Fuzzy automata and languages: theory and applications

Author: John N. Mordeson | Davender S. Malik

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FUZZY AtTOMATA and

LANGUAGES

Theory and Applications

© 2002 by Chapman & Hall/CRC

COMPUTATIONAL MATHEMATICS SERIES Series Editor Mike J . Atallah

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

Mathematics of Quantum Computation Ranee K. Brylinski and Goong Chen

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

Forthcoming Titles Cryptanalysis of Number Theoretic Ciphers Samuel s. Wagstaff

© 2002 by Chapman & Hall/CRC

FUZZY AUTOMATA and

LANGUAGES

Theory and Applications John N. Mordeson Davender S . Malik

4ffl

CHAPMAN & HALL/CRC Boca Raton

© 2002 by Chapman & Hall/CRC

A CRC Press Company

London

NewYork Washington, D.C .

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

2002017475

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

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

© 2002 by Chapman & Hall/CRC

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

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

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

© 2002 by Chapman & Hall/CRC

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

© 2002 by Chapman & Hall/CRC

CONTENTS Preface

v

Authors

ix

List of Symbols

xvii

1

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

2

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

© 2002 by Chapman & Hall/CRC

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

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

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

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

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

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

CONTENTS

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91 91 95 98 102 108 116 120 125

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

CONTENTS 5.16 5.17 5.18 5.19

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

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

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

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

6

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

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

7

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

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

8

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

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

© 2002 by Chapman & Hall/CRC

xiv

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

9

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

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

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

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

CONTENTS

xv

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

© 2002 by Chapman & Hall/CRC

529

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

E

C C _D 01 U

n

A\B A A x B

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

PDA rp

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

© 2002 by Chapman & Hall/CRC

xviii

List of Symbols

S2" 0* Qi + Qz Al x AZ ti

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

© 2002 by Chapman & Hall/CRC

List of Symbols ffsm E(M)

p . 237 p . 238 p . 239

11-11

E(M) xM SM ~ x >Ml <_ M2 MIWM2 Ml o M2 Ml - M2 M/7r = Ml ~_ M2 ® MI ~_,

M2

-A

0 FR(X") fqfa L (G) L(c) _~ ffa QlQ2 ~Q 1

FG pN°

© 2002 by Chapman & Hall/CRC

p . 239 p . 240 p . 240 p . 240 covering of Ml by M2, p . 253 cascade product, p . 259, 294 wreath product, p . 261 Cartesian composition, p . 289 quotient fuzzy finite state machine, p . 298 p . 304 strong covering of Ml by M2, p . 305 direct sum, p . 305 sum, p . 305 weak covering of Ml by M2, p. 310 p . 341 p . 356 p . 367, 393, 494 family of F-regular languages, p . 368 finite quasi-fuzzy automata, p . 384 regular fuzzy language, p . 384 L(c) = {x I x E X", tt(x) > c}, p . 385 p . 394 p . 394 p . 404 p . 406 p . 406 p . 482 . 500 end of proof

Chapter 1

Introduction 1 .1

Sets

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

I

x E S, P(x)}

or A= {x E S

I P(x)}

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

2

1 . Introduction

Definition 1.1.1 The union of two sets A and B, written AUB, is defined to be the set AUB={x I xEAorxEB} . Definition 1.1.2 The intersection of two sets A and B, written A n B, is defined to be the set An B={x I xEAandxEB} . The union and intersection for any finite number of sets can be defined in a similar manner . That is, suppose that A1, A2 , . . . , An are n sets. The union of Al , A2, . . . , An, denoted by UZ 1AZ or A1 U A2 U . . . U An , is the set of all elements x such that x is an element of some AZ, where 1 <_ i <_ n . The intersection of Al, A2 , . . . , An , denoted by n2 ,AZ or A1 n A2 n . . . n An , is the set of all elements x such that x E AZ for all i, 1 <_ i <_ n . We say that a set I is an index set for a collection of sets A if for any i E I, there exists a set AZ E A and A = {AZ I i E I} . The union of the sets AZ, i E I, is defined to be the set {x I x E AZ for at least one i E I} and is denoted by UZEIAZ . The intersection of the sets AZ , i E I, is defined to be the set {x I x E AZ for all i E I} and is denoted by njEZAj . Definition 1.1.3 Given two sets A and B, the relative complement (or set difference) of B in A, written A\B, is the set A\B={x I xEA, but x~B} . We sometimes write A for the relative complement of a set A in the universal set . If A and B are sets, we let A x B denote the Cartesian cross-product of A and B, i.e., A x B is the set of all ordered pairs (a, b), where a E A and b E B . We write An for the set of all ordered n-tuples of elements from A, where n E N . We let P(X) denote the power set of a set X, i .e., P(X) is the set of all subsets of X. 1 .2

Relations

An important and fundamental concept in mathematics is the notion of a relation. Definition 1.2.1 A binary relation or simply a relation R from a set A into a set B is a subset of A x B. Let R be a relation from a set A into a set B. If (x, y) E R, we sometimes write xRy or R(x) = y. If A = B, then R is called a binary relation on A. © 2002 by Chapman & Hall/CRC

1.2. Relations

3

Definition 1.2.2 Let R be a relation from a set A into a set B. Then the domain of R, written Dom(R), is defined to be the set

{x I x E A and there exists y E B such that (x, y) E R} . The image of R, written Im(R), is defined to be the set {y I y E B and there exists x E A such that (x, y) E R} .

Definition 1.2.3 Let R be a binary relation on a set A. Then R is called (1) reflexive if for all x E A, xRx, (2) symmetric if for all x, y E A, xRy implies yRx, (3) transitive if for all x, y, z E A, xRy and yRz imply xRz .

Definition 1.2.4 A binary relation E on a set A is called an equivalence relation on A if E is reflexive, symmetric, and transitive .

Definition 1.2.5 Let E be an equivalence relation on a set A. For all x E A, let [x] denote the set

[x] = {y E A I yEx} . The set [x] is called the equivalence class (with respect to E) determined by x.

We prove some basic properties of equivalence classes in the following theorem. Theorem 1.2.6 Let E be an equivalence relation on the set A. Then the following properties hold: (1) for all x E A, [x] :?~ 0, (2) for all x, y E A, if y E [x], then [x] = [y], (3) for all x, y E A, either [x] = [y] or [x] n [y] = 01, (4) A = UxEA[x], i.e ., A is the union of all equivalence classes with respect to E . Proof. (1) Let x E A. Since E is reflexive, xEx . Hence x E [x] and so (2) Let y E [x] . Then yEx and by the symmetric property of E, xEy. Let u E [y] . Then uEy. Since uEy and yEx, the transitivity of E implies that uEx. Hence u E [x] . Thus [y] C_ [x] . Now let u E [x] . Then uEx. Since uEx and xEy, uEy by transitivity and so u E [y] . Hence [x] C [y]. Consequently, [x] = [y] . (3) Let x, y E A. Suppose [x] n [y] z,4 01. Then there exists u E [x] n [y] . Thus u E [x] and u E [y] . By (2), [x] = [u] = [y] . © 2002 by Chapman & Hall/CRC

4

1 . Introduction

(4) Let x E A . Then x E [x] UxEA[x] C A . Hence A = UxEA[x] .

C_ UxEA[x] . 0

Thus A C_

UxEA[x] . Also,

One of the objectives of this section is to study the relationship between an equivalence relation and a partition of a set . We now turn our attention to partitions . Definition 1.2.7 Let A of A. Then P is called a

be a set and P be a collection of nonempty subsets partition of A if the following properties hold: (1) for all B, C EP, either B = C or B n C = 01,

(2) A = UBEVB . m

The next theorem follows easily from Theorem 1 .2 .6. Theorem 1.2 .8

Let E be an equivalence relation on the set P = fl x]

is a partition of

A.

I

x E

A.

Then

A}

0

Given an equivalence relation E on a set A, the set of all equivalence classes forms a partition of A by Theorem 1 .2.8. We now prove that corresponding to any partition, we can associate an equivalence relation . Theorem 1.2.9 Let P be A by for all x, y E A, xEy

a partition of the set A. Define a relation E on if there exists B E P such that x, y E B . Then E is an equivalence relation on A and the equivalence classes are precisely the elements of P .

Proof. Since P is a partition of A, A = UBEVB . First we show that reflexive . Let x be any element of A. Then there exists B E P such that x E B . Since x, x E B, we have xEx . Hence E is reflexive . We now show that E is symmetric. Let xEy . Then x, y E B for some B E P . Thus y, x E B and so yEx . Hence E is symmetric. We now establish the transitivity of E . Let x, y, z E A. Suppose xEy and yEz . Then x, y E B and y, z E C for some B, C E P. Since y E B n C, B n C :?4 01 . Also, since P is a partition and B n C :?4 01, we have B = C so that x, z E B . Hence xEz . This shows that E is transitive. Therefore, E is an equivalence relation . We now show that the equivalence classes determined by E are precisely the elements of P. Let x E A. Consider the equivalence class [x] . Since A = UB E pB, there exists B E P such that x E B . We show that [x] = B . Let u E [x] . Then uEx and so u E B since x E B . Thus [x] C_ B . Also, since x E B, we have yEx for all y E B and so y E [x] for all y E B . This implies that B C_ [x] . Hence [x] = B . Finally, note that if C E P, then C = [u] for all u E C. Thus the equivalence classes are precisely the elements of P. m E is

© 2002 by Chapman & Hall/CRC

1.2 . Relations

5

The relation E in Theorem 1 .2 .9 is called the equivalence relation on A induced by the partition P . Given a relation R from a set A into a set B and a relation S from B into a set C, there is a relation T from A into C that arises in a natural way as follows : (a, c) E T for a E A and c E C if and only if there exists b E B such that (a, b) E R and (b, c) E S. This relation T is called the composition of R and S and is denoted by S o R. More formally we have the following definition. Definition 1.2.10 Let R be a relation from a set A into a set B and S be a relation from B into a set C. The composition of R and S, denoted by S o R, is the relation from A into C defined by x(S o R)z if there exists y E B such that xRy and ySz for all xEA, zEC.

It is important to note that composition is associative . Let R be a relation on a set A. We define recursively relations R' on A, n E N, as follows : Let Rl = R.

Suppose Rn has been defined for n E N. Define Rn+1 = R o Rn .

Definition 1.2.11 Let R be a relation from a set A into a set B. The inverse of R, denoted by R-1 , is the relation from B into A defined by xR - 'y if yRx for all xEB, yEA .

We are now interested in defining a lattice . Definition 1.2.12 A relation R on a set S is called a partial order on S if it satisfies the following conditions : (1) (x, x) E R for all x E S (i.e., R is reflexive. (2) V'x, y E S, if (x, y) E R and (y, x) E R, then x = y (i .e ., R is

antisymmetric) . (3) V'x, y, z E S, if (x, y) E R and (y, z) E R, then (x, z) E R (i.e ., R is transitive. A partial on a set S is usually denoted by < . We often write x <_ y rather than (x, y) E < . Let R be a partial order on S. Then R is called a total order or linear order if b'x, y E S, either (x, y) E R or (y, x) E R. © 2002 by Chapman & Hall/CRC

6

1 . Introduction

Definition 1.2 .13 A set S together with a partial order is called a partially ordered set (poset). A set S together with a total order is called a totally ordered set or a chain. Definition 1.2 .14 Let (S, <) be a poset and {x, y} C_ S. Then z E S is called an upper bound of {x, y} if x <_ z and y < z . An element w E S is called a lower bound of {x, y} if w < x and w < y . Definition 1.2 .15 Let (S, <) be a poset and {x, y} C_ S. Then z E S is called a least upper bound of {x, y} if z is an upper bound of {x, y} and if z' E S is an upper bound of {x, y}, then z <_ z' . An element w E S is called a greatest lower bound of {x, y} if w is a lower bound of {x, y} and if w' E S is a lower bound of {x, y}, then w' < w. Definition 1.2 .16 Let (L, <) be a poset. Then (L, <) (or simply L) is called a lattice if for every x, y E L, x and y have a least upper bound, denoted x V y, and greatest lower bound, denoted x A y . A lattice (L, <) is called distributive if x A (y V z) = (x A y) V (x A z) for all x, y, z E L.

1 .3

Functions

In this section, we review some of the basic properties of functions. Definition 1.3 .1 Let A and B be nonempty sets . A relation f from A into B is called a function (or mapping) from A into B if (1) Dom(f) = A and (2) for all (x, y), (x', y') E f, x = x' implies y = y' . When (2) is satisfied by a relation f, we say that f is well defined or single-valued. We use the notation f : A ~ B to denote a function f from a set A into a set B. For (x, y) E f, we usually write f(x) = y and say that y is the image of x under f and x is a preimage of y under f. If f is a relation from A into B such that (2) of Definition 1 .3 .1 holds, but (1) is not required, then f is called a partial function from A into B . Definition 1.3 .2 Let f be a function from a set A into a set B . Then (1) f is called one-one if for all x, x' E A, f (x) = f(x') implies x = x' . (In this case, f is sometimes called injective.) (2) f is called onto B (or f maps A onto B) if Im(f) = B. (In this case, f is sometimes called surjective .) (3) If f is both injective and surjective, it is called b ective .

© 2002 by Chapman & Hall/CRC

1.3. Functions

7

Let A be a nonempty set. The function i : A ~ A defined by i(x) = x for all x E A is a one-one function of A onto A. i is called the identity map on A. Let A, B, and C be nonempty sets and f : A ~ B and g : B ~ C. By Definition 1 .2 .10, the composition o of f and g, written g o f, is the relation from A into C defined as follows : g o

f

=

I x such that

{(x, z)

E A, f (x)

z E C, there exists y = y and g(y) = z} .

E

B

Let f : A----> B and g : B ~ C and (x, z) E g o f , i .e ., (g o f) (x) =z . Then by the definition of composition of functions, there exists y E B such that f (x) = y and g(y) = z. Now z = g(y) = g(f(x)) . Hence (g o

f ) (x) =

g (f (x)) .

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

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

8

1 . Introduction

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

Fuzzy Subsets

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

of X is a function of X into the closed

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

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

c} .

We call

gc

c-cut.

a

> 0}.

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

Let

/t

!,(x)

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

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

=

= =

1 - ft(x),

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

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

© 2002 by Chapman & Hall/CRC

tt n v,

inter-

1.5. Semigroups

9

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

=

=

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

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

=

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

(UiEIN'i)( x )

=

N'1 (x)

. . nll n

and V

w2 (x)

V .. . V

wn(x),

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

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

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

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

1 .5

Semigroups

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

10

1 . Introduction

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

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

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

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

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

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

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

© 2002 by Chapman & Hall/CRC

1.5. Semigroups

11

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

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

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

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

© 2002 by Chapman & Hall/CRC

12

1 . Introduction

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

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

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

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

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

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

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

S.

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

1.6. Finite-State Machines 1 .6

13

Finite-State Machines

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

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

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

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

f (qi, b)

= = = =

qo qi qi qi

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

= = = =

0 1 1

0

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

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14

1 . Introduction

obtained.

b/0 Figure 1.1 If S is a set such that 1 <_ ~S1 < oo, let S* denote the free monoid generated by S with operation, concatenation. The elements of S* are called strings or words. The empty string, A, is identity of S* . Let S+ _ S* \{A} . Definition 1.6 .4 Let M = (Q, X, Y, f, g, q) be a finite-state machine. An input string for M is a string over X. The string y1' . .4Jn

is the output string for M corresponding to the input string

if there exists states qo . . . qn qo qZ

= =

yZ

=

E

Q such that

q f (qZ-1 , x2) g(qz-1, x2)

for for

n2 n.

Example 1 .6 .5 We now determine the output string corresponding to the input string (read from left to right aababba

(1 .1)

for the finite-state machine of Example 1 .6 .2 . We start in the state qo . The symbol a is inputted and a 0 is outputted. We stay at qo . The symbol a is again inputted and once again we remain in state qo . Next the symbol b is inputted, 1 is outputted, and the machine goes to state q1 . Continuing this process until the last a is inputted, we obtain 0011001 as the corresponding output sequence .

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1.7. Finite-State Automata

1 .7

15

Finite-State Automata

A finite-state automaton is a special kind of finite-state machine. It is of special interest due to its relationship to languages as shown in Section 1.10. Definition 1.7.1 A finite-state automaton A = (Q, X, Y, f, g, s) is a finite-state machine such that the set of output symbols Y is {0,1} and where the current state determines the last output. Those states for which the last output was 1 are called accepting states . Example 1 .7.2

The transition diagram of the finite-state machine A is defined by the table . The initial state is qo . A is a finite-state automation and the set of accepting states is {ql, q2} .

The transition diagram is shown in Figure 1 .2. If the finite-state machine is in state qo, the last output was 0 . If the machine is in either state q l or q2, the last output was 1 . Consequently, A is a finite-state automaton. The accepting states are ql and q2 .

a/0

Figure 1 .2

We see from Example 1.7.2 that the finite-state machine defined by a transition diagram is a finite-state automaton if the set of output symbols is {0,1} and if, for each state q, all incoming edges to q have the same output label. The transition diagram of a finite-state automaton is often drawn with the accepting states in double circles and with the output symbols omitted . © 2002 by Chapman & Hall/CRC

16

1 . Introduction

If we draw the transition diagram of Figure 1 .2 in this way, we obtain the transition diagram of Figure 1.3.

Figure 1 .3

In Figure 1.3, accepting states are in double circles and output symbols are omitted .

Example 1.7.3

We now draw the transition diagram of the finite-state automaton of Figure 1.4 as a transition diagram of a finite-state machine.

Figure

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1.4

1 .7. Finite-State Automata

17

a/0

Figure 1 .5

Since q2 is an accepting state, we label all its incoming edges with output 1 as in Figure 1 .5. Since the states qo and ql are not accepting, all their incoming edges are labelled with output 0 . The transition diagram of Figure 1 .5 is obtained .

An alternative to Definition 1 .7 .1 can be obtained by regarding a finitestate automaton A as consisting of (1) a finite set Q of (2) (3)

states,

a finite set X of input symbols, a next-state function f from Q x X into

(4) a subset A of Q of accepting (5) an initial state q E Q.

states,

In this case, we write A = (I, S,

f, A, q) .

Example 1.7.4 The transition diagram of the finite-state automaton A = (Q, X, f, A, g), where Q = {go, qi, q2}, X = {a, b}, A = {q2}, q = qo, and f is given by the following table,

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18

1 . Introduction

is shown in Figure 1 .6.

Figure 1 .6

If a string is input to a finite-state automaton, we end at either an accepting or a nonaccepting state . The status of this final state determines whether the string is accepted by the finite-state automaton . Definition 1 .7.5 x = xl . . . xn be a (1)

Let A = (Q, X, f, A, q) be a finite-state automaton. Let string over X . If there exists states qo, . . . , q,, satisfying

qo = q ;

n; f (qZ-i, xi) = qZ for (3) qn E A ; then x is said to be accepted by A . The empty string is accepted if and only if q E A . Let Ac(A) denote the set of strings accepted by A . Then A is said to accept Ac(A) . (2)

Let x = xl . . . x n be a string over X . Define states qo, . . . , q n by conditions (1) and (2) above. Then the (directed) path (qo, . . . , qn ) is called the path representing x in A . It follows from Definition 1 .7.5 that if the path P represents the string x in a finite-state automaton A, then A accepts x if and only if P ends at an accepting state . The next two examples illustrate design problems . Example 1 .7.6

We design a finite-state automaton that accepts precisely those strings over {a, b} that contain no a's. We use two states : A : There exists an a . B : There does not exist an a .

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1 .7. Finite-State Automata

19

The state B is the initial state and the only accepting state . The finitestate automaton in Figure 1 .7 correctly accepts the empty string.

Figure 1 .7 Example 1 .7 .7 We design a finite-state automaton that accepts precisely those strings over {a, b} that contain an odd number of a's . We use two states : E : An even number of a's are in the string. O : An odd number of a's are in the string . The initial state is E and the accepting state is O . We obtain the transition diagram shown in Figure 1 .8.

Figure 1 .8 A finite-state automaton is essentially an algorithm to decide whether or not a given string is accepted. Definition 1 .7.8 Finite-state automata A and A' are called equivalent if Ac(A) = Ac(A') . Example 1 .7 .9 It can be shown that the finite-state automata of Figures 1 .7 and 1 .9 are equivalent.

Figure 1 .9

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20 1 .8

1 . Introduction Languages and Grammars

A derivation within a grammar proceeds as follows : Starting with the current string with the starting symbol s, we search the productions for a rule whose left-hand side x is a substring of the current string. Any rule that is matched in this manner can be applied by replacing the substring with the right-hand side y of the rule. A derivation is complete when there are only terminal symbols in the current string . Grammars are classified according to the form of the production rules used (Chomsky hierarchy) . These grammars are sometimes described as follows : Unrestricted or type 0 grammars have no restrictions on the form of the rules. Context-sensitive or type 1 grammars require that if x ~ y is a production, then the length of the string x, denoted by Ixl , must be not greater than the length jyj of the string y . A grammar is said to be context-free or type 2 if every production is of the form A ~ y, where A E N. A grammar is said to be regular (left linear) if all its productions are of the form A ~ B, A ~ b, or A ~ bB, where A, B E N and b E T . It is sometimes useful to write a grammar in a particular form. Two forms for context-free grammars have been commonly used: Chomsky normal form and Greibach normal form. A context-free grammar is said to be in Chomsky normal form if every production rule is one of the form A ~ BC or A ~ a, where A, B, C E N and a E T. Also, if A E L(G), the language generated by G, then s ~ A is a production and s does not appear on the right-hand side of any production . A context-free grammar is said to be in Greibach normal form if every production rule is of the form A ~ az, where A E N, a E T, and z E N* . Also, if A E L(G), then s ~ A is a production and s does not appear on the right-hand side of any production. Definition 1.8.1 Let A be a finite set. A (formal) language L over A is a subset of A*, the set of all strings over A. Example 1.8.2 Let A = {a, b} . The set L of all strings over A containing an odd number of a's is a language over A. As we saw in Example 1.7.7, L is precisely the set of strings over A accepted by the finite-state automaton of Figure 1.8.

One method to define a language is to give a list of rules that the language must obey. Definition 1 .8.3 A phrase-structure grammar or type 0 grammar (or, simply, grammar) G is a 4-tuple G = (N, T, P, s), where (1) N is a finite set whose elements are called nonterminal symbols, (2) T is a finite set whose elements are called terminal symbols, where

NnT=0, (3) P is a finite subset of [(N U T)*\T*] x (N U T) *, called the set of

productions,

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1.8 . Languages and Grammars

21

(4) s E N and is called the starting symbol.

Example 1.8.4 Let N = {s, S}, T = {a, b}, and P = {s --+ bs, s ---> aS, S ---+ bS, S i b} . Then G = (N, T, P, s) is a grammar.

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

z2

zn.

We call zl

z2

~> zn

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

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

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

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22

1 . Introduction

The only derivations from s are given as follows: s

==~>

bs

bnab--1 S bnabm

n > 0,

m > 1.

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

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

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

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

A E N,

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

(1 .3)

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

xET,

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

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

1.8 . Languages and Grammars

23

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

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

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

A ~ aAC, CD , CE,

s aB, D b, Cc ~ Dcc,

A ~ aC, CE , DE,

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

aAB aaaDCCc aaaDDDccc

aaACB aaaDCDcc aaabbbccc.

aaaCCDc aaaDDCcc

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

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

fanbnCn

In= 1,2. . . .

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

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24

1 . Introduction

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

N = {s},

with productions s ~ asb,

s ~ ab

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

==~>

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

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

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

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

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

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

Nondeterministic Finite-State Automata

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

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1.9 . Nondeterministic Finite-State Automata

25

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

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

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

O - bO .

(1 .4)

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

(1 .5)

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

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

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

The derivation s ==~> A

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

(1 .6)

26

1 . Introduction

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

~

xiS2

for i = 2, . . . , n.

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

xl . . . xnSn xl . . . xn,

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

N = {s, q}

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

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1.9 . Nondeterministic Finite-State Automata

27

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

s~aq,

q~bq

give the graph shown in Figure 1.10.

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

FAA,

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

s ~ aq,

q - bq,

q - bF

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

Figure 1.11

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

© 2002 by Chapman & Hall/CRC

28

1 . Introduction

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

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

X = {a, b},

. A={F}

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

f

a {q}

{s}

Ql

Ql

{q, F}

0

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

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

A = {q, p}

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

8

q p

-7{s,

f

q}

{q, p}

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

Figure 1 .1 2

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1.9. Nondeterministic Finite-State Automata

29

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

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

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

The string x = abba is not

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

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

30

1 . Introduction

1 .10

Relationships Between Languages and Automata

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

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

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

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

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

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

UQex f (q, x) = Y.

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1 .10. Relationships Between Languages and Automata

31

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

{s, q, F}, {F}

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

b Figure 1 .14 Example 1 .10 .2

The finite-state automaton equivalent to the nondeter-

© 2002 by Chapman & Hall/CRC

32

1 . Introduction

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

Figure 1.15

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

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

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

A.

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

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

for i = 1, . . . , n;

Set Yo = {qo} and YZ

= f'(YZ-1, x2)

for i = 1. . . . , n.

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

1.10. Relationships Between Languages and Automata we have that

q1

33

E Y1 . Now

q2

E

f(ql, x2)

C Uscyl f (S, x2)

= P(Yl, x2) =

Y2

q3

E

f(q2, x3)

C

= P( Y2, x3) =

Y3 .

Also, USEY2 f (S, x3)

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

= P(YO,xl)

f'(q',xl)

P(Yl, x2)

=

Y2

=

Y1

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

and there exists a state Since qn

there exists

qn- 1

qn-1

= =

Yo (Ya-1, xa)

E Yn

qn

E Yn n

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

n,

,A.

= f (Yn-1, xn)

E Yn-1 such that

E Yn-1

=

s, Yi

qn

= E

USEY,-1 f(S, xn),

f(qn-1,

x n ). Similarly, since

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

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

qn-1

E

f(qn-2, xn-1).

for i = 0, . . . ,

E YZ

Continuing in

n,

such that qZ

E

f (qZ-1,

xi)

for i = 1. . . . , n .

In particular, qo

E Yo = {s} .

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

34

1 . Introduction

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

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

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

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

© 2002 by Chapman & Hall/CRC

1 .10. Relationships Between Languages and Automata

35

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

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

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

© 2002 by Chapman & Hall/CRC

36

1 . Introduction

obtained.

a Figure 1.18

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

1 .11

Pushdown Automata

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

1.11 . Pushdown Automata

37

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

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

38

1 . Introduction

Definition 1.11 .2

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

y) .

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

(q, x, y)

-W (q, x, yu) .

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

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

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

language recognized

© 2002 by Chapman & Hall/CRC

by M is the set

of

accepted strings .

1 .11 . Pushdown Automata

39

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

Program of M,,. 11

21 31 41 51 61

scan write write scan read read

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

State Diagram of Load Stack

Mc. Compare

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

© 2002 by Chapman & Hall/CRC

40

1 . Introduction

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

s s s s s

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

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

(6, abcb, ab)

(5, abcba, a)

R ~ R >

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

[accept.

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

S

_~

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

[stop and reject.

Thus Lcm

=

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

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

S~A A aAa A bAb A c.

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

1 .11 . Pushdown Automata

41

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

L rZ =

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

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

2] 3] 4] 5] 6]

scan write write scan read read

Load Stack

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

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

Compare

Figure 1 .20 Example of a nondeterministic pushdown acceptor.

© 2002 by Chapman & Hall/CRC

42

1 . Introduction

1 .12

Exercises

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

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

LZ

AZ =

f ((qi, q2),

x)

=

s

=

Show that A,(A) = 9. Let

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

=

E ,A2},

Ll rl L2 .

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

LZ

AZ =

f((gi,q2),x)

= =

s Show that AJA) =

Ll

= U

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

E ,A2},

L2-

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

L = {anbnck I

G'

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

1.12. Exercises

43

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

aabaabbb

in Example

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

M,Z

Ll

L is

and

L2

a regular

of Example 1.11.4 for

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

© 2002 by Chapman & Hall/CRC

L be a context-free L, ~zj >_ n implies 1, vwx j <_ n, and

Chapter 2

Max-Min Automata 2 .1

Max-Min Automata

This chapter is based on the results of [193] and [203] . A general formulation of automata is given which is similar to that of sequential machines introduced in [212] . We show how deterministic, nondeterministic, and probabilistic automata fit into this formulation . The max-min automata include both deterministic and nondeterministic automata as special cases . Moreover, its state transition function as well as initial distribution may be interpreted as grade of membership functions of fuzzy subsets . Despite its generality, most of the basic concepts of existing automata, e.g., equivalences, reduction, behaviors, etc. may be carried over to maxmin automata with appropriate modifications . Moreover, results related to these basic concepts can be generalized to max-min automata . There is a strong resemblance between max-min automata and probabilistic automata [165] . Hence one can introduce similar concepts definable for probabilistic automata, but not present in deterministic and nondeterministic automata. Recall that effective procedure means algorithm. It is within this context that we use the terminology "effectively constructed ."

2 .2

General Formulation of Automata

Definition 2.2.1 A

pseudoautomaton A is a 5-tuple (Q, X, /t, F, a) such that the following properties hold: (1) Q is a finite nonempty set (states), (2) X is a finite nonempty set (inputs), (3) /t is a function from Q x X x Q x T into [0,1] where T is a subset of R (state transition function with members of T representing time, 45 © 2002 by Chapman & Hall/CRC

46

2. Max-Min Automata (4) F is a subset of Q (final states, (5) a is a function from Q x T into [0,1] (initial distribution .

If T = N, the set of all natural numbers, then A is said to be discrete. If ft and a are independent of T, then A is said to be stationary. Elements of X are called input symbols . Finite sequences of input symbols are called input strings, words, or tapes. Recall that the empty tape A has the property that xA = x = Ax for every tape x . The collection of all input tapes is denoted by X* . Let x be a tape. Then lxl denotes the length of x. The tape y is a k-suffix of the tape x if x = zy for some tape z and jy j = k . Definition 2.2.2 An automaton A* is a 5-tuple (Q, X, tt*, F, a) where Q, X, F, and a are the same as that given above. /t* is the same as ft given in Definition 2.2 .1 with X* replacing X .

To each automaton A* there is associated in a natural manner a pseudoautomaton A, i.e., ft is the restriction of /t* to Q x X x Q x T. A pseudo automaton is said to be of class C(C) if ft and a satisfy a set C of constraints . Definition 2.2.3 If the pseudo automaton A associated with the automaton A* is of class C(C) and if A* can be obtained from A by a rule of extension R, consistent with C, which extends ft uniquely into tt*, then A* is said to belong to class C(C, R) . 2 .3

Classes of Automata

We now present several classes of automata. 1 . Probabilistic automata C(Cp, RP) : The constraints in Cp are as follows : for every q E Q, u E X, and n E N, q'EQ

ft(q, u, q, n) = 1 and

E a(q , n) = 1.

q'EQ

The rule of extension Rp is as follows: /t* is defined recursively on lxl,

xEX*,

w* (q, A, q, n) ft * (q, xu, q, n) =

1 0

if q= q1 if q q1

1: ft * (q, x, q", n) ft (q", u, q, n + Ix 1) .

q"EQ

2. Deterministic automata C(CD, RD) : CD is the same as Cp with the additional constraints that Im(y) C {0,1} and Im(a) C_ {0,1} . RD is the same as RP.

© 2002 by Chapman & Hall/CRC

2.3. Classes of Automata

47

3. Max-min automata C(CA, RA) : CA is an empty set, i.e., there are no constraints . 1 * fI (q~A, q~n)={ 0 * [I (q, xu, q', n) = q EQ

x,

n {ft* (q,

if ' if q :?~ q'

q",

n), fI(q", u, q, n +

Ix I) }.

An interesting subclass of C(CA, RA) is the class of all max-min automata satisfying the constraints V [I(q, u, q , n) =land V a(q, n) = 1 q' EQ q' EQ

for all q E Q, u E X, and n E N. Automata belonging to this subclass are called restricted max-min automata. 4. Nondeterministic automata C(C,v , RN) : The only constraint in C,v is that Im(p) C {0,1} and Im(a) C_ {0,1}. RN is the same as RA . 5. Min-max automata C(CI, RI) : CI is empty. * fI (q, A, q~ n)

~

0 1

if ' if q :?~ q'

fI* (q, xu, q, n) = q , EQ V {fI*(q, x, q", n), tI(q", u, q, n + l xl )} .

6. Composite automata C(Cc, Rc) :Cc is empty. [IC =aft* + bpl, where a and b are nonnegative real numbers such that a + b = 1, and the subscripts C, A, and I refer to C (Cc, Rc), C (CA, RA), and C(CI, RI), respectively. 7. Max-product automata C(CT, RT) : CT is empty. 1

fI * (q~A, q~n)={ 0

w* (q,

if ' if q :?~ q'

xu, q, n) = , (q, x, q", n)tI(q ", u, q, n + lx l) . q EQ't

One may consider, in a similar manner, restricted nondeterministic, min-max, composite, and max-product automata. 8. Fixed automata C(Cx, Rx ) : Cx is empty . * [I (q, ux, q~, t) = ft (q, u, q, t)

© 2002 by Chapman & Hall/CRC

48 2 .4

2. Max-Min Automata Behavior of Max-Min Automata

In this and the following sections, we confine our discussion to stationary max-min automata . The symbol A*, with or without subscripts, always denotes a max-min automaton . All automata are assumed to have the same input set X . Thus an automaton is represented by (Q, tt*, F, a) with X being suppressed. We note that max-min automata include both deterministic and nondeterministic automata as special cases . The response function of A* is defined to be rp,(A*, x, q) = V{a(q) A w* (q , x, q) I q E Qj.

We simply write rp(x, q) rather than rp(A*, x, q) when A* is understood . If a(q) is concentrated at qk, i.e., a(qk) = 1 and a(q) = 0 for q :?~ qk, then * rft(x, q) = [t (qk, x, q) .

Let q E Q and 0 <_ c < 1 . Then q is called an accessible state of _ A* with threshold c if there exists x E X* such that rp(x,q) > c. Let Q, denote the set of all such states . The state q is an accessible state of A* if and only if for all 0 <_ c < 1, q is an accessible state of A* with threshold c. Let Q denote the set of all such states . Clearly, Q = n{Qc~ 0 < c < 1}. Theorem 2.4.1 If q is an accessible state of A* with threshold c, then there exists x E X* such that rp(x, q) > c and Ix1 < IQI . Proof. By hypothesis, there exists xo E X* such that rp(xo, q) > c. Let xo = UZl UZ2 . . . u2,, . If n < Qj, the result follows trivially. Thus assume n >_ IQ1. From the definition of rp, there exist qi l , qi l , q22, . . . , q2n such that a(gjo) > c and p* (q2,,+1, u2,, , q2,, ) > c for k = 1, 2, . . . , n, where Since n >_ I Q I , at least two of qz o , q2. . . . . . qi,, are equal, say qj,, = q. qz, = qi,, where j < l . Let xl = u2 1 . . . u2j uil+1 . . . u2 . Then rp(xl, q) > c. If Ix11 > IQI, repeat the same argument. In this manner, the desired x is obtained . Theorem 2.4.2 If q is an accessible state of A*, then there exists x E X* such that rp(x, q) = 1 and IxI < IQI . Proof. By Theorem 2.4.1, there exists xn E X* such that rtt(xn, q) > 1-(l/n) and x n I < QI for n = 1, 2 . . . . . Since the set of all x, where IxI < IQI , is finite, there exists a j such that rp,(xj , q) > 1 - (1/n) for infinitely many n. Thus rp(xj, q) = 1 . The above theorems provide a practical method for determining Qc and Q A generalization of the concept of accessible states is the concept of accessible initial distributions . An initial distribution a is said to be an © 2002 by Chapman & Hall/CRC

2.4 . Behavior of Max-Min Automata

49

accessible initial distribution of A* with threshold c if there exists x E X* such that for every q E Q, a(q) > c ~? ry(x, q) > c. a is an accessible initial distribution of A* if for every 0 <_ c < 1, a is an accessible initial distribution of A* with threshold c. a is called a strictly accessible initial distribution of A* if there exists x E X* such that a(q) = ry(x,q) for all q E Q. Clearly, if a is a strictly accessible initial distribution of A*, then a is an accessible initial distribution of A* . Example 2.4 .3 Let Q = {gi,q2} and X = {u} . Define ft : Q x X x Q [0,1] and a : Q ----> [0,1] as follows: ft (qi , ft (qi , ft(g2, ft(g2,

u, qi ) u, q2) u, qi) u, q2)

0 1

a(qj) a(q2)

Now rtt(A*, un, q2) = V{a(q) A ft(q, un, q2) I q' E Q} = 2 and rtt(A*, un, ql) 2 b'n E N. It follows that Q'_ = Q for 0 <_ c < z and < _ 01 for c < 1 . Hence ql and q2 are accessible with threshold c for QC 2 0
=

2~ 1,

rft(A * ,u n~gi) rft(A*, u~ : q2)

_ = 2.

is not accessible with threshold z . In fact, a is c for 0 <_ c < z, but not accessible with threshold a is not strictly accessible . a(q2) = 1, we define a(q2) = 2, then a is strictly

Theorem 2.4.5 If a is an accessible initial distribution of A* with threshold c, then there exists x E X* such that x1 < 21Q1 and for every q E Q, a(q) > c if and only if ry,(x, q) > c. Proof. The proof is similar to that of Theorem 2.4 .1 . m It is sometimes convenient to consider what are called in [193] max-min tables and max-min pre-tables . Max-min tables are max-min automata without an initial distribution . Max-min pre-tables are max-min tables without the final states . (We use different terminology in subsequent chapters .) They are well defined since the rule of extension RA does not depend

© 2002 by Chapman & Hall/CRC

50

2. Max-Min Automata

on the initial distribution nor the final states . The symbols 52* and 0*, with or without subscripts, are used to denote max-min tables and pre-tables, respectively. A max-min automaton having 52* as its max-min table and Q its initial distribution is denoted by (52*, Q) . If Q is concentrated at q, we also write (52*, q) for (52*, Q) . A max-min table having 0* as its pre-table and F its final states is denoted by (0*, F) . Let Q1 = (Q1, 1 F'1) and 522 = (Q2, /t2, F2), where Q1 n Q2 = 0 . The direct sum of 521 and 522 is a max-min table 521+522 defined by (Q, tt*, F), where Q = Q1 U Q2, F = Fl U F2 , and for every u E X, y1( ,q u, q~) ft2 (g~ u~ q~)

ft (q, u, q) =

0

if q, q~ E Q1, if q, q' E Q2, otherwise .

Let A1 = (Q1, ftl, Fl, al) and A2 = (Q2, ft2, F2, a2) . The direct product of A1 and A2 is the max-min automaton A1 x A2 defined by (Q, ft * , F, Q) where Q = Q1 x Q2, F = Fl x F2, and for every ql, ql E Q1, q2, q2 E Q2, x E X, ft((gl,q2),x,(qi,q2)) c,((gl,q2))

= =

ftl(gl,x,qi)nft2(q2lx,q2) ,,(gl) AQ2(g2) . c

The behavior of A* with threshold c is defined to be the set B(A*, c) = {x E X*

Theorem 2.4.6 B(A*, c) that

Ixl

< Q1-

I

rft(x, q) > c} . qEF V

Ql if and only if there exists x E B(A*, c) such

Proof. The proof follows from Theorem 2.4 .1

n B(A2, c) 01 if and only if there exists x E B(A2, c) such that I xl < ~Q1 I . IQ21 .

Theorem 2.4.7 B(A*, c) B(A*, c)

n

Proof. Let A* = A1 x A2 . The desired result follows from Theorem 2.4 .6 and the fact that x E B(A*, c) if and only if x E B(A*I, c) n B(A2, c) . Theorem 2.4.8_For every A* and 0 <_ c < 1, there exists a nondeterministic automaton A1- such that B(A*, c) = B(AC, 0) .

Proof. Let A* = (Q, y*, F, Q) and define A~c = (Q, ~*, F, ac) by b'q, q' E

Q and Vu E X,

f~~(g~ u, q) =

© 2002 by Chapman & Hall/CRC

1 0 ~

if if

ft (q, u ' >c ft (q, u, q~) < c

2.5. Equivalences and Homomorphisms of Max-Min Automata ac (g)

1 0

_

if if

51

a(q) > c a(q) < c .

Clearly, A, is nondeterministic . Moreover, B (A*, c) = B(A~, 0) . Let Y C_ X* . Then Y is said to be regular if Y = B(D*, 0) for some deterministic automaton D* . Theorem 2.4.9 For every A* and 0 < c < 1, B(A*, c) is regular. Proof. Let A* = (Q, /t*, F, a) and define D* = = P(Q), the power sets of Q,

QD

FD = {Q'EQD

and

I

Q'nF7~

(QD ,

yD, FD, aD)

where

Of

for every u E X, Q' C Q, Q" C Q,

ftD (Q

1,

u'

Q11)

1 0

aD(Ql)

if Q"= {gEQI/t(q',u,q)>cforsome otherwise j a(q) >_ otQ/rwi{e. E Q

_ { 0

q'EQ'}

c}

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

2 .5

Equivalences and Homomorphisms Max-Min Automata

of

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

I

VgEF ry(x, q) > c

and l xl < k} .

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

2.

x

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

© 2002 by Chapman & Hall/CRC

Otherwise, we write

52

2. Max-Min Automata 3.

A1

A1

x A2 .

k

cNk

A2 ~ A1

A2 for all 0 < c < 1 .

Otherwise, we write

4. A1 - A2 ~ A1 -c A2 for all 0 <_ c < 1 . Otherwise, we write A1 x A2 . The following lemmas are immediate consequences of the above definitions. Lemma 2.5 .1 A1 c A2 if and only if B(A*, c) = B(A2, c) . Lemma 2.5 .2 A1

k

A2 if and only if V rft(A*, x, q) =

for all x E X* where Ixl

qEF

< k. m

Lemma 2.5 .3 A1 - A2 if and only if V rft(A*, x, q) = qEF

for all xEX* . "

m

V rft(A2, x, q)

qEF

V rft(A2, x, q)

qEF

A* is said to be connected with threshold c if Q = Qc. A* is connected if for every 0 <_ c < 1, A* is connected with threshold c. Clearly, A* is connected if and only if Q = Q. A* is totally connected with threshold c if every initial distribution of A* is accessible with threshold c. A* is totally connected if for every c, A* is totally connected with threshold c. A* is strictly connected if every initial distribution of A* is accessible . Clearly, A* is totally connected if and only if every initial distribution of A* is accessible . Moreover, if A* is strictly connected, then A* is totally connected and connected. Example 2 .5 .4 Let A = (Q, X, ft, a), where Q = {qi, q2}, X = {u}, and ft : Q x X x Q [0,1] is defined as follows: ft(gi, u, q2) = 1 = ft(g2, u, qi) and ft (gi, u, qi) = 0 = ft (q2, u, q2) . Then ry,(A*, an, qi) = (ft* (qi, an, qi) A a(qi)) V (ft* (g2,an,qi) n a (q2)) = (1 n a(qi)) V (0 n a (q2)) = a(qi) and ry(A*, an, q2) = (ft* (qi, an, q2) n a(gi))V (ft* (q2 , an, q2) n a(q2)) = (0 n a(qi)) V (1 Aa(g2 )) = a(q2) . Hence a is accessible. Since a is arbitrary, A* is strictly connected. Theorem 2.5 .5 For every A* and 0 <_ c < 1, there exists a max-min automaton T that is connected with threshold c and is such that A* -c Ac . Proof. Let A* = (Q, ft*, F, a) . Define T = (Q c , y*, Fc, ac) as follows: Fc = Qc rl F and b'q, q' E Q and Vu E X, Tic (q, u, g ) _

ft(q, u, q~)

~0

if if

a(g) 0 Then T has the desired properties .

© 2002 by Chapman & Hall/CRC

m

if if

ft(q, u, q~) > c ft (q, u, q') < c

a(g) > c a(q) < c.

2.5. Equivalences and Homomorphisms of Max-Min Automata Theorem 2.5.6 ever

= 1 2

If (S2*, a) k - 1

c,

c't1

x

(S2*, a') and (S2*, a) c

there exist o

53

(S2*, a), then for

such that (S2* a .) c,jti

and a'

1

(S2*, a') and (S2*, o j) ° r; (Q* , aj) .

Proof. By hypothesis, there exists x o E X* such that Ixol = k and say V gE Qrp(Al*xo, q) > c, but VgE Qrp(A'*, xo, q) <_ c, where A* _ (Q * , a) and A'* = (S2*, a') . Let xo = UZlu22 . . . u2,k and for 1 <_ j <_ k - 1, let xi = UZ lu2 2 . . .UZ,,, where r = k- j. Define oj (q) = rp(A*,xj,q) and a'. (q) = rlt(A'*, xj, q) for all q E Q. Then the initial distributions aj and l a~ have the desired properties. Two initial distributions al and a2 are said to be indistinguishable with threshold c if for every q E Q, al(q) > c if and only if a2 (q) > c. In symbols, al c a2 . Clearly, al = a2 if and only if for every 0 < c < 1, al c a2 .

Given 1.

c,k

al

S2*

cNk

and initial distributions al and a2 define : cNk a2 if and only if (SZ*, al) (SZ*, a2) . Otherwise, we write

al ,w a2 . 2.

write

al - a2

al

x a2 .

if and only if

al

cNk

a2

for

k=0,1,2. . . . .

Otherwise, we

3. al - a2 if and only if al c a2 for every 0 <_ c < 1. Otherwise, we write al x a2 . If al and a2 are concentrated at ql and q2, respectively, then we also cNk cNk c c write ql q2, ql - q2, and ql - q2 for al a2, al - a2, and al - a2, respectively. Clearly, al c a2 implies al - a2 .

Theorem 2.5.7

If (S2*, al)

(S2*, a2) .

c'N

(S2*, a2), where m = 21Q1-1, then (S2*, al) N

Proof. Let H denote the set of all initial distributions . Let Pk and P denote, respectively, the equivalence classes of H under cNk and C . Clearly Pk <_ IPk+1 <_ P for every k. Since al c a2 implies al - a2, P is finite. In fact, IPI <_ 21Q1 = m+l. By the Theorem 2 .5 .6, if Pk = Pk+1 for some k, then Pk = Pk+j = P for all nonnegative integers j . Thus Pr = P +1 = P and the desired result holds. Corollary 2.5 .8

If (S2*, al) N (SZ*, a2) where m = 21QI - 1, then (SZ*, al)

Theorem 2.5.9

A1 -c A2 if and only if A1

- (SZ*, a2) . 0

c-T

A2, where r = 21QII+IQ21-1 .

Proof. We assume without loss of generality that and

Al = (Qi, a), A2 = (Q2, a2), Q* = Qi +Q2, 01(q) ai (q) _ ~

© 2002 by Chapman & Hall/CRC

if q E Q

2

Ql

n Q2

= 01 .

Let

54

2. Max-Min Automata a2 (q) _

02(q)

if q E

Ql

Since (52*, ai) - (521, a,) and (52*, a2) - (522, a2), the desired result follows from Theorem 2.5 .7 . Corollary 2.5.10 A1 - A2 if and only if A1 N A2, where r = 21Q11+IQ21 1.0 Let A1 = (Q2, [t2*, F2, a2) . A homomorphism from A1 onto A2 with threshold c is a function f from Q1 onto Q2 such that for every q', q" E Q1 and u E X, the following conditions hold: (1) ft, (q~, u, q") > c if and only if ft2(f(q~), u, f (q")) > c; (2) aI(q') > c if and only if a2 (f (q')) > c; (3) q E Fl implies f(q) E F2 . f is called a strong homomorphism with threshold c if f is a homomorphism with threshold c and q E Fl if and only if f (q) E F2 . f is called an isomorphism with threshold c if and only if f is a strong homomorphism with threshold c that is one-one . f is a homomorphism (strong homomorphism, isomorphism) if and only if for every 0 <_ c < 1, f is a homomorphism (strong homomorphism, isomorphism) with threshold c. The symbols --c-->, ==cz>, and ~ will denote homomorphic, strong homomorphic, and isomorphic with threshold c, respectively. The symbols ==~>, and <--~ are similarly defined . Theorem 2 .5 .11 A1

c

A2 implies A1 N A2 .

c

Proof. Clearly, B(A*, c) C_ B(A2, c) . Let x E B(A2, c) . Then there exists q E F such that rtt(A2, x, co(q)) > c. Since A1 A2, q E Fl . Thus x E B(A1, c) . Corollary 2.5.12 A1 ~ A2 implies A1 - A2 . m Theorem 2.5.13 If A1 is totally connected with threshold c and A1 A2, then A1 N A2 implies A1 ~ A2 .

Proof. Let qo E Fl and Q the initial distribution concentrated at qo . Since A1 is totally connected with threshold c, there exists x E X* such that ry,(A1, x, qo) > c and ry,(A2, x, f (q)) <_ c for q ~ qo . Moreover, x E B(A1, c) = B(A2, c) . Hence f (qo) E F2 . Corollary 2.5.14 If A1 is totally connected and A1 ~ A2, then A1 A2 implies A1 ===> A2 . m

© 2002 by Chapman & Hall/CRC

2.6. Reduction of Max-Min Automata 2 .6

Reduction

of Max-Min

55 Automata

Two max-min tables 521 and 522 are called statewise equivalent with threshold c ~ for every q E Q1, there exists q' E Q2 such that (521, q) N (522, q') and vice versa . In symbols, 521 N 522 . Otherwise, 521 x 522 . 521 ti 522 ~ for every 0 <_ c < 1, 521 N 522 . Otherwise, 521 x 522 . 521 and SZ2 are distributionwise equivalent with threshold c ~ for every al, there exists Q2 such that (521, al) N (522, a2) and vice versa. In symbols, Qi ^ Q* . Qi ^ SZ2 ~ for every 0 < c < 1, 521 . 522.

A max-min table 52* is statewise (distributionwise) minimal with threshold c if it is not statewise (distributionwise) equivalent with threshold c to any max-min table with a fewer number of states . 52* is statewise irreducible with threshold c if q', q" E Q such that q' N q" implies q' = q" . 52* is distributionwise irreducible with threshold c if al N 1.2 implies al c Q2 . 52* is statewise (distributionwise) minimal (irreducible) if and only if for every 0 <_ c < 1, 52* is statewise (distributionwise) minimal (irreducible) with threshold c. Clearly, distributionwise irreducible (with threshold c) implies statewise irreducible (with threshold c) . Theorem 2.6 .1 52* is statewise minimal with threshold c if and only if 52* is statewise irreducible with threshold c.

Proof. Let 52* = (Q, tt*, F) . Suppose 52* is not statewise irreducible with threshold c. Then there exist qo, ql E Q such that qo :?~ ql and qo N ql. Define 521 = (Ql, 1a1, Fl) where Q1 = Q\{qo}, Fl = F\{qof and for every q, q'EQ1,UEX, N'1 (q, u, q) = N'(q, u, q) if q zA ql

wl (q,

u, ql) = V{w(q, u, qo), w(q, u, ql)f .

Clearly, 52* N 521, where IQ, I < IQ 1 . Thus 52* is not statewise minimal with threshold c. Conversely, suppose 52* is not statewise minimal with threshold c. Then 52* N 521 for some 521 with IQ, I < IQ 1 . Thus there exist q E Ql and q', q" E Q with q' :?~ q" such that (52*, q') N (521, q) and (52*, q") ti (521, q) . Hence (52*, q') ti (52*, q") or q' N q" . Consequently, 52* is not statewise irreducible with threshold c.

Corollary 2.6 .2 52* is statewise minimal if and only if 52* is statewise irreducible . Theorem 2.6 .3 If 52* is distributionwise irreducible with threshold c, then 52* is distributionwise minimal with threshold c.

© 2002 by Chapman & Hall/CRC

56

2. Max-Min Automata

Proof. Let 52* = (Q, p*, F) . Suppose 52* . 521 for some 521 such that Let H and Hl denote, respectively, the collections of all initial distributions of 52* and 521 . Also, let P and Pl denote, respectively, the equivalence classes of H and Hl under Clearly, I P, I < I PI . Thus there IQ II < IQ I .

c.

exist al E Hl and 0'', 0'" E H with 0'' 7~ 0'" such that (52*, 0'') N (521, 0'1) and (52*, 0'") N (521, 0'1) . Therefore, (52*, 0'') N (52*, or") or or' -c ,//, contradiction .

Corollary 2.6.4 If 52* is distributionwise irreducible, then 52* is distributionwise minimal. 0

Theorem 2.6.5 521 N 522 implies 521^.. 522 . Proof. For every q E Q1, define T(q) = {q' E Q2 I (Q*, q) -c (Q*, q) Then T(q) 0 by hypothesis. Given al , define 0'2 as follows : 0'2

_

1

(q) - { 0

if for some q' E 0'1, 0'1(q') > c and q E T(q') otherwise .

Clearly, (521,x 1 ) N (522,0'2 ) . In a similar manner, it follows that for every 0'2, there exists 0'1 such that (521, 0'1) N (522, 0'2) . Thus 521 522 . Corollary 2 .6.6 521 - 522 implies 521 ~ 522 . m Corollary 2.6.7 If 52* is distributionwise minimal (with threshold c), then 52* is statewise minimal (with threshold c) . m 2 .7

Definite Max-Min Automata

Let p > 0 and Y C_ X* . Then Y is said to be weakly p definite if for all x E X* such that Ix I >_ p, x E Y if and only if the p-suffix of x belongs to Y. For p > 1, Y is p definite if Y is weakly p definite but not weakly p - 1 definite. Y is 0 definite if and only if Y is weakly 0 definite. Y is definite if Y is p definite for some p. A max-min automata A* is said to be definite (p definite, weakly p definite) with threshold c if B(A*, c) is definite (p definite, weakly p definite) . A* is said to be definite (p definite, weakly p definite) if for all 0 <_ c < 1, A* is definite (p definite, weakly p definite) with threshold c. A max-min table 52* is said to be weakly p definite with threshold c if for every initial distributions 0'1 and 0' 2 and x E X* with 1X I > p, VgEFrtt(A*, x, q) > c ~ Vg EF rtt(A2, x, q) > c,

where A1 = (52*, 0'1) and A2 = (52*, 0'2) . A max-min pre-table 0* is said to be weakly p definite with threshold c if for all initial distributions 0'1 and 0'2 and x E X*, with I xI > p, q E Q, V ng'EQ {al (q), ft * (q , x, q)} > c ~ Vq'EQ A{0'2 (q), ft * (q , x, q)} > c. © 2002 by Chapman & Hall/CRC

2.7. Definite Max-Min Automata

57

For p > 1,52*(0*) is p definite with threshold c if and only if Q* (0*) is weakly p definite with threshold c, but not weakly p - 1 definite with threshold c. 52*(0*) is 0 definite with threshold c if and only if 52*(0*) is weakly 0 definite with threshold c. 52*(0*) is definite with threshold c if and only if 52*(0*) is p definite with threshold c for some p. 52*(0*) is definite (p definite, weakly p definite) if and only if for 0 <_ c < 1, Q* (0* ) is definite (p definite, weakly p definite) with threshold c. Clearly if 0* is weakly p definite (with threshold c), then 52* = (0*, F) is weakly p definite (with threshold c). Theorem 2.7.1 If 52* = (0*, F) is distributionwise irreducible with threshold c and weakly p definite with threshold c, then 0* is weakly p definite with threshold c .

Proof. Suppose 52* is weakly p definite with threshold c. Then for every al, a2, and xo, x E X* such that 1XI > p, VgEFrtt(A*, xox, q) > c ~ Vg EF rtt(A2, xox, q) > c,

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

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

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

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

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

58

2. Max-Min Automata

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

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

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

> C ~?

VgEFrN(A * , x2x, q) >

C.

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

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

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

c 92,

but al

C,1

a2 .

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

Reduction

of Max-Min

Machines

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

2.8 . Reduction of Max-Min Machines

59

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

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

VZ i(az

n xi),

(2 .1)

where a2 is a nonnegative real number, i = 1, 2. . . . , n. If 0 <_ a2 < 1 for i = 1, 2, . . . , n, then (2.1) is called a convex max-min combination of X.

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2. Max-Min Automata

Definition 2.8.3 Let Tx be the set of all distinct terms of x. Then x is called admissible if Tx is finite and Tx can be effectively constructed from x. X is called admissible if every x in X is admissible .

Proposition 2 .8.4 Let x = VZ jai A x2 . If Tx is finite, then = VZ 1'"i A x'i, where b2 E Tx , i = 1, 2, . . . , n.

Proof. Choose b2 to be the largest number in Tx such that b2 < a2 . m Example 2.8.5 Let xl = (1, 1 4,

1), x 2 = (1 ,1,1), x3 = (1 ,1, 4), al = 1, a2 = 2, and a3 = 3 . Then al A xl = (1, 41 1) 1 a2 A x2 = ($, 2, 2), and a 3 A x3 = (31 3, 1) . Thus x = (al A xl) V (a2 A x2) V (a3 A x3) _ (1, 2,1) . Hence Tx = {1, 2 } . Let b2 = V{b E Tx I b< _ a2}, i = 1,2,3. Then bl = 1 and b2 = z . However, there does not exist b E Tx such that b <_ 3 <_ a3 . Since VO1 = 0, we take b3 = 0. Hence x = (b l A x1) V (b2 A x2) V (b3 A x3) .

Proposition 2 .8.6 Suppose that Tx is admissible and X is finite . Then it is decidable whether or not x is a (convex) max-min combination of X.

Proof. The proof follows from Proposition 2.8 .4. Definition 2.8.7 The (convex) max-min span of X is the collection of all (convex) max-min combinations of finite subsets of X.

Let C(X) denote the convex max-min span of X. Note that if X = {(a, b)} with a > 1, then X Z C(X) since 1 A (a, b) :?~ (a, b) and so (a, b) C(X) .

Proposition 2.8.8 (1) X C C(X) if x E X implies 0 < xk < 1 for each (2) If X1 C X2 , then C(X1) C_ C(X2) ; (3) C(C(X)) = C(X) . 0

We can consider C as a function from P (V) into P(V), where V = [0,1]n for some n E N . Then C is said to satisfy the Exchange Property if b'X C_ V and b'x, y E V; if y E C(X U {x}) and y ~ C(X), then x E C(X U {y}) . This is a key property that holds in some situations . It is mainly due to the Exchange Property that one obtains the existence of bases of unique cardinality for certain algebraic structures. It is important to note that it does not hold here . Example 2.8.9 Let X = {(1, $)}, x = ($,1), and y = (2 , 4) . Then y

C(X) . Now y E C(X U {x}) since y = (2 A (1, $)) V (4 A x) . However, x ~ C(X U {y}) . Thus the Exchange Property does not hold .

© 2002 by Chapman & Hall/CRC

2.8 . Reduction of Max-Min Machines

61

Definition 2.8.10 Y is called a convex max-min set if for every yl, y2

E Y, all convex max-min combinations of

{yl, y2}

are also in Y.

Proposition 2.8 .11 Y is a convex max-min set if and only if Y = C(Y) . Proposition 2.8 .12 For every X, C(X) is a convex max-min set. Proof. The proof follows from Propositions 2.8 .11 and 2.8.8(2) . In the rest of this section, Y always denotes a convex max-min set. Definition 2 .8.13 Let X C_ Y.

(1) X is called a set of generators of Y if Y = C(X) . (2) If X does not contain any proper subset which is itself a set of generators of Y, then X is called a set of vertices of Y.

Clearly, X is a set of generators of C(X) . Note that if x E C(X\{x}), then X C C(X\{x}) and so C(X) C C(C(X\{x})) = C(X\{x}) . Proposition 2.8 .14 Let X C_ Y. Then X is a set of vertices of Y if and only if (1) Y = C(X) and (2) if x E X, then x ~ C(X\{x}) .

0

Proposition 2.8 .15 Suppose that X is a finite set of generators of Y. Then there exists X' C_ X such that X' is a set of vertices of Y. can be effectively constructed provided X is admissible .

Moreover, X'

Proof. The proof follows from Propositions 2.8 .14, 2.8 .6, and the fact that X is finite . m Definition 2.8.16 Y is called finitary if it contains a set of generators that is finite.

Proposition 2.8 .17 Every finitary convex max-min set has at least one set of vertices .

Proof. The proof follows from Proposition 2.8 .15. Proposition 2.8 .17 is not true in general for arbitrary convex max-min sets. Proposition 2.8 .18 Suppose that Y is finitary. Then every set of generators of Y contains a finite subset that is a set of vertices of Y.

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2. Max-Min Automata

Proof. Let X' be a set of generators of Y. By Proposition 2.8 .17, there exists a set X of vertices of Y which is finite . Now V'x E X, x is a convex max-min combination of a finite subset of X', say Xx . Let X" = UxExXX . Clearly, X" is finite and Y = C(X) C C(X") C_ Y. Thus X" generates Y. The result now follows from Proposition 2.8 .15. Proposition 2 .8.19 Suppose that Y is finitary . Then every set of vertices of Y is finite.

Proof. Let X be a set of vertices of Y. By Proposition 2.8 .18, there is a finite subset X' of X such that X' generates Y. By the minimality of X, X' = X and so X is finite . m Definition 2.8.20 Let X be a set of vertices of Y. X is called funda-

mental if for every x E X, there does not exist y E Y such that x :?~ y = a A x for some 0 < a < l and (X\{x}) U {y} is a set of vertices of Y.

We write x <_ a if xk <_ a for all k, and x = a means xk = a for all k.

XI

< x 2 means xl < x2 for all k,

Lemma 2.8.21 Let X be a set of vertices of Y. Then X is fundamental if and only if for every x E X,

where y E C(X\{x}) and 0 < a < 1 implies x = a A x.

Proof. Let x E X be such that x = (a A x) V y, where y E C(X\{x}) and 0<_a<_l,butx :?~ aAx .Lety'=aAxandX'=(XVx})U{y'} . Then x E C(X\{x},y') and so X C C(X\{x},y') . Thus Y = C(X) = C(C(X')) = C(X) and so it follows that X' is a set of vertices of Y. Thus X is not fundamental . Conversely, suppose that X is not fundamental. Then there exists x E X, y' E Y such that x :?~ y' = a A x for some 0 <_ a <_ 1 and (X\{x}) U {y'} is a set of vertices of Y. Thus x is a convex max-min combination of a finite subset X' of (X\{x}) U {y'} . By Proposition 2.8 .14, x E C(X'), i.e ., x = (b A y')V y, where y E C(X'\{y'}) and 0 < b <_ 1. Let c = a A b. Then x = (b A (a Ax)) V y = (c Ax)V y. Since cAx=bAy'
Example 2.8.23 Let X = {(1, 0), (z, 4)} . Then X is a set of vertices of Y = C(X) . Now (2, 4) E C({a A x, (1, 0)}), where a = 4 and x = (2, 4) since (2, 4) _ (4 A (2 , 4)) V (z A (1,0)) . Thus letting y = a Ax, we see by Definition 2 .8 .20 that X is not fundamental. If we let y = z A (1, 0), then we see that X is not fundamental by Lemma 2 .8.21.

© 2002 by Chapman & Hall/CRC

2.8 . Reduction of Max-Min Machines

63

Proposition 2.8 .24 Suppose that X is a fundamental set of vertices of

C(X) and x E X. Then X\{x} is a fundamental set of vertices of C(X\{x}) .

Proof. The proof follows from Proposition 2.8 .14 and Lemma 2.8 .21 . Proposition 2.8 .25 Let X' and X" be fundamental sets of vertices of Y and .x E X'

n X" . Then C(X'\{ .x}) = C(X"\{Jc}) .

Proof. Let x E X'\{Jc} . Now x = VZoai A x2',

(2 .2)

where for i = 0, 1, 2, . . . , n, x2' E X" and 0 <_ a2 < 1 . If Je does not occur among the x2', then x E C(X"\{Jc}) . Suppose that . appears among the x2', say xo = .7c. Now for i = 1, 2 . . . . , n, // / m x 2 = Vj=obzj n x~ ,

(2 .3)

where for j = 0, 1, 2. . . . m, x' E X' and 0 < b2j < 1 . Thus x = (ao Ax) V (V jm=o c; n x,1 ),

where cj = VZi (ai n b2j) .

Since X' is a set of vertices of Y and x E X'\{x}, x must appear among the x', say x'o . Also, since X' is fundamental, by Lemma 2 .8 .21,

Therefore, x < co = V{a2 n bio I i = 1, 2, . . . , n} = ak n bko

for some 1 < k < n . Thus x < ak and x < bko. By (2 .2) and (2.3), x>aknxk>aknbkonxo=cony=x .

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

© 2002 by Chapman & Hall/CRC

64

2. Max-Min Automata

Similarly,

Consequently,

Let {x} and X' be sets of vertices for Y. Then Vx' E X', there exists E [0,1] such that x' = a' A x. Thus Vx', x" E X', either x' <_ x" or x" <_ x' . Hence it follows that ~X'I = 1, say X' = {x'}. Then there exists a, b E [0,1] such that x' = a A x and x = b A x' . Thus x' <_ x and x <_ x' . Hence x = x'. a'

Theorem 2.8.26 If Y is finitary, then the fundamental set of vertices of Y is unique . Proof. Let X' and X" be fundamental sets of vertices of Y. By Proposition 2.8 .19, X' and X" are finite. Let X' = {xl, x2, . . . , xn} and X" _ {x'1', x2, . . . ' X" }, where n <_ m. We prove the result by induction on n. The n = 1 case follows from the comments preceding the theorem. Suppose that the theorem is true for all n < p. We show that it is also true for n = p. Let / =

Xi

V7m-1ai7

and x~ = Vk-1bik n xk . Then n = Vk=1cik

/ n Xk,

Cik = V jm=1aij

n bjk .

/ Xi

where

Since X' is fundamental, by Lemma 2.8 .21, x'1 = c11 A x'1 . Hence xl < for some

j1 .

c11

= V~ 1(a19

n bj 1)

=

a1 j1

n bj1 1

This implies that x'1 < a 1j1 and xl < bj11 . Thus

© 2002 by Chapman & Hall/CRC

2.8 . Reduction of Max-Min Machines

x~l > bill n X I

65

=

xl .

Similarly, there exists j2 such that x~2 >_ x~l . By repeating this process, we obtain a sequence jl, j2, . . . , j,,, . . . , such that x/72~-i <x'.72~ <X172~+1 <x' . . 72k+2 Since X" 1S finite, we must have xj"2k -1 = x21-1, for some k and l, where l > k. Thus = x~ x~2k-1 2k . Let x = x~2k-1 = x~2k . Now consider X'\{x} and X"\{x} . By Propositions 2.8 .24 and 2.8 .25 and the induction hypothesis, X'\{x} = X"\{x} . Hence X' = X" . m Proposition 2.8 .27 Let Y be admissible and X = {xl, x2, . . . , xnJ be a set of vertices of Y. Then there exists a fundamental set X' of vertices of Y such that X' I = X I . Furthermore, X' can be effectively constructed from X. Proof. Define x2, i = 1, 2 . . . . n, recursively as follows : Let A1

=

{a I xl = (a A xl ) V (VZ>lai n xZ for some 0 < a2 < 1, i :?~ 1},

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

=

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

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

= nA k

xk = (ak A xk) V y,

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

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

m

66

2. Max-Min Automata

Example 2.8 .28 Let X = {(1, 0), (2, 4)} . Let xl = (1, 0) and x 2 = (2, 4) . Then Al = {a I (1, 0) = (an(1, 0))V (a2n(2, 4)) for some a2 , 0 <_ a2 < 1} _ {1}. Thus al = A{1} = 1 and xl = 1 A (1,0) = (1,0). Now A2 {a < A A (1, 0)) for some al , 0 <_ al 1} _ . Hence [4,1] (2 , 4) _ (a (2 , 4)) V (a, a2 = A [4 ,1] = 4 and x'2 = 4 A (2, 4) _ (4 , 4) Thus {(4 , 4), (1,0)} is fundamental. Now let xl = (2 , 4) and x2 = (1,0) . Then A1 = {a (2, 4) _ (a A (2 4)) V (a2 n ( 1, 0)) for some a2 , 0 < a2 < 1} _ [4,1] . Thus a' = /~[4,1] _ 4 andxl=4/x(2,4)=(4,4) . Now A2 ={a (1,0)=(aA(1,0))V(alA (2 , 4)) for some al , 0 <_ al <_ 1} = {1} . Hence a2 = A{1} = 1 and x2=1A(1,0)=(1,0) . Theorem 2.8 .29 Suppose that Y is finitary and admissible . Then every set of vertices of Y has the same number of elements. Proof. The proof follows from Propositions 2.8 .19, 2.8 .27, and Theorem 2.8 .26. Proposition 2 .8 .30 Let Y be finitary and admissible and let Xl, X2 be sets of generators of Y. If IX, I > IX2I, then there exists x E Xl such that x E C(XI\{x}) . Proof. By Proposition 2.8 .18, there exist Xl C Xl and X2 C X2 , such that XI and X2 are sets of vertices of Y. By Theorem 2.8 .29, IX' I = IX2I . Thus IX'I = IX2I < IX2I < X, I . This implies tha_ Xl is a proper subset of Xl . The result now follows from Proposition 2.8 .14. m Proposition 2 .8 .31 Let X be a collection of sequences consisting of 0 and 1. If X is a set of vertices of C(X), then X is fundamental. Proof. Let x E X and x = (a A x) V y, where 0 < a< _ 1 and y E C(X\{x}) . Suppose that a < 1. Let xo = a A x. If xk = 0, then y k = 0 . If x k = 1, then since xo = a < 1, it follows that y k = 1. Thus x = y, which is a contradiction. Hence a = 1 and x = xo. By Lemma 2.8 .21, X is fundamental. m Definition 2.8 .32 Y is called fundamental if every set of vertices of Y is fundamental. It follows that not every convex max-min set is fundamental. Theorem 2.8 .33 Let Y be finitary . Then Y is fundamental if and only if Y contains only one set of vertices . Proof. The proof follows from Proposition 2.8 .17 and Theorem 2.8 .26.

© 2002 by Chapman & Hall/CRC

2.9 . Equivalences

67

Definition 2 .8 .34 Let Xl, X2 C_ Y. Then X l is called a basis of X2 if every x E X2 can be expressed uniquely as a convex max-min combination of a unique finite subset of Xl . Let Y = [0,1] x [0,1] . Then clearly {(1, 0), (0,1)1 is a basis of Y. Proposition 2 .8 .35 Let X be a basis of Y. Then X is a fundamental set of vertices of Y. Proof. The proof follows from Proposition 2 .8 .14 and Lemma 2 .8 .21 . Proposition 2 .8 .36 Let Y be finitary and have a basis . Then Y is fundamental . Proof. Let X = {x l , x2, . . . , xnJ be a basis of Y. By Proposition 2 .8 .35, X is a fundamental set of vertices of Y. Suppose that Y is not fundamental . Then by Theorem 2 .8 .33, Y has a set X' of vertices such that X' :?~ X. By Proposition 2 .8 .27, we may assume, without loss of generality, that X' = ~xl ~ x2 ~ . . . , xn_1, xnf ~ where

xn :?~ xn .

Furthermore,

x n =aAxn, for some 0 < a < 1 and

Xn - Xn V Y,

where y E C(X\{x n }) .

Thus

Xn = (a A xn) V (a A y) .

Since X is a basis of Y, a = 1 . Hence Therefore, Y is fundamental . m

2.9

xn

= xn, which is a contradiction .

Equivalences

Definition 2 .9 .1 A max-min sequential-like machine (MSLM) is a quadruple (Q, X, Y, /t), where Q, X, Y are finite nonempty sets and /t is a function from Q x X x Y x Q into [0,1] .

© 2002 by Chapman & Hall/CRC

68

2. Max-Min Automata

The set X is the input alphabet . The set Y is the output alphabet . Q is the set of internal states and ft is the transition function. We may interpret ft(q, u, v, q') as the grade of membership that the MSLM will enter state q' and produce output v given that the present state is q and the input is u. As before, finite sequences of elements of X(Y) are called input (output) tapes. The collection of all input (output) tapes is denoted by X*(Y*) . We denote the set {(X,y)

I

X

E X*, y E Y*,

1XI = ly1f

by (X x Y)* . In the following, the symbol M, with or without subscripts, always denotes an MSLM. We assume that all MSLMs have the same input set X and output set Y. Therefore, by suppressing X and Y, an MSLM may be represented by (Q, ft) . Also, (X x Y) * is ordered in such a way that Ix1I < I x21 implies (XI, YO < (x2, y2) . The order of (X x Y)* is the same for all MSLMs and will be kept fixed throughout the remainder of the chapter . Definition 2.9.2 Let M = (Q, /t) be a MSLM. The extended transition function /t* of M is a function from Q x (X x Y)* x Q into [0,1] defined recursively on 1x1 , x E X*, as follows: 1 0

if q' = q11 if q' zA q",

ft* (q, ux, vy, q") = V eEQ{ft(q~, u, v, q)

n

ft* (q, x, y, q") },

where u E X, v E Y, and y E Y* with xI = Iy I . Furthermore, the overall transition function qM of M is a function from Q x (X x Y)* into [0,1] defined as follows: q ' (q, x, y) = Ve'EQft * (q, x, y, q~) .

If no confusion arises, we write q for q " . For ease of notation, we assume that Q = {ql, q2, . . . , q n } . Furthermore, (1) P M (x, y) _ [b 2j ] is a matrix, where b2j = ft * (qj, x, y, qj) ; (2) QM(x, y) _ [a2] is the column matrix, where the i-th row a2 = q M (qj, x, y) ; (3) E is the column matrix with all entries 1. Proposition 2 .9.3 Let M = (Q, ft) be an MSLM. Then for every (xl, yi), (x2, y2) E (X x Y)*, (1 ) PM (xlx2, Y1 Y2) PM (X 1, yi) P M(X2, y2), and = 0 (2) Q M(xlx2, yly2) = PM (x l, YO 0 Q M(x2, y2) = PM ( x lx 2, yly2) 0E . © 2002 by Chapman & Hall/CRC

2.9 . Equivalences

69

Proof. The proof follows immediately since 0 is associative . m (1) Let A' denote the matrix whose columns are Q`v (x, y) arranged in the order of (X x Y)* .

(2) For any nonnegative integer n, let Ant denote the submatrix of A" consisting of only those columns corresponding to QM (x, y) with Ix I < n.

