(A) = #(Z)

=

61

and v(L) = mm{v(G) | L — L(G), G is regular} , p(L) = min{p(G) | L = L(G), G is regular} , nv(L) — min{v(G) | L = L(G), G is a regular grammar in normal form} , np(L) = min{p(G) | L = L(G), G is a regular grammar in normal form} , s(L) = min{s(A)

\ L = T(A), A is a finite automaton} .

A conditional grammar is a quadruple G = (N, T, P, S), where: • N and T are disjoint alphabets of nonterminals and terminals, respectively, • P is a finite set of pairs p = (Ap —> zp,Rp) and Rp is a regular set called condition, •

where Ap £ N, zp G (N U T)*

SEN.

We say that y is directly derived from x by application of p — (Ap —> zp, Rp), denoted by x ==> y, iff x — xiApX2, y = x\zpX2 for some xi,X2 G (N U T)* and x £ Rp. We write cc =^> 1/ iff a; = > y for some p. By =^> we denote the p

reflexive and transitive closure of ==>. The set S(G) of sentential forms and the language L(G) generated by G are defined as

S{G) =

{w\S^w}

and L(G) = {w | S =^> w and

weT*}.

We say that a conditional grammar G — (N,T,P,S) is A-free if any rule (A -+ w,R) £ P satisfies w / A with the only possible exception (S —> A, {£}) if S does not occur in a right-hand side of any production. For c G {v,p,nv,np,s} and a positive integer i, by £(c, i) ( £ A ( C , i), resp.) we denote the family of languages which can be generated by (A-free) conditional grammars G = (N, T, P, S) where, for any (A -> z,R) e P, c(R) < i. By definitions we get immediately the following statement. Lemma 1 i) For any c G {v,p, nv, np, s} and any integers i and j with i < j , C{c,i) CC{c,j)

CC{RE)

and £x{c,i)

C£A(c,j)

ii) For any d G {v,p} and i G N, C(nd, i) C C(d, i) and C\(nd, i) C £\{d, i) . iii) For c G {v,p, nv, np, s} and i G N, £\{c,i)

C

£(c,i).

CC{CS).

62

In this paper we shall investigate whether or not the hierarchies from Lemma 1 i) are finite and whether the inclusions of Lemma 1 ii) and iii) are proper. 3

Restrictions by the Number of Nonterminals

We first prove that with respect to the number of nonterminals and states, respectively, we obtain only finite hierarchies. Theorem 2 i) £(v,2) = C{nv,Z) = £(s,3) = £{RE). ii) Cx{v, 2) = £x{nv, 3) = £x(s, 3) = C(CS). Proof. i) The proof in [2] that any recursively enumerable language can be generated by a conditional grammar requires only conditions of the form X* {ab}X* and X*, where X is the set of all nonterminals and terminals of the grammar. Obviously, X*{ab}X* can be generated by the regular grammar G = {{S, A},X, {S -> abA, S -+ ab} U ( J {S -> xS, A -»• xA, A ->• x}, S) and the regular grammar G = {{S, A, B},X, {S -^aB,B-^

bA, B -»• 6}U [j {S -»• xS, A -> xA, A -> x},S) x£X

in normal form. Moreover, X* {ab}X* is accepted by the finite automaton given in Figure 1. X

—( zo

X

-M zi

Figure 1. Automaton accepting

-*•((

Z

2

X*{ab}X*

Moreover, as can be seen from the proof above X* can be generated by a regular grammar in normal form with exactly one nonterminal and can be accepted by a finite automaton with exactly one state. ii) can be shown analogously. • Because C(CS) is a proper subset of C(RE) we immediately get the following statements

63

Corollary 3 i) For i > 3, £(nv,i) = £(v,i). ii) For i > 3 and c £ {nv, s}, £X(c, i) C £(c, i). in) For j > 2, £x{v,j) C C[v,j). We now consider the case of one nonterminal and one state. Theorem 4 i) £{nv, 1) = £x{nv, 1) = C(ETQL). ii) £{ET0L) C £{v, 1) and C(ETOL) C £x{v, 1). in) £{s, 1) = £ A (s, 1) = £{ETQL). Proof. We only prove the relations in the general case; those for the A-free case follow by the same arguments. i) It is easy to see that any language L C V* generated by a regular grammar in normal form with one nonterminal is of the form L — U* B with U C V and B C V. Especially, any combinational language over V, i.e. L = V*B can be generated. By [1], Corollary 5, any ETOL language can be generated by a conditional grammar where all conditions are combinational languages. Thus C(ETOL) C£(nv,l). The converse inclusion £(nv, 1) C £(ETQL) can be shown as in the proof of [1], Lemma 4. ii) follows from i) and Lemma 1. iii) Obviously, any language L CV* accepted by a finite automaton with one state is a submonoid U* of V*. By [1], Corollary 5, the set of languages generated by conditional grammars whose conditions are monoids coincides with the family of ETOL languages. • By the known relation £(ET0L) C £(CS) C £(RE), we get immediately the following corollary. Corollary 5 For c 6 {nv, s}, £(c, 1) C £(c, 3) and £\{c, 1) C £\{c, 3). We mention that it is an open question whether or not the inclusions £{c, 1) C £(c, 2) and £x{c, 1) C £x(c, 2) for c G {nv, v, s} and £{d, 2) C £(d, 3), £x(d, 2) C £x(d, 3) and £x(d, 2) C £(d, 2) for d £ {nv, s} are proper. 4