(3) Let BM and BM denote, respectively, the matrix obtained from AM and An by omitting all columns that are convex max-min combinations of previous columns.

(4) Let A be a (finite or infinite) matrix . Then JAI denotes the number of columns of A and p(A) denotes the set of distinct rows of A.

Example 2 .9.4 Let M = (Q, X, Y, ft), where Q = Y = {0,1}, and

{q1, q2, q3},

X = {u, v},

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

Define

ft(q, w,

PM (u,1)

w(qi,

u,

l, q2)

w(qi,

u,

l, q3)

w(q2,

u, l, qi)

w(q2,

u,

w(q3,

u, 0, qi )

w(q3,

u, 0, q2)

3 4~

w (q1, v, 0, q2)

2,

w(ql, v,1, q3)

_3 4~

ft (q2, v, 0, qi)

1

2~

w(q2, v,1, q3)

5

1 4~

ft (q3, v, 0, qi)

1 4~

_1 8'

w(q3, v,

1

l, q3)

0, q2)

_3 8'

5 8,

_3 8'

_1 8'

z, q') = 0 for all other (q, w, z, q') . Then

=

0 0

© 2002 by Chapman & Hall/CRC

3

4 0 0

1

2

0

PM(u, 0)

=

0

0

4

4

0

,

70

2. Max-Min Automata q

(gl,

q

(q2,

q' (q3, gM (g1, gM (g2, gM(q3,

u, l) u, l)

u,1) u, 0) u, 0) u, 0)

gM (g1, v, 1 ) M g (g2,v, 1 ) q

(q3,

q

(q1,

q

(q2,

q

(q3,

v, l) v, 0) v, 0) v, 0)

=

ft(gl,

u,

l, ql)

=

ft(g2,

u,

l, q1)

= = = =

ft(g3,

q2)

ft(g3,

u,1, q1) V ft(g3, u,1, u, 0, g1)Vft(g1, u, 0, u, 0, g1)Vft(g2, u, 0, u, 0, q1) V ft(g3, u, 0,

=

ft(g1,

v, 1,

g2)Vft(g1,

=

ft(g2,v,1,g1)Vft(g2,v,1,g2)Vft(g2,v,1,g3)

= =

ft(g3,

ft(g1, ft(g2,

V N'(gl, u, V ft(g2, u,

g1)Vft(g1,

v, l, v, 0,

q1)

ft(g1,

=

ft (q2,

v, 0,

q1)

=

ft(g3,

v, 0,

q1)

q1)

l, q2) l, q2)

v, 1,

V ft(g3, v, l, V ft(g1, v, 0, V ft (q2, v, 0, V ft(g3, v, 0,

V N'(gl, u, V ft(g2, u,

l, q3) l, q3)

V ft(g3, u,1,

q3)

g2)Vft(g2,

u, 0, V ft(g3, u, 0,

g3)

q2)

q3)

v, 1,

g3)

g2)Vft(g1, u~0~g3)

Q

M

(u, 1)

YM (u, 0)

-

= I

0 0

3

q2)

V ft(g3, v, 0,

q3)

s'

1)

-

5 8 5 8

0

yM(v,

3

0)_

=

=

=

1 VLOA0 3 A 3 8~ z n 11 8 3 1 A 1} V LO A 38~ 4 A0 2 8 V{0 A 0, 4 A 0, 2 AOf

ft* (g2,uv,10,g1)

=

=

1

ft* (g2,uv,10,g2)

=

V{3/~00/~ 3 4 8~

=

3

[* (q2 ,

=

(g1,uv,10,q1)

=

uv, 10, q2)

[* (q1,

[* (q1 ,

uv,10, q3 )

uv,10, q3 )

1 2

/~

V{ 3 4 A 3 8> 0A0 > 1 2 A

81 } 81 }

=

3 8~

1

8~

0,

8~ 8~

V{ 4 A 0, 0 A 0, 2 A 0}

=

$}

= 0,

0 n 0, 0 n .1 }

= 0,

,t*

(q3 , uv,10,

q1)

=

V{o n 0, 0 n 1, 0 n

,t*

(q3 , uv,10, q2 )

=

V{o n 1,

,t*

(q3 , uv,10, q3 )

=

V{o n $ , o

© 2002 by Chapman & Hall/CRC

0,

q3)

Now N*

5

q2)

q3)

3

,

1 V 5

q3)

q2)

QM(v,

0

0, 0, 0,

V ft(g3, v, l, V ft(g1, v, 0, V ft (q2, v, 0,

q2)

Thus 3 4 3 4

3 V 3 V

n o, o n o}

=

0,

0.

3

1

2.9 . Equivalences qna

(q1, uv,10)

71 = =

V ft * (q1, uv,10, q2) V ft* (q1, uv,10, q3) A ft(q, v, 0, q1) I q E Q})V (V{ft(g1, u,1, q) A ft(q, v, 0, q2) I q E Q})V (V{ft(g1, u,1, q) A ft (q, v, 0, q3) I q E Q}) (V 0A0,1A1,n})V(V{0A1,1A0,-A V 4 8 2$ 8 4 2 8

/t* (q1,

uv,10,

q1)

(V{ft(g1, u,1, q)

(V{0 n 0, 4 n 0, z n 0}) 3 8' qM (q2

uv,10)

= =

_ qM

(q3, uv,10)

_ _

10, q1) V ft * (q2, UV, 10, q2) V ft* (q2, UV, 10, q3) (V{ft(g2, u,1, q) A ft(q, v, 0, q1) I q E Q})V (V{ft(g2, u,1, q) A ft(q, v, 0, q2) I q E Q})V (V{ft(g2, u,1, q) A ft (q, v, 0, q3) q E Q}) (V{3n00 n3 1A11)V(V{3n3 0A0 1A 4 2 V $~ (V{4 no,ono,~2 no})

/t* (q2, UV,

3 $.

10, q1) V ft * (q3, UV, 1 0, q2) V ft* (q3, UV, 10, q3) A ft(q, v, 0, q1) I q E Q})V (V{ft(g3, u,1, q) A ft(q, v, 0, q2) I q E Q})V (V{ft(g3, u,1, q) A ft (q, v, 0, q3) I q E Q}) (V{on0,0 A38 ,0A11)V(V{o n38 OAO,OA$})V 8 ft* (q3,

UV,

(V{ft(g3, u,1, q)

(V{on 1,0A0,0A0}) 0.

0

0

0

Since uv,10)

=

q' (q2, UV, 10 )

= =

q-'4 (q 1 ,

q' (q3, UV, 1 0)

3 3

0,

Q If (X x Y)* is ordered so that the first four members are (u, 0) < (u, l) < (v, 0) < (v,1), then 3 4 3 4

0

© 2002 by Chapman & Hall/CRC

3 8 3 8

$

5 8 5 8

0

72

2. Max-Min Automata

Since 3

3

3~ ] =( 3, [ ~31 8 4

0 A[ 0

V

and

we have 0 0 1 4

If the first four elements of (X x Y)* are such that (v, 0) < (v,1) < (u, 0) < (u,1), then 3 8 3 8

5 8 5 8

8 0

0 0

4

3 4 3 4 1

In this case, B1

3 8 3 8 1 8

5 8 5 8

0

0 0 1 4

3 4 3 4 I .

Proposition 2 .9.5 If Bn 1 = Bn for some n, then BX = Bn 1 for all m>n-1. Proof. It suffices to show that Bn 1 = Bn. Let (x, y) E (X x Y)*, where Ixl = n. Then QM (x, y) is a convex max-min combination of columns of BM 1 . Since Qlvi

(ux, vy) =

plvi

(u v) 0

Qlvi ( X

the result now follows from Proposition 2 .9 .3 and the fact that 0 is associative . m Theorem 2.9.6 Let M = (Q, p) be an MSLM. Then B m < d1Q1 - 1 and d is the number of distinct entries in A © 2002 by Chapman & Hall/CRC

= B,M, where

2.9. Equivalences

73

Proof. Clearly, I BM I <_ I Bn 1 I <_ I BM I < d1Q1 for every n. Thus Bm 1 = B,M, where m = dl QI -1 . The result now follows from Proposition 2 .9.5 . Proposition 2.9 .7 The matrix B" can be constructed effectively from M. Proposition 2.9 .8 C[p(A')] is admissible and finitary.

0

Definition 2.9.9 Let M = (Q, /t) be a MSLM. A state distribution (sd) of M is a function rl from Q into [0,1] . rl is said to be concentrated at q E Q if 97(q) = 1 and 97(q~) = 0 b'q' E Q\{q} . We will also use the symbol rl to denote the row matrix whose i-th row 97(g2) is . Definition 2 .9.10 An initialized max-min sequential-like machine (IMSLM) is an ordered pair (M, 97), where M is an MSLM and rl is an sd of M. If rl is concentrated at

q,

we write (M,

q)

for (M, rl) .

Definition 2.9.11 Let I = (M, 97) be an MMSLM. The response function rI of I is a function from (X x Y)* into [0,1] such that r, (x, y) =

VgEQ{71(q)

A q'(q, x, y)} .

Let RI = 97 0 AM . Then RI is a row matrix whose entries are r , (x, y). Furthermore, RI is a convex max-min combination of p(AM) . Definition 2 .9 .12 Let 11 and equivalent H if rI, = rI2 . Definition 2.9.13 Let 97, and 972 are called M-equivalent

12

A

be IMSLMs . Then 11 and

Proposition 2.9 .14 Let 97, and 972 if and only if

© 2002 by Chapman & Hall/CRC

are called

be state distributions of M. Then if (M, 971) - (M, 972) .

972

If no confusion arises, we use the symbol - for M . If 97, is concentrated at q, then we write q - 972 for

97,

12

972

0 B'v'

97,

and

97, - 972 .

be state distributions of M . Then 97,

= 972

0 13" .

74

2. Max-Min Automata

Definition 2.9 .15 Let Ml = (Q1, N1) and M2 = (Q2, [t2) be MSLMs. (1) Ml and M2 are called statewise equivalent (-) if for every q' E Q 1 , there exists q" E Q2 such that (Ml, q') - (M2 , q") and vice versa. (2) Ml and M2 are called compositewise equivalent (-) if for every q E Q1, there exists an sd rl of M2 such that (Ml, q) - (M2,97) and vice versa. (3) Ml and M2 are called distvibutionwise equivalent (;z:~) if for every sd rll of Ml, there exists an sd 972 of M2 such that (Ml, rll) - (M2,972) and vice versa. Theorem 2.9 .16 Let Ml and M2 be MSLMs. p(A-142) . (1) Ml - M2 if and only if p(AM M (2) Ml - M2 if and only if p(A C C[p(AM2)] and p(AM2 ) C ml C[p(A )] . (3) Ml ;zz~ M2 if and only if C[p(Aml)] = C[p(Amz)] . 1)

=

I)

Proposition 2 .9 .17 Let Ml and M2 be MSLMs such that Ml - M2 . Then M1 - M2 . Proof. 2.9 .16.

The proof follows from Propositions 2.8 .8(1) and Theorem

Proposition 2 .9 .18 Let Ml and M2 be MSLMs. Then Ml - M2 if and only if Ml ;zz~ M2 . Proof. It follows from Theorem 2 .9 .16(2) and (3) that Ml M2 implies Ml - M2 . That Ml - M2 implies Ml zz~ M2 follows from Proposition 2 .8 .8(2) and (3) and Theorem 2 .9 .16. m ;Z:~

Example 2.9 .19 Let M2 = (Q, X, Y, v), where Q = {s o , s3}, X = {u, v}, Y = {0,1}, and v :QXX XYXQ~ [0,1]

is defined as follows: V ( s 0, uJ, S0) V(S0, u, 1, 83) V(83, u, 0~ 8 0) v(s, w, z, s')

for all other (s, w, z, s') .

© 2002 by Chapman & Hall/CRC

3 4~ 1

2~

1 4~

0,

V ( S 0, V, 0 1 80) V ( S 0, V, 1, 83) V ( 8 3, V, 0 1 80)

_3 8' 5 8~ _1 8~

2.10. Irreducibility and Minimality

75

Let 972 : S ----> [0,1] . Then r12 (u,1)

=

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

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

972 (so) n 4*

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

=

=

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

_

2 .10

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

Irreducibility

and

Minimality

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

_

1 3

P1 (q2, u, 1, q1)

_

2 1 2

P2 (82, u, 1, sl)

0

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

2 3

0

2 3

0.

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

_ _ _

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

© 2002 by Chapman & Hall/CRC

76

2. Max-Min Automata

and r12(u,1)

_ (972(81)

n

2) V (972 (82)

n 3) .

In fact, it follows that rII

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

n

2) V (97, (q2)

n ) 2

n

2) V (972 (82)

n

and r12 (u n ,

11 ) = (972(sl)

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

i

i

2 _2 3

(u,1)

3 0

,

1 (ql, u,1)

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

q2)

=

1 (q2, u,1)

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

q2)

=

1 z 2

QM' (u, 1) _

=

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

=

uu,11, ql)

=

fti (q2,

2, 2,

~

I

© 2002 by Chapman & Hall/CRC

2 2

3

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

(ql, uu,11)

=

1 (ql, uu,11)

=

1

1 3 1 3

2 1 2

1

2

V

1

3

=

V0=

-

Thus PMl

1

,

3 1 2 2 3

2)

V

1

3

=

,

1

2,

= =

3

s

1

2, 2

3,

2.10. Irreducibility and Minimality 1

1

11~=2V

1(g2,uu,

QMl (uu, ll) _

1

3= 2'

2 1

A"' (uu,11) _

1

2

2

1 2 1

_1 3 1 3

2 3

1 2

It is easy to see that PIVI i (un

1n) l

2

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

1

2

2

2

2

1

1

and

Now PMz

(u,1)

=

0

2 3

1

2

0

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

2(82,u,1~

© 2002 by Chapman & Hall/CRC

=

2 3

'

1

= 2'

V0=

2

3~

78

2. Max-Min Automata

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

= =

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

2, 0,

Thus

1

PM2 (uu,11) =

8 1 , uu,11)

0

2 0

1

=

0 2.

,

2

= 2 V 0 = 2, 2=

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

Q

=

(uu,11)

1

=

1

2'

,

A212 (uu,11) _

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

1 z

0

0

1 2

0

1 z

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

1 2

0

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

and

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

© 2002 by Chapman & Hall/CRC

2.10. Irreducibility and Minimality

79

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

1

2

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

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

= =

r'(uu,01)

=

rI(uu,00)

=

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

0,

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

= = = = = = = =

0, 0,

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

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

0 3

0 0

2 0

0 2

0 0

0 0

0 0

2 0

0 0

0 0

0 2

0 0

0 0

and

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

© 2002 by Chapman & Hall/CRC

80

2. Max-Min Automata

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

m)] .

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

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

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

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

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

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

ft(gi, u, v, qn)

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

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

y) _

© 2002 by Chapman & Hall/CRC

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

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

2.10. Irreducibility and Minimality

81

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

f t (gZ, x, y, gj)

if i = 3 if .j = 2.

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

y, y, y, y,

q3) q3) q2) q2)

= = = =

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

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

=

It' (q2, v,1, q3)

=

It'(q2, u,1, q2)

=

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

=

'4

=

8

It' (q3, v, 0, q2)

=

2~ OV4 OV V

'4 8

3

4~

3

1

4~ 1

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

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

© 2002 by Chapman & Hall/CRC

82

2. Max-Min Automata

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

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

2J)

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

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

UI v,

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

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

_

qm,

(qi, x, y)

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

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

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

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

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

© 2002 by Chapman & Hall/CRC

2.10. Irreducibility and Minimality

83

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

© 2002 by Chapman & Hall/CRC

v)

= plvi (u

v)

84

2. Max-Min Automata

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

(2.4)

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

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

Construct by j' and Theorems M' - M.

If M

is

dis-

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

Suppose that distributionwise irreducible, then M

M is compositewise minimal . is compositewise simple .

If M

is

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

M = (Q, X,

Let

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

Y, p), where Q =

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

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

w, z, q') .

© 2002 by Chapman & Hall/CRC

= 2, =

ft(g2, u, 0, qi)

_

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

2.10. Irreducibility and Minimality

85

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

_ _

g m (ql, nun, 101)

_

I 2~ 1

q

i

(q2, uuu, 010)

q

2~

3 41 1

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

q

=

i

21

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

0 4

0 0

2 0

= [ i

2 0

0 2

0 0

0 0

0 1 . 4

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

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

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

_ _

rI(uu, 01)

_

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

_ = = = =

= = =

= =

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

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

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

© 2002 by Chapman & Hall/CRC

=

_

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

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

86

2. Max-Min Automata

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

0 0

1

0

0 0

3

0 ]'

4 0 0

0 0

PM (uu,10)

2 0

0 0 ]'

PM (uu, 01)

0 0

0

PM (uu, 00)

0 0

PM (uu,11)

'

z 0 0

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

_

{(o, 2)

[

3 0

z 0

(4~ )

(0, _,)[

o

0

,

3 ] , (0, 0) 4

[ 01

4

3

(0 0)

and p(BM)

_

{(2

0) (0'

4)}.

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

Nondeterministic and Deterministic Case

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

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

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

2.11 . Nondeterministic and Deterministic Case

87

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

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

0

Proposition 2.11 .4

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

Proposition 2.11 .5 mal if and only if M

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

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

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

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

a DSLM. Then minimal .

M

is compositewise minimal

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

= V {xi I

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

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

88

2. Max-Min Automata

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

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

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

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

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

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

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

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

Exercises

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

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

© 2002 by Chapman & Hall/CRC

2.12. Exercises

89

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

= 97(g2) _

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

© 2002 by Chapman & Hall/CRC

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

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

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

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

Definition 3.1 .1 is applicable even if

n

or

m is

infinite.

Proposition 3.1 .2

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

© 2002 by Chapman & Hall/CRC

91

92

3. Fuzzy Machines, Languages, and Grammars

that is, the operation (~) is associative .

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

VZlaix2,

(3.1)

where a2 is a nonnegative real number, i = 1, 2. . . . , n. If 0 <_ a2 < 1 for i = 1, 2, . . . , n, then (3 .1) is called a convex MPcombination of X.

Definition 3.1.4 The (convex) MP-span of X is the collection of all (convex) MP-combinations of finite subsets of X.

We let C(X) denote the convex MP-span of X. The situation here differs with that of convex max-min combinations since for 0 < a, b < 1, a :?~ a- b b.

Proposition 3 .1 .5 (1) X C C(X) ; (2) If Xl C_ X2, then C(XI) C_ C(X2) ; (3) C(C(X)) = C(X) . 0

We can think of C as a function of P(S) into P(S) . When doing so, recall that an important property to consider is the Exchange Property : if x E C(X U {y}) and x ~ C(X), then y E C(X U {x}) . In the next example, we show that the Exchange Property does not hold for C. Example 3.1 .6 Let X = {(1, 1 )}, x = ($,1), and y = (2, 4) . Then y C(X) . Since y = z (1, $) V (s, l), y E C(X U {x}) . However x C(X U {y}) . Thus the Exchange Property does not hold .

© 2002 by Chapman & Hall/CRC

3.1 . MaxProduct Machines

93

Definition 3.1.7 Y is called a convex MP-set if for all yi, y2 E Y all convex MP-combinations of {yl,

y2}

are also in Y.

Proposition 3.1 .8 Y is a convex MP-set if and only if Y = C(Y) . Proof. Suppose that Y is a convex MP-set . Then Y = C(X) for some subset X of Y. Thus C(Y) = C(C(X)) = C(X) = Y. The converse follows by the definitions. Proposition 3.1 .9 For every X, C(X) is a convex MP-set.

0

Proof. The proof follows from Propositions 3.1 .8 and 3.1.5(2) . Definition 3.1.10 Let Y denote a convex MP-set and let X C_ Y. Then X

is called a set of generators of Y if Y = C(X) . If X does not contain any proper subset which is itself a set of generators of Y, then X is called a set of vertices of Y.

Proposition 3.1 .11 Let Y denote a convex MP-set and let X C_ Y. Then X is a set of vertices of Y if and only if (1) Y = C(X) and (2) if x E X, then x ~ C(X\{x}) . 0

Proposition 3.1 .12 Let Y denote a convex MP-set and let X be a set of generators of Y such that X is finite . Then there exists X' C_ X such that X' is a set of vertices of Y.

Proof. The proof follows from Propositions 3.1 .11 and the fact that X is finite. m We write xl > x2 if xl >

x2

for all k.

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

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

x2 E X'

and 0 <

a2 <

(3.2)

1 . Moreover

x2 = V~ obij xj ,

where for j = 0, 1, 2, . . . , m, xj E X and 0 < b2j © 2002 by Chapman & Hall/CRC

(3.3) <

1 . Thus

94

3. Fuzzy Machines, Languages, and Grammars

x =

V.7=0 c7 x3,

where cj = VZ laib2j .

(3.4)

By Proposition 3.1 .11(2), x must be one of the xj , say x = x 0 . Since for all c, b E [0,1], cb < b if c < 1 it follows that co = 1 else x = VZl cj xj E C(X\{x}) which contradicts the assumption that X is a set of vertices of Y. Therefore, by (3.4), there exists k such that ak = 1 and bk0 = 1 . It follows from (3.2) and (3.3) that x>xk

and

xk>x0 =x .

Thus x = xk E X' . Consequently, X C_ X'. m In the proof of Proposition 3.1 .13, the conclusion that co = 1 in the last paragraph followed partially from the fact that b'c, b E [0,1], cb < b if c < 1 . This property for product does not hold for minimum, i .e., c A b < b cannot necessarily be concluded if c < 1 . Theorem 3.1 .14 Let Y be a convex MP-set . Then the set of vertices of Y is unique if it exists .

Proof. The proof follows from Proposition 3.1 .13. Proposition 3 .1 .15 Let Y be a convex MP-set . Let X be the set of vertices of Y. If Xl and X2 are sets of generators of Y such that IX, > IX2I, then there exists x E Xl such that x E C(XIVx}) .

Proof. From Proposition 3.1 .13, X C_ Xl and X C_ X2 . Therefore, IXI <_ IX2 I < IX,1 . Thus X is a proper subset of Xl . The desired result follows from Proposition 3.1 .11 . Definition 3.1.16 Let Xl,

X2 C_ Y. Xl is called a basis of X2 if every x E X2 can be expressed uniquely as a convex MP-combination of a finite subset of Xl .

Proposition 3 .1 .17 Let Y be a convex MP-set . Let X be a basis of Y. Then X is a set of vertices of Y.

Proof. Clearly, Y = C(X), i.e., (1) of Proposition 3.1 .11 holds . Suppose there exists x E X such that x E C(X\{x}) . Then there exists xl , x2, . . . , xn E X\{x} such that x is a convex MP-combination of x1, x2, . . . , xn . Now x = 1 - x is another combination of elements of X. However, this contradicts the fact that x has a unique representation as a convex MPcombination of elements of X. Thus (2) of Proposition 3.1 .11 holds . m

© 2002 by Chapman & Hall/CRC

3.2. Equivalences

95

Example 3.1 .18 Let X = {(4, 4), (1,0){ and X' _ {(z, 4), (1,0){ . Then X and X' are sets of vertices for C(X) with respect to max-min convex combinations . In fact, X is fundamental. However, for max-product convex combinations, X' does not generate C(X) . This follows since ( 1 , 4) C(X') with respect to max-product convex combinations . X is not a basis for C(X) here for suppose (a, b) = x - (14 , 14 ) V y - (1, 0) . Then a = 4x V y and b = 4x . Thus x = 4b . Hence a = b V y . Thus either y = a > b (in which case the representation of (a, b) is unique or a = b and y can be any element in [0, b] . Clearly, X is a set of vertices of C(X) . In this section, we presented a max-product algebra. We note in the Exercises that this algebra can be extended by replacing product with a t-norm, which we now define . Definition 3.1 .19 Let T be a binary operation on [0,1] . Then T is called a t-norm if the following conditions hold b'a, b, c E [0,1] (1) T(a,1) = a; (2) b <_ c implies T(a, b) < T(a, c) ; (3) T(a,b) =T(b,a) ; (4) T(a,T(b,c)) =T(T(a,b),c) .

3.2

Equivalences

Definition 3 .2 .1 A quadruple (X, Q, Y, /t) is called a max-product sequential-like machine (MPSM) if Q, X, Y are finite nonempty sets and ft is a function from Q x X x Y x Q into [0,1] . We note that Definition 2 .9 .1 and Definition 3.2 .1 are the same . The use of "max-min" in the first and "max-product" in the second refers to the type of extended transition function tt* . The sets X and Y are the input and output alphabets, respectively. Q is the set of internal states and /t is the transition function . We can regard ft(q, u, v, q') as the grade of membership that the MPSM will enter state q' and produce output v given that the present state is q and the input is u. The reader should note the differences between the above definition and the one given in [195]. As before we call finite sequences of elements of X input tapes or strings. Similarly, we call finite sequences of elements of Y output tapes or strings. The collection of all input (output) tapes will be denoted by X* (Y*) . Let (X x Y)* = {(x, y) I x E X* , y E Y*, Ixl = y1} . © 2002 by Chapman & Hall/CRC

96

3. Fuzzy Machines, Languages, and Grammars

Unless stated otherwise, the symbol M, with or without subscripts, denotes an MPSM. All MPSMs are assumed to have the same input and output sets. Hence for ease of notation, we represent an MPSM by (Q, /t) . In addition, we assume that (X x Y)* is ordered in such a way that Ix, I < x21 implies (xi,yi) < (x2,y2) . The order of (X x Y)* will be kept fixed and assumed to be the same for all MPSMs under consideration . Definition 3.2.2 Let M = (Q, /t) be an MPSM. The extended transition function /t* of M is a function from Q x (X x Y)* x Q into [0,1] defined recursively on Ixl, x E X*, as follows:

~

1 0

if q' = q11 if q' zA q" ;

* ft (q, ux, vy, q") = VeEQ{ft(q, u, v, q) . ft * (g, x, y, q") }, where u E X, v E Y, and y E Y*, IxI = I yI . Moreover, the output function qM of M is the function from Q x (X x Y)* into [0,1] defined as follows: q ' (q, x, y) = Ve'EQft * (q, x, y, q) .

If no confusion arises, we sometimes write q for q " . Assume that Q = {ql, q2, . . . , qn } . Furthermore, let (1) PM (x,y) _ [p2j] be the matrix, where p2j = ft*(gj,x,y,qj) ; QM (x, y) _ (2) [qz] be the column matrix, where the i-th row gM (qj, x, y) ; (3) E be the column matrix with all entries 1 .

qZ =

Proposition 3 .2.3 Let M = (Q, ft) and (XI, yi), (x2, y2) E (X XY)* . Then (1) PM (xix2, Y1 Y2) = PM (X 1, YO O PM (X2, y2), (2) Q M (xlx2, yly2) = PM ( x l, YO Q M (x2, y2) = PM (xlx2, yly2) O E . Proof. The proof follows from Proposition 3.1 .2 . Definition 3.2.4 Let M = (Q, y) . (1) An initial distribution (i.d.) of M is a function

t from Q into [0,1] . (2) t is said to be concentrated at q E Q if t(q) = 1 and 0 elsewhere.

We also use the symbol

t

to denote the row matrix whose i-th row is

t(qj) .

Definition 3.2.5 Let M be an MPSM and t be an i.d. of M. Then the ordered pair (M, t) is called an initialized max-product sequential-like machine (IMPSM) . If

t

is concentrated at

© 2002 by Chapman & Hall/CRC

q,

we write (M, q) for (M, t) .

3.2 . Equivalences

97

Definition 3.2 .6 Let I = (M, L) be an MMPSM. The response function r I of I is a function from (X x Y)* into [0,1] defined as follows: rI (x, y) = VeEQ{t(q) - qM(q, x, y) 1 . We let Ana denote the semi-infinite matrix whose columns are Q' (x, y) arranged in the order of (X x Y)* . If A is a matrix, we let p(A) denote the collection of all rows of A. If RI = L O Am, then RI is a row matrix whose entries are r I (x, y) . Moreover, RI is a convex MP-combination of p(A M) . Definition 3.2 .7 Let h and 12 be IMPSMs . Then equivalent, denoted by 11 - 12, if r i l = r12 .

h and 12 are called

Definition 3.2 .8 Let Ll and L2 be iAs of an MPSM M. Then Ll and L2 are called M-equivalent, denoted by Ll - L2, if (M, Ll) - (M, L2) . If Ll is concentrated at

q,

then we also write

q -

L2 for Ll - L2 .

Proposition 3.2 .9 Let Lland L2 be i.d.s of an MPSM M. Then Ll ' L2 if and only if L10A`vl

=L20A

Definition 3.2 .10 Let Ml = (Q1, /q) and M2 = (Q2, ft2) (1) Ml and M2 are called statewise equivalent, written Ml - M2, if for all q' E Q1, there exists q" E Q2 such that (Ml, q') - (M2, q") and vice versa. (2) Ml and M2 are called compositewise equivalent, written Ml M2, if for all q E Q1, there exists an i.d . L of M2 such that (Ml, q) - (M2, L) and vice versa. (3) Ml and M2 are called distributionwise equivalent, written Ml M2 , if for every i.d. Ll of Ml , there exists i.d . L2 of M2 such that (Ml, L l) (M2, L2) and vice versa. Theorem 3.2 .11 Let Ml and M2 be MPSMs. Then the following assertions hold. (1) Ml - M2 if and only if p(AMI ) = p(AM2) ; (2) Ml N M2 if and only if p(AMI) C C[p(AM2)] and p(AM2) C l C[p(A )] ; (3) Ml ~ M2 if and only if C[p(Al)] C C[p(A l )] . 0 Proposition 3.2 .12 Let Ml and M2 be MPSMs. If Ml - M2, then Ml M2 . Proof. The proof follows from Proposition 3.1 .5(1) and Theorem 3.2 .11.

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Proposition 3 .2 .13 Let Ml and M2 be MPSMs. Then Ml - M2 if and only if Ml ;z:~ M2 . Proof. The proof follows from Proposition 3.1 .5 and Theorem 3.2 .11 .

3 .3

Irreducibility and Minimality

In this section, we study the irreducibility and minimality of an MPSM . Definition 3.3 .1 Let M = (Q, ft) . (1) M is called statewise irreducible if for all q', q" E Q, q' - q" implies q' = q" . (2) M is called compositewise irreducible if for every q E Q and i.d. t of M, q - t implies t(q) > 0. (3) M is called distnibutionwise irreducible if for every i.d. t1 and L2 of M, tl - t2 implies t1 = L2 . Theorem 3.3 .2 Let M be an MPSM. Then the following assertions hold. (1) M is statewise irreducible if and only if no two rows of AM are identical. (2) M is compositewise irreducible if and only if p(A M ) is a set of vertices of C[p(A M )], i .e., no row of AM is a convex MP-combination of the other rows of AM . (3) M is distributionwise irreducible if and only if p(AM ) is a basis of C[p(A

m )] .

Proof. (1) follows from Proposition 3.2 .9, (2) follows from Propositions 3.2 .9 and 3.1 .11, and (3) follows from Proposition 3.2 .9 and the definition of basis . m Example 3.3 .3 Let M = (Q, X, Y, p), where Q, X, and Y are defined as in Example 2 .10 .24 and p = p . Then q M (q1, u,1)

q -'4 (q1, an, 10)

-_

gm(ql, u2n , (10) n )

q

13 24~

(2) n (4) n ,

(ql, 21,2n+1 , (10) n l)

(2)n+1(4)n, 3 4~ 31 42~

q M (q2, u, 0)

g M (q2 , an, 01) g M (q2,U2n ,(Ol) n )

=

gM(q2,u2n+1, (Ol)no)

=

© 2002 by Chapman & Hall/CRC

1

(4) n (2) n , ) n + 1 (2) n .

(4

3.3 . Irreducibility and Minimality

99

The nonzero columns of A"1 are [

I ]

0

2 8

0

,

[ 0

,

4

3

0

0 ]

,

8

,. .. ,

[

(

On

,

8

n

'

Clearly, no two rows of Ana are identical. In fact, p(A') is clearly a set of vertices of C[p(A M )] . It also follows that p(A M) is a basis of C[p(AM)] .

Proposition 3.3 .4 Let M be an MPSM. If M is distributionwise irreducible, then M is compositewise irreducible .

0

Proposition 3.3 .5 Let M be an MPSM. If M is compositewise irreducible, then M is statewise irreducible .

0

Definition 3.3.6 Let M be an MPSM. Then M is called statewise (compositewise minimal if M is not statewise (compositewise equivalent to any MPSM with a fewer number of states .

Let M = (Q, lt) . Then the cardinality of M is defined by I MI = IQI . Theorem 3.3.7 Let M be an MPSM. Then M is statewise minimal if and only if M is statewise irreducible .

Proof. Let M = (Q, ft) . Suppose that M is not statewise minimal. Then there exists an MPSM M' such that IM'I < IMI and M' - M. By Theorems 3.2 .11(1) and 3.3.2(1), M is not statewise irreducible . Conversely, suppose that M is not statewise irreducible . Then there exist q', q" E Q such that q' - q" and q' :?~ q" . By renumbering the elements of Q, if necessary, we may assume that q' = q,,_1 and q" = qn , where n = IQI . Let M' = (Q', /t'), where Q' = Q\{q, } and for i = 1, 2. . . . , n - 1, f~ (gZ, u~ v, gj) _ { ft (gj, u, u v, v qn) 1) V ft(gi, u, v, qn)

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

It follows that p(AM ) = p(AM ) . Hence by Theorem 3.2 .11(1), M' . Thus M is not statewise minimal. m

M

Theorem 3.3.8 Let M be an MPSM. Then M is compositewise minimal if and only if M is compositewise irreducible .

Proof. Let M = (Q, ft). Suppose that M is not compositewise minimal. Then there exists an MPSM M' such that IM'I < IMI and M' - M. By Theorems 3.2 .11 and 3.3.2 and Proposition 3.1 .15, M is not compositewise irreducible . Conversely, suppose that M is not compositewise irreducible. Then there exists q' E Q and an i.d. t of M such that q' - t and t(q) = 0. © 2002 by Chapman & Hall/CRC

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3. Fuzzy Machines, Languages, and Grammars

By renumbering the elements of Q, if necessary, we may assume that q' _ qn, where n = I Q1 . Since qn - t and t(qn) = 0, g1V1

glvl (qn, X, Y) = V {t(qi) (qi, x, 2J)

I

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

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

It'(qi, u, v, qj) = ft (qi, u, v, qj) V t(qj) . ft (qi, u, v, qn) .

Since for all (x, y) E (X x Y) * and i = 1, 2. . . . , n - 1, gM(qn, x, y) = m gm (qi,x,y), we have C[p(A )] = C[p(A m' )] . Hence by Theorem 3.2 .11, M'-- M. Thus M is not compositewise minimal . m Proposition 3 .3.9 Let Ml and M2 be MPSMs. Suppose that Ml is state-

wise minimal and Ml - M2 . Then M2 is statewise minimal if and only if 1M11 <- IM21 . .

Proposition 3 .3.10 Let Ml and M2 be MPSMs. Suppose that Ml is state-

wise minimal and Ml - M2 . Then M2 is compositewise minimal if and only if IM1 I < IM2 1 . 0

Proposition 3 .3 .11 Let Ml and M2 be MPSMs that are compositewise minimal. Then Ml - M2 if and only if Ml - M2 .

Proof. The proof follows from Proposition 3.2 .12 and Theorems 3.1 .14,

3.3 .2, and 3.3 .8

Proposition 3 .3.12 Let Ml and M2 be MPSMs. Suppose that Ml and M2 are statewise minimal and Ml - M2 . Then for all (x, y) E (X, Y)*, P"'

Al" = P1v12 (x, y) O (x, y) (~) A

after an appropriate rearrangement of states.

Proof. From Theorems 3.2 .11(1), 3.3 .2(1), and 3.3 .7, Amt = AMz after an appropriate rearrangement of states. 0 Definition 3.3.13 Let Ml and M2 be MPSMs.

Then Ml and M2 are called isomorphic () if they are equal up to a permutation of states .

Definition 3.3.14 Let M be an MPSM. Then M is called simple if there

exists no MPSM M' such that I M' j <_ I M1, M' - M, but M' and M are not isomorphic.

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3.3 . Irreducibility and Minimality

101

Let M be an MPSM. Then we let P(M) = Uvex U'EY P[P

(u, v)]

and p (M) = {t O

A"1 I t E p(M)j .

Theorem 3.3.15 Let M be an MPSM such that M is statewise minimal. Then M is simple if and only if p(AM) is a basis of p (M) .

Proof. Suppose that p(A') is a basis of P (M) . Let M' be an MPSM such that M' I <_ I MI and M' - M. Then by Proposition 3.3.9, M' is statewise minimal. It follows from Proposition 3.3.12 that PM (x, y) = P M'(x, y) for all (x, y) E (X x Y)* after an appropriate rearrangement of states. Thus M - M' . Hence M is simple . Conversely, suppose that p(AM) is not a basis of P (M) . Then there exists j E p(M) such that j OA

=

Y

for some j' :?~ j. Let j be the i-th row of the matrix Pl"4 (u, v) . Construct M' from M by replacing the i-th row of the matrix PM (u, v) by j' and leaving the rest unaltered. By (3.5), AM = AM' . Thus IM'I = IMI and M' - M. However, M' * M. Hence M is not simple. m Proposition 3.3 .16 Let M be an MPSM such that M be statewise (compositewise minimal. Then there exists no statewise (compositewise minimal MPSM that is statewise (compositewise equivalent, but not isomorphic to M if and only if p(AM) is a basis of P (M) . Proof. If M is statewise minimal, the desired result follows immediately from Theorem 3.3.15 . The case when M is compositewise minimal follows from Theorem 3.3.15 and Propositions 3.3.10 and 3.3.11. Proposition 3.3 .17 Let M be an MPSM such that M is distributionwise

irreducible . Then there exists no statewise (compositewise minimal MPSM that is statewise (compositewise equivalent, but not isomorphic to M.

Proof. The proof follows from Theorem 3.3.2(3) and Proposition 3.3.16.

© 2002 by Chapman & Hall/CRC

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3 .4

3. Fuzzy Machines, Languages, and Grammars

Max-Product Grammars and Languages

The remainder of the chapter is based on [208] . In [204], Santos introduced a mathematical formulation of probabilistic grammars and random languages generated by probabilistic grammars . (See Chapter 5 also. In that formulation, the probability associated with each "derivation" of a probabilistic grammar is taken to be the sum of the probabilities of all distinct "paths" of the "derivation." This leads to the sum interpretation which is customary in the theory of probabilistic automata . The sum interpretation is one of the two interpretations considered in [188] . The other interpretation is the maximal interpretation, which leads naturally to the mar-product operation introduced in [193, 202] . (See Chapter 2 and Sections 3.1 and 3.2.) A new type of grammar arises naturally from the application of the maximal interpretation to the probabilistic grammars defined in [204] . This grammar, which is readily obtained from a probabilistic grammar by replacing the "sum" operation by the "mar-min" or "least upper bound" operations, is called a normalized mar-product grammar here. Normalized mar-product grammars are a special case of mar-product grammars. In mar-product grammars, the functions involved are allowed to assume any nonnegative value . In view of [81, 255], mar-product grammars may be considered as a model of fuzzy grammars . Another formu lation of mar-product grammars is given from this point of view. It is shown that the two formulations are equivalent as far as the family of fuzzy languages generated are concerned . By imposing restrictions on the fuzzy productions, certain more restricted types of mar-product grammars can be obtained. Two of these are regular and weak regular mar-product grammars . It turns out that regular and weak regular mar-product grammars are closely related to mar-product automata [193, 207] and asynchronous mar-product automata . This allows for the establishment of many interesting properties of languages generated by regular and weak regular mar-product grammars with cut point . An interesting family of languages that arises from the consideration of regular and weak regular mar-product grammars is the family of languages I generated by finitary weak regular mar-product grammars with cut point. The family I may also be characterized by regular mar-product grammars, asynchronous mar-product automata, mar-product automata, and type 3 weighted grammars under maximal interpretation [188] . The family I contains the family of regular languages as a proper subfamily. Moreover, there are context-free and stochastic languages [235] that are not in 1. A formulation of mar-product grammars is given in this section . An alternative formulation of mar-product grammars is also given . This formulation is motivated by the fact that mar-product grammars may be viewed © 2002 by Chapman & Hall/CRC

3.4 . MaxProduct Grammars and Languages

103

as a model of fuzzy grammars. The two formulations are shown to be equivalent as far as the family of fuzzy languages generated is concerned . Let S denote a nonempty set . Let R~° denote the collection of all nonnegative real numbers and g'-O oo =Z>° U {oo} . Definition 3.4.1 A fuzzy production p over S is a function from S* x S* into Z>° such that the set p = {s E S* p(s, t) > 0 for some t E S*} is finite and A ~ p. If, in addition, for all s E S*, the set {t E S* I p(s, t) > 0} is

finite, then p is called finitary.

Definition 3.4.2 A fuzzy production p over S is strict if the range of p is a subset of [0,1] .

The concept of a fuzzy subset requires that the grades of membership function assume values in [0,1]. This concept is extended in [81] to allow the grades of membership function to assume values in any partially ordered set . Since R> O, as well as R> O, is a partially ordered set, we may interpret p(s, t) as the grade of membership that s will be replaced by t. If p is normalized, i.e., Etes= p(s, t) S 1 for all s E S*, then p(s, t) may also be interpreted as the probability that s will be replaced by t. Definition 3.4.3 A (finitary) mar-product grammar is a quadruple

G = (T, N, P, h), where (1) T and N are disjoint finite nonempty sets; (2) P is a finite collection of (finitary) fuzzy productions over T U N such that for all p E P, p(s, t) > 0 implies s E (T U N)*N(T U N)*; and (3) h is a function from N into R> O . If in addition, every p E P is strict and h is a function from N into [0,1], then G is said to be strict. Moreover, if every p E P is normalized and EAEN h(A) S 1, then G is said to be normalized.

In the above definition, T and N are, respectively, the terminals and the nonterminals . h(A) is the grade of membership that A is the start symbol of G. It is important to note that in Definition 3.4 .3(2), that s must contain a nonterminal . The following concepts are needed to define the fuzzy languages generated by mar-product grammars . Recall that N is the collection of all positive integers . Definition 3.4 .4 Let t, s E S* and k E N . m(t, s) = k if there exist u2 , v2 E S*, i = 1, 2, . . . , k, where v2 :?~ vj for i :?~ j, such that (1) t = uisv2

for i = 1, 2, . . . , k, and (2) t = usv implies u = u2 and v = v2 for some i such that 1 _ i _ k . Define m(t, s) = 0 if and only if t usv for all u,VES* .

Note that m(t, s) = k if and only if t can be expressed in the form t = usv in exactly k distinct ways. © 2002 by Chapman & Hall/CRC

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3. Fuzzy Machines, Languages, and Grammars

Definition 3.4.5 Let G = (T, N, P, h) be a max-product grammar. A re-

placement function of G is a function S from (T U N)* into the set {s E (TUN)* I p(s,t) > 0 for some p E P and t E (TUN)*} such that S(r) = s implies m(r, s) > 0.

It follows that S (r) = s if and only if some occurrence of s in r will be replaced . Let R(G) denote the collection of all replacement functions of G. at, Example 3.4.6 Let N = {so, t}, T = {a, b}, and P = {so bs o , so t ~ bt, t a b} . Then m(bt, t) = 1 since bt = utv implies u = b and y = A . Also, S(bt) = t since p(t, bt) > 0.

Example 3.4.7 Let N = {so, A, B}, T = {a, b}, and P = {so ~ B,

B ~ BAB, B ~ b, A a} . Then m(BABABAB, BAB) = 3 since BABABAB = BABA(BAB)A = BA(BAB)AB = A(BAB)ABAB . However, there does not exist t E (T U N) * and p E P such that p(BAB, t) > 0 .

Definition 3.4.8 Let r, w, s, t E S* and k E N . r

k

w mod(s, t) if there exist u, v E S* such that r = usv, w = utv, and m(us, s) = k. k We have that r w mod(s, t) if and only if w can be obtained from r by replacing the kth occurrence of s in r by t.

Definition 3.4.9 Let G = (T, N, P, h) be a max-product grammar and S =

TUN . (1) With all p E P, S E R(G), and k E N, associate the function from S* x S* into R> O, where A(P,6,k) (r, w) _

P (6(r), t) 0

A(P

s k)

k if r w mod (S (r) , t) otherwise.

(2) With all z E (P x R(G) x N)*, associate the function A z from S* x S* into R>o defined recursively as follows: AA (r, w)

1 -{ 0

if r = w ifr :?~w

and Az(P,6,k) (r, w) = V,ES* [Az(P,6,k) (v, w) A, (r, v )] . (3) Define the function AG from T* into [0,1] by b'r E T* AG(r) = VzEZ* VAEN h(A)A,(A,r), where Z = P x R(G) x N.

© 2002 by Chapman & Hall/CRC

3.4. MaxProduct Grammars and Languages

105

Example 3.4.10 Let N = {so, A, B, C, D}, T = fa, b}, and P = {so oAC, C H AD, A ~ B, A ~ a, since oAA = aA(A)A = a(A)A . z Also, aAAb aABbmod(A, B) (Note oAA = aA(A)A = a(A)A.) p(A, B) > 0. Hence

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

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

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

(aAAD, aABb)

=

=

(aAAD) = D. We have

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

Now AG(aABb)

=

= =

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

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

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

where s E S* .

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

into [0,1] .

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

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

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

V PEPp(s, t)

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

0

© 2002 by Chapman & Hall/CRC

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

3.4 . MaxProduct Grammars and Languages

107

and 1 h(A) _ -{ 0

if A = Ao if A7~ A0 .

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

0

Moreover, it can be verified that

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

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

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

_ 0 F(s, t)

if F(s, t)

0,

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

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3. Fuzzy Machines, Languages, and Grammars

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

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

(3.6)

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

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

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

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

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

3 .5

Weak Regular Max-Product Grammars

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

3.5 . Weak Regular Max-Product Grammars

109

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

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

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

~

1 0

ifs=tEN

otherwise

and i

Pn+1 (s, t)

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

=

~O

if s, t E N

otherwise .

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

_ { o {Po(s, t),

Define G = (T, N, p,

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

AO)

otherwise .

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

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

if sEN,t~N

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

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

otherwise .

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

110

3. Fuzzy Machines, Languages, and Grammars

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

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

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

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

E p(q, u, y, q) <_ 1

VEY* q'EQ

and E geQ h(q) S 1.

In the above definitions, Q, X, and Y are, respectively, the state, input, and output sets . p(q', y, u, q) is the grade of membership, or conditional probability in the case of normalized AMA, that the next state of M is q' and output string y is produced, given that the present state of M is q and input symbol u is applied. h(q) and g(q) are, respectively, the grade of membership, or probability in the case of normalized AMA, that q is the initial state of M and that q is a final state of M. Definition 3.5.5 Let M = (Q, X, Y, p, h, g) be an AMA . (1) pm is the function from Q x X* x Y* x Q into all q',q"EQ,uEX,xEX*, and yEY*, pM(q,A,A,q")

such that for

1 if q' = q11 =~ 0 if q'zAq"

and pM (q~, ux, y, q")

=

© 2002 by Chapman & Hall/CRC

V{p(q~, u, yi, q)p' (q, x, y2, q") y = yly2, Y1, Y2 E Y*} .

I q E Q,

3.5 . Weak Regular Max-Product Grammars

111

(2) FM is the function from X x Y into Ilgo'-oO such that for all x E X* and y E Y*, FM (x, y) = V {h(q)p ' (q, x, y, q')g(q') I q, q' E Q} . (3) D

is the function from X* into D lvl (x )

= VIFlvl

(x, y) I y E Y* }.

(4) RM is the function from Y* into R"

such that for all x E X*,

0° such that for all y E Y*,

(y) = V{F` (x, y) I x E X' }.

Clearly, D and R" are fuzzy languages over X and Y, respectively. Using the terminology given in [216], the class of all F is the class of fuzzy functions computable by AMA, the class of all D is the class of fuzzy languages acceptable by AMA, and the class of all R is the class of fuzzy languages that can be generated by AMA . It follows from the above definitions that a synchronous strict AMA reduces to a max-product machine if g(q) = 1 for all q E Q . Example 3.5.6 Let M = (Q, X, Y, p, h, g), where Q = {q1, q2}, X = {u}, Y = {0,1}, h : Q ----> [0,1], g : Q ----> [0,1], and p : Q x X x Y* x Q is defined as follows: p(q1, u, ll, q2) p(q2, u, 0, q1)

= =

[0,1]

1

2~

3 4~

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

q1)

uu, 011, q2)

1

2

3 8~ _3 4 3 8'

3 4 _1

2

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

© 2002 by Chapman & Hall/CRC

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

3. Fuzzy Machines, Languages, and Grammars

112 nEN,n>2. Then

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

= = = =

=

= _ _ _

(2 4 2(I)n(3)n-1 ¢

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

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

= =

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

=

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

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

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

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

=

RM(znn)

= =

RM( zn,n-1)

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

-

DM(u2n-1

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

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

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

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

g(A)={

1 0

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

otherwise

if A = Ao if AzAAO

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

© 2002 by Chapman & Hall/CRC

if A

E Ao .

3.5. Weak Regular Max-Product Grammars

113

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

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

if t = yq' otherwise .

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

0(q)

PO(go,t) _

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

h(q) _ It follows that A =

AG

if q = qo if gzAqo .

and G has the desired properties.

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

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

_ 1 g(`4) - { 0

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

ifA=A 1 orA=A2 otherwise

and h(A) =

P(AO, A) 1 0

if A = A2 if A = AO otherwise .

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

114

3 . Fuzzy Machines, Languages, and Grammars

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

if x E X otherwise .

The converse follows in a similar manner .

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

1 0

by for every

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

and p'

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

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

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

I

q E Q} .

by for every x E X*,

I

q, q E Q} .

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

Z

© 2002 by Chapman & Hall/CRC

3.5 . Weak Regular Max-Product Grammars

115

Proof. Let A = Am , where M = (Q, X, p, h, g) is an MA. Define M' _ (Q, X, Y, p', h, g), where Y = {vo} and for all q, q' E Q, u E X, y E Y*, _ p(q, u, q~) p (q' u ' y' q ) - { 0

if y = vo otherwise .

Clearly, M' is asynchronous AMA with a single input, and A = D" . Thus (1) implies (2) . That (2) implies (3) is immediate. Now, suppose A = Dm, where M = (Q, X, Y, p, h, g) is an AMA. Define M' = (Q, X, p', h, g), where for every q, q' E Q and u E X, p'(q, u, q') = V,Ey*p(q, u, y, q') . It follows that A = Am' . Hence (3) implies (1) . Theorem 3.5.13 The following statements are equivalent:

(1) A is a finitary strict weak regular fuzzy language ; (2) A = AG for some strict max-product grammar G = (T, N, p, AO), --> A; where D(p) contains only regular discrete fuzzy productions and AO -P (3) there exists a strict regular fuzzy language A' such that A(r) = A(r) for all r :?~ A; (4) A = Rm for some finitary strict AMA M; (5) A = Rm for some synchronous strict AMA M with a single input; (6) A = Dm for some strict AMA M; (7) A = Dm for some synchronous strict AMA M with a single output; and (8) A = Am for some strict MA M.

Proof. The proof follows from Theorems 3.5 .1 and 3.5 .12. Theorem 3.5 .14 The following statements are equivalent:

(1) A = AG for some max-product grammar G = (T, N, p, AO), where D(p) contains only regular discrete fuzzy productions and AO --P-+ A; (2) there exists a regular fuzzy language A' such that A(r) = A(r) for all r :?~ A; (3) A = Rm for some synchronous AMA M; (4) A = DM for some synchronous AMA M with a single input; (5) A = Dm for some AMA M where the range of the transition function p is a subset of [0, m] for some m E R'o ; (6) A = Dm for some synchronous AMA M with a single output; and (7) A = Am for some MA M.

By Theorem 3.5 .13, many interesting properties of finitary strict weak regular fuzzy languages can be obtained from the results of Section 5.16. Similar results can be established for the family of fuzzy languages characterized by Theorem 3.5 .14. © 2002 by Chapman & Hall/CRC

116

3 .6

3. Fuzzy Machines, Languages, and Grammars

Weak Regular Max-Product Languages

In this section, we study the family of languages generated by weak regular mar-product grammars with cut point . Definition 3.6.1 Let A be a fuzzy language over T and c E L(A, c, >) = {r E T*

I

00 .

Define

A(r) > c} .

L (A, c, _), L(A, c, =), etc. are defined in a similar manner.

Note that if G = (T, N, p, AO) is a mar-product grammar, then r E N)* such that AO =:g=> r L(AG, c, >) if and only if there exists w E (D(p) x mod(w) and p > c. Theorem 3.6.2 For every L C_ T*, there exists a strict weak regular fuzzy

language A over T such that L = L(A,1, _) = L (A, c, >) for all c such that O<_c<1 .

Proof. Let G = (T, N, p, AO) be the mar-product grammar such that N = {Ao} and D(p) contains all discrete fuzzy productions of the form AO --P-> r, where r E L. Then AG has the desired properties. m Definition 3.6.3 A phrase-structure grammar is a quadruple G =

(T, N, P, AO), where T is the terminal set, N is the nonterminal set, P is a finite collection of productions, and AO E N is the start symbol . Derivations according to G, the language L(G) generated by G, and the type i (i = 0, 1, 2, 3, ) grammars in the Chomsky hierarchy are defined in the usual manner . (See [77] or Chapter 1 .)

Theorem 3.6.4 L is regular if and only if L = L(A, 0, >) for some finitary weak regular fuzzy language A.

Proof. Suppose A = AG, where G = (T, N, p, AO) is a finitary weak regular mar-product grammar . Define a phrase structure grammar G' = (T, N, P, AO), where P contains all productions of the form s --> t where p(s, t) > 0. Clearly, the language L(G) is regular and L(A, 0, >) = L(G'). The converse follows easily. Theorem 3.6.5 Let 0 S c < 1 .

L = L(A, c, >) for some finitary strict weak regular fuzzy language A if and only if L\{A} = L(A', c, >) for some strict regular fuzzy language A' .

Proof. The proof follows from Theorem 3.5 .13 Theorem 3.6.6 L is regular if and only if L = L(A, c, >) for some finitary strict weak regular fuzzy language A and 0 <_ c < 1 .

© 2002 by Chapman & Hall/CRC

3.6. Weak Regular Max-Product Languages

117

Proof. Suppose L = L(A, c, >) for some finitary strict weak regular fuzzy language A and 0 5 c < 1. By Theorem 3.6 .5 L\{A} = L(AG, c, >), where G = (T, N, p, AO) is a strict regular max-product grammar . Let D' be the collection of all discrete fuzzy productions of D(p) of the form I s - -+ t, t ~ T* . Let D" be the collection of all discrete fuzzy productions of D(p) of the form s -P-+ t, t ~ T* and p < 1 . Clearly, r E L(AG, c, >)

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

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

p'(s, t) - {

p(s, t)

0

if p(s, t) > c otherwise .

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

In

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

AO) be a weak regular max-product grammar where the range of p is a subset of [0, m] with m < 1 . Then L(AG, c, >) is finite for all 0 <_ c < 1 .

Proof. It is clear that r E L(AG, c, >) implies lrl S n, where n is the largest integer such that mn > c. m Note that Theorems 3.6.5 to 3.6.8 hold if > is replaced by >, or =. © 2002 by Chapman & Hall/CRC

118

3. Fuzzy Machines, Languages, and Grammars

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

<) is regular for all finitary weak regular fuzzy

Theorem 3.6 .11 If L = L(A, c >) for some finitary weak regular fuzzy language A and c >, 0, then L = L(AG, c, >) for some finitary max-product grammar G = (T, N, p, AO), where D(p) contains only regular discrete fuzzy productions and AO --P-+ A. Proof. Let L = L(AG,, c, >), where Gl = (T, Nl, pl , Al) is a finitary weak regular max-product grammar and c >, 0. By Theorem 3.6 .5 we may assume that 0 <_ c < 1. It follows from the proof of Theorem 3.5.1 that we may assume that pl satisfies the conditions given in Theorem 3.5.1 as long as we allow pl to assume the value oo . By Theorem 3.6.9, L(AG,,oo,=) = L(G2), where G2 = (T, N2, P, A2) is a phrase structure grammar such that P contains only productions of the form A2 ---+ A, A ---+ a, and A ---+ aB, where A, B E N2 and a E T. Without loss of generality, we assume that N1nN2=Q1 . LetAo~TUNjUN2andN=NjUN2U{Ao} . LetD be the collection of discrete fuzzy productions over T U N consisting of all discrete fuzzy productions of the following forms: (i) s --P-> t, where (s --P-> t) E D(p) and p < oo; (ii) Ao -P--> t, where (Al -P--> t) E D(p) and p < oo ; (iii) s -I -> t, where (s ---> t) E P ; (iv) AO - I -+ t, where (A2 ---> t) E P . Clearly, D is finite and contains only regular discrete fuzzy productions, and AO --P-+ A. Moreover, it can be shown that L = L(AG, c, >), where © 2002 by Chapman & Hall/CRC

3.6. Weak Regular Max-Product Languages

11 9

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

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

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

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

Theorem 3.6.14 Let c >, 0.

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

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

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

© 2002 by Chapman & Hall/CRC

E T*, and t = (n

120

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

O(q,

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

Properties of £

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

3.7. Properties of X

121

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

E

I, L2

E

1Z implies Ll

n L2 E -C-

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

Thus Ll

Am'( x

_

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

0

)

Hence Ll

if x if x

rl L2

E L

L.

E -C .

0

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

E

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

n L2 = {an bm

I n >, 1} .

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

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

12 2

3. Fuzzy Machines, Languages, and Grammars

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

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

0

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

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

AAA

'-->

A.

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

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

© 2002 by Chapman & Hall/CRC

3.7. Properties of X

123

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

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

n L z,4 Ql}.

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

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

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

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

d (rw)T = WTrT

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

12 4

3. Fuzzy Machines, Languages, and Grammars

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

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

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

yx E L) .

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

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

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

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

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3.8. Exercises 3 .8

125

Exercises

1. Let T be a norm on [0,1] . Prove that Va E A, aTa <_ a . Prove also that aTa = a Va E [0,1] if and only if T = A. 2. Let V = {(al, . , an) I a2 E [0,1], i = 1, . . . , n}, where n E N. Let T be a t-norm on [0,1] . Assume that Va, b E [0,1], a < 1 implies aTb < b. For all a E [0,1] and Vx, y E V, let aTx denote (aTal . . . . , aTan), where x = (a,, . an) and let x V y denote (al V bl . . . . , an V bn), where y = (bl, . . . , bn ) . Define the function C : 'P(V) ~ 'P(V) by VX E 'P (V), C(X) = {Vm l aZTx2 I ai E [0, 1], x2 E X,

2 = 1, . . . , n2;

m E N} .

Prove that the following assertions hold: (a) VX

E P(V),

(b) VX,Y (c) VX

X C C(X) ; X C Y implies C(X) C C(Y) ; C(C(X)) = C(X) .

E P(V),

E P(V),

3. Let X,Y E P(V) be such that X C_ Y. X is said to generate Y if Y = C(X), where C is defined in Exercise 2. If X generates Y and there does not exists X' C X which generates Y, then X is called a set of vertices of Y. Prove that X is a set of vertices of Y if and only if Y = C(X) and Vx E X, x ~ C(XV X}) . 4. Let X,X',Y E P(V) . If X is a set of vertices of Y and X' generates Y, prove that X C X'. Conclude that a set of vertices is unique . 5. Let X, Y E 'P(V) be such that X C Y. If every element of Y can be expressed uniquely in the form VT_ laiTx 2 , a2 E [0,1], x2 E X, prove that X is a set of vertices of Y. 6. Complete the proof of Theorem 3.4 .16 . 7. Show that Theorems 3.5 .1 and 3.5 .2 do not hold in general for weak regular fuzzy languages . 8. Prove that Theorems 3.6.5 to 3.6.8 hold if > is replaced by > or = .

© 2002 by Chapman & Hall/CRC

Chapter 4

Fuzzy Languages and Grammars 4.1 Fuzzy Languages

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

NU

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

Vx E T* .

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

n A2)

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

Vx E T* .

(4.1)

n A2 (4.2)

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

© 2002 by Chapman & Hall/CRC

127

(4.3)

128

4. Fuzzy Languages and Grammars

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

w)

= C,

C

> 0,

(4.4)

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

__>

srt --C---> swt

(4.5)

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

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

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

(4.6)

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

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

© 2002 by Chapman & Hall/CRC

(4.7)

4.1 . Fuzzy Languages

129

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

f

G

PG

(x)

(x)

=

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

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

(x)

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

1)

Then, on (4.8)

,

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

= = = =

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

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

= = = =

,t (A,

0.s 0.6 0.4 0.2

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

0 .8

->

PG

(0)

0.4

->

->

->

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

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

->

PG

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

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

PG

(x)

© 2002 by Chapman & Hall/CRC

13 0 4 .2

4. Fuzzy Languages and Grammars Types of Grammars

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

C

c

__ c__~

__ c__~

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

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

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

the subchain

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

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

. .

, CZ,

CZ+1 ~

. . .

,

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

© 2002 by Chapman & Hall/CRC

n{Cl, . . .

, CZ,

C7+1~

. .

.

, C, ,,+l

f

4.3. Fuzzy Context-Free Grammars

131

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

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

(4.9)

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

if j > i.

(4.10)

Let IT U NJ = k. Since there are at most k' distinct strings in (T U N)* length l and since the derivation chain is loop-free, it follows from (4.10) of that the total length of the chain is bounded by 1o=1+k+ . . .+klxl . We next provide a method for generating all finite derivation chains from S to x of length <_ lo. We start with S and using P generate the set Q1 of all strings in (T U N)* of length <_ IxI that are derivable from S in one step. We then construct Q2, the set of all strings in (T U N)* of length <_ IxI, that are derivable from S in two steps . This is accomplished by noting that Q2 is identical with the set of all strings in (T U N) * of length <_ IxI that are directly derivable from strings in Q1 . Continuing this process, we construct consecutively Q3, Q4, . . . , Qk until k = to or Qk = 0, whichever happens first. Since the QZ, i = 1, . . . , k, are finite sets, we are able to find in a finite number of steps all loop-free derivation chains from S to x of length <_ to and thus to compute PG (x) by the use of (4 .9) . This, then, constitutes an algorithm, though not necessarily an efficient one, for the computation of PG (x) . Hence G is recursive . m 4 .3

Fuzzy Context-Free Grammars

Many of the basic results in the theory of formal languages can easily be extended to fuzzy languages . As an illustration, we now sketch, without giving proofs, such extensions in the case of the Chomsky and Greibach normal forms for context-free languages . We first consider the Chomsky normal form for fuzzy context-free languages . We recall the following result from the theory of crisp grammars. Any context-free language without A is generated by a grammar in which all © 2002 by Chapman & Hall/CRC

132

4. Fuzzy Languages and Grammars

productions are of the form A ~ BC or A ~ a, where A, B, C are nonterminals and a is a terminal, [96, Theorem 4.5, p. 92] . This form is known as Chomsky normal form. Let G be a fuzzy context-free grammar . Then, any such grammar is equivalent to a grammar G' in which all productions are of the form A -c > BC or A -c > a, where c > 0 and A, B, C are nonterminals and a is a terminal. We construct G' in three stages . First, we construct a grammar Gl equivalent to G in which there are no productions of the form A B, A, B E N. Suppose that in G we have productions of the form A --~ B, which lead to derivation chains of the form A~B1~B2~ . . .~B~~B~r, where r ~ N. Then we replace all such productions of the form A C-+ Bl, Bl C2 B2, . . . , Br B in G by single productions of the form A -c -> r, in which c= p, (A===> B)Aft(B,r),

(4.11)

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

BI)

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

(4.12)

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

(4 .13)

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

© 2002 by Chapman & Hall/CRC

4.3 . Fuzzy Context-Free Grammars

133

replace this production by the productions A

__c___>

BD,

Dl I B2 D2

D,-2

i

I

(4.14)

Bm-,Bm,

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

S

0 .6

S

bA

Bib 0.3 A -> bSA

aB

Ana

B~aSB

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

0 .8

0 .6

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

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

0.8

CIA

Cl imb 0.6

0.3

C3D1

D1 ~SA

C2~a

C3 -1-> b 0.5 B C4D2

Ana

D2 ~SB

Bib

C4~a

S

C2 B

A

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

13 4

4. Fuzzy Languages and Grammars

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

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

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

(4 .15)

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

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

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

j =1 . . . , m

Z-----> ri z,

i=1, . . .,k,

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

i = 1, . . . , k .

Z-----> r2 ,

(4 .16) (4 .17)

where (4 .18)

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

(4 .19)

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

© 2002 by Chapman & Hall/CRC

1 >, k + 1.

(4 .20)

4.3 . Fuzzy Context-Free Grammars

135

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

Ak --~ Al s,

Ak -----> as,

aET

(4 .21) (4 .22)

Zk

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

Al ~ A2A3 A2 -4 A3A1 A2 ~ b.

3 ~ AIA2 3 ~ a

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

Al

0 .8 0 .6

A2A3 b

A2 A3 ~ b

A2 ~ A3A1 A3 ~ A2A3A2

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

© 2002 by Chapman & Hall/CRC

136

4. Fuzzy Languages and Grammars

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

Al A2

0.8

A2A3

0.6

b

2 ~ A3Al

A3 ~ A3AlA3A2

A3 ~ bA3A2

A

3 ~ a.

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

2 ~ A3Al 0.2 A3 ~ bA3A2

0.6

A3

0.5 ->

a

Z3 ~ AlA3A2Z3

A3

0.5 ->

A3

0.2

bA3A2Z3

aZ3

Z3 ~ AlA3A2

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

A3 A3

0.2 0.5

bA 3A2 a

A2 ~ bA 3A2Al

A2 ~ aA l A2 A2 ~ b

A Z

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

3 ~ bA3A2Z3

A3 ~ aZ3

2 ~ bA3A2Z3Al ~ aZ3Al

Al ~ bA3A2AIA3 l ~ aAIA3

3 ~ AjA3A2Z

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

© 2002 by Chapman & Hall/CRC

4.4. Context-Free Max-Product Grammars

137

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

3 ~ bA3A2AlA3A3A2 3 ~ bA3A2Z3AlA3A3A2

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

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

The final

~ 0.5 ~ ~ 0.5 ~ 0.2

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

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

4 .4

Context-Free Max-Product Grammars

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

13 8

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

form A

(3)

4 x,

71

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

4

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

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

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

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

k2 =

1 and

z2_1 = uAZv,

where u E T*,

L y(mod w) .

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

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

and © 2002 by Chapman & Hall/CRC

= nwEW*

VAEN 97(A)Aw(A, x)

4.4 . Context-Free Max-Product Grammars

AG(x) -

139

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

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

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

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

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

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

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

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

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

4

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

14 0

4. Fuzzy Languages and Grammars

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

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

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

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

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

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

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

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

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

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

if (sot) EP otherwise .

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

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

1

0

if s=tEN, otherwise,

4.5 . Context-Free Fuzzy Languages

Gs' t) - {

0AE

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

_

14 1

if s,t E N,

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

into

U N) * x (T U N) *

0 {~(sh)'wise

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

such that for all

if s E N, t E N,

Let G = (T, N, P, A O) be the CMG, where P is the set of all fuzzy productions s -P > t in which ~'(s, t) = p > 0. It follows that G has the desired properties. Let x E T* . For all w E (P x N)+ and E > 0, there exists W O E (Po x N)+ such that Au, (A, x) > A,, o (A, x) - E . Moreover, for every W O E (Po x N) +, there exists w E (P x N)+ such that A,,,, (A, x) <_ Au, (A, x) . Thus A = AG. The converse follows from the fact that

AG(X)

=

A{v{n(A)aw(A, x) I A E N} I w E (P

Iw1 <- IXI} .

x N)*,

By Theorem 4.4 .10, it can be seen that the usual procedures, [96], can be modified to obtain the following normal form theorems for context-free max-product grammars . Theorem 4.4.11 (Chomsky normal form) A is a finitary CFFL if and only if A = AG for some reduced CMG G = (T, N, P, AO), where P contains only fuzzy productions of the forms A, B,CEN, and aET.m

AO 4 A,

A

4 a,

and A

4 BC,

where

Theorem 4.4.12 (Greibach normal form) A is finitary CFFL if and only if A = AG for some reduced CMG G = (T, N, P, AO), where P contains only fuzzy productions of the forms aET, zEN*, Iz1 <2 .0

AO 4

A and A

4

az, where A E N,

Theorems 4.4 .11 and 4 .4 .12 are, in general, not valid if A is not finitary, unless we extend the definition of fuzzy productions to include A -P-> x, where p = oo . 4 .5

Context-Free Fuzzy Languages

In this section, we study the set of languages generated by context-free max-product grammars with cutpoints . Let A be a fuzzy language over T and r E Let L(A, r, >) denote the set {x E T* I A(x) > r}. © 2002 by Chapman & Hall/CRC

14 2

4. Fuzzy Languages and Grammars

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

if and only if L =

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

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

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

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

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

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

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

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

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

4.5.

143 (3) (4) Let

.

and {ql and Lo in occurrence L(A)

x,

. For . .

GA Clearly, that Ll, r

. .

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

. . ., .. .

.

.

::3k

. :::3k

::3k .

.

.

Theorem .5.5 c all .

.

.

. .

. Note

.5 .4 .5.6

Proof. remarks © 2002 by Chapman & Hall/CRC

::3k

.

Proof.

Theorem

For

; Lo,

AG be xI

.

L2

. .

.5 .5

.

. .4.11

.4 .12,

. .4 .12,

.

14 4

4. Fuzzy Languages and Grammars

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

a, a, qi) I a

E T, q E Q} .

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

E

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

r, >) I A is a CFFL}

and G = U,ER>oL, .

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

LO = C .

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

E

R>o . Then C C G, . .

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

E

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

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

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

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

4.5 . Context-Free Fuzzy Languages

14 5

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

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

m

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

© 2002 by Chapman & Hall/CRC

146

4. Fuzzy Languages and Grammars

Theorem 4.5.13

(anbncn

I

n

> 1} ~ G.

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

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

s2

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

I

n

>_ 1} is a context-sensitive language.

There exists a context-sensitive language that

is

not in

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

4 .6

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

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

147

4 .7. Trees and Pseudoterms2

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

Trees and Pseudoterms l

As before, let N be the set of natural numbers and N* be the set of all strings on N including the null string A. A finite closed subset U of N* is called a finite tree domain if the following conditions hold: (1) w E U and w = uv implies u E U, where u, v, w E N* ; (2)wnEUandm<_nimplies wmEU,where wEN*, m,nEN. Let U be a finite tree domain . Then the subset U = {w E U I w - 1 ~ U} is called the leaf node set . A pair (N; T) of finite alphabets N and T, where N n T = 0, is called a partially ranked alphabet . A tree t on a partially ranked alphabet (N; T) is a function from a finite tree domain U into N U T, written t : U_~ (N ; T), such that t(w) E N for w E_ U\U, t(w) E T for w E U. Of course, a finite tree t : U ---> (N ; T) can be represented by a finite set of pairs (w, t(w)), i .e., {(w, t(w)) I w E U} . Trees on (N; T) can be represented graphically by constructing a rooted tree (where the successors of each node are ordered), representing the domain of the mapping, and labeling the nodes with elements of N U T, representing the values of the function . Thus in the following figures there are two examples ; as a mapping, the left-hand tree has the domain U = {A,1, 2,11,12} and the value at 11 is a. Note also that U = {2, 11, 12} .

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

© 2002 by Chapman & Hall/CRC

14 8

4. Fuzzy Languages and Grammars

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

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

This correspondence can be made precise in the following manner : (1) If a pseudoterm tP E D(NST) is atomic, i.e., tP = a E T, then the corresponding tree t has domain {A} and t{A} = a. (2) If tP = A(tP . . . tPP), then t has domain UZ<_ .{iw I w E domain (t2)} U {A}, t(A) = A, and for w = iw' in the domain of t, t(w) = t2(w') . We denote the set of trees on (N; T) by D(iv,T), its element by t, and the pseudoterm corresponding to a tree t by p(t) or tP . A fuzzy set T of trees is defined by a membership function PT D(iv,T) ~ [0,1] . The set of all fuzzy set of trees is denoted by F(D(w,T)) .

4.8

Fuzzy Dendrolanguage Generating Systems

We present a fuzzy system that generates fuzzy sets of trees, as an extension of the dendrolanguage generating system of [101] . Definition 4 .8.1 A fuzzy context-free dendrolanguage generating

system (F-CFDS) is 5-tuple, S = (No, N, T, P, AO), such that the following conditions hold : (1) No is a finite set of symbols whose elements are called

nonterminal node symbols, (2) N is a finite set of symbols whose elements are called node symbols, (3) T is a finite set of symbols whose elements are called leaf symbols, (4) P is a finite set of fuzzy rewriting rules of the form

which are also represented by

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4.8. Fuzzy Dendrolanguage Generating Systems

149

t E D(w;NO UT) ; c E [0,1] or equivalently by A -c~

p(t) ;

p(t) is a pseudoterm corresponding to a tree t. (5) Ao E No is an initial nonterminal node

symbol.

Define the fuzzy relation ==cz> on the set D(iv ;ivo UT) of trees as follows : For all cti,,3 E D(iv ;ivo UT) cti =c=~> 3 if and only if (i) p(cti) = xAy, (ii) = xp(t)y, and (iii) A c~ t is in P, where x, y E [N U No U T U p(l3) A E No and t E D(w;NO UT) . Moreover, define the transitive closure =*=c~> of fuzzy relation ==cz> as follows : (1) cti =*=z> cti for all cti E D(,v ;,voUT), (2) cti ~ 3 if and only if c = V cED(N,NO UT) {c' A c"

a ~ y, y

C

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

dendrolanguage (F-CFDL) generated by

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

97},

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

A

A

a A

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

© 2002 by Chapman & Hall/CRC

150

4. Fuzzy Languages and Grammars n

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

a a

a

\

A a

a

a a

Figure 4.3 or equivalently,

A

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

Normal Form of F-CFDS

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

or

Figure 4 .4

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

4.9 . Normal Form of F-CFDS

151

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

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

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

Figure 4 .5

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

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

. . .