Restrictions by the Number of Productions

We first prove that - in contrast to Theorem 2 - we obtain an infinite hierarchy with respect to the number of productions. Theorem 6 Let c £ {p, np], and let n be an arbitrary integer. Then £(c, n) C £(c, n + 2) and £x(c, n) C £X{c, n + 2). Proof. For given n, we consider the language Ln = afa^ . . -a^+i o v e r the alphabet T = {a\, ai, • • •, a n +i}- Assume that L = L{G) for some conditional

64

grammar G = (N,T,P,S). For q = {Aq -> zq,Rq), let Gq = (Nq,Tq, Pq, Sq) be a regular grammar (in normal form) generating Rq and r = max{|v| | ^ 4 - > » £ P g , g £ P } . We consider the word w = a\r+la\r+l .. . a^ r+1 . Then there is a word u/ such that w' = > u; using a rule (Aq —» zq,Rq). Obviously, w' contains only one nonterminal and has to be contained in Rq. Thus

«,' = ar+1a^

. ..a^a^A^a^a^

.

..a^

with s + t > 2r + 1 (and thus s > r + l o r t > r + l ) o r ,,./ _

n3r

+ l 3r- + l

3r + l s 4

t

3r+13r +l

3r + l

w — al a2 •••a i _i a !'^? a i+i a i+2 a ;+3 • • • an with s > 2r + 1 and t > 2r + 1. Hence, for any i, 1 < i < n+ 1, w' contains a subword a^ +1 . In order to generate these subwords the set Pq of rules of Gq has to contain rules of the form A -> af B or A -> af for 1 < i < n + 1. Therefore p(Rq) > n + 1 (np(Rq) > n + 1) and L g £(p, n) (L £ C(np, n)). On the other hand, L is generated by ({Al,A2,...,An+l},T,P,Al) with P = {(Ai + l —• an + l, Rn + l)

U

[J

{(Ai-KnAi,Ri)}

l w for some p G P. p

Obviously, w' = vp and w' = u>iApW2, w = w\wpW2- Thus L(G) C {w \w = wiwpu>2,vp = wiApw2,wi,W2

G T*, (Ap ->• top, {^p}) G P } ,

which proves that L(G) is finite. This implies £(c, 1) C C(FIN). On the other hand, let L = {iti, U2,..., us} C T* be a finite language over T. Since Z, = L(G') for the conditional grammar G' = ({S'},T, {(5' -> U i ) {5'}) | 1 < i < where np({S'}) = 1, the converse inclusion C(FIN)

s}},S')

C C(c, 1) holds, too.

D

Lemma 8 Force {p,np}, £{FIN) C Cx{c,2) CC{REG). Proof. By Lemmas 1 and 7, C(FIN) C £A(C, 2). The properness of the inclusion follows from the conditional grammar G = {{S}, {a}, {{S -> a5, a*5), {S -+ a, a*S)},S) which produces the infinite language a + and whose condition a* S is generated by the regular grammar H = ({5'}, {a, 5 } , {S* -> aS', 5' -»• 5 } , 5') in normal form with only two productions. By the proof of Theorem 6 there is a regular language which is not contained in£A(c, 2). Thus it is sufficient to show the inclusion C\(c,2) C C{REG) in order to complete the proof. We do this only for c = np; the proof for c = p follows the same lines. Let H be a regular grammar in normal form with at most two productions. Then by a discussion of all possible cases we obtain that one of the following cases holds for the language generated by H: (a) (b) (c)

L(H) is empty or L{H) consists of a single word of length at most 2 or L{H) = {«}*{&} for two letters a and b.

66

Now let L be a language in £\(np, 2). Then L = L(G) for some conditional grammar G = (N, T, P, S) where all productions of P are of the form (A->w,{v})

or

(A ->•

w,{a}*{b})

for some word v of length at most 2 and some letters a and b. Let k = max{\w\ | (A —>• w,R) £ P}. As in the proof of Lemma 7 we can show that using the rules of the first given form generate a finite set K whose words have at most the length k + 1. Let w =>• w' be a derivation where |tu| = r + 1 > 4 and p = (A —> p u>, {a}*{6}). By definition w = arb. Thus ^4 = a or /I = b. In the former case we obtain w' = aswatb with s + t = r — 1. Since a is a nonterminal we have to continue the derivation. Thus w' = aswatb has to be in a language {a'}*{6'} which implies w — a", u > 1, a = a', 6 = 6', i.e. #N(w) < #N(W'). Thus we can only terminate a derivation by p if A = b and a G T. Moreover, in order to get words which allow a termination a £ T has already to hold. Therefore the rule has to have the form (A —> a u B , {a}*{yl}) for some integer u or (A -» w, {a}*{A}) with w e T*. From this it follows that the regular grammar G" = (TV', T, P ' , 5') with N' = {£"} U {Aa \(A->w,

{a}*{A}) € P for some w},

P' = {5' -> w | v G 1(G), |«| < ife + 1} U{S" -> a M a | a" A G S(G), 3 < u < k + 4} U{Aa -»• a"fl a | (A -+ a u 5 , {a}*{A}) G P } U{,4a -> w | (A -> w, {a}*{A}) eP, weT*} generates L, too. Hence L is regular.