Wy

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

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

== cz>

Hence xAy we have

c z> ==

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

For

xAy ==c=> xp(t)y

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

© 2002 by Chapman & Hall/CRC

152

4. Fuzzy Languages and Grammars

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

A

(1)

x0

a

or

(2)

Figure

k>_1

4.6

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

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

U No

4 .7

Figure

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

Figure

where

~Z

= Xi if Xi E No and

~Z

4 .8

is a new symbol if Xi E T, and rules

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

© 2002 by Chapman & Hall/CRC

4.9 . Normal Form of F-CFDS

153

Example 4.9.4 Consider the following set of rules:

Figure 4 .9

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

The normal form for this F-CFDS

X

Figure 4 .10

© 2002 by Chapman & Hall/CRC

1 .0

P-

b

154

4. Fuzzy Languages and Grammars

4 .10

Sets of Derivation Trees of Fuzzy ContextFree Grammars

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

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

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

= QS

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

where Ua _Z_1 is the leaf node set of

ctiwi

- 1 and by c' = c A c2 .

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

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

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

Yk

Figure 4.11

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

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

155

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

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

Example 4.10 .3 0 .5 S_ _>

Consider the F-CFG given by the following rules :

AB, A

0.8

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

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

b

b Figure 4 .13

For the derivation of F-CFG, SznABzUABAzMaBAznabAzaabb

© 2002 by Chapman & Hall/CRC

156

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

Figure 4.14

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

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

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

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

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

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

157

(1) If

Figure 4.15

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

is in

PG,

(2)

(~2, X2) . . .

(~k, Xk)

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

If A

--->

(A, a)

__c___>

(f

a)

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

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

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

(A, A)

(A, a) (A, A)

(A, A)

(A, a) (A, a)

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

(f, a)

158

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

has a derivation tree <X, A>

<X, a >

Figure 4 .17

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

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

4 .11

Fuzzy Tree Automaton

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

© 2002 by Chapman & Hall/CRC

4.11 . Fuzzy Tree Automaton

159

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

S

S

2

k

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

For a tree

t E D(iv,T),

define a fuzzy transition function

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

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

t

be

X(t1t2 . . .tk)

ctit (8182 . . . Sn, s)

_

=

in terms of pseudoterm.

Then for

f =

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

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

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

Let Define ce as follows :

S=

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

© 2002 by Chapman & Hall/CRC

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

sn 1 .0

cti(A)

sA

s~

sn

8a

1 .0 0 0 0 .2 0

0 1 .0 0 0 .8 0

0 0 1 .0 0 1 .0

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

{sn}.

160

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

A

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

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

a2 (A)

06 (a)

06 (a)

06 (a)

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

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

QS

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

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4.11 . Fuzzy Tree Automaton

161

following one :

a

a

a

A

S

Figure 4 .

Theorem 4.11 .3

Any

F-CFDL

is recognizable by an

F-TA.

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

Any fuzzy set of trees recognizable by an

© 2002 by Chapman & Hall/CRC

F-TA

is an

F-

162

4. Fuzzy Languages and Grammars

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

trees

of

any F-CFG is a

fuzzy

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

Fuzzy Tree Transducer

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

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

I

ft4~ (t1,t2)

= c,

t2 E D(w2 ;T2 )1 .

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

I

ft(ti,t2)

=

ufrb

of D(N2 ;T2) . The

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

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

4.12. Fuzzy Tree Transducer

163

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

7-

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

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

C,

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

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

~

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

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

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

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

==c,

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

i.e., -b(M) is

x

D(N2 ;T2)

1,

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

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

© 2002 by Chapman & Hall/CRC

164

4 . Fuzzy Languages and Grammars

Example 4.12 .2

x

Consider the following set of rules :

1. 0

a, ) Figure 4 .23

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

© 2002 by Chapman & Hall/CRC

4.12. Fuzzy Tree Transducer

165

be determined as follows:

)L

)L

Figure 4.24

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

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

© 2002 by Chapman & Hall/CRC

166

4 .13

4. Fuzzy Languages and Grammars

Fuzzy Meaning Languages

of

Context-Free

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

A

© 2002 by Chapman & Hall/CRC

4.13. Fuzzy

Meaning

of

Context-Free

Languages

O o

f~

0 .9 very

c

o f

Y

C

© 2002 by Chapman & Hall/CRC

167

168

4. Fuzzy Languages and Grammars C (10)

1 .0

~' Y

~' Y

C (11)

0 .6

~S

O (12)

1 .0 old

f old

Y (13)

1 .0

f

young

young

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

0 laNo (old k) - { [1 +

(k

50 )-2]-i 5

for k < 50 for k > 50

and laNo (young, k)

- {

1

[1 + ( k

25)2]-l 5

fork < 25 for k > 25,

respectively, where k E K.

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

4.13. Fuzzy

Meaning

of

Context-Free

Languages

169

(A)

old

young Figure 4 .25

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

old

young

old

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

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

© 2002 by Chapman & Hall/CRC

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

170

4. Fuzzy Languages and Grammars

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

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

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

4 .14

Exercises

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

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

A~CaS, B~CaD2, Ca a,

A~Cb Dj , Bib, Ca b,

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

->

5. Use induction on

Ixl

to finish the proof of Theorem 4.4 .4.

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

4.14. Exercises

171

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

© 2002 by Chapman & Hall/CRC

Chapter 5

Probabilistic Automata and Grammars 5 .1

Probabilistic Automata Approximation

and

Their

The presentation in Sections 5.1-5 .7 is essentially from [166] . We consider probabilistic automata and their approximation by nonprobabilistic automata . Let X be a finite alphabet and let X* be the set of all words over X. Suppose that a preassigned function f : X* ~ [0,1] is given. We consider the construction of a physical device (black box) that is capable of reading words fed into it and such that after a word x is read, an output q(x) is produced such that the following properties hold V'x E X* : (1) q(x) provides full information as to the exact value f(x); (2) q(x) provides full information as to whether f(x) > c or f(x) < c for a given real number c, 0 < c < 1 ; (3) q(x) provides enough information to approximate f(x) within any preassigned E; (4) q(x) provides enough information to approximate within any preassigned E whether f(x) > c or f (x) < c for a given c, 0 <_ c < 1 . The solution to the above problems has potential applications in areas such as pattern recognition, communication of information, and analysis of noise processes in sequential networks. The study of functions f of the above form, but over general spaces (not necessarily X*), has been introduced in [255] with a similar purpose. Functions over X*, which may result from noise processes, have been studied in [21] while the study of special kinds of functions of the above type, namely, functions induced by probabilistic automata, have been studied in [165, 180] . The opinion is expressed in [166] that the solution to problems (1) and © 2002 by Chapman & Hall/CRC

173

17 4

5 . Probabilistic Automata and Grammars

(2) above involves requirements that are too stringent . The following reasons are given . In connection with the second problem, it was proved in [180, Theorem 2] that even the set of functions f over X* induced by probabilistic automata is an uncountable set . However, the set of Turing machines (which is the most powerful set of sequential devices having a finite number of internal states is a countable set . It was also shown in [180] that if a function induced by a probabilistic machine is such that there is a number d > 0 with If (x) - cl > d for all x E X*, then f can be realized by a finite state automaton, but the above condition (that If (x) - cl > d) seems to hold for only degenerate cases . In addition to the above-mentioned theorem in [180], there are other theorems on probabilistic automata that seem to imply that the nature of the cutpoint c and its relation to the range of the function f (x) is related to the equivalence or nonequivalence of probabilistic automata to nonprobabilistic automata . The following are such theorems . Theorem 5 .1 .1 (165 There exists a probabilistic automata A such that the set of words T(A, c) (e.g ., the set of all words x such that f (x) > c, where f (x) is the function induced by A) is regular if and only if c is a rational number. Theorem 5 .1 .2 (165 There exists a probabilistic automata A such that the set of words T(A, c), with a given rational number c, is not a regular set. Theorem 5 .1 .3 (186 For any probabilistic automata A with a single input letter, there are only finitely numbers c such that T (A, c) is not regular.

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

© 2002 by Chapman & Hall/CRC

5.1 . Probabilistic

Automata

and

Their

Approximation

175

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

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

function (fsf) .

Definition 5.1.5 A fuzzy star acceptor (fsac) is a pair (0, c), where is afsfand0
Definition 5.1.6 If (0, c) is an fsac, then the set of tapes accepted or defined by (0, c), written T((O, c)), is defined as follows: T((O, c)) = {x

I x E X*, O(x) > cf .

We now define and make distinctions between probabilistic, deterministic, automata acceptors, and machines . Definition 5.1.7 A finite automaton (fa) over the alphabet X is a triple

A = (Q, /t, qo), where Q is a finite nonempty set (the internal states of A), qo is an element of Q (the initial state, and /t is a function from Q x X into Q (the transition function .

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

accepted by

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

set.

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

17 6

5. Probabilistic Automata and Grammars

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

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

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

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

i

1

3

3

Q =

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

A(b)

2

1

2

2

4 ] , and

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

z4 ] . Also,

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

- ~ 1

0

if q,EF otherwise.

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

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

5.2 . E-Approximating by Nonprobability Devices

177

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

0

97 F(ab)

.

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

7r

Also = =

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

12 5 9

~

5

12

4 9

j

~

0 1

1

5

12 4 9

Furthermore, i p(ab) _ [ 4

5

4

12 4 9

_ 7

16

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

A is a pa and 0 <_ c < 1 . The set of words defined or accepted by a pac (A, c) is the set

Clearly, a pac gives rise to an fsac. Definition 5.1 .14 Two acceptors are said to be equivalent if they accept the same set of words.

Definition 5.1.15 An fa B is said to be equivalent to a pa A (to an fsf 0) if there is a function 0 from the set of states Q of B into the interval [0,1] such that for all x E X*, O(B(qo, x)) = p(A, x), (O(B(qo, x)) =O(x)) . If B is a fa that is equivalent to a pa A (fsf 0), then for any cutpoint c, it is possible to transform B into an fac that is equivalent to the pac (A, c) (fsac (0, c)) .

5 .2

E-Approximating by Nonprobability Devices

The importance of Turing machines follows from the widely held belief that any function that can be computed by a digital computer can be computed © 2002 by Chapman & Hall/CRC

17 8

5. Probabilistic Automata and Grammars

by a Turing machine. This assertion is known as Turing's hypothesis or Church's thesis. Church's thesis implies that a Turing machine is the correct abstract model of a digital computer. The formal definition of an algorithm follows from these ideas, namely, an algorithm is a Turing machine that eventually stops after given an input string. Definition 5.2.1 A Tuning automaton over the alphabet X is a triple A = (Q, /t, qo), where Q is a finite nonempty set (the internal states of A), qo E Q (the initial state, and ft is a function from Q x X into Q x X x {-1, 0,1} (the transition function .

The operation of a Turing machine is described as follows : An infinite tape is fed into the machine. The machine begins scanning the input word on the tape, starting at the left-most input symbol in its initial state and reading a symbol at a time. If the scanned symbol is x2 and the machine is in state qj, then it replaces x2 by another symbol xk, changes its state into a new state qi, and moves the tape either left or right or keeps it stationary depending on the value m in ft(qj, x2) = (q1, xk, m), where m = 1 means left move, m = -1 means right move, and m = 0 means no move. In particular, the input string can be changed. A Turing machine A accepts a string x if A halts in an accepting state, where x is input . It can be shown that a language L is generated by a phase-structure grammar if and only if there is a Turing machine that accepts L. Example 5.2 .2 Consider the Turing automaton A = (Q, ft, qo) over X =

{xl, x2, x3 , x4, 0}, where Q = {qi, q2, q3, q4, q5, q6, q7} and where ft is defined as follows: ft (qo, 0) = (0, q4, 1 ) ft (go,x1) = (x1,gi,-1) ft (go, x2) = (x2, qo, -1 ) ft (go, x3) = (x3, qo, -1 ) ft (go, x4) = (x4, qo, -1 ) ft (gi, 0)= (0, q3, 1 ) ft (gi,x1) _ (x1,gi,-1) ft (gi, x2) _ (x2, qo, -1 ) ft (gi, x3) _ (x3, qo, -1 ) ft(gi, x4) _ (x3, q2, 1 ) ft(g2,XI) _ (x4,g7,1) ft(g3, xi) = (xi, q5, 1 )

ft (q4, 0) = (0, q7, -1) ft(g4, xi) _ (xi, q4, 1) ft(g4, x2) _ (x2, q4, 1) ft(g4, x3) - (x3, q4, 1) ft(g4, x4) _ (x4, q4, 1) ft (q5, 0) = (x2, q6, -1 ) ft(g5, XI) _ (xi, q4, 1 ) ft (q5 , x2) _ (X2, q4, 1 ) ft (q5 , x3) - (X3, q4, 1) ft (q5 , x4) _ (X4, q4, 1 ) ft(g6, XI) _ (x3, q6, 1 ) ft(g6, x2) _ (X2, q7, 1 )

and for any other (q, x) E Q x X, ft(q, x) = (q, x, 0) . The symbol 0 is a special symbol called a blank. Let p* and q* be arbitrary expressions . Suppose ft(gi, xj) = (xk, qi, Then expressions of the form p*gixjsq* are transformed into expressions of the form p*xkglsq* . Expressions of the form p*qixj are transformed into expressions of the form p*xkgl0. Other expressions are not transformed.

© 2002 by Chapman & Hall/CRC

5.2 . E-Approximating by Nonprobability Devices

179

Suppose ft(gi, xj ) = (xk, q1,1) . Then expressions of the form p*x'gixjq* are transformed into expressions of the form p*glx~xkq* . Expressions of the form gixjq* are transformed into expressions of the form glOxkq* . Other expressions are not transformed. Note that in the definition of /t, when ft (q, x) :?~ (q, x, 0), q is never q7 . Hence q7 is, in effect, a "stop" state . Consider the expression g o x 2 x 3 x l x 4 x3 . The machine will transform this successively into x2gOx3xlx4x3, x2x3goxlx4x3, x2x3xlglx4x3, x2x3g2xlx3x3, and x2g7x3x4x3x3, at which point it will stop . Definition 5 .2 .3 A linear bounded automaton is the same as a Turing automaton, but its input tape is finite . The length of the input tape of a linear bounded automaton is a linear function of the length of the input word printed on the tape . It is known that a language is context sensitive just if it is recognized by some nondeterministic linear-bounded acceptor . However, it is not known whether each such language can be recognized by a deterministic linear bounded acceptor . It is worth noting that the theory of context-sensitive languages has had little impact on programming languages . Any tape of nonprobabilistic automata having finitely many states and capable of reading tapes is called a nonprobabilistic (np) device . Definition 5 .2 .4 An np device B is said to E-approximate an fsf 0 if there is a function qi from the states of B into the interval [0,1] such that for all xEX*,

10(x) -

O(B(go, x)) I < E .

If for given 0 and E there is a B satisfying the above properties, then be E-approximated .

0

can

The motivation of this criterion can be seen by the proposition and corollary below. Definition 5 .2 .5 A Turing automaton (or a linear bounded automaton) B is said to E-approximate the fsf 0 if B when fed with a tape x performs a finite number of operations, at the end of which a number d is printed on the tape such that I d - 0(x) < E . We assume here that the symbols X are numerals so that a word x = xl . . . . , xk represents the number xl . . . xk . Proposition 5 .2 .6 Let an fsf 0 and E be given. Then 0 can be E-approximated by a Turing automaton (or linear bounded automaton) B if and only if 0 is z E-computable by B .

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18 0

5. Probabilistic Automata and Grammars

The proof of this proposition follows easily from the definitions . Corollary 5.2.7 The fsf induced by pa's can be E-approximated by Turing automata. Clearly, Turing automata, having infinite tape, can compute the fsf's induced by pa's for any tape x within any arbitrary given E . However, this may not be true for linear bounded automata. The set of fsf's induced by pa's is not a countable set, but the set of Turing automata is countable .

5 .3

E-Approximating by Finite Automata

In this section, we characterize fsf's that can be E-approximated by finite automata . Definition 5.3.1 Let 0 be an fsf and E > 0 . An E-cover induced by 0 is a finite set {Ci}k o, where the Ci are subsets of [0,1] satisfying the following conditions. (1) Uk OCi = {d I O(x) = d, x E X*} . (2) dl, d2 E Ci ==> Idl - d2 l <_ E, i = 0, 1, 2, . . . , k. (3) Let Ciz for z E X* be defined as Ciz = {d I O(xz) = d, O(x) E Ci}.

It follows that for any i and any z there exists a

j

such that Ciz C Cj .

Theorem 5.3.2 Let 0 be an fsf and E > 0. Then 0 can be E-approximated by a fa B if and only if there is an 2E-cover induced by 0 .

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

where j Set

is

the smallest index satisfying this relation. 1

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

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

5.4. Applications

181

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

2 [VdEG d

+ ndEc,d] I

<

E

since O(x) E CZ . Suppose that 0 can be E-approximated by an npa B . Let the states of B be qo, qi, . . . , qk and define the sets Ci = {d I O(x) = d, B(qo, x) = qZ},

i =0,1, . . . , k . It follows that the set 5 .4

{CZ

I i = 0,1, . . . , k} is a 2E-cover . m

Applications

Definition 5.4.1 An fsf 0 is said to be quasi-definite if for all E > 0 there is k(E) such that for any x with Ixj > k(E) the inequality I0(x) - 0(y) I < E holds, where y is the k(E)-suffix of x . Quasi-definite pa's have been introduced in [165] by a similar definition. In that paper, a decision procedure was given for determining whether a given pa is quasi-definite. A theorem similar to the next theorem has been proved in that paper for pac's . Theorem 5 .4.2 Let E > 0. Any quasi-definite fsf 0 can be E-approximated by an fa .

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

Ct+i

= =

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

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

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

Clearly, Ui1,C

Z

=

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

If 0(u) and O(v) are in the same set Cn+i (the sets Ci for i < n are one-sets and hence are out of consideration), then u = ulzi and v = vlzi . Thus I0(ulzi)-O(zi)I <_ 2E

d

I0(vlzi) - O(zi)I <_ 2E

since 0 is quasi-definite . Hence I 0(u) -OMI = I0(ulzi) -O(vlzi)I <- E . © 2002 by Chapman & Hall/CRC

182

5. Probabilistic Automata and Grammars

Finally, it follows from the definitions that C+Zw

0(uw) = d, O(u) E C +Z} O(xziw) = d, x E X*} O(xwlzj) = d, x E X*}

= = =

{d {d {d

C

C' .+7 ,

where zj is the k(2 E) suffix of ziw. The set {C+Z}Z_i' is thus an E-cover . Hence by Theorem 5.3 .2, the desired result follows . A definite fa can be defined as an automaton B such that there is an integer k with the property that B(qo, xy) = B(qo, y) for all y with I y~ > k and for all x. Then from the proof of Theorem 5.4 .2, it follows that if A is a quasi-definite automaton, then A can be E-approximated by a definite automaton B. The converse is also true. The following example shows that there are fsf's that can be E-approximated by npa's, but are not quasidefinite . Example 5.4.3 Consider the fsf 0 induced by the pa A defined as follows: X = {a, b}, the initial vector is 7r = [ 0

Q = {qo, qi },

1 ] , the final vector is

and the

transition matrices are P(a) _ -

1

1 0

0 1 ]

and

P(b)

_

i

1

i 0

Then O(x) is the (1,1) entry in the matrix P(x) . Clearly, 0 is not quasidefinite, for O(x) = 0(bk), where k is the number of b's in the word x (e .g., - 4'(a O(ban) = O(b) for any n) . Therefore, for any n, I0(ban) n )I 2 contradicting Definition 5.4 .1 . To show that 0 can be E-approximated for any E, consider the following cover induced by 0 . Divide the interval [0,1] by k-1 points dl, d2 . . . . , dk_1 (do = 0, dk = 1) so that d2 - d2_1 = 2 E, i = 1, 2, . . . , k - 1 and 1 - dk_1 <_ -'EDefine the cover {CZ}ko as follows:

=

CZ={d

I

d2
I

O(x)=d, xEX*},

i = 1, 2, . . . , k - 2 . Then (1) and (2) of Definition 5 .3 .1 clearly hold . We now show that (3) of Definition 5 .3 .1 holds. Let O(x), 0(y) E CZ . Then 0(xa), 0(ya) E CZ . If 7r(x) = (u, 1 - u) and 7r(y) = (v,1 - v),

© 2002 by Chapman & Hall/CRC

5.5 . The PE Relation

183

then u-v for u =

O(x)

and v 7r(xb) 7r(yb)

= 0(y) = =

<

E,

with O(x), 0(y) E Ci, and

(-In + 1 - u, n) ( 2 v + 1 - v, i 2v)

_ _

(1 - -In, in), (1- 2v, 2v) .

Hence l0(xb)

-

0(yb)l

= _ <

(1 - 2u) - (1 - 2v) I ~2(v-u)I 2E .

By induction, it follows that for all z, either

O(xz) = O(x) and 0(yz) = 0(y) or

l0(xz) - O(yz) l <- E 1 provided that

10(x) - O(y)

l

<-

E.

Thus for any z and i, either Ciz = Ci or the maximal distance between any two points in CZz is not greater than 2E . Now let c and d be two points in Ci such that c < d . Suppose that dj < c <_ dj+1 . Then since l c - dl < 2 , we have that d. <_ d <_ dj+2 and, therefore, c and d are both in Cj . Hence Ciz C Cj for some j. Thus the function 0 can be E-approximated .

5 .5

The PE Relation

Definition 5.5.1 Let 0 be an fsf and E > 0. A PE-relation induced by 0 is a relation over X* satisfying the following properties : (1) PE is symmetric and reflexive; (2) PE is right invariant, i.e., xPE y ==> xzPEyz for x, y, z E X* ; (3) xpEy ==> l0(x) - O(y) E. l

<

Definition 5.5 .2 A relation over a set X is said to be of finite index if there is an integer k such that in any subset of k + 1 elements of X there are at least two elements that are related. The following lemma is easily proved.

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184

5. Probabilistic Automata and Grammars

Lemma 5.5 .3 Given 0 : X* ---> [0,1] and E > 0. Define the relation PE on X* by V'x, y E X*, xPEy ~b'z I0( xz ) - O(yz) I < E. Then any PE -relation induced by 0 is a refinement of PE .

0

Lemma 5.5 .4 For given fsf 0 and E, if 0 can be E-approximated by an fa, then there is a P2E relation of finite index induced by 0. Proof. Consider the machine B that E-approximates P. As in the proof of Theorem 5 .3.2, let {CZ}k o be the E-cover defined by B. Then O(x) and (y) are in the same set CZ if B(qo, x) = B(qo, y) = q2 . Define the relation PE over X* by xPEy ~ B(qo, x) = B(qo, y) . Then PE is clearly symmetric, reflexive, right-invariant, and of finite index, by definition . Condition (3) in Definition 5.5 .1 is also satisfied since xPE y implies that O(x) and 0(y) are in the same set CZ, and as in the proof of Theorem 5.3 .2, this implies that

10(x) - 0(y)I < 2E . Combining Lemma 5.5 .3 and Lemma 5 .5 .4, we have the following result . Proposition 5 .5 .5 Let 0 be an fsf and E > 0. If 0 can be E-approximated by an fa, then the explicit relation P2E defined in Lemma 5.5 .3 is of finite index. Consider the set Pn={C=(CI, " . . ,Cn ), CZ

> 0,

~CZ=1} . i=1

Let E > 0. Let UE be any set such that UE C Pn for any pair of vectors c and d = (dl, .dn ) in UE,

I c2 - dZ I

>

E.

Lemma 5.5 .6 If Pn and UE are as above, then UE is a finite set containing at most k(E) elements, where k(E) = (1 + 2)n-1 for n > 1 . E

© 2002 by Chapman & Hall/CRC

5.5. The PE Relation

185

Proof. The proof of this lemma is implicit in the proof of Theorem 3 in [180] . m The bound in Lemma 5.5 .6 is not sharp and from a practical point of view, it would be desirable to have a sharp bound . We can prove the following result. Theorem 5.5.7 Let A be a pa and E > 0. If P E is induced by A, then PE is offinite index k, where

k < (1 + 1)n-1

E

and n here is the number of states of A.

Proof. Let x1, . xk be tapes that are pairwise nonrelatives by PE. Then for all 1 E

or

IIr(xi)97( 7J) - Ir(x7) 97(y)I

Now t=1

7rt xi

7rt x j

Therefore,

=

t=1

= 1( 7r (xi)

7rt xi

-7r(xj))t7(y)I

t=1

E.

rt(x7) = 1 - 1 = 0. 7

(5 .2) rt(xi) - 7 rt(xi) - 7rt(xj)), rt(xj)) = -Xt (7 Xt (7 where Xt and Xt are summations over indices t for which 7rt (xi) - 7r t (xj ) is nonnegative or negative, respectively. Combining (5 .1) and (5.2), we have that

E

< <

_ <

IEt( 7rt( xi) - 7rt(x7))nt(y)I 7t(xj)) Vt t7t(y)+ I E+(7rt(xi) - r (7rt(xi) - 7 rt(x7)) At t7t(y)I E+(7rt(xi) -7 rt(xj))(Vtt7t(y) - ntt7t(y)) rt(xi) -7rt( x7) E+7

Applying (5.2), we have

2E

n

< 1: 17rt (xi) - 7rt (x7) t=1

and this inequality implies by Lemma 5.5 .6 that the set of vectors 7r(xl), 7r(x2), . . . , 7r(xk)

is finite with

© 2002 by Chapman & Hall/CRC

18 6

5 .6

5. Probabilistic Automata and Grammars

Fuzzy Stars Acceptors and Probabilistic Acceptors

Definition 5.6 .1 Let A be an fsac, B an fac, and T(A) and T(B) the languages defined by A and B, respectively. Let E > 0. Then T(B) Eappnoxi-mates T(A) if (T(B)\T(A)) U (T(B)\T(A)) C {x I x E X*, I0(x) - cI < E}, where T(A) and T(B) are the complements of T(A) and T(B), respectively, in X* . Proposition 5 .6 .2 Let 0 be an fsf and B an fa . If B E-approximates 0, then for any c, B can be transformed into an fac that E-approximates the fsac (0, c) . Proof. Let the final states of B be the states such that O(g2) > c. If x E T(B), then B(qo, x) = qZ with O(qi) > c and since

I0(x) - O(qj) I < E, it follows that

If x ~ T(B), then B(qo, x) = qj with qi(qj) <_ c and thus O(x) <_ c + E . Now x E T(A) implies that O(x) > c and x E T(B) implies that O(x) < c. Hence the desired result follows. The following proposition is a converse of Proposition 5.6 .2 and is related to Theorem 5 in [180] . Proposition 5 .6 .3 Let 0 be an fsf such that for any c, (0, c) is an fsac that can be E-approximated by some fac B. Then there is an fa that 2Eapproximates 0 . Proof. Divide the interval [0,1] into k parts c1 , c2 . . . . , ck- 1 (co = 0, ck = 1) so that c2-c2-1 <_ E, i = 1, 2, . . . , k and let BC,, i = 1, 2, . . . , k-1 be the corresponding E-approximating acceptors for (0, c2) . Define the machine B as follows: B = (Q, qo, y), where Q = {(gzl( eo)I gz2(el),- . . ,gzk (ek-1)) I gik( ej) EQC;}, qo = (go(co), go(cl), . . . , go(ck-1)), ft((gj l (eo), q22(CI)I . . . , qi k (ek-1)), a)

© 2002 by Chapman & Hall/CRC

=

([Qgjl, a'), Y"Cl (qi2 a), . . . , [tCk l (qik , a)),

5.7. Characterizations and the R E -Relation

187

with B, =

(Qc,

~

go(cz)~ftc, ~ Fcj*

Set O(q) =4'(gil(co), gi,(cl), . . . ,gi,a(Ck-1)) =Vj{cj I gi j 1 (cj) EF~j } _

so that 4'(B(go, x)) = cj =~, x E T(Bcj ) and x ~ T(Bcj+1) . Then 0(B(go, x)) = cj implies that O( X )

>

cj - E

and O (x) < cj+l + E < cj + 3E . Thus I4'(B(go, x)) - O(x) I <_ 2E . It follows from Propositions 5 .6 .2 and 5 .6 .3 that E-approximation of an fsf by an fa is possible if and only if E-approximation of the fsac derived from that function, with any given c, is possible . We shall show, however, by an example that there are fsf's that cannot be E-approximated by fa's, but the derived fsac with some c can be Eapproximated by an fac. Corollary 5 .6 .4 The pac's are E-approximable by Turing acceptors. Proof. The proof follows from Corollary 5 .2 .7 and Proposition 5 .6 .2 . It was shown in [180, Theorem 2] and [165, Theorem 12] that the set of pac's is not a countable set even if the transition matrices have only rational entries . However, the set of Turing machines is countable . 5 .7

Characterizations and the R E -Relation

Definition 5 .7.1 Let (0, c) be an fsac and E > 0 . An E-cover induced by (0, c) is a finite set {CZ I i = 0, . . . , k}, where the CZ are subsets of [0,1] satisfying the following conditions:

© 2002 by Chapman & Hall/CRC

18 8

5. Probabilistic Automata and Grammars (1)

Uk OC2 either

= {d

I O(x) = d, x E X*}; CZC{dl d>c-E}

or CZC{d l d
Theorem 5.7.2 Let (0, c) be an fsac and E > 0. Then (0, c) can be E-

approximated by an fac if and only if there is an E-cover induced by the fsac (0, c) . If there exists an E-cover induced by 0, then there exists an E-cover induced by (0, c) . 0 The proof of the first assertion of Theorem 5.7.2 is similar to the proof Theorem 5.3 .2 and is omitted. The machine B here would be defined as in Theorem 5 .3 .2 and the final states of B would be those Ci that satisfy of

the relation

Ci C{dld>c-E} . Clearly, any E-cover satisfying the conditions of Definition 5.3 .1 also satisfies the conditions of Definition 5.7 .1 .

Definition 5.7.3 The relation RE on X* induced by an fsac ('b, c) is defined as follows: b'x, y E X*, xR Ey

l0(yz) - cl [0(xz) > c ~ O(yz) > c]) .

b'z([l0(xz) - cl > E and

> E]

It follows that the relation PE defined in Lemma 5.5 .3 is_ a refinement of the relation RE here. Thus PE of finite index implies that RE is of finite index. The following result thus holds.

Corollary 5.7.4 The relation RE induced by a pac is of finite index. Proof. The proof follows from Theorem 5.5 .7 and the above remark . Corollary 5.7.5 If the relation RE is induced by an fsac such that the corresponding fsf can be E-approximated, then R2E is of finite index.

Proof. The proof follows from Proposition 5.5 .5 and the above remark . The next result follows by using the same in the proof of Proposition 5.5 .5 .

© 2002 by Chapman & Hall/CRC

kind of reasoning as that used

5.7. Characterizations and the RE -Relation

189

Proposition 5.7.6 Let (0, c) be an fsac and E >_ 0. If (0, c) can be Eapproximated by an fac, then the induced relation RE is of finite index.

0

Definition 5.7.7 An fsac (0, c) is called quasi-definite if for any E > 0

there is k(E) such that for all x with xI >_ k(E), the following holds for all yEX* : O(x) > c (O(x) E T(O, c)) ==> 0(yx) > c - E; O(x) <- c (O(x) ~ T(O, c)) ==> 0(yx) <- c + E .

Clearly, if 0 is a quasi-definite fsf, then (0, c) is a quasi-definite fsac. The converse is, however, not true. For example, suppose that the matrices of a pa are A(a)

_

0 1

and

A(b) =

3

1

4

4

I

and O(x) is the (1,1) entry in A(x) . Then with c = 2, T((O, c)) = X*. Therefore, (0, c) is a quasi-definite acceptor. However, for any n, I0(ban) - O(an) I =

Hence 0 is not quasi-definite . We thus have the following corollary.

4

.

Corollary 5.7.8 If 0 is a quasi-definite fsf, then for any c and E, then the fsac (0, c) can be E-approximated and the approximating acceptor may be chosen to be a definite acceptor. Proof. The proof follows by the remarks following Theorem 5.4 .2 and the above remark. Corollary 5.7.8 follows, also, from Theorem 5.7 .9 below, which can be proved in the same manner as Theorem 5 .4.2 . Theorem 5.7.9 Let (0, c) be a quasi-definite acceptor and E > 0 . Then (0, c) can be E-approximated by a definite finite acceptor.

0

In [180, Theorem 2] and [165, Theorem 12] the fsac's are quasi-definite so that the set of quasi-definite acceptors is not a countable set . We now present a counterexample based on an example of H. Kesten and on an idea of R.E. Stearns (private communication to A. Paz). Consider the following pa A = (Q, 7r, {A(a)f, F) over X = {0,1} with Q = {qo, q1, q2, qs}, 7r = [ 1

0

0

0 ] ,

2 0

0

2 0

0

A(0) -

© 2002 by Chapman & Hall/CRC

0 0

1

0 0

1 0

2z

0 1

and A(1) _

1 z 1

2 1 0

1 z 1

2 0 1

0

0 0 0

0

0 0 0

19 0

5. Probabilistic Automata and Grammars

By straightforward computations, it can be shown that the following relations hold: n=0,1, . . ., 0°=A if x=0n (z)i on, 10n21 . . . Onk 1 if x = n >_ 0, j = 1, 2, . . . , k = z and there is i with n2 = 0 AX) = > i if x = on, 10n21 . . . Onk 1 n > 0, j = 1, 2, . . . , k < 2 if x = on, 10n21 . . .OnkjOnk+1 n >_ 0, j = 1,2, . . .,k nk+1 > 0, where p(x) is the (1,1) entry in A(x) . Consider now the fsf defined by A, PA, and let (PA, c) be the pac with c = 2 . Then T((PA, c)) = {x I PA (x) > } . 2

It follows from the above inequalities that T((PA, c)) for c = 2 is the set of tapes x such that x = A or x begins with a zero, ends with a one, and contains no subtapes of two or more consecutive ones. It follows that this set of tapes is a regular set since there exists an fac accepting it . Therefore, it can be E-approximated (even for E = 0) by an fac. We now show that by the use of Proposition 5.6 .2 there exists c such that (PA, c) cannot be E-approximated by an fac. Hence it is for that c with the result that PA is an fsf that cannot be approximated by an npa . Let xn be the word xn = (on l)m. It follows easily that p(xn) -

1 + [1 - (2)n ] m 2

Thus limn-,, p(xn) = 1 for fixed m > 0, while lim y-,, p(xn) - z for fixed n > 0. Now let c be a real number such that z < c < 1, say c = 4, and let E > 0 be a real number such that E < 4 . Suppose that (PA, c) can be E-approximated for the given c and E. Let the approximating machine have k states . Choose no sufficiently large so that p(x o) >c+E, m=1,2, . . . ,k+1 . The first k + 1 applications of the input sequence x must send the approximating machine B through a sequence of states qo, ql, . . . , qk+l, which are all final states of B. However, B has only k states . Hence qk+1 = qZ for i < k + 1. Consequently, all the tapes of the form x m = 1, 2, . . . , are in T(B) . Thus B cannot E-approximate PA since there is an mo with p(x o°) < c - E, i .e.,

o,

o° ) - cl > E,

lp(q

while xoo E T(B) and x oo ~ T(A) . The following are direct consequences of the above example . © 2002 by Chapman & Hall/CRC

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191

(1) There is a pac that cannot be approximated by an fac . (2) There is an fsf that cannot be approximated by an fa . This follows from the example and Proposition 5 .6 .2 . (3) There is a pa that cannot be approximated by an fa, but the acceptor defined by the pa with some c (c = z in the example) can be approximated by a fa. Therefore, the two concepts of approximation are not equivalent . (4) The class of pa's is stronger than the class of fa's . This is a consequence of the intrinsic nature of the probabilistic automata and not of the actual properties of the cut point . (5) There is an fsf and E > 0 such that there is no E-cover induced by this fsf. (See Definition 5 .3.1 .) (6) There is a fsac and E > 0 such that there is no E-cover induced by this fsac . (See Definition 5 .7 .1 .) We have shown that fsf's that are computable (e .g ., pa's) can be approximated by Turing automata . However, as shown by the previous example, there are fsf's defined by pa's that cannot be approximated by fa's. We pose in the Exercises some rather difficult problems concerning the ability of certain types of automata to approximate fsf's and pc's . We have shown that Turing automata can approximate pa's, but fa's cannot . We ask in the Exercises if push-down automata can approximate pa's . We have shown in Section 5 .5 that the condition that the relation P E defined in Lemma 5 .5 .3 induced by a given fsf is of finite index is a necessary condition for the function to be approximated by a fa . This condition is however not sufficient since, by Theorem 5 .5 .7, that relation is of finite index for any pa, but the previous example is a pa that cannot be approximated by a fa.

0

0

The previous example also shows that there is a pa which cannot be approximated by fa's, but the acceptor defined by that pa with some c can be approximated by a fa . The two concepts are therefore not identical and the above remarks and the problems posed in the Exercises may be stated and posed also for the case of approximating fuzzy star acceptors separately.

5.8

Probabilistic and Weighted Grammars

In the next several sections, we consider the work in [188] . The customary Chomsky hierarchy of formal languages is obtained by imposing restrictions on the form of the rewriting rules, i .e ., productions . In certain grammars, an application of some production determines which productions are applicable on the next step . These are called programmed grammars . In ordered grammars, some productions can never be applied if some others are applicable. In matrix grammars, only certain previously specified strings of productions can be applied or, more generally,

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192

5. Probabilistic Automata and Grammars

in a grammar with a control set, the string of productions corresponding to a derivation must belong to a set of strings previously specified . See, for example, [185, 61, 1, 80] . We consider time variant and probabilistic grammars. In a probabilistic grammar, there is given together with each production a stochastic vector whose i-th component indicates the probability that the i-th production is applied after the production . Furthermore, there is given an initial probability distribution over the set of productions . Thus each derivation is assigned a probability. The language generated by a probabilistic grammar consists of all words P generated by the grammar such that the probability assigned to the derivations of P is greater than some previously chosen cutpoint. We consider the following two interpretations. In the maximal interpretation, it is required that there is at least one derivation of P with probability greater than the cutpoint. In the sum interpretation, it is required that the sum of the probabilities assigned to the distinct derivations of P is greater than the cutpoint. The latter interpretation is customary in connection with probabilistic automata . A probabilistic grammar is a special case of a weighted grammar . In a weighted grammar, vectors with arbitrary nonnegative components are considered instead of stochastic vectors . This situation can be viewed as having a reward or punishment associated with the application of each production. Some restrictions on probabilistic as well as on weighted grammars may be imposed in order to guarantee the effectiveness of the procedures. For example, one may assume that the probabilities and the weights are rational. Let G = (N, T, Xo , F) be a phrase structure grammar, where N is the set of nonterminals, T is the set of terminals, Xo is the initial symbol, and F is the set of productions . Derivations according to G, the language L(G) generated by G, as well as type i(i = 0, 1, 2,3) grammars in the Chomsky hierarchy obtained by imposing restrictions on F, are defined in the usual fashion . Let { fl , . . . , fk} be a set of distinct labels for the productions in F. Let D : Xo = PO

===>

fj

===> cl> Pl

fj cz>

2

P

===>

fj c3>

.

. .

==

:~>fjcT>

PT

(5 .3)

be a derivation according to G, where in the transition from Pi to PZ+1 (0 <_ i < r) the production labeled by fj(Z+1) with 1 <_ j(i + 1) < k is applied. Then the word f7(1) f7(2) . . . f7(,)

over the alphabet { fl, . . . , fkI is termed a control word of the derivation (5.3) . If r = 0, then the control word is defined to be the empty word A. A derivation from Xo determines a unique control word, provided the productions in F are distinct. However, the existence of two identical productions in F is not excluded in the following discussion. © 2002 by Chapman & Hall/CRC

5.8. Probabilistic and Weighted Grammars

193

By a language, we mean any set of words. Let C be a language over the alphabet {fl, . fk} . Then the language Lc(G) is defined to be the subset of L(G) consisting of words that possess at least one derivation whose control word is in C. Lc (G) is called the language generated by G with control language C. Example 5.8.1 Consider the grammar G = ({X, Y, Z}, {x, y, z}, X, F), where the productions of F are as follows: fl f2 f3 f4 fS fs f7

: X XYZ : X xX Y ~ YY :Z~ZZ :X ~x :Y~y :Z -----> z

Assume that C consists of all words of the form fl (f2 f3f3)Zf5f6f7,

i = 0, 1, 2, . . . .

Then Lc(G) = {x'y'z2 I i > 1} .

(5 .4)

The language (5.4) is also generated by the grammar G = ({X, Z}, {x, y, z}, X, F), where the productions of F are as follows: fl f2 f3 f4 f5

: X XZ :X~XXy :Z~ZZ xy : X :Z~z,

with the control language consisting of words of the fo fl (f2f3)Zf4f5,

i = 0, 1, 2, . . . .

Control languages provide a uniform way of describing grammars with restrictions on the use of productions . The notion of a control language used here differs from the notion of a control set used in [80], where only leftmost derivations were considered. © 2002 by Chapman & Hall/CRC

19 4

5. Probabilistic Automata and Grammars

Let G be a type i grammar (i = 0, 1, 2,3) whose productions are labeled by the labels in the set { fl, . . . , fk}. (See Definitions 1 .8 .3 and 1.8 .7.) Let co be a function of the set { fl, . . . , fk} into the set of k-dimensional row vectors with nonnegative components, and let S be a k-dimensional row vector with nonnegative components. Then the triple (G, S, cp) is called a weighted grammar of type i . If, in addition, S as well as the values of co are stochastic vectors, then (G, S, cp) is a probabilistic grammar of type i. In a probabilistic grammar, S is referred to as the initial distribution of the productions . The u-th component of the vector cp(f,), where 1 <_ u, v <_ k, is referred to as the probability of applying the production labeled by f~ after applying the production labeled by fv . Consider a weighted grammar G = (G, S, co) . We assign a numerical value O(D) to each derivation (5.3), where r > 0. If r = 1 then O(D) is defined to be the j(1)-th component of S. Suppose that O(D) has been defined for the derivation (5.3), where r > 1. Then, for the derivation Dl : Xo =

PO ===>j(1)

Pl

==~>f,(z) . . . ==>fj(T)

PT

==>fj(T+,)

P,+,,

O(DI) is defined to be cp(D)[~P(f~(T))]~(~.+1), where the second factor is the j (r + 1)-th component of the vector cp(fj (,)) . Hence for a probabilistic grammar Gp = (G, S, co), the number O(D) can be interpreted as the probability of the derivation D. Let c be a nonnegative number . We now define two languages L,., (Gu c) and L, (G,,, c) generated by the weighted grammar G,, = (G, S, cp) with cutpoint c. The former is defined to be the subset of L(G) consisting of all words that possess at least one derivation D such that O(D) > c.

(5 .5)

The latter is defined to be the subset of L(G) consisting of all words P such that J:O(D) > c,

(5 .6)

D

where D ranges over all distinct derivations of P. Thus two derivations are distinct if they possess distinct control words . Suppose that L = L, (Gw, c), for some weighted grammar G,, = (G, S, cp), where G is a type i (i = 0, 1, 2,3) grammar and c is some cutpoint . Then L is said to be generated by a weighted grammar of type i under maximal interpretation or, merely, L is mi .m. Similarly, if L = L,(G u c), L is said to be generated by a weighted grammar of type i under sum interpretation or, merely, L is mi .s . Moreover, if G,, is a probabilistic grammar we say that L is p.i .m. or p.i.s., respectively. Clearly, for a probabilistic grammar GP, both L, (GP, c) and Ls (GP , c) are empty whenever c > 1 . By a rational weighted (probabilistic) grammar, we mean a weighted (probabilistic) grammar, where the components of S as well as the components of each value of co are rational. If a language L is mi .m. and also © 2002 by Chapman & Hall/CRC

5.8. Probabilistic and Weighted Grammars

195

the corresponding weighted grammar and cutpoint are rational, then L is said to be r.w. i. m. The abbreviations r.w.i.s., r.p.i.m. and r.p. i. s. are defined similarly. It will be seen that the sum interpretation corresponds to the interpretation customary in connection with probabilistic automata . Following the customary definition in automata theory, we have assumed a strict inequality in (5 .5) and (5.6) . A more difficult problem arises in characterizing the language families if in (5 .5) and (5.6) the symbol > is replaced by the symbol > . Example 5.8.2 Consider the type 2 probabilistic grammar Gp with nonter-

minals X and Z, terminals x, y, and z, initial symbol X, and the following labeled productions: X~XZ f2 :X~XXy f3 :Z~ZZ f4 :X~xy A :Z~z

~0 ~0 0 ~0 0

2 0 0 1 2 0 0 0 0 0

2 0~ 0 0~ z 0 0 1~ 0 1

The values of co are given together with the productions. Let the initial distribution be [ 1 0 0 0 0 ] . It is easy to see that (5.4) is the language generated by Gp with cutpoint 0 under both maximal and sum interpretation. Thus (5.4) is p.2.m. and p.2 .s. and, hence, also w.2.m. and w.2.s . Since all components involved are rational, the language (5.4) is also r.p.2.m, r.p.2.s., r.w.2 .m ., and r.w.2 .s. In a probabilistic grammar, the probabilities tend to 0 with the length of derivations . Consequently, c = 0 is the only interesting cutpoint for probabilistic grammars, whereas the structure of weighted grammars is much richer. We first prove that the family of type i languages in the Chomsky hierarchy is contained in each of the families involving i introduced above, where i = 0, 1, 2,3. Theorem 5.8.3 For i = 0, 1, 2, 3, any language L of type i is r.p.i .m, and

r.p.i .s. Furthermore, L is r.w.i.m., r.w .i .s., p.i .m ., p.i .s., w.i.m., and w.i.s.

Proof. Assume that L = L(G), where G is a type i grammar whose productions are labeled by the elements of the set {fi, . . . , fk}. Define

6= ~P(fj) = [ k

. ..

k I,

j=1, . . .,k.

Consider the probabilistic grammar Gp = (G, S, co) . Clearly,

L = Ls (GP, 0) = L~ (GP , 0) . © 2002 by Chapman & Hall/CRC

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5. Probabilistic Automata and Grammars

Thus L is r.p.i.m. and r.p.i.s. That L satisfies the remaining properties follows easily from definitions . In the statement of the next theorem, PQ* denotes the language consisting of all words PQ', i = 0, 1, 2, . . . . Thereby, P and Q are words. Theorem 5.8.4 Let G be a grammar. If the control language C is a finite sum of languages of the form PQ*, then LC (G) is finite . Proof. Since clearly Lc+D (G) = Lc (G) + LD (G) ,

it suffices to consider the case where the control language C is of the form PQ* . If Q is the empty word, then the desired result holds . Hence assume that Q is not empty. Clearly, only a finite number of derivations from the initial symbol possess the control word P. Thus there is only a finite number of last words in these derivations . It suffices to consider one such last word R and show that, starting from R, control words in Q* do not lead to an infinite number of terminal words. Consider the productions labeled by the letters of Q. Let u(v) be the total number of nonterminals appearing on the left (right) sides of these productions, where each nonterminal is counted as many times as it occurs. If u <_ v, then the control words Qj , j > 1, do not lead to any terminal words. If u > v and t is the number of nonterminals in R, then the control words Q j , where j > t, are not applicable. Hence the desired result holds . As was seen in the examples above, Theorem 5.8 .4 does not remain valid for control languages of the form PIQ*P2 . It is clearly not valid even for control languages of the form Q*P . Theorem 5.8.5 Let Gp be a probabilistic grammar of type i, i = 0, 1, 2, 3, and c > 0. Then the language L , (GP , c) is finite. Proof. Let Gp = (G, S, cp) . Then we show that L, (GP , c) = Lc (G),

(5 .7)

where the control language C is a finite sum of languages of the form PQ* . Since c > 0, the derivation of any word belonging to the left side of (5.7) contains at most u transitions with probability < 1, where the bound u depends on c and on the greatest probability < 1 occurring in GP. On the other hand, if a sequence of transitions with probability 1 does not constitute a loop, then this sequence cannot contain more productions than the total number of productions . It is clear that if a loop (with probability 1) is entered, it is impossible in the derivation to leave this loop. Thus if there are k distinctly labeled productions in G, then the control words of © 2002 by Chapman & Hall/CRC

5.9 . Probabilistic and Weighted Grammars of Type 3

197

the derivations of the words in the language on the left side of (5.7) are of the form PQj, j = 0, 1, 2, . . . , where the lengths of P and Q possess a finite upper bound. In fact, the length of P does not exceed ku + k + u, and the length of Q does not exceed k. Hence (5.7) holds with C of the form mentioned and the desired conclusion follows by Theorem 5.8 .4 . Theorem 5.8.6 For i = 0, 1, 2, 3, the family of p.i .m . languages is a subset

of the family of p.i .s . languages. Similarly, the family of r.p .i .m . languages is a subset of the family of r.p .i .s . languages.

Proof. For any probabilistic grammar Gp, Lm (Gp, 0) = Ls (Gp, 0) .

By Theorem 5 .8 .3, all finite languages are r.p .3 .s. The desired result now follows by Theorem 5 .8 .5 . We ask in the Exercises whether or not the family of w.i .m . languages is a subset of the family of w.i.s. languages . The same problem can be stated also for the corresponding rational families . We show next how one of the decidability results concerning ordinary grammars can be extended to weighted grammars . For simplicity, we restrict ourselves to the rational case. By length-increasing productions, we mean productions of the form P ---> Q, where the length of P is less than or equal to the length of Q. Theorem 5 .8.7 Let G,, be a rational weighted grammar whose productions

are length-increasing and c a rational number. Then there is an algorithm for deciding whether or not a word P belongs to the language L, (G7 , c) and an algorithm for deciding whether or not a word P E L, (G.,,, n) .

Proof. An algorithm can be obtained by modifying the well-known algorithm for length-increasing grammars, [187, pp . 171-172] . In fact, the difference is that the occurrence of loops Xo==~> P1==~> . . .==~> P.==~> . . .==~> P.==~> . . .==~> P cannot be ignored . The existence of such a loop with weight > 1 guarantees that P belongs to both languages under consideration. Loops with weight <_ 1 can be ignored in the case of maximal interpretation. Under sum interpretation, the existence of a loop with weight 1 guarantees that P belongs to the language, and the effect of a loop with weight < 1 can be determined through summation .

5 .9

Probabilistic and Weighted Grammars of Type 3

A stochastic language is a language that is acceptable by a finite probabilistic automaton with some cutpoint. See [189, pp .73-77] . The au© 2002 by Chapman & Hall/CRC

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5. Probabilistic Automata and Grammars

tomaton considered may possess an initial state or an initial distribution of states. This does not affect the family of stochastic languages . A language is called rational stochastic if it is acceptable by a finite probabilistic automaton, where all of the probabilities involved are rational, with some rational cutpoint . In this section, interrelations between stochastic languages and languages generated by weighted and probabilistic grammars of type 3 are studied . Theorem 5.9.1 Let L be a stochastic language . Then L is w .3 .s . Proof. Let L be accepted with cutpoint c by the finite probabilistic automaton A = (Q, X, qo, Q1 , M), where Q is the set of states, X the set of input symbols, qo the initial state, Ql the final state set, and M the set of transition matrices. Consider the type 3 grammar G = (Q, X, qo, F), where F consists of the following productions: qu

xqv,

qv

E

Qw,

qu - A,

qv

qu

E

E

Qw,

Q1 .

x E X;

(5.8) (5 .9)

Let k denote the number of these productions . The productions are labeled by the elements of the set { fl, . . . , fk} . The grammar G is extended to a weighted grammar G,, = (G, S, co) in the following manner . The vector S consists of 0's and 1's with 1's in the positions indicating the productions with qo on the left side. The vector associated by cp to productions of the form (5.9) consists of 0's only. The vector v associated to a production of the form (5.8) is defined in the following manner . A component of v corresponding to a production with qv on the left side equals the probability of A entering qv , after being in qu and receiving the input x. The other components of v equal 0. Then the transition probabilities of A are preserved in the grammar G,, and L = Ls (Gu c) .

We call a language L stochastic under maximal interpretation if there is a finite probabilistic automaton A and a cutpoint c such that L consists of words that move A from the initial state to a final state through at least one path whose probability is greater than c. Then every language stochastic under maximal interpretation is w .3.m . This is established exactly as in Theorem 5.9 .1 . Theorem 5.9.2 Let G,, be a type 3 weighted grammar. If G,, does not contain productions of the form X ---> Y, where X and Y are nonterminals, then for any c, the language L, (G,,, c) is stochastic .

© 2002 by Chapman & Hall/CRC

5.9 . Probabilistic and Weighted Grammars of Type 3

199

Proof. We may assume, without loss of generality, that the productions of G,, are of the form X -----> XY

(5 .10)

X -----> A,

(5 .11)

and

where X and Y are nonterminals and x is a terminal . This can be seen as follows. A production of the form X --->

xlx2 . . .

xr.,

r > 1,

(5 .12)

where the xi's are terminals, is replaced by the sequence of productions X ~ x 1 X, X ~ x 2 X, . . . , X ~ x,X, X ~ A.

(5 .13)

The weights are adjusted so that in the sequence (5.13), the transition to the next production is given weight 1 and transitions elsewhere are given weight 0. Furthermore, the transition to the first production of (5.13) is given the weight originally associated with the transition to the production (5 .12) . A production of the form X ----->

xlx2 . . . x,Y,

r > 2,

(5 .14)

is replaced by the sequence of productions X

~ X I X,

X ~ x2 X, . . . , X ~ x,_ 1 X, X ~ x,Y,

(5 .15)

where the weights are adjusted as above, the vector associated with the last production of (5 .15) corresponding to the vector originally associated with (5.14) . (Note that in a weighted grammar of type 3, the vector associated with a production X -----> P, where P is a word over the terminal alphabet, does not influence the language generated .) Clearly, these changes do not affect the language Ls(Gw,c) . Suppose that the productions of Gw = (G, S, co) are of the forms (5 .10) and (5.11). We construct a "generalized probabilistic automaton" A, in the following manner . The states of A are the labels of the productions of G. The initial distribution is S. The final state set consists of the labels of the productions (5.11) . Consider a production of the form (5.10) whose label is fv . Then the "probability" of A entering the state fv, after being in fv and receiving the input x, equals the v-th component of the vector associated with fv or 0, depending on whether Y or some other nonterminal appears on the left side of the production labeled by fv. The probability of all other transitions equals 0. Hence Ls(Gw, c) = L(A, c),

© 2002 by Chapman & Hall/CRC

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where the right side denotes the language accepted by A with cutpoint c. Note that this language is defined for A by exactly the same matrix product for ordinary probabilistic automata. Although A is not a probabilistic automaton, it follows by a result in [235] that L(A, c) is stochastic . Thus the desired result holds . It follows from Theorems 5.9.1 and 5 .9.2 that the family of stochastic languages equals the family of languages generated under sum interpretation by type 3 weighted grammars that do not contain productions of the form X -----> Y. It may be possible to remove this restriction on the form of the productions . To accomplish this, it would suffice to prove that a language obtained from a stochastic language by deleting all occurrences of one letter is stochastic. This again is a special case of the conjecture that the family of stochastic languages is closed under homomorphism . It follows from Theorem 5.9.1 that the family of w.i.s. languages is nondenumerable and therefore contains languages that are not of type 0. This reflects the fact that no computability assumptions are made in the definitions about the real numbers involved. The following theorem is established exactly as are Theorems 5.9.1 and 5.9 .2 . Theorem 5.9.3 Every rational stochastic language is r.w.3 .s. Let G7 be a type 3 rational weighted grammar that does not contain productions of the form X -----> Y, where X and Y are nonterminals . Then, for any rational number c, the language L, (G,,, c) is rational stochastic. 0 Theorem 5.9.4 For a type 3 probabilistic grammar Gp and c > 0, the language L,(Gp, c) is finite . The family of p.3.s. languages, as well as the family of p.3.m. languages, equals the family of type 3 languages. Proof. First replace the grammar Gp by a grammar whose rules are of the forms (5.10), (5.11), and X ---> Y, where X and Y are nonterminals . The new grammar is then rewritten as an automaton, exactly as was done in the proof of Theorem 5.9 .2. Productions of the form X ---> Y corre spond to transitions caused by the empty word. The first assertion of the theorem now follows since every loop with an exit to a final state possesses a probability less than 1. The second assertion follows from Theorems 5.8 .3 and 5.8 .5 and the fact that the languages LS (GP , 0) and Lr (G P, 0) are of type 3. This holds since these languages are accepted by finite nondeterministic automata . Clearly, the families of r .p .3.s. and r .p .3.m. languages also equal the family of type 3 languages . However, as was seen in Section 5 .8, the family of r.p.2 .s. languages, as well as the family of r.p .2.m. languages, properly includes the family of type 2 languages . The families ofp .3.m ., p .3.s ., and w .3.s . languages have been considered. We now consider the remaining family of w .3.m . languages . Theorem 5.9.5 The family of r.w.3.m. language s properly includes the family of type 3 languages. © 2002 by Chapman & Hall/CRC

5.9 . Probabilistic and Weighted Grammars of Type 3

201

Proof. The inclusion follows, by Theorem 5.8 .3. Consider the rational weighted grammar Gw of type 3 with the productions f1 :X~XX

[2

f2 :X~YX

[0

f3 : X ~A

and with S = [ 1

0

[ 0

1 z

0

0] 1 ]

0 ]

0 ] . Clearly, Lr., (Gw, 1) = ~ x'y7 I i > j > 1} .

Consequently, the inclusion is proper . Theorem 5.9.6 There is an algorithm for deciding whether or not the

language Lr (Gw, c) is empty, where G7 is a rational weighted grammar of type 3 and c is a rational number .

Proof. We first determine all of the finitely many parts of the derivations according to G,, whose control word begins and ends with the same letter, but does not have proper subwords with this property. Suppose there exists such a loop with weight > 1 that is also a part of a terminat ing derivation with weight > 0. Then the language under consideration is not empty. Suppose no such loop exists with the described properties. Then the emptiness can be decided by checking all of the (finitely many) derivations without loops since, in this case, loops do not increase the total weight of a derivation. m It can be shown that there is also an algorithm for deciding whether or not the language Lr (Gu c) is infinite. The same problems are undecidable for the languages LS (Gu c), with G,, and c as above since the existence of a decision method would imply the decidability of the emptiness and infinity problems for rational stochastic languages . Consequently, not every r.w.3.s . language is r .w.3.m . An example is given in our next theorem. Theorem 5.9.7 The language W b' I i > 1}

(5 .16)

is nw.3.s ., but not r.w.3.m.

Proof. The first assertion follows, by Theorem 5.9 .3, and results in [234] . To prove the second assertion, we assume on the contrary that {a z bz I i >_ 1} equals Lr (Gu c), for some rational weighted grammar G,, of type 3 and rational cutpoint c. Without loss of generality, we may assume that the productions of G,, are of the forms (5.10), (5.11), and X ---> Y . Consider a word ajbj, where j exceeds the number of productions of Gw . There is a derivation of this word with weight greater than c. Furthermore, in this © 2002 by Chapman & Hall/CRC

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derivation there is a loop that begins and ends with the same production and possesses r >_ 1 occurrences of productions of the form X --~ aY . If the weight associated with this loop is > 1, then the word aj+''b~ belongs to the language L, (Gu c). If the weight is <_ 1, then the word aj `b~ belongs to the language . Thus we have a contradiction in both cases . Hence the desired result holds.

5 .10

Interrelations with Programmed and Time-Variant Grammars

The family of w.i.s. languages is a subset of the family of w .j .s . languages, for i > j, by definition . By Theorem 5.9.1, the family of w.3.s. languages contains all stochastic languages . It is very difficult to give examples of languages that are not stochastic and so it is also difficult to solve the problem of whether the family of w.i.s. languages, i = 0, 1, 2, 3, constitute a proper hierarchy. A detailed definition of a programmed grammar can be found in [185] . In a programmed grammar, the productions are labeled and together with each production there are given two sets of labels, namely, the success field and the failure field. After the application of some production f, only productions with labels in the success field of f are applicable on the next step of the derivation. If f is not applicable, the next production applied must have its label in the failure field of f . It is shown in [185] that all recursively enumerable (i.e., type 0) languages are generated by programmed grammars with context-free (i .e., type 2) core productions . From the point of view of weighted (and time-variant) grammars, programmed grammars with context-free core productions and with empty failure fields are of special interest. Theorem 5.10 .1

Every language generated by a programmed grammar G with context-free core productions and with empty failure fields is r.p .2.m . and r.p .2.s.

Proof. Let G be a programmed grammar . Then G is transformed into a rational probabilistic grammar Gp of type 2 in the following manner . If the success field of a production f contains r >_ 1 labels, then the transition from f to each of these labels is given probability 1/r, and the transition from f to all other labels is given probability 0. If the success field of f is empty then a "sink" production X ---> X is added to the grammar and the transition from f to this sink production is given probability 1. (A common sink production may be used for all productions of G with empty success fields .) The initial distribution S is defined similarly. Then the languages L S (GP , 0) and L, (GP , 0) equal the language generated by G and hence the desired result holds . © 2002 by Chapman & Hall/CRC

5.10. Interrelations with Programmed and Time-Variant Grammars203 Consider a type i (i = 0, 1, 2, 3) grammar

G = (N, T, Xo , F) . The grammar G together with an infinite sequence Fl, F2, F3 . . . .

(5 .17)

of subsets of F forms a time-variant grammar GZ, of type i. (See [187] for a corresponding recognition device of type 3.) The language L(GZ,) generated by GZ , is defined to be L C (G), where the control language C is the union of all languages of the form FIF2 . . . F , u >_ 1, where juxtaposition denotes catenation. Such type 2 time-variant grammars, where the sequence (5.17) is periodic, are of special interest. Languages generated by these grammars are called periodically time-variant context-free languages (ptvcf) . Actually, it can be shown that the same languages are obtained by assuming that the sequence (5 .17) is almost periodic. For an example, consider the type 2 grammar with the initial symbol Xo and the productions Xo ~ XXIYX2 XXl y X1 f3 : X1 ~A f4 Y2 A X2 -----> ZX2 A : X2 fl : f2 :

Time-variance is specified in such a way that, for u = 1, 2, . . . , the productions fl, f2, f3, f4 belong to the set F2 _1 and f5, f6 belong to the set F2 . Then the language generated is that of (5 .4) and thus is ptvcf. (Note that Y2 is introduced to prevent the derivation of words x2+1 g2 +lz 2 .) Derivations according to ptvcf grammars are easy to describe and perform . Neverthe less the generative power of these grammars is remarkable. It follows that the family of ptvcf languages includes all languages generated by contextfree matrix grammars . Thus the family of ptvcf languages contains all context-sensitive languages, provided the result in [1] is correct . On the other hand, ptvcf languages form a subset of the family of languages generated by programmed grammars with context-free core productions and empty failure fields . This subset is obtained by imposing on programmed grammars the additional restriction that whenever two labels fi and f2 are in the success field of a production, then the productions labeled by fi and f2 possess identical success fields. Throughout, we have assumed in considering a step Pi==~>fP2

(5 .18)

of a derivation that the production f is actually applied, i.e., Pl = QJQQ2, P2 = QjRQ2, and f is the label of the production Q -----> R, for some Q 1 © 2002 by Chapman & Hall/CRC

20 4

5 . Probabilistic Automata and Grammars

and Q2 . Another possibility is to specify a subset Fl of productions such that the notation (5 .18 may be used also in case f E Fl is not applicable, i .e ., Pl does not contain an occurrence of Q and Pl = P2 . We assume that this possibility is included in (5 .3) when control words are defined. Everything concerning weighted and probabilistic grammars is defined now as it was defined before using this new interpretation, the so-called checking interpretation, of control words . Then Theorem 5 .10 .1 can be strengthened as shown in the next result . Theorem 5 .10 .2 Every recursively enumerable language is r.p .2.m . np .2 .s ., under checking interpretation.

and

Proof. By the result in [185], any recursively enumerable language L is generated by a programmed grammar G with context-free core productions . G is first written so that for every production f, either the success field of f or the failure field of f is empty. This is accomplished by making two copies of f, one with the success field and the other with the failure field of the original f, and putting the labels of both copies into all fields that P contain the label of the original production f . All productions X

are then replaced with an empty success field by productions X Y, where Y is a new nonterminal (which, thus, does not appear on the left side of any production . The remainder of the proof proceeds like the proof of Theorem 5 .10 .1 . It is always possible in Theorem 5 .10 .2 to choose the required rational probabilistic type 2 grammar G p = (G, S, co) in such a way that the language L(G) generated by the basic grammar G is of type 3 .

5.11

Probabilistic Grammars and Automata

The material in the remainder of the chapter is from [204] and [207] . In Sections 5 .8-5 .10, the concept of a probabilistic grammar is formulated and random languages generated by probabilistic grammars are defined . See also [188] and [57] . According to Santos, none of these formulations were broad enough to encompass the conventional deterministic grammars and still preserve its probabilistic character . In Sections 5 .11-5 .15, another formulation of probabilistic grammars is presented . This formulation reduces to the conventional deterministic grammars if all functions involved are deterministic . Various types of probabilistic grammars are considered, namely, type0, context-sensitive, context-free, weakly regular, and regular probabilistic grammars . They correspond to the various types of grammars in the conventional theory. Most of the study here concerns the relations between the various types of probabilistic grammars and the corresponding types of probabilistic automata.

© 2002 by Chapman & Hall/CRC

5.12. Probabilistic Grammars

205

In Section 5.13, the concept of asynchronous probabilistic automata (APA) is defined . It is shown that every weakly regular random language can be generated by an APA, and vice versa . It is also shown that every regular random language can be generated by a synchronous APA, and vice versa. In Sections 5 .14 and 5 .15, we define the concepts of probabilistic Turing machines (PTM) and probabilistic pushdown automata (PPA) . It is shown that every bounded type-0 (leftmost-bounded context-free) random language can be generated by a bounded PTM (PPA), and vice versa . It is shown in [136] that every language generated by any grammar using only leftmost derivations is context free. A similar result for random languages is given in Section 5.15. 5 .12

Probabilistic Grammars

In this section, we present a mathematical formulation of probabilistic grammar and we define the class of random languages generated by probabilistic grammars . Definition 5 .12 .1 Let X and Y be nonempty sets. A random function from X into Y is a function F from X x Y into [0,1] such that

F (x, y) <_ 1 for all x E X. If EYE1' F (x, y) = 1 for all x E X, then F is called a total random function. EYE1'

F (x, y) is the probability that the value of the function at x is y.

Let S be a nonempty set .

Definition 5.12 .2 A (total) probabilistic production p over S is a (total) random function from S* into S* such that p (A, A) = 1 and the set p = {s E S* I p (s, t) > 0 for some t E S*, where t :?~ s} is finite . If, in addition, for all s E S*, the set {t E S* I p (s, t) > 0} is finite, then p is called bounded . An element of p is called a genetrix of p-

p (s, t) is the probability that s will be replaced by t.

Definition 5.12 .3 A (bounded) probabilistic grammar is a quadruple G = (T, N, P, h) such that the following conditions hold: (1) (2) TU N (3)

T and N are disjoint finite nonempty sets; P is a finite collection of (bounded) probabilistic productions over such that s E P = UPEPp implies s E (T U N)* N (T U N) * ; h is a function from N into [0,1] such that EAEN h(A) S 1 .

© 2002 by Chapman & Hall/CRC

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5. Probabilistic Automata and Grammars

In the above definition, T and N are, respectively, the sets of terminal and nonterminal symbols . Furthermore, h(A) is the probability that A is the start symbol of G. Recall that N is the collection of all positive integers . We now restate some definitions and observations quite similar to those in Section 3.4 for max-product grammars and languages . Definition 5.12 .4 Let r, s E S* and k E N. Then m(r, s) = k if there exist u2, v2 E S*, i = 1, 2, . . . , k, where v2 :?~ vj for i :?~ j, such that (1) r = uisv2 for all i = 1, 2 . . . . , k, and (2) r = usv implies u = u2 and v = v2 for some i, where i = 1, 2, . . . , k. Define m (r, s) = 0 if r z,4 usv for all u,v E S* .

Note that m (r, s) = k if and only if r can be expressed in the form usv in exactly k distinct ways. Definition 5.12 .5 Let G = (T, N, P, h) be a probabilistic grammar.

A replacement function of G is a function S from (T U N) * into P such that S (r) = s implies m(r, s) > 0. If, in addition, S (r) = s implies r = usv, where u E T* and v E (T U N) * , then S is called leftmost .

Let D(G) denote the collection of all replacement functions of G, and let DL(G) denote the collection of all leftmost replacement functions of G. Note that S (r) = s if and only if some occurrence of s in r will be replaced .

Then define r ,k w mod(s, t) if there exist u, v E S* such that r = usv, w = utv, and m (us, s) _

Definition 5.12 .6 Let r, w, s, t, E S* and k E N. k.

Note that r , k w mod(s, t) if and only if w can be obtained from r by replacing the k-th occurrence of s in r by t. Definition 5.12 .7 Let G = (T, N, P, h) be a probabilistic grammar and

S=TUN. (1) For all p E P, S E D(G), and k E N, associate the function from S* x S* into [0,1] , where

f(P

s k)

if r , k w mod (S (r) , t) otherwise.

(2) For all z E (P x D(G) x N)* , associate the function into [0,1] defined recursively as follows:

and fz(P,s,k) (r,

© 2002 by Chapman & Hall/CRC

w)

= E fz (r, x)f(P,s,k) (x, w) . XES*

fz

from S* x S*

5.12. Probabilistic Grammars

20 7

It is clear that fz is a random function from (T U N)* into (T U N)*. Furthermore, fz(r,w) is the probability that w can be obtained from r by the "derivation" z. Definition 5.12 .8 A random language f over S is a function from S* into [0,1] .

f(r) is the probability that r is a member of the language.

Definition 5.12 .9 A random control set 7r of a probabilistic grammar G = (T, N, P, h) is a random language over P x D(G) x N . 7r (z) is the probability that the derivation z will be applied . This concept of a control set is similar to, but distinct from that introduced in [80].

Definition 5.12 .10 Let G = (T, N, P, h) be a probabilistic grammar and 7r a random control set of G. The random language fG,, generated by G under 7r is the random language over T defined by b'r E T*,

fG,,(r) = V,Ez*7r (z) {

1: h(A)f, (A, r)},

AEN

where Z = P x D(G) x N. If 7r (z) = 1 for all z E Z*, then we write fG for fG,, . In this case, fG is said to be the random language generated by G. Furthermore, if (z)

_

1 0

if z E (P x DL(G) x {1}) * otherwise,

then we write fG,L for fG,, . In this case, fG,L is said to be the random language generated by G with leftmost derivations only. fG,, (r) is the probability that r will be generated by G under 7r . Example 5.12 .11 Let G = (N, T, p, rl) be the grammar defined in Example 9 8 7 3.4 .10. Let rl = (so ' ~ oAC), r2 = (C ' > AD), r3 = (A ' ~ B), r4 = (A a), r5 = (B

4 b),

and r6 = (D ~ b) . Then

fG ,,(aABb)

= = = =

V,Ez7r(z) - h(A) fz (oAAb, oABb) 7r(p, S, n)h(A)f(P,s,n) (aAAb, aABb) 7r(p, S, n)h(A)p(A, B) 7r (p, S, n)h(A) (.7) .

The probabilistic grammars defined above are referred to as type-0 probabilistic grammars . Other more restrictive types of probabilistic grammars can be obtained by imposing certain restrictions on the probabilistic productions . We define four such probabilistic grammars below. Three of them correspond to the conventional context-sensitive, context-free, and regular grammars . © 2002 by Chapman & Hall/CRC

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5. Probabilistic Automata and Grammars

Definition 5.12 .12 Let G = (T, N, P, h) be a probabilistic grammar.

(1) G is called context-sensitive if for all p E P, p (s, t) > 0 implies <_ I tI . (2) G is called context-free if P C N.

(3) G is called weakly regular if for all p E P, p C N and for all s E p, p (s, t) > 0 implies t = uA, where u E T* and A E N U {A} . (4) G is called regular if for all p E P, p = N and for all s E p, p (s, t) > 0 implies t = aA, where a E T and A E N U {A} .

The grammar in Example 5.12.11 is context-free. Clearly, every language generated by a conventional type-0, contextsensitive, context-free, or regular grammar in the conventional manner may be associated with fG for some type-0, context-sensitive, context-free, or regular probabilistic grammar G = (T, N, P, h), where h and every p E P, are deterministic, and vice versa. From the above definition, it follows that every regular probabilistic grammar is weakly regular and every weakly regular probabilistic grammar is context-free . Furthermore, if G is a context-free probabilistic grammar, then DL(G) contains exactly one replacement function of G. If f = fG for some specified (type-0), context-sensitive, context-free, weak-regular, and regular probabilistic grammar G, then we say that f is that specific random language, and vice versa . Theorem 5.12 .13 Let f be a (bounded) (type-0, context-sensitive, contextfree, weakly regular, regular random language over T. Then f = fG for some (bounded) (type-0, context-sensitive, context-free, weakly regular, regular) probabilistic grammar G = (T, N, P, h) such that (1) every p E P is a total probabilistic production, (2) h(Ao) = 1 for some AO E N, and (3) N C P. Proof. Let f = fG, where Go = (T, No, Po, ho) is a (bounded) (type-0, context-sensitive, context-free, weakly regular, regular) probabilistic grammar . Let Ao, A1 ~ T U No and ao an arbitrarily fixed element of T. Let N = No U {Ao, A1 } and Nl = N\P. For every p E PO, we associate the probabilistic productions p' and p" over T U N such that for all S'

t E (TUN)*,

p (s, t) P '(S' t)

1 - Ew1(TUN,,)* lJ (sl w) 1

0

© 2002 by Chapman & Hall/CRC

if 8EP, tE(TUNo) * if s E P, t = a0A, if s=t PUN or s E Nl , t = a0A, otherwise

5.13. Weakly Regular Grammars and Asynchronous Automata

209

and p"(s t) =

EAEwo h(A) p (A, t) 1 - EAENO h(A)p (A, t) 1 0

if s = Ao, t E (T U No)* if s = Ao, t = a0A, if s=tz,4 AO

otherwise .

Let G = (T, N, P, h), where P is the collection of all p' and p" defined above and 1 h(A) _ -{ 0

if A = Ao if A7~ A0 .