•

L e m m a 9 For c G {p,np}, £(c, 3) (anc? £A(C> 3), respectively) is incomparable withC(REG) and C(CF). Proof. Again, by the proof of Theorem 6, there is a regular language which is not contained in £(c, 3). On the other hand the conditional grammar G = ({S,A,B}, {a}, {pi,P2,P3, PA], S) with

= p2 = p3 = p4 = Pl

(S^A2,{S,A}*{S}), (A^B,{A,BY{A}), (B^S,{S,B}*{B}), (S^a,{S,a}*{S}),

67

generates the non-context-free language {a 2 " | n > 0} since starting from S2 (i = 0 gives the axiom), essentially, we only have the following derivations PI

M2)2'_15=»

=» PI

(A2)2'=^2'+1

PI

^i,1_85(2'+1-fc'1,1-i) —^ nl + 1

B2

1

^v^vB^'+'-^.i-'v-^) O' + l

B2

M=^

5fci',i5_g(2'+1_'si',i_:L) —s ol + l

52 S2'

>

1

X

Qk2,iSBk'^2SB^2'+1~k'^1~k'^2~2'> nl+ 1

S2

5 ^

•, S^2'1 aS^2'2 aS^2'~k2,1~k2'2~2^

c^i.ig c(2'-ki,i-i)

P4

P4

, P4

k

P4

2'-lc

>

2*+1

P4

Moreover, the language {a, b}*{a} can be generated by the regular grammar H = {{S',A'},

{a, b}, {S' -»• aS',S' -»• 65', 5' -»• a}, 5')

in normal form which proves that {a2 Corollary 10 Force

| n > 0} £ £(c, 3) .

•

{np,p}, Cx{c, 1) C C\{c, 2) c £ A ( c , 3 ) .

References 1. J. Dassow, H. Hornig, Conditional grammars with subregular conditions, Proc. Internat. Conf. Words, Languages and Combinatorics II (M. Ito, H. Jiirgensen, eds.), World Scientific, 1994, 71-86. 2. J. Dassow, Gh. Paun, Regulated Rewriting in Formal Language Theory, EATCS Monographs on Theoretical Computer Science 18, Springer-Verlag, Berlin, 1989. 3. I. Fris, Grammars with partial ordering of the rules, Inform. Control, 12 (1968), 415-425. 4. A. Gill, L.T. Kou, Multiple-entry finite automata, J. Comp. Syst. Sci., 9 (1974), 1-19. 5. J. Krai, A note on grammars with regular restrictions, Kybernetika, 9 (1973), 159-161. 6. E. Navratil, Context-free grammars with regular restrictions, Kybernetika, 2 (1970), 118-126.

68

7. Gh. Paun, On the generative capacity of conditional grammars, Inform. Control, 43 (1979), 178-186. 8. M. Perles, M. O. Rabin, E. Shamir, The theory of definite automata, IEEE Trans. Electronic Computers, 12 (1963), 233-243. 9. G. Rozenberg, A. Salomaa, The Mathematical Theory of L Systems, Academic Press, New York, 1980. 10. G. Rozenberg, A. Salomaa, eds., Handbook of Formal Languages, Volume II, Springer-Verlag, Berlin, 1997. 11. A. Salomaa, Formal Languages, Academic Press, New York, 1973.

69 ON COMPLETE CLASSES OF DIRECTED

GRAPHS

PAL DOMOSI Institute

of Mathematics and Informatics, Debrecen Egyetem ter 1, 4032 Debrecen, Hungary E-mail: d o m o s i @ m a t h . k l t e . h u

University

We introduce a new type of directed graph completeness and we characterize the class of digraphs having this property. Some consequences of this characterization are also discussed.

1

Introduction and Basic Notions

In this article we study the situation, where, in a network of size n, the ith member of the network wants to send a message to the r(i) member, where r is a transformation over { 1 , . . . , n} such that it is compatible with the digraph representing the communication link. (In this case every member of the network will send exactly one message to another member choosing one possibility among the communication links.) It may be also important if we can send appropriate messages using some consecutive applications of appropriate transformations. This model can be considered as a special type of finite automata networks or other certain structures generating transformation monoids. (See, for example [2], [6],[7],[8],[13], etc.) Considering this model, we give methods how to realize full transformation semigroups and full permutation groups by products of certain special type of transformations. Algorithms to determine permutation groups by their generators goes back to [12]. Furthermore, there are a number of software packages for symbolic computation on groups (CAYLEY, MAPLE, GAP, etc). Description of finite monoids by their structure is given in [1], [10]. Algorithms to compute finite semigroups by using a set of generators is presented in [5]. (Some other connections see also [11].) We are going to study finite transformation semigroups and finite permutation groups realized by products of special transformations such that these transformations are compatible with a given directed graph. In particular, we give a constructive proof how to get these special type of products. By these results using methods in [5] we can construct products realizing appropriate full transformation semigroups and permutation groups. In this framework, we can also give a new proof of results in [4]. Now we need to fix some standard terminology. A (finite) directed graph (or, in short, a digraph) V = (V, E) is a pair of the nonempty finite sets of vertices V and edges E C V x V. If \V\ = n (> 0) then we also say that V is a digraph

70

of order n. In addition, we put 2 ? ^ = (V,EU{(v,v) \ v £ V}) for every digraph V = (V,E)._ A walk in 2? = (V, E) to be a sequence of vertices v\,..., vn, n > 1, such that (v{, Vi+i) £ E,i = 1,.. .,n — 1. A walk is closed if v\ = vn. By a path from a vertex a to a vertex b ^ a we shall mean a sequence Vi . . . vn, n > 0 of distinct vertices such that a = v\, b = vn and (v{, v;+i) G E for every i = 1,. .., n— 1. The positive integer n is called the length of the path. Thus a path is a walk with all n vertices distinct. Allowing v\ = vn above, we speak about a cycle of length n. Two cycles of a graph will be called disjoint if they have no vertex in common. In the opposite case we say the cycles intersect. If (v, v') G E and v = v' then (v, v') is called (self-) loop edge. A branch in a digraph is a pair of nonloop edges (v,v'),(v,v") with v' ^ v" (and u ^ {v',v"}). Moreover, if (v,v') £ E then it is said that (v,v') is an outgoing edge of v, and simultaneously, (U,D') is an incoming edge for v'. (In this way, a loop edge (v, v) has both of these properties concerning the vertex v.) The digraph V = (V, E') is a subdigraph otV if V is a non-void subset of V, and E' C E. V is said to be connected for u £ V if every vertex i/ £ V has a (directed) path from v to u'. V is called strongly connected if it is connected for all of its vertices. A transformation semigroup is a pair T = (A,S), where A is a finite nonempty set and S is a semigroup of mappings A —• A. If 5 is a group of permutations on A, then T is a permutation group. The following concept is proposed in [3]. Let T = {A,S) and T' = (A', S') be transformation semigroups. 7" divides T', in notation T ) = (V,G(V)) is the permutation group ofQ. In addition, let C(V) = {F : V -»• V | aw'.v" £ V, («',«") G £ , « ' # w" : F(u') = F(«") = v", F{v) = v, v £ V \ {«',«"}, {(«, t>) |, u £ K \ {^'}} C £?}. Then C(2?) is the set of all p( £ )-compatible elementary collapsings.

71

We shall use the following statements. Theorem 1.1 [9] For every homomorphism (f of a finite semigroup S onto a group G, there exists a subgroup (i.e., a subsemigroup which is a group) of K of S such that f(K) = G. Theorem 1.2 [9] Let G be a semigroup of mappings of a finite set H and let K be a subgroup of G. Then there exists a subset H' of H such that the restrictions of the elements of K to H1 are permutations forming a group isomorphic to K. Theorem 1.3 [9] Let S be a semigroup of mappings of a finite set H, and assume that there exists a subset H' of H such that some elements of S when restricted to H1 are permutations. Then there exists in S a subgroup G such that the permutation group K generated by the above-mentioned permutations of H', is a homomophic image of G. 2

Penultimately Permutation Complete Digraphs

Take a digraph V — (V, E) with an ordered set V = {«i, ...,vn), n > 1 of vertices. Place a coin c; onto Vj for every i = 1,.. ., n — 1 such that c,- ^ Cj whenever i ^ j for some 1 < i, j < n — 1. Moreover, let us say that a vertex is free if it is not covered by any coin. (Thus the last vertex is free before the coins are moved.) Suppose that we are allowed to move the coins according to the following two types of rules. (1) If vtlt.. .,Vim, 1 < m < n form a cycle (i.e., (vil, u,-2), (v i2 , via),..., (vim-i >vim), ivim>vii) S E) then we can move the coins such that after this step v i,+i mod m i s covered by Ck3 whenever DS-. was covered by c^j, and" in addition, Vi •,, j is free whenever vi was free before (7 = 1 , . . . , m). *j + l mod m

*j

\J

J

)

/

(2) A coin c; can be moved to a vertex Vj if there is an edge (vk,Vj) G E such that Vk is covered by c,- moreover Vj is free. After this moving we assume that Vj is covered by the coin c; and vk is free. These rules are called allowed steps. We say that V = (V, E U {(v, v) | v £ V}), V — {i>i,. .., vn} penultimately realizes the permutation p : {1,... ,n — 1}—> {1,. . ., n — 1} with respect to Vi,..., vn if we can reach after one or more number of allowed steps that V{ is covered by cp^,i — I, ... ,n — I (such that vn should become free). In addition, if V penultimately realizes all permutation a

For arbitrary integers fc, m (m > 2), k mod m denotes the least positive integer k' such that m divides k — k'. (In particular, 0 mod m = m mod m = m.)