It follows that G has the desired properties. m We also write G = (T, N, P, AO) if it satisfies condition (2) above. Moreover, we say that G is total if it satisfies conditions (1)-(3) above. We conclude this section by introducing the concepts of random domains and ranges of random functions, which are needed later. Definition 5.12 .14 Let F be a random function from X into Y. (1) The random domain D(F) of F is the function from X into [0,1] such that D(F)(x) = E,Ey F(x, y) for all x E X. (2) The random range R(F) of F is the function from Y into [0,1] such that R(F) (y) = VxExF (x, y) for all y E Y.

5 .13

Weakly Regular Grammars and Asynchronous Automata

In this section, we study weakly regular probabilistic grammars and their relationship with asynchronous probabilistic automata. Proposition 5.13 .1 If G is a weakly regular probabilistic grammar, then .fG = fG,L . 0

Proposition 5.13 .2 If

f is a bounded weakly regular random language, then f = fG for some total weakly regular probabilistic grammar G = (T, N, P, h) such that for all p E P and A E N, p (A, t) > 0 implies t = aB, whereaETU{A} andBENU{A} .m

Definition 5.13 .3 An asynchronous probabilistic automaton (APA)

is a sextuple M = (Q, X, Y, p, h, g), where Q, X, and Y are finite nonempty sets, p is a random function from Q x X into Y* x Q, and h and g are functions from Q into [0,1] such that h(q) _ 1 . EeeQ In Definition 5.13.3, Q, X, and Y are, respectively, the state, input, and outputs sets. p(q, u, y, q') is the conditional probability that the next state of M is q' and output string y is produced given that the present state of M is q and input symbol u is applied. h(q) and g(q) are, respectively, the probabilities that q is the initial state of M and q is a final state of M.

© 2002 by Chapman & Hall/CRC

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5. Probabilistic Automata and Grammars

Definition 5.13 .4 Let M = (Q, X, Y, p, h, g) be an APA .

(1) Define the function pm from Q x X* x Y* x Q into [0,1] by for all q',q"EQ,uEX,xEX*,andyEY*, pM(q, A,A,q") and pm(q,

1 =~ 0

if q' = q11 if q'zAq"

= 1: p(q, u, yl, q)pM(q, x, y2, q1),

ux, y, q")

where the summation ranges over all q E Q and yi, y2 E Y* such that y = yiy2(2) Define the function FM from X* x Y* into [0,1] by for all x E X* and y E Y*,

F"1 (x

y) =

q,q'EQ

h(q)p' (q, x, y, q)g(q) .

The function pM has the following interpretation . p M (q', x, y, q") is the conditional probability that M will be in state q" and produce output string y given that the present state of M is q' and input string x is applied. FM (x, y) is the probability that M will produce y when x is applied . The above model of an APA is a generalization of the model introduced in [221] . Clearly, FM is a random function from X* into Y*, and D(FM) and R(FM) are random languages over X and Y, respectively. Using the terminology introduced in [216], the class of all FM is the class ofall random functions computable by an APA, the class of all D(FM) is the class of all random languages acceptable by an APA, and the class of all R(FM) is the class of all random languages that can be generated by an APA . Definition 5.13 .5 Let M = (Q, X, Y, p, h, g) be an APA .

(1) M is called bounded if for all q, q' E Q and u E X, p(q, u, y, q') = 0 except for finitely many y E Y* . (2) M is called synchronous if for all q, q' E Q and u E X, p(q, u, y, q) > 0 implies y E Y.

A synchronous total APA reduces to a stochastic sequential machine if It follows that a random language is acceptable by a synchronous APA if and only if it is realizable by a conventional probabilistic automaton, Section 5.16 or [207] .

g(q) = 1 for all q E Q .

© 2002 by Chapman & Hall/CRC

5.13. Weakly Regular Grammars and Asynchronous Automata

211

Theorem 5.13 .6 The following statements are equivalent. (1) f is a weakly regular random language; (2) f = R (FM) for some APA M = (Q, X, Y, p, h, g), where (i) p is a total random function from Q x X into Y* x Q, (ii) h(q°) = 1 for some qo E Q, and (iii) there exists ql E Q such that g(qi) = 1 and g(q) = 0 for all q7~ qi ; (3) f = R(F M) for some APA M. Proof. Suppose f is a weakly regular random language. Then, by Theorem 5 .12 .13, f = fG for some total weakly regular probabilistic grammar G= (T, N, P, AO) . Let Al , A2 ~ TUN and N° = NU{A l , A2}. Define M = (P, N° , T, p, h, g) such that for all p E P, r E T*, and A, A' E N°, p(A, aA') p(A, a) 1 0

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

g(A) =

~

if A, A' E N if A E N, A' = Al if A' = A2 and A E {Al, A2} otherwise

0

if if

A=A1 A 7~ Al

and h(A) =

1 ~ 0

if A = A° if A 7~ A° .

It follows that M has the desired properties . Thus (1) implies (2) . That (2) implies (3) is immediate. Now suppose that f = R(FM) for some APA M = (Q, X, Y,p, h, g). We may assume without loss of generality that Q n Y = 0. For all u E X, we associate the probabilistic production p. over Y U Q, where p. = Q and p(q, u, y, q~) 0

p~ (q' t)

if t = yq otherwise .

Moreover, let po be the probabilistic production over Y U Q, where = Q and po p°(q't)

-

g(q) 0

if t = A otherwise .

Define G = (Q, Y, P, h), where P = {p°} U that f = fG . Thus (3) implies (1) . m © 2002 by Chapman & Hall/CRC

{pv I u E X} .

It follows

212

5. Probabilistic Automata and Grammars

Theorem 5.13 .7 The following statements are equivalent:

(1) f is a bounded weakly regular random language ; (2) f = R(F') for some APA M = (Q, X, Y, p, h, g) satisfying conditions (i)-(iii) of Theorem 5.13.6(2) and (iv) for all q, q' E Q and u E X, p(q, u, y, q') > 0 implies y E Y U {A} ; (3) f = R(FM) for some bounded APA M.

Proof. The proof follows from Proposition 5.13.2 and the proof of Theorem 5.13.6. Theorem 5.13 .8 (1) If f is a regular random language, then f = R(F' ) for some synchronous APA M satisfying conditions (i)-(iii) of Theorem 5.13 .6 . (2)

If f = R(F') for some synchronous APA M, then there exists a regular probabilistic grammar G such that f (r) = fG(r) for all r :?~ A .

Proof. Assertion (1) follows from the proof of Theorem 5.13.6. Now, suppose f = R(F'), where M = (Q, X, Y, p, h, g) is a synchronous APA . We assume without loss of generality that Q n Y = 0. For all u E X, we associate the probabilistic productions pU and pu, where p~ = p~ = Q and P~ (q, t) _

PU (qt) -

p(q, u, v, q~)

q'EQ

0

if t = vq~ otherwise

p(q, u, v, q')g(q')

0

if t = v otherwise

{pu

for all q

E Q, v E Y, and t E (YUQ)* . Let G = (Q,Y P, h), where P = u E X} U {p2 I u E X} . It follows that f (r) = fs (r) for all r E Y+ .

Hence assertion (2) holds. The above result states that every regular random language can be generated by a synchronous APA, and vice versa . For deterministic languages, it is known that every regular language is also acceptable by a finite automaton, and vice versa. This does not hold in general for random languages . However, a sufficient condition is given below. Theorem 5.13 .9 If f = fG for some regular probabilistic grammar G = (T, N, P, h), where P contains exactly one element, then f is acceptable by some synchronous APA.

Proof. It follows from the proof of Theorem 5 .13.6 that f = R(F' ) for some synchronous APA M = (Q, X, T, p, h, g), where X = {uo} . Let M' = (Q, T, {uo}, p', h, g), where for all q, q' E Q and v E T, p(q, v, no, q) = p(q, vo, v, q) .

Clearly, M' is a synchronous APA.

D (F m') .

© 2002 by Chapman & Hall/CRC

Moreover, it follows that

f =

5.14. Type-0 Probabilistic Grammars and Probabilistic Turing Machines213

5 .14

Type-0 Probabilistic Grammars and Probabilistic Turing Machines

In this section, we study type-0 probabilistic grammars and their relationship to probabilistic Turing machines . Definition 5.14 .1 A probabilistic Turing machine (PTM is a sex-

tuple M = (Q, X, Y, Wp, h), where Q, X, Y, and W are finite nonempty sets, X U Y C_ W Q n W = O, p is a random function from Q x W into (W* U {+, -1) x Q, where +, - W and h is a function from Q into [0,1] such that EgEQ h(q) _ 1 .

In Definition 5 .14 .1, X, Y, and W are, respectively, the set of input, output, and tape symbols . Q denotes the set of states . h(q) is the probability that q E Q is the initial state. p(q, w, z, q') is the conditional probability of the "next act" of the PTM given that its present state is q and the tape symbol w is scanned . Like the conventional Turing machines, the "next act" of the PTM may be one of the following : (1) z E W* : replace w by z and go to state q~ . (2) z = +: move one square to the right and go to state q'. (3) z = -: move one square to the left and go to state q'. In the following, if M = (Q, X, Y, W p, h) is a PTM, then we assume that b E W and b ~ X U Y . The symbol b stands for blank .

W p, h) be a PTM and r E (W U Q) * . r is called an instantaneous description of M if (1) r contains exactly one q E Q and q is not the rightmost symbol of r, (2) the leftmost symbol of r is not b, and (3) the rightmost symbol of r is not b unless it is the symbol immediately to the right of q . Definition 5.14 .2 Let M = (Q, X, Y,

Let I(M) denote the set of all instantaneous descriptions of M.

W p, h) be a PTM. (1) Define the function pm from I(M) x I(M) into [0,1] by for all r, s E I(M), Definition 5.14 .3 Let M = (Q, X, Y,

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

p(q, w, -, q~)

0

© 2002 by Chapman & Hall/CRC

if r = ugwv, s = uq~zv, zv :?~ A orr =uqw, s = uq~b, z = A if r = ugww~v, s = uwq~w~v, uw z,4 b or r = gww~v, s = q~w~v, w = b or r = uqw, s = uwq'b, uw b orr=qw, s=q'b, w=b if r = uw~gwv, s = uq~w~wv, wv :?~ b or r = uwlqw, s = UV w1, w = b or r = qwv, s = q'bwv, wv :?~ b orr=qw, s=q'b, w=b otherwise,

214

5. Probabilistic Automata and Grammars

where u, v E W*, q, q' E Q, w, w' E W and z E W* . (2) For all k E hY U {0}, define the function pk from I(M) x I(M) into [0,1] by for all r, s E I(M), PO,

0

pki (r, s)

= 1:

UEI(M)

if

pk (u,

r z,4 s s)PM

(

(3) For all k E N, define the function qk from I(M) x I(M) into [0,1] by for all r, s E I(M),

gk(r, s) = PM, (r, s) [1

-

q'EQ zEZ

(q, w, z, q)] I

where Z = W* U {+, -}, q is the state symbol contained in s, and w is the symbol contained in s, which is immediately to the right of q. (4) Define the function FM from X* x Y* into [0,1] by for all x E X* and y E Y*, F

M (x, y) = 1: 1: h (q) =y

qEQ

~qk (qx, r) k=1

where < r > is the output string obtained from r by deleting all symbols in r not belonging to Y .

The functions in Definition 5 .14 .3 have the following interpretations . (1) pm(r, s) is the conditional probability that the "next" instantaneous description is s given that M "starts" with instantaneous description r. (2) qk (r, s) is the conditional probability that the instantaneous description of M is s "after k steps" given that M "starts" with r . (3) qk (r, s) is the conditional probability that M will "terminate" with s "after k steps" given that M "starts" with r. (4) FM (x, y) is the conditional probability that the output of M is y given that input x is applied . It follows that FM is a random function from X* into Y* . Definition 5.14 .4 Let M = (Q, X, Y, Wp, h) be a PTM.

(1) M is called bounded if for all q, q' E Q and w E W p(q, w, z, q') = 0 except for finitely many z E W* U {+, -} . (2) M is called synchronous if for all q, q' E Q and w E W, p(q, w, z, q') > 0 implies z E W U {+, -} .

The above model of a PTM differs slightly from those given in [198] and

[201] .

© 2002 by Chapman & Hall/CRC

5.14. Type-0 Probabilistic Grammars and Probabilistic Turing Machines215

Proposition 5.14 .5 Every random function that is computable by a bounded PTM is computable by a synchronous PTM. 0

Proposition 5.14 .6 If F is a random function computable by a PTM,

then F = FM for some PTM M = (Q, X, Y, Wp, h) satisfying the following conditions: (1) There exists ql E Q such that h(qi) = 1 . (2) For all q E Q and w E W, p(q, w, z, q') > 0 implies z E W*, p(q, w, z, q~) > 0 implies z = +, or p(q, w, z, q~) > 0 implies z = - . (3) There exists q2 E Q such that (i) p(q2 , w, z, q) = 0 for all q E Q, w E W and z E W* U (ii) for all w E W and q E Q, where q :?~ q2, E Ep(q, w, z, q) = 1~

q'EQzEZ

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

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

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

P4(s' t) =

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

0

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

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

It also follows

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

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

21 6

5. Probabilistic Automata and Grammars

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

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

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

if m(ri, si) >, k.

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

© 2002 by Chapman & Hall/CRC

5.15. Context-Free Probabilistic Grammars and Pushdown Automata 217

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

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

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

5 .15

Context-Free Probabilistic Grammars and Pushdown Automata

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

fG(r) =

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

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

© 2002 by Chapman & Hall/CRC

In fact,

21 8

5. Probabilistic Automata and Grammars

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

N)* ,

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

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

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

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

1: 1:

qEQ wEW

h(q, w) = 1,

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

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

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

© 2002 by Chapman & Hall/CRC

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

5.15. Context-Free Probabilistic Grammars and Pushdown Automata 219

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

1 -_ ~ 0

ifr=s if r :?~ s

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

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

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

q,q'EQ-EW

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

q, q' EQw EW z EW *

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

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

(x, y) =

q,q'EQwEW*

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

and GM(x, y) _

q,q'EQwEWzEW

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

It follows from the above definition that PPAs are stochastic generalizations of conventional pushdown automata. FM is the random function computed by M with empty store, while GM is the random function computed by M with final states . Theorem 5.15 .6 If f is a leftmost(bounded) context-free random language, then f can be generated with empty store by some total (bounded) PPA having a single state and such that p(q, w, u, y, z, q~) > 0 implies y E T U {A}. © 2002 by Chapman & Hall/CRC

22 0

5. Probabilistic Automata and Grammars

Proof. Let f = fG,L, where G = (T, N, P, h) is a (bounded) contextfree probabilistic grammar . By Proposition 5.15 .1, we may assume that G is total and for all p E P and s E p, p(s, t) > 0 implies t = av, where a E T U {A} and v E N* . Let M = (P, {qo}, T, N, p, h', g), where h'(qo, A) = h(A) and p(qo, w, y, p, z, qo) = p(w, yz) for all z E N* and y E T*, p E P, and w E N. Clearly M is a PPA with the desired properties. Moreover, it can be shown that f = R(FM) . Theorem 5.15 .7 If

f is a leftmost (bounded) context-free random language, then f can be generated with final states by some total (bounded) PPA having two states and such that p(q, w, u, y, z, q') > 0 implies y E T U {A} .

Proof. Let f = fG,L, where G = (T, N, P, h) is a (bounded) contextfree probabilistic grammar . By Proposition 5.15 .1, we may assume that G is total and for all p E P and s E p, p(s, t) > 0 implies t = av, where a E T U {A} and v E N* . Moreover, we may assume, without loss of generality, that there exist uniquely Ao, Al, A2 E N and p o , pi E P such that (1) h(AO) = 1, (2) -PO = {Ao} and po(Ao,A2Ai) = 1, (3) pi = {Ai} and pi (A i , A) = 1, (4) for i = 0, 1, Ai E p implies p = pi, (5) for all p E P and s E p, p(s, t) > 0 implies t ~ (T U N) * {Ao}(T U N) *, and (6) for all p E P, s E p, and p zh p o , p(s, t) > 0 implies t (TUN)*{Ai}(TUN)* . That is, we modify G in such a way that A1 serves as an endmarker . Let M = (Q, P, 2', N, p, h~ , g) where Q = {qi, q2}, h' (qi , AO) = 1, g(q2) = l and for all q,q'EQ,pEP,wEN,zEN*, and yET*,

p(q, w, p, y, z, q) =

p(w, zy) 1 0

if q = q' = qi, w 7~ Ai or p 7~ pi if q=qi, q= q2, w=z = Ai, p = pi, y = A or q=q'=q2, w=z, y=A otherwise .

Clearly, M is a PPA with the desired properties. Moreover, it can be shown that f = R(GM). Let S be a nonempty set and k E N. Let S k = {r E S* I Irl S k}. Theorem 5.15 .8 If

f = R(F") for some bounded PPA M, then leftmost bounded context-free random language .

f

is a

Proof. Let M = (Q, X, Y, W p, h, g) . Since M is bounded, there exists k E hY such that for all q, q' E Q, u E X, w E W y E Y*, and z E W*, p(q, w, u, y, z, q') > 0 implies IzI <_ k. Let No = {(q, w, q') I q, q' E Q and w E W} and N = NOU{Ao}, where Ao ~ No . For ease of notation, we write (q, q2 . . . qn, WI W2 . . . wn, giq2 . . q,,) for (qi, wi, qi) (q2, w2, q2) . . . (qn, wn, q"'), where (qj, w2, qZ) E No, i = 1, 2, . . . , n. Let F be the set of all functions f from Wk into Qk such that f(z) = r implies Irl = IzI-1. For all u E X and © 2002 by Chapman & Hall/CRC

5.15. Context-Free Probabilistic Grammars and Pushdown Automata 221 f E F, let p.,f be the probabilistic production over Y UN, where p. f = No and for all A E No, t E (Y U N)*, P., f (A, t) =

p(q, w, u, y, z, q') 0

if A = (q, w, q') and t = y(qf(z), z, f (z)q~) otherwise .

Moreover, for all q E Q, let pe be the probabilistic production over YUN where pq = {Ao} and _ h(q, w) P', (A, to) - { 0

if t = (q, w, q') otherwise .

Let G = (Y, N, P, h'), where h'(Ao) = 1 and P = {p .j I u E X, f E .F} U {pq I q E Q}. Clearly, G is a bounded context-free probabilistic grammar . Furthermore, it can be shown that f = fG,L . Theorem 5.15 .9 If f = R(G') for some bounded PPA M, then f is a leftmost-bounded context-free random language . Proof. By Theorem 5.15 .6, it suffices to show that f = R(F'° ) for some bounded PPA Mo . Let M = (Q, X, Y, Wp, h, g) and Mo = (Qo, Xo, Y, Wo, po, ho, go), where Qo = Q U {go, qi }, go, qi ~ Q, Xo = X U {uo, ul }, no, ui ~ X, Wo = W U {wo' $1, wo, $ W ho(go, wo) = 1, and for all q, q'EQo,uEXo,wEWO,yEYo,zEWo, h(q, w') p(q, w, u, y, z, q') Po(q, w, u, y, z, q) =

g(q) 1 0

if q = qo, w = wo, u = uo, q'EQ, z=w~$,y=A if q, q' E Q, w E W u E X, YEY*,zEW* if q E Q, q' = qi, w E WO, u=ui, z=y=A if q=q'=qi, wEWo, u=ui, z=y=A otherwise .

Clearly, Mo is a bounded PPA . Furthermore, it can be shown that f = F(FM°) . m Combining Theorems 5.15 .6, 5.15.7, 5.15.8, and 5.15.9 yields the following result. Theorem 5.15 .10 The following statements are equivalent: (I) f is a leftmost-bounded context-free random language . (2) f = R(FM) for some bounded PPA M. (3) f = R(GM) for some bounded PPA M. (4) f = R(FM) for some total bounded PPA M, where p(q, w, u, y, z, q') > 0 implies y E Y U {A} . (5) f = R(GM) for some total bounded PPA M, where p(q, w, u, y, z, q') > 0 implies y E Y U {A} . m © 2002 by Chapman & Hall/CRC

222

5. Probabilistic Automata and Grammars

It is known, [136], that languages generated by any grammar using only leftmost derivations are context free. We show next that a similar result holds for random languages . Theorem 5.15 .11 If f = fG,L for some bounded probabilistic grammar G = (T, N, P, h) such that P = {p}, then f is leftmost-bounded context-

free .

Proof. For all s E p, let AZ denote the i-th symbol of s. Moreover, for all s E p, we associate the symbols u2, q~, i = 1, 2, . . . , IsI - 1. Let M = (Q, X, T, W p, h', g), where X = {p} U {u2 I s E p and i = 1, 2, . . . , Isl -1}, Q={qo}U{gsIsEpandi=1,2, . . .,Isl -1},W=TUN,andforall q,q'EQ,uEX,wEW yET*, and zEW*,

1

if q=gk_1,w=A%, u=p, q1= qo1 z=t, y = A,k = Isl if (u, w, q) _ (uk Ak qk-1), q' = q%

0

or q=q'=_qo, wET and w~A', for any s E P, z = A, y = w or (u, w, q) is not equal to any of those given above, q' = q, z = w, y = A otherwise,

p(s,t)

p(q, w, u, y, z, q) =

h'(q, w) _

z=y=A, k<

h(w) 0

Isl

if q = qo, w E N otherwise .

Clearly, M is a bounded PPA . Moreover, it can be shown that f = Hence by Theorem 5 .15 .10, f is leftmost-bounded context-free.

R(Fm) .

5 .16

Realization of Fuzzy Languages by Various Automata

The remainder of the chapter concerns the results in [207]. Consider a probabilistic automaton A over an input set X. For all x E X*, we associate a number r'(x) representing the probability that x is accepted by A. Then rA may be viewed as a fuzzy language realized by A . In a similar manner, one may consider fuzzy languages realized by maxproduct and max-min automata . In the remainder of the chapter, we examine the class of fuzzy languages realized by probabilistic, max-product, and max-min automata, denoted by LI, L2 , and L3, respectively. Related results for Ll and L3 can be found in [158] and [196] . © 2002 by Chapman & Hall/CRC

5.16. Realization of Fuzzy Languages by Various Automata

22 3

We give characterizations of Lk, k = 1, 2, 3. They reduce to Nerode's Theorem [181] in the case of deterministic automata . We show that neither Ll is a subclass of L2, nor L2 is a subclass of L1 . We also show that L3 is generated by the class of all probabilistic, maxproduct, or mar-min automata with deterministic transition function. This leads to the interesting result that L3 = Ll n FL = L2 n FL, where FL is the class of all fuzzy languages with a finite image. The results obtained in the next two sections can be applied to the study of input-output relations [30] or sequential functions [196] realized by stochastic, mar-product, and mar-min machines and to the study of languages accepted by probabilistic, mar-product, and mar-min automata. Definition 5.16 .1 Let X be a finite nonempty set. A pseudoautomaton

over X is a quadruple A = (Q, p, h, g), where Q is a finite nonempty set, p is a function from Q x X x Q into [0,1], and h and g are functions from Q into [0,1] . If, in addition, EgeQ h(q) = 1 and EgeQ p(q, u, q') = 1 for all q E Q and u E X, then A is called a probabilistic pseudoautomaton.

In Definition 5 .16.1, Q is the set of states, X is the set of input symbols, and p(q, u, q') is the grade of membership that the next state of A is q' given that the present state of A is q and input u is applied. h(q) is the grade of membership that q is the initial state of A and g(q) is the grade of membership that q belongs to the final states of A. For probabilistic pseudoautomaton, grade of membership is equivalent to probability. For ease of notation, the input set X is assumed fixed throughout the remainder of the chapter . Hence any pseudoautomaton is understood to be a pseudoautomaton over X. As usual X* is the free monoid with identity A generated by X. Given a pseudoautomaton A = (Q,p, h, g), various response functions rA from X* into [0,1] may be associated with A. For x E X*, rA(x) is the grade of membership that x is accepted by A. The manner in which rA is related to p, h, and g determines the type of automaton generated by A. Definition 5.16 .2 Let A = (Q, p, h, g) be a probabilistic pseudoautomaton.

(1) Define the function pi from Q x X* x Q into [0,1] by for all q', q" E Q,UEX,XEX*, 1 i(q, A, q~) _ { 0 pi (q, xu, q") _

qEQ

if q' = q1~ if q' zA q"

p(q, u, q" )P (q, x, q) .

(2) Define the function qi from Q x X* into [0,1] by for all q E Q and XEX*,

q' E Q

© 2002 by Chapman & Hall/CRC

PA (q, x, q~)g(q~) .

22 4

5. Probabilistic Automata and Grammars

(3) Define the function rl from X* into [0,1] by for all x E X*, rt (x) _

qEQ

h(q)gi (q, x) .

A probabilistic pseudoautomaton A together with the response function rl defined above constitute a probabilistic automaton . This model of probabilistic automata reduces to the conventional model [189] if Im(g) C {0,1} . Definition 5.16 .3 Let A = (Q, p, h, g) be a pseudoautomaton.

(1) The functions p2, q2 , and r2 are defined in exactly the same manner asp i , gi , and rl , respectively, with summation replaced by maximum. (For example, p2 (ql, xu, q') = V{p(q, u, q')p2(q, x, q) I q E Q} .) (2) The functions p3, q3 , and r3 are defined in exactly the same manner as p2, q2, and r2, respectively, with product replaced by minimum. (For example,

p3 (q, xu, q') = V{p(q, u, q') A p3 (q, x, q) I q E Q} .)

A pseudoautomaton A together with the response function r2 defined above constitute a max-product automaton . Similarly, a pseudoautomaton A together with the response function r3 defined above constitute a maxmin automaton . As usual, a fuzzy language (over X) is a function from X* into [0,1] . Let L denote the set of all fuzzy languages over X. Let L1 denote the set of all response functions rl , where A is a probabilistic pseudoautomaton . Let Lk denote the set of all response functions rk , where A is a pseudoautomaton, k = 2,3. The following operations of fuzzy languages are considered in the remainder of the chapter . If A,, A2 E L, then A, A2, Ai U A2, and Ai rl A2 are fuzzy languages such that for all x E X*, (A, A2) (x) = A, (x) - A2 (X), Ai U A2 (x) = A, (x) V A2 (X), and Ai r1 A2 (x) = A, (x) AA2 (x) . In general, if AZ E L, for i = 1, 2, . . . , n, then 1p,2=Z ,Aa = (A1 . . . An-1)An, U ,Ai = (A1 U . . . U An-1) U An , and nZ 1 AZ = (A, n . . . n An-,) n An . Moreover, Ai + A2 and E', AZ are defined similarly provided Ai + A2 and E', AZ are fuzzy languages . If AO E L is constant, i.e., there exists 0 <_ c <_ 1 such that AO(x) = c for all x E X*, then we write cA, c U A, and c n A for AOA, AO U A, and AO rl A, respectively, where A E L. Definition 5.16 .4 Let

AZ E

© 2002 by Chapman & Hall/CRC

L, i = 0, 1, 2, . . . , n.

5.16. Realization of Fuzzy Languages by Various Automata

22 5

(1) Ao is a convex combination or 1-combination of {A1, A2 . . . . , An} En if there exists 0 < 1 c2 = l and - c2 < - 1, i = 1, 2, . . . , n such that AO = EZ 1 c2A2 . (2) AO is a max-product combination or 2-combination of 01~A2~ . . . ,An} if there exists 0 <_ c2 < 1, i = 1, 2, . . . , n such that AO = UZ 1ciAZ . (3) AO is a max-min combination or 3-combination of {A1, A2, . . . , An} if there exists 0 < c2 < 1, i = 1, 2, . . . , n such that Ao = UZ 1 (c2 n AZ) .

Let A E L and x E X* . Let Ax denote the fuzzy language over X such that Ax(y) = A(xy) for all y E X* . Moreover, let T(A) = fAx I x E X*} .

Definition 5.16 .5 Let k = 1, 2, 3, and AZ E L, i = 1, 2, . . . , n. Then

{A1, A2, . . . , An} is called a k-admissible set if for all i = 1, 2. . . . n and u E X, AZ is a k-combination of {A1, A2, . . . , A n }. Moreover, let A E L. Then A is said to have a k-admissible set if there exists a k-admissible set {A1, A2, . . . , An J such that A is a k-combination of {A1, A2, . . . , An} . In this case, {A1, A2, . . . , An} is called a k-admissible set of A.

Proposition 5.16 .6 Let k = 1, 2,3. If {A1, A2, . . . , An} is a k-admissible

set of A E L, then for all x E X*, {A1, A2, . . . , An} is a k-admissible set of Ax .

In the remainder of the chapter, the symbol A denotes a probabilistic pseudoautomaton if the case k = 1 is under consideration and a pseudoautomaton if the case k = 2 or 3 is under consideration . Theorem 5.16 .7 Let A E L . Then for k = 1, 2, 3, A E Lk if and only if A has a k-admissible set.

Proof. We prove the result for the case k = 1 only. The other cases can be proved in a similar manner . Suppose A E L1 . Then A = rl for some A = (Q, p, h, g), where Q = {q1, q2, . . . , qn}. For all i = 1,2, . . . , n and x E X*, let Ai(x) = q1 (qj, x) . Then it follows that {A1, A2, . . . , An} is a 1-admissible set of A. Conversely, suppose {A1, A2, . . . , An} is a 1-admissible set of A. Then for all i = 1, 2, . . . , n and u E X, AZ = E~ 1 a2j (u)Aj for some a2j (u) > - 0 and Enj- 1 a2j (u) = 1. Moreover A _ EZ 1 cjAZ for some En c2 >_ 0 and 1 cz = 1 . Let A = (Q, p, h, g), where Q = {ql, q2, . . . , qn} and for all 1 < i, j < n and u E X, p(qj, u, qZ) = cti2j (u), h(qj) = c2, and g(qz) = AZ(A) . It follows that A = rl . We show in Section 5.17 that the above characterization for Lk, k = 1, 2, 3, reduces to Nerode's Theorem [181] in the case of deterministic automata. Corollary 5.16 .8 If A E L1, then the vector space V(A) spanned by T(A) is finite-dimensional . m © 2002 by Chapman & Hall/CRC

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5. Probabilistic Automata and Grammars

The converse of Corollary 5 .16.8 does not hold in general . A counterexample may be constructed by adopting a method described in [45]. Nevertheless, it follows from the proof of Theorem 5.16.7 that V (A) is finite dimensional if and only if A is realized by a generalized automaton [196] . Moreover, the dimension of V (A) is a lower bound for the number of states of any probabilistic automaton realizing A . Let A E L and x E X* . Let A(x) denote the fuzzy language over X such that A(x)(y) = A(yx) for all y E X*. Moreover, let To(A) = {A(x) I x E X*} . It is clear that, for k = 2 or 3, a result similar to Theorem 5 .16.7 can be established using A(x) instead of Ax . However, for k = 1, some modifications are needed. Theorem 5.16 .9 Let A E L . Then A E L1 if and only if there exist AZ E L, i = 1, 2, . . . , n, such that the following conditions hold : (1) for all x E X*, E', AZ(x) = 1;

(2) for all 1 0 and E'i cti2j (u) = 1 ; < < (3) A = E7,=1 cjAZ, where for all 0 c2 2, . . . , n .

Proof. Suppose A = rl , where A = (Q, p, h, g) and Q = {q1, q2, . . . , q,,j . For all i = 1, 2, . . . , n and x E X*, let AZ(x) = E'_ 1 h(qj )pA(qj , x, qj) . It follows that {A1, A2, . . . , AnJ has the desired properties. Conversely, sup pose there exists AZ E L, i = 1, 2, . . . , n, such that conditions (1), (2), and (3) hold. Let A = (Q, p, h, g), where Q = {q1, q2, . . . , qnI and for all i = 1, 2, . . . , n and u E X, p(qj, u, qj ) = ctij2(u), h(qj) = AZ(A), and g(qi) = c2 . It follows that A = rl . Corollary 5.16 .10 If A E L1, then the vector space VO(A) spanned by TO (A) is finite dimensional. m

Suppose that X* is well ordered . Consider the infinite matrix M(A) whose (i, j)-th entry is A(xixj) where x2 is the i-th element of X* . Then the rows and columns of M(A) are precisely T(A) and To(A), respectively. Thus, the dimension of V(A), the dimension of VO(A), and the rank of M(A) are equal . Suppose the dimension of V(A) is n . Then there exist x2, x~ E X*, 1 <_ i, j < n, such that the matrix MO, whose (i, j)-th entry is A(XZX~ ), is nonsingular and has rank n. It follows from the proof of Theorem 5.16.7 that a generalized automaton that realizes A can be constructed from the knowledge of M and A(x'ux"), 1 <_ i, j < n, and u E X. Since x2, x2' can always be chosen [30] such that Ix'j < n and I x"~ < n for all i = 1, 2, . . . , n, we have proved the following theorem. Theorem 5.16 .11 If the dimension of V (A) is <_ n, then A is uniquely determined by A(x), where xl < 2n - 1 . m © 2002 by Chapman & Hall/CRC

5.16. Realization of Fuzzy Languages by Various Automata

227

An elaborate procedure, which is essentially the same as that given above, can be found in [30] for stochastic sequential machines . A similar procedure was given in [196] for max-min sequential machines . We conclude this section by showing that for Lk, k = 1, 2, 3, there is no loss of generality by considering only certain more restrictive classes of pseudo automata, e.g., the conventional probabilistic, max-product, and max-min automata, respectively. For k = 3, additional results along this direction can be found in [196] . Theorem 5.16 .12 Let k = 1, 2,3. If A (Q, p, h, g), where Im(g) C {0,1}.

E Lk, then A = rk for some A =

Proof. By Theorem 5 .16.7, A has a k-admissible set {A1, A2, . . . , An}. For all i = 1, 2, . . . , n and j = 1, 2, define Aij (x) =

b'x

Ai (x)

1 0

if x A if x = A and j = 1 if x = A and j = 2,

X* . It is clear that (a) if A E L and A = E', ciAi, then

E

A (b)

if A

EL

n

= I: Ci Ai (A) Ail + i=1

i=1

ci[1

- Ai(A)]Ai2,

and A = U,'= 1ciAi, then A = (U

1eiAi(A)Ai1) U (U

1eiAi2),

and

(c) if A E L and A = Ui 1 (ci n Ai), then (Ui 1( ci n Ai(A) n Ail)) U (Ui 1( ci n Ai2))

Thus {Aid I i = 1, 2, . . . , n and j = 1, 2} is a k-admissible set of A. The desired result follows from Theorem 5.16.7 . Theorem 5.16 .13 For k = 1, 2, 3, if A (Q, p, h, g) such that Im(h) C {0,1} .

E Lk, then A = rk for some A =

Proof. The proof follows from Theorem 5 .16.7 and the fact that if {A1, A2, . . . , An} is a k-admissible set of A, then so is {A, Al, A2, . . . , An}. Theorem 5.16 .14 For k = 2, 3, if A E Lk, then A = rk for some A = (Q, p, h, g), where V{h(q) I q E Q} = 1, V{g(q) I q E Q} = 1, and V{p(q, u, q~) I q' E Q} = 1 for all q E Q and u E X . Proof. By Theorem 5.16.7, A has a k-admissible set {A1, A2, . . . , An}. For all x E X*, let AO (x) = 0 and An+ 1 (x) = 1 . Since {Ao, A1, A2, . . . , An, An+ 1 } is a k-admissible set of A, the conclusion follows from the proof of Theorem 5.16.7 . © 2002 by Chapman & Hall/CRC

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5 .17

5. Probabilistic Automata and Grammars

Properties of Lk, k = 1, 2, 3

In this section, properties of Lk, k = 1, 2,3 are derived using the characterizations established in the previous section. Theorem 5.17.1 If A E L is constant, then A E Lk for k = 1, 2,3. Proof. The proof follows from Theorem 5.16.7 and the fact that if A is constant, then {A} is a k-admissible set of A. Theorem 5.17.2 For k = 1, 2, 3, if A E Lk, then for all x E X*,

Ax

E Lk .

Proof. The proof follows from Proposition 5.16.6 and Theorem 5.16.7 . The next theorem gives a converse of Theorem 5.17.2 . Theorem 5.17.3 Let A E L. For k = 1, 2, 3, if there exists a positive Ax Lk integer m such that E for all x E X* such that I xl = m, then A E Lk . Proof. For all x E X* such that xl = m, let 17(x) be a k-admissible set of Ax . Let ro = 4F(x) I x E X* and Ixl = m} and I7 = ro U fAx x E X* n and Ixl < m} . It follows that I7 is a k-admissible set of A. This A E Lk . Theorem 5.17.4 Let A E L. For k = 1, 2, 3, if Ao E Lk and {x E X* A(x) z,4 AO (x) I is a finite set, then A E Lk . Proof. There exists a positive integer m such that A(x) _ Ao(x) for all x E X* and Ixl >_ m. Thus by Theorem 5.17.2, for all x E X* where xl = m, Ao E Lk . By Theorem 5.17.3, A E Lk . Definition 5.17.5 A Rabin (probabilistic) pseudoautomaton is a (probabilistic) pseudoautomaton A = (Q, p, h, g) such that Im(h) U Im(g) C_ {0,1} . Theorem 5.17.6 Let A E L . For k = 1, 2, 3, A = rk for some Rabin pseudoautomaton A if and only if (1) A(A) E {0,1} and (2) there exists Ao E Lk such that A(x) = Ao(x) for all x E X*\{A} . Proof. The proof follows from Theorems 5.16 .12, 5.16.13, and 5.17.4 . Theorem 5.17.7 Let A E L. For k = 1, 2, 3, if A is a k-combination of {A1, A2, - - - , An}, where Aa E Lk, i = 1, 2, . . . , n, then A E Lk . Proof. For all i = 1, 2, . . . , n, let 172 be a k-admissible set of AZ . Let I7 = UZ 1172. Then it follows that I7 is a k-admissible set of A. Thus A E Lk .

© 2002 by Chapman & Hall/CRC

5.17. Properties of Lk,

k = 1, 2,3

Corollary 5.17.8

Let

k = 1, 2.

If A E Lk and 0 < c < 1, then cA E Lk . 0

Corollary 5.17.9

Let

k = 2,3-

If A1, A2 E Lk, then A1 U A2 E Lk . 0

Theorem 5.17 .10

Let

k = 1, 2.

229

If A1, A2 E Lk, then AIA2 E Lk .

Proof. Let IZ be a k-admissible set of AZ, i = 1, 2 and I' = {A/ A/ It follows that I' is a k-admissible set of A, A2 . Thus

AZ E FZ, i = 1, 2} . AIA2 E Lk .

Theorem 5.17 .11

A E L1 if and only if A E L1, where A = 1 - A .

Proof. The proof follows from Theorem 5 .16 .7 and the _ fact that if a 1-admissible set of A, then {A1, A2, . . . , An} is a 1admissi-ble set of ~. m {A1, A2, . . . , An} is

Theorem 5.17 .12

If A1, A2, A3 E LI, then A, A2 + A, A3 E L1 .

Proof. Let IZ be a 1-admissible set of AZ, i = 1, 2, 3. It can be shown that {AA' + NA" I A E Fl , A' E I2, A" E I3 } is a 1-admissible set of AIA2 +VA3 . Thus AIA2 +VA3 ELI . Let B C X* . Recall that XB is the characteristic function of B. Proposition 5.17.13

If A1, A2 E L and

{x E X*IA1(x) > A2(x)} C B C {x E X*IA1(x) > A2(X)I, then A1 U A2 = XB A1 + XBA2 and A1 n A2 = XB A1 + XBA2 . 0

Theorem 5.17 .14

Let A1, A2 E L1 . If XB E Ll for some B such that

{x E X* IA1(x) > A2(x)} C B C {x E X*IA1(x) > A2(X)I, then A1 U A2, A1 n A2 E L1 .

Proof. The proof follows from Proposition 5.17.13 and Theorems 5.17 .11 and 5.17 .12. Theorem 5.17 .15 A,nA2EL1 .

Proof. Let

If A1, A2 E Ll and Im(A2) C_ {0,1}, then A1 U A2 and

B = {x E X* I A2(x) = 0} .

Clearly,

{x E X* I A1(x) > A2(x)} C B C {x E X* I A1(x) > A2(x)}-

Thus by Theorem 5.17.14, A1 U A2 and A1 n A2 E L1 . Let A E L . Let AT denote the fuzzy language over X such that for all x E X*, AT (x) = A(X T ), where AT = A and (xu)T = UXT for all u E X and x E X* . © 2002 by Chapman & Hall/CRC

23 0

5. Probabilistic Automata and Grammars

Theorem 5.17 .16 Let k = 2,3. Then A E Lk if and only if AT E Lk . Proof. The proof follows from Theorem 5 .16 .7 and a remark made in the previous section . The following operations of functions are needed in the proof of the next theorem. The reader is referred to [158] for a detailed treatment . Definition 5.17 .17 Let g be a real-valued function with domain Q, a finite nonempty set. (1) A real-valued function g' with domain P(Q) is called a column necomposition of g if for all Q' E P(Q), gc(Q') = E,EQ, g(q) . (2) A real-valued function gT with domain P(Q) is called a now necomposition of g if for all q E Q, g(q) = ~Q'ESZ(q) gT(Q'), where Q(q) = {Q' E P(Q) I q E Q} .

The following lemma was proved in [158]. Lemma 5.17 .18 (1) If gi and 92 are real-valued functions with domain Q, then

qEQ

91(q) ' 92 (q)

= 1:

Q'EP(Q)

gi (Q') ' g2 (Q~) .

(2) If g is a function from Q into [0,1], then a row recomposing gT of g may be constructed from g such that EQ , EP(Q) gT (Q') = 1 and gT(Q') > 0 for all Q'CQ .m

Theorem 5.17 .19 A E L1 if and only if AT E Ll . Proof. There exists AZ E L, i = 1, 2, . . . , n, satisfying conditions (1), (2), and (3) of Theorem 5.16.9. For all x E X*, define gx (i) = AZ(x) for all i = 1,2, . . . , n. Then gx is a function from Qo = {1,2, . . . , n} into [0,1] . For all Q E P(Qo) and x E X*, let AQ(x) = gx(Q) . It follows from Lemma 5.17.18 and conditions (1), (2), and (3) of Theorem 5.16.9 that {AQ I Q E P(Qo)} is a 1-admissible set of AT . Thus AT E L. We conclude this section by constructing two examples showing that neither Ll is a subclass of L2 nor L2 is a subclass of Ll . Example 5 .17 .20 Let X = {uo, ulI and for all x E X*, A(x) = .b b _1

. . .

b2 b1

written in binary expansion, where x = ubl ub2 . . . ub, . It is well known 11801 that A E Ll . Suppose A E L2 . By Theorem 5.16.7, A has a 2-admissible set {Ai, A2, . . . , An} . Hence for all x E X*, there exists 0 <_ /3Z (x) <_ 1, i = 1, 2, . . . , n, such that for all x' E X*, A(xx') = VZ i/3j(x)AZ(x') . For all x E X*, let Ix = {i I 32 (x) = 1} and,3(x) = V{/3j(x) I i ~ Ix} < 1 . © 2002 by Chapman & Hall/CRC

5.18. Further Properties of L3

23 1

Clearly, there exists an I C_ {1, 2, . . . , n} such that X*(I) = {x E X* I Ix = I} is infinite. Let XO*(I) be a subset of X*(I) containing exactly n + 1 elements. By the definition of A, there exists yo E X* such that A(yo) > v{/3(x) I x E X*(I)} . Thus for all x E XO* (I), A(xyo) = VZEIA(yo). This is impossible since Xo (I) contains more than n elements and A is one-one. Hence A ~ L . Thus L1 z L2 .

Example 5.17 .21 Let X = {ul, u2} and for all x E X*, let A(x) = 2 m(x), where m(x) = ml (x) A m2 (x) and ml (x) and m2 (x) are, respecLively, the number of occurrences of ul and u2 in x. First we show that A E L2 . Consider A = (Q, p, h, g), where Q = {qi, q2} and p(qj, uk, qj) =

h(qZ)

1 _{ 0

1 i 2 0

if i=j=k if i=j k otherwise, ifi = 1 ifi = 2,

g(qi) = 1 for i = 1, 2 . It follows that A = r2 . Suppose A E Ll . By Corollary 5.16.8, V (A) is finite-dimensional. For all x E X*, let mo(x) = ml (x) - m2 (x) . Clearly, 2M2W>A x' = 2m2W'>Ax" . Thus there exist positive mo(x') = mo(x") implies integers k2, i = 1, 2, . . . , m, and l2 , i = 1, 2, . . . , n, such that Ax', AX2' . . . , Ax ,n Axi Ax2 . Axis a basis of V(A) where x = uk` i = 1,2. . . . Tn, = u2, i = 1, 2, . . . , n. Consequently, for all x E X*, xo = A and x2 +1 = xx 2 . We assume that k2, as well as l2, are arranged in ascending order. Hence there exists constants a2, i = 1, 2, . . . , r, b2, i = 1, 2. . . . , t, where r < m, t < n, and a,. :?~ 0, bt z,4 0 such that for all x E X*, A(x) = i a2A(xix) + EZ-i b2A(x'x) . Clearly, not both r and t are 0. Suppose r > 0. Let x'o = uk, and xo = uThen A(xix'o) = A(xixo) for all i = 1,2 . . . . , r and A(xZxo) = A(xZxo) = 1 for all i = 1, 2, . . . , t . Moreover, A(xo) = A(xo) = 1 and A(x,xo) = 2A(x,x") . Thus a,. = 0, a contradiction. Suppose r = 0. Then t > 0. By a similar argument, it can be shown that bt = 0, a contradcition . Hence A ~ L2 . Consequently, L2 Z Ll .

5 .18

Rirther Properties of L3

In this section, we show that L3 is generated by the classes of all probabilistic, max-product, or max-min automata with deterministic or nondeterministic transition functions . For k = 1, 2, 3, let NLk = {A E Lk I A = rk for some A = (Q,p, h, g), where Im(p) C {0,1}} . © 2002 by Chapman & Hall/CRC

23 2

5. Probabilistic Automata and Grammars

Theorem 5.18 .1 Let A E L . For k = 1, 2, 3, A E NLk if and only if T(A) is finite .

Proof. We only prove the case for k = 1. The other two cases can be proved similarly. Suppose A E NL1 . It follows from the proof of Theorem 5.16.7 that there exists a 1-admissible set I' = {A1, A2, . . . , An} of A such that for all i = 1, 2, . . . , n and u E X, AZ E I' . Since A = E', c2A2 for some c2 >_ 0 and E', c2 = 1, we have for all x E X* that Ax = E', c2AZ, where AZ E F for all i = 1, 2, . . . , n. Thus T(A) is finite . Conversely, suppose that T(A) is finite . Clearly, T(A) is a k-admissible set of A. The desired follows immediately from the proof of Theorem 5.16.7 . For k = 3, Theorem 5.18.1 can be strengthened as follows : Theorem 5.18 .2 Let A E L. Then A E L3 if and only if T(A) is finite . . Proof. Suppose A = r3 , where A = (Q, p, h, g) and Q = {q1, q2, . . . , q,,} For all i = 1, 2, . . . , n and x E X*, define A2 (x) = q3(x, q2) and /32(x) _ V~ 1(h(qj) np3(qj,x,g2)) Then for all x E X*, Ax = U=I (/32(x) n A2) . Let R = Im(p) U Im(h) . Clearly, R is finite and for all i = 1, 2, . . . , n and x E X*,/3 2 (x) E R. Thus T(A) is finite . The converse follows from Theorem 5.16.7.

Theorem 5.18 .3 Let k = 1, 2,3. Then A E NLk if and only if A = rk for some A = (Q, p, h, g), where (1) Im(p) C_ {0,1}, and for all q E Q and u E X, there exists a unique q' E Q such that p(q, u, q') = 1, and (2) Im(g) C {0,1} or Im(h) C {0,1} and there exists a unique q E Q such that h(q) = 1 . Proof. The proof follows from Theorems 5.16.12, 5.16.13, and 5.18.1 . Theorem 5.18 .4 Let k = 1, 2. If A E NLk, then Im(A) is finite. Proof. We prove only the case k = 1 . The other case can be proved in a similar manner . Let = rl , where A = (Q, p, h, g) and Im(p) C {0,1} . Then for all q, q' E Q and x E X*, pi (q, x, q') E {0,1} . Since rl (x) _ Eq q, EQ h(q)pi (q, x, q')g(q') for all x E X*, we have that Im(A) = Im(rl) is finite. Let FL = {A E L I Im(A) is finite} . Theorem 5.18 .5 NL 1 = L1 rl FL . Proof. Let A E L1nFL. By Corollary 5.16.8, V(A) is finite-dimensional. Let x2 E X*, i = 1, 2, . . . , n, be such that ~Axl, Axe , . . . , Axe } is a basis of V(A) . Then for all x E X*, there exist unique /32(x), i = 1, 2 . . . . I n, such that Ax = E'1,32(x)A` .Clearly there exists a positive integer m such that for all x E X*,/32(x), i = 1, 2, . . . 1 n, is completely determined by © 2002 by Chapman & Hall/CRC

5.18. Further Properties of L3

23 3

A'(y), where y E X* and jyj < m. Since Im(A) is finite, we have that T(A) is finite . Thus by Theorem 5.18.1, A E NL l . Therefore, Ll n FL C_ NL 1. By Theorem 5.18.4, NLl C Ll rl FL . Hence NL l = L1 n FL . m Theorem 5 .18 .6 Let A E L2 . If there exists c > 0 such that for all x E X*,

A(x) = 0 or A(x) > c, then T(A) is finite .

Proof. Let A = r2 , where A = (Q, p, h, g) and Q = {ql, q2, . . . , qn}. For all i = 1, 2, . . . , n and x E X*, define Ai(x) = g2(qj, x) and 3i (x) _ Ax V~1(h(qj )p2(qj ,x,g2)) . Then for all x E X*, = UZ i(/3Z(x)AZ), where

13 Z (x) =

if there exists y E X* such that 0 7~ Ax (y) = l3i (x) AZ (y)

,3Z(x) 0

otherwise .

Thus for all i = 1, 2, . . . , n and x E X*, /3'(x) = 0 or /3'(x) > c. Let R = Im(p) U Im(h) U Im(g), Fn = {IZo c2 E R, i = 0, 1, 2. . . . I n, and

o c2 > c} and F = (U'OF ) U {0} . Clearly, F is finite and for all i = 1, 2, . . . , n and x E X*, l3'(x) E F. Thus T(A) is finite. m Theorem 5 .18 .7 NL2 = L2

n FL .

Proof. The proof follows from Theorems 5.18.1, 5.18.4, and 5.18 .6 . The next result follows from Theorems 5.18.1, 5.18.2, 5.18.5, and 5.18.7 . Theorem 5 .18 .8 L3 = L1

rl FL

= L2

rl FL . m

The following closure property of L3 follows from the results given in the previous section and Theorem 5.18.8 . Theorem 5 .18 .9 (1) If A E L3, then X and AT E L3(2) If A,, A2 E Ls, then A, A2, Ai U A2 and Ai A2 E L3 . (3) If Al, A2, A3 E L3, then Ai A2 + Ti A3 E L3 . (4) Let k = 1, 2,3. If AZ E L3, i = 1, 2, . . . , n, and A is a k-combination of {Ai, A2, - . . , A n }, then A E L3 . 0

rl

Closure properties of L3 with respect to other operations can be found in [196] . Definition 5 .18 .10 A deterministic pseudoautomaton is a probabilis-

tic automaton (Q, p, h, g), where Im(p) U Im(h) U Im(g) C {0,1}.

Note that if A is a deterministic pseudoautomaton, then rl = r2 = r3 . For simplicity, we write rA = rk for k=1,2,3 . Let DL = {rA I A is a deterministic pseudoautomaton} . Clearly, A E DL if and only if Im(A) C {0,1} and {x E X* I A(x) = 1} is a regular language [189] .

© 2002 by Chapman & Hall/CRC

23 4

5. Probabilistic Automata and Grammars

Theorem 5.18 .11 Let A E L. Then A E DL if and only if Im(A) C {0,1} and T(A) is finite . Proof. The proof follows from Theorem 5.18.2 . The above theorem is Nerode's Theorem [181] . Theorem 5.18 .12 Let k = 1, 2,3. Then A E DL if and only if A E Lk and Im(A) C {0,1} . Proof. The proof follows from Theorem 5.18.8 . Theorem 5.18 .13 For k = 1, 2, 3, Lk is a proper subclass of L . Proof. Let A E L be such that Im(A) C_ {0,1} and {x E X* I A(x) = 1} is a nonregular language. By Theorem 5.18 .12, A ~ Lk, k = 1, 2, 3. Thus Lk, k = 1, 2, 3, is a proper subclass of L . m Theorem 5.18 .14 Let A E FL . Then A E L3 if and only if all 0 <_ c < 1, {x E X* I A(x) > c} is a regular language . Proof. Suppose A E L3 . It follows from Theorems 5 .18.2 and 5.18 .11 that for all 0 <_ c < 1, {x E X*I A(x) > c} is a regular language. Conversely, < c < 1, {x E X*IA(x) > c} is a regular language. Let suppose for all 0 _ Im(A) = {CI, C2, . . , cn } . Then A = UZ 1cjAZ, where for all i = 1,2 . . . . , n, 1 0

if A(x) > c2 otherwise .

By hypothesis, AZ E L3 for every i, i = 1, 2, . . . , n. Hence by Theorem 5.18.9(4), A E L3 . Theorem 5.18 .14 was first proved in [195]. Clearly, {x E X* I A(x) > c} may be replaced by {x E X* I A(x) = c}, {x E X* I A(x) >_ c}, {x E X*I A(x) < c}, or {x E X*I A(x) < c} in Theorem 5 .18 .14 . 5 .19

Exercises

1. Let A = (Q, 7r, {A(u) }, F) be a probabilistic automaton. Prove that 7r(xy) = 7r(x)A(y), where x, y E X* . 2. Prove that p(xy) = 7r(x)9F (y), where x, y E X* . 3. Prove Proposition 5.2 .6 . 4. Let A be a probabilistic automaton . If A can be E-approximated by

a definite automaton, prove that A is quasi-definite .

© 2002 by Chapman & Hall/CRC

5.19. Exercises

235

5. Characterize the fsf's (the pa's) that can be approximated by (a) linear bounded automata, (b) pushdown automata, (c) sequential automata . 6. Determine the most powerful class of np devices that suffices for approximating the pa's. 7. Determine whether pushdown automata can approximate pa's. 8. Determine whether or not the family of w.i.m. languages is a subset of the family of w.i.s. languages . 9. Prove that every stochastic language under maximal interpretation is w .3.m. 10. Prove Theorem 5.16.7 for the cases k = 2,3 . 11. For k = 2, 3, state and prove a result similar to Theorem 5.16.7 replacing Ax with P) . 12. Prove Theorem 5 .18.1 for the cases k = 2,3 . 13. Prove Theorem 5.18.4 for the case k = 2. 14. Prove Theorem 5.18.14 with > in {x E X* I A(x) > c} replaced by =, >, and < .

© 2002 by Chapman & Hall/CRC

Chapter 6

Algebraic Fuzzy Automata Theory 6 .1

Fuzzy Finite State Machines

A sequential machine consists of two main structures, the transition structure and the output structure . The transition structure is an internal part of the machine while the output structure is the external part . Consequently, the output structure is of more interest for practical applications than the input structure. The output structure is dependent on the transition structure while the transition structure is independent of the output structure. Hence the input structure can be studied separately. In this chapter, we study the transition structure of fuzzy machines . A fuzzy finite state machine (ffsm) is a triple M = (Q, X, ft), where Q and X are finite nonempty sets and /t is a fuzzy subset of Q x X x Q, i.e ., /t : Q x X x Q ~ [0,1] . As usual X* denotes the set of all words of elements of X of finite length. Q is called the set of states and X is called the set of input symbols. Let A denote the empty word in X* and Ixl denote the length of x b'x E X*. X* is a free semigroup with identity A with respect to the binary operation concatenation of two words.

6 .2

Semigroups of Fuzzy Finite State Machines

Definition 6.2.1 Let M = (Q, X, /t) be a ffsm. Define /t* : Q x X* x Q [0,1] by

*(q, A, p) _ { 0 if q 237 © 2002 by Chapman & Hall/CRC

zAp

238

6. Algebraic Fuzzy Automata Theory

and w* (q, xa, p) = V{ft* (q, x, r) A ft(r, a, p) I r E Q} VxEX*,aEX. Let X+ = X*\{A} . Then X+ is a semigroup. For /t* given in Definition 6.2 .1, we let y+ = ft* restricted to Q x X+ x Q.

Lemma 6.2 .2 Let M = (Q, X, /t) be a ffsm. Then * ft (q, xy, p) = V {ft*(q, x, r) n ft * (r, y, p) Ir E Q} Vq, p E Q and Vx, y E X* .

Proof. Let q, p E Q and x, y E X* . We prove the result by induction on lyl = n . If n = 0, then y = A and hence xy = xA = x . Thus V{ft* (q, x, r) n ,a* (r, y, p) I r E Q} = V{ft* (q, x, r) n ,a* (r, A, p) I r E Q} = ft * (q, x, p) = ft * (q, xy,p) by the definition of ft * . Thus the result is true for n = 0. Suppose the result is true for all u E X* such that lul=n-1,n>0.Lety=uawhere uEX*, aEX,andJul=n-1,n>0 . Now y,* (q, xy, p) = y,* (q, xua, p) = V {/t* (q, xu, r) A y,(r, a, p) I r E Q} = V{(V{,a* (q , x, s) n,a* (s, u, r) I s E Q}) n,a(r, a, p) I r E Q} = V{V{ft* (q, x, s) n ft*(s,u,r)Aft(r,a,p)}I r,s E Q}=V{,a*(q,x,s)A(V{,a*(s,u,r)A,a(r,a,p)Ir E Q}) Is E Q} = V{w* (q, x, s) Aw* (s, ua, p) I s E Q} = V{w* (q, x, s) Aw* (s, y, r) s E Q} . Thus the result is true for Iyl = n . m Define a relation - on X* by Vx, y E X*, x - y if and only if ft* (q, x, p) _

ft * (q, y, p) V q, p E Q. Clearly - is an equivalence relation on X* . Let z E X* and let x y . Then Vp, q E Q, /t* (q, xz, p) = V{/t* (q, x, r) A /t* (r, z, p) I r E Q} _ V{ft* (q, y, r) ntt* (r, z, p) I r E Q} = ft * (q, yz,p) . Thus xz - yz. Similarly zx - zy. Thus - is a congruence relation on the semigroup X* . We have thus proved the following result .

Theorem 6.2.3 Let M = (Q, X, ft) be a ffsm . Define a relation - on X*

by Vx, y E X*, x - y if and only if ft * (q, x, p) = ft * (q, y, p) V q, p E Q . Then - is a congruence relation on X* . Let x E X*, [x] = {y E X* Ix - y}, and E(M) = {[x] Ix E X*} .

Theorem 6.2.4 Let M = (Q, X, ft) be a ffsm. Define a binary operation * on E(M) by V [x], [y] E E(M), [x] * [y] = [xy] . Then (E(M), *) is a finite semigroup with identity.

Proof. Clearly * is well defined and associative. Now [x] * [A] = [xA] _ [x] = [Ax] = [A] * [x] V [x] E E(M) . Thus [A] is the identity of (E(M), *) .

© 2002 by Chapman & Hall/CRC

6.2. Semigroups of Fuzzy Finite State Machines

239

Hence (E(M), *) is a semigroup with identity. Let x E X* and let x = xlx2 . . . xn, where xl, X2 . . . . , x n E X. Then b' q, p E Q, N' * (q,x,p) = V{ft(q,xl,gl) AN(gl,x2,q2) n gl, q2, . . . , qn-1 E Q} .

. . . ANt(gn-l,xn,p)

Hence since Im(y,) is finite, Im(y,*) is finite . Thus (E(M), *) is a finite semigroup with identity. Example 6.2.5 Let M = (Q, X, p) be a ffsm, where Q = {q} and X = {a, b} . Define p : Q x X x Q ~ [0,1] by ft(q, a, q) = z = ft(q, b, q) . Then b'x, y E X+, x - y since /t* (q, x, q) = 2 = / t* (q, y, q) . Hence E(M) _ {[A], [x]}, where x E X+ . Clearly, [A] is the identity of E(M) and [x]2 = [x] .

Example 6.2.6 Let M = (Q, X, y) be a ffsm, where Q = {q} and X = {a, b} . Define y : Q x X x Q ~ [0,1] by ft(q, a, q) = 3 and ft(q, b, q) = s . Then V'x E X*, y* (q, x, q) = s if and only if x contains a b. Hence Vx, y E X+, x - y if and only if x and y both contain a b. Hence E(M) _ {[A], [x], [y]}, where x contains a b and y does not contain a b, x, y E X+ . Here [A] is the identity of E(M), [x] 2 = [x], [y]2 = [y], and [x] * [y] _ [x] = [y] * [x] .

We now define another type of congruence relation on X* . Let x, y E X* . Define x ~ y if and only if (Vs, t E Q, ft * (s, x, t) > 0 ~ ft* (s, y, t) > 0) . Clearly ; is an equivalence relation on X* . Let z E X* and x s: y. Then * * Vs, t E Q, ft (s, zx, t) = V{Nt* (s, z, r) A ft (r, x, t) lr E Q} > 0 if and only if * * 3 r E Q such that ft (s, z, r) A ft (r, x, t) > 0 if and only if 3 r E Q such that ft* (s, z, r) A ft* (r, y, t) > 0 if and only if ft* (s, zy, t) = V{Nt* (s, z, r) A y* (r, y, t) I r E Q} > 0 . Hence zx s: zy. Similarly xz s: yz . Thus s: is a congruence relation on X* . Hence, we have the following theorem. Theorem 6.2.7 Let M = (Q, X, y) be a ffsm. Let x, y E X* . Define a relation ; on X* by x ,: y if and only if Vs, t E Q, /t* (s, x, t) > 0 [t* (s,

y, t) > 0. Then ; is a congruence relation on X* .

Let x E X* and let Qx~ = {y E X*Ix ~ y} . Let E(M) = {Qx~lx E X*} . Theorem 6.2.8 Let M = (Q, X, /t) be a ffsm . Define a binary operation ~ on E(M) by b' Qx~, Qy~ E E (M), Qx~~Qy~ = Qxy~ . Then (E (M), ~) is a finite semigroup with identity and [x] ----> Qx~ is a homomorphism of E(M) onto E(M) .

Proof.

Cle arly (E(M), *) is a semigroup with identity . Define f E(M) - E(M) by f ([x]) = Qx~ b' [x] E E(M) . Let x, y E X* and [x] = [y]. Then Vs, t E Q, /t* (s, x, t) = y,* (s, y, t) . Thus Vs, t E Q, /t* (s, x, t) > 0 ~ y* (s, y, t) > 0. Hence x s: y or Qx~ = Qy~ . Thus f is well defined . Clearly f is an onto homomorphism. Now since E(M) is finite E(M) is finite . m © 2002 by Chapman & Hall/CRC

24 0

6. Algebraic Fuzzy Automata Theory

Example 6.2.9 Let M = (Q, X, /t) be the ffsm as defined in Example 6.2.5 . Then Vx, y E X*, x ,~ y . Thus i(M) _ {[[x]]}, where x E X+ . Now [[x]]2 = [[x]] .

Definition 6.2.10 Let M = (Q, X, /t) be a ffsm. For all x E X* define the fuzzy subset xM of Q x Q by xM (s, t) = y* (s, x, t) Vs, t E Q.

Theorem 6.2.11 Let M = (Q, X, y) be a ffsm . Let Sm = {x` I x E X*} .

Then (1) xM o ym = (xy) M Vx, y E X*, (2) (Sm, o) is a finite semigroup with identity, where o is defined as in Definition 1.4 .4 .

Proof. (1) Let s, t E Q . Then (xy) M (s, t) = y,*(s, xy, t) = V{/t* (s, x, q) A Ixm ym (xm w* (q, y, t) I q E Q} = V (q, t) I q E Q} = (s, q) A o yM) (s, t) . Thus xM yM . (xy)M -_ o (2) Clearly (Sm, o) is a finite semigroup with identity, where Am is the identity element . Sm is finite since Q and Im(y) are finite . Theorem 6.2 .12 Let M = (Q, X, /t) be a ffsm. Then Sm - E(M), i.e., Sm and E(M) are isomorphic as semigroups .

Proof. Define f : Sm ~ E(M) by f (x') = [x] VxM E Sm. Let XM, ym E Sm . Then x M = ym if and only if xm (s, t) = ym(s, t) Vs, t E Q if and only if ft * (s, x, t) = ft* (s, y, t) Vs, t E Q if and only if [x] = [y] . Thus f is single valued and one-one . Now .f (X

IV[

o ylvl) = f ((xy),V,) = [xy l = [x] * [y] = f (X IV[ ) * f (y IV[ ) .

Thus f is a homomorphism. Clearly f is onto. Hence Sm - E(M) . m Let M = (Q, X, ft) be a ffsm. The index of an equivalence relation is the number of distinct equivalence classes. Let - be a congruence relation of finite index on X* . Let x E X* and ~ x _ {y E X*Ix - y} . Let Q={-<x>- IxEX*} . Define a :QxXxQ~ [0,1] by V-<x>- EQ and V a E X, a(-< x _ >-, a, -< xa >-) an arbitrary fixed element in (0,1] and

V-<x>-,-<w>- EQ,

Q(~x>-,a,~xa>-)ifw-xa

0 otherwise.

Let ~ x >-, ~ y >-, ~ u >-, ~ v >- E Q and a, b E X. Suppose that (fix>-,a,~u>-)=(may>-,b,~v>-) .

Then ~ x >-

v - ya . Thus

y >-, a = b, ~ u >-

v >- . Now u - xa if and only if

a(~x>-,a,~u>-)=a(~y>-,b,~v>-) .

© 2002 by Chapman & Hall/CRC

6.2. Semigroups of Fuzzy Finite State Machines

241

Hence a is single valued. Thus M = (Q, X, a) is ffsm. Now extend Q to Q* as /t was extended to /t* in Definition 6.2.1 . Lemma 6.2 .13 Let M be as above and let ~ z >-, -< w >-E Q . Then the following assertions hold. (1)VXEX*, if a*(-0, then -0VZ,XEX* . Proof. (1) Let x E X* and Ixj = n. If n = 0, then x = A. Hence if a*(~z~--,x,~w~--)>0,then ~z~-- =~w~or~zx~=-<w~-- . Suppose the result is true b' y E X* such that I yI = n-1, n > 0. Let x = yo; where y E X*, a E X, and I y I = n - 1 . Let a* (~ z ~, ya, ~ w ~) = a*(~ z ~, x, ~ w ~) > 0 . Now

>

I -
Hence a* (~ z ~--, y, ~ q ~--) > 0 and a* (~ q ~--, a, w ~--) > 0 for some-< q _ -< q and -< qa ~-- _ E Q. Thus by the induction hypothesis, -< zy ~w~-- .Hence-0. Suppose the result is true b' y E X* such that I yI = n - 1, n > 0. Let x = ya where y EX*, a EX, and jyj =n-1 . Now a* (~ z >-, x, ~ zx ~)

=

a* (~ z >-, ya, ~ zya >-)

>_ >

I ~ q ~EQ} a* (-< z ~, y, -< zy 0.

~) n c,(-<

zy ~, a,

zya ~)

The result now follows by induction . m Let x, y E X* . Suppose x - y . Let ~ z ~--, ~ w ~-- E Q . Suppose Q*(~z~,x,~w~)>0 . Then ~ zx ~ = ~ w ~ . Since x - y and - is a congruence relation, zx ~ = -< zy ~ . Thus zy a = -< w ~ . Hence a* (-< z ~, y, -< w ~) > 0. Similarly if a* (-< z ~, y, -< w ~) > 0, then Q*(~z~,x,~w~)>0 . © 2002 by Chapman & Hall/CRC

242

6. Algebraic Fuzzy Automata Theory

Hence x y . Conversely, suppose x ,~ y . Let ~ z >- E Q . Now c,*(-< z x,-0.Hence a*(~z~,y,~zx~--)>0.Thus -
1 0

T(p,a,q)=

a, q')

E gEQ T (p, a,

1

q)

if if if if

p', q' E Q, p' E Q and q' = z, p'=zandq'EQ, p'=zandq'=z,

for all a E X. Let X+ = X*\{A} . Let T be a t-norm on [0,1] . Define T+ :QxX+xQ~[0,1]by T + (p, a1

. . . an , q)

=

V {T(p, a1, r1)TT(r1, a2, r2)T . . . TT(rn_ rZ E Q, i=1,2, . . . ,n-1},

where p, q E Q and a,, . an E X. M is called a T-generalized state machine, when T+ is defined in terms of T. A theory for T-generalized state machines is developed in [113] . We present the main results of [113] in the Exercises . 6 .3

Homomorphisms

Definition 6.3.1 Let Ml = (Q1, X1, fq) and M2 = (Q2, X2, P2) be ffsms . A pair (a,,3) of mappings, a : Q1 ----> Q2 and 3 : X1 ----> X2, is called a homomorphism, written (a,,3) : Ml - M2 , if /t l (q, x,p) <_ P2 (a(q),,3(x), a (p)) d q, p E Q1 and b' x E X1 . The pair (a,,3) is called a strong homomorphism if w2 (a(q),,3(x), a(p)) = V{lt1(q, x, t) I t E Q1, a(t) = a(p)} V q, p E Q1 and V x E X1 . © 2002 by Chapman & Hall/CRC

6.3 . Homomorphisms

an

243

A homomorphism (strong homomorphism) (a,,3) : Ml ~ M2 is called isomorphism (strong isomorphism) if a and,3 are both one-one and

onto .

Example 6.3 .2 Let Ml = (Q1, X1 , p l) and M2 = (Q 2 , X2 , p2 ) be ffsms, where Q1 = {q1, q2, q3}, X1 = {a, b}, Q2 = {qi, q2, q3}, X2 = {a, b}, and p 1 and P2 are defined as follows:

1

P1(g1,a,g1)

3 2 3

P1(gl,b, g2)

ft, (q2,

1

a, g1)

3 2 3

ft, (g2,b, g3)

1

P1 (q3, a, q3) P1 (q3,

3 2 3

b, q2)

P2 (q', a, q') f'2(gl, b, q2) P2 (g', a, qi )

1

3 2 3

1

P2(g2,b,qi)

3 2 3

P2(ga, a, qi)

2

P2 (q3,

2'

b,

q2)

1 1

For all other ordered triples, (q, x, s) and (q', x', s') define ft, (q, x, s) = 0 and p 2 (q', x', s') = 0 . Define a : Q1 ----> Q2 and 3 : X1 ----> X2 as follows: x(q1) = cti(g3) = ql, cti(g2) = q'2 ,,3(a) = a, and,3(b) = b. We note that (a,,3) is a strong homomorphism of equalities . P2(a(gi),,3(a),c,(qi))

w2 (a(qi), /3 (b), a(q2))

Q1

=

into

Q2 .

This follows from the following

P2 (q', a, q')

=

P2(gl,b, q2)

P2(a(g2),,3(a),c,(qi))

=

P2 (q2 , a, qi )

w2 (a(q2), /3(b), a(q3))

=

P2(g2,b, ql) P2 (q', a, qi)

w2 (a(g3), ,3(a), c ,(g3)) w2 (a(q3), /3(b), a(q2))

(1)

=

P2(gl,b, q2)

1

3 2 3

P1 (g1, a, q1)

3 2 3

P1 (q2, a, q1)

3 2 3

P1 (q3, a,

1 1

P1 (g1, b, q2) P1 (q2, b, q3)

q3)

P1 (q3, b, q2) .

6.3 .1, if X1 = X2 and 3 is the identity map, then we M1 ~ M2 and say that a is a homomorphism or strong homomorphism accordingly. (2) If (a,,3) is a strong homomorphism with a one-one, then In Definition

simply write a :

P2 (a(q),,3(x),a(p)) =ft1(q,x,p) Vq,pEQ1andVxEX1 .

Lemma 6.3 .3 Let M1 = (Q1, X1, ft1) and M2 = (Q2, X2, p2) be two ffsms. Let (a,,3) : M1 ----> M2 be a strong homomorphism . Then dq, r E Q1, b'x E X1, if ft2 (a(q), l3(x), cti(r)) > 0, then 3 t E Q1 such that ft1(q, x, t) > 0 and a(t) = a(r) . Furthermore, b'p E Q if a(p) = a(q), then ft1(q,x,t) > P1 (p, x,

0

© 2002 by Chapman & Hall/CRC

244

6. Algebraic Fuzzy Automata Theory Proof. Let p, q, r E Q1, x E X 1 , and ft2(a(q), /3(x), a(r)) > 0 . Then V{ftl (q, x, s) I s E Q1, a(s) =a ( r )I > 0.