72

of the form p : { 1 , . . . ,n - 1 } ->• { 1 , . . . , n — 1} then we say that V is penultimately permutation complete with respect to vn. T> is called penultimately permutation complete if it is penultimately permutation complete with respect to every vt £ V. In other words, V is penultimately permutation complete if considering an arbitrary permutation fp(i), •. •, Vp(n) of vertices, for every permutation p : { 1 , . . . , n — 1} —> { 1 , . . . , n — 1} we can reach after one or more number of allowed steps that fp(i) is covered by cp(,-),i = l , . . . , n — 1 (such that t>p(n) should become free). Now we are going to characterize the class of penultimately permutation complete digraphs. To the simplicity, for every digraph we will identify the vertices with their serial number during this chapter.Therefore, we assume that a digraph of order n has the (ordered) set of vertices V = {1,.. .,n}. We define the concept of an allowed transformation (with respect to V) in the following way. (1) If F is a ©(^-compatible permutation then F is allowed. (2) If F is allowed having F(j) = n and (i, j) G E,i ^ j then F' : { 1 , . . . , n) -+ {1,. . ., n) is also allowed whenever F'(i) = F'{j) = F(i) and F'(k) - F(k),k l,...,n,k A fcp(i), • • ' . ^ ( m - i ) ) with ^ £ V . ) If Z> = Z>' then we are ready with the proof. Otherwise, by the strongly connectivity of V, there are vertices i 6 V, j G V\V with an egde (i,j) 6 i? and a p a t h j , ki, . . ., ks, i. To the simplicity, we assume j = 1, k\ = 2, . . ., fts = i — 1. Let £ be the minimal index with < £ V " . It is clear t h a t for an appropriate v £ V', F[(l,..., n) = (i, 1 , . . . , i — 1, i + l , . . . , n - l , v ) is allowed with respect to V" = (\/' U { 1 , . .. ,£ - 1 } , £ " U {(z, 1), (1, 2), . . ., (i — 1, i)). Now we show t h a t for an appropriate v' £ V', ^ ( l , • . ., n) = (1, • • •, i— 1,« + 1, i, « + 2, .. ., n— 1, n') is also allowed with respect to V". Because V has a branch, V \ {n} has at least two vertices. In addition, V' is assumed to be penultimately permutation-complete. Therefore, F(£,..., n) = (£,..., i — 1, i+ 1, i, i + 2 , . . . , n — 1, v) is allowed for a suitable v £ V' with respect to £>'. Thus, of course, F2 i s allowed with respect to V". B u t then considering Fi = F ^ ' with F ! ( l , . . . , n ) = (i+ 1 , 1 , . . . , i - l,i,i + 2,...,n - l,v), and ^ £ { l , . . . , n } and F 2 = ( F 1 ) ! ' - 1 F ^ ( F 1 ) 2 w i t h F 2 ( l , . . . , n ) = ( 2 , 1 , 3 , . . . , n - l , V ' ) and v' £ { l , . . . , n } , by Proposition 2 we get t h a t V" is also penultimately permutation-complete. If V" = V then we are ready. Otherwise we repeat this procedure. Repeating this treatment finally we receive t h a t V is also penultimately permutation-

76

complete. 3

D

Complete Classes of Directed Graphs

Let T be a nonempty class of digraphs. Consider the following definitions: T is isomorphically complete if every transformation semigroup can be embedded in the transformation semigroup of a digraph in T. T is homomorphically complete if every transformation semigroup divides the transformation semigroup of a digraph in V. V is complete if every finite semigroup divides the semigroup of a digraph in F. Similarly, T is isomorphically group complete if every permutation group can be embedded in the transformation semigroup of a digraph in T. T is homomorphically group complete if every permutation group divides the transformation semigroup of a digraph in T. Finally, T is group complete if every finite group divides the semigroup of a digraph in F. Using our results for the class of penultimately permutation complete digraphs, in this chapter ve give a quite simple new proof of the following two results. Corollary 3.1 [4] Let F be a nonempty class of digraphs containing all loop edges. The following conditions are equivalent. (3.1.1.) T is isomorphically group complete. (3.1.2.) T is homomorphically group complete. (3.1.3.) T is group complete. (3.1.4.) For every integer n > 3 there is a digraph in T which has a strongly connected subdigraph of order at least n and contains a branch. Corollary 3.2 [4] Let T be a nonempty class of digraphs containing all loop edges. The following conditions are equivalent. (3.2.1.) T is isomorphically complete. (3.2.2.) T is homomorphically complete. (3.2.3.) T is complete. (3.2.4.) For every integer n > 3 there is a digraph in T which has a strongly connected subdigraph of order at least n and contains a branch. Proof of Corollary 3.1. First we prove that (3.1.4) implies (3.1.1). Indeed, by (3.1.4) for every positive integer n, there exist a positive integer m and a digraph having a stongly connected subdigraph V = (V, E) of order m containing all loop edges and a branch. Theorem 2.5 implies that this subdigraph is penultimately permutation complete. Therefore, the (m — l)-degree full permutation group

77

can be embedded isomorphically in T{V^>). (See also Theorem 1.1, Theorem 1.2 and Theorem 1.3.) Clearly, then for every 1 < k < m, the fc-degree full permutation group can be embedded in the transformation semigroup of V^>. (3.1.1) immediately implies (3.1.2) and (3.1.3). Of course, (3.1.2) is a simple consequence of (3.1.3). To the end of our proof now we show that (3.1.2) implies (3.1.4). Suppose the contrary and assume that (3.1.1) holds without the validity of (3.1.4). Then there exists an integer n > 2 such that for every strongly connected subdigraph V — (V, E) of an arbitrary digraph in T, either \V\ < n or V does not have a branch. Consider a fc-degree symmetric group Sk for some k > 0. If k > n and \V\ < n then Sk does not divide S(V). On the other side, if V is strongly connected having no branch then it forms a cycle (up to the loop edges). In this case Sk < S(V) implies that Sk is an abelian group which is impossible unless k < 2. Consequently, the fc-degree full permutation group can not be embedded in the transformation semigroup of£>W.Then (3.1.2) impl les (3.1.4). • Proof of Corollary 3.2. It is enough to prove that (3.1.4) (i.e (3.2.4)) implies (3.2.1). Let V = (V, E) be a strongly connected subdigaph of order n+l > 2 of an arbitrary digraph in T having all loop edges and a branch. Then using Theorem 2.5, we can find an appropriate arrangement v\,..., vn+i of vertices such that for every permutation p : { 1 , . . . , n — 1} —> { 1 , . . ., n — 1} there exists a product Fp of T'W-compatible transformations having Fp(v\,..., v n +i) = (f p (i), • • •, fp(n)i v) with v G V. Let (v{,Vj) G E,i ^ j , i, j < n. (By n > 1 we should have such an edge.) Then, by our definition, F(vi,.. .,v„+i) = (vi,.. .,Vj-i,Vi,Vj+i,.. .,vn+i) is a X>^-compatible transformation. Then for appropriate permutations p, q there is a pair Fp, Fq of X>^-compatible transformations such that the transformation -F3 = FpFFq has the property F${vi,.. .,vn+i) = (vi, vi, v3 .. ., v„, v) for a suitable v G V. On the other side, by Theorem 2.5, for appropriate products Fi,F2 of V^-compatible transformations, Fi(vi,..., v n +i) = (vn, vi, • • • 1 vn-i> v') with v' G V and F2(vi,..., f„+i) = {v2,vi,v3 ... ,vn,v") with v" G V. By Proposition 2.2 this ends the proof. •

4

Summary and Further Problems

In this paper we studied a class of digraphs having special completeness properties with respect to the permutation groups. By their characterization we gave a new proof of certain results of characterizations given in [4], Similarly to [13], for an arbitrary digraph V = (V, E) we considered a mapping F : V —> V to be compatible with V if {(v,F(v)) | v G V} C E U {(v,v) \ v G V). We can give

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a natural extension of this definition considering a mapping F : V —> V to be compatible with V if {(v, F(v)) \ v £V} C E.ltis an open problem how can we extend our results considering this generalized form of the discussed concept. Acknowledgements. This work has been supported by LIAFA - University Paris-7 - Denis Diderot, by EGIDE (Center of French International Exchanges, Grant No 28025L), and by the Hungarian National Foundation for Scientific Research (OTKA T030140). The author is grateful to EGIDE (Center of French International Exchanges) and LIAFA (Laboratoire d'Informatique Algorithmique: Fondements and Application) - University Paris 7 - Denis Diderot for providing excellent working conditions. Moreover, the author would like to thank Professor Jean-Eric Pin for his invitation and kind hospitality during the visit. References 1. C. Cousineau, J.F. Perrot, R.L. Rifflet, APL programs for direct computation of a finite semigroup, APL Congress 73, Amsterdam, North Holland, 1973,67-74. 2. C. Choffrut, An introduction to automata network theory, in Automata Networks (C. Choffrut, ed.), Argeles-Village, France, 1986, Lecture Notes in Computer Science, 316, Springer-Verlag, 1988, 1-18. 3. S. Eilenberg, Automata, Languages and Machines, Vol. B., Academic Press, New York, 1976. 4. Z. Esik, A note on isomorphic simulation of automata by networks of twostate automata, Discrete Appl.Math., 30 (1991), 77-82. 5. V. Froidure, J.-E. Pin, Algorithms for computing finite semigroups, Foundations of Computational Mathematics (F. Cucker, M. Shub, eds.), Lecture Notes in Computer Science, Springer-Verlag, 1997, 112-126. 6. Z. Fiilop, S. Vagvolgyi, A complete rewriting system for a monoid of tree transformation classes, Information and Computation, 86 (1990), 195-212. 7. F. Gecseg, Products of Automata, Springer-Verlag, Berlin, 1986. 8. F. Gecseg, B. Imreh, Finite isomorphically complete systems, Discrete Applied Mathematics, 36 (1992), 307-311. 9. A. Ginzburg, Algebraic Theory of Automata, Academic Press, New York, London,1968. 10. H. Jiirgensen, Computers in semigroups, Semigroup Forum, 15 (1977), 1-20. 11. P. Maryse, J. Sakarovich, Easy multiplications II. Extensions of rational semigroups, Information and Computation, 88 (1990), 18-59. 12. S. Piccard, Sur les bases du groupe symetrique et les couples de substitutions qui engendrent un groupe regulier, Mem, Univ. Neuchdtel, vol. 19, Librarie