Since Q1 is finite, 3 t E Q1 such that a(t) = a(r) and ftl (q, x, t) _ V{ftl(q, x, s)I s E Q1, a(s) = a(r)} > 0 . Suppose a(p) = a(q) . Then

wl (q, x, t) = w2(a(q), /3(x), a(r)) = w2 (C(p), /3(x), C(O) >- wl (p, x, r)- . Definition 6.3.4 Let Ml = (Q 1 , X1, ftl) and M2 = (Q2, X21 [t2) be two ffsms. Let (a,,3) : Ml ----> M2 be a homomorphism. Define /3* : Xl ----> X2 by /3* (A) = A and /3* (ua) = /3* (u),3(a) b' u E X*, a E X1 . Lemma 6 .3.5 Let Ml, M2, (a,,3), and /3* be as above. Then 3*(uv) _ * '3 (u)/3* (v) b' u, v E Xl . Proof. Let u, v E Xl and Iv I = n . If n = 0, then v = A and hence /3* (uv) = /3* (u) = /3* (u),3* (v) . Suppose now the result is true b' y E Xl such that IyI = n - 1, n > 0. Let v = ya where y E X*, a E X1, and /3* (uv) = /3* (uya) = /3* (uy),3(a) _ /3* (u),3* - 1 . Then Iy (y),3(a) _ /3*I = n (ya) /3* = (v) . The result now follows by induction. m (u),3* (u),3* Theorem 6.3.6 Let Ml, M2 be as above. Let (a,,3) : Ml ----> M2 be a homomorphism. Then ft*, (q, x, p) <_ [t2*2 (a(q), /3* (x), a(p)) b' q,p E Q1 and xEXl . Proof. Let q, p E Q1 and x E X* . We prove the result by induction on IxI = n. If n = 0, then x = A and 3* (x) _ 3* (A) = A. Now if q = p, then y* (q, A, p) = 1 = ft2(a(q),A,a(p)) . If q p, then fti(q,A,p) = 0 < lt2(a(q), A, a (p)). Suppose now the result is true b' y E X* such that IyI = n-l,n>O .Letx=yawhereyEXl,aEXI,andlyl=n-1 . Now fti (q, x, p)

= = < < = =

fti (q, ya, p) V{fti (q, y, r) n fti (r, a, p) Ir E Ql V{ft2(a(q),,3*(y),a(r)) A[t*(a(r),,3(a),a(p))Ir E Ql} V{P2*(a(q), 3* (y), r~) nft2* W, 3* (a), a (p) I r' E Q2} ft2 (a(q), ,3* (y),3(a), a (p)) ft2(a(q),,3*(ya),a(p)) ft2(a(q),,3*(x), a (p)). m

Theorem 6.3.7 Let Ml, M2 be as above. Let (a,,3) : Ml ----> M2 be a strong homomorphism. Then a is one-one if and only if fti (q, x, p) = ft2(a(q),,3*(x), a (p)) V q, p E Q1 and x E Xl . Proof. Suppose a is one-one . Let p, q E Q1 and x E X* . Let Ix I = n . We prove the result by induction on n. Let n = 0. Then x = A and /3* (A) = © 2002 by Chapman & Hall/CRC

6.4 . Admissible Relations

245

A. Now a(q) = a (p) if and only if q = p. Hence fti (q, A, p) = 1 if and only if ft(a(q),,3*(A), a(p)) = 1. Suppose the result is true b' y E Xl, IyI = n- 1, 2 n > O . Let x = ya, Iy j =n-1, y E Xl , a E Xl . Then ft2 (a(q),

/3*

/3* ft2 (a(q), (ya), a(p)) /3* ft2 (a(q), (y),3 (a), a (p)) V{ft2(c,(q),,3*(y),a(r)) AIt2(C(r),,3(a),a(p)) IrEQ1}IrEQ1}

(x), a(p))

V{fti(q,y,r) Afti(r,a,p) fti (q, ya, p) fti (q, x, p) .

I r E Qi}

Conversely, let q, p E Qi and let a(q) = a(p) . Then 1 = y,2 (a (q), A, a (p)) = fti (q, A, p) . Hence q = p, i.e ., a is one-one . m

6 .4

Admissible Relations

Definition 6.4 .1 Let M = (Q, X, ft) be a ffsm and let - be an equivalence

relation on Q . Then - is called an admissible relation if and only if V p, q, r E Q, b'a E X, if p - q and ft(p, a, r) > 0, then 3 t E Q such that ft(q, a, t) > ft(p, a, r) and t - r.

Theorem 6.4.2 Let M = (Q, X, ft) be a ffsm and let - be an equivalence relation on Q. Then - is an admissible relation if and only if V p, q, r E Q, V'x E X*, if p - q and ft* (p, x, r) > 0, then 3 t E Q such that ft* (q, x, t) >_ p,* (p, x, r) and t - r.

Proof. Suppose - is admissible. Let p, q E Q be such that p - q. Let x E X*, r E Q be such that /t* (p, x, r) > 0. Suppose Ixj = n. If n = 0, then x = A . Thus W(p, x, r) > 0 ==~, p = r and /t* (p, x, p) = 1. Now ft* (q, x, q) = 1 = ft* (p, x, p) and q - p. Thus the result is true for n = 0. Suppose now the result is true b' y E X* such that Iyj = n - 1, n > 0. Let x = ya where y E X*, a E Xl, and Iyj =n-1 . Now y*(p, x, r) = p,* (p, ya, r) = V{ft* (p, y, ql) A /t* (ql, a, r) I ql E Q} > 0 . Let s E Q be such that y,* (p, y, s) A /t* (s, a, r) = V {/t* (p, y, qi) A /t* (qi, a, r) I qi E Q} . Then ft* (p, y, s) > 0 and ft* (s, a, r) > 0. By the induction hypothesis, 3 is E Q such that y,* (q, y, ts) >_ y,* (p, y, s) and is - s. Now ft(s, a, r) > 0 and is - s. Since - is admissible, 3 t E Q such that ft(ts, a, t) >_ ft(s, a, r) and t - r. Thus 3 t E Q such that t - r and y* (q, y, ts) A ft (t,, a, t) > ft* (p, y, s) A /t* (s, a, r) . Thus * ft (p, x, r)

= < = =

© 2002 by Chapman & Hall/CRC

ft * (p, y, s) n ft* (s, a, r) ft * (q, y, ts) n ft(ts, a, t) V{ft* (q,y,ri) Aft(rl,a,t)Iri E Q} ft * (q, ya, t) ft * (q, x, t)

24 6

6. Algebraic Fuzzy Automata Theory

and t - r. The result now follows by induction . The converse is trivial . m Let M = (Q, X, ft) be a ffsm and let - be an admissible _ relation on Q. For q E Q, let [q] denote the equivalence class of q . Let Q = Q1 = { [q] Iq E Q} . Define the fuzzy subset ~ of Q x X x Q by w([q], x, [p])=V{ft(q,x,t) It E [p]} Vq, p E Q, x E X. Suppose that [q] = [q~], x = y, and [p] = [p~], q, q~, p, p' E Q and x, y E X. Then q - q' . Now ~([q] , x, [p]) = V {ft(q, x, r) I r E [p]

and

w([q ], y, [p])

= w([q ], x,

[p]) = V{ft(q, x, t) It E p]} .

Let r E [p] be such that ft(q, x, r) > 0. Then since - is admissible, 3 t E Q such that ft(q~, x, t) >_ ft(q, x, r) > 0 and t - r. Now since t - r, t E [p] = [p'] . Thus 3 t E [p~] such that ft(q~, x, t) >_ ft(q, x, r) > 0. Similarly if ft(q~, x, t) > 0 for some t E [p~], then 3 r E [p] such that ft(q, x, r) >_ ft(q~, x, t) > 0. Hence ~([q], x, [p]) = ~([q], x, p]) . Thus %~ is single-valued . Hence (Q, X, ~) is a ffsm. Define a : Q ---> Q by a(q) _ [q] b' q E Q. Clearly, a maps Q onto Q . Let 3 : X ---> X be the identity map. Let q, t E Q and x E X. Then ~(a(q), x, a(t)) = ~([q], x, [t]) = V{ft(q, x, r) Ir E [t] j > ft(q, x, t) . Hence (a,,3) is a homomorphism. Definition 6.4.3 Let Ml = (Q1, X, ftl) and M2 = (Q 2 , X, ft2) be two ff-

sms. Let a : (Q1, X, ftl) (Q2, X, ft2) be a strong homomorphism . The kernel of a, denoted Ker a, is defined to be the set Ker a = {(p, q) I a(p) = a(q)1 .

Lemma 6 .4 .4 Let a be as defined in Definition 6.4 .3 . Then Ker a is an admissible relation.

Proof. Now clearly Ker a is an equivalence relation . Let p, q E Q1 and (p, q) E Ker a . Then a(p) = a(q) . Let a E X, r E Q1, and ftl(p, a, r) > 0 . Then y 2 (a(q), a, a(r)) = y 2 (cti(p), a, a(r)) > ftl(p, a, r) > 0. By Lemma 6.3 .3, 3 t E Q1 such that ft, (q, a, t) >_ ft, (p, a, r) > 0 and a(t) = a(r) . Since a (t) = a (r), (t, r) E Ker a. Thus Ker a is admissible. m Theorem 6.4.5 Let Ml = (Q1, X, ftl) and M2 = (Q2, X, [t2) be two ftsms and let a : (Q 1 , X, ftl) - (Q 2 , X, ft2) be an onto strong homomorphism. Then 3 an isomorphism 'Y : (Q1/(Ker a), X, f~l) -' (Q2, X, ft2) such that a = -Y o a.

© 2002 by Chapman & Hall/CRC

6.5. Fuzzy Transformation Semigroups

247

Proof. Define y : Q,I(Ker a) ~ Q2 by y([q]) = a(q) . Let p, q E Q1 be such that [p] = [q] . Then (p, q) E Ker a and hence a(p) = a(q) or y([q]) = y([p]) . Now, let q, p E Q1 and x E X. Then wl([q],x, [p])

V~fq(q,x,r)Ir E [p]f V{ftl (q, x, r) I a(r) = a(p), r E Q11 ft2(a(q), x, a(p)) ft2('Y([q]), x, -y ([p])) .

Thus y is a homomorphism. Clearly y maps (Q 1I(Ker cti), X, fil) oneto-one onto (Q 2, X, p 2 ) . Example 6.4.6 Let MZ = (QZ, XZ , p 2 ) be the ffsm of Example 6.3.2, i = 1, 2, 3. Let M2 = M, X2, fez) be the ffsm, where Q'2 = {ql, q2}, X2 = {a, b},

and ft'2 = [t2 IQ', X 2, 2 . Let a and,3 be defined as in Example 6.3.2. Then (a,,3) is a strong homomorphism of Ml onto M2 . Ker a = {(ql, q1), (q2, q2), (_g3, q3), (g1, g3), (g3, q1)} . Then [ql] = {g1, g3} = [q3] and [q2] = {q2} . Also Q1 = Q1lKer a = {[q1], [q2]} and N'1([q1], a, [q1]) fa'1l[q1], b, [q2])

fa'1l[q2], a, [q1]) f71l[q2], b, [q1])

1

3 2 3

1

3 2 3'

It is easily seen that there exists an isomorphism y : (Q1/Ker a, X, fit ) ----> (Q2, X2, fez) such that cti = y o cx, where X = X2 . 6 .5

Fuzzy Transformation Semigroups

A transformation semigroup is a pair (Q, S), where Q is a finite nonempty set and S is a finite semigroup with an action S of S on Q, i.e ., a partial function S of Q x S into Q such that (1) S(S(q, s), s') = S(q, ss') b'q E Q, s, s' E S, and (2) S(q, s) = S(q, s') b'q E Q implies s = s', where s, s' E S, [92, p. 33]. Definition 6.5.1 A fuzzy transformation semigroup (fts) is a triple

(Q, S, p), where Q is a finite nonempty set, S is a finite semigroup, and p is a fuzzy subset of Q x S x Q such that (1) p(q, uv, p) = V{p(q, u, r) A p(r, v, p) I r E Q} Vu, v E S and b'q, p E Q; (2) If S contains the identity e, then p(q, e, p) = 1 if q = p and p(q, e, p) _

0ifgzAp,dq,pEQ . If, in addition, the following property holds, then (Q, S, p) is called faithful . (3) Let u, v E S. If p(q, u, p) = p(q, v, p) b'q, p E Q, then u = v.

© 2002 by Chapman & Hall/CRC

248

6. Algebraic Fuzzy Automata Theory

Let M = (Q, S, p) be a fts . This fts may not be faithful . Define a relation R on S by b' u, v E S, uRv if and only if b' q, p E Q, p (q, u, p) = p(q, v, p) . Clearly R is an equivalence relation on S. Suppose that u, v, x E S and uRv. Then p(q, ux, p)

=

V{p(q, u, r) n p(r, x, p) I r E Q} V{p(q, v, r) n p(r, x, p) I r E Q} p(q, vx, p)

b' q, p E Q. Similarly, p(q, xu, p) = p(q, xv,p) b' q, p E Q. Hence R is a congruence relation on S. Let [u] denote the equivalence class of R induced by u. Let SIR = { [u] I u E S} . Define p :QxSIRxQ----> [0,1]

by p(q, [x],p) = p(q,x,p) b' q, p E Q and b' [x] E SIR. Clearly, p is single-valued . Now P(q, [x][y],p)

=

P(q, [xy],p) p(q, xy,p) V{p(q, x, r) n p(r, y, p) r E Q V{P(q, [x], r) n p(r, [y], p) I r E Q

Also _ _ p(q' [e]'p)

l ifp=q 0 otherwise .

Suppose that p(q, [x], p) = p(q, [y], p) Vq, p E Q. Then p(q, x, p) = p(q, y, p) Vq,p E Q . Hence xRy and so [x] = [y] . Thus (Q, SIR, p) is a faithful fts . We call (Q, SIR, p) the faithful fuzzy transformation semigroup represented by the triple (Q, S, p) . Theorem 6.5.2 Let M = (Q, X, p) be a ffsm . Let E(M) be defined as before. Then (Q, E(M), p) is a faithful fts where p(q, [x], p) = / t* (q, x, p) b' q,pEQ,xEX* .

Proof. By Theorem 6.2 .4, E(M) is a finite semigroup with identity [A] . Clearly p is single-valued . Let q, p E Q and [x], [y] E E(M) . Then p(q, [x] * [y],p)

© 2002 by Chapman & Hall/CRC

= =

p(q, [xy],p) [t*(q, xy,p) V{[t*(q,x,r) n[t* (r,y,p)Ir E Q} V{p(q, [x], r) n p(r, [y],p) Ir E Q} .

6.5. Fuzzy Transformation Semigroups

249

Now P (q, [A], p) _ { p,* (q, A, p) = 0 if q z,4p,

by the definition of tt*. Suppose p(q, [x], p) = p(q, [y], p) b' q, p E Q. Then ft * (q, x, p) = ft * (q, y, p) dq, p E Q. Thus x - y or [x] = [y] . Hence (Q, E(M), p) is a faithful fuzzy transformation semigroup . m Let M = (Q, X, p,) be a ffsm. Then by Theorem 6.5 .2 (Q, E(M), p) is a fuzzy transformation semigroup that we denote by FTS(M) . We call FTS(M) the fuzzy transformation semigroup associated with M. Let M = (Q, X, ft) be a ffsm. Define the relation -+ on X+ by V'x, y E X+, x -+ y if and only if ft+(q, x, p) = ft+(q, y,p) dq,p E Q. Then -+ is the restriction of - to X+ x X+ . Let S(M) denote the set of all equivalence classes induced by -+ . Then S(M) = E(M)\{A} and S(M) is a subsemigroup of E(M) . In view of the crisp case, it would have been reasonable to define (Q, S(M), p) as the fuzzy transformation semigroup associated with M. Example 6.5.3 Let M = (Q, X, p) be the ffsm such that Q = {q1, q2, q3}, X = {a, b}, and ft : Q x X x Q ----> [0,1] is defined as follows: ft (q1 , a, q1) ft(g1, b, q2)

ft(g2, a, q1) ft(g2, b, qj) ft(g3, a, q3) ft(g3, b, q2)

1

3 _2 3 1

3 0 1

3 2 3

for j = 1, 2,3 and ft(q, x, q~) = 0 for any other triple (q, x, q~) E Q x X x Then E(M) = {[[All, [[all, [[b]], [[ab]], [[ball, [[aba]]} U {[[bell} and E(M) = {[A], [a], [b], [ab], [ba], [aba], [bab]} U {[[b2ll} . In the crisp case, ft would be considered a "partial" function since ft(g2, b, qj) = 0, for j = 1, 2, 3 and the equivalence classes [[b2]] and [b2 ] would be designated as an empty relation 0. Hence if we set 0 = [[b2]] and then 0 = [b2], we have that the following tables give the semigroup operations for E(M)\{0} and E(M)\{0}, respectively. Note that 0*x = x;~d = 0 for all

© 2002 by Chapman & Hall/CRC

25 0

6. Algebraic Fuzzy Automata Theory

xEE(M) andO*x=x*8=0 forallxEE(M) . *

[[A]]

[[a]]

[[a]]

[[b]] [[ab]] [[ba]] [[aba]]

[[b]] [[ab]] [[ba]] [[aba]]

*

[A]

[a] [b] [ab] [ba] [aba] [bab]

[[a]]

[[a]] [[a]]

[[ba]] [[aba]] [[ba]] [[aba]]

[[b]]

[[ab]]

[[b]] [[ab]]

[[ab]] [[b]] [[ab]]

[[b]] [[ab]] 0 0

[a] [b] [ab]

[a] [a] [a] [ba] [aba]

[b] [b] [ab] 0 0

[ab] [ab] [ab] [b] [ab]

[ba] [aba] [bab]

[ba] [aba] [ba]

[b] [ab] 0

[b] [ab] [bab]

[[ab]] [[ab]] [[b]]

[ba]

[ba] [aba] 0 0 [ba] [aba] 0

[[ba]] [[ba]] [[aba]] 0 0

[[aba]] [[aba]] [[aba]]

[[ba]] [[aba]]

[[ba]] [[aba]]

[aba] [aba] [aba] [ba] [aba] [ba] [aba] [ba]

[[ba]] [[aba]]

[bab] [bab] [ba] 0 [bab] 0 [ba] 0

The fts (Q, E(M), p) is now easily determined.

Definition 6.5.4 Let (Q, S, p) be a fts. Let - be an equivalence relation on

Q. Then - is called an admissible relation if and only if dp, q, r E Q, Vu E S, if p - q and p(p, u, r) > 0, then 3 t E Q such that p(q, u, t) >_ p(p, u, r) and t-r.

Theorem 6.5.5 Let M = (Q, X, /t) be a ffsm and let - be an equivalence relation on Q. Then - is an admissible relation for M if and only if - is an admissible relation for the fuzzy transformation semigroup, FTS(M) _ (Q, E(M), p) .

Proof. Suppose - is admissible for M. Let p, q E Q be such that p - q and [u] E E(M) . Let p(p, [u], r) > 0 for some r E Q. Then [t* (p, u, r) > 0 . Hence by Theorem 6.4 .2, 3 t E Q such that /t* (q, u, t) >_ /t* (p, u, r) and t - r. Thus p(q, [u], t) = p,* (q, u, t) >_ p,*(p, u, r) = p(p, [u], r) . Hence is admissible for FTS(M) . Conversely, suppose that - is admissible for FTS(M) . Let p, q E Q be such that p - q and u E X. Let /t* (p, u, r) > 0 for some r E Q . Then p(p, [u], r) > 0. Then 3 t E Q such that p(q, [u], t) >_ * p(p, [u], r) and t - r. Now /t* (q, u, t) = p(q, [u], t) >_ p(p, [u], r) = [t (p, u, r) and t - r. Hence - is admissible for M. m Definition 6.5.6 Let (Q1, Sl, pl ) and (Q2, S2, p2) be two fts's. A pair (f, g) of mappings, where f : Q 1 -' Q2 and g : Sl - S2, is said to be a homomorphism from (Q1, Sl, pl ) to (Q2, S2, p2) of (1) g(xy) = g(x)g(y) d X, y E Si, © 2002 by Chapman & Hall/CRC

6.5. Fuzzy Transformation Semigroups

251

(2) If el is the identity of Sl and e2 is the identity of S2, then g(el) = e2,

(3) pi (q,x,p) <_ p2(f(q),g(x),f(p)) Vq,p E Q1, x E Sl . (f , g) is called a strong homomorphism if it satisfies (1), (2), and p2 (f (q), g(x), f (p)) = V{pi (q, x, t) It E Q1, f(t) = f (p)}

dq,pEQ1,xES1 . A homomorphism (strong homomorphism) (f, g) : (Q1, Sl , pi) ~ (Q2, S2, p2) is called an isomorphism (strong isomorphism) if f and g are both one-one and onto .

Let S be a semigroup with identity. Let A = (Q, S, S) be a faithful fts . Define the ffsm M = (Q, S, /t) by taking ft = S . Consider FTS(M) = (Q, E(M), p), where E(M) = S*1 - and p(q, [u], p) = p,* (q, u, p) . Let e be the identity element of S and A the empty word in S* . Now p(q, [e], p) = p,* (q, e, p) = S(q, e, p) = 1 if p = q and 0 if p z/4 q. Hence p(q, [e], p) = p(q, [A], p) Vp,q E Q. Thus [e] _ [A] . Theorem 6.5.7 Let S be a semigroup with identity . isomorphic to A = (Q, S, S) .

Then FTS(M) is

Proof. Define f : Q ~ Q by f(q) = q Vq E Q and g : S ~ E (M) by g(x) = [x] V x E S. Let x, y E S be such that g(x) = g(y) . Then [x] = [y]. Thus /t* (q, x, p) = ft * (q, y, p) dq, p E Q . Hence ft(q, x, p) = ft(q, y, p) V q, p E Q. This implies that 6(q, x, p) = 6(q, y, p) Vq, p E Q. Since A is faithful, we find that x = y. Hence g is injective. Let - denote the binary operation of the semigroup S. Let a, b E S. Then a - b E S and ab E S* . Let q, p E Q . Now * ft (q, a - b, p)

= = = = =

ft (q, a b, p) S(q, a b, p) V{6(q,a,r) AS(r,b,p)Ir E Q} V{ft(q,a,r) Ap,(r,b,p)Ir E Q} ft * (q, ab, p) .

Hence [a - b] = [ab] . Thus g(ab) = [a b] = [ab] = [a][b] = g(a)g(b) . By induction it can be shown that if c2 E S, 1 <_ i <_ n, then ICI c2 - . . . cn] = [cic2 . . . cn] . Let [u] E E(M) . If u = A, then [A] = [e] and g(e) = [A] . Suppose u = ala2 . . . an, a2 E S, 1 < i <_ n. Then g(al a2 - . . . an) = [al - a2 - . . . - an] = [al a2 . . . an] = [u]_Thus g is surjective. Finally, p(f (q), g(x), f (p)) = p(q, [x],p) = ft * (q, x,p) = ft(q, x,p) = b(q, x, p) . .

Theorem 6.5.8 Let Ml = (Ql, Xl, p l) and M2 = (Q2, X2, p2 ) be two ffsms and let (a,,3) : Ml M2 be a strong homomorphism with a oneone and onto. Then 3 a strong homomorphism (fa,goo) from FTS(MI) to FTS(M2) .

© 2002 by Chapman & Hall/CRC

25 2

6. Algebraic Fuzzy Automata Theory

Proof. Define fa : Qi Q2 by fa(q) = a(q) b' q E Qi and goo E(M2) by go([x]) _ [/3* (x)] d [x] E E(MI) . Let [x], [y] E E(MI) and [x] _ [y] . Then g* (q, x, p) = g* (q, y, p) b' q, p E Q . Now E(MI)

Nt2 (a(q),

3*

(x), a(p))

= =

Nti (q, x, p) Nti (q, y, p) lt2(a(q),,3*(y), a(p))

b' q, p E Q1 . Thus since a is onto, [,3*(x)] = [,3*(y)] . Hence goo is well defined. Now goo ([x] * [y]) = go ([XYD = [X (xy) I = [X (x),3* (y)] = IX (x)] * IX (y)] = go ([x])

*

go ([y])

and

go ([A])

P, (q, [x] , p)

= [,3* (A)] = [A] . Also = =

Nti (q, x, p) l~2(a(q),,3*(x), a(p)) P2(fa(q), go Qx]), f.(p)) .

Hence by definition (fa, goo) is a strong homomorphism . Definition 6.5.9 A polytnansfonmation semignoup is a triple (Q, S, v), where Q is a finite nonempty set, S is a finite semigroup, and v : Q x S P(Q)\{O1} such that (1) v(v(q, u), v) = v(q, uv) b' q E Q, u, v E S. It is being understood for any subset P of Q, v(P, u) = U Epv(p, u) . (2) If S contains identity e, then v(q, e) _ {q} b' q E Q . If, in addition, the following holds, then (Q, S, v) is called faithful. (3) Let u, v E S. If v(q, u) = v(q, v) b' q E Q, then u = v.

Let M = (Q, X, /t) be a ffsm. For q E Q and x E X*, we define Sx (q) = {p E Qllt* (q, x, p) > 0} .

Theorem 6.5.10 Let M = (Q, X, y) be a ffsm. Let E(M) be defined as before. If b'x E X* and b'q E Q, Sx (q) :?~ 01, then (Q, E(M), v) is a faithful polytransformation semigroup with identity.

Proof. Define v : Q x E(M) ~ P(Q)\{O} by v(q, y* (q, x, p) > 0} . Suppose that -< x ~-- = -< y ~-- . Then x v(q, --<x ~--)

= = =

x >-) = {p E Q

y . Thus

{p E Qjft * (q,x,p) > 0} {p E Qjft * (q,y,p) > 0} v(q, -< y ~)

Thus v is single-valued. We write q --< x for v(q, -< x ~) . Then, by definition, (q -< x ~--) -< y >- = UpEq~x} p l y >- . Let p E q -< xy >* * = {p E Q I ft (q, xy, p) > 0} . Now /t* (q, xy, p) > 0 ==> /t* (q, x, r) n ft (r, y, p) > 0 for some r E Q . Thus y* (q, x, r) > 0 and y* (r, y, p) > 0. Thus © 2002 by Chapman & Hall/CRC

6.6. Products of Fuzzy Finite State Machines

253

rEq~x>-andpEr~y>- .Hence pEUt Eq ~x} t-- .Thus q~xy>-C(q-<x>-)(~y~) .Let p E (q --< x>-)(--< y>-) = UtEq~x>-t ~ y >- . Then p E t ~ y >- for some t E q ~ x >- . Thus y* (t, y, p) > 0 and y* (q, x, t) > 0. Hence y* (q, xy, p) > 0. Thus p E q -< xy . Hence (q-<x~--)(- 0 iff /t* (q, y, p) > 0 b' p E Q. Thus -< x _ y . Hence (3) of Definition 6.5 .9 holds.

6 .6

Products of Fuzzy Finite State Machines

The concepts of transformation semigroup, covering, cascade product, and wreath product play a prominent role in the study of automata [92] . In this chapter, we examine these ideas for fuzzy finite state machines . Some severe and interesting complications arise when introducing these ideas to the fuzzy setting. One of the concepts we introduce to overcome some of the problems that arise is that of a polysemigroup . Let P be a nonempty set and P(P) be the power set of P. Let * be a function of P x P into P(P)\{Q)} . Then (P, *) is called a polysemigroup, [39, 102], if and only if x * (y * z) = (x * y) * z b'x, y, z E P. The obvious abuse of notation is explained as follows: If x E P and A, B C_ P, then x * A denotes {x} * A, A * x denotes A * {x}, and A * B = UaEA,bEBa * b.

Definition 6.6.1 Let MZ = (QZ, XZ, /t 2 ) be a ffsm, i = 1, 2 . Let 97 be a function of Q2 onto Q1 and let ~ be a function of Xl into X2 . Extend to a function ~* of Xl into X2 by ~* (A) = A and b' x E X*, ~* (x) _ ~(xl)~(x2) . . . S (xn) where x = xlx2 . . . xn and x2 E X1, i = 1, 2, . . . , n. Then (t7, ~) is called a covering of Ml by M2, written Ml <_ M2 , if and only if b' q2 E Q2, ql E Q1, and x E X* ' ftl* (t7(g2), x, ql) = V{ft2 * (q2, ~* (x), r2) I t7(r2) = ql, r2 E Q2} . Clearly (r7, ~) is a covering of Ml by M2 if and only if b' q2 E Q2, ql E Q1, and x E Xi, /ti(rl(g2), x, ql) > tt2(g2, ~ * (x), r2) V r2 E Q2 such that t7(r2) = ql and 3 r2 E Q2 such that 9(r2) = ql and /t l* (t7(g2), x, ql) _ ft2* (q2, C(x), r2)-

Example 6.6.2 Let Ml = (Ql, X1, yl ) and M2 = (Q2, X2, [t2) be ffsms

such that Q1 = {ql, q2}, Xl = {a, b}, Q2 = {qi, qz, q3}, X2 = {a, b}, and © 2002 by Chapman & Hall/CRC

254

6. Algebraic Fuzzy Automata Theory

ft, and P2 are defined as follows: P1(gi, a, q1)

=

P1(gL,b, q2)

=

P1(g2,a,q2)

=

P1(g2,b, q1)

=

P2 (q', a, q3)

1 3 _2 3 1 3 2 3 1 3

2 3 _1 3 2 3 1 3 1 4

w2(gi,b, q') P2 (q21 a, q2) P2(g2,b,gs) w2 (qs, a, qi) w2(gs,b, q2)

and ft, (q, x, p) = 0 for all other (q, x, p) E Q1 x X1 x Q1 and p,2 (q', x, p') = 0 for all other (q',x,p') E Q2 x X2 x Q2 . Define 97 : Q2 ----> Q, by 97(q') = rl(q'3 ) = q1 and rl(g2) = q2 . Let ~ be the identity map on X1 = X2 . Now for all xEXl =X2 fti (q1, fti (q1, I t* (g2, f ti (g2,

x, q1) x, q2) x, g1) x, g2)

= = =

=

ft2 (qi, x, qi) V ft2 (qi, x, q3) P2 (gi, x, g2) p2 (q2, x, qi) V ft2 (g2, x, g3) ft2 (q2, x, q2)

Thus (97, ~) is a covering of M1 by M2 .

Let MZ = (QZ, Xi, pi) be a ffsm, i = 1, 2 . Let X be a finite set and f a function from X into X1 x X2 . Let 7r 2 be the projection map of X1 x X2 onto Xi, i = 1, 2. Definition 6.6.3 Define ft f : (Q1 x Q2) x X x V(q1, q2), (p1, p2) E Q1 x Q2 and V'a E X,

(Q1

x

Q2) ----> [0,1] as follows:

ftf((g1,q2),a, (PI, P2)) =w1 x w2((g1,g2), (7r1(f(a)),7r2(f(a))), (p1,p2))-

Then (Q1 x Q2, X, ftf) is called the general direct product of M1 and M2 and we write M1 * M2 for (Q1 x Q2, X, tt f) .

Recall P1 x P2 ((g1, q2), (a1, a2), (PI, P2)) = ft, (g1, a1, p1)

n P2 (q2, a2, p2)

V(q1, q2), (PI, P2) E Q1 x Q2 V(a1, a2) E X1 x X2 . If X = X1 x X2 and f is the identity map, then M1 * M2 is called the full direct product of M1 and M2 and we write M1 x M2 for M1 * M2 . If X1 = X2 , X = {(a 1 , a 2 ) ja i E Xi , i = 1, 2, a 1 = a 2 }, and f is the identity map, then M1 * M2 is called the restricted direct product of M1 and M2 and we write M1 A M2 for M1 * M2 . (We could also let X = X1 = X2 and f : X ~ {(a1, a2) I a2 E Xi, i = 1, 2, a1 = a2} where f (a) = (a, a) to

obtain the restricted direct product .) For every result concerning M1 * M2 , there is a corresponding result for M1 AM2 . We see this by making the identifications (a, a) ~ a V'a E X1 = X2 ~ and (x1, x1) . . . x n , xn x 1 . . . x n for x2 E X1, 2 = 1, . . . , n. © 2002 by Chapman & Hall/CRC

6.6. Products of Fuzzy Finite State Machines

255

Example 6.6.4 Let Ml = (Q1, X1 , fq) and M2 = (Q 2 , X2, ft2) be ffsms, where Q1 = {q1, q21, X1 = {a}, Q2 = {qi, q2}, X2 = {a}, and ft, and It2 are defined as follows: wL(gi, a, qi) wL(gj, a, q2) wL(g2 , a, qi) wL(g2, a, q2)

= 0 = CI > 0 = c2 > 0 = 0

w2 (gi, a, qi) w2 (gi, a, q2 ) w2(g2,a,gi) w2 (g2 , a, g2 )

=

0

=

0

Then x w2((gi, qi), (a, a), (g2, q2)) x w2 ((g2, qi), (a, a), (gi, q2)) x lt2((gj, q2), (a, a), (g2, q2)) x w2 ((g2, q2), (a, a), (qj, q2))

wL(g1, a, q2) cl Ad,

dl

>0

d2

>0 .

n w2 (qi, a, q2)

wL(g2, a, g1) c2 n dl

n w2 (gi, a, q2)

wL(g2, a, qL) c2 n d2 .

n w2 (q2, a, q2)

wL(gj, a, q2) cl n d2

n w2 (q2, a, q2)

Suppose that c1 = c2 and dl = d2 . Then an - 1 am if and only if n and m are both even or both odd. Hence E(M1 ) _ {[A], [a], [a2 ]} . For M2 , an - 2 am b'n, m E N . Thus E(M2 ) = {[A], [a]} . (The reader is asked to determine E(M1 ) and E(M2 ) in the Exercises when c 1 :?~ c2 and d l :?~ d2 .) Since c1 = c2 and dl = d2, cl A dl = c2 A dl = cl A d2 = c2 A d2 . Hence an -12 am if and only if n and m are both even or both odd. Thus E(M1 x M2 ) = {[A], [a], [a 2 ]}, where [a3 ] = [a] . In this example, E(M1 A M2) = E(M1 x M2 ) .

Example 6.6.5 Let M1 = (Q1, X1, y 1 ) and M2 = (Q2, X2, y 2 ) be ffsms, where Q1 = {q1, q2 }, X1 = {a}~ Q2 = {qi, q2 }, X2 = {a, b}, and ft, and It2 are defined as follows: ft1(g1, It, (g1, ft1(g2, ft, (q2,

a, q1) a, q2)

a, q1) a, q2) It2 (qi, a, qi) It2 (qi, a, q2)

= = =

= = =

0 c1 > 0 c2 > 0 0 0

d1 > 0

ft2(gi, b, qi) ft2(gi, b, q2) ft2(g2, a, qi) ft2(g2, a, q2) It2 (g2 b, 1) It2 (q2, b, q2)

Then ft1 ftl ftl f~1 ftl ft1 ftl ftl

x x x x x x x x

ft2((g1, ft2((g2, ft2((g1, f~2((g2, ft2((g2, ft2((g1, ft2((g1, ft2((g2,

© 2002 by Chapman & Hall/CRC

qi), qi), qi), q2), qi), q2), q2), q2),

(a, b), (a, b), (a, a), (a, b), (a, a), (a, b), (a, a), (a, a),

(q2, q2)) (q1, qi)) (q2, q2)) (q1, qi)) (q1, q2)) (q2, qi)) (q2, q2)) (q1, q2))

_ = _

=

= = = = = =

d2 > 0

0 0

ds > 0 d4 > 0

0.

n d2 c2 A d2 c1 Ad, c2 n d4 c2 n d1 c1 n d4 c1 n d3 c2 A d3 . c1

25 6

6. Algebraic Fuzzy Automata Theory

3

4

I

Suppose that cl = c2 and dl = d2 = d = d . Then E(M ) = {[A], [a], [a 2 ]} and ({[a],[a2]},*) is a group with identity [a2 ] . E(M2 ) = {[A], [a], [b]}, where [a] = [a 2 ], [b] = [b 2 ], [ab] = [b], and [ba] = [a] . Thus S(M ) = {[a], [a2]} and S (M2 ) = {[a], [b]} . S (MI) x S (M2) = {([a], [a]), ([a], [b]), ([a 2 ], [a]), ([a2 ], [b])} . It follows that S(MI x M2) = {[(a, a)], [(a, a) 2 ], [(a, b)], [(a, b) 2 ]} . The operation tables of S(Mi ) x S(M2) and S(MI x M2 ) are given below.

([a], [a]) 2 ([a ], [a]) ([a], [b]) 2 ([a ], [b])

I

S(M ) x S(M2 ) ([a 2] , [a]) ([a], [b]) ([a2 ([a], [a]) ], [b]) ([a ([a], [a]) 2] , [a]) ([a], [b]) 2 2 ([a ], [a]) ([a ], [b]) ([a], [a]) ([a ([a], [a]) 2] , [a]) ([a], [b]) ([a], [a]) ([a2 ], [a])

I

2 ([a ], [b]) ([a], [b]) 2 ([a ], [b]) ([a], [b]) 2 ([a ], [b])

S(MI x M2 ) [(a, a)]

[(a, a)] [(a, a) 2 ] [(a, b)] [(a, b) 2 ]

[(a, a)'] [(a, a)] [(a, a) 2 ] [(a, a)]

[(a, a) 2 ] [(a, a)] [(a, a) 2 ] [(a, a)] [(a, a) 2 ]

[(a, b)]

[(a, b)L] [(a, b)] [(a, b) 2 ] [(a, b)]

[(a, b) 2 ] [(a, b)] [(a, b) 2 ] [(a, b)] [(a, b) 2 ]

We see that S(MI) x S(M2) - S(MI x M2 ) under ([a], [a]) ----> [(a, a)], 2 2 ([a ], [a]) - [(a, a) 2 ], ([a], [b]) [(a, b)], ([a ], [b]) - [(a, b) 2 ] . Lemma 6 .6 .6 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 . Consider the general direct product Ml * M2 . Then

ft f ((ql, q2), A, (pl, p2)) = ft~ (ql, A, pl) A ft2 (q2, A, P2) Vgl,pl E Q1, Vg2,P2 E Q2(2) [t*

where

a2

I

n , (PI, P2)) = fti (ql, 7r1(f (al)) . . . 7r (f (an)),pl)A q2), al . . .a ft2 (q2, 7r2 (f (al)) . . . 7r2 (f (an)), P2)) E X, i = 1, . . . , n, Vgl,pl E Q1, Vg2,P2 E Q2-

Proof. (1) The proof is straightforward . (2) Then /rf((gl,g2), TI

- -j., (PI, P2)) = V{ftf((gl,q2), al, (ri lf ,r21f ))A

flf((ri lf ,r2 lf ),a2,(ri2f ,r22f ))A . . . Apf((rin-lf ,r2n-lf ),an,(Pl,P2))

(ri ll ,

. .,n-1f =V{ftlxft2((gl,q2),(7r1(f(al)),7r2(f(al)), x r1(f(a2)),~2(f(a2)), (ri2f,r22f)) ' . . . . A (ri lf ,r2 lf )) Apl ft2((ri ll ,r2lf ),(7 ftl x ft2((rin-lf ,r2n-lf ), (7r 1(f(an)), 7r2(f(an),(Pl,M) (ri lf ,r2 lf ) E Q1 x r21f)EQ1xQ2,a=1,

© 2002 by Chapman & Hall/CRC

6.6. Products of Fuzzy Finite State Machines

257

Q2, a = 1, . . n- 1} = V{ft1(g1, 7r1(f(a1)), ri 1) ) AItA2, 7 2(f(a1), r21)) A 7 ft1(ri 1f , r1(f(a2)),ri2f )nf, 2(r (1) 7r2(f(a2),r2 2f )n . . . Aft 1(rin-~,7fa)P1 r21) n- 1} r1 1f E Q1, nft2(r2n-1> 7r2(f(an)), P2) E Q2, n-1) = V {ft1(g1, 7r1(f (a1)) rilf) n ft1(ri 1) , 7r1(f (a2)) r (2) ) n . . . n p1(r( > 7r1(f(an)), P1) I r11f E Q1, a = 1, . . , n- 1} AV{ftA2, 72(f(a1)), r21f ) n >

ft2(r21f , 72(f(a2)),r22f) n . . . n p2(r2n-1> 7r2(f(an)), P2) I r2 1f E Q2,i = 1, . . . , n-1} = pi(g1, 7r1(f(a1)) . . .7r1(f(an)), nft2(g2, 7r2(f(a1)) . . 7r2(f(an)), P2) . >

p1)

0

Corollary 6 .6 .7 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 . Then (1) ft2) * = ft2, (2) n p2) * = pr o p* . (p1 X

pi

X

(p1

Proposition 6 .6 .8 Let MZ = (QZ, X, pi) be a ffsm, i = 1, 2 . Define the relation -12 on X* by x -12 y if and only if x -1 y and x -2 y where -Z is the congruence relation on MZ defined in Theorem 6.2 .3, i = 1, 2 . Then -12 is a congruence relation on X* . 0 Definition 6 .6 .9 Let MZ = (QZ, X, pi) be a ffsm, i = 1, 2 . b'x E X*, let <x>={YEX*jy-12x} andletT={<x> IxEX*} . Let FTS(M1) n FTS(M2) = (Q1

X

Q2, T, P1 A P2)

where P1 n P2((g1, q2), < x >, (p1,p2)) = P(q1, [x]1, P1) n P(q2, [x]2,P2)We wish to point out that the coverings of the fuzzy transformation semigroups that appear in Theorems 6 .6 .10 and 6 .6 .11 are coverings as fuzzy finite state machines as stated in Definition 6 .6 .1 . Theorem 6 .6 .10 Let MZ = (QZ, X, pi) be a ffsm, i = 1, 2 . Then the following assertions hold.

FTS(M1 n M2) > FTS(M1) n FTS(M2) . Proof. Let x E X* and (g1, q2), (PI, P2) E Q1 X Q2 . Then P ((q1, q2), [x], (PI, P2))

= = = =

(ft, n ft2) * ((q1, q2), x, (P1,P2)) pi(g1,x,P1) np2(g2,x,P2) P1(g1,[x]1,P1) n P2(g2,[x]2,P2) (P1 n P2)((g1,q2), < x >, (PI, P2))

Let T be as in Definition 6 .6 .9 . Define C : T ~ E(M1 A M2 ) by C(<x>)=[x]vxEX* .

© 2002 by Chapman & Hall/CRC

25 8

6. Algebraic Fuzzy Automata Theory

Then C is single-valued since < x > = < y > ~ x -12 y <#~ x - 1 y and x -2 y =~> x - y (w. r. t. M1 A M2 ) <#~ [x] = [y] <#~ C(< x >) C(< y >) where the implication holds since x - y <#~ d(q1, q2), (PI, P2) E

Q1 x Q2, (ft, Aft2) * ((q1, q2), x, (P1,P2)) = (ft, nft2) * ((q1, q2), y, (P1,P2)) <#~ d(g1,q2), (PI, P2) E Q1 x Q2, fti(g1, x, P1) n [t* (q2, x, P2) = fti(g1, y, P1) n [t *2 (q2, y, P2) . (The latter does not imply x -1 y and x -2 y .) If we take 97 to be the identity map of Q1 x Q2, then we have ( 97, C) is a covering of FTS(M1)AFTS(M2) by FTS(M1 A M2 ) .

Theorem 6.6.11 Let MZ =

(QZ, X, p 2 ) be a ffsm, i = 1, 2 . Then the fol-

lowing assertions hold:

FTS(M1 x M2) > FTS(M1) x FTS(M2) > FTS(M1) n FTS(M2) .

Proof. Let a2 E X1 and b2 E X2 , i = 1, . . . , n. Then (a1, b1) . . .(an, bn) _ (a1 . . . an , b 1 . . . bn ) = (x 1 , x2) where x1 = a 1 . . . an and x2 = b1 . . . bn . Now M1 x M2= (Q1 x Q2, X1 x X2 , ft, A [t2) and so FTS(M1

x

M2 ) = (Q1

x

x

Q2, E(MI

M2), P)

and FTS(M1 )

Let

x1 E

x

x

FTS(M2 ) = (Q1

Xl and

x2 E X2 .

x

Q 2 , E(MI )

E(M2 ), P1 n P2)

Then

P((g1,q2), [(XI, X2)], (PI, P2))

=

(ft1 x ft2)*((g1,q2), (XI, X2), (PI, P2)) fti(g1,x1,P1) nft2(g2,x2,P2) P1 (g1, [x1 ], P1) AP2(g2, [x2], P2) (P1 n P2)((g1,q2), ([x1], [x2]), (P1,P2))-

Now define C : E (M1)

x

E (M2 ) ----> E (M1

x

M2 )

by C(([x1], [x2])) = [(x1, x2)]

X* . Then ([x1], [x2]) = ([y1], [y2]) <#~ [x1] = [y1] and [x2] = [y2] fti(g1, x1,P1) = ft1 * (g1, y1,P1) dg1,P1 E Q1 and ft2(g2, x2,P2) = f~2(g2, y2, P2) dg2,P2 E Q2 fti(g1, x1,P1)nft2 * (g2, x2, P2) = fti(g1, y1, P1) Aft * (q2, y2,P2) dg1,P1 E Q1, dg2,P2 E Q2 <#~ (ft1 x ft2) * ((q1, q2), (x1,x2), (P1,P2)) = (ft1 x ft2) * ((q1, q2), (y1,y2), (P1,P2)) d (g1,q2), (P1,P2) E Q1 x Q2 <#~ [(x1, x2)] = [(y1, y2)] <#~ C(([x1], [x2])) = C(([y1], [y2])) . Thus C is single-

Vx1,x2 E

valued .

© 2002 by Chapman & Hall/CRC

6.6. Products of Fuzzy Finite State Machines

259

Let 97 be the identity map of Q1 x Q2 . Then (9 7, C) is a covering of FTS(M1) xFTS(M2) by FTS(M1 x M2 ) . Let T be the set of all equivalence classes of E(M1 x M2 ) given by the relation -12 on (X1 x X2 )* where (X1, X2) =12 (y1, y2) ~ x 1 =1 y1 and x 2 -2 y2 . Define T :

T ---->

E(M1)

x

E(M2 )

by T(< (X1,X2) >) = ([X1], [X2])

b' < (x 1 , x 2 ) > E T. Then < (x 1 , x2 ) > = < (y1, y2) > ~ (X1, X2) =12 (y1, y2) ~ x1 -1 y1 and x2 -2 y2 ~ [x1] = [y1] and [x2] = [y2] ([X1], [X2]) = ([y1], [y2]) . Thus T is single-valued and one-to-one. It follows that (9, T) is a covering of FTS(M1) A FTS(M2) by FTS(M1) x FTS(M2) . Let M1 = (Q1, X1, ft1) and M2 = (Q2, X2, ft2) be fuzzy finite state machines . Let w be a function of Q2 x X2 into X1 . Let Q = Q1 x Q2 . Define ft' :QXX2XQ----> [0,1]

as follows :

d((g1,q2), b, (PI ,P2))

ft' ((g1, q2), b, (PI, P2))

=

E

Q

x X2 x

Q,

n ft2(g2, b,P2) .

ft, (g1, w (q2, b), P1)

Then M = (Q, X2, ft') is a ffsm. M is called the cascade product of M1 and M2 and we write M = M1wM2 . ft' is called separable if d (g1,q2),(P1,P2) E Q and b' y = y1 . . . y. E X2, ft-* ((g1,q2),y,(P1,P2)) = q (-1f ,y .),P1)Att* (g2,y,P2) for some q2W fti(g1,w(g2,y1)w(g21f ,y2) ,n-1 .

Q2,

Let M1 = (Q1, X1, fq) and M2 = (Q 2 , X2, [t2) be ftsms, where Q1 = {q1, q2}, X1 = {a, b}, Q2 = {qi, q2}, X2 = {a}, and ft, and ft2 are defined as follows:

Example 6.6.12

ft1(g1, a, q1) ft1(g1, a, q2) ft1(g1, b, q1) It, (g1, b, q2) ft1(g2, a, q1) ft1(g2, a, q2) Let

w :

Q2

x

X2

= = = = = =

~ X1

0 c1 > 0 c2 > 0 0 0 cs > 0 be defined as

w(qi,

© 2002 by Chapman & Hall/CRC

a) = a,

ft1(g2, b, q1) ft, (q2, b, q2) ft2 (qi, a, qi) ft2 (qi, a, q2) ft2 (q2, a, qi) ft2 (qz, a, qz)

follows : w(q2,

a) =

b.

c4>0 0 0 d1 > 0

d2 > 0 0.

260

6. Algebraic Fuzzy Automata Theory

Let Q

=

Q1 x Q2 . Then ftw : Q x X2 x Q ~ [0,1] is such that

ft - ((ql, qi), a, ft - ((q2, q2)a, ft - ((ql, q2)l a, ft - ((q2, qi)l a,

(q2, (ql, (ql, (q2,

q2)) qi)) qi)) q2))

= = = =

ftl(gl, w(qi, a), ftl(g2, w(q2, a), ftl(gl, w(q2, a), ftl(g2, w(qi, a),

q2) ql) ql) q2)

n n n n

ft2(gi, a, q2) ft2(g2, o;, q1 ) ft2(g2, a, qi) ft2(gi, a, q2)

= = = =

cl c4 c2 cs

Ad,, A d2 , A d2, A d1,

and ft' is 0 elsewhere . Note that (ft-)*((ql, q2), aa, (q2, q2))

= = =

tt * (ql, w(q2, a)w(gi, a), q2) n ft2(g2, aa, q2) fti (ql, ba, q2) A [t* (q2, aa, qz) ftl(gl,b,ql) Aftl(gl,a,q2) A[t * (gz,aa,qz) c2ncjnd2ndl .

It follows that ft' is separable.

Proposition 6 .6.13

Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2. Let M = M1wM2 for some w . Then b' (q1, q2), (PI, P2) E Q and b' y = yl . . . yn E X2* , ft w* ((ql, q2), y, (Pl,P2)) =V{fti(gl, w(g2,y1)w(q2 1) , y2) . . . W(g2 n-1) ,yn), PI) n-1) yn,P2) I q22) E Q2, nft2(g2, y1,g2 1) ) n ft2(g21) , y2,g2 2) ) A . . . A ft2(g2 ,n

Proof.

ft'* ((ql, q2), y, (Pl,P2)) = V{ft w ((gl, q2), yl, (gil),q21))) n q2 1) n-1) ), yn, (pl,P2)) wW ((gi 1) , g2 ), y2, (g i2) ~ g22))) n . . . A ft-((qin-1)' n-1) n-1 ( 1 ) , q(')), (q . . . . E yl), gi 1) )nft2(g2, yl, ))~ ~ V{ftl(gl, w(g2, (gi~ g2 1 g2 )) Q} = (1) (1) (2) (1) (2) (n-1) (n-1) . . .nftl(g y2~ g2 ) w(g2 q2(1)) nftl(gl , w(q2 , y2)~ ql )nft2(g2 l (n -1 ) ( 1 ) q(')), . . . , ( n-1 ) (~'-1)) E Q yn), pi) yn, P2) ft2(g2 (gl (gl g2

= V{ftl(gl, w(q2, yl), qi1) ) n ftl(gi 1) , w(g2 1) ,y2), q (2) ) n . . . n ftl(q(n-1) n-1) , 1 1 q(2) n-1 yn), PI) n ft2(g21 yl~ g2 )) n f~2(g2 )~ y2~ )~ w(g2 ) n . . . A l~2(g2 -1)' n-1) ) g21)), . . . yn, P2) (gi1)' (gin q2 E Q} =V{ftl(gl, w(g2,y1) w(g2 1) 1 ( n-1) y2) . . .co(q 2n-1) ,yn),Pl)nf~2(g2,yl,g21) )nft2(q21) ,y2,q2 2) )A . . . nft2(g2 yl , P2) I q22) E Q2, i = 1, 2, . . . , n - 1} .

Proposition 6 .6.14

Let MZ = (QZ, XZ, pi) be a ffsm, i = {0,1}, then ft' is separable .

1, 2. If Im(ft 2 )

_

Proof. Let

(g1, q2), (PI, P2) E Q and y = y1 . . .Y n E X2* . Now ft' * ((ql, . .. (q2n-1 ) , yn), PI) n ft2 W21 P2)) = V {fti (ql , w(q2, yl)w(g21) , y2) q2), y, (Pl, , (1) (1) g2(Z) E Q2> i = 1 > 2. . . . I nyl, g2 )nft2 (g2 , y2, g2(2) )n . . . Aft2 (g2(n-1) yl, P2)

1f by Proposition 6.6.13 . Case 1 : ft2(g2,y,P2) = 1Now v {ft2 ( q2, yl, q2(1)) A f~2 (q2('), y2 g2(2) ) n . . . n f~2 ((n g2 -1 ) = 1,2, . . . , n - 1} = ft2(g2, y,P2) = 1 . Hence 3 q2l ), © 2002 by Chapman & Hall/CRC

(Z)

yl,P2 )I q2 n-1) . . . , q2(

E E

6.6. Products of Fuzzy Finite State Machines

261

Q2 such that ft2(g2,y1,g21)) n ft2(g21)~Y2~g22)) n . . . n ft2(g2n-1)~Y1,P2) _ 1 . Thus tt' * ((glq2), y, (P1,P2)) = V {[t1 (qi, w(g2, yi)w(q (I) ' Y2) . . . c~(g2n-1) ~ n-1) ' Yn), PI) I ft2(g2, Y1, q21)) n ft2(g2 1) ~ Y2, q22)) A . . . A ft2(g2 Y1, P2) = 1~ q2(1), E Q2}Since ~ ~ ~ ~ q2 fti is finite valued, the supremum is at2n-1) E Q2 . q2n-1) tained at some q21), . . . , q Hence 3 q21), . . . , E Q2 such that tt'*((glq2),y,(P1,P2)) = fti(gl,w(q2, Yl) w(g21) ~ Y2) . . . w(g2n-1)' n-1 ) Yn), Pl) n 1 = fti(gl, Yn), PI) = tt * (ql, w(q2, yl) w(q(l), 2J2) . . .w(g2 w(g2 n-1 ), Yn),PI)n ft2(g2, Y,P2) . w(g2,y1)w(q21) ,Y2) . . . . .. n-1) Case 2 : N'2 (g2 , y, P2) = 0 . Then y g2l) E Q2, ft2 (q2, Y1, q21)) A , g2 ft2(g21) ~Y2 g22))n . . .n/'2(q2n-1) Y1 P2)=0 . Thusf~W* ((glg2) (Pl P2))_ 0 = fti(gl, w(q2, Yl)w(g21) , Y2) . . . W(g2n-1) Yn), Pl) n 0 = fti(gl, w(q2, yl) n-1) n-1) E Q2 w(g21) , Y2) . . . w(g2 Yn), Pl)A ft*(g2, Y, P2) V g21)~ . . . , g2

Let Ml = (Q1, Xl, ftl) and M2 = (Q2, X2, ft2) be fuzzy finite state machines . Let f be a function from Q2 into Xl . Define /t° : Q x (XQ2 x X2) xQ ----> [0,1] as follows : d ((ql, q2), (f, b), (PI,P2)) E Q x (XQ2 xX2) xQ,

w ° ((ql, q2), (f, b), (P1,P2)) = ftl(gl, f (g2),pl) n w2(g2, b,P2) .

Then M = (Q, XQ2 x X2, It°) is a ffsm . M = Ml o M2 is called the wreath product of Ml and M2 . ft o is called separable if b' (ql, q2), (pl, P2) E Q and

d (fl, bl) . . . (fn, bn) E (XQ2 XX2)*, w°*((ql, q2), (fl, bl) . . . (fn, bn), (P1,P2)) = tt * (ql, fl(g2)f2(q2 1) ) . . . fn (q2n-1)), pl) n ft2(g2, b l . . . bn , P2) for some g21)EQ2,i=1,2, . . .,n-1 .

Example 6 .6 .15 Let Ml and M2 be the ffsms defined in Example 6.6.12. Let XQ 2 = {fl, f2, f3, f4}~ where fl(gi) = fl(g2) = a, f2(gi) = a, f2(g2) _ b, f3(gi) = b, f3(g2) = a, and f4(gi) = f4(g2) = b . Then ft o ((gl, qi), (fl, a), (q2, q2))

= =

ftl (ql, fl (qi), q2) n ft2(gi, a, q2) cl n d l

ft O ((g2, qi), (fl, a), (q2, q2))

= =

ftl (q2, f1(gi), q2) n ft2(gi, a, q2) c3 A d1

ft o ((gl, q2), (fl, a), (q2, qi))

= =

ftl (ql, fl (q2), q2) A It2(g2, a, qi) cl n d2

ft O ((g2, q2), (fl, a), (q2, qi))

= =

ft o ((gl, qi), (f2, a), (q2, q2))

= =

[t o ((g2, qi), (f2, a), (q2, q2))

= =

ftl (q2, f1(g2), q2) A It2 (q2, a, qi) c 3 n d2 ftl (ql, f2(gi), q2) n ft2(gi, a, q2) cl n dl ftl (q2, f2(gi), q2) A It2(gi, a, q2) c3 A d 1

[t o ((gl, q2), (f2, a), (ql, qi))

= =

ftl (ql, f2(g2), ql) A It2(g2, a, qi) c2 n d2

ft o ((g2, q2), (f2, a), (ql, qi))

= =

ftl (q2, f2(g2), ql) n ft2(g2, a, qi) c4 n d2

© 2002 by Chapman & Hall/CRC

262

6. Algebraic Fuzzy Automata Theory w°((q1, q~), (f3, a), (q1, q2))

=

w°((q2, q~), (f3, a), (q1, q2))

=

w°((q1, q2), (f3, a), (q2, q~))

=

w°((q2, q2), (f3, a), (q2, q~))

=

It' ((g1, qi), (f4, a), (q1, q2)) ft'((g2, qi), (f4, a), (q1, q2)) It'((gl, q2), (f4, a), (q1, qi)) ft'((g2, q2), (f4, a), (q1, qi))

= = =

wl(gl, f3(gi), q1) c2 n dl wl(g2, f3(gi), q1) c4 n d l wl(gl, f3(g2), q2) c l A d2 wl(g2, f3(g2), q2) c3 n d2 ftl (q1, f4 (qi), q1) c2 Ad, ftl(g2, f4 (q'), q1) c4 Ad, ftl(gl, f4 (q2), q1) c2 n d2 ftl(g2, f4 (q2), q1) c 4 n d2 .

n ft2 (qi, a, q2) n ft2 (qi, a, q2) n ft2 (q2, a, qi) n ft2 (q2, a, qi) n ft2 (qi, a, q2) n ft2 (qi, a, q2) n ft2(g2, a, qi) n ft2(g2, a, qi)

In the case that cl = c2 = c3 = c4 and dl = d2 , it follows that the semigroup of this ffsm has eight elements and the transformation semigroup is isomorphic to the wreath product 2 0 7L 2 , [92, p . 59] . Example 6 .6 .16 Let Ml and M2 be the ffsms defined in Example 6.6.15. Let X2 1 = { f }, where f (q 1 ) = f (q2 ) = a . Then It'o ((q', [to ((g2, [t ((gi, fto ((g2, [to ((gi, [too ((g2, [to ((gi, [t ((g2,

q1), (f, a), (q2, q2)) q1), (f, a), (qi, q2)) q2), (f, a), (q2, q2)) q2), (f, a), (qi, q2)) q1), (f, b), (q2, q1)) q1), (f, b), (qi, q1)) q2), (f, b), (q2, q1)) g2), (f, b), (g1 q1))

ft2 (q', a, q2) ft2 (q2, a, qi) ft2 (q', a, q2) ft2 q2 I a, qi) ft2 (qi, a, q2) ft2 (q2, a, qi) ft2 (qi, a, q2) ft2 (q2, a, qi)

n ft, (g1, n ft, (g1, n ft, (g2 , n ft, (g2 , n ft, (g1, n ft, (g1, n ft, (q2, n ft, (q2,

a, q2) a, q2) a, q2) a, q2) b, q1) b, q1) b, q1) b, q1)

dl d2 dl d2 dl d2 dl d2

n c1 n c1 n c3 n c3 n c2 n c2 n c4 n c4 .

Suppose that d l = d2 and cl = c2 = c3 = c4 . Then it follows that S(M1 oM2 ) has four elements while S(M1) o S(M2) has eight elements, [92, p .59] . Proposition 6 .6 .17 V(ql, q2), (PI, P2) E Q and b' (fl, bl) . . . (fn, bn) E (XQ' x X2 ) *, w°* ((q1, q2), (fl, bl) . . . (fn, bn), (P1,P2)) = V{w*(ql, f1(g2), n-1) n-1) f2 (q2"') . . . fn (q2 , bn, P2) ) pl)A ft2(g2, bl, q2 1) ) n . . . n ft2(g2 (1) (n-1) E . . . , q2 , Q21g2 Proof. [t o* ((q, q2), (fl, bl) . . . (fn, bn), (Pl,P2)) = V{ttl(gl, fl(g2), g1 1) )n n-1) ft2(g2, b1, q21)) Aft, (gi1), f2 (q2(')), qi2) ) Ail (q21) , b2, g22) ) n . . . n ft1(gi , (n-1) (n-1) (1) (n-1) (1) (n-1) . . . . . pi) A " fn(g2 f'2 (g2 , bn, p2) g1 , , 1 E Qli g2 , , g2 ), 1 1 2 n-1 . . . n f, 1(g~ E Q2} _ {w1(g1, f1(g2), gi )) ), w1(g~ ), f2(g (') ), 1 ))

© 2002 by Chapman & Hall/CRC

6.6. Products of Fuzzy Finite State Machines

263

n-1) bn, fn( q (-1) ), pl) nft2(g2, bl, q2 1) ) nft2(g2 1) ~ b2, g22) ) A . . . AIt2(g2 P2) n-1) 2n-1) E Q2} =V{ftl*(ql, fl(g2)f2(g2 1) ) . . . qi 1) , . . . , qi E Q1 ; q2( , ), . . . , q n-1) bn, fn(g2 n-1) ), pi) n f~2(g2, bl, q21) ) n ft2(g2 1) , b2, q22)) n . . . n IA'2(g2 P2) Ig21), . . .,q2n-1)EQ2} . "

Proposition 6.6 .18

Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, {0,1}, then [to is separable .

Proof. Let (ql, q2), (P1,P2) E Q and (fl, Now [to * ((glq2), (fl, bl) . . . (fn, bn), (P1,P2))

2. If Im(p 2 )

_

bl) . . . (fn, bn) E (XQ2 x X2) * .

= V{fti(gl, fl(g2), f2(g2 1)) . . . . . . , q2 n-1) E fn (q2 n-1) ), pl) n ft2(g2, b1, q2 1) ) A . . . AIt2(g2n-1) , bn,P2) I q ('), 2 Q2} by Proposition 6.6.17. Case 1 : [t 2* (q2 , bl . . . bn,p2) = 1 . Now V{ft2(g2, bl, q21) ) n . . . n ft'2(g2n-1) , bn,P2) Ig2 1) , . . . ,q2n-1) E Q2} _ n-1) [t 2* (q 2 , bl . . . bn,P2) = 1 . Hence 3 q21) , . . . , q2 E Q2 such that ft2(g2, n-1) ' n . . . n 1 . Thus ( P2) = [t°*((glq2), bl, q2 1) ) ft2(g2 bn, (fl, bl) . . . (fn, bn), (PI, P2)) = V{ft*(ql, fl(g2) f2(g2 1 )) . . . fn (g2n-1 )), PI) w2 (g2, bl, n-1) n-1) A . . . A . . . . Since , E ft2 (g2 > bn, P2) = 1, g2 1) > Q2} fti is g21)) g2 ,q2n-1) EQ2 finite valued, the supremum is attained at some q2(,), . . . .

Hence 3

q2(,) , . . . , q2n-1)

such that

tto* ((ql, q2), (fl, bl) . . . (fn, (q2n-1)), pl) = bn), (PI, P2)) wl(gl, fl(g2), f2(g21 )) . . . fn [ti* (ql, fl(g2) . . fn (g2n -1 )), n 1 . . fn (q2n-1 )), pl)n . = . f2(g21) ) pl) wl(gl, fl(g2)f2(q2 1) ) . . . bn, P2) . w2(g2, bl Case 2 : [t2* (q2, bl . . . bn,P2) = 0. Then b' q21) , . . . , q2n-1) E Q2, ft2(g2, bl,

=

E Q2

q2 1) ) n . . . nf t 2(g2n-1) , bn, P2) = 0. Thus ,t 0* ((ql, q2), (fl, bl) . . . (fn, bn), (q2(1)) (PI, P2)) = 0 wl(gl, f, (q2) f2 . . . fn(g2 n-1) ), pl) A0 wl(gl, f, (q2) pi) n [t2 * (q2, f2(g2 bl . . . bn, P2) . ) . . fn(g2 ),

=

=

We are now interested in obtaining covering results involving the cascade and wreath product of fuzzy finite state machines . The method that is immediately suggested is to suitably relate elements of E(M) with those of E(Ml)Q2 x E(M2) . However it turns out that these relations are not necessarily single-valued. Hence we replace E(M) with X* in the following definition. Definition 6.6.19 Let M = (Q, X, p) and MZ = (QZ, XZ, p 2 ), i = 1, 2, be ffsms . Let 97 be a function of Q1 x Q2 onto Q and C a function of X* into E(M1)Q2 x E(M2 ) . Then ( 97, C) is said to be a weak covering of FTS(M) by FTS(M1),FTS(M2) if p(97(gl, q2), [X], 97(P1,P2))

b' x

V{p ° ((ql, q2), C(x), (rl, r2)) I97(rl, r2) (PI , P2) , (rl, r2) E Q1 x Q2}

E X* and (gl,q2), (PI, P2) E Q1 x Q2-

© 2002 by Chapman & Hall/CRC

264

6. Algebraic Fuzzy Automata Theory

Theorem 6 .6 .20 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 . If /t° is separable, then FTS(MI o M2) < (weakly) FTS(MI) o FTS(M2) . Proof. Now Ml o M2 = (Q 1 x Q2, XQ 2 x X2 , /t°), where [to ((gl, q2), (9, b), (PI, P2)) = ft, (g1,9(q2),PI)

n ft2(g2, b, P2)

and FTS(MI o M2) = (Q1 x Q2, E(MI o M2), po), where p ° ((ql, q2), [(91, bl) . . .(9k, bk) ], (PI, P2)) = [to* ((ql, q2), (91, bl) . . . (9k, bk), (PI, P2)) . Also, FTS(MI ) o FTS(M2 ) = (Q1 x Q2, E(Ml) Q2 x E(M2), ~°), where ~0 ((ql, q2), (f, [bl . . .bk]), (PI, P2)) = pl(gl, f (q2), pl) AP2(g2, [bl . . .bk], P2) = fti(gl, x1, pl) n [t* (q2, bl . . . bk, P2) where f (q2) = [xl] and xl E Xi is selected below . Let 97 be the identity map of Ql x Q2 . Define (XQ2 x X2)*

E(Ml)Q2 x E(M2)

as follows : C((91, bl) . . . (9k, bk)) = (f, [bl . . . bk])Now (91,b1) . . .(9k,bk) = (h i , a,) . . .(hj,aj) if and only if k = j and (gi, b2) = (h2, a2) i = 1, . . . , k if and only if k = j and gZ = h 2 , b2 = a2 , i = 1, . . . , k . Thus C is single-valued . po((g l , q2), [(gl, bl) . . . (9k, bk) ], (PI, P2)) = [t o * ((gl,q2), (91, bl) . . . (9k, bk), (PI, P2)) = V{fti(gl, k-1) 91(g2)92(q2 1) ) . . . 9k(g2k-1) ), pl) n ft2(g2,bl,q2 1) ) n . . . n ft2(g2 , bk,P2) cc = q2 1) , . . . ,q2 k-1) E Q2} ftl(gl, xl, pl) n ft2(g2, x2, P2) = pl(gl, f (q2), pl) n p2(g2, [x2], P2) _ ~0 ((ql, q2), (f, [x2]), (PI, P2)) = V{~O((ql, q2), C((91, bl) . . .(9k, bk)), (rl, r2))1 9 7(rl, r2) = (PI, P2), (rl, r2) E Ql x Q2} since 97 is the identity map . Select xl = gl(g2)92(g21) ) . . .gk(q2k-1) ) so that (1) (k-1) Hence (~7, C) is a weak covering of FTS(M1oM2) q2 , . . . , q2 ggives by FTS(MI) o FTS(M2 ) . Theorem 6 .6 .21 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 . If ft' is separable, then FTS(M1wM2) < (weakly) FTS(MI) o FTS(M2) .

© 2002 by Chapman & Hall/CRC

6.6. Products of Fuzzy Finite State Machines Proof. Now

and

M1wM2 = (Q 1 x Q2, X2 , ftw ),

265 where w :

Q 2 x X2 ----> Xl

ft' ((gl, q2), b, (PI, P2)) = ft, (gl,w(q2, b), PI) n ft2(g2, b, P2) d(ql, q2), (pi, P2) E Q1 x Q2, b E

X2 and

FTS(M1wM2) = (Q1 x Q2, E(M1wM2), p),

where p((ql, q2), [x2], (P1,P2)) = ft'*((ql, q2), x2, (P1,P2)), V(ql, q2), (PI, P2) E Q1 x Q2, V x2 E X2 . Also FTS(MI ) o FTS(M2 ) = (Q1 x Q 2 , E(Ml ) Q 2 x E(M2), ~0),

where P ° ((ql, q2), (f, [x2]), (P1,P2)) = pl(gl, f(g2),Pl) Ap2(g2, [x2],P2) d(ql, q2), (pl, P2) E Q1 x Q2, Vx2 E X2* . Let 97 be the identity map of Q1 x Q 2 . Since ft' is separable we can define C in the following manner .

Define

C : X2 - E(Ml ) Q 2 x E(M2 )

by C(x2) =

where fx2 (q2) = 2-1) , and w(gi bz)

(fx21 [x2])

b'x2 E X2,

[xl] for x2 = b l . . .b,, and xl = a l . . . a n and w(ql, bl) = al = a2, i = 2, . . . , n for those q(' - 1) so that

V{ft* (ql, w(q2, bl)w(g2 1) , b2) . . . w(g2n-1), bn ), pl) n ft2(g2, bl, g2 1) )n n- 1} ft2(gzl> > b2, g22)) n . . .n ft2(g2n-1>> bn , P2) 1 q21) E Q2, = fti(gl, al . . . an , PI) A ft2(g2,x2,P2) . Now C is single-valued since for y2 = d l . . . dj , x2 = y2 if and only if n = j and d2 = b2 for i = 1, . . . , n .

Then p((gl,g2), [x2], (P1,P2)) = ft-* ((ql, q2), x2, (PI, P2)) =V{fti(gl, w(q2, w( q2 1) , b2) . . . w(g2n-1) , bn), pl)n ft2(g2, bl, g2 1) )n ft2(g21) , b2, g22))A . . . A bl) n-1) ' bn,P2) lt2(g2 q21) E Q2, i = 1, 2, . . . , n - 1} _ /t1 (g1, a1 . . . an~~P1) n ft2(g2, x2, P2) = pl(gl, f (g2),pl) AP2(g2, [X21, P2) = ~ O ((ql, q2), (f, [x2]), (PI, P2)) = ~0((gl,q2), C(x2), (P1,P2))- 0 If A = (Q, S) and A' = (Q', S') are transformation semigroups, Section 6.5, then in the wreath product A o A' = (Q x Q', SQ' x S') one has that SQ' x S' is a semigroup where we recall that S and S' are finite semigroups . © 2002 by Chapman & Hall/CRC

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6. Algebraic Fuzzy Automata Theory

We now consider the fuzzy case. Here the difficulty that arises is that the natural way to define a binary operation on SQ' x S' fails since V{ft'(q', s', r') I r' E Q'} may not attain its maximum uniquely. Let (Q, S, /t) and (Q', S', /t') s' E S', V q' E Q', let .A4 (q', s') r') r' E Q'} . Let Q' = {qi . . . .

be fuzzy transformation semigroups. V = {m' E Q' Iw' (q', s', m') = V {ft' (q', s', . qk} . Define : (SQ' x S') x (SQ' x S') P(SQ' x S')\{O} as follows : V (f, s), (g, s') E SQ' x S', (f, s) ' (g , s') _ {(hs,m, ss') hs,m(gZ) _ .f (gZ)9(mZ) i = 1, . . . , k, m = (mi . . , mk) E .M(qi, s) x . . . x .M(qk, s)} Theorem 6.6.22 Suppose that V q' E Q', V s, s' E S', M(q , ss)

=

U.'EM(q',s)M(m , 8') '

Then SQ' x S' is polysemigroup.

Proof. Let (f, s), (g, s'), (k, s") E SQ' x S' . Then (f, s) ' (g, s~)

=

(hs,m, ss') I hs,m(gi) = .f (gi)g(mi), i = 1, . . . , k, m= I .m%) E .M(qi, s) x . . . x .Nt(gk, s)}

Thus s//) ((f, s) ' (9, s')) - (k,

sll) UmE.M(gi,s)x .. .x .M(qb,s)(hs,m, ss') ' (k, s// UmE .M(gi,s)x .. .x .M(q',s) L(s,mlss',r, lss' s,mlss',r(gi) = hs,m(gi)k(ri), r = (r /l . . . . . r/k.). E M(qi, ss /) x . x M(qk, ss')

and so s,mlss',r(gi) = hs,m(q')k(r') = .f (gj')9(m')k(r'), i = 1, . . . , k.

Now {(js , ,u, s's") Ljs, ,u(gj') = g(q')k(u'), i =1, . . . , k, u = (ul, . . . , u%) E A4 (q',, s~) x . . . x .M(qk, s')} .

Thus UuE .M(qi,s')x .. .xM(gm,s')(f,8)(js',u,ss ) UuE .M(gi,s')x ...x .M(q',s') L(s',uws,v, 8(8/ 8 ) s',u ws,v(ga) = f (ga)js,,u(vi), v = (vi/ . . . . . v/k) E .Nt(gi, s) x . . . x .Nt(gk, s)} (6.2)

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267

and so S',U w s,v(gi)

= f(q')js,,u(Vi') = f (gj')9(v')k(u(v')),

where u (v2)

E A4 (v', s~), i = 1, . . . , k . Now m2, v2 E A4 (q', s), r2 E A4 (q', ss~), u (v2) E A4 (v', s') for i = 1, . . . , k . Since .A4 (q, ss) = U . , E

.M(q',S)M(m', s),

i = 1, . . . , k,

by hypothesis, the sets in (6.1) and (6.2) are equal. Thus - is associative and so SQ' x S' is polysemigroup . m Definition 6.6.23

Let (Q, X, /t) be a ffsm. Define the fuzzy subset ~ of Q x (P(X)\{0}) x Q as follows: b' q, p E Q and b' Y E P(X)\{O}, w(q, Y p) = V{ft(q, y,p) l y E Y} .

Definition 6.6.24

Let (Q, X, /t) be a ffsm. By [to ((q, q'), {(h, ,, n , ss') h,, (gi) = f (gZ)h(mZ), i = 1, . . . , k, m = (ml, . . . , m'k) E .M(qi, s) x . . . . x .M(qk, s)}, (p, p')), we mean ft(q, {hs, .(q') I m E A4(q~, s) x . . . x A4(gk, s)}, p) A ft'(q,

ss,p) .

Let (Q, S, /t) and (Q', S', /t') be fuzzy transformation semigroups. Assume (SQ' x S', -) is a polysemigroup . Recall that we have the ffsm (Q x Q" SQ, x S" p) where V(q, q), (p, pl ) E Q x Q' and b' (f , s) E SQ' x S~, p((q, q), (f, s), (p,p))

= ft(q, f (q),p) A ft (q, s,p) .