79 Vuibert, Paris, 1946, 223 pp. 13. M. Tchuente, Computation on finite networks of automata, in Automata Networks (C. Choffrut, ed.), Argeles-Village, France, 1986, Lecture Notes in Computer Science, 316, Springer-Verlag, 1988, 53-67.

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81 CIRCULARITY A N D OTHER INVARIANTS OF G E N E A S S E M B L Y IN CILIATES

ANDRZEJ EHRENFEUCHT University of Colorado, Department of Computer Boulder CO 80309-0347, USA E-mail: andrzej9cs.colorado.edu

Science

ION PETRE Turku Centre for Computer Science and Department of Mathematics University of Turku, FIN 20520, Finland E-mail: ipetre9cs.utu.fi DAVID M. PRESCOTT University of Colorado Department of Molecular, Cellular and Developmental Boulder CO 80309-0347, USA E-mail: prescotdOspot. Colorado . edu

Biology

GRZEGORZ ROZENBERG Leiden Institute of Advanced Computer Science Leiden University Niels Bohrweg 1, 2333 CA Leiden, The Netherlands E-mail: rozenberQliacs.nl Ciliates (an ancient group of single cell organisms) have two sorts of nuclei with different functionalities: the micronucleus and the macronucleus. After the cell mating the micronuclear genes are converted into the macronuclear genes in the process called gene assembly. This is one of the most complex examples of DNA processing known in any organisms, and it is fascinating from the computational point of view. This paper continues the investigation of gene assembly in the framework of three molecular operations: Id-excision, hi-excision/reinsertion, and dlad-excision/reinsertion. In general, for a given micronuclear gene there exists many strategies for using these three operations to accomplish gene assembly. Since it is not known yet which strategies are actually used by ciliates, it is important to study the invariants of gene assembly, i.e., those properties of gene assembly that are common to all these strategies. A macronuclear gene (before its excision and capping with telomeres) can be assembled either in a linear or in a circular molecule. We prove in this paper that the circularity property (whether or not a given gene will be assembled in a circular molecule) is an invariant. We give a simple decision algorithm for the circularity property, and discuss a number of other related invariants.

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1

Introduction

Ciliates are an ancient group of organisms (their origination is estimated at about two billion years ago), comprising at least ten thousand genetically different organisms. They have evolved one of the most intricate DNA processings in living organisms. Each ciliate has two nuclei: the micronucleus - essentially used for storing the genetic information, and the macronucleus - used to provide the RNA transcripts to operate the cell. When ciliates are starved, they proceed to sexual reproduction (or make cyst, or... die), and in this process, after cell mating, a micronuclear genome is transformed into a macronuclear genome for the new cell. The DNA in the micronucleus is hundreds of thousands of base pairs long, with genes occurring individually or in groups, dispersed along the DNA molecule, and separated by stretches of spacer DNA, while the DNA molecules in the macronucleus are gene-size, on average about 2000 base pairs long. The gene assembly process, leading from the micronuclear to the macronuclear form of a gene is very interesting from the computational point of view. One of the amazing features of this process is that ciliates apparently know the linked list data structure and use it in a very elegant pattern matching process. Computational aspects of gene assembly were brought to the attention of DNA computing community by L.Landweber and L.Kari - see, e.g., [5] and [6]. They have proposed a set of operations for intermolecular processing in ciliates, and they have investigated the computational power of these operations (in the sense of computability theory) - e.g., they have shown that their operations are Turing-universal. A different set of operations was postulated in [4] and [11]. The processing model based on these operations is intramolecular, and the investigation of this model has focused on the process of gene assembly itself- see, e.g., [1], [2], [3], [4], and [11]. The usefulness of this model has been demonstrated by applying it to all known cases of gene assembly in ciliates, obtaining in this way a uniform explanation of all cases. Moreover, it has been proved in [2] that each micronuclear gene can be assembled using three molecular operations postulated by this formal system. Prom the theoretical point of view, the gene assembly process is nondeterministic: there may be many strategies to assemble its macronuclear version. Since it is not known yet what specific strategies ciliates use in gene assembly, it is important to study invariants of these strategies, i.e., properties that hold for all of them. The assembly process may produce the gene (before it is excised, and telomeres are added at the ends) either in a linear or in a circular molecule (see [4]).