The following result examines condition (1) of Definition 6.5 .1 . Theorem 6.6.25 V(q, q~),

(p, p') E Q x Q' and V (f, s), (g, s~) E SQ' x S~, V{w°((q, q'), (f, s), (r, r')) A p((r, r'), (y, s'), (p, p)) I (r, r') E Q x Q'} > [to ((q, q'), (f, s) - (9, s'), (p, p)) . o [t ((q, q'), (f, s)-(g, s'), (p, p)) =w°((q, q'), {(hs, ., ss') I h,, . (q') = f (gi)h(mi), i = 1, . . . , k, m = (mi, . . . , m%) E M(qi, s) x . . . x .M(gk, s)} (p,p')) = ft (q, {hs, .(g2) m E A4 (q, s) x . . . x A4(q%, s)}, p) n [t'(q', ss', p') (by Definition 6.6 .24) = ft(q, {f (q')9(m'), p) I rn' E A4 (q, s')}nft'(q', ss',p') < V{ft(q, f (q')9(r'), p) I r' E Q'} n fft'(q', ss', p') (by Definition 6.6 .23) = v{ft(q, f(q'), r) Aft(r, g (r'), p) I r E Q, r' E Q'}AV{ft'(q', s, r') Aft'(r', s, p) r' E Q'} = V{ft(q, f(q'), r) A ft'(q', s, r') A ft (r,g(r'),p) A ft' (r ' , s, 1') I r E Q, r' E Q'} = v{w°((q, q'), (f, s), (r, r')) n [to ((r, r'), (y, s'), (p, p)) I r E Q, r'EQT 0

Proof.

© 2002 by Chapman & Hall/CRC

26 8

6. Algebraic Fuzzy Automata Theory

6 .7

Submachines of a Fuzzy Finite State Machine

In this section, we continue our study of a fuzzy finite state machine utilizing algebraic techniques. Definition 6.7.1 Let M = (Q, X, /t) be a ffsm . Let p, q E Q. p is called an immediate successor of q if 3a E X such that ft(q, a, p) > 0. p is called a successor of q if 3x E X* such that /t* (q, x, p) > 0.

Proposition 6 .7.2 Let M = (Q, X, /t) be a ffsm. Let q, p, r E Q . Then the following assertions hold. (1) q is a successor of q . (2) if p is a successor of q and r is a successor of p, then r is a successor of q. Proof. (1) Since /t* (q, A, q) = 1 > 0, q is a successor of q. (2) Now 3x, y E X such that [t* (q, x, p) > 0 and [t* (p, y, r) > 0 . Thus [t * (q,xy,r) > [t * (q, x, p)

n [t* (p, y, r)

by Lemma 6.2 .2 . Hence r is a successor of q.

> 0,

m

Definition 6.7.3 Let M = (Q, X, /t) be a ffsm and let q E Q. We denote by S(q) the set of all successors of q.

Definition 6.7.4 Let M = (Q, X, /t) be a ffsm and let T C_ Q. The set of all successors of T, denoted by SQ(T) in Q, is defined to be the set SQ(T) = U{S(q)Iq E T} .

If no confusion arises, then we write S(T) for SQ(T) . Theorem 6.7.5 Let M = (Q, X, y) be a ffsm . Let A, B C_ Q. Then the following assertions hold. (1) If A C B, then S(A) C S(B) . (2) A C S (A) . (3) S(S(A)) = S (A) . (4) S(A U B) = S(A) U S(B) . (5) S(A n B) C S(A) n S(B) .

Proof. The proofs of (1), (2), (4), and (5) are straightforward . (3) Clearly S(A) C S(S(A)) . Let q E S(S(A)). Then q E S(p) for some p E S(A) . Thus p E S(r) for some r E A . Now q is a successor of p and p is a successor of r. Hence by Proposition 6.7 .2 , q is a successor of r. Thus q E S(r) C S(A) . Hence S(S(A)) = S(A) . .

© 2002 by Chapman & Hall/CRC

6.7. Submachines of a Fuzzy Finite State Machine

269

Exchange Property : Let M = (Q, X, p) be a ffsm. Let q, p E Q and let T C Q . Suppose that if p E SITU {q}), p ~ S(T), then q E SITU {p}) . Then we say that M satisfies the Exchange Property. Proposition 6.7.6 Let M = (Q, X, /t) be a ffsm . Then the following assertions are equivalent . (1) M satisfies the Exchange Property. (2) b'p, q E Q, q E S(p) if and only if p E S(q) .

Proof. (1)x(2) : Let p,q E Q and p E S(q) . Now p ~ S(O) . Hence q E S(p) . Similarly if q E S(p) then p E S(q) . (2)x(1) : Let T C Q, p, q E Q . Suppose p E S(T U {q}), p ~ S(T) . Then p E S(q) . Hence q E S(p) C S(T U {p}) . m Definition 6.7.7 Let M = (Q, X, /t) be a ffsm . Let T C_ Q. Let v be a fuzzy subset of T x X x T and let N = (T, X, v) . The fuzzy finite state machine N is called a submachine of M if (1) pjTXXXT = v and (2) SQ (T) C T. We assume that 0 = (0, X, v) is a submachine of M. Clearly, if K is a submachine of N and N is a submachine of M, then K is a submachine of M.

Theorem 6 .7.8 Let M = (Q, X, p) be a ffsm . Let Mi = (Qi, X, pi), i E

I, be a family of Submachines of M, where Qa C_ Q . Then the following assertions hold . (1) niEIMi = (niEIQi, X, njEllaa) is a submachine of M. (2) UiEIMi = (UiEIQi, X, v) is a submachine of M, where v = [tIUiCIQiXXXUiCIQi-

Proof. (1) Let (q, x, p) E niEIQi x X x niEIQi . Then (niElp'i)(q, x, p) = njElp'i(q, x, p) = njEll7 (q, x, p) = p(q, x, p) . Also S(ni(E IQi) c niEls(Qi) c niEIQi

Thus niEIMi is a submachine of M. (2) Since S(UiEIQi) = UiEIS(Qi) = UiEIQi' UiEIMi is a submachine of M. m

Definition 6.7.9 Let M = (Q, X, p) be a ffsm. Then M is called strongly connected if b'p, q E Q, p E S(q) .

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6. Algebraic Fuzzy Automata Theory

Definition 6.7.10 Let M = (Q, X, p) be a ffsm and let N = (T, X, v) be a submachine of M. N is called proper if T :?~ Q and T :?~ 01 . Theorem 6.7.11 Let M = (Q, X, /t) be a ffsm. Then M is strongly connected if and only if M has no proper submachines. Proof. Suppose M is strongly connected . Let N = (T, X, v) be a submachine of such that T :?~ 01. Then 3q E T. Let p E Q. Since M is strongly connected p E S(q) . Hence p E S(q) C_ S(T) C_ T. Thus T = Q and so M = N. Conversely, suppose M has no proper submachine. Let p,q E Q and let N = (S(q),X,v) where v = wjs(q )XXXS(q) . Then N is a submachine of M and S(q) :?~ 01 . Hence S(q) = Q. Thus p E S(q) . Hence M is strongly connected . m Proposition 6 .7.12 Let M = (Q, X, /t) be a ffsm. Let R C_ Q. Then N = (S(R), X, NR) is a submachine of M where NR = / t1 s(R)XXXs(R)- 0 Definition 6.7.13 Let M = (Q, X, y) be a ffsm. Let R C_ Q and {NZ i E A} be the collection of all submachines of M whose state set contains R. Define < R > = njEA{NZ I i E A} . Then < R > is called the submachine generated by R. In Definition 6.7.13, it is clear that < R > is the smallest submachine of M whose state set contains R. Proposition 6 .7.14 Let M = (Q, X, /t) be a ffsm. Let R C Q. Then

= (S(R), X, N'R) .

Proof. Now < R > = (niEAQZ, X, niEAfti), where {NZ I i E A} is the collection of all submachines of M whose state set contains R and Ni = (QZ, X, fti), i E A . It suffices to show that S(R) = niEAQZ . Since (S(R), X, NR) is a submachine of M such that R C_ S(R), we have that S(R) _D niEAQZ . Let p E S(R) . Then 3 r E R and x E X* such that /t* (r, x, p) > 0. Now r E niEAQZ, and since < R > is a submachine of M, p E niEAQZ . Thus S(R) C niEAQZ . Hence S(R) = niEAQi . Definition 6.7.15 Let M = (Q, X, ft) be a ffsm. M is called singly generated if 3q E Q such that M =< {q} > . In this case q is called a generator of M and we say that M is generated by q. Theorem 6.7.16 Let M = (Q, X, y) be a ffsm. Let R, T C_ Q. Then the following assertions hold. (1)=U . (2) C n . © 2002 by Chapman & Hall/CRC

6.7. Submachines of a Fuzzy Finite State Machine

271

Proof. (1) By Theorem 6.7.5, S(R U T) = S(R) U S(T) . Now l's(R)US(T)

= wI (S(R)US(T)) .X .(S(R)US(T)) NI S(RUT)xX xS(RUT) NS(RUT)

Hence =U . (2) By Theorem 6.7.5, S(R rl T) C S(R) rl S(T) . Now and Hence

ltS(RnT) =

lt S(R)nS(T) =

wIS(RnT)xXxS(RnT)

wI(S(R)nS(T))xXx(S(R)nS(T))-

NS(RnT) = NS(R)nS(T)IS(RnT)xXxS(RnT)-

HenceCrl .m Definition 6.7.17 Let M = (Q, X, p) be a ffsm. Let MZ = (QZ, X, pi), i = 1, 2, be Submachines of M. If M = < Q1 U Q2 >, then we say that M

is the union of Ml and M2 and we write M = Ml U M2 . If M = Ml U M2 and Q1 n Q2 = 01, then we say that M is the (internal) direct union of Ml and M2 and we write M = Ml U M2 .

Suppose M = Ml U M2 . Then S(QZ) = QZ in M, i = 1, 2, since MZ is a submachine of M, i = 1, 2 . Now S(Q1 U Q2) = S(Q1) U S(Q2) = Q1 U Q2 . Definition 6.7.18 Let M = (Q, X, /t) be a ffsm. Let T C_ Q. T is called free if b't E T, t ~ S(T\{t}) . Definition 6.7.19 Let M = (Q, X, y) be a ffsm. Let T C_ Q. If T and M = < T >, then T is called a basis of M.

Theorem 6.7.20 Let M = (Q, X, y) be a ffsm . Let T C_ Q . Then the

following (1) T (2) T (2) T

assertions are equivalent. is a minimal system of generators of M. is a maximally free subset of Q. is a basis of M.

Theorem 6.7.21 Let M = (Q, X, /t) be a ffsm. Suppose that M satisfies the exchange property. Then M has a basis and the cardinality of a basis is unique. 0

Theorem 6.7.22 Let M = (Q, X, /t) be a ffsm. Suppose that M satisfies the exchange property . Let {ql, q2, . . . , qn} be a basis of M. Then M = 00 . . .0 .

Proof. We have < qZ > = (S(g2), X, y, 2), where pi = ltI S(qi)XX XS(qi) Now if i :?~ j, then S(g2)nS(q;) = Ql since the exchange property is equivalent to the statement that b'p, q E Q, p E S(q) if and only if q E S(p) . Since M = ,itfollows that M=0U . . .0 .~ © 2002 by Chapman & Hall/CRC

272

6 .8

6.

Algebraic Fuzzy Automata Theory

Retrievability, Separability, and Connectivity

Definition 6.8 .1 Let M = (Q, X, /t) be a ffsm. M is said to be retrievable if b'q E Q, b'y E X* if 3t E Q such that p* (q, y, t) > 0, then 3x E X* such that ft * (t, x, q) > 0. Definition 6.8 .2 Let M = (Q, X, /t) be a ffsm. M is said to be quasiretrievable if b'q E Q, b'y E X* if 3t E Q such that [t* (q, y, t) > 0, then 3x E X* such that y* (q, yx, q) > 0. Definition 6.8 .3 Let M = (Q, X, /t) be a ffsm. Let q, r, s E Q . Then r and s are said to be q -related if 3y E X* such that lt*(q, y, r) > 0 and /t* (q, y, s) > 0. If r and s are q-related, then r and s are said to be q-twins if S(s) = S(r) . Remark 6 .8 .4 Let M = (Q, X, /t) be a ffsm. The following assertions are equivalent. (1) b'q, r, p E Q, b'x, y E X* if /t* (q, y, r) > 0 and /t* (q, yx, p) > 0, then p E S(r) . (2) b'q, r, s E Q, if r and s are q-related, then r and s are q-twins. Proof. (1)x(2) : Let q, r, s E Q be such that r and s are q-related. Then 3y E X* such that y* (q, y, r) > 0 and y* (q, y, s) > 0. Let p E S(s) . Then 3x E X* such that y* (s, y, p) > 0. Then y* (q, yx, p) > 0 . Thus by the hypothesis p E S(r) . Similarly if p E S(r) then p E S (s) . (2)x(1): Let q, r, p E Q, x, y E X* be such that y,* (q, y, r) > 0 and [t * (q, yx, p) > 0. Now /t* (q, yx, p) = V{[t* (q, y, s) n [t* (s, x, p) I s E Q} > 0 . Hence 3s E Q such that y* (q, y, s) > 0 and y* (s, x, p) > 0. Then r and s are q-twins and p E S(s) . Thus by the hypothesis p E S(r) . Proposition 6 .8 .5 Let M = (Q, X, /t) be a ffsm. Then the following assertions are equivalent : (1) M is retrievable . (2) M is quasi-retrievable and b' q, r, s E Q, if r and s are q-related, then r and s are q-twins. Proof. (1)x(2) : It is immediate that retrievability implies quasiretrieva-bility. Let q, r, p E Q and x, y E X* . Suppose that /t* (q, y, r) > 0 and ft* (q, yx, p) > 0. Since M is retrievable, 3z E X* such that ft * (p, z, q) > 0. Thus p E S(r) . Hence (2) holds by Remark 6.8 .4. (2)x(1): Let q E Q and y E X* . Suppose 3t E Q such that /t*(q,y,t) > 0. Then 3x E X* such that ft * (q, yx, q) > 0 since M is quasi-retrievable. By Remark 6.8 .4, q E S(t) . 0

© 2002 by Chapman & Hall/CRC

6.8. Retrievability, Separability, and Connectivity

273

Theorem 6.8.6 Let M = (Q, X, /t) be a ffsm. The following assertions are equivalent. (1) M satisfies the Exchange Property, (2) M is the union of strongly connected submachines, (3) M is retrievable. Proof. (1)x(2) : By (1), M = UZ1 < qZ >, where {ql, q2, . . . , qn} is a basis of M. Also S(qj) n S(q;) = 01 if i :?~ j. Let p, q E S(qj) . Then qZ E S (p) and so q E S(p) . Thus < qZ > is strongly connected. (2) x(1) : Now M = UZ1 M2, where each MZ = (QZ, X, pi) is strongly connected. Let p, q E Q . Suppose p E S(q) . Now 3i such that q E QZ . Then p E S(q) C S(QZ) = Qz. Thus p, q E QZ . Since MZ is strongly connected, q E S(p) . Hence M satisfies the Exchange Property by Proposition 6.7 .6 . (2) x(3) : Now M = UZ 1 M2, where each MZ = (QZ, X, pi) is strongly connected. Let q E Q, y E X* be such that y* (q, y, t) > 0 for some t E Q. Now q E QZ for some i. Thus t E S(q) C_ S(QZ) . Since MZ is strongly connected, q E S(t) . Hence 3x E X* such that tt*(t,x,q) > 0. Thus M is retrievable . (3)x(2) : Let q E Q and let r, t E S(q) . Then 3y, z E X* such that * (q, y, r) > 0 and * (q, z, t) > 0. Since M is retrievable 3x E X* such that ft ft tt*(r, x, q) > 0. Hence q E S(r) . Thus t E S(r) . Hence < q > is strongly connected. Now M = UeEQ < q > . 0 Definition 6.8.7 Let M = (Q, X, y) be a ffsm. Let N = (T, X, v) :?~ Q be a submachine of M. Then N is said to be separated if S(Q\T) n T = 0. Theorem 6 .8.8 Let M = (Q, X, y) be a ffsm. Let N = (T, X, v) :?~ Ql be a submachine of M. Then N is separated if and only if S(Q\T) = Q\T. Proof. Suppose N is separated . Let q E S(Q\T). Now S(Q\T) nT = 0. Hence q ~ T. Thus q E Q\T . Hence S(Q\T) C Q\T. Thus S(Q\T) = Q\T. Conversely, suppose that S(Q\T) = Q\T . Clearly then S(Q\T) n T = 0. Thus N is separated. m Theorem 6.8.9 Let M = (Q, X, y) be a ffsm. Let N = (T, X, v) :?~ Ql be a submachine of M. If N is separated, then so is C = (Q\T, X, a) where a = ft I (Q\T)XxX(Q\T) Proof. Now 0 :?~ T :?~ Q and so Q\T :?~ 0 . By Theorem 6.8 .8, S(Q\T) = Q\T . Hence C is a submachine of M. Now S(Q\(Q\T)) = S(T) = T . Thus S(Q\(Q\T)) n (Q\T) = T n (Q\T) = 01. Hence C is separated . m Definition 6.8.10 Let M = (Q, X, /t) be a ffsm. Then M is said to be connected if M has no separated proper submachine .

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6. Algebraic Fuzzy Automata Theory

Theorem 6.8.11 Let M = (Q, X,

/t) be a ffsm. Then the following assertions are equivalent . (1) M is strongly connected. (2) M is connected and retrievable. (3) Every submachine of M is strongly connected.

Proof. (1)x(2) : By Theorem 6.7 .11, M does not have any proper submachines and so M has no proper separated submachines . Thus M is connected . We now show that M is retrievable . Let q, t E Q and y E X* be such that ft* (q, y, t) > 0. Since M is strongly connected, q E S(t) . Then 3x E X* such that f~* (t, x, q) > 0. Hence M is retrievable. (2) x(3): Let N = (T, X, v) be a submachine of M. Suppose p, q E T are such that p ~ S(q) . Then S(q) :?~ Q and so K= (S(q),X,ltIS(a)XXXS(q)) is a proper submachine of M. Since M is connected, S(Q\S(q)) n S(q) :?4 01 . Let r E S(Q\S(q)) nS(q) . Then r E S(t) for some t E Q\S(q) and r E S(q) . Now 3 y E X* such that W (t, y, r) > 0. Since M is retrievable, 3z E X* such that /t* (r, z, t) > 0. Thus t E S(r) . Hence t E S(r) C_ S(q), a contradiction. Thus p E S(q) b'p, q E T. Hence N is strongly connected . (3)x(1) : Obvious. m

6 .9

Decomposition of Fuzzy Finite State Machines

Definition 6.9.1 Let M = (Q, X, /t) be a ffsm. Let P be a submachine of M. Then P is called a primary submachine of M if (1) 3q E Q such that P = ; (2) VsEQifPC<s>,then P=<s> .

Theorem 6.9.2 (Decomposition Theorem) Let M = (Q, X, /t) be a ffsm . Let p = {P1, P2 , . . . , P,,} be the set of all distinct primary submachines of M. Then (1) M = Uz 1 Pi ; (2) M :?~ U' l, Z0jPZ for any j E {1, 2, . . . , n} . Proof. (1) Let qo E Q. Now b'q2 E Q, either (a) < qZ >E p or (b) 3qZ+1 E Q\S(g2) such that < qZ > C < qZ+1 >. Since Q is finite, either < qo > E p or there exists a positive integer k such that < qo > C < qk > E p. Thus Q = UZ 1S(pi) where Pi = < pi >, i = 1, 2 . . . . , n . Hence M = U2 1PZ . (2) Let N = UZ 1, Zo,pi and let pi _ < pj > . If pj E UZ 1, Z0jS(p2), then pj E S(pi) for some i :?~ j. Hence Pj _ < pj > C Pi . However this contradicts the maximality of Pj since Pj Pi . Thus pj ~ ~ UZ 1, Z0jS(p2) . Hence M :?~ N. Corollary 6.9.3 Let M = (Q, X, /t) be a ffsm . Then every singly generated submachine of M :?~ 0 is a submachine of a primary submachine of M.

© 2002 by Chapman & Hall/CRC

6.9 . Decomposition of Fuzzy Finite State Machines

Corollary 6.9 .4 Let M = Theorem 6.9 .2 are unique .

m

(Q, X, p)

be a

275 Then P1 , P2 , . . . , P. in

Definition 6.9.5 Let M =

(Q, X, /t) be a ffsm. Then rank of M, rank(M), is the number of distinct primary submachines of M.

Theorem 6.9.6 Let M =

(Q, X, /t) be a ffsm . The following assertions are equivalent. (1) M is retrievable . (2) Every primary submachine of M is strongly connected.

Proof. (1)x(2) : Let P be a primary submachine of M. Then P = for some p E Q. Then as in the proof of (3)x(2) of Theorem 6.8 .6, is strongly connected. (2)x(1) : Now M = UZ 1 P2 where Pi are primary submachines of M. Then the Pi are strongly connected . Thus M is the union of strongly connected submachines . By Theorem 6.8 .6, (1) holds . m Lemma 6.9.7 Let M = nected submachine .

(Q,

X, y) be a ffsm. Then M has a strongly con-

Proof. We prove the result by induction on ~Q1 = n. If n = 1, then the result is obvious . Suppose the result is true for all ffsms N = (T, X, v) such that ~Tj < n, n > 1 . Let q E Q. Then M' = (S(q),X,yjS(q)XXXS(q)) is a submachine of M. If M' is strongly connected, then the result follows. Suppose that M' is not strongly connected . Then 3p E S(q) such that q ~ S(p) and hence S(p) C S(q) . Now ~S(p)j < n. Hence by the induction hypothesis the ffsm M" = (S(p),X,ltjS(p)XXXS(p)) has a strongly connected submachine. Since M" is a submachine of M, M has a strongly connected submachine. Theorem 6.9.8 Let M =

(Q, X, /t) be a ffsm . The following assertions are equivalent. (1) M is retrievable . (2) Every singly generated submachine of M is primary. (3) Every nonempty connected submachine of M is primary.

Proof. (1)x(2) : Now M = UZ 1 P2 where the Pi are primary submachines of M. By Theorem 6.7.11, the Pi are strongly connected . Let N = < q > be a singly generated submachine of M. Then < q > C_ Pi for some i . Hence < q > = Pi by Theorem 6.7 .11. Thus N is primary. (2)x(1) : Since every singly generated submachine of M is primary, every singly generated submachine of M is strongly connected . Thus every primary submachine of M is strongly connected. By Theorem 6.9 .6, (1) holds. (2) x(3) : Let N = (T, X, v) be a nonempty connected submachine of M. Let q E T. Suppose S(q) :?~ T. Since N is connected, S(T\S(q))nS(q) :?~ © 2002 by Chapman & Hall/CRC

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0. Let r E S(T\S(q)) n S(q) . Then r E S(t) for some t E T\S(q) and rES(q) .NowC_andC .Since isprimary, < t > = < r > = < q > . Hence t E S(q), which is a contradiction. Hence N = < q > and so N is primary. (3)x(2) : Let N = < s > be a singly generated submachine. By Lemma 6.9.7, N has a strongly connected submachine B = < r >, say. Then B is connected and hence primary. Thus < r > _ < s > = N. Hence N is primary. m Lemma 6.9.9 Let M = (Q, X, p) be a ffsm and let N = (T, X, v) be a separated submachine of M. Then every primary submachine of N is also a primary submachine of M. Proof. Let < q > be a primary submachine of N. Suppose < q > is not a primary submachine of M. Then 3p E Q\S(q) such that < q > C . Clearly p ~ T. Thus p E Q\T . Since q E S(p), q E S(Q\T). Thus q E S(Q\T) n T, which is a contradiction since N is separated . Hence < q > is a primary submachine of M. m Theorem 6.9.10 Let M = (Q, X, y) be a ffsm and let Ni = (Ti , X, vi ), i = 1, 2, . . . , n, be the primary submachines of M. Then a proper submachine N = (T, X, v) of M is separated if and only if for some J C {1, 2, . . . , n}, J zA 0, Q\T = UiEJT,,. Proof. Suppose N = (T, X, v) be a proper separated submachine of M . Then S(Q\T) = Q\T. Since N is proper, the submachine < Q\T > is nonempty. Thus < Q\T > is the union of all its primary submachines . Since < Q\T > is separated every primary submachine of < Q\T > is a primary submachine of M . Thus S(Q\T) = UiEJTi for some J C_ {1, 2, . . . , n}, J :?~ 01 . Since Q\T = S(Q\T), Q\T = UjEJTj for some J C_ {1,2, . . . , n}, J :?~ 01. Conversely, let N = (T, X, v) be a proper submachine of M such that Q\T = UiEJTi for some J C {1,2, . . . , n}, J :?~ 01. Then S(Q\T) _ S(UiEJTi) = UiEJS(Ti) = UiEJTi = Q\T. Hence N is separated. m Corollary 6.9.11 Let M = (Q, X, ft) be a ffsm . Then M is connected if and only if M has no proper submachine N = (T, X, v) such that Q\T is the union of the sets of states of all primary submachines of M. m Definition 6.9.12 Let M = (Q, X, y) be a ffsm and let N = (T, X, v) be a submachine of M. A subset R C_ Q is called a generating set of N, and is said to generate N if N = < R > . Lemma 6.9.13 Let M = (Q, X, y) be a ffsm and let Ni = (Ti, X, vi), i = 1, 2, . . ., n be the primary submachines of M. Let R C Q . Then R generates M if and only if b'i, l . © 2002 by Chapman & Hall/CRC

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Proof. Suppose that R generates M. Then M = < R > = U TE R < r > . Let qZ E TZ be such that TZ = < qZ >, i = 1, 2, . . . , n . Then qZ E U,ER < r > and so qZ E for some r E R. Thus < qZ > C_ . Since < qZ > is primary, < qZ > = < r > . The converse is immediate. m Definition 6.9.14 Let M = (Q, X, /t) be a ffsm. Let R C_ Q be a generating set of M. Then R is said to be a minimal generating set of M if (1)M= , and (2) Vr E R, < R\{r} > 7~ M. Theorem 6.9.15 Let M = (Q, X, p) be a ffsm. Let R C_ Q be a generating set of M. Then R is a minimal generating set of M if and only if ~R1 =rank(M) . Proof. Let n = rank(M) and let Ni = (Ti, X, v2) be a primary submachine of M, i = 1, 2, . . . , n . By Lemma 6.9 .13, since R is a generating set of M, 3r2 E R such that < r2 > = Ti, i = 1, 2, . . . , n. Since the TZ are distinct, the r2 are distinct. Thus ~R1 >_ rank(M) . Now assume that R is a minimal generating set . Suppose ~R1 > rank(M) . Then 3r E R such that r ~ {rl, r2, . . . , rn}. Thus < R\{r} > = Q. Hence R is not minimal, a contradiction . Thus ~R1 = rank(M) . Conversely suppose that ~R1 = rank(M) . Then R = {rl, r2, . . . , rn} . Hence < R\{r2} > = UjOZNj z,4 M. Thus R is minimal. Theorem 6.9.16 Let M = (Q, X, /t) be a ffsm. Then M is not connected if and only if 3 a generating set R of M with nonempty subsets Rl and R2 such that n=01 and M=U . Proof. Suppose that M is not connected . Then 3 a proper submachine N = (T, X, v) of M that is separated ., i.e., S(Q\T) n T = 0. Let Rl be a generating set of < S(Q\T) > and let R2 be a generating set of N. Then < Rl > n < R2 > = 01 and M = < Rl > U < R2 > . Conversely, suppose that Rl and R2 exists . Let N = < R2 > . Then < S(Q\T) > = < Rl > . Hence N = (T, X, /tjTXXXT) is a proper submachine of M that is separated.

6 .10

Subsystems of Fuzzy Finite State Machines

In this and the next section, we introduce the notion of subsystems and strong subsystems of a ffsm in order to consider state membership as fuzzy . Definition 6.10 .1 Let M = (Q, X, /t) be a ffsm. Let S be a fuzzy subset of Q. Then (Q, S, X, /t) is called a subsystem of M if b'p, q E Q, b'a E X, b(q)

© 2002 by Chapman & Hall/CRC

>

b(p)

n ft(p, a, q) .

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6. Algebraic Fuzzy Automata Theory

If (Q, S, X, ft) is a subsystem of M, then we simply write S for (Q, S, X, ft) . Theorem 6.10 .2 Let M = (Q, X, p) be a ffsm and let S be a fuzzy subset of Q . Then S is a subsystem of M if and only if Vp, q E Q, Vx E X*, 6(q) > 6(p) n ft * (p, x, q) .

Proof. Suppose S is a subsystem . Let q, p E Q, and x E X* . We prove the result by induction on IxI = n. If n = 0, then x = A. Now if p = q, then 6(q) A [t* (q, A, q) = 6(q) . If q p, then 6(p) A [t* (p, A, q) = 0 < 6(q) . Thus the result is true if n = 0. Suppose the result is true Vy E X* such that lyl=n -1 ,n>O . Letx = ya,Iyl = n -1 ,y EX*, aEX. Then b(p) A w* (p, x, q)

= = = < <

6(p) A w* (p, ya, q) b(p) A (V{ft* (p, y, r) A ft(r, a, q) I r E Q}) V{6(p) nw*(p,y,r) nft(r,a,q)Ir E Q} V{6(r) Aft(r,a,q)Ir E Q} 6(q) .

Hence S(q) > S(p) A ft * (p, x, q) . The converse is trivial. m Theorem 6.10 .3 Let M = (Q, X, y) be a ffsm. Let S, 61, and 62 be subsystems of M. Then the following assertions hold. (1) 6, n62 is a subsystem of M. (2) 61 U 62 is a subsystem of M. (3) N = (Supp(S), X, v) is a submachine of M, where v = ft I SUPP(S) xX x SUPP(S) (4) Let Nt = (St, X, v( t)) where v(t) = ftl6, xxxst , t E [0,1] . If Vt E [0,1], Nt is a submachine of M, then S is a subsystem of M.

Proof. The proofs of (1) and (2) are straightforward . (3) Let p E S(Supp(S)) . Then p E S(q) for some q E Supp(S) . Then 6(q) > 0 . Since p E S(q), 3x E X* such that /t* (q, x, p) > 0. Hence since S is a subsystem, S(p) >_ S(q) A /t* (q, x, p) > 0. Thus p E Supp(S) . Hence S(Supp(S)) C Supp(S) . Thus N is a submachine of M. (4) Let q, p E Q, x E X* . If S(p) = 0 or y,* (p, x, q) = 0, then S(q) > 0 = S(p) A tt* (p, x, q) . Suppose S(p) > 0 and y,* (p, x, q) > 0. Let S(p) A /t* (p, x, q) = t. Then p E St . Since Nt is a submachine of M, S(St) = St . Hence q E S(p) C S(St) = St . Hence S(q) > t = S(p) Aft* (p, x, q) . Thus S is a subsystem . m Example 6.10 .4 Let Q = {p, q}, X = {a}, ft(r, a, t) = z Vr, t E Q . Let S(q) = 4 and S(p) = 1 . Then

b(q) n ft(q, a, p) = 2 = b(p) © 2002 by Chapman & Hall/CRC

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and b(p) n ft(p, a, q) = 2 < 4 = 6(q) . Thus S is a subsystem. Let z < t <_ 4 . Let Nt = (St, X, v(t)) where v(t) _ [tJstxxxst . Now S(q) >_ t. Thus q E St . Also ft(q,a,p) = 2 > 0. Hence p E S(q) . Thus p E S(St) . But S(p) = 2 < t. Hence p ~ St . Thus Nt is not a submachine of M.

Definition 6.10 .5 Let M = (Q, X, ft) be a ffsm and let S be a fuzzy subset of Q. For all x E X*, define the fuzzy subset Sx of Q by (6x)(q) = V{b(p) Att* (p,x,q)Ip E Q} Vq E Q .

Proposition 6.10 .6 Let M = (Q, X, ft) be a ffsm. Then V fuzzy subsets S of Q and Vx, y E X*,

(sx)y = 6(xy) . Proof. Let S be a fuzzy subset of Q and let x, y E X* . We prove the result by induction on Iy1 = n. If n = 0, then y = A . Let q E Q. Now ((bx)A)(q)

=

V{(bx)(P) nft* (p,A,q) I p E Q} (sx) (q) .

Hence (Sx)A = Sx = S(xA) . Suppose now the result is true for all u E X* such that Jul = n-1, n > 0 and for all S . Let y = ua, where a E X, u E X*, and Jul = n - 1. Let q E Q. Then (b(xy))(q)

= = = = = = =

(6(xua))(q) (b((xu)a))(q) V{(S(xu))(r) A/t*(r,a,q)Ir E Q} V{(V{(&)(p) Aw * (p,u,r)Ip E Q}) Aw*(r,a,q)Ir E Q} V{(&)(p) A (V{ft*(p,u, r) Aw*(r,a,q)Ir E Q})Ip E Q} V{ (bx) (p) n ft * (p, ua, q) I p E Q} ((bx)y)(q) .

Hence S(xy) = (6x)y. The result now follows by induction . 0 Theorem 6.10 .7 Let M = (Q, X, y) be a ffsm and let S be a fuzzy subset of Q. Then S is a subsystem of M if and only if Sx C S Vx E X* .

Proof. Let S be a subsystem of M. Let x E X* and q E Q . Then (6x) (q) = V{6(p) A ft * (p, x, q) Ip E Q} < 6(q) . Hence Sx C S. Conversely, suppose Sx C S Vx E X* . Let q E Q and x E X* . Now 6(q) > (6x) (q) = V {6 (P) n ft * (p, x, q) I P E Q} > 6(P) n ft * (p, x, q) Vp E Q . Hence 6 is a subsystem of M.

© 2002 by Chapman & Hall/CRC

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Algebraic Fuzzy Automata Theory

Definition 6.10 .8 Let M = (Q, X, p) be a ffsm. Let t E (0,1] and q E Q . Define the fuzzy subset qtX of Q by

(qtX) (p) = V{t h ft (q, a, p) l a E X VP E Q.

Definition 6.10 .9 Let M = (Q, X, y) be a ffsm. Let t E (0,1] and q E Q . Define the fuzzy subset qtX* of Q by

(qtX*) (p) =V{t n ft* (q, y, p) ly E X*} VP E Q.

Theorem 6.10 .10 Let M = (Q, X, y) be a ffsm. Let t E (0,1], q E Q . Then the following assertions hold. (1) qtX* is a subsystem of M. (2) Supp(gtX * ) = S(q) .

Proof. (1) Let r, s E Q and x E X* . Now ((qtX*)x)(r)

= = = = < =

V{(qtX * ) (P) nft* (p,x,r)lP E Q} V{(V{ft*(q, y, P) Atly E X*}) Aw* (p,x,r)IP E Q} V{ft*(q,y,P) A[t*(p,x,r) Atly E X* , p E Q} V{ft* (q,yx,r) Atly E X* } {ft* (q,u,r) lulu E X*} (qtX*)(r) .

Hence (qtX*)x C qtX* . Thus qtX* is a subsystem by Theorem 6.10.7. (2) p E S(q) 3x E X* such that ,t* (q, x, p) > 0 V{t n tt * (q,x,p)lx E X*} > 0

(qtX*)(p) > 0

p E Supp(gtX*) . .

Theorem 6.10 .11 Let M = (Q, X, y) be a ffsm and let S be a fuzzy subset of Q . The following assertions are equivalent. (1) S is a subsystem of M. (2)gtX*C6,Vgt C6,qEQ,tE(0,1] . (3)gtXC6,Vqt C6,qEQ,tE(0,1] .

Proof. (1) x(2) : Let qt C S, q E Q, t E (0,1]. Let p E Q and y E X* . Then ft* (q, y, p) h t = ft* (q, y, p) h qt (q) < y,* (q, y, p) h S(q) < S(p) since S is a subsystem . Hence qtX* C_ S. (2) ===>(3): Obvious. (3) x(1): Let p, q E Q and a E X. If 6(q) = 0 or ft (q, a, p) = 0 then 6(p) > 0 = 6(q) h ft (q, a, p) . Suppose 6(q) :?~ 0 and ft (q, a, p) 0. Let S(q) = t. Then qt C_ S. Thus by the hypothesis, qtX C_ S. Thus S(p) >_ (qtX) (p) = V {t h ft(q, y, p) l y E X} > t u ft(q, a, p) = b(q) n ft(q, a, p) . Hence 6 is a subsystem of M. m © 2002 by Chapman & Hall/CRC

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281

Definition 6.10 .12 Let Ml = (Q1, Xl , fq) and M2 = (Q2, X2, ft2) be two ffsms. Let (f , g) : Ml ----> M2 be a homomorphism. Let S be a fuzzy subset of Ql . Define the fuzzy subset f(S) of Q2 by f (b) (q) _ dq/

V{6(q) I q E Q1, f (q) = q~} if f -1 (q~) zA 0 0iff - (q) - 01

E Q2 -

Theorem 6.10 .13 Let Ml = (Q1, X, gl) and M2 = (Q2, X, g2 ) be two ffsms. Let f : Ml ---> M2 be an onto strong homomorphism. If S is a subsystem of Ql , then f(S) is a subsystem of Q2 . Proof. Let p', q' E Q2 and a E X. Then f(b)(p') nft2(p',a,q')

= =

(V{b(p)Ip E Q1, f (p) = p'}) nft2(p',a,q') V{6(p) Aw2(p',a,q')Ip E Qj, f(p)=j'} .

Let p, q E Q1 be such that f(p) = p' and f(q) = q' . Then 6(p) A w2 (p', a, q')

= = = <

b(p) A w2 (f (p), a, f (q)) b(p) n (V {fti (p, a, r) r E Q1, f (r) = f (q) = q~}) V{6(p) n ft, (p, a, r) r E Q1, f(r) = f (q) = q~} V{6(r) I r E Q1, f (r) = q'}

f(6)(q') .

Hence f(b)(p') Aw2(p',a,q')

<_ =

V{f(b)(q') I p E Q1, f (p) =p'} f(6)(q') .

Thus f (S) is a subsystem of M2 . m The following example shows that the above result need not be true if f is not onto. Example 6.10 .14 Let Q = {p, q} = Q1 = Q2, X = {a}, and ft = ft, = ft2,

where g (r, a, s) = 1 b'r, s E Q. Then M = (Q, X, ft) is a fuzzy finite state machine. Let f : Q ---> Q be a mapping such that f(p) = f(q) = p. Then f is not onto . Clearly, f is a strong homomorphism . Let S be a fuzzy subset of Q such that S(p) = S(q) = z . Then S(r) = z = S(s) Ag(s, a, r) b'r, s E Q. Thus S is a subsystem of M. Now f(S) (p) A tt(p, a, q) = 2 > 0 = f (S) (q). Thus f (S) is not a subsystem of Q .

Definition 6.10 .15 Let M = (Q, X, ft) be a ffsm and let S be a subsystem of M. Then S is called cyclic if 3qt C_ S, q E Q, t E (0,1] such that S = qtX* . In this case we call qt a generator of S .

© 2002 by Chapman & Hall/CRC

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Theorem 6.10 .16 Let M = (Q, X, p) be a ffsm and let S be a subsystem of M. Suppose 3q E Q, t E (0,1] such that S = qtX* . Then (1) S(q) = t, (2) d p E Q, b(q) > b(p), (3) b' subsystems S' of M such that S' C S, if S'(q) > S'(p) Vp E Q, then 61 = gs'(q) X*

Proof.

Now

b(q) _ (qtX*) (q) = (V {w* (q, x, q) I x E X*}) A t =1 n t = t . (2) Let p E Q. Then b(p) _ (gs(q)X*)(p) = (V{w*(q,x,p)Ix E X*}) Ab(q) <_ b(q) . (3) Let p E Q. Then b'(p)

= = _

<

_

b~(p) n 6(p) V{b'(p)nb(q)n[t*(q,x,p)IxEX*} V{b'(p) n w* (q, x,p)I x E X*} v{s'(q) Aw*(q,x,p)Ix E X*} (gs,(q)X*)(p)-

Hence S' C gs,( q)X* . Thus S' = gs,(q)X* . 0 Definition 6.10 .17 Let M = (Q, X, /t) be a ffsm and let S be a subsystem of M. Then S is called super cyclic if and only if S = gs(q)X * b'q E Q.

By Theorem 6.10.16, if S is super cyclic, then S is constant . Example 6.10 .18 Let Q = {p, q}, X = {a}, S(q) = S(p) = 1 and ft(q, a, q) = ft (p, a, p) = 1 and ft (q, a, p) = ft (p, a, q) = 2 . Then S is a subsystem of M = (Q, X, /t) and S is constant. Now

(q,X*) (p) = V{1 A [t* (q, x, p) I x E X*}

= 2 < 1= 6(p) .

Similarly (p 1 X*)(q) < S(q) . Hence S is not cyclic .

Theorem 6.10 .19 Let M = (Q, X, y) be a ffsm and let S be a subsystem of M. Suppose Supp(S) = Q. If S is super cyclic, then M is strongly connected.

Proof. Let p, q E Q. Then (gs(q)X*)(p) =V{b(q) Aw * (q,x,p)Ix E X*} > 0

since S = gs(q)X * and Supp(S) = Q. Hence /t* (q, x, p) > 0 for some x E X* . Thus p E S(q) . Hence M is strongly connected. m © 2002 by Chapman & Hall/CRC

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283

Theorem 6.10 .20 Let M = (Q, X, p) be a ffsm and let S be a subsystem of M. Then S is super cyclic if and only if b'p, q E Q, 3x E X* such that [t * (p, x, q) > 6(p) .

Proof. Suppose that S is super cyclic . Then S is constant by Theorem 6.10.16. Suppose 3p, q E Q such that V'x E X*, y* (p, x, q) < t, where t = S(r) b'r E Q. Then (ps(P)X*)(q) =V{b(p) Aw*(p,x,q) I x E X*} < t=6(q) .

Hence ps(P)X * :?~ S. Thus S is not super cyclic, a contradiction . Conversely, suppose that b'p, q E Q, 3x E X* such that y* (p, x, q) > S(p) . Then b'p, q E Q, 3x E X* such that S(q) > S(p) Aft* (p, x, q) = S(p) . Similarly S(p) > S(q). Hence S is constant . Now (ps(P)X*)(q) = V{6(p) A w*(p, x, q) I x E X*} = 6(p) = b(q) .

Thus ps(P)X* = S. Hence S is super cyclic. 6 .11

m

Strong Subsystems

Definition 6.11 .1 Let M = (Q, X, /t) be a ffsm. Let S be a fuzzy subset

of Q. Then (Q, S, X, /t) is called a strong subsystem of M if and only if b'p, q E Q, if 3a E X such that ft(p, a, q) > 0, then S(q) > S(p) . If (Q, S, X, ft) is a strong subsystem of M, then we simply write S for

(Q, 6, X,

w)

Theorem 6.11 .2 Let M = (Q, X, y) be a ffsm and let S be a fuzzy subset of Q. Then S is a strong subsystem of M if and only if b'p, q E Q, if 3x E X* such that [t* (p, x, q) > 0, then S(q) > S(p) .

Proof. Suppose S is a subsystem. We prove the result by induction Ixj =n . If n =0, then x =A. Now if p = q, then /t*(q, A, q) = 1 and S(q) = S(q) . If q zA p, then ft * (q, A, p) = 0. Thus the result is true if n = 0. Suppose the result is true b'y E X* such that IyI = n-1, n > 0. Let x = ya, I y I = n - 1, y E X*, a E X. Suppose that /t* (p, x, q) > 0. Then on

V{w* (p, y, r) A w(r, a, q) I r E Q}

=

_ >

ya, q) [t * (p, x, q) 0. w* (p,

Thus 3r E Q such that y,* (p, y, r) > 0 and y,(r, a, q) > 0. Hence S(q) > S(r) and S(r) > S(p) . Thus S(q) > S(p) . The converse is trivial . m Theorem 6.11 .3 Let M = (Q, X, y) be a ffsm and let S be a fuzzy subset of Q. If 6 is a strong subsystem of M, then 6 is a subsystem of M. m

© 2002 by Chapman & Hall/CRC

28 4

6. Algebraic Fuzzy Automata Theory

The following example shows that in general, the converse of the above theorem is not true. Example 6.11 .4 Let 6, Q, X, /t be defined as in Example 6.10.4 .

Then a, p) = > 0, but 6(q) = > = Thus 6 is subsystem of M, 6(p) . ft(q, 2 1 2 which is not a strong subsystem.

Theorem 6.11 .5 Let M = (Q, X, y) be a ffsm . Let 6 1 and 62 be strong

subsystems of M. Then the following assertions hold. (1) 6, n62 is a strong subsystem of M. (2) 61 U 62 is a strong subsystem of M. (3) Let 6 be a strong subsystem of M. Then N = (Supp(6), X, v) is a submachine of M, where v = ftjSupp(s)XxxSupp(s) . (4) Let 6 be a strong subsystem of M. Let Nt = (6t, X, v( t)) where v(t) = ft IbtXXXbt, t E [0,1] . Then b't E [0,1], Nt is a submachine of M. (5) Let 6 be fuzzy subset of Q. Let Nt = (6t, X, v( t)) where v( t) _ ft Ibt xxxb t , t E [0,1] . If b't E [0,1], Nt is a submachine of M, then 6 is a strong subsystem of M.

Proof. The proofs of (1) and (2) are straightforward . (3) Let p E S(Supp(6)) . Then p E S(q) for some q E Supp(6) . Then 6(q) > 0. Since p E S(q), 3x E X* such that /t* (q, x, p) > 0. Hence since 6 is a strong subsystem, 6(p) > 6(q) > 0. Thus p E Supp(6) . Hence S(Supp(6)) C Supp(6) . Thus N is a submachine of M. (4) Let q E S(6t) . Then q E S (p) for some p E St . Thus 6(p) > t . Now 3x E X* such that y* (p, x, q) > 0. Then 6(q) >_ 6(p) >_ t. Thus q E 6t . Hence Nt is a submachine of M. (5) Let q, p E Q, x E X* be such that y* (p, x, q) > 0. Suppose 6(p) > 0 . Let 6(p) = t. Then p E St . Since Nt is a submachine of M, S(6t) = St . Thus q E S(p) C S(6t) = St . Hence 6(q) > t. Thus 6 is a strong subsystem . 0 Theorem 6 .11 .6 Let M = (Q, X, y) be a ffsm . Let N = (T, X, v) be a submachine of M. Then XT is a strong subsystem of M.

Proof. Let p, q E Q, a E X, and ft(p, a, q) > 0. Then q E S(p) . If p E T, then q E S(p) C_ S(T) C_ T. Hence XT(q) = 1 = XT(p) . If p ~ T, then XT(p) = 0 < XT(q) . Thus XT is a strong subsystem of M. Theorem 6 .11 .7 Let M = (Q, X, /t) be a ffsm. Then M is strongly connected if and only if every strong subsystem of M is constant .

Proof. Suppose M is strongly connected . Let 6 be a strong subsystem of M. Let p, q E Q . Then p E S(q) and q E S(p) . Hence 6(p) > 6(q) and 6(q) >_ 6(p) . Thus 6(p) = 6(q) . Hence 6 is constant. Conversely, suppose that every strong subsystem is constant . Let p, q E Q . Suppose q ~ S(p) . Then Ql z,4 S(p) z,4 Q. Let 6 be a fuzzy subset of Q such that 6(r) = 1 if r E S(p) and 6(r) = 0 if r ~ S(p) . Let r, s E Q be such that lt* (r, x, s) > 0 © 2002 by Chapman & Hall/CRC

6.11 . Strong Subsystems

285

for some x E X* . If S(r) = 0, then S(s) >_ 0 = S(r) . Let S(r) = 1 . Then r E S(p) . Hence s E S(r) C S(p) . Thus S(s) = 1 = S(r) . Hence S is a strong subsystem . Now S(p) = 1 and S(q) = 0. Thus S is not constant, which is a contradiction. Hence q E S(p) . Thus M is strongly connected . m Definition 6.11 .8 Let M = (Q, X,

/t) be a ffsm and let S be a strong subsystem of M. Suppose Q has at least two elements . Then S is called simple if (1) S is not constant, and (2) for all strong subsystems Q of M, X0 :?~ Q C S ==~, Supp(Q) _ Supp(S) .

Theorem 6.11 .9 Let M = (Q, X, y) be a ffsm and let S be a strong subsystem of M. Suppose ~Q1 > 2 . If S is simple, then Im(S) = {0, t}, where 0
Proof. Since S is not constant, ~Im(S) j >_ 2 . Suppose ~Im(S) j > 2 . Then

3 tl , t2 , t3 E [0 ,1], 0 < tl < t2 < t3 < 1, and 3 rl , r2, r3 E Q such that S(r2) = t2 , i = 1, 2, 3 . Let m E [0,1] be such that t2 < m <_ t3. Let Q be a fuzzy subset of Q such that b'r E Q, Q(r)

_~ mif6r >m 0 if S(r) < m.

Let p, q E Q be such that y* (p, x, q) > 0 for some x E X* . Then S(q) > S(p) since S is a strong subsystem. If S(p) >_ m, then S(q) >_ m. Hence a(q) = m = a(p) . Suppose s(p) < m . Then a(q) > 0 = a(p) . Thus Q is a strong subsystem. Clearly, Q C_ S. Now S(r2) = t2 z,4 0 and a(r2) = 0 . Thus X0 :?~ Q C_ S and Supp(Q) :?~ Supp(S), which is a contradiction since S is simple. Hence ~Im(S) j = 2 . Let Im(S) = {t2, t3}, 0 <_ t2 < t3 < 1. Suppose t2 :?~ 0. Let Q be as defined previously. Then X0 :?~ Q C_ S and Supp(Q) 7~ Supp(S), which is a contradiction . Hence t2 = 0. 0 Theorem 6.11 .10 Let M = (Q, X, y) be a ffsm and let S be a fuzzy subset

of Q . Suppose Im(S) = {0, t}, 0 < t <_ 1. Then S is a simple strong subsystem if and only if N = (Supp(S),X,v), where v = ltjSupp(s)XxxSupp(s), is a strongly connected submachine of M.

Proof. Suppose N is strongly connected . Let p, q E Q be such that tt* (p, x, q) > 0. If S(p) = 0, then S(q) > 0 = S(p) . Suppose S(p) > 0. Then p E Supp(S) . Now q E S(p) C_ S(Supp(S)) = Supp(S), since N is a submachine. Hence S(q) = t = S(p) . Thus S is a strong subsystem . Let Q be a strong subsystem of M such that X0 :?~ Q C S. Then by Theorem 6.11 .5(3), K = (Supp(c,),X,ltlsupp(a)Xxxsupp(a)) is a submachine. Now K C_ N. Since N is strongly connected, N has no proper submachine. Hence K = N. Thus Supp(Q) = Supp(S) . Hence 6 is simple. Conversely, © 2002 by Chapman & Hall/CRC

28 6

6. Algebraic Fuzzy Automata Theory

suppose S is simple. Let K = (T, X, 97) C N, Ql :?~ T C_ Supp(6) C Q, be a submachine. Let Q be a fuzzy subset of Q such that b'r E Q, Q(r)

-

S(r) if r E T Oifr~T.

Then Q C S. Let p, q E Q be such that p* (p, x, q) > 0 . Then q E S(p) . If a(p) = 0, then a(q) >_ 0 = a(p) . Let a(p) > 0 . Then p E T. Thus q E S(p) C S(T) = T . Hence a(q) = S(q) > S(p) > a(p) . Thus Q is a strong subsystem . By the definition of Q, Q z,4 XO . Since S is simple, Supp(Q) = Supp(S) . Hence Supp(Q) C_ T C_ Supp(S) = Supp(Q) . Thus T = Supp(S) . Hence K = N. Thus N has no proper submachine. Hence N is strongly connected . Theorem 6.11 .11 Let Ml = (Q1, X, pl) and M2 = (Q2, X, p2) be two ffsms. Let f : Ml -----> M2 be an onto strong homomorphism. If S is a strong subsystem of Q1, then f (S) is a strong subsystem of Q2 . Proof. Let p, q Now

E Ql

and a

E

X be such that y,2 (f (p), a,

f (b) (f (q)) = V{b(r)

f (q)) > 0 .

r E Q1, f (r) = f (q)

and f (b) (f (p)) = V

Let s

E Q1

V(s)

be such that S(s) > 0 and

s E Q1' f (s) = f (p* f (s) = f (p) .

Now

ft2 (f (s), a, f (q)) = ft2 (f (p), a, f (q)) > 0.

Hence V {ftl (s,

a, r) jr

E Qj, f (r) = f (q) } >0 .

Thus 3 r E Q1 such that ft, (s, a, r) > 0 and f (r) subsystem, 6(r) > 6(s) > 0. Hence f (S) (f (q)) > strong subsystem.

= f (q) .

Since S is a strong Thus f (S) is a

f (S) (f (p)) .

The following example shows that the above result need not be true if not onto.

f is

Example 6.11 .12 Let Q, X, y, S, and f be defined as in Example 6.10.14.

Then S is a strong subsystem and f is a strong homomorphism such that f is not onto . Now ft(p, a, q) = 1 > 0, but f (S) (p) = 2 > 0 = f (S) (q) . Hence f (S) is not a strong subsystem.

© 2002 by Chapman & Hall/CRC

6.12. Cartesian Composition of Fuzzy Finite State Machines

6 .12

287

Cartesian Composition of Fuzzy Finite State Machines

Here and the next section, we study a new product of two fuzzy finite state machines Ml and M2 , written Ml - M2 , and called the Cartesian composition of Ml and M2, as in [48] . We show that Ml, M2, and Ml - M2 share many similar structural properties, e.g., those of singly generated, retrievability, connectedness, strongly connectedness, commutativity, perfectness, and state independence . This is important since a fuzzy finite state machine, which is a Cartesian composition of submachines can thus be studied in terms of smaller machines . Definition 6.12 .1

Let M = (Q, X, /t) be a ffsm . Then M is said to be connected if and only if M has no separated proper submachine .

Theorem 6.12 .2

Let M = (Q, X, /t) be a ffsm . Then M is connected if and only if b' proper submachines N = (T, X, v) 3 s E Q\T and t E T such that s(s) n S(t) z,4 0 .

Proof. Suppose M is connected. Let N = (T, X, v) be a proper submachine. Then S(Q\T) n T :?~ 01 since M has no separated proper submachine. Thus 3 r E S(Q\T) n T . Now T = S(T) . Hence r E S(s) for some s E Q\T and r E S(t) for some t E T. Thus s(s) n S(t) z,4 0 . Conversely, let N = (T, X, v) be a proper submachine. Then 3 s E Q\T and t E T such that s(s) ns(t) :?~ 0 . Hence 0 :?~ s(s) ns(t) C s(Q\T) ns(T) = s(Q\T) nT. Thus N is not separated. Hence M has no proper separated submachine. Thus M is connected . m Definition 6.12 .3 Let M = (Q, X, /t) be a ffsm . Let p, q E Q . Then q and p are called connected if either q = p or 3 qo, q l , . . . , qk E Q, q = qo, p = qk and 3 al, a2 . . . . , ak E X such that b' i = 1, 2, . . . , k either ft(gi-1, a2, qZ) > 0 or ft (gj, aZ, qZ-1) > 0 . r

Clearly, if q and p are connected and p and are connected .

Definition 6 .12 .4

r

are connected, then

Let M = (Q, X, /t) be a ffsm. For all q E Q, let

C(q) = {p E Q

Ip

and q are connected}

Vq E Q .

Clearly, Let

b'q, p E Q

M = (Q, X, y)

if p

E C(q),

then C(p) = C(q) .

be a ffsm. For all T C

Q,

C(T) = UpETC(p) " © 2002 by Chapman & Hall/CRC

let

q

and

28 8

6. Algebraic Fuzzy Automata Theory

Lemma 6.12 .5 Let M = (Q, X, p) be a ffsm . Let U, V C_ Q. Then the following properties hold. (1) If U C V then C(U) C C(V) . (2) U C C(U) . (3) C(C(U)) = C(U) . (4) C(U U V) = C(U) U C(V) . (5) C(U n V) c C(U) n C(V) . (6) Let q, p E Q. If q E C(U U {p}) and q ~ C(U), then p E C(U U {q}) .

SM C- C(Q) . (8) S(C(U)) = C(U) . ('7)

Proof. The proofs of (1), (2), (4), (5), and (7) are straightforward. Consider (3) . Now C(U) C C(C(U)) by (2) . Let q E C(C(U)) . Then 3 p E C(U) such that q E C(p) . Hence q E C(r) for some r E U. Thus q E C(r) C C(U) . Consider (6) . Suppose that q E C(UU{p}) and q ~ C(U) . Since C(U U {p}) = C(U) U C(p), q E C(p) . Thus p E C(q) C C(U U {q}) . Consider (8) . Let p E S(C(U)) . Then p E S(q) for some q E C(U) . Now q E C(r) for some r E U. Hence p E C(r) C C(U) . Thus S(C(U)) C C(U) and so S(C(U)) = C(U) . m Since Q is a finite set, it is clear that b' q E Q and U C_ Q, if q E C(U), then q E C(U) for some finite subset U' of U. This fact together with properties (1), (2), (3), and (6) give C the basic spanning properties in [264, p. 50] that are used for various algebraic structures to obtain the existence of bases and the uniqueness of their cardinalities. Definition 6 .12 .6 Let M = (Q, X, /t) be a ffsm and let T C_ Q . Then T is

called a connected component if b' s, t E T, s and t are connected. T is called a maximal connected component when b' p E Q, if p is connected to t for some t E T, then p E T.

Theorem 6.12 .7 Let M = (Q, X, p) be a ffsm and let q E Q. Then C(q) is a maximal connected component of Q.

Proof. The proof is obvious . m Theorem 6.12 .8 Let M = (Q, X, p) be a ffsm and let q E Q. Let N = (C(q),X, ltja(q)XXXC(q)) . Then N is a submachine of M. m Theorem 6 .12 .9 Let M = (Q, X, /t) be a ffsm. Then M is connected if and only if b'q E Q, C(q) = Q.

Proof. Suppose M is connected and let q E Q . Suppose 3p E Q such that p ~ C(q) . Then N = (C(q),X, /tjc(q)XXXC(q)) is a proper submachine of M. Hence by Theorem 6.12.2, 3s E Q\C(q) and t E C(q) such that S(s) nS(t) z,4 0. Let r E S(s) nS(t) . Then s and r are connected and r and t are connected . Hence s and t are connected . Thus s E C(t) = C(q), which is a contradiction . Hence C(q) = Q . Conversely, suppose that b'q E Q, © 2002 by Chapman & Hall/CRC

6.13. Cartesian Composition

289

= Q. Let N = (T, X, v) be a proper submachine of M. Suppose that N is separated. Then S(Q\T) n T = Ql and S(Q\T) = Q\T. Let q E Q\T and t E T. Then C(t) = Q = C(q) . Hence q and t are connected . Thus 3 qo, ql, . . . , qk E Q, q = qo, t = qk, and 3 al , a2 , . . . , ak E X such that C(q)

b' i = 1, 2, . . . , k either ft(gi_1, a2, qZ) > 0 or tt(q a qZ-1) >O .Now3 i such that q2_1 E Q\T and qZ E T. Hence either qZ E S(g2_1) C_ S(Q\T) or q2_1 E S(qi) C_ T, which is a contradiction . Thus N is not separated. Hence M is connected . m

Corollary 6.12 .10 Let M = (Q, X, ft) be a ffsm. Then M is connected if and only if b' p, q E Q, p and q are connected. m

6 .13

Cartesian Composition

Definition 6.13 .1 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 and let Xl n X2

= 01 . Let

M1 -

M2 = (Ql X Q2,

X1 U X2, w1

'

w2),

where

(Nt1 ' ft2)((pl,p2), a, (ql, q2))

=

ft, (pl, a, ql) Nt2(p2, a, q2)

0 otherwise,

if a E Xl and p2 = q2 if a E X2 and pl = ql

E Q1 X Q2, a E Xl U X2 . Then Ml the Cartesian composition of Ml and M2 .

d(pl,p2), (ql, q2)

M2

is a ffsm, called

Theorem 6.13 .2 Let MZ = (QZ, XZ , p2 ) be a ffsm, i = 1, 2 and let Xl rl = 0. Let Ml - M2= (Ql X Q2, XI U X2, ftl,ft2) be the Cartesian composition of Ml and M2 . Then V'x E Xl U X2, x z,4 A, X2

(ftl ' ft2)*((pl,p2),

d(pl,p2), (ql, q2)

E

x,

(ql, g2))

=

fti (pl, x, ql)

if x E Xl and p2 = q2 x, g2) if x E X2 and pl = gl It * (P2, 0 otherwise,

Q1 X Q2-

Proof. Let x E Xl U X2, x z,4 A and let Ix I = n . Suppose that x E Xl . Clearly the result is true if n = 1. Suppose the result is true b'y E X*, © 2002 by Chapman & Hall/CRC

290

6. Algebraic Fuzzy Automata Theory

yI =n-1, n > 1 . Let x = ay where a E X1 and y E Xl . Now (ftl ' w2)*((P1,P2), ay, (qj, q2))

= =

V{(ftl ' w2)((Pl,P2), a, (rl, r2)) n(ftl ' ft2) * ((rl, r2), y, (ql, q2)) l (rl, r2) E Q1 X Q2} V{ftl(pl,a,rl) n (ft, ' ft2)*((rl,P2),y, (gl,q2))l r l E Q1}

V {ftl(pl, a, rl) n fti (rl, y, r l E Q1} if P2 = q2 0 otherwise fti (pl, ay, ql) if P2 = q2 0 otherwise .

= __

The result now follows by induction. The proof is similar if x

ql)

E X2 . m

Theorem 6.13 .3 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 and let Xl X2 = 0. Then V'x E X*, b'y E X2 (pl'p2)*((P1,P2),xy,(gl,q2))

= _

rl

fti(pl,x,gl)np2(P2,y,q2) (pl ' p2)*((P1,P2), yx, (qj, q2))

V(Pl,P2), (gl,q2) E Q1 X Q2-

Proof. Let x E Xl, y E X2, (PI, P2), (qj, q2) E Q1 X Q2 . If x = A = y, then xy = A. Suppose (PI, P2) _ (qj, q2) . Then pl = ql and P2 = q2 . Hence (pl ' p2) * 01,P2),xy, (gl,q2)) = 1 = 1 n 1 = fti(pl,x,gl) n ft2(P2,y,g2) " If (pl, p2) 7~ (qj, q2), then either pl ql or P2 :?~ q2 . Thus pi (pl , x, ql) n . Hence 0 ' p2 (P2, y, q2) = (It, p2) * 011 P2), xy (ql, q2)) = 0 = pi (PI, x, q1) n If x = A and y :?~ A or x :?~ A and y = A, then the result [t2* (P2, y, q2) . follows by Theorem 6.13.2 . Suppose x :?~ A and y :?~ A. Now I

(pl '

p2) * 01,P2), xy, (qj, q2))

=

V{(pl '

=

p2)*((P1,P2), x, (rl, r2))n (pl ' ft2) * ((rl, r2), y, (qj, q2)) l h, r2) E Q1 X Q2} V {V{(ftl ' p2)*((P1,P2), x, (rl, r2))n (pl ' ft2) * (h, r2), y, (qj, q2))

=

Similarly

I r2EQ21 I r1EQ1I

V{(pl '

p2)*((P1,P2), x, (rl,P2))n (pl ' p2)*((r1,P2), y, (qj, q2)) l rl E Q1} pi (pl, x, gl) n p2 (P2, y, q2) .

(ftl ' p2) * 011P2), yx, (qj, q2)) = pi(pl, x, ql) n [t* (P2, y, q2) .

Theorem 6.13 .4 Let MZ X2 = 01. Then b'w E (XI U (pl '

= (QZ, XZ, pi) be a ffsm, i = 1, 2 X2 )* 3 u E X*, v E X2 such that

p2)TP1,P2), w, (qj, q2)) =

V(Pl,P2), (gl,q2) E Q1 X Q2-

© 2002 by Chapman & Hall/CRC

(pl '

and let Xl

p2)TP1,P2), UV, (qj, q2))

rl

6.13. Cartesian Composition

291

Proof. Let w E (X1 U X2)* and (pl, p2), (ql, q2) E Q1 X Q2 . If w =A, then we can choose u = A = v. In this case the result is trivially true. Suppose w :?~ A. If w E Xl or w E X2*, then again the result is trivially true. Suppose w ~ Xl and w ~ X2 . Case 1: If w = xy, x E X+, y E X2+ , then the result follows by Theorem 6.13.3. Case 2: Suppose w = xlyx2, X1, X2 E Xl and y E X2, x2 and y are nonempty strings, i = 1, 2. Let u = xlx2 E Xl and v = y . Now by Theorem 6.13.3, (ftl ' ft2) *((rl, r2), x2y, (ql, q2)) = (ftl ' ft2) * ((rl, r2), yx2, (ql, q2)) E Q1 X Q2 . Thus (ftl ' ft2) * ((pl,p2),xlyx2,(gl,q2)) _ n (ftl'ft2) * ((r1Ir2),yx2,(gl,q2)) I (rl,r2) E ((pl,p2),xl,(rl,r2)) V{(ftl'ft2) * Q1 XQ2} = V{(ftl'ft2)*((pl,p2),xl, (rl,r2)) n (ftl'ft2) * ((rl,r2),x2y, (gl,q2))

V(rl,r2),(gl,q2)

(rl,r2) E Ql X Q2} = (ftl ' ft2) * ((pl,p2),xlx2y, (gl,q2))-

Case 3: Suppose w = ylxy2, yl, y2 E X2 and x E X*, yZ and x are nonempty strings, i = 1, 2. Let v = yly2 E X2 and u = x . The proof of this case is similar to Case 2. Case 4: Suppose w = xlylx2y2, X1, X2 E Xl, Y1, Y2 E X2, xZ and yZ are nonempty strings . Let u = xlx2 E Xl and v = yly2 E X2 . Then (ft, ft2) * ((pl,p2),xlylx2y2,(gl,q2)) = VWtl ' ft2) * ((pl,p2),xl,(rl,r2))A (ft, '

ft2) * ((rl~r2),ylx2y2,(gl,q2))I (rl,r2) EQ1XQ2}=VWtl'ft2) * ((pl,p2),xl, (rl, r2)) A (ftl ' ft2) * ((rl, r2), x2yly2, (ql, q2))

I

(rl, r2) E Ql X Q2}

(by Case

3) = (ftl ' ft2)*((pl,p2), xlx2yly2, (ql, q2)) = (ftl ' ft2)*((pl,p2), uv, (ql, q2)) . Case 5: Suppose w = ylxly2x2, X1, X2 E Xl , Y1, Y2 E X2* . Let u = xl x2 E Xl and v = yly2 E X2 . The proof of this case is similar to Case 4. Case 6: Let w E (XI U X2)* . Then w = xlylx2y2 . . . xnyn or w = ylxly2x2 . . . ynxn, xZ E Xl, yZ E X2, x2 and yZ are nonempty strings, i = 1, 2, . . . , n - 2. To be specific, let w = xlylx2y2 . . . xnyn . The proof of the second case is similar. If n = 0, 1, or 2, then the result is true by the previous cases . Suppose the result for all z = xlylx2y2 . . . xn-lyn-1 E (XI U X2)*, n > 2 . Let ul = xlx2 . . . x n-1, vl = yly2 . . . yn-1, u = ulxn, and v = vlyn . Now (ftl ' ft2)*((pl,p2), xlylx2y2 . . . xnyn, (ql, q2)) _ V{(ftl ' ft2) * ((pl,p2),xlylx2y2 . . . xn -1 yn-1, (rl,r2))n (ftl ' ft2) * ((rl1 r2), xnyn (ql, q2)) I (rl, r2) E Q1 XQ2} = V{(ftl ' ft2) * ((pl1 p2), ulvl, (rl, r2))n

(ftl ' ft2) * ((rl, r2), xnyn, (ql, q2)) (rl, r2) E Q1 X Q2} = (ftl ' ft2)*((pl, p2), ulvlxnyn, (ql, q2)) = (ftl ' ft2) * ((PI, P2), UV, (gl, q2)) . The result now

follows by induction. m

In Theorem 6.13.4, u consists of all the elements from w that are in X l (in the given order) and v consists of all the elements from w that are in X2 (in the given order) . We write w* = uv and call w* the standard form of w.

© 2002 by Chapman & Hall/CRC

29 2

6. Algebraic Fuzzy Automata Theory

Theorem 6.13 .5 Let

MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 and let Xl rl X2 = 0. Then the Cartesian composition Ml - M2 is cyclic if and only if Ml and M2 are cyclic.

Proof. Suppose Ml and M2 are cyclic, say Ql = S(qo) and Q2 = S(po) for some qo E Q1, PO E Q2 . Let (q, p) E Q1 x Q2 . Then 3 x E Xl and y E X2 such that pi (qo, x, q) > 0 and p2 (po, y, p) > 0. Thus (pl ' p2) * ((qo,po), xy, (q, p)) = pi (qo, x, q) n p2 (POI y, p) > 0 . Hence (q, p) E S((go,po)) . Thus Q1 x Q2 = S((go,po)) . Hence Ml - M2 is cyclic . Conversely, suppose Ml - M2 is cyclic . Let Q1 x Q2 = S((qo, po)) for some (qo, po) E Q1 x Q2 . Let q E Q1 and p E Q2 . Then 3 w E (XI U X2)* such that (pl'p2N(go,po), w, (q,p)) > 0 . By Theorem 6.13.4, 3u E X1 and v E X2 such that pi(go,u,q)np2(po,v,p) _ (pl'p2)*((go,po),w, (g,p)) > 0 . Hence 3u E X1 and v E X2 such that pi (qo, u, q) > 0 and p2 (po, v, p) > 0 . Thus q E S(qo) and p E S(po) . Hence Q1 = S(qo) and Q2 = S(po) . Thus Ml and M2 are cyclic . m Theorem 6.13 .6 Let

MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2 and let Xl rl X2 = 0. Then the Cartesian composition Ml - M2 is retrievable if and only if Ml and M2 are retrievable.

Proof. Suppose that Ml and M2 are retrievable . Let (q, p), (t, s) E Q1 x Q2, and w E (XI U X2)* be such that (pl ' p2)*((q, p), w, (t, s)) > 0 . Let w* = uv be the standard form of w, u E X*, v E X2* . Then (ft, p2)*((q,p), w, (t, s)) = (pl ' p2)*((q,p), uv, (t, s)) = pi(q, u, t) n p2 (p, v, s) . Thus pi (q, u, t) > 0 and p2 (p, v, s) > 0. Since Ml and M2 are retrievable, El u' E X * ' v' E X2 such that pi (t, u', q) > 0 and p2 (s, v', p) > 0. Thus (pl ' p2)*((t, s), u'v', (q, p)) > 0 . Hence Ml - M2 is retrievable. Conversely, suppose that M1-M2 is retrievable. Let q, t E Q1 and y E Xl be such that pi (q, y, t) > 0. Then b' s E Q2, (pl ' p2)* ((q, s), y, (t, s)) = pi (q, y, t) > 0. Thus 3 w E (XI UX2)* such that (It, ' p2)*((t, s), w, (q, s)) > 0. Let w* = uv be the standard form of w, u E X*, v E X2 . Then 0 < (pl ' p2)* ((t, s), w, (q, s)) = pi (t, u, q) np2 (s, v, s) . Thus pi (t, u, q) > 0. Hence Ml is retrievable . Similarly M2 is retrievable . m The following corollaries follow from Theorems 6.8 .6 and 6.13.6. Corollary 6.13 .7 Let MZ

= (QZ, Xi, pi) be a ffsm, i = 1, 2, and let Xl rl X2 = 0. Then the Cartesian composition Ml - M2 is the union of strongly connected submachines if and only if Ml and M2 are the union of strongly connected submachines.

Corollary 6.13 .8 Let MZ

= (QZ, XZ, pi) be a ffsm, i = 1, 2, and let Xl rl X2 = 0. Then the Cartesian composition Ml - M2 satisfies the Exchange Property if and only if Ml and M2 satisfies the Exchange Property . m

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6.13. Cartesian Composition

293

Theorem 6.13 .9 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let X1 rl

X2 = 0 . Then the Cartesian composition Ml - M2 is connected if and only if Ml and M2 are connected.

Proof. Suppose that Ml and M2 are connected. Let (q,q'),(p,p') E Q1XQ2 " NowElg0,q1, " . . ,qn EQ1,q - q0,p=gnand3a,,a2, " . . ,an E X1 such that b' i = 1, 2, . . . , n either p1 (qZ-1, aZ, gZ ) > 0 or ft, (gZ, aZ, gZ-1) > 0 and 3 qo, q. . . . . . gm E Q2, q' = q9, p~ = qm and 3 b1, b2, " . . , bm E X2 such that d i = 1, 2, . . . , m either p2(gZ-1, bZ, qZ) > 0 or pz(gZ, b2, qZ-1) > 0. Consider the sequence of states (g, q) _ (g0, g0), (g1, g0), . . . , (gn, g0), (gn, gl), . . , (gn, gm) _ (p,p ' ) E Ql X Q2 and the sequence a 1, a2, " . . , a n, bl1,, 2, bm E X1 U X2 . Then Vi = 1,2, . . . , neither (It,' p2) ((qj-1, q0), a2, (q2, qo)) > 0 or (p1 ' p2) ((qi, q0), a2 , (qZ-1, qo)) > 0 and b' j = 1, 2, . . . , m either (p1 ' p2)((gn, '-1), bj, (qn, g;)) > 0 or (p1 ' p2)((gn, g;), bi, (gn, g;-1)) > 0. Hence (q, q) and (p, p) are connected. Conversely, suppose that M1 - M2 is connected . Let q, p E Q1 and let r E Q2- If p = q then p and q are connected. Suppose p :?~ q.Then 3 (q, r) = (g0,p0), (gl,pl), " . . , (gn,pn) = (p, r) E Q1 X Q2 and a1, a2, " . . , an E X1 U X2 such that Vi = 1, 2, . . . , n either (p1 ' p2 ) ((g2-1, pj-1), aZ, (qi, p2)) > 0 or (p1 ' p2)((gi,pi),aj, (qZ-1, pi-1)) > 0. Clearly, if qZ-1 :?~ qZ, then pi- 1 = pi and if pi- 1 :?~ pi, then qZ-1 = qj d i = 1, 2, . . . , n. Let {q = qz l , qz . . . . . . qj,, = p} be the set of all distinct qj E {qo, q1, . . , qn} and let a21, a22 , . . . , a2,, be the corresponding a2's . Then a21, ail , . . . , a2, E XI and b' j = 1, 2, . . . , k either ft, (gi, 1 , a 2j , qi;) > 0 or ft, (gi, , a2j , qZ, ,) > 0. Thus q and p are connected. Hence M1 is connected . Similarly, M2 is connected . 0 The following theorem follows from Theorems 6.8 .11, 6.13.6, and 6.13.9. Theorem 6.13 .10 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let X1 rl

X2 = 0 . Then the Cartesian composition M1 - M2 is strongly connected if and only if M1 and M2 are strongly connected. m

Definition 6.13 .11 Let M = (Q, X, p) be a ffsm. Then M is said to be commutative if b'a, b E X and b'q, p E Q,

p* (q, ab, p) = p* (q, ba, p) .