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We prove in this paper that this ending in either a linear or a circular molecule is an invariant (of all assembling strategies), and we provide a simple decision algorithm for the circular vs. linear property. Moreover, we prove a number of other invariants, mostly concerned with "the fate" of IESs during the assembly process - IESs are multiple, non-coding segments that interrupt coding segments of micronuclear genes. 2

D N A Molecules

We recall in this section some basics about DNA molecules. The reader is referred to [7] and [8] for more background. DNA molecules are polymers composed of simple monomers called nucleotides. Each nucleotide consists of three components: a pentose sugar molecule, a phosphate group and a base. The sugar molecule has five points (carbons) where other molecules can attach to, numbered 1' through 5'. The phosphate group is attached to 5' (carbon) and the base is attached to 1' (carbon). There are 4 different bases (adenine, cytosine, guanosine, and thymine) denoted by A, C, G, T] since two nucleotides may differ only in their bases, there are four possible nucleotides, also denoted by A, C, G, T. Two nucleotides can join together through a strong (covalent) bond which binds the 3' of one with the 5' of the other (through the phosphate on the 5', and the hydroxyl group on the 3'). A single stranded DNA molecule is a strand of nucleotides connected in this way. In such a strand, the nucleotide at one end of the strand has the 5' attachment point free (i.e., available for bonding), while the nucleotide at the other end of the strand has the 3' attachment point free. Since the chemical properties of these ends are very different, a single stranded DNA molecule has polarity: one can read it from the 5' end to the 3' end, or from the 3' end to the 5' end. Since the 5' - 3' direction is "preferred" by nature, the single stranded DNA molecules are read in the 5' — 3' direction. For example, TGCATC denotes the single stranded DNA molecule with the nucleotide T at its 5' end and the nucleotide C at its 3' end. Hence, single stranded DNA molecules are denoted by strings over the alphabet {A, C, T, G} of nucleotides, where the left-to-right reading orientation of a string corresponds to the 5' — 3' polarity of the denoted molecule. Nucleotides can also bind through their bases by forming weak (hydrogen) bonds. However, such bonds may be formed only between complementary nucleotides: A with T, and C with G (this is known as the Watson-Crick complementarity). Double stranded DNA molecules are formed when two single stranded, complementary DNA molecules bind through their bases. Such a bond may be formed only if the joined single stranded molecules are of opposite polar-

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ity, meaning that the first nucleotide at the 3' end of one strand binds to the first nucleotide at the 5' end of the other strand, the second nucleotide at the 3' end of one strand binds to the second nucleotide at the 5' end of the other strand, etc. The same alphabet of nucleotides {A, C, G, T} is used to denote the double stranded DNA molecules, by simply writing the strings denoting the two strands of a given DNA molecule above each other. In this notation, the upper string is in 5' — 3' direction, and the lower string is in the 3' — 5' direction. For example, TACGA denotes the double stranded molecule such that one of its strands is TACGA and the other is TCGTA. The molecule in our example is "perfect", in the sense that all nucleotides in one strand have a "matching" nucleotide in the other strand. There are also imperfect molecules, e.g., T(~,pAr,

is such a molecule - it has the end CAG

with missing complementary nucleotides in the other strand. Such an end is called sticky: other molecules with the sticky end complementary to CAG can bind to this one by forming the hydrogen bonds between the sticky ends. For a single stranded DNA molecule a, the molecule that binds to it to form a perfect double stranded DNA molecule is the inversion of a, denoted by a. The inversion of a is obtained by taking the Watson-Crick complement of the mirror image of a. For example, for the single stranded DNA molecule ATCGA, its inversion is TCGAT. In the notation for the double stranded DNA molecules, the upper string denotes one of the strands in the 5' — 3' direction. Hence, the same molecule can be written in two ways depending on which strand is denoted by the upper ACCTC1 CACGT string. For example, both m^,^, A/-, and „mnn A denote the same molecule we say that „ „ „ „ . is the inversion of mp^Afi-

Hence, for a double string

a (representing a double stranded molecule), its inversion is obtained by first exchanging the two single strings of a for each other, and then taking the mirror CAOOT image of the resulting double string. Thus, e.g., m(~* + 1 ) an their inverses are denoted as M; = (i + 1,/Z^i). Thus, we use the alphabets A = {2,3,...} and A = {2,3,...} and we introduce II - A U A to denote the set of pointers. For any z € II, z is the partner of z, where z = z. Consequently, an MDS M = (p, /z, q) is denoted as (p, q), and its inverse M = ( ~P,P) as (