The following result is immediate from Theorem 6.13.4. Theorem 6.13 .12 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let X1 rl

X2 = 0. Then the Cartesian composition M1 - M2 is commutative if and only if M1 and M2 are commutative. 0

Definition 6.13 .13 Let M = (Q, X, p) be a ffsm. If M is commutative and strongly connected, then M is said to be perfect.

Theorem 6.13 .14 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let X1 rl

X2 = 0 . Then the Cartesian composition M1 - M2 is perfect if and only if M1 and M2 are perfect. m

© 2002 by Chapman & Hall/CRC

294

6. Algebraic Fuzzy Automata Theory

Definition 6.13 .15 Let M = (Q, X, /t) be a ffsm. Then M is said to be state independent if Vq',p' E Q, Vx,y E X+, (/t*(q',x,p') > 0 and p* (q', y, p') > 0) ==~, (dq, p E Q, p* (q, x, p) > 0 ~ /t* (q, y, p) > 0) .

Theorem 6.13 .16 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let Xl rl X2 = 0 . Then the Cartesian composition Ml - M2 is state independent if and only if Ml and M2 are state independent. Proof. Suppose that Ml and M2 are state independent . Suppose that

(pl ' p2)*((gi,q2),wl,(pi,p2)) > 0 and (pl ' p2)* ((gi,q2),w2,(piIp2)) > 0, where (gi, q2), (pi,p2) E Q1 x Q2, and WI, w2 E (XI U X2)* . Now 3 ul, u2 E Xi and VI, v2 E X2 such that (pl p2)* ((qi, q2), wl, (pi, p2)) ~ ~ = * pi(gi,ul,pi)np2(q2,vl,p2) and (pl'p2 * ((gl~g)~ q2), W2, 2~ (P, P2)) p1 1 u2~ [t* (q' ) . Thus [t* (q, u 0 ' ) l~ > V2, p' g1 ~ 1 ~ 1p') vi, p1 p2 g21 2 1 1 p2 (g2, p2) > 0, pi(gi, u2~ and 0. Hence E 0, p2(g2,v2,p2) > dgl,pl Q1, pi(gl,ul,pl) >0 pi) > pi(gl,u2,p1) > 0 and dg2,p2 E Q2, p2(g2,vl,p2) > 0 p2(g2,v2,p2) > 0. Hence pi(gi, ui, pi)n [t2* (q2, VI, p2) > 0 U2,pi)A p2 (g2, v2, p2) pi( > 0, dql, pl E Q1, dq2, p2 E Q2 . Thus (pl ' p2)*((gl,q2),wl, (PI, P2)) > 0 (pl'p2)*((gl,q2),w2, (PI, P2)) > 0, dgl,pl E Q1, dg2,p2 E Q2 . Hence Ml - M2 is state independent.

Conversely, suppose that Ml - M2 is state independent . Suppose that pi(gi,ul,pi) > 0 and pi(gi,u2,pi) > 0 where U1, U2 E Xl and q', p' E Q1 Then VS E Q2, (pl ' p2)*((gi"s),ul,(pi,s)) = p1 1 ul,pi) > 0 and (pl ' p2)*((gi,8),u2,(pi,s)) = pi(gi,u2,pi) > 0. Thus dq,p E Ql, s E Q2, (pl ' p2) * ((q, s), ul, (p, s)) > 0 (pl ' p2)* ((q, 8), u2, (A s)) > 0 . Hence b'q, p E Q1, pi (q, ul, p) > 0 pi (q, u2, p) > 0. Thus Ml is state independent . Similarly M2 is state independent. m

Theorem 6.13 .17 Let MZ = (QZ, XZ, pi) be a ffsm, i = 1, 2, and let Xl rl X2 = 01 . Let Ni = (Ti, XZ, v2) be a submachine of MZ, i = 1, 2 . Then Nl - N2 is a submachine of Ml - M2 . Conversely, if N = (TI x T2 , Xl U X1 , v) is a

submachine of Ml - M2 , then there exist submachines Nl of Ml and N2 of M2 such that N = Nl - N2 .

Proof. Let Ni = (Ti , XZ , v2) be a submachine of MZ , i = 1, 2 . Now N1-N2 = (TI XT2, XI UX2, VI 'V2) . Let (r, s) E S(TI xT2) . Then 3 w E (XI U X2)* , (p, q) E Tl x T2 such that (pl ' p2)*((p, q), w, (r, s)) > 0. Let w* = uv be the standard form of w, u E XI* , v E X2* . Now [t* (p, u, r) A [t*(q, v, s) (pl'p2) * ((p, q), w, (r, s)) > 0. Thus pi (p, u, r) > 0 and p2 (q, v, s) > 0. Hence rES(p)CS(Tl)=TlandsES(q)CS(T2)=T2 .Thus (r,s)ET1 xT2 .

© 2002 by Chapman & Hall/CRC

6.13. Cartesian Composition Hence S(Tj x T2 )

C Tl x T2 .

295 Let (p, q), (r, s)

E Tl x T2 , a E X l U X2 . Now

(VI - V2) ((p, q), a, (r, s) )

v l (p,a,r) if a E X1, q =s v2 (q, a, s) i£ a E X2, p=r

0 otherwise

P1 (PI a, r) if a E X1, q =s p,2 (q, a, s) if a E X2, p=r

0 otherwise

(P1 ' P2) ((p, q), a, (r, s)) .

Hence (PI 'N2)I(T1xT2)x(XIUX2)x(T1xT2) =vl-v2 . Thus Nl -N2 is asubmachine of Ml - M2 . Conversely, let N = (TI x T2 , Xl U X1 , v) be a submachine of Ml - M2 . Let vl = /t1IT1 xX 1 xT1 , v2 = /t2lT2XX2XT2, N1 = (TI IX1IV1)1 and N2 = (T2, X2, v2) . Let p E T1, x E Xl , r E Q1 be such that pl (p, x, r) > 0. Let t E T2 . Then (p,l'Nt2)*((p, t), x, (r, t)) = p,l(p, x, r) >0. Thus (r, t) E S(Tj x T2) = Tl x T2 . Hence r E Tl and so S(TI ) C_ T1 . Thus Nl is a submachine of Ml . Similarly N2 is a submachine of M2 . Let (p, q), (r, s) E Tl x T2 , aEXIUX2 .Now

v((p, q), a, (r, s))

=

Gt1 ' ft2)((p, q), a, (r, s)) P 1 (p, a, r) if a E X 1 , q = s N'2 (q, a, s) if a E X2, p = r

0 otherwise

vi (p,a,r)ifaEX1, q=s v2 (q, a, s) if a E X2, p = r

0 otherwise

(VI - v2) ((p, q), a, (r, s)) .

Hence N = Nl - N2 . m Let M = (Q, X, ft) be a ffsm and let - be an equivalence relation on Q . Recall from Definition 6.4 .1 that - is called an admissible relation if and only if b' p, q, r E Q, b'a EX, if p - q and lt(p, a, r) > 0, then 3 t E Q such that ft(q, a, t) >_ ft(p, a, r) and t - r . Let M = (Q, X, ft) be a ffsm and let - be an equivalence relation on Q . By Theorem 6.4.2, - is an admissible relation if and only if d p, q, r E Q, V'x E X*, if p - q and /t* (p, x, r) > 0, then 3 t E Q such that /t* (q, x, t) >_ p,* (p, x, r) and t - r . Let Ml = (Q1, X1, p l ) and M2 = (Q2, X2, p 2 ) be two ffsms and let Xl n X2 = Ql . Let p i be an admissible relation on MZ, i = 1, 2. Define a relation p l ' P2 on Ml - M2 by (PI IP2)P l ' P2(gl,q2) if

and only

if p1Plg1

and

p2P2g2

d(pl,p2), (ql, q2) E Q1 x Q2Clearly pl'P2 is an equivalence relation on Ml .M2 . Let (PI, p2), (ql, q2) E Q 1 x Q2 be such that (pl,p2)Pl ' P2(gl,g2) . Let a E Xl U X2 and (ft,

© 2002 by Chapman & Hall/CRC

29 6

6. Algebraic Fuzzy Automata Theory

p2)((pl,p2), a, (r l , r 2 )) > 0 for some (r l , r2 ) E Q1 x Q2 . Suppose that a E X1 . Then ft, (PI,a,rl) = (ftl' P2)((pl,p2),a,(rl,r2)) > 0 . Thus p2 = r2 . Now plplgl and ft, (PI, a, rl) > 0 . Since p l is admissible, 3 t l E Q1 such that ftl(gl, a, tl) >_ ft, (pl, a, rl) and tlplrl . Hence (It, ' P2)((gl, q2), a, (tl, q2)) = ftl(gl,a,tl) >_ ft, (pl,a,rl) = (ftl'ft2)((pl,p2),a, (rl,p2)) . Also, since tj p, r, and g2p2p2, (tl,g2)pl ' p2(rl,p2) . Thus p l ' p2 is an admissible relation on Ml - M2 . We have thus proved the following theorem . Theorem 6 .13 .18 Let MZ = (QZ, XZ, p 2 ) be a ffsm, i = 1, 2, and let Xl rl X2 = 0 . Let Pi be an admissible relation on MZ, i = 1, 2 . Then pl 'P2 is an admissible relation on Ml - M2 . 6 .14

Admissible Partitions

In this section, we introduce the concept of a covering of a ffsm by another, admissible partitions and relations of a ffsm, ft-orthogonality of admissible partitions, irreducible ffsm, and the quotient of a ffsm induced by an admissible partition of the state set . We derive results concerning ft-orthogonality and covering, Theorems 6.14 .14 and 6 .14 .15 . We show that an admissible partition 7r of Q is maximal if and only if the quotient M17r is irreducible, Theorem 6 .14 .20 . The paper culminates with a result showing that a ffsm can be covered by irreducible ffsms, Theorem 6 .14 .21 . These results allow us to study fuzzy finite state machines via coverings of products of simpler machines . Irreducible finite state machines seem to arise naturally in some applications, e .g ., biology. An example is given in [92] of a finite state machine arising from a metabolic pathway. Recall from Definition 6 .6 .1 that (9, ~) is a covering of Ml by M2, written Ml <_ M2 , if and only if b' q2 E Q2, ql E Q1, and x E Xl, fri(97(g2), x, ql) > tt2(g2, ~ * (x), r2) d r2 E Q2 such that 9(r2) = ql and 3 r 2 E Q2 such that 9(r2 ) = ql and frl*(?7(q2), x, ql) = ft2 * (q2, ~ * (x), r2)Let Ml = (Q1, Xl, ftl) and M2 = (Q 2 , X2, ft2) be fuzzy finite state machines . Let w be a function of Q2 x X2 into X1 . Let Q = Q1 x Q2 . Define ft' :QXX2XQ----> [0,1] as follows : d((ql, q2), b, (PI, P2)) E Q x X2 x Q, ft' ((gl, q2), b, (PI, P2)) = ft, (gl,w(q2, b), PI) n P2 (q2, b,p2) . Then M = (Q, X2, ft') is a ffsm . M is called the cascade product of Ml and M2 and we write M = M1wM2 . Let M = (Q, X, ft) be a ffsm and let - be an equivalence relation on Q . Recall that ti is called an admissible relation on Q if and only if d p, q, r E Q, b'a E X, if p - q and ft (p, a, r) > 0, then 3 t E Q such that ft(q, a, t) > ft(p, a, r) and t - r .

© 2002 by Chapman & Hall/CRC

6.14. Admissible Partitions

297

Theorem 6.14 .1 [3] Let M = (Q, X, /t) be a ffsm and let - be an equiv-

alence relation on Q . Then - is an admissible relation on Q if and only if V p, q, r E Q, Vx EX*, if p - q and /t* (p, x, r) >0, then 3 t E Q such that * * ft (q, x, t) > ft (p, x, r) and t - r. m

Definition 6.14 .2 Let M = (Q, X, /t) be affsm and P = ~Q1, Q2, . . . , QkI

be a partition of Q . Then P is called an admissible partition of Q if the following holds: Let a E X, then Vi, 3j, 1 <_ i, j <_ k such that VPl, P2 E QZ, if ft(P,, a, r) > 0 for some r E Q, then 3 t E Q such that ft(P2, a, t) > ft(Pl, a, r) and t, r E Qj .

Proposition 6 .14 .3 Let M = (Q, X, ft) be a ffsm.

(1) Let 1Q = {{q} I q E Q} . Then 1Q is an admissible partition of Q . (2) {Q} is an admissible partition of Q . m

Theorem 6.14 .4 Let M = (Q, X, ft) be a ffsm and P = {Q1, Q2, . . . , Qk}

be a partition of Q . The following are equivalent: (1) P is an admissible partition of Q . (2) Let x E X* . Then Vi, 3j, 1 <_ i, j < k such that VP1,P2 E QZ, if y* (pl, x, r) > 0 for some r E Q, then 3 t E Q such that y* (P2, x, t) >_ y*(PI,x,r) and t, r E Qj .

Proof. (1)x(2) : Let x E X* and Ixj = n. Let pl, p2 E QZ and y * (pl, x, r) > 0 for some r E Q. If n = 0, then x = A and ft*(pl, x, r) > 0 implies that pl = r. Thus [t* (P2, x, P2) = 1 = ft*(pl, x, pl) . In this case i = j. Hence the result is true for n = 0. Suppose the result is true for all y E X* such that I y I = n -1, where n > 0. Let x = ya, where a E X. Now y* (p l , x, r) = * * ft (pl, ya, r) = V{ft* (pl, y, s) A ft (s, a, r) I s E Q} > 0. Since Q is finite, * there exists t E Q such that ft (PI , y, t) A ft * (t, a, r) = ft* (p l , ya, r) . Thus * * ft (pl, y, t) > 0 and y,(t, a, r) = ft (t, a, r) > 0 . By the induction hypothesis, there exists j and there exists s E Q such that ft * (P2, y, s) >_ y* (PI, y, t) and s, t E Qj . Now s, t E Qj and ft(t, a, r) > 0 . Hence by (1), there exists l and there exists q E Q such that y (s, a, q) >_ y(t, a, r) and r, q E Qi . Now /t* (P2, x, q) = ft* (P2, ya, q) >_ ft * (P2, y, s) n ft(s, a, q) > ft * (Pl, y, t) n ft (t, a, r) = y* (pl, ya, r) = y* (pl, x, r) and q, r E Ql . The result now follows by induction .

m

Corollary 6.14.5 Let M = (Q, X, ft) be a ffsm. Then every admissible partition P of Q induces an admissible relation - on Q such that the set of all equivalence classes of - is P. Conversely, the set of all equivalence classes of an admissible relation on Q is an admissible partition of Q. 0 Lemma 6.14 .6 Let M = (Q, X, ft) be a ffsm and

7r = {HZ i E 11 be an admissible partition of Q . Let i, j E I. Then Vq, q' E HZ, Va E X,

V{ft(q,a,r)

© 2002 by Chapman & Hall/CRC

I

r E Hj } = V{ft(q ,a,r)

I

r E Hj }.

29 8

6. Algebraic Fuzzy Automata Theory

Proof. Let q, q' E HZ , a E X, A = {ft(q, a, r) I r E Hj}, and B = ft(q, a, r) I r E Hj }. Suppose ft(q, a, r) > 0 for some r E Hj . Since 7r is an admissible partition, there exists r' E Q such that N(q', a, r') >_ ft(q, a, r) . Again by the admissibility of 7r, r' E Hj . Similarly if ft (q, a, p) > 0 for some p E Hj,then there exists p' E Hj such that ft(q,a,p') > N(q',a,p) . Also, by the admissibility of 7r, it follows that ft(q, a, r) = 0 b'r E Hj if and only if N(q', a, r) = 0 b'r E Hj . Hence V{ft(q,a,r) I r E Hj }=V{ft(q,a,r) I r E Hj } . Theorem 6.14 .7 Let M = (Q, X, /t) be a ffsm . Let it = an admissible partition of Q . Define

{HZ I i

E 11 be

y" : 7r X X x 7r ~ [0,1] by

ft' (Hi,a,Hj) = V{ft(q,a,r) I r E Hj I VHZ , Hj E 7r and a E X, where q E HZ . Then M17r = (7r, X, ft") is a ffsm, called the quotient fuzzy finite state machine with respect to 7r .

Proof. By Lemma 6 .14 .6, ft" is well defined . m Proposition 6 .14 .8 Let M = (Q, X, ft) be a ffsm. Let 7r = be an admissible partition of Q. Then for q E HZ ft'*

{HZ I i

E 11

(Hz, x, Hj) < V{ft * (q, x, r) I r E H;

VHZ , Hj E 7r and x E X* .

Proof. Let HZ, Hj E 7r and x E X*. Let Ixj = n . If n = 0, then x = A. If HZ = Hj then y"* (HZ , x, Hj ) = 1 and V{y* (q, x, r) I r E Hj } = V{ft*(q,x,r) I r E HZ} = y*(q,x,q) = 1, where q E HZ . If HZ Hj, then ft'* (Hi, x, Hj) = 0 and HZ n Hj = 0 . Since HZ n Hj = Ql and q E HZ, V{/t*(q,x,r) I r E Hj I =0. Hence ft'* (Hi,x,Hj) = V{/t*(q, x, r) I r E Hj } . Suppose that the result is true b'y E X* such that I yI = n - 1, n > 0. Let n>0andx=ya,whereyEX*,aEX,andjyl=n-1 .NowforgEHi © 2002 by Chapman & Hall/CRC

6.14. Admissible Partitions

299

ands E Hk, [t"*(HZ x H;)

= = = <

p,"*(HZ,ya,Hj) V{ft"* (HZ, y, Hk) A ft"* (Hk a Hj) I Hk E 7rf V{(V{ft*(q,y, r) I r E Hkf) A (V{ft*(s,a,p) pEHjf)IHkE7f V{V {ft* (q, y, r) n w* (s, a, p) I r E Hk , p E H; f I Hk E 7r f VfV{ft* (q, y, r) n ,a* (r, a, p) I r E Hk , p E H; f I Hk E 7r f Vf V{ft* (q, y, r) A tt* (r, a, p) I r E Hk, Hk E 7rf

IpEH;f V{V {ft* (q, y, r) A w* (r, a, p) I r E Qf I p E H; f V{ft* (q, ya, p) I p E H;f V{ft* (q, x, p) I p E H; f . m

Proposition 6.14.9 Let M = (Q, X, ft) be a ffsm . Let 7r = fH2 I i E If be an admissible partition of Q. Then for all x E X* ;), ft * (q, x, p) < ft"* (Hi , x, H where gEHZ,pEHj . Proof. Let q E HZ , p E Hj . Let x E X* and Ixj = n. If n = 0,

then x = A. If q = p, then HZ = Hj and p* (q, x, p) = 1 = ft" * (HZ, x, HZ) . Suppose q :?~ p. Then p* (q, x, p) = 0 <_ ft"* (Hi, x, Hj) . Suppose n = 1. Then x= aEXand

ft * (q, a, p)

= < = =

ft (q, a, p) V{ft(q,a,r) I r E H;f ft' (Hi, a, Hj) ft"* (HZ a Hj) .

Hence the result is true for n = 0 and n = 1. Suppose that the result is true b'y E X* such that Iyj = n - 1, n > 0. Let n > 0 and x = ya, where y E X*, a E X, and Iyj =n-1 . Then ft * (q, x, p)

= = = <_ = =

ft* (q, ya, p) V{[t* (q, y, r) A ft(r, a, p) I r E Q} VfV{/t* (q, y, r) A y,(r, a, p) I r E Hkf Hk E 7rf Vfv{ft"*(Hi, y,Hk) Aft"(Hk,a,H;) r E Hk f Hk E 7f (by induction and n = 1 case) V{ft"*(HZ,y,Hk) A ft"(Hk ,a,H;) I Hk E 7rf ft"* (HZ x Hj) .

The result now follows by induction . m

© 2002 by Chapman & Hall/CRC

30 0

6. Algebraic Fuzzy Automata Theory

{HZ I i E 11 be an admissible partition of Q . Then for all x E X*, for all HZ , Hj E 7r,

Corollary 6.14 .10 Let M = (Q, X, p) be a ffsm . Let 7r = (Hi ,x,Hj V{ft*(q,x,p) I p E Hj I < ft'* ), where q E

HZ . 0

Theorem 6.14 .11 Let M = (Q, X, p) be a ffsm. Let 7r = Hi I i E 11 be an admissible partition of Q . Then for q E Hi ft"*

(Hz, x, Hj) = V{ft* (q, x, r) I r E Hj

VHZ ,Hj E7randxEX* .m

Definition 6 .14 .12 Let M = (Q, X, /t) be a ffsm. Let 7r and T be admissible partitions of Q. 7r and T are called /t-orthogonal if (1) 7r n T = 1Q, and

(2) VHZ, H E 7r, VK3 , K, E T, b'a E X, if HZr1Kj = {qo} and H r1K, _ {po}, then ft (go,a,po) = V{ft(go,a,p) Aft(go,a,p) I p E H., p' E K, 1 .

Theorem 6 .14 .13 Let M = (Q, X, ft) be a ffsm. Let 7r and T be admissible partitions of Q . Then 7r and T are ft-orthogonal if and only if (1) 7r n T = 1Q, and (2) VHZ,H E 7r, VK3 ,K, E H rl K, = {po}, then

T,

V'x E X*, if

HZ

rl K. = {qo} and

ft * (go,x,po) =V{ft* (qo, x, p) nft*(go,x,p) I p E H., p' E K, }.

Proof. Suppose 7r and T are called ft-orthogonal. Then clearly (1) holds. (2) Let HZ, H. E 7r, Kj, K, E T, x E X*, HZr1Kj = {qo}, and H r1K, _ {po} . Suppose Ixj = n. If n = 0, then x = A. Now ft*(go ,x,p o ) = 0 if po . Then either qo 7~ po and /r* (qo, x, po) = 1 if qo = po . Suppose qo H and qo ~ K, HZ rl H. = Ql or K, rl K, = 0, say, HZ n H = 0 . Thus qo Hence V{/t*(go,x,p) oft* (go,x,p~) I P E H.,1~ E K, } = V{0A/t*(go,x,p') I p E H., p' E K, } = 0. Suppose qo = po . Then Hi = H and Kj = K, Thus V{/t*(go,x,p) n ft*(go,x,p~) I P E H.,p' E Kv } = ft * (go,x,po) n * ft * (qo, x, po) = 1 . Hence if n = 0, then ft (qo, x, po) = V{ft* (qo, x, p) A y*(go,x,p') I p E H.,p' E Kv}. Suppose that the result is true b'y E X* suchthat I y l=n-1,n>O.Letn>Oandx=ya,whereyEX*,aEX, © 2002 by Chapman & Hall/CRC

6.14. Admissible Partitions

301

and I y I = n - 1 . Then ft * (qo, x, po)

ft * (qo, ya, po) V {ft*(qo, y, r) n ft(r, a, po) I r E Q} = V{w*(qo, y, r) A (V{ft(r, a, p) A ft(r, a,p) I p E H~, p'EKv}) I rEQ} = V{(V(ft*(go,y,r) Aft(r,a,p))) A (V(ft*(go,y,r)A ft(r, a, p))) I p E H., p' E K, r E Q} = V {(V {ft* (qo, y, r) A w(r, a,p) I r E Q}) A (V{w*(go,y,r) Aft(r,a,p') I r E Q}) I p E H.,p' E Kv} V{ft* (go,ya,p) nft * (go,ya,p') p E H., p' E K, } = V{ft* (go,x,p) nft* (go,x,p') I p E H., p' E K, }. =

The result now follows by induction . The converse is trivial . Let M = (Q, X, ft) be a ffsm. Let 7r and T be admissible partitions of Q. Consider the ffsms M/7r = (7r, X, ft") and MIT = (T, X, ft') . Define It ^ : (7r x T) x X x (7r x T) ----> [0,1] by It A ((HZ, Kj), a, (H., K,)) = ft' (Hi, a, H.) n ft' (Kj, a, VHZ, H. E 7r, Kj , K, E T, and a E X. Then a fuzzy finite state machine. Note that VHZ ,H

E

7r,Kj ,K,

M17r n MIT = (7r x T, X,

(H., Kv)) =,t"*(Hz, X, H.)

It ^*((H,,Kj), X, E

T,

Kv)

[r^) is

Aft' * (Kj,x, Kv)

and x E X* .

Theorem 6.14 .14 Let M = (Q, X, ft) be a f'sm . Let 7r and T be admissible partitions of Q that are ft-orthogonal. Then M < M17r n MIT. Proof. Define 97 : 7r xT ----> Q by 97((HZ, Kj)) = qo, where Hi nKj = {qo} . Since 7r and T are ft-orthogonal, 97 is one-to-one. Let C be the identity map on X. Let HZ, H. E 7r, Kj , K, E T, and x E X* . Suppose HZ n Ki = {qo} and H n K, = {po } . Then w* (97 ((Hi, Kj)), x, 97((H., K,))) = w* (qo, x,po) .

Also, It ^ * ((HZ, K;), x,

(H.,

Kv))

n ft' (Kjx Kv) = (V{ft* (qo, x, p) p E H })n (V{ft* (qo, x, p) p' E K, }) V{ft * (go,x,p)nft * (go,x,p') I pEH., p' EKvj = ft*(go,x,po), =

ft"* (HZ a

where the last inequality holds since [t* (97((H,,

7r

and

T

H.)

are ft-orthogonal . Thus

Kj)), x, 97((H., K,))) = w^ * ((HZ,

© 2002 by Chapman & Hall/CRC

K;), x, (H., K,)) .

30 2

6. Algebraic Fuzzy Automata Theory

Now ItA*((H,,Kj),X, (H., K')) =V{Nn*((H,,Kj),x, (H" K')) 197((H"KS)) = r7((H., K,)), (H,, Ks) E 7rxTj since 97 is one-to-one. Hence /t*(97((Hi, Kj)), x, 97((H.,Kv))) = V{ft^*((Hi, Kj), X, (HT, Ks)) 197((H" Ks)) = 97((H., K,)), (H,, Ks) E 7r x T} . Consequently, M < M17r n MIT . 0

Theorem 6.14 .15 Let M = (Q, X, /t) be a ffsm . Let 7r be an admissible partition of Q. If there exists a partition T of Q such that 7r and T are ft-orthogonal, then there exists a ffsm N such that M < NwM17r . Proof. Let

7r = {Hj}jEI and T = {Kj}jEj be ft-orthogonal partitions Let N of Q. = (T, 7r x X, ft'), where

ft'(Kj, (Hi, a), Kv) = V{ft(go, a, p) I p E K, }, and {qo} = Hi n Kj . Since T is admissible, ft' is well defined. Define w : 7r x X 7r x X to be the identity map. Define 97 : T x 7r ----> Q by r7((Kj, Hi)) = qo, where {qo} = Hi n Kj . Then 97 is one-to-one and onto. Let C be the identity map on X. Then for {po} = H n K, l~(n7((K7 Hi)), a, n7((Kv, Hv)))

_ ft (qo, a, po) V{ft(go, a, p) n ft (go, a, p) (p' ,p) E Kv x Hv } (V{ft(go,a,p') I p' E K, })n (V{ft(go, a,p) I p E H.}) ft'(Kj, w(Hi, a), K,) n ft"(Hi, a, H.) pW((Kj, Hi), a, (Kv, H.)).

Thus 97 N'* (97((Kj, Hi)) , x, ((K,, H.))) = ft'* (( Kj , Hi) , x, (Kv, H.))

(6.3)

for x E X* such that IxI = 1. Suppose that (6.3) is true if IxI = n-1, n > 0, where x E X* . Now p,*(97((Kj,Hi)),xa,97((Kv,H.))) = V{(97((Kj, Hi)), x, 97 ((K, H))) A ft (97((K, H)), a, 97 ((K,, H.))) I 97 (K, H) E 97 (Tx7r)I (since 97

is onto) =V{ft-*((Kj, Hi), x, (K, H))Apw((K, H), a, (K, H.)) I (K, H) E (T x 7r)} = ttw*((Kj,Hj), xa, (Kv, H.)) . Now ,t*(?7((Kj, Hi)), A, 97((Kv, H.))) = 1 if and only if 97((Kj, Hi)) = r7((K, H.)) if and only if (Kj , Hi) _ (K, H ) (since 97 is one-to-one) if and only if f rW * ((Kj , Hi), A, (K, H")) _ 1. From this, it follows that (6 .3) holds for x = A. Hence (6 .3) holds b'x E X* . Thus by induction, (r7, C) is a covering of M by NwMl7r . m

Definition 6.14 .16 Let Q be a nonempty set and 7r and T be partitions of Q. Then 7r < T if VA E 7r, there exists B E T such that A C B. The proof of the following lemma is straightforward.

Lemma 6.14 .17 Let Q be a finite nonempty set. Let

7r = {Hi}Z i and T = {Kj},T=1 be partitions of Q such that 7r <_ T . Then Vj, 1 <_ j <_ m, Kj = Hjl U Hj2 U . . . U Hj,,,, for some Hj l , Hj . . . . . . Hj,, j E 7r and m <_ n . If m = n, then 7r = T . m

© 2002 by Chapman & Hall/CRC

6.14. Admissible Partitions

303

Definition 6.14 .18 Let M = (Q, X, /t) be a ffsm. Let 7r be an admissible partition of Q. Then 7r is called maximal if 7r is nontrivial and if T is any admissible partition of Q with 7r < T < {Q}, then either T = 7r or T = {Q} . Definition 6 .14 .19 Let M

= (Q, X, /r) be a ffsm. Then M is called irreducible if IQI > 1 and 1Q and {Q} are the only admissible partitions of Q.

Theorem 6.14 .20 Let

M = (Q, X, y) be a ffsm . Let 7r_ {HZ}Z 1 be an admissible partition of Q . Then 7r is maximal if and only if M17r is irreducible .

Proof. Suppose 7r is maximal . Now M/7r = (7r, X, ft") where ft" (HZ, a, Hj) = V{ft(q, a, r) I r E Hj }VHZ, Hj E 7r and a E X, where q E HZ. Since

is maximal, 7r {Q} . Thus I7rl > 1 . Let 7r be an admissible partition 7r . Suppose 7r 1" . Then there exists T C_ 7r such that T E 7r and ITI > 1 . Suppose that T z/4 7r . Without loss of generality, we may assume that T = {Hl, . , H, }, where 1 < m < n. Let 7r' = {Hl U . . . UH,, H,+1 U . . . U H }. Then 7r < 7r' and 7r' is a partition of Q . We now show that 7r' is admissible. It suffices to consider q, p E Hl U . . . U Hm with q E Hl and p E H2 . Suppose ft (p, a, r) > 0 where r E HZ . Then ft" (H2, a, HZ) = V{ft(p, a, r') I r' E HZ} > 0 . Since 7r is admissible, there exists K E 7r such that ft" (Hl, a, K) > ft" (H2, a, HZ) and K and HZ belong to the same element of 7r . Hence V{ft(q, a, t') I t' E K} >_ V{ft(p, a, r') I r' E HZ}. This implies that there exists t E K such that ft(q, a, t) >_ ft(p, a, r) > 0. Now HZ E T if and only if K E T since K and HZ belong to the same element of 7r . If K, HZ E T, then t, r E Hl U . . . U H, and if K, HZ ~ T, then t, r E Hm+1 U . . . U Hn , i.e ., t and r belong to the same element of 7r' . Hence 7r' is admissible. Since 7r is maximal, it follows that 7r' = {Q} and so T = {Q} . This implies that 7r = {7r} . Thus M17r is irreducible . Conversely, suppose that M17r is irreducible . Let T be an admissible partition of Q such that 7r <_ T <_ {Q} . Suppose that 7r :?~ T . By Lemma 6.14.17, there is no loss of generality in assuming that T = {Hl U . . . U Hm, Hm+1, . . . , Hn }, 1 < m < n. Since T :?~ 7r, 7r

of

T=. {{Hl, Hm }, {Hm+1}, . . . , {Hn }f 7~ 1" .

We now show that T is admissible. It suffices to consider Hl, H2 . Suppose y"(H j , a, HZ ) > 0 for some HZ E 7r . Then V{ft(q, a, r') I r' E HZ } > 0 where q E Hl . Let r E HZ be such that ft(q, a, r) = V{ft(q, a, r') I r' E HZ } > 0 . Since T is admissible, Vp E H2, there exists tp E Q such that ft(p, a, tp) >_ ft(q, a, r) and r, tp are in the same element of T Vp E H2 . Now if HZ ~ {Hl, . , Hm}, then r, tp E HZ Vp E H2 and ft" (H2, a, HZ) > ft (p, a, tP) > ft (q, a, r) = ft"(Hl, a, HZ) . Suppose HZ E {Hl, . . . , Hm} . Then r, tp E Hl U . . . U Hm for all p E H2 . Let ft (p, a, tp,) = V{ft(p, a, tp) I p E H2} . Since tp, E Hl U . . . U Hm , tp, E Hk for some k, 1 < k < m. © 2002 by Chapman & Hall/CRC

30 4

6. Algebraic Fuzzy Automata Theory

Hence /t" (H2 , a, Hk) = V{ft(q, a, r')~ r' E HkI > ft (p, a, tp,) > ft (q, a, r) = ft" (H1, a, HZ ) and Hk, HZ E {H1, . . . , H, } . Consequently T is an admissible partition of 7r . Since M17r is irreducible, T = {7r} and so T = {Q}. Hence 7r is maximal . m Theorem 6 .14 .21 Let M = (Q, X, p) be a ffsm and IQ I = n > 2 . Then M < N1w1N2w2 . . .w ._1N., where N1 , N2 , . . . , N, are irreducible ffsms and the state sets Qj of Ni are such that I Qj I < n.

Proof. Since Qj >_ 2, we can choose a maximal admissible partition 7r of Q. Clearly I7rI < IQI . By Theorem 6.14.15, there exists a ffsm N such that M <_ NwMl7r for some suitable w . Since 7r is maximal, M17r is irreducible . Also 7r is the state set of M17r and I7rI < I QI . If R is the state set of N, then IRI < Qj by the construction of N. We can now apply Theorem 6.14.15 to N, to obtain N <_ NV N17' such that N17r' is irreducible and the number of states in N' is less than the number of states in N. Thus, we have M < N'wN/7r wM/7r .

Continue this process and apply Theorem 6 .14.15 to N'. Since Qj is finite, this process must stop after a finite number of steps . Hence M <_ N1w1N2w2 . . . W.-IN., where N1, N2 , . . . , N, are irreducible ffsms and the state sets Qj of Ni are such that I Qj I < n.

6 .15

Coverings of Products of Fuzzy Finite State Machines

We are now interested in the notion of a covering that is more in line with the crisp notion. The next three sections are based on [119] . We present several ways of constructing products of fuzzy finite state machines and their relationship through isomorphisms and coverings . We prove that the wreath product of two fuzzy finite state machines covers their cascade product . Similar relationships hold for the direct sum and sum of two fuzzy finite state machines . A distributive property of the cascade product over the sum of fuzzy finite state machines in relation to coverings of fuzzy finite state machines is established . Let M1 = (Q1, X1, fq) and M2 = (Q 2 , X2 , /r 2 ) be fuzzy finite state machines . Let cx : Q1 ~ Q2 and,3 : X1 ~ X2 be functions . The homomorphism (a,,3) : M1 ~ M2 (Definition 6.3 .1) is called a monomorphism (epimorphism, isomorphism), if both the functions a,,3 are injective (surjective, bijective, respectively) . If (a,,3) is an isomorphism, we write M1 -- M2 . © 2002 by Chapman & Hall/CRC

6.15. Coverings of Products of Fuzzy Finite State Machines

305

Definition 6.15 .1 Let Ml = (Q1, X1, fq) and M2 = (Q 2 , X2, ft2) be fuzzy finite state machines . Let 97 : Q2 ~ Ql be a surjective partial function and ~ : Xl ~ X2 be a function. Then the pair (rl, ~) is called a strong covering of Ml by M2 , written Ml ~ M2, if ( 97(p~), x, 97(q~)) < ft' (1~, ~+ (x), q~) for all x E Xl and p', q' belong to the domain of 77, where + : Xl X2 is defined by ~ + (xlx2 . . . xn) = ~( xl)~(x2) . . . S(xn) for x l, X2 . . . . , x n E X1-

Clearly, the strong covering relation is reflexive and transitive, but not symmetric. Definition 6.15 .2 Let M = (Q, X, ft) be a fuzzy finite state machine. A fuzzy finite state machine MS = (Q s , Xs, fps) is called a subfuzzy finite state machine of M, written MS C_ M, if (1) QS C Q, XS C X, and (2) fts =ftJQ .Xx.XN.. We point out that a subfuzzy finite state machine of M and a submachine of M (Definition 6.7.7) differ in that for the latter, the set of input symbols remains X. Identifying X with the diagonal elements of X x X, the restricted direct product of M and M' may be considered as the subfuzzy finite state machine of the direct product of M and M' . The restricted direct product of M and M' is a special case of their cascade product, where X = X' and w : Q' x X ~ X is a projection mapping . We now introduce two more ways of connecting fuzzy finite state machines . Definition 6.15 .3 Let M = (Q, X, ft) and M' = (Q', X', ft') be fuzzy finite

state machines such that Q n Q' = 01 and X n X' = 0. Then the fuzzy finite state machine M() M' _ (Q U Q', X U X', ft () ft') is called the direct sum of M and M', where ft ft' : (Q U Q') x ( X U X') x (Q U Q') ~ [0,1] is defined as follows:

(ft () f') (p, x, q) =

ft(p, x, q) N'' (p, x, q) 1 0

if p, q E Q and x E X if p, q E Q' and x E X' if either (p, x) E Q x X and q E Q' or (p, x)EQ'xX'andgEQ otherwise.

Definition 6.15 .4 Let M = (Q, X, ft) and M' = (Q', X', ft') be fuzzy finite

state machines such that Q n Q' = 01 and X n X' = 0. Then the fuzzy finite state machine M + M' = (Q U Q', X U X', ft + ft') is called the sum of M © 2002 by Chapman & Hall/CRC

30 6

6. Algebraic Fuzzy Automata Theory

and M', where p + /t' : (Q U Q') x ( X U X') x (Q U Q') ----> [0,1] is defined as follows: (/t + /t') (p, x,

6 .16

q) _

ft(p, x, q) /t' (p, x, q) 0

if p, q E Q and x E X if p, q E Q' and x E X' otherwise.

Associative Properties of Products

The direct and restricted direct products of fuzzy finite state machines are associative. As proved in the following theorem, the wreath product, sum, and cascade products of fuzzy finite state machines are also associative . However, it can easily be shown that the direct sum of fuzzy finite state machines is not associative . Theorem 6.16 .1 Let M, M', and M" be fuzzy finite state machines. Then

the following properties hold: (1) (M0M')0M"- M0(M'0M") (2) (M+M~)+M"-M+(M' +M") and (Mw1M')w2M" - Mw3(M'w4M"), where (3) W3 and W4 are defined naturally in terms of wl and w2 .

Proof. Let M = (Q, X, /t,), M' = (Q', X', /t,'), and M" = (Q", X", W") . (1) Then (MoM')oM"= ((QxQ') xQ",(XQ,

and M o (M' o M") = (Q x (Q' x

xX,)Q"

x X", (it oit')oN," )

X,Q11 Qii), (XQ x Q11) It//)) . x x X", /t o (N,' o I

Let cx : (Q x Q') x Q" - Q x (Q' x Q") be the natural map . Let pl XQ , x X' , XQ' and p2 : XQ' x X' -, X' be the natural projection mappings . Given XQ ' x X', let fi = pl o f and f2 = P2 o f . _ a function f : Q" ~_ Q" Q' x Define : X by ft fi ((q',q")) = fi(q")(q') . Define 3 : (XQ' x X,)Q x X , XQ , XQ x (X,Q x X by x")) = (TI, x 1 l)) . Now )

l3((f,

(f2,

13((fIx")) =,3((9,y")) _(T (f2, x")) = (9i, (92, y")) ft = 9i, f2 = 92, and x" = y" .

Thus fi (q") (q') = gl (q") (q'), f2 = g2, and x" = y" . This implies that fi = 91, f2 = 92, and x" = y", which implies that pl o f = PI 0 9, P2 o f = P2 0 9, and x" = y" . Therefore, f = g and x" = y" and so (f, x) = (g, y) . Thus /3 is injective.

© 2002 by Chapman & Hall/CRC

6.16. Associative Properties of Products

307

XQ"Q" Let (g, (h, x")) E x (X'Q" x X") . Define f : Q" ----> XQ' )) X' by f (q") = (gq , h(q")), where gq (q') = g(q', q") . Then,3((f, x (g, (h, x")) . Thus 3 is surjective.

x

It follows that (a,,3) is a required isomorphism. (2) Let cx and,3 be identity mappings on Q U Q' U Q" and X U X' U X", respectively. (3) Consider

and

wwIft

/

Mw 1 M' =

(Q x Q', X',

pwIft'),

where w 1

((p,p ), x', (q, q)) = ft (p, w1 (p, x), q) n ft (p, x', q)

and (Mw1M')w2M" X" ----> X' and

=

((Q x Q') x Q", X",

(ftWIft')w2ft"(((P,P'),P"),x ,((q,q'),q"))

(ftw1ft')w2ft"),

= =

It follows immediately that

M'w4M" =

=

w1 (P'

(Q' x Q", X",

w2 (P" x"))

It 'w4ft")

and

q1) A ft // (P // , X // , q11) .

(Q x (Q' x Q"), X", gw3(g,'w4g,")) and

Ntw3(ft'w4ft")((P,(p',p")),x",(q,(q',q")))

© 2002 by Chapman & Hall/CRC

w2

ft(P,w1(P',w2(P",x")),q) Aft' (P', w2 (P", x"), q) Aft// (P'/x//q11) .

It'w4N-"((p',p"),xii (q', q " )) =ft'(PI,w4(P",x"),

Mw3(M'w4M") =

where

ftWIft'((P,P'),w2(P",x"), (P " , x " , q") (q, q')) n N'"

Define w3 : (Q' x Q") x X" - X by w3((P',P"),x11) and set w4 = w2 .

Moreover,

'xX'----> X

=

ft(P,w3((P',P"),x ),q)A (ft'w4ft"((p', p"), x~~ (q', q"))) ft (P, w3((P',P"), x"), q) A (It'(P', w4(P", x"), q') Aft// (pii x// q11)) .

30 8

6. Algebraic Fuzzy Automata Theory

Let cx : (Q x Q') x Q" - Q x (Q' x Q") be the natural mapping and 3 be the identity mapping on X" . Then (f~wif~')w2f~")(((p,p'),p"),~ ,((q,q'),q"))

Thus (Mw1M')w2M" - Mw3(M'w4M") . 6 .17

=

ft(p,wi(p',w2(p",x")), q) Aft/(p',w2(p",x q') A ft"(p", X", q") ft (p, wi (p~, w2 (p", X // )), q) n (W (p, w2 (p", X q1) n ftii(pii, xii, q11)) ft (p, w3((p~,p"), x"), q) n(ft /(p/ , w4(p// , x// ), q1) Aft// (p ii X // q11) ftw3(ft'w4ft")((p, (p~, er p )), x", (q, (q~, q"))) ftw3 (ft'w4ft") (gy(((p, p') , per)),,3(x"), o(((q, q~),

0

Covering Properties of Products

The following theorem is a direct consequence of the definition of the direct sum and sum of two fuzzy finite state machines . Theorem 6.17 .1 Let M and M' be fuzzy finite state machines. following properties hold: (1)M~M+M', (2)M'~M+M', (3)M~M+M', (4)M'~M+M' .

Then the

Proof. We only prove (1) . Let 97 : Q U Q' ~ Q be a partial surjective function defined by n(q) = q for all q E Q and let ~ : X ~ X U X' be the inclusion function . Clearly, (r7, ~) is a required strong covering of M by M+M' .

Theorem 6.17 .2 Let M = (Q, X, ft) and M' = (Q', X', ft') be fuzzy finite state machines. Then the following properties hold: (1) MWM' ~ M o M', (2) M+M' ~ M+M' .

Proof. (1) Let w x, : Q' ~ X be the function defined by wx, (p') = w(p', x') for all p' E Q' and x' E X' . Define ~ : X' ~ XQ' x X' by ~(x') = (wx,, x') and let 97 be the identity map on Q x Q'. © 2002 by Chapman & Hall/CRC

6.17. Covering Properties of Products

309

(2) Let 97 and ~ be the identity mappings on Q U Q' and X U X', respectively. If p, q E Q and x E X, then (/t+ft~)(97(p), x, 97(q)) _ (ft (@ft~)(p, (x), q) = ft (p, x, q) . If p, q E Q' and x E X', then (/t+/t') (97(p), x, 97(q)) _ (ftoft') (p, (x), q) = ft'(p, x, q) . If (p, x) E Q x X and q E Q' or (p, x) E Q' x X' and q E Q, then (ft + ft') (97(p), x, 97(q)) = 0 < 1 = (ft () ft') (p, ~(x), q) . In all other cases, the values of ft + ft' and ft ft' are 0. Theorem 6.17 .3 Let M = (Q, X, p), M' = (Q', X', p'), and M" = (Q", X", ft") be fuzzy finite state machines such that M ~_ M'. Then the following assertions hold: (1)MXM"~M'XM" ; (2)M"xM~M"xM' ; (3) Given wl : Q" x X" ~ X, there exists co t : Q" x X" ~ X' such that Mw1M" _< M'w2M" ; (4) If (97, ) is a covering of M by M' and is surjective, then for all X", there exists co t : Q' x X' ~ X" such that M"w1M wl : Q x ;X M W2M (5) M0M"~M'0M" ; (6) M + M11 MI + Mii ; (7) M// +M M"+M' ; (8) M+M" M'+M"; (9) M" + M M" o M/ . Moreover, if X = X' = X", then (10) M AM" M' AM"; (11) M" A M ~ M" A MI . Proof. Since M --< _ M', there exists a partial surjective function 97 Q' ----> Q and a function ~ : X ----> X' such that ft+(?7(p), x, r7(p')) <_ y,'+(p,~(x),p'), where x E X+ . (1) Define 97' : Q' x Q" ----> Q x Q" by 97i (p~,p") = (97(p~),p") and ' X x X' ----> X' x X" by ~'(x, x") _ ( (x), x") . Clearly (97', ') is the required covering . (2) The proof is similar to that of (1) . (3) Given wl : Q" x X" X, set W2 = o wl and let ' be the identity map on X". (4) Given w l : Q x X X", let cot : Q' x X' ----> X" be such that Since ~ is surjective and X is finite, such W2 w2(p',~(x)) = wi(?7(p'),x) . exists . Clearly, W2 is not unique . Define 97' : Q" x Q' - Q" x Q by ?Ap",p~) = (p", 97(p)) and set ~' = ~ . 97' : Q' x Q" ~ Q x Q" by 97i (p~,p") = (97(p~),p") and ' XQ~~(5)x Define X ----> (XI)Q" x X 11 by ~/(f, x") = (~ o f I x") .

© 2002 by Chapman & Hall/CRC

310

6. Algebraic Fuzzy Automata Theory (6) Recall that M + M" = (Q U Q", X U X", ft + ft"), where

(ft + ft") (p, x, q) =

ft(p, x, q) ft" (p, x, q) 0

if p, q E Q and x E X Q" and x E X" if p, q E

otherwise

and M' + M" = (Q' U Q", X' U X", p' + p"), where (ft' + ft ") (p, x, q) _

ff' (p, x, q) /t" (p, x, q) 0

if p, q E Q' and x E X' if p, q E Q" and x E X"

otherwise.

Define 9' :Q'UQ"----> QUQ"by i7/ (p~)

if p'EQ'

otherwise

and ~' :XUX"~X'UX"by (x) -

(x)

x

otherwise .

Since 9 is a partial surjective function, so is 9' . we now note that (,~+w")+(~'(p'), x, ~'(q')) <_ (w'+,t )+(p~, (~')+(x), q') . If p', q' E Q' and x E X or p', q' E Q" and x E X", then clearly (ft + ft") +(n'(p), x, 97(q)) <_ (ft' + ft") +(p , (C')+(x), q) . In all other cases, (ft + ft") +(i7'(p'), x, 9'(q')) = 0 . The proofs of (7), (8), and (9) are now obvious. (10) Define 97' : Q' X Q" - Q X Q" by 97'(P/' p") = (97(p'), p") and set ' = ~ the identity map on X. Then (9', ~') is the required strong covering . (11) Follows easily. It follows that M" o M', in general, does not strongly cover M" o M, even though M --< M' . Hence we introduce a weaker notion of covering . Definition 6.17 .4 Let M = (Q, X, ft) and M' = (Q', X', ft') be fuzzy finite

state machines . Let 9 : Q ~ Q' be a partial surjective function and ~ : X ~ X' be a partial function. The ordered pair (9, ~) is called a weak coveving of M by M', urritten M --<w M, if

ft(97(p), x, 97(q)) << ft (p, ~(x), q) for all p', q' in the domain of 9 and x in the domain of ~.

A weak covering differs from a strong covering only in that ~ in Definition 6.17.4 is a partial function, while ~ in Definition 6.15.1 is a function. Thus every strong covering is a weak covering . © 2002 by Chapman & Hall/CRC

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311

Theorem 6.17 .5 Let M, M', and M" be fuzzy finite state machines and MOM" . Then M"oM~,M"0 M' . Proof. Since M --< _ M', there exists a partial surjective function 97

Q' ----> Q and a function ~ : X ----> X' such that /t+(?7(p'), x, ?7(q')) <_ It '+ (p', ~+ (x), q') . We have M" o M = (Q" X Q, (X")Q X X, /t" o N-) and M//OM' = (Q" X Q', (X")Q' X X', /t" o N') . Define 97' : Q" X Q' ----> Qii X pi) = (p" ,97(p)) and ' : (X")Q X X ----> (X")Q' X X' Q by 97'(p"

by ~'(f, x) = (f o 97, ~(x)) . Clearly, 97' is a partial surjective function and ~' is a partial function. Now (ft" 0 tt)(97(p",p'), (f, x), r7'(q", q')) = (ft" o ft)((p", 9J(p')), (f, x), (q",?J(q'))) = ft"(p", f (9J(p')), q") n ft(9J(p'), x, 9J(q')) . However, since [t+ (97(p), x, r7(q')) < ft'+(p', ~+(x), q'), it follows that (w" o ft)+(97(p",

p'), (f, x), 97'(q", q'))

<_

(w")+(p", f (?7(p')), q")n ft/+ (1 ~, ~+ (x), q~) ii 0 ii, (ft Iti)+0 pi), (f o 97, ~+ (x)), (q", q~)) ii 0 (ft It') +o", pi), ~I+ (f, x), W1 ,gl ))

The following theorems are easy consequences of the transitive property of strong coverings of fuzzy finite state machines and Theorem 6.17.3. Theorem 6.17 .6 Let M, M', and M" be fuzzy finite state machines and M ~_ M" . Then (1)MAM"~M'AM", (2) M" n M M" n M', (3) MWM" M' o M", M" o M', (4+) M"WM (5) M + M" M' o M", (6) M11 +M M"+M' . We next prove a distributive property for strong coverings . Theorem 6.17 .7 Let M, M', and M" be fuzzy finite state machines and M ~ M" . Then MO (M/ +M//) ~ Mom'+Mom". Proof. Let M = (Q, X, /t,), M' = (Q', X', W), and M" _ (Q", X", /t,"). Define 97 : (QXQ')U(QXQ")-QX(Q'UQ") by 97 (p, p~) _ (p, p~) and XQ' UQ" X (X' U X") ----> (XQ' X X') U © 2002 by Chapman & Hall/CRC

(XQ"

X X")

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6. Algebraic Fuzzy Automata Theory

by (91 Q" X,) (91 Q" , X,)

(9~ X') _

if X' E X' if x' E X" .

In the case of finite state machines, different types of products and their coverings play a crucial role in their decomposition. We have established fuzzy analogs of different products and their coverings . Therefore, these results may be useful in the decomposition of fuzzy finite state machines .

6 .18

Fuzzy Semiautomaton over Group

a Finite

In the remainder of the chapter, we assume that the reader is familiar with the basic results of group theory. The group semiautomaton has been studied extensively in [90] and [60] . We now consider fuzzy semiautomata over a finite group, [41]. A group semiautomaton is a quadruple (Q, *, X, S) such that (Q, *) is a finite group, where Q is called the set of states, X is a finite set, called the set of inputs, and S is a function from Q x X into Q. We present the notion of a fuzzy semiautomaton over a finite group. The fuzzy kernel and fuzzy subsemiautomaton of a fuzzy semiautomaton over a finite group are defined using the notions of a fuzzy normal subgroup and a fuzzy subgroup [184] of a group. We let (G, *) denote a group. We sometimes write G for (G, *) when the operation * is understood. Definition 6.18 .1 A fuzzy subset A of G is called a fuzzy subgroup of G

if the following properties hold: (1) A(x * y) > A(x) A A(y), (2) A(x) = A(x_1) for all x, y E G.

Definition 6.18 .2 A subgroup of G if

fuzzy

subgroup A of G is called a fuzzy normal

A(x * y * x-1 ) > A(y) for all x, y E G.

Definition 6.18 .3 Let A and /t be fuzzy subsets of G. The product of A and /t is defined by

(A * ,t) (x) = v{a(y) for all x E G.

© 2002 by Chapman & Hall/CRC

n ,t(z) I y, z E

G such that x = y * z}

6.18. Fuzzy Semiautomaton over a Finite Group

313

Definition 6.18 .4 Let A and ft be fuzzy subgroups of G such that A C_ ft . Then A is called a fuzzy normal subgroup of ft if A(x * y * x-1 ) > A(y) n ft(x) for all x, y E G.

Let A and ft be fuzzy subgroups of G such that A is a fuzzy normal subgroup of ft . Then Supp(A) is a normal subgroup of the group Supp(p). Define the fuzzy subset ft/A of the quotient group Supp(y)/Supp(A) by b' cosets gSupp(A) E Supp(y,)/Supp(A), ft/A(q Supp(A)) = V{ft(p) I p E q Supp(A)} . Then ft/A is a fuzzy subgroup of Supp(tt)/ Supp(A) . Definition 6.18 .5 Let A and ft be fuzzy subsets of the groups G and H,

respectively . Let f : G ~ H be a group homomorphism . The fuzzy subsets f(A) of H and f-1 (ft) of G are defined as follows: f (A) (h)

-

V {A(g) I g E G such that f(g) = h} 0

if f-1 (H)

Ql

if f-1 (H) _ 0,

for all h E H, and f ` (it) (g) = it (f W) for all g E G .

Definition 6.18 .6 Let A and ft be fuzzy subgroups of the groups G and H,

respectively . A function f : G ~ H is called a weak homomorphism from A into ft if the following conditions hold: (1) f is an epimorphism, (2) f(A) C ft . If f is an isomorphism from G onto H, then f is called a weak isomorphism from A into ft .

Definition 6.18 .7 A fuzzy semiautomaton (fsa) over a finite group

(Q, *) is a triple (Q, X, ft), where X is a finite set and ft is a fuzzy subset of QXX XQ.

Let ft* be the extension of ft to Q x X* x Q as defined in Definition 6.2.1 . Definition 6.18 .8 Let S = (Q, X, ft) and T = (Q 1 , X1, ft1) be fsa's over a finite group. A pair of functions (f, g), where f : Q ~ Q1, g : X ~ X1, is called a homomorphism from S into T, written (f, g) : S ----> T if the following conditions hold: (1) f is a group homomorphism, (2) ft (p, x, q) <_ ft, (f (p), g (x), f (q)) for all p, q E Q, x E X. © 2002 by Chapman & Hall/CRC

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6. Algebraic Fuzzy Automata Theory

The pair (f , g) is called a strong homomorphism from S into T if it satisfies (1) of Definition 6.18.8 and the added condition, ft1(f (p), g(x), f (q)) = V{ft(p, x, r) I r E Q, f (r) = f (q)}

for all p,gEQ,xEX. In Definition 6.18.8, if X = X1 and g is the identity map of X, then we write f : S ~ T and say that f is a homomorphism or strong homomorphism from S into T.

Let S = (Q, X, ft) be a fsa over a finite group in the remainder of the chapter .

Definition 6.18 .9 A fuzzy subset A of Q is called a fuzzy kennel of S if the following conditions hold :

(1) A is a fuzzy normal subgroup of Q, (2) A(p * r -1 ) > ft(q *_ k, x, p) n ft(q, x, r) n A(k) for all p, q, k, r E xEX.

Definition 6.18 .10 A fuzzy subset A of Q is called a fuzzy subsemiautomaton of S if the following conditions hold: (1) A is a fuzzy subgroup of Q, (2) A(p) >- ft(q, x,p) A A(q) for all p, q E Q, x E X.

Theorem 6.18 .11 A fuzzy normal subgroup A of Q is a fuzzy kernel of S if and only if

A(p * r-1 ) > ft * (q * k, x, p) n ft * (q, x, r) n A(k) for allp,q,k,rEQ,xEX* .

Proof. Let A be a fuzzy kernel of S. We prove the theorem by induction . Then x =A. Let p, q, k, r E Q. If p = q * k, r =q, on Ixl =n . Let n=0 then ft*(q*k, x, p) Aft* (q, x, r) AA(k) = A(k) <_ A(q*k*q-1 ) since A is a fuzzy normal subgroup. Ifp :?~ q*k or r :?~ q, then g* (q*k, x, p) ntt* (q, x, r) AA(k) = 0 <_ A(p * r-1 ) . Thus the result holds for n = 0. Suppose that the result holds for all y E X*, where jy j = n - 1, n > 0. Let x E X* be such that © 2002 by Chapman & Hall/CRC

6.18. Fuzzy Semiautomaton over a Finite Group

315

x=ya,yEX*,aEX, lyl=n-l,n>O.Then w* (q * k, x, p) A w* (q, x, r) A A(k)

_

(V {w* (q * k, y, u) A w(u, a, p) l u E Q}) n (V {ft * (q, y, v)n

ft(v, a, r) l v E Q}}) n A(k) V{V{,t* (q * k, y, u) n ft(u, a, p)n * ft (q, y, v) n ft(v, a, r) n A(k) <_ <_

<

IUEQ} I VEQ} V{V{a(u * v -1 ) A y,(u, a, p) nft(v, a, r) l u E Q} l v E Q} V{V{a(v -1 * u) A y(v * v -1 * u, a, p) A y,(v, a, r) l u E Q} l v E Q} (since A is a normal fuzzy subgroup) A(p * r-1) .

Hence the desired condition holds. The converse follows easily. Theorem 6.18 .12 A fuzzy subgroup of S if and only if

A of Q is a fuzzy subsemiautomaton

A (p) > ft * (q, x, p) n A (q)

for all p,gEQ,xEX* .

Proof. The proof is similar to that of Theorem 6.18.11 . Theorem 6.18 .13 Let T = (Q 1 , X, [t1) be an fsa over a finite group and let f be a homomorphism from S into T. If A is a fuzzy subsemiautomaton (fuzzy kernel of T, then f -1 (A) is a fuzzy subsemiautomaton (fuzzy kernel of S.

Proof. The proof follows easily. Theorem 6.18 .14 Let T = (Q1, X, [t1) be an fsa over a finite group and let f be a strong homomorphism from S onto T. If A is a fuzzy subsemiautomaton (fuzzy kernel of S, then f (A) is a fuzzy subsemiautomaton (fuzzy kernel of T.

Proof. Let A be a fuzzy kernel of S. Then A is a fuzzy normal subgroup of Q. Since f is an epimorphism from Q onto Q1, f (A) is a fuzzy normal subgroup of Q1 . Let p1, q1, r1, k1 E Q1, x E X. Then ft1(g1 * k1,x,p1) nft1(g1,x,r1) n f(A)(k1)

© 2002 by Chapman & Hall/CRC

=

ft, (q, * k1, x,p1)A

ft1(g1,x,r1) n (V{A(k) l k E Q, f (k) = k1}) V{ft1(g1 * k1,x,p1)A ft1(g1,x,r1) n A(k) l k E Q, f (k) = k1} .

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6. Algebraic Fuzzy Automata Theory

Now let p, q, r, k E Q be such that f (p) = pl, f (q) = q1, f (r) = r 1 , and f (k) = k 1 . Then fti(q1*k1,x,p1)nft1(g1,x,r1)nA(k)

= =

<

ft, (f(q*k),x,f(p))A fti (f (q), x, f (r)) n A (k) (V {ft (q * k, x, a) I a E Q, f (a) = f (p)}) n (V {ft(q, x, b) j b E Q, f (b) = f (r)}) A A(k) V {V {ft (q * k, x, a) n ft (q, x, b) A A(k) I a E Q, f (a) = f (p)} bEQ,f(b)=f(r)} V{V{a(a * b -1 ) I a E Q, f (a) =f(p)} I bEQ,f(b)=f(r)} f (A) (p1 * r1 1 ) .

Hence f (A) (p1 * ri 1 )

>_ =

V{ft1(g1 * k1,x,p1) n ft1(g1,x,r1) n A(k) I k E Q, f (k) = ki} fti(q1*k1,x,p1)nft1(g1,x,r1)nf(A)(k1) .

Thus f (A) is a fuzzy kernel of T. As in Theorem 6 .18 .14, it can be shown that f (A) is a fuzzy subsemiautomaton of T if A is a fuzzy subsemiautomaton of S. For the remainder of the chapter, unless otherwise stated, S = (Q, X, ft) denotes an fsa over a finite group Q for which the following condition is satisfied : ft (p * q, x, r) < ft (p, x, k) n ft (q, x, k) for all p, q, r,kEQ,xEX . Also, e will denote the identity element of the group (Q, *) . Theorem 6 .18 .15 Let A be a fuzzy kernel of S. Then A is a fuzzy subsemiautomaton of S if and only if A (p) > ft (e, x, p) n A (e) for all pEQ,xEX . Proof. Suppose that the given condition is satisfied . p,q,rEQ,xEX, A(p)

= >

=

Then for all

A(p * r -1 * r) A(p * r -1 ) A A(r) ft(q, x,p) n ft(e, x, r) n A(q) n A(r) ft(q, x,p) n ft(e, x, r) n A(e) A A(q) (by the given condition) ft(q, x,p) n A (q) (since A (e) > A (q) and ft (e, x, r) > ft (e * q, x, p)) .

Thus A is a fuzzy subsemiautomaton of S. The converse is immediate .

© 2002 by Chapman & Hall/CRC

m

6.18. Fuzzy Semiautomaton over a Finite Group

317

Corollary 6.18 .16 If A is a fuzzy kernel and v is a fuzzy subsemiautoma-

ton of S such that v C_ A and A(e) = v(e), then A is a fuzzy subsemiautomaton of S. m

Theorem 6.18 .17 Let A be a fuzzy kernel and v be a fuzzy subsemiautomaton of S. Then A * v is a fuzzy subsemiautomaton of S.

Proof. Since A is a fuzzy normal subgroup and v is a fuzzy subgroup of Q, it follows that A * v is a fuzzy subgroup of Q and A * v = v * A. Now

(A

v) (p)

> =

A(p * r -1 ) A v(r) (ft(a * b, x, p) n ft(a, x, r) n A(b)) A (ft (a, x, r) A v (a)) ft(a * b, x, p) A A(b) A v(a) (since y (a * b, x, p) < y (a, x, r) )

for all a, b, p E Q, x E X. Thus for all p, q E Q, x E X, V~ft(a*b,x,p)AA(b)Av(a) I a,bEQ, a*b=q} ft(q, x, p) A (V{A(b) A v(a) I a, b E Q, a * b = q}) ft (q, x, p) n (v * A) (q) ft (q, x, p) n (A * v) (q) .

Hence A * v is a fuzzy subsemiautomaton of S. Theorem 6.18 .18 If A and v are fuzzy kernels of S, then A * v is a fuzzy kernel of S.

Proof. Since A and v are fuzzy normal subgroups of Q, it follows that A * v is a fuzzy normal subgroup of Q and A * v = v * A . Now (A * v)(p * r-1 )

> > =

A(p * q-1 ) A v(q * r-1) (y(a * b * c, x, p) n ft(a * b, x, q) n A(c))n (ft (a * b, x, p) A y, (a, x, r) A v (b)) ft(a * b * c, x, p) A y(a, x, r) A A(c) A v(b) (since ft (a * b * c, x, p) < ft (a * b, x, p))

for all a, b, c, p, r E Q, x E X. Thus for all p, q, r, k E Q, x E X, (A * v)(p * r-1 )

>_

_

V{ft(q * b * C, x,P) A ft(q, x, r) A A(c) n v(b) I b,cEQ, b*c=k} (ft(q * k, x, P) n ft (q, x, r)) A (V{A(c) n v(b) I b, c E Q, b * c = k}) ft(q * k, x, P) n ft(q, x, r) n (A * v) (k) .

Thus A * v is a fuzzy kernel of S. Let A be a fuzzy kernel of S. Let C(A) = {v I v is a fuzzy kernel of S . such that A(e) = v(e)f © 2002 by Chapman & Hall/CRC

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6. Algebraic Fuzzy Automata Theory

Theorem 6.18 .19 (C(A), C) is a lattice. Proof. Let v, S E C(A) . Then v n S E C(A) and so v n S is the glb of v and S for the inclusion relation " C_ ." By Theorem 6.18.18, v * S E C(A), and so v * S is the lub of v and S for the inclusion relation " C_ ." Thus (C(A), C) is a lattice . Definition 6.18 .20 Let A be a fuzzy subsemiautomaton of S. A fuzzy sub-

set v of Q is called a fuzzy kennel of A if (1) v C_ A and v is a fuzzy normal subgroup of A and (2) v(p r-1 ) > ft(q * k, x, p) A ft(q, x, r) A v(k) for all p, r, k E Q, q E Supp(A), x E X.

Theorem 6.18 .21 Let A be a fuzzy subsemiautomaton of S and let v be a fuzzy kernel of S such that A(e) = v(e) . Then A * v is a fuzzy subsemiautomaton of S, v is a fuzzy kernel of A * v, and A n v is a fuzzy kernel of A . Moreover, the weak isomorphism from A/A n v into A * v1 v gives an into homomorphism from the fsa over a finite group (G, X, fq) into the fsa over a finite group (H, X, ft2), where G = Supp(A/An v) and H = Supp(A*vlv) . Proof. By Theorem 6.18.17, A * v is a fuzzy subsemiautomaton of S. Since A(e) = v (e), it follows that v C_ A* v and v is a fuzzy normal subgroup of A * v. It also follows that A is a fuzzy kernel of A * v since A is a fuzzy kernel of S. In addition, A n v C_ A and A n v is a fuzzy normal subgroup of A. Now for p, r, k E Q, q E Supp(A), x E X, (an v)(p * r-1 )

>

(w(q * k, x,p) A w(q, x, r) A v(k))n (ft(k, x, p * r-1 ) A A(k)) ft(q * k, x, p) n ft(q, x, r) n (A n v) (k) (since ft(q * k, x, p) < y,(k, x, p * r-1)) .

Thus A n v is a fuzzy kernel of A. Now, since A and v are, respectively, a fuzzy subgroup and a fuzzy normal subgroup of Q, Supp(A) and Supp(v) are, respectively, a subgroup and a normal subgroup of Q . Also Supp(A n v) = Supp(A) n Supp(v) and Supp(A* v) = Supp(A)* Supp(v) . Thus by the second isomorphism theorem of groups, there is an isomorphism f : Supp(A)/Supp(A n v) ~ Supp(A * v)lSupp(v)

given by f(gSupp(A n v)) = gSupp(v)

for all gSupp(A n v) E Supp(A)/Supp(A n v) . Define A/A n v : Supp(A)/Supp(A n v) ~ [0,1]

© 2002 by Chapman & Hall/CRC

6.18. Fuzzy Semiautomaton over a Finite Group

319

by (A/A n v)( g Supp(A n v)) = V{a(p) I p E

for all gSupp(A n v)

E Supp(A)/Supp(A n v)

g Supp(A

n v)}

and

A * v/v : Supp(A * v)/Supp(v) ~ [0,1]

by (A* v/v)(gSupp(v)) =V{(A* v)(p) I p E gSupp(v)}

for all gSupp(v) E Supp(A*v)/Supp(v) . Then A/Anv and A*v/v are fuzzy subgroups of Supp(A)/Supp(A n v) and Supp(A * v)/Supp(v), respectively. Also, f is a weak isomorphism from A/A n v into A * v/v. Now, let G = Supp(A)/Supp(Anv) and H = Supp(A*v)/Supp(v) . Then G and H are subgroups of Supp(A)/Supp(A n v) and Supp(A * v)/Supp(v), respectively. Define ft, : G x X x G ~ [0,1] by p, l (gSupp(A n v), x,pSupp(A n v))

=

V{ft(a, x, b) I a E gSupp(A n v), b E pSupp(A n v)}

for all gSupp(Anv),pSupp(Anv) E G, and x E X. Clearly, ft, is well defined and (G, X, fq) is an fsa over a finite group. Define N2 : H x X x H ~ [0,1] by p, 2 (gSupp(v), x,pSupp(v))

=

V{ft(a, x, b) I a E gSupp(v), b E

pSupp(v) I

for all gSupp(v), pSupp(v) E H, and x E X. Then (H, X, P2) is an fsa over a finite group. Since f is a weak isomorphism from A/A n v into A * v/v, f (A/A n v) C A * v/v. Let gSupp(A n v) E G. Then (A * v/v)(gSupp(v))

>

= >

f (A/A n v)(gSupp(v))

(A/A n v)(gSupp(A n v)) 0.

Hence gSupp(v) = f (gSupp(A n v)) E H . Thus f induces a monomorphism g : G ----> H given by g(gSupp(A n v)) = gSupp(v) for all gSupp(A n v) E G. It follows that g is a homomorphism from (G, X, fq) into (H, X, P2). Theorem 6 .18 .22 Let T = (Q, X, S) be an fsa over a finite group and let A be a fuzzy kernel of T with Supp(A) = Q. Then there is an onto homomorphism from T into the fsa over a finite group (G, X, a), where

G = Supp(Q/A) .

© 2002 by Chapman & Hall/CRC

32 0

6. Algebraic Fuzzy Automata Theory

Proof. Now A is a fuzzy normal subgroup of Q. Define AP : Q ~ [0,1] by Ap(q) = A(p * q-1 ) for all q E Q . Let H = {Ap I p E Q} . Then H is a group with respect to the binary operation AP *A, = Ap*q for all AP, A q E H. Define Q/A : H ~ [0,1] by (Q/A)(Ap) = A(p)

for all AP E H. Then Q/A is a fuzzy subgroup of H, called the quotient subgroup of Q by A. Let G = Supp(Q/A) . Define a : G x X x G ~ [0,1] by a(Ap, x, Aq) = V{6(a, x, b) I a, b E Q, Aa = Ap, Ab = Aq}

for all AP, A q E G, x E X. Clearly, a is well defined and (G, X, a) is an fsa over a finite group. Define f : Q ----> G by f(q) = A q for all q E Q. Then it follows that f is an onto homomorphism from T into (G, X, a) . Let T = (Q, X, S) be an fsa over a finite group. An element xo E X is called an e-input if S(e, xo , e) > 0. Definition 6.18 .23 An fsa T = (Q, X, S) over a finite group is called mul-

tiplicative if there exists an e-input xo E X having the following properties: (1) S(q, x, p * r) = S(q, xo,p) A S(e, x, r) for all p, q, r E Q, x E X . (2) 6(p1 *p2 1 , xo, q1 * q2 1) = 6(p1, xo, q1) Ab(p2, xo, q2) for all p1, p2, q1, q2 E Q.

Theorem 6.18 .24 Let T = (Q, X, S) be a multiplicative fsa over a finite

group with e-input xo E X . Let A be a fuzzy subsemiautomaton of T and let v be a fuzzy kernel of A . If v is a fuzzy normal subgroup of Q, then v is a fuzzy kernel of T.

Proof. Since T is multiplicative, for all p, q, r, k b(q * k, x,p) n b(q, x, r)

Now for any q we have

=

(6(q * k, xo,p) n 6(e, x, e))n (S(q, xo , r) A S(e, x, e)) 6(q, xo,p) n 6(k -1 , xo, e)n S(e, x, e) A S(q, xo, r) .

E Q, b E Supp(A), q = b * (b * q' -1 ) -1

b(q * k, x,p) n b(q, x, r)

=

> = =

for some q'

Thus (6.4)

E Supp(A), x E X,

S(b * k, x, p) A S(b, x, r) A v(k) S(b * k, xo , p) A S(b, xo , r) A S(e, x, e) A v(k) 6(b, xo,p) n 6(k -1 , xo, e) n 6(b, xo, r)n 6(e, x, e) A v(k) .

© 2002 by Chapman & Hall/CRC

E Q.

6(b, lo,p) n 6(b * q' -1 , xo, e)A S6(k - ', xo, e) A S(e, x, e) A S(b, xo, r) .

Since v is a fuzzy kernel of A, for all p, r, k E Q, b v(p * r -1 )

E Q, x E X,

(6.5)

6.18. Fuzzy Semiautomaton over a Finite Group

321

From (6.4) and (6.5), it follows that v(p * r-1 ) > 6(q * k, x, p) n 6(g, x, r) n v(k)

for all p, q, r, k E Q, x E X. Since by assumption, v is a fuzzy normal subgroup of Q, v is a fuzzy kernel of T. Theorem 6.18 .25 Let T = (Q, X, 6) be a multiplicative fsa over a finite

group with e-input xo E X . Then the following statements are equivalent : (1) A is a fuzzy kernel of T. (2) A is a fuzzy normal subgroup of Q and A (q) > 6(p, xo, q) A A (p) for p,gEQ all .

Proof. (1)x(2) . Since A is a fuzzy kernel of T, A(q)

= > =

A(q * e-1) 6(e * p, xo, q) n 6(e, xo, e) n A(p) 6(p, xo, q) n A(p) (since 6(p, xo , q) < 6(e, xo, e))

for all p, q E Q . (2)x(1) . Since T is multiplicative, for all p, q, r, k E Q, x E X, 6(q * k, x, p * r) A 6(g, x, p) n A(k)

=

< < <

6(q * k, xo, p * r) A 6(g, xo, p) n 6(e, x, e), A(k) (6(q, xo,p) n 6(k-1 , xo, r-1 )n 6(e, x, e)) A A(k) 6(q, xo, P) A 6(k, xo, r) A 6(e, x, e) AA(k) (since 6(k-1 , xo, r-1 ) _ 6(e, xo, e) A 6(k, xo, r) = 6(k, xo, r)) 6(k, xo, r) A A(k) (by (6.5)) A (r) A(p * r * p- 1 ) (by (6 .5)) .

(6.6)

Now let PI ,P2, q, k E Q, x E X. Since Q is a group, there exists a unique element r E Q such that p1 = P2 * r. Thus A(p1 * p2 1 )

= > >

A(p2 * r * P21) 6(q * k, x, p2 * r) A 6(q, x, p2) n A(k) 6(q * k, x,p1) A 6(q, x,p2) A A(k) .

(by (6 .6)

Hence A is a fuzzy kernel of T. Theorem 6.18 .26 Let T = (Q, X, 6) be a multiplicative fsa over a finite group with e-input xo E X. There exists a semigroup homomorphism f Q ~ .FP (Q) and a function g : X ~ FP (Q) such that 6(q,x,p) = f(q)(p) n 9(x)(e) for all p,gEQ,xEX.

© 2002 by Chapman & Hall/CRC

32 2

6. Algebraic Fuzzy Automata Theory

Proof. Define f : Q ----> .FP(Q) by f(q) = Sq for all q E Q, where Sq : Q ----> [0,1] is defined by 6q (P) = S(q, xo , p) for all p E Q . Since T is multiplicative,

bq,*qz (pi * p2)

=

= =

b(qi * q2, xo,pi * p2) b(qi, xo,pi) n b(q2 1 , xo,p2 1 ) b(qi, xo,pi) A S(q2, xo,p2) bq, (PI) n Sgz(p2)

for all P1, P2, ql, q2 E Q . Hence (bq, * bqZ) (p)

=

V{6q, (q) S qi*q2(p)

n bqz (r) I

q, r E Q, q * r = p}

for all p E Q. Thus f (qi) * f (q2) = f (qi * q2) .

(6.7)

It is well known that,F'P(Q) is a semigroup with respect to the product of fuzzy subsets of Q. Hence, it follows from (6.7) that f is a semigroup homomorphism from Q into .FP(Q) . For x E X, define the fuzzy subset Ax of Q by Ax (q) = 6(e, x, q) for all q E Q. Define g : X ----> .FP(Q) by g(x) = Ax for all x E X. Since T is multiplicative, it follows that S(q, x,p) = f(q) (p) A g(x) (e) for all p,gEQ,xEX. 6 .19

Exercises

1. Let M = (Q, X, ft) be the ffsm, where Q = {ql, q2}, X = {a}, and ft is defined by ft(ql , a, qi) = s, ft(ql , a, q2) = 0, ft(g2, a, qi) = 0, and ft(g2, a, q2) = 3. Determine E(M) and E(M) . 2 . Let M = (Q, X, ft) be the ffsm, where Q = {ql , q2 }, X = {a}, and

ft is defined by ft(ql , a, gi) = 0, ft(gi a, g2) = 3, ft(g2, a, qi) = 0, and ft(g2, a, q2) = 3. Determine E(M) and E(M) .

3. Let M = (Q, X, ft) be the ffsm, where Q = {ql , q2 }, X = {a}, and

ft is defined by ft(ql , a, qi) = 0, ft(ql , a, q2) = 3, ft(q2 , a, qi) = 3, and ft(g2, a, q2 ) = 0. Determine E(M) and E(M) .

4. In Example 6.6.4, determine E(MI), E(M2), E(MI xM2), and E(MIA M2) when cl :A c2 and/or dl :?~ d2 . 5. Prove (2), (3), and (4) of Theorem 6.17.1. 6. Let A and ft be fuzzy subgroups of G such that A is a fuzzy normal subgroup of ft . Prove that ft/A is a fuzzy subgroup of Supp(ft)/ Supp(A) . © 2002 by Chapman & Hall/CRC

6.19. Exercises

323

7. Prove Theorems 6.18.12 and 6.18.13. 8. (113) Let (Q, X, T) be a T-generalized state machine. Prove that T + (p, xy, q) = V {T+ (p, x, r) T

T+(r,

y, q) I r E Q},

for all p, q E Q and x, y E X+ . 9. (113) Let (Q, X, T) be a T-generalized state machine . Define - on X+ by Vx, y E X+, x - y if T+ (p, x, q) = T+ (p, y, q) Vp, q E Q. Prove that - is a congruence relation on X+ . 10. (113) Let M = (Q, X, T) be a T-generalized state machine . Let -be a congruence relation defined in Exercise 6. Let [x] = {y E X+ x - y} Vx E X+ and S(M) = {[x] I x E X+} . Prove that S(M) is a semigroup, where the binary operation on S(M) is defined by [x] [y] = [xy] V [x], [y] E S(M) . 11. (113) Let M = (Q, X, T) be a T-generalized state machine, where T is the ordinary product on [0,1] . Show by example that S(M) need not be finite, where S(M) is defined in Exercise 10 . 12. (113) Let M = (Q, X, T) be a T-generalized state machine. Prove that since Q is finite, S(M) is finite if and only if Im(T+) is finite . 13. (113) Construct an example of a T-generalized state machine (Q, X, T) such that EgEQ T+ (p, x, q) > 1 and x E X+ . 14. (113) A T-generalized state machine (Q, S, p) is called a T-generalized transformation semigroup if S is a finite semigroup such that (a) p(p, uv, q) = V{p(p, u, r) T p(r, v, q) I r E Q} Vp, q E Q and Vu, v E S and (b) Vu, v E S, p(p, u, q) = p(p, v, q) Vp, q E Q implies u = v . A t-norm T is said to be T-generalized transformation semigroup inducible if S(M) is finite and EgEQ T+ (p, x, q) <_ 1 Vp E Q and x E X+ for every T-generalized state machine (Q, X, T) . Prove that there exists a T-generalized transformation semigroup inducible tnorm T.

© 2002 by Chapman & Hall/CRC

Chapter 7

More on Fuzzy Languages 7.1

Fuzzy Regular Languages

In this chapter, we study the concepts of max-min (FLv (M)) and min-max (FL A (M)) fuzzy languages recognized by a type of fuzzy automaton M. We show that if Ai and A2 are Fv-regular languages, then so are Ai U A2 and Ai rl A2 .We prove a type of fuzzy pumping lemma. We use this result to give a necessary and sufficient condition for FLv (M) to be nonconstant . In this section, fuzziness is introduced through the initial and final states. Definition 7.1.1 A partial fuzzy automaton (pfa) is a 5-tuple M = (Q, X, S, t, T), where (1) Q is a finite nonempty set of states,

(2) X is a finite nonempty set of input symbols, (3) S : Q x X ~ Q is a function, called the transition function, (4) t is a fuzzy subset of Q, i.e., t : Q ~ [0,1], called the initial fuzzy

state, (5)

T is a fuzzy subset of Q, called the

fuzzy subset of final states .

The transition function S can be extended to S* : Q x X* ----> Q by 6* (q, A) S* (q, ua)

= =

q S(S* (q, u), a)

b' q E Q, b' u E X*, b' a E X. It can be easily verified that S* (q, uv) _ S*(S*(q,u),v) Vu, v E X* . Definition 7.1.2 Let M = (Q, X, S, t, T) be a pfa. The max-min fuzzy language recognized by M is the fuzzy subset FL v (M) of X* defined by FL v (M) (u) = VaEQ (t(q) 325 © 2002 by Chapman & Hall/CRC

n T(6* (q, u)))

7. More on Fuzzy Languages

32 6

and the min-max fuzzy language recognized by M is the fuzzy subset FLA (M) of X* defined by FLA(M)(u) = AIEQ(t(q) V T(6*(q,u))) .

Definition 7.1.3 Let X be a nonempty finite set. A fuzzy subset A of X* is called F v -regular (F A -regular if there exists a pfa, M = (Q, X, S, t, T), such that A = FL v (M) (A = FLA(M)) .

Theorem 7.1.4 Let L be a regular language on a nonempty finite set X. Then the characteristic function XL of L is a Fv -regular language .

Proof. Since L is a regular language on X, there exists a deterministic partial finite automata (dpfa), M' = (Q, X, S, qo, F), such that the language L(M') recognized by M' is L. Consider the pfa, M = (Q, X, S, t, T), where t : Q ~ [0,1] is defined by t(qo) = 1 and t(q) = 0 if q z,4 qo and T : Q ~ [0,1] is defined by T(q) = 1 if q E F and T(q) = 0 if q ~ F. Let u E X*. Then FL v(M) (u)

Thus FL v (M) = XL .

VgEQ(t(q) AT(6*(q,u))) T ((S* (qo, u)) 1 if S* (qo , u) E F F 0 if S* (qo , u)

0

Theorem 7.1.5 Let L C_ X* . Suppose the characteristic function is a Fv -regular language . Then L is a regular language on X.

XL of L

Proof. Since XL is a Fv -regular language, there exists a pfa, M = (Q, X, S, t, T), such that FL v (M) = XL . Thus V4EQ( L(q) nT(6*(q,u))) _ { 0 if u

L.

Now u E L if and only if there exist q' E Q such that T(S * (q', u)) = 1 . Let

1 and

Qo = {q E Qjt(q) = 1}

and F = {S * (q, u) E QIT(S* (q, u)) = 1 for some u E L} .

Then Qo :?~ Ql and F :?~ 0. Let Mq = (Q, X, S, q, F), q E Qo . Let Lq be the language recognized by Mq. Then Lq C_ L . Hence U gEQo Lq C_ L. Let u E L. Then VgEQ(t(q) AT(S*(q,u))) = 1. Hence there exists q E Qo such that t (q) AT(S*(q,u)) = 1. Then T(S*(q,u)) = 1 and so S* (q, u) E F. Thus u E Lq. Hence U gEQ,,Lq = L. Since each Lq is regular, L is regular . m © 2002 by Chapman & Hall/CRC

7.1 . Fuzzy Regular Languages

327

Theorem 7.1.6 Let L be a regular language on a nonempty finite set X. Then the characteristic function XL of L is a FA-regular language.

Proof. Since L is a regular language on X, there exists a dpfa, M' = (Q, X, S, qo, F), such that the language L(M) recognized by M' is L. Consider the pfa, M = (Q, X, S, t,T), where t : Q ~ [0,1] defined by t(qo) = 0 and s(q) = 1 if q :?~ qo and T : Q ~ [0,1] defined by T(q) = 1 if q E F and T(q)=0 if q~F.Now FLA(M)(u)

= ngEQ(t(q) V T(6*(q,u))) .

Let u ~ L. Then XL(u) = 0 and 6*(qo,u) ~ F. Hence t(qo) = 0 and T(6* (qo, u)) = 0. Thus FL A (M) (u) = 0. Suppose u E L. Then 6* (qo, u) E F and hence T(6*(qo,u)) = 1 . Thus t(qo) V T(6*(qo,u)) = 1 . If q :?~ qo, then t (q) = 1 . Thus t (q) V T(6* (q,u)) = 1 . Hence FL A(M)(u) = 1 . Thus FLA (M) = XL . 0 Theorem 7.1.7 Let L C_ X* . Suppose the characteristic function XL of L in X* is a FA-regular language. Then L is a regular language.

Proof. Since XL is a FA -regular language, there exists a pfa, M = (Q, X, 6, t, T), such that FL A (M) = XL . Thus for all u E X*,

=

XL(u)

ngEQ(t(q) V T(6*(q,u))) ~ 1ifuEL 0ifa~L .

Now u E L if and only if for all q E Q, either t (q) = 1 or T(6*(q, u)) = 1.

Let

Qo = {q E Q I t(q) = 1} and for all q E Q\QO, F'q = U.EX*{6* (q, x) I T( 6 *(q,u)) = 1} .

For all q E Q\QO, let Mq = (Q, X, 6, q, Fq ) and let Lq denote the language recognized by Mq. Then y E ngEQ\QoLq if and only if for all q E Q\QO, y E Lq if and only if for all q E Q\QO, 6* (q, y) E Fq if and only if for all q E Q\QO, T(6*(q, y)) = 1 if and only if y E L . Hence L = ngEQ\QoLq and XL(u)

= ngEQ\QoT(6*(q,u)) . 0

Example 7.1.8 Let Q = {qi, q2}, X = {0,1}, 6(qi, 0) = qi = 6(q2, 0), 6(qj,1) = q2 = 6(g2,1) , t(qi) = 0 = t(q2), T (qi) = 0, and T(q2) = 1 . Thus in the proof of Theorem 7.1 .7, Qo = 01 and Fq2 = {1,11,111, . . . } . Theorem 7.1.9 Let A be a Fv -regular language on X. Im(A), A, is a regular language on X.

© 2002 by Chapman & Hall/CRC

Then for all c E

328

7. More on Fuzzy Languages

Proof. Since A is Fv-regular, there exists M = FLv (M) = A. Thus A(u)

(Q, X, S, t, T)

such that

=VgEQ(t(q) AT(6*(q,u)))

for all u E X* . Let c E Im(A) . Let u E Ac . Then A(u) = VgE Q(L(q) A T(S*(q,u))) >_ c. Since Q is finite, there exists qv E Q, such that t(qv) A T(S*(qv,u)) > c. Hence for all u E Ac there exists qv E Q such that t(qv) A T(S*(qv,u)) > c. For all u E A,, let Mq . = (Q, X, S, q., Tc.) where T, _ {q E Q T(q) >_ c} . Let L(Mq.) be the language of Mq . . Let x E L(Mq.) . Then S * (q.,x) E T c and so T(S*(qv,x)) >_ c. Since t(qv) >_ c, t(qv) AT(S*(q.,x)) >_ c. Hence VgE Q(L(q) A T(S*(q, x))) >_ t (q.) A T(S* (q., x)) >_ c. Thus x E A, . Hence L(Mq.) C_ A c . Conversely, let x E A, . Then there exists qx E Q such that t(qx) AT(S*(qx,x)) > c. Thus T(S*(qx,x)) > c and so S * (qx,x) E T, . Hence x E L(Mq,) . Thus A, C UL(Mq,) . Hence A, = UL(Mq,) . Since each L(Mq,) is regular, A, is regular . Theorem 7.1.10 Let A be a finite valued fuzzy subset ofX* . If A, is regular for all c EIm(A), then A is a Fv -regular language on X* . Proof. Let Im(A) =

{CI,

C, . . . ,

A,, C

x`12

ck}

where cl > C > . . . >

ck .

Then

C . . . C Alk .

Let us denote AZ = Ac j , i = 1, 2, . . . , k. Let Mi = (QZ, X, SZ , qZ, FZ) be the dfa for Ai - AZ_ 1, i = 1, 2, . . . , k where Ao = 01 . Let TZ be the characteristic function of FZ U {d} in QZ U {d}, where d is a state such that d ~ QZ, i = 1, 2, . . . , k. Let M = (Q, X, S, t,T), where ~

Q = (Q1 U {d}) X (Q2 U {d}) X . . . b((pl,p2,

x

(Qk U {d}),

. . . , Pk),a) = (61(pl,a), 62(p2,a),

where we set 6i (pi, a) T(pl,p2,

= d

if pi

. . . , Sk(Pk,a)),

2, . . . , k,

=d, i = 1,

. . . , Pk) = T , (pl) AT2(p2) n . . . ATk(pk)

and t(pl, p2,

. . . , Pk) = ~

Ci

. .

if (PI, " 0 otherwise .

,

Pk) = (d,

Let u E X* and A(u) = c2. Then u E Ai VgEQ(t(q) AT(6* (q, u)))

© 2002 by Chapman & Hall/CRC

AZ_1 .

. . . , d, qi, d, . . . , d)

Now

= ci n 1 =

c2

7.1 . Fuzzy Regular Languages

329

since t (q) AT(6* (q u)) _ '

Hence A(u) X* . 0

cZ if q = (d,

~ 0 otherwise.

=VaEQ(t(q) AT(6*(q,u))) .

. . . , d, qZ, d, . . . , d)

Thus A is a Fv-regular language on

We illustrate Theorems 7.1 .9 and 7.1 .10 in the next two examples . Example 7.1.11 Let G = (N, T, P, s) be the grammar of Example 1.8.4 .

Then L(G) = L bm abn I m = 0,1. . . . ; n = 1,2. . . . } (see Example 1.8.6 . Define A : {a, b}* ~ [0,1] by Vbmabn E L(G), A(bn'abn) _ .5 if m > 0, A(bmabn) = .9 if m = 0, and A(x) = 0 for all other x E {a, b}* . Then A.5 = L(G) is regular. Now A.9 = {ab n I n = 1, 2. . . . } is regular since it is generated by the productions s aS, S ~ bS, and S ~ b.

Example 7.1.12 Let A be the fuzzy language of Example 7.1 .11.

Then A.9 = {abn I n = 1,2 . . . . } and A .5\A .9 = {bm abn I m, n = 1,2. . . . f . Let M1 = (Q1, X, 61, s, F) and M2 = (Q2, X, 62, s, F), where Q1 = {s, S, 01, F}, Q2 = {s, Sl, S2, 0, F}, X = {a, b}, and Sl and 62 are defined as follows. 61 (s, a)

6 1 (s, b) 61 (S, a) 6 1 (S, b) 61(01, a) 6 1 (0, b)

61 (F, a) 61 (F, b)

= S _

_ = F _ = 01 - 01 = F

62 (s, a)

S1 S2 S1 01

62 62 (SL, a) 62 (SL, b) 62 (S2, a) 6 2(S2, b) 62 (0, a)

F 01

2(0,b) 62 (F, a) 62 (F, b)

01

F.

Then Ml accepts A.9 and M2 accepts A.5\A .9 . Now (Ml U {d}) x (M2 U {d}) has 30 elements . Hence we list only a few of the images of S, T, and t . 6((s, d), a) 6((S, d), b) 6((d, s), a) b ((d, 0), b)

= = = =

(61 (s, a), 62 (d, a)) (61 (S, b), 62 (d, b)) (61 (d, a), 62 (s, a)) (6 1 (d, b), 62 (0, b))

T(F, d) T(d, F) ~(s, d)

(d, s)

© 2002 by Chapman & Hall/CRC

=

= _ _

1

1 .9 .5 .

= = =

(S, d) (F, d) (d, 01)

=

(d, 0)

and is=Then a(Ql, FLv by Fv-regular FLv(M2) A1 Theorems 7(M) X, Now Since Ubl,A2= =ti, AT(S*(q,bab))) (A1 A1 = AT(6*(q,ab))) is language A1 Let (Ql Tl) A2and 7aUrl(P,q)EQI (u) A1 A1 Fv-regular XXA2), A2 and Let n (S2 T2((bi(A Q2, A2 and A2)(u)nand and (q, 0A2 M2 on = are X, XQ2 XQ2(tl(p) u)) A2 (u) A2 7Al, X* =Fv-regular b1 be )(tl ATl(6i(p,u))))n AT2(62(q,u)))) AFLv(M2)(u) be language u), (Q2, XU(p) Fv-regular finite A2, Xb2, b2 m(nt2)(p,q) X, (q, tl (s, (d, t2(q)n t2(q) for AT(6*((d, AT(S*((0, AT(O, A0)V( nT(S*((d, AT(F, A1)V( 62, Xvalued u))) d) s) ATS*((d, s) on languages all t2, t2, AT(6*((s,d),ab))V AT(S*((s,d),bab))V AT(S*((d, AT1(61(Au)) X* languages AT((S1 TT2) More F)) cb)) =Fv-regular E T1 Vsuch Sl), S2), 0), V S2), [0,1] (on (on Xb)) Xs), s), ab)) b)) T2) Fuzzy AT(d, b)) on that AT(d, b2)*((p,q),u))) VThe X*, ab)) bab)) V X* languages FLv(MI) there 0)) Languages F)) result Thenexists now A1 on _ rl

330

7.

Now

-

VgEQ(t(q)

_ _ _ Then

_

VgEQ(t(q)

_ _ _ _ _ Theorem X* .

.1.13

Proof. follows Theorem A2 Ml A1

(t(s,d) (t(d, ( .9 ( .5 ( .9 ( .9

.5 .5A0)

(t (t .9AT(S*((O1,S1),ab))V ( .5 ( .9 V( .5 .5 ( .9 .5A1) ( .9 .5 . .

.1.9

.1 .10 .

.1.14

. .

.

Proof. . M

Now FLv(M)(u)

. -

V(p,q)EQ,XQ2((tl V Tl V(P,q)EQ, AT2 (VPEQ,(tl(p) (VgEQz(t2(q) FLv(MI)(u) At (al

Hence © 2002 by Chapman & Hall/CRC

.

7.1 . Fuzzy Regular Languages

331

Theorem 7.1.15 Let A be a Fv -regular language on X* . Then is a FA -regular language on X* . (Q,

Proof.

= 1 -A

Since A is a Fv-regular language on X*, there exists M = = A . Let M' = (Q, X, 6, t', T) where T =

X, 6, t, T) such that FLv(M) 1-T and T=1-t . Now

=

FL A(MC)(u)

= = for all u X* . 0

E

ngEQ(T (q) V T(6*(q,u))) ngEQ(( 1 - t (q)) V ( 1 - T(6* (q, u)))) 1 - VgEQ(t(q) AT(6 * (q, u)))

1_ A(u) A(u)

X* . Hence FLA (MC)

Thus _X is a FA-regular language on

Let M = (Q, X, 6, t,T) be a pfa. Let Q = {ql, g2, - . . , qnj and for all 1 [0,1] by (P-1 T) ( x) for all x

E

X* . For all pi

E

(PI,

t(qi) =

ti

= T(6*(p,x))

Q, 1 [0, 1]

by 01, P2,

" . . ,Pn)-1T)(x)

=Vi--1(Ci

n (Pi

1T)(x))

for all x E X* . Let/C= {(p1, p2, . . . , Pn) -1 T lpi E Q, 1 <_ i <_ n} . Since Q is finite, K is finite. Let A be a fuzzy subset of X* . Let u E X* . Define u-1A : X* ----> [0,1] by (u-1A)(v) = A(uv) for all v

E

X* . Let p

E

Q. Then (P -1 T)(v)

for all v

E

=T(6*(p,v))

X* . Suppose that A = FLv (M) . Let x

(u-1A)(x)

= =

E

X* . Then

A(ux) VgEQ(t(q) AT(6*(q,ux))) (t(ql) AT(6*(6*(gl,u),x))) V . . . V (t(gn)A T(6*(6*(qn, U), X))) (C1 n T ( 6* ( 6* (ql, u), x))) V . . . V (Cn n T(6*(6*(gn,U),x)))-

© 2002 by Chapman & Hall/CRC

332

7. More on Fuzzy Languages

Let S* (qj, u) = pi for all 1 < i < n. Thus (U'-1A)(x)

= = =

(cl AT(6*(pl, x))) V . (cl n (P1 'T) (x)) V . . .

. . V (Cn AT(6*(pn, x))) V (c n A (Pn1T)(x)) . ((pl~p2~ . .~pn)-1 T)(x) .

Hence u-1 A = (pl,p2, . . . ,pn) -1 T E K . Thus the set contains only a finite number of distinct elements .

F= {u -1A

I

u E X*}

Theorem 7.1.16 Let A be a fuzzy subset of X* . The following are equiv-

alent. (1) A is a Fv -regular language of X* . (2) 1Z = {(u,v) E X* x X* I u-1 A = v-1 A} is a right congruence of finite index. (3) P = {(u, v) E X* x X* I A(xuy) _ A(xvy) for all x, y E X*} is a right congruence of finite index.

Proof. (1)x(2) : Clearly 1Z is an equivalence relation on X* . Since I u E X* } is a finite set, the index of 1Z is finite . Let u, v, x E X and uRv . Then u-1 A = v-1A. Hence A(uw) = A(vw) for all w E X* . Thus A(uxy) = A(vxy) for all y E X* . Hence (ux)-1 A = (vx)-1A. Thus R is a right congruence. (2) x(1): Let A denote the empty word in X* . Let M = (Q, X, S, t, T) be a pfa, where Q is the set of all distinct R-equivalence classes and S, t, and T are defined as follows : ,F = {u -1 A

6 :QXX----> S([x], a) = [xa]

for all [x]

E Q, a E X, t :Q----> [0,1]

tQxD

for all [x]

E Q

1 if [x] _ [A] 0 if [x] [A]

and T :Q----> [0,1] T([x]) = A(x)

for all [x] E Q . The function S can be extended to S* : [xy] for all [x] E Q, y E X* . © 2002 by Chapman & Hall/CRC

Q x X* ~ Q,

where S* ([x], y) _

7.1 . Fuzzy Regular Languages

333

Let [x], [y] E Q and a E X. Suppose [x] = [y] . Then xRy . Since 1Z is a right congruence, xaRya . Thus [xa] = [ya] and so 6([x], a) = 6([y], a) . Hence 6 is well defined . Now if [x] = [y], then x -1 A = y-1A . Thus (x -1 A)(A) = (y-1A)(A) and so A(x) = A(y) . This implies that T([x]) = T([y]) . Hence T is well defined . Let x E X* . Then FLv(M)(x)

= =

VgEQ(t(q) AT(6*(q,x))) T(6*([A],x))

Hence FL v(M) = A. Thus A is a Fv -regular language of X* . (1)x(3) : The proof of this part is similar to (1)x(2).

m

Lemma 7.1.17 (Pumping Lemma) Let M = (Q, X, 6, t, T) be a pfa. Suppose IQ I = n. Then for all x E X*, I x I >_ n, there exist u, v, w E X such that x =uvw, Iuv I 0 and

FL v (M) (uv'w) = FL v (M) (x) for all m>0.

Proof. Let x = al a2 . . . aP , ai E X, 1 n. Now FLv(M)(x) = VgEQ(t(q) AT(6*(q,x))) .

Let q E Q . Let 6(q, al) = ql, 6(ql, a2) = q2, . . . , 6(gp_1, aP) = qP . Since j, 0 0. Consider uv2 w . Now uv 2w = ala2 . . . a2_laiai+1 . . . aj_laiai + 1 . . . aj _ l ajaj+l . . . aP .

Clearly 6* (q, u) = q2, 6* (q2, v) = qj = q2, and 6* (qj, w) = qP . Now 6* (q2, v2) _ 6* (6 * (qj, v), v) = 6* (qj , v) = qj = qi . By induction it follows that 6 * (qj, vm) = 6* (6* (qj, vm-1 ), v) = 6 * (qj, v) = qj = qa

for all m > 1. Hence 6* (q2, uv mw)

= = =

6* (6* (q, u), vmw) 6* (qi , vmw) 6* (6* (qj, vm), w) 6* (qj, w) qP 6* (qj, uvw)

© 2002 by Chapman & Hall/CRC

334

7. More on Fuzzy Languages

for all m > 0. Thus FLv (M)(uv -w)

= =

=

VaEQ(t(q) AT(S*(q,uv-w))) VgEQ(t(q) AT(6*(q,uvw)))

FLv (M) (uvw) FLv(M)(x) . m

Theorem 7.1 .18 Let M = (Q, X, S, t, T) be a pfa. Suppose IQ I = n . Then FLv (M) is nonconstant if and only if there exist W1, W2 E X* such that Iw1I < n and FLv (M)(w2) < FLv(M)(wl) . Proof. Suppose that FLv (M) is nonconstant . Then there exist u, v E X* such that FLv (M) (u) :?~ FLv (M) (v) . Let w E X* be such that Iw I >_ n . Now either FLv (M) (w) zA FLv (M) (u) or FLv (M) (w) zA FLv (M) (v) . Suppose FLv (M) (w) zA FLv (M) (u) . Case 1 : FLv (M) (w) > FLv (M) (u) . Let S= {x E X*I IxI > n and FLv (M)(x) > FLv(M)(u)}. Since w E S, S z,4 0. By the well ordering principle there exists wo E S such that Iwol is smallest. Now Iwol > n and FLv (M)(wo ) > FLv (M)(u) . By the pumping lemma, there exist x, y, z E X* such that wo = xyz, I xy I < n, Iy I > 0, and FLv (M) (xy'z) > FLv(M) (wo) for all i >_ 0. Let wl = xz. Since Iw1I < IwoI , wi ~ S. Since FLv (M)(xz) >_ FLv (M)(wo) > FLv (M)(u) it follows that Iw1I < n. Hence wl and u are the required words . Case 2: FLv (M) (w) < FLv (M) (u) . In this case if Jul < n, then u and w are the required words. Suppose that Jul >_ n. Then proceeding as in Case 1, we can show that there exists words wl and w2 such that Iw1I < n and FLv (M)(w2) < FLv(M)(wl) . The converse is immediate. m

7 .2

On Fuzzy Recognizers

In this section, we define and examine the concept of a fuzzy recognizer . If L(M) is the language recognized by an incomplete fuzzy recognizer M, we show that there is a completion M' of M such that L(M') = L(M), Theorem 7.2.14. We also show that if A is a recognizable set of words, then there is a complete accessible fuzzy recognizer MA such that L(MA) = A, Theorem 7.2.20. Our long-term goal is to determine rational decompositions of recognizable sets. In fact, we wish to determine a decomposition that gives a constructive characterization of a recognizable set . We lay groundwork for this determination by proving Theorems 7.2.28 and 7.2.29 . © 2002 by Chapman & Hall/CRC

7.2. On Fuzzy Recognizers

335

Let M = (Q, X, /t) be a fuzzy finite state machine, where Q and X are nonempty sets and /t is a fuzzy subset of Q x X x Q . Q is called the set of states and X is called the set of input symbols. Let X* denote the set of all strings of finite length over X. Let A, B C_ X* . Then AB = {uv I u E A, v E B} . For x E X*, xA = {x}A and Ax = A{x} . Definition 7.2 .1 Let M = (Q, X, /t) be a ffsm . Let q E Q, x E X*, and ACX* . Define q*x :Q~[0,1] and q*A :Q----> [0,1] by * (q * x) (p) = [t (q, x, p) b'p E Q and q*A = UxEAq*x .

Definition 7.2 .2 Let M = (Q, X, /t) be a ffsm. Let a : Q ----> [0,1], x E X*, and A C X* . Define (1) a * x : Q ----> [0,1] by

(a * x) (p) = V{a(q) A [t * (q, x, p) I q E Q} dpEQ, (2)cx*A :Q~[0,1] a*A = UxEAa*x, (3) a * x-1 : Q ~ [0,1] by (a * x -1 ) (p) = V{o(q) n tt* (p, x, q) I q E Q} dpEQ, A-1 : Q ----> [0,1] (4) a *

(k * A-1 = UxEAC' * x-1

Lemma 7.2 .3 Let M = (Q, X, y) be a ffsm. Let a be a fuzzy subset of Q and A C X* . Then

(a * A) (p) = V{V {a(q) A w* (q, x, p) I q E Q} I x E A},

(a * A- 1) (p) = V {V {a(q) A w* (p, x, q) I q E Q} I x E A} vpEQ .0

Lemma 7.2 .4 Let M = (Q, X, y) be a ffsm and let a : Q ~ [0,1] . Let S = {a * x I x E X*}. Then S is a finite set.

© 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

336

Proof. Let x E X* . Then (a*x)(p) = V{a(q)nft* (q, x, p) I q E Q} . Since ft* is finite valued . Also, a is finite valued. is ft finite valued, it follows that It now follows that the number of mappings of the form a * x : Q ~ [0,1] is finite. Hence S is finite . m

Theorem 7.2.5 Let a be a fuzzy subset of Q, x, y E X* and A, B C_ X* . Then

(1) (a * x) * y = a * (xy), (2)(a*A)*y=a*(Ay), (3) (a*A) *B=a* (AB), (4) (a (5) (a (6) (a

x -1 ) y -1 = a (yx) -1 , A-1 )

y-1 = a

A-1 )

B-1 = a

(yA) -1 , (BA) -1 .

Proof. (1) Let p E Q . Then ((a * x) * y) (p) _ = _

V~(a*x)(q) nw * (q,y,p) I qE Qj V{VQa(r) Aw*(r,x,q) I r E Q}) Aw*(q,y,p) gEQI V{a(r) A (V {[t* (r, x, q) A w* (q, y, p) I q E Q}) I rEQ} V{a(r) n ft * (r, xy, p) I r E Q} (a * (xy))(p)-

Hence (a * x) * y = a * (xy) . (2) Let p E Q. Then ((a * A) * y) (p)

V~(a*A)(q) nft* (q,y,p) I qE Qj V{(V{(a* x) (q) I x E A}) A[t*(q,y,p) I qEQ} V{V{(a * x) (q) A w* (q, y, p) I q E Q} I xEA} V{((a * x) * y)(p) I x E A} V{(a * (xy))(p) I x E A} (UxEACI * (xy))(p) (a * (Ay)) (p).

Thus (a*A)*y=a*(Ay) . (3) (a * A) * B = UVEB((a * A) * y) = UVEBa * (Ay) = a * (AB) . © 2002 by Chapman & Hall/CRC

7.2 . On Fuzzy Recognizers

337

(4) Let p E Q. Then

((a * x-1) * y -1 )(p)

_

V{(a * x-1)(q) n ft * (p, y, q) I q E Q} V{(V{a(r) Aw*(q,x,r) I r E Q}) Aft* (p, y, q) I q E Q} V {a(r) A (V {ft* (q, X, r) A w* (p, y, q) IgEQ}) I rEQ} V {a(r) A (V {ft* (p, y, q) A w* (q, x, r) IgEQ}) I rEQ} V {a(r) n ft * (p, yx, r) I r E Q} (a * (yx)-1)(p)

Hence (a * x-1) * y-1 = a * (yx)-1 . (5) Let p E Q. Then ((a * A-1) * y - 1) (p)

=

A-1) V{(a * (q) A w* (p, y, q) I q E Q} x-1) * V {(V{(a (q) I x E A}) (p, y, q E Aft* q) I Q} V{V{(a * x -1 ) (q) A w* (p, y, q) I q E Q} I xEAl V {((a * x-1 ) * y -1 )(p) I x E A} V {(a * (yx) -1 )(p) I x E A} (UxEAOI * (Jx)-1)(p) (a * (yA) -1 ) (p) .

Hence (a * A-1) * y-1 = a * (yA)-1 .

A-1) * B-1 = A-1) * UVEB((a * (6) (a * y-1) = UVEB(a * (yA)-1) -1 . a * (BA) m

Definition 7.2.6 M = (Q, X, ft) be a ffsm. Let a : Q ----> [0,1] and T Q ----> [0,1] . Define

a-1 o T = {x E X* Ia(q) A /t*(q, x, p) A T(p) > 0 for some q, p E Q} .

Definition 7.2.7 Let Q and X be finite subsets. A fuzzy recognizes is a five tuple M = (Q, X, ft, t, C), where

(1) Q is a finite nonempty set of states, (2) X is a finite nonempty set of input symbols, (3) ft : Q x X x Q ~ [0,1] is a function, called the fuzzy transition function, (4) t is a fuzzy subset of Q, i.e ., t : Q [0,1], called the initial fuzzy state, and (5) C is a fuzzy subset of Q, i.e ., C : Q [0,1], called the fuzzy subset of final states .

Clearly, if M = (Q, X, ft, t, C) is a fuzzy recognizer, then N = (Q, X, ft) is a fuzzy finite state machine . We call N the fuzzy finite state machine associated with the fuzzy recognizer M. © 2002 by Chapman & Hall/CRC

338

7. More on Fuzzy Languages

Definition 7.2 .8 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Let x E X* . Then x is said to be recognized by M if VgEQ(t(q) A(VPEQ{w * (q,x,p) AC(p)})) > 0. Lemma 7.2 .9 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Let x E X* . Then x is recognized if and only if there exists p, q E Q such that t(q) A ft * (q, x, p) n C(p) > 0 . 0 Definition 7.2 .10 Let M = (Q, X, /t, t, C) be a fuzzy recognizer . Let L(M) = {x E X* I x is recognized by M} . L(M) is called the language recognized by the fuzzy recognizer M. Lemma 7.2 .11 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Then L(M) = {x E X* I t(q) A /t*(q, x, p) A C(p) > 0 for some q, p E Q} . m Definition 7.2 .12 Let M = (Q, X, /t) be a fuzzy finite state machine. M is called complete if for all q E Q, a E X, there exists p E Q such that ft (q, a, p) > 0. Definition 7.2 .13 Let M = (Q, X, ft, t, C) be a fuzzy recognizer. Then M is called complete if the fuzzy finite state machine associated with M is complete . Let M = (Q, X, ft, t, C) be a fuzzy recognizer such that M is not complete. Let Qc = Q U {t}, where t is an element such that t Q. For all qEQ,let0<mg <1 .Let0<m<1.Define yc :QxXxQ~[0,1]by for all p, q E Q, a E X, such that ft(p, a, q) :?~ 0, ftc(p, a, q) = ft (p, a, q), for all pEQ,aEX ,t c(p, a, t)

-~ m

if V {ft(p, a, q) I q E Q} = 0 if V {ft(p, a, q) I q E Q} > 0

and m if t=p 0 if t :?~ p.

ftc (t, a, p) _ Define ,c : Qc ----> [0, 1] by ~c

© 2002 by Chapman & Hall/CRC

(p) -

t (p) 0

if p t if p=t .

7.2. On Fuzzy Recognizers

339

Define C' : Qc ----> [0,1] by OP) _

C(p) if p 0

EQ

otherwise .

It is easy to see that the fuzzy recognizer Mc = (Qc, complete. Qc is called a completion of M.

X, /tc, T, Cc)

is

Theorem 7.2.14 Let M = (Q, X,

/t, t, C) be an incomplete fuzzy recognizer. Let Mc be a completion of M. Then L(M) = L(Mc) .

Proof. Let x E L(M) . Then 3q, p E Q such that t (p) A /t* (q, x, p) A ~(p) > 0. This implies that tc (p) A ft* (q, x, p) AC(p) > 0. Hence x E L(Mc). Thus L(M) C_ L(Mc) . Now let x E L(Mc). There exists q, p E_ Q such that tc(p) A ft* (q, x, p) A C(p) > 0. This implies that tc(p) > 0. Thus p E Q and so t(p) = tc(p) . Suppose /t* (q, x, p) = 0. Since f,* (q, x, p) > 0 and /t* (q, x, p) = 0, we must have p = t . This is a contradiction since p E Q. Hence /t* (q, x, p) > 0. Since p E Q, C(p) = ~(p) > 0 . Thus t(p) A/t* (q, x, p) A C(p) > 0 and so x E L(M) . Consequently, L(M) = L(Mc) . m Definition 7.2.15 Let M = (Q, X,

/t, t,

C)

be a fuzzy recognizer. Let

S = {q E Q I V {V{L(p) Aw*(p,x, q) I p E Q} I x E X*} > 0}.

Definition 7.2.16 Let M = (Q, X, is called accessible if S = Q.

/t, t, C) be a fuzzy recognizer . Then M

Theorem 7.2.17 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Then M is accessible if and only if (t * X*) (q) > 0 b'q E Q . m

Theorem 7.2.18 Let M = (Q, X,

/t, t, C) be a fuzzy recognizer. Then = 1 t1 o . C ( ) L(M) (2) Let x E X* . Let A = L(M) . Then

x-'A = (t * x) -1 0 ~, where x-'A = {y E X* I xy E A} .

Proof. (1) The proof is straightforward . (2) Let y E (t * x) -1 o C. Then (t * x) (q) A /t* (q, y, p) A C(p) > 0 for some q, p E Q. This implies that (t * x) (q) > 0 and ft * (q, y, p) A C (p) > 0. Now (t * x) (q) = VTEQ{t(r) A /t* (r, x, q) } > 0 and so t(r) A /t* (r, x, q) > 0 for some r E Q . Thus we have t(r) A /t* (r, x, q) A /t* (q, y, p) A C (p) > 0 for some q, p, r E Q . This implies that t(r) A ft * (r, xy, p) A C (p) > 0 for some p, r E Q. Thus xy E A and so y E x- 'A . Now let y E x-'A . Then xy E A and so there exists p, r E Q such that t(r) n[t* (r, xy, p) AC(p) > 0. This implies that [t*(r,xy,p) > 0. Now y,*(r,xy,p) =VaEQ{[t*(r,x,q)nft*(q,y,p)} > 0. Thus there exists q E Q such that /t* (r, x, q) A /t* (q, y, p) > 0, i.e ., /t* (r, x, q) > 0 and ,t* (q, y, p) > 0 . Now (t * x) (q) = v,EQ{t(s) n w* (s, x, q)} >_ t (r) n y,* (r, x, q) > 0. Hence (t * x) (q) A /t* (q, y, p) A C (p) > 0 for some q, p E Q. Thus y E (t * x) -1 o C. Hence x -'A = (t * x) -1 o C. m © 2002 by Chapman & Hall/CRC

complete Now a=is Let Lemma MA complete QA {(L let well LA(A) =(xa)-'A and xEy-'A (x-'A, *= xXX A QA Ex)-1 defined M Esuch x, 7(QA, aaccessible A A There 77L(MA) Let x-'A) XQA~ = [t* Esuch accessible Define oT Then a,that QA, QA X, = by-'A) (A, IS exists Now y-'A that Let By ltA, xx, Let L(MA) = aThere C(x-'A) Cfuzzy =a, E x-1A) E[0,1] Theorem A {x-1A LA, A X*} A n y-'A) recognizer QA = X (xa)-'A {L ifabe 0mAAmAA Na C =C)(u-'A, recognizer *and fuzzy exist by Define = ~ L(M) ais X* (y-1A, x= AC(x-1A) _ Next AIrecognizable = finite = [0,1] Ixonly 7al, CA {y-1A, Let xThen recognizer = Eb,LA 0a1 >E we X*} x, v-'A) by a-1(x-'A) x-'A, if Hence 0MA 1A) an-1)-'A, X*} z-1A) Ax-'A QA show (ua)-'A z-1A >Let otherwise isLet otherwise, nsuch 0a3, x-1A xsubset ~ y-'A, called is Then (xa)-'A QA NA(al xM This En that E=(a,a2a3)-'A) 0[0,1] finite = A C(z-1A) QA that = < (L* More = = is = aja2 an, recognizable u-'A, of implies MA, MA x-'A a1A, (Q, b-1(u-'A) v-'A A by x)-1 such X* L(MA) finite (a1a2 =on X, It a2, isnA, y-'A v-1A > oT =now an, that aFuzzy Then /t, that (a1a2)-1A) This set 0u-'A, complete CA t,=a2an) for A T) A follows xEClearly = <E there shows all Languages if3 QA, E such AX -'A) 1(ub)-'A xy-'A L(MA) aDefine Ea, for exists fuzzy fuzzy that that X* MA ballE

340

7.

Definition .2.19 recognizer Theorem a

.

.

.2.20

.

Proof. L(M) . : NA

. MA 0

ItA(X-'A, b'x-'A,

.

.

nA

.

if otherwise,

CA

if

.

.

~O .

.

.2 .18,

.2.4,

.

. >_ = > . LA(y- 1A)

© 2002 by Chapman & Hall/CRC

= .

.

.

ftA(A,

Thus Now

if

: C(x_1A)

Set recognizer X. v-1A, Hence ltA By D is Let i.

.

:

LA(X_'A) b'x-'A

.

.

.

. .

. .. .

[tA(A, ANA((a,a2)-1A, NA((aia2 . . . . . .AmA mA .

. ..

.

. ..

. .

and p [t* general, and finite (u-1A, EX*, u,[x] all all Erelations on xso On if(u-'A, xvQ) so x, Thus LA(y-'A) uyv x, denote x-'A A M only u-'A, X*1 is Eto EX* with Now Fuzzy on yyxyX*) index is finite i=1,2, Then is of y, L(M) = constructed the if7 EEX* Ethe E(1)x(2) by the y, v-1A) recognizable -A respect LA, by and if A = A (QA, X*, Suppose X*, Since A v-lA relation the v-'A) Recognizers converse for Then uyv = z-'A by -Clearly, for the union itis Consequently, and ifx, only x>xis *U{ X, follows Theorem equivalence all finite, and Let all > X*1 -A y(A, 0, definition to aEEV'x, ft, -A [x] Since by ,0> congruence ifn} Ex, of yu, lta(y-'A, A in -A A -A x, QA, yA t,Hence only 0may (ft* for yif I-A is vthe X* congruence be C_ z-'A) C) the if are where xfor that Eaand Let EThus A X* aall and (q, 6(ux)-'A E definition X*, is congruence The if X*) not proof X*, and equivalent recognizable all is of Q x, A}, class y-1A xyA finite, x, only only X*1 recognizable, = The p) ltA, x, xu-'A, = Erelation be yrelation (ux)-'A xxIt X*1 [t* of where yz-1A) EL(M) X*I L(MA) > true, if with -A is-A following classes if = ltA(u-'A, -A y-A (y-'A, X*, = The 0(ft* Theorem of easily (uxv -A A v-'A if ifof yv-'A laA yrespect denotes iby subset and is Thus and if [x] > (u-'A, Thus if converse Let finite = =aof is 0E defined xThen x, Theorem and 0, seen is Hence Econgruence for finite {[x] ifonly A only (uy)-'A aassertions z-'A) 7A x, QA) if and X*1 the xof congruence index y,the if x, v-'A) only the to C_Iand that -A X* Vu, and if then ifv-'A) xis on C(x-'A) equivalence C(z-'A) X* --A= xDefine fuzzy ft* Eset y, 6>von trivial X* if -A Let Define only -A only X*} (q, Vu 0> xLet then Eof Define (uxv are X* relation -A MA by is > 0yy,X*, recognizer X*1 all Eif if relation Also, -A = 0ap) if m A equivalent Sincexfor >uyv For y(uy)-'A Eand congruence be X* C(z-'A) if congruence = > class a0uxv tion However, all A the and yxifU{[aj] 0EBy relation of only xEfor if X* Hence x,Since of A fuzzy finite Eis with MA, only X*, and yX* the for all y, by of >Eif A =

7.2.

341

Hence definition

.

. [

and 0

.

.

Let X*, . q, then -A only relation in

.2 .7. .

. .

.

Theorem recognizer for if for all

.2.21

Proof. if for v-'A . [t*

Let

Theorem (1) (2) classes (3) of

.2.22

Proof. the X*1 (2)x(3) : index . respect (3)x(1) : let ai

.

.

.

.e., .2.20.

.

.

. .

. .

. .

.

.

.

.

.

.

. . : .2 .8,

.

.

.

. .. .

.

© 2002 by Chapman & Hall/CRC

.

. .

7. More on Fuzzy Languages

342

finite index, Q is a finite set. Let 0 < m, n, c <_ 1. Define ft : Q x X x Q [0,1] by m

ft([x], a, [y]) _

0

b'[x], [y] E Q, a E X. Define t : Q L

x

-

if [xa] = [y]

otherwise,

[0,1] by n

if [x] =[A]

0

otherwise,

V[x] E Q. Define C : Q ----> [0,1] by if xEA

otherwise, V[x] E Q. Set M = (Q, X, ft, t, C) . Then M is a fuzzy recognizer . Let x E A. Now t([A]) = n > 0 and C([x]) = c > 0. Let x = aja2 . . . an, a2 E X for all i . Now ft*([A], x, [x]) > ft([A], a1, [al])A ft([a1], a2 , [a la2])A ft([aja2], a3, [aja2a3])n . . . A ft([ala2 . . . an-111 any [aja2 . . . an-Ian]) = m A . . . A m = m > 0. Thus i([A]) A [t* QA], x, [x]) A C ([x]) > 0. This implies that x E L(M) . Now let x E L(M) . There exist [y], [z] E Q such that t([y]) A [t * ([y],x, [z]) n C([z]) > 0 . Thus t([y]) > 0, ft * ([y],x, [z] > 0, and C([z]) > 0. By the definition of t, it follows that [y] = [A] . Thus [t* QA], x, [z]) = [t* ([y], x, [z]) > 0 and so [x] = [z] by the definition of ft* . Hence C ([x]) = C([z]) > 0 and so x E A. Consequently, A=L(M) . m

Definition 7.2 .23 Let M1 = (Q1, X,

p1, t1, CI) and M2 = (Q2, X, ft2, t2, C2) be fuzzy recognizers . Then M = M1 UM2 = (Q 1 x Q2, X, ft, nft2, t1 nt2, C1 V

Theorem 7.2.24 Let M1 = (Q1, X, be fuzzy recognizers . Then

p1,

t1, CI) and M2 = (Q2, X, ft2, t2, C2)

(ft, nft2) * ((g1,q2),x,(PI,p2)) =fti(g1,x,PI) nft2(g2,x,p2)

V((g1,q2),x, (PI, P2)) E (Q1 x Q2) x X x (Q1 x Q2) .

Proof. Let ((q1, q2), x, (p1, p2)) E (Q1 x Q2) x X x (Q1 x Q2) and xI = n. If n = 0 or n = 1, then the result is true by definition. Assume it is true for xI =n . Let a E X. Now (ftLnft2) * ((gL,q2),xa,(P1,p2))

= V~(ftLn ft2)*((gL,q2), x, (rL, r2)) A (ft, n ft2) ((r1 , r2), a, (PI, p2)) (r1, r2) E Q1 x Q2} V{fti (q1 , x , r1) n ft2 (q2 , x , r2) n ft1(r1,a,p1) AIt2(r2,a,p2) (r1, r2) E Q1 x Q2} V {fti(g1, x, r1) n ft1(r1, a,p1) I r1 E Q1} n {ft2(g2,x,r2)A ft1(r2,a,p2) I r2 E Q1} fti(g1,xa,p1) nft2(g2,xa,p2) .

© 2002 by Chapman & Hall/CRC

0

7.2 . On Fuzzy Recognizers

34 3

Theorem 7.2 .25 Let Ml = (Q1, X, pl, t i , C1) and M2 = (Q 2 , X, ft2, L2, C2) be complete fuzzy recognizers . Then L(MI U M2 ) = L(MI) U L(M2) . Proof. Let x E L(MI U M2 ) . Then 3(ql, q2), (PI, P2) E Q1 X Q2 such that (LI AL2)(gl,g2) n (ft, A w2)*((gl,q2),x, (PI, P2)) A (C1 V C2)(P1,P2) > 0 . Thus El (ql, q2), (pl, p2) E Q1 X Q2 such that n ft2(g2, x,P2) n (C I (PI) V C2(P2)) > 0tl(gl) n L2(g2) n ft(gl,x,pl) 1 This implies that 3(gl,q2), (PI, P2) E Q1 X Q2 such that (tl(gl)

o ft(gl,x,pl) 1

n C1(Pl)) V (t2(g2) n ft2(g2,x,P2) n C2(P2)) > 0-

Hence EI(gl, q2), (P1,P2) E Q1 X Q2 such that either (t1(gl) n fti(gl, x, pl) n ~1(pl)) > 0 or (t2(g2)nft2(q2,x,P2)AC2(P2)) > 0 . Thus x E L(MI) UL(M2 ) . Hence L(MI U M2 ) C_ L(MI) U L(M2 ) . Now let x E L(MI) U L(M2 ) . Then x E L(MI) or x E L(M2) . Suppose x E L(MI) . Then 3gl,pl E Q1 such that tl(gl) nfti(gl,x,pl) nC 1 (pl) > 0 . Now 3q2 E Q2 such that L2(g2) > 0 . Since M2 is complete, 3P2 E Q2 such that ft2 (q2, x, p2 ) > 0 . Thus we have tl(gl) nL2(g2) > 0, fti(gl,x,pl) nft2(g2 X, P2) > 0, and CI (PI) V C2 (P2) > 0 . Hence (tl n L2)(gl,q2) n (ft l n ft2)*((gl,q2),x, (PI, P2)) n (C1 V C2)(P1,P2) > 0 and sox E L(MIUM2) . Thus L(Ml)UL(M2) C_ L(MIUM2) . Consequently, L(MI U M2 ) = L(MI) U L(M2) . m Definition 7.2 .26 Let Ml = (Q1, X, pl, t i , C1) and M2 = (Q 2 , X, ft2, L2, C2) be fuzzy recognizers. Then M = Ml n M2= (Ql X Q2, X, ft, A ft2, t j n Theorem 7.2 .27 Let Ml = (Q1, X, pl, t i , C1) and M2 = (Q 2 , X, y 2 , t2, C2) be fuzzy recognizers . Then L(M, Proof. Now x E L(M, such that

n

M2) = L(MI)

n m2)

n

L(M2) .

if and only if 3(gl,q2), (PI , P2) E Q1 X Q2

(LI n L2)(gl,g2) n (ft, n ft2)*((gl,q2),x, (P1,P2)) n (C1 n C2)(p1,P2) > 0 if and only if 3(gl,q2), (PI, P2) E Ql X Q2 such that tl(gl) n L2(g2)

o fti(gl, x,pl) n ft2(g2, x,P2)

© 2002 by Chapman & Hall/CRC

n (C I (pl) n C2(P2)) > 0

344

7. More on Fuzzy Languages

if and only if 3(qi, q2), (PI, P2) E Qi X Q2 such that (ti(gi)

o fti(gi,x,PI)

n C,(Pj)) n (L2 (q2) n ft2(g2,x,P2) n C2 (P2)) > 0

if and only if x E L(Mi) n L(M2) . Hence L(MI n M2) = L(MI) n L(M2) . In the proofs of the next two theorems, we use the fact that if M = (Q, X, p) is a ffsm, q, p E Q, and x = xix2 . . . xn, where xi E X, 2 = 1,2 . . . . n, then ft* (q, x,p)

=

V {ft(q, xi, ri) n ft(ri, x2, r2) A . . . nft(rn-1, xn,P) I ri E Q, i = 1, 2 . . . . n - 1} .

The above fact follows from Lemma 6 .2 .2 . Theorem 7.2 .28 Let A, B C_ X* . If A and B are recognizable, then A - B is recognizable. Proof. Let Mi = (Qi, X, pi, ti, C I ) and M2 = (Q2, X, [t2, L2, C2) be recognizers of A and B, respectively. Let Ii = {q E QS I ti(q) > 0 } and TZ = {q E Q2 I Ci (q) >

for i = 1, 2 . Define t ° : Qi x P(Q2) ~ [0,1] by V (q, P) E Q i o

)-~ 0

(gP

ti (q)

P(Q2),

if P~0 if P = Ql .

Define C ° : Qi X P(Q2) ----> [0,1] by V(q, P) E Qi X P(Q2) C'~'(q, P) = V {C2 (P) I P E P} . Define p l Ap 2 : (Q i X P(Q2)) X X X (Q i X P(Q2)) ----> [0,1] as follows :

wiAw2((q, P), c, (q', s2 (P)) _

fti (q~ c, q') n V {ft2 (P1 c1 P') I if q' P E Pp' E s2 (P)} fti (q, c, q')

w1Aw2((q,P),c,(q',s2(P)UI2))

=

Ti, P z,4 0

if q' ~ Ti, P = 0; wi(q,c,q')AV{ft2(P,C,P') IPEP,p'Es2(P)UI2}

if q E Ti, where s2 (P) = {P' E Q2 I lt2(p, C, P) > 0, p E P} ; and fti0ft2 is 0 elsewhere .

© 2002 by Chapman & Hall/CRC

7.2 . On Fuzzy Recognizers

34 5

Let aEAandbEB.Then 3gE11,q'ET1,pE12,p'ET2such that lti (q, a, q~) > 0 and Nt2 (p, b, p~) > 0 . Now ab = ul . . . u21 u2 1 +1

. . .

UZ2 . . . u2j u 2j +1

. . .

u2,k v

where ul E X, v E X*, ij is the smallest such that ij > . . . > il > io and ul . . . u2j E A for j = 1, . . . , k, io = 1 and there does not exist ik+1 such that ul . . . u2,k+1 E A. Let o = ul . . . u2 1 and uj = u2j +1 . . . u2j+1, j = 1, . . . , k - 1 . For all x E X* and VP E P(Q2), let sx (P) = {p E Q2

I

N2(AX,p) > 0,P E P} .

Let Rab

=

s2 (s2

-1 (s2 -2 (. . . s2° (I2) . . . )))U s2 (s2'°-1( s2b-2 ( . . . . .. . . . U s2 (12) . SU1(12) ))) U

(7.1)

Let q" E Q1 be such that [ti (q', b, q") > 0. Now a = ul . . . u2 j for some ij. Let (12)a = s2j-1 (s2j-2( . . .s2°(12) . . .)) . Then (wIAw2)* ((q, 0), ab, (q",Rab))

=

>

V{(wIAw2)*((q, 0), a, (v, R)) n(N'1AN'2)*((v, R), b, (q",Rab)) (v, R) E Q1 X 'P(Q2)1 (wlow2)*((q, 0), a, (q' , (12) a)) n(N'10N'2)*((q~, (12)a), b, (q", Rab)) 0

by the definition of (12)a and since Rab is a continuation of (12)a, i.e ., Rab = ((12)a)b " Conversely, let x E L(M10M2) . Then L ° (q,

O) n (wlow2)*((q, 0), x, (q ", (12)x)) > 0,

where fti(q, x, q") > 0 and (12)x is defined as follows: we have that x = uv, where u = i o . . . ukll and where the uj and v are defined as above (k exists since x E L(M1 0M2 )) . Then (12)x is the right-hand side of equation (7 .1) . Since x E L(M10M2), (ft1 0ft2) * ((q, 0), x, (q", (12)x)) > 0 and so 3j, EIP E 12, s2 2 Thus

~+1 . . .U1~+1v'Tj+1 . . .v,mv(p)

u2j+ 1 . . . u2j+1 u2j+1

. ..

ukv

E (12 ) x

nT2 "

E B . Since ul . . . u2 , E A, x E A " B.

m

Theorem 7.2 .29 Let A C X* be recognizable . Then A* is recognizable.

© 2002 by Chapman & Hall/CRC

34 6

7. More

Proof. Let

M = (Q, X, /t, t,

X x P(Q) ----> [0,1]

,t'(P, ,t'(P,

as follows :

on Fuzzy Languages

C) be a recognizer of A. Define

/t' : P(Q) x

c, s'(P)) = V{w(p, c, q)

I p E

P,

q E s'(P) I if s , (P) n T = 0

c, q)

I p E

P,

q E

c, sc(P) U I)

= v{w(p, It'(P,

sc(P)

U I} if

sc(P)

n T z,4

0

c, P') = 0 otherwise,

where c E X, P, P E P(Q), T = {q E Q I C(q) > 0}, I = {q E Q I t(q) > 0}, and sc(P) = {q E Q I ft(p, c, q) > 0, p E P} . Define C' : P(Q) ----> [0,1] by C'(P) = V{C(p) I p E P} and t' : P(Q) [0,1] by t'(I) = V{ t(q) I q E I}, L' is 0 otherwise, T' = {P E P(Q) I C(P) > o} = {P E P(Q) I P n T :A 0} . Let M' = (P(Q), X, ft', t', C') . Let x E A* . Then x = al a2 . . . a , where a2 E A and if a2 = a21 . . . ail,, for a2,, . , ail,, E X, 3" m2 < n2 such that . . . . . . , n. Let Ix = sa,, ( . . . sae (sal (I) U I) U . . . ) U I . = a21 a2-E A, i 1, 2, > l,s a l ( Now p' (I, x, Ix) ft* (I,a I)UI)A N'*(,a , (I)UI,a2 ,sa2(sal(I)UI)U I)n . . . n t'* (8 a,,-1 (. . . sae (,a1 (I) U I) U . . . ) U I, an , Ia) > 0 since 3i E I, 3p E sat (I) n T such that /t* (i, al, p) > 0 (because al E L(M)), 3i' E I, 3p' E sae (sal (I) U I) n T such that /t* (i', a2,p') > 0 (because a2 E L(M)) and so on. Thus x E L(M) . Hence A* C_ L(M) . Let y E L(M) . Then 3P E T' such that y'* (I, y, P) > 0. Now suppose 3k such that y = ul . . . UZ1UZ1+1 . . . UZj u2j +1 . . . UZkv,

where ul . . . uz1 E A with i l smallest and the ii are the smallest such that u2j-1+1 . . . u2j A, j = 2, . . . , k. Let ul = ul . . . uz1 and uj E u2j-1+1 . . . u2j, j = 2, . . . , k. Then ul . . . u2j-1 E A* and P = (I) ., where (I)~ = 81 (8'k( . . . s -2 ( 8 -1 (1) U I) U . . . I) U I) U I . There exists j, 3q E I such that /t* (q, uj . . . ukv, p) > 0 for some p E (i) y nT . Thus ft'*((I)~1 .. .~--1, uj . . . ukv, (I) y) > 0. Hence uj . . . ukv E A* . If no such k exists, then a similar argument shows that y E A. Thus y E A* . Hence L(M') C A* . Consequently L(M) = A* . m If L(M) is the language recognized by an incomplete fuzzy recognizer, we showed that there is a completion Mc of M such that L(Mc) = L(M) . We also showed that if A is a recognizable set of words, then there is a complete accessible fuzzy recognizer MA such that L(MA) = A. If A and B are recognizable sets of words, then A - B and A* are recognizable . These results are significant in that they lay the groundwork for determining methods of decomposing recognizable sets and thus for giving a constructive characterization of recognizable sets. In particular, we hope to show an analog of Kleene's result for fuzzy finite state machines, namely, that the class of recognizable subsets of X* equals the class of all regular subsets of X* . © 2002 by Chapman & Hall/CRC

7.3 . Minimal Fuzzy Recognizers 7 .3

34 7

Minimal Fuzzy Recognizers

In this section, we show that for any fuzzy recognizer M there is a deterministic fuzzy recognizer Md with the same behavior, Theorem 7.3 .8 . Then we show that there is complete accessible deterministic fuzzy recognizer MdA with the same behavior as M and Md and that is minimal, Theorem 7.3.11. Our long term goal is to develop methods of decomposing a recognizable set of a fuzzy finite state machine . One method would be to follow along the lines of Kleene to give a constructive characterization of a recognizable set . In this section, we lay the foundation for the accomplishment of our goal . Clearly, if M = (Q, X, ft, t, C) is a fuzzy recognizer, then N = (Q, X, ft) is a fuzzy finite state machine . We call N the fuzzy finite state machine associated with the fuzzy recognizer M. Definition 7.3 .1 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Let

S = {q E Q I V {V{L(p) n [t * (p, x, q) I p E Q} I x E X*} > 0} . Definition 7.3 .2 Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Then M

is called accessible if S = Q.

Definition 7.3 .3 Let M = (Q, X, /t) be a ffsm. Let a : Q ----> [0,1], x E X*,

and A C X* . Define (1) a * x : Q ----> [0,1] by

* (a * x) (p) = V{a(q) n ft (q, x, p) I q E Q} VP E Q, (2)a*A :Q----> [0,1] a*A=UxEAa*x. Let M = (Q, X, /t, t, C) be a fuzzy recognizer. Then by Theorem 7.2 .17, M is accessible if and only if (t * X*) (q) > 0 b'q E Q . Definition 7.3 .4 Let A C_ X* . Then A is called recognizable if 3 a fuzzy

recognizer M such that A = L(M) .

Let Q and X be finite nonempty sets and let /t : Q x X x Q ---> [0,1]. /t is called a fuzzy function of Q x X into Q if for all q E Q, a E X, if ft(q, a, p) > 0 and ft(q, a,p~) > 0 for some p, p' E Q, then p = p~ . Theorem 7.3 .5 Let M = (Q, X, /t, t, T) be a fuzzy recognizer. Then /t is

a fuzzy function of Q x X into Q if and only if /t* is a fuzzy function of Q x X* into Q.

© 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

34 8

Proof. Suppose ft is a fuzzy function. Let q E Q and x E X*. Suppose p* (q, x, p) > 0 and /t* (q, x, p') > 0 for some p, p' E Q. If x = A, then p = q = p' . Suppose x z,4 A. Let x = ala2 . . . an E X*, a2 E X. There exists gi~g2~ . . . ,qn-l~gi~q2, . . . q'n -1 E Q such that ft(g, al, gl) nft(gl, a2, g2) n . . .A /t(gn-l, an,p) >Oand/ (g, ai, gi)nft(gi, a2, g2)n . . .h/t (gn-1, an, p~) > 0. This implies that ft(q, al, ql) > 0, ft(gi, a2+1, qZ+1) > 0, i = 1, 2, . . . , n-2, 1, 2, . . . , n - 2, ft(gn-1, an,p) > 0, ft(q, a,, qi) > 0, ft(q', ai+l, q2+1) > 0, i 0. Now 0 and ft (q, 0. Since ft is a ft(gn_1, an,p') > P(q, al, ql) > al, qi) > fuzzy function, ql = ql . Suppose qj = qj , j = 1, 2, . . . , i, i < n - 2. Now ft(gi, a2+1, qZ+1) > 0 and ft(gi, a2+,, q2+1) > 0 implies that qZ+1 = qZ+1 since ft is a fuzzy function . Hence by induction qj = qj , j = 1, 2, . . . , n-1 . Hence * f (qn-1, an,p) > 0 and f (qn-1, an, P') > 0 implies that p = p' . Thus ft is a fuzzy function . The converse is trivial. m

Definition 7.3.6 A deterministic fuzzy necognizen is a fuzzy recog-

nizer Md = (Qd, X, ft, t, T) such that (1) there exists a unique so E Qd such that t(so) > 0 ; so is called the initial state, (2) ft is a fuzzy function of Q x X into Q, and * (3) for all x E X*, there exists a unique qx E Qd such that ft (so, x, qx ) > 0.

Let Md = (Qd, X, ft, t, T) be a deterministic fuzzy recognizer. Let Cd = I T(d) > 0} . Cd is called the set of final states of Md .

{q E Qd

Theorem 7.3.7 Let M = (Q, X, ft, t, T) be a fuzzy recognizer. Suppose M is complete and ft is a fuzzy function of Q x X into Q. Let so E Q. Then the following are equivalent. (1) For all a E X, there exists a unique qa E Q such that ft(so, a, qa) > 0 . (2) For all x E X*, there exists a unique qx E Q such that ft * (so, x, qx) > 0.

Proof. (1)x(2) : Let x E X* and Ixl = n. If x = A, i .e., n = 0, then * * ft (s o , A, s o ) = 1 > 0 and if ft (s o , A, p) > 0, then by the definition of ft * , so = p. Suppose the result is true for all y E X* such that jyj < Ixl , where Ixl = n >_ 1 . Let x = ya, where y E X*, a E X, jyj = n - 1. By the induction hypothesis, there exists a unique q. E Q such that ft * (so, y, q,) > 0. Since M is complete, there exists p E Q such that ft (q., a, p) > 0. Thus * * ft (s o , x, p) >_ ft (s o , y, q,) A ft (q., a, p) > 0. Since ft is a fuzzy function of Q x X into Q, ft * is a fuzzy function of Q x X* into Q by Theorem 7.3.5 . Thus if ft * (so, x, p) > 0 and ft * (so, x, p') > 0 for some p,p' E Q, then p = p' . It now follows that there exists a unique qx E Q such that ft* (so, x, qx) > 0 . (2)x(1) : Immediate . m

Theorem 7.3.8 For each fuzzy recognizer Mn = (Q, X, ft, t,

one can construct a deterministic fuzzy recognizer Md = (Qd, X, ft, t, ~) such that L(Md) = L(M.) .

© 2002 by Chapman & Hall/CRC

7.3 . Minimal Fuzzy Recognizers

34 9

Proof. For all x E X*, set Qx =

{q'

E Q I Elq E Q such that t(q) A /t* (q, x, q') > 0} .

Then QA = {q E Q I t(q) > 0} .

Let Qd

= LQx I x E X*} .

2 l`wx) _

Define i : Qd ~ [0,1] by VQx E Qd, V{t(q) I q E QA}

0

if x = A if x :?~ A

Let Cd = {Qx E Qd I C(q) > 0 for some q E Qx }. Define T : Cd VQx E Cd, T(Qx) = V{C(q) I q E Qx} . Define v : Qd X X X Qd d(Q,, a, Qx) E Qd X X X Qd, v(Q,, a, Qx) =

[0,1] by [0,1] by

V{ft* (q, y, q') A ft(q', a, r) I q E Qy, q' E Q, r E Qya} if x = ya

0

otherwise .

Let Md = (Qd, X, v, i,T) . We now show that L (M.) = L(Md) . Now x E L(Mn) if and only if t(q)nft* (q, x, q')AC(q') > 0 for some q, q' E Q if and only if C (q') > 0 for some q' E Qx if and only if T(Qx) > 0. It suffices to show that v* (QA, x, Qx) > 0 for then x E L(Md) if and only if i(QA) A v*(QA,x,Qx) AT(Qx ) > 0 if and only if T(Qx) > 0 (since i(QA) > 0 and v* (QA, x, Qx) > 0) if and only if x E L(Mn) . We show v* (QA, x, Qx ) > 0 by induction on Ix I . Suppose Ix I = 0 . Then x = A and v* (QA, A, QA) = 1 > 0. Suppose Ixj > 1 and the result is true for all y E X* such that Iyj < Ixj . Let x = ya, where a E X. Then v* (QA, x, Qx) = V{v* (QA, y, r) A v(r, a, Qx) I r E Qd } and v* (QA, y, Qv) > 0 by the induction hypothesis. Hence it suffices to show that v(QV, a, Qx) > 0, but the latter inequality is true by the definition of v since x = ya . m Let Md = (Qd, X, ft, t, T) be a complete accessible deterministic fuzzy recognizer and let so denote the initial state of Md . For x E X*, we let qx E Qd denote the unique state such that ft* (so , x, qx) > 0 . Theorem 7.3.9 Let Md = (Qd, X, ft, t, T) be a complete accessible deter-

ministic fuzzy recognizer. Let so be the initial state of Md . For all q E Qd, let q-1 o T = {y E X* I y,* (q, y, p) A T(p) > 0 for some p E Qd}. Let A= L(Md) . (1) For all x E X*, x-1 A= qx 1 oT . (2) Let q E Qd . Then there exists x E X* such that q-1 oT = x-1 A and q=qx (3) Let q E Qd be such that T(q) > 0. Then q-1 o T = x -1 A for some xEA . (4) A=A-1 A=so 1 o T . (5) Let x = a1a2 . . . an E X*, where a2 E X, i = 1, 2, . . . , n. Then

/a'*(so,x,gx) = ft(so,a,,gal) Aft(gal,a2,qalaz) A . . . Aft(ga l . . .a,,_l ,an,qx) .

© 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

35 0

Proof. (1) Let y E x-1A . Then xy E A and so t(so) A /t* (so , xy, p) A > 0 for some p E Qd . This implies that ft * (so, xy, p) > 0. Thus there exists q E Qd such that /t* (so, x, q) A /t* (q, y, p) > 0. Hence /t* (so , x, q) > 0 . Since Md is deterministic, q = qx . Hence t(so) A/t*(so, x, qx)A /t*(qx , y, p) A 1 1 1 T(p) > 0. Thus y E qx o T. Hence x-1A C_ qx o T. Now let y E qx o T . Then ft * (gx,y,p) AT(p) > 0 for some p E Qd. This implies that t(so)A la' * (qx, y, p) n T (p) > 0. Now [t* (so, x, qx) > 0. Hence t(so)A [t* (80, x, qx) A y,* (qx , y, p) AT(p) > 0 for some p E Qd. Thus t(so) A /t* (so , xy,p) AT(p) > 0 for some p E Qd. Hence xy E A or y E x-1A . It now follows that x-1A = qx 1 o T. (2) Let q E Qd . Since Md is accessible, there exists x E X* such that * (so, x, q) > 0. Since Md is deterministic, it follows that q = qx . By (1), ft x-1A = qx 1 o T = q-1 o T . (3) By (2), q-1 oT = x -1A for some x E X* . Since T(q) > 0, /t* (q, A, q) A T(q) >0andsoAEq-1 oT . Thus AEx-1A and so xEA . (4) Let y E A. Then t(so) A/t*(so, y, q) AT(q) > 0 for some q E Qd. This implies that ft* (so , y, q) A T(q) > 0 for some q E Qd and so y E so1 o T . Thus A C sot o T. Now let y E sot oT. Then W (so, y, q) AT(q) > 0 for some q E Qd. Thus t(so) A [t* (so, y, q) AT(q) > 0 for some q E Qd and so y E A . It now follows that A = so t o T. (5) Now N'* (s0, x, qx) = V {ft(s0, al, ql) n ft(gl, a2, q2) . . . Aft (qn-1, an, qx) ql ~ q2, . . . ~ qn-1 E Qd} . Since Qd is finite, there exists q1, q2, . . . , qn-1 E Qd such that y,*(so, x, qx ) = y,(so, al, ql)n ft(gl, a2, q2) . . . n ft (gn-1, an, qx) . Since /t*(so, x, qx) > 0, ft(so, al, ql) n ft(gl, a2, q2) n ft(g2, a3, q3)n . . . A N(qn-1, an, qx) > 0. This implies that ft(so, a1, ql) > 0 and so q1 = qal since Md is deterministic . Now ft(so, a1, qa l )A f (qal, a2, q2) > 0 implies that [t* (so, a1a2 , q2 ) > 0 and since Md is deterministic q2 = gala, . We see that an argument by induction will yield qZ = gal ...ai for all i = 1, 2, . . . , n - 1 . T(p)

Theorem 7.3.10 Let A be a recognizable subset of X* . Then there exists a complete accessible deterministic fuzzy recognizer MdA such that L(MdA) _ A. Proof. Let Md = (Qd, X, ft, t, T) be a deterministic fuzzy recognizer such that A = L(Md) . Since Md is deterministic, there exists a unique so E Qd such that t(so) > 0. Let QdA = {x-1 A I x E X*}. Let MA =V{ft(q,a,p) I q,p E Q, a E X} . Then 1 >_ MA > 0. Let 0 < to <_ 1. Define ydA : QdA x X x QdA ~ [0,1] by ma ftda(x-1A, a, y-'A) _ ~ 0

© 2002 by Chapman & Hall/CRC

if (xa) -1A = y-1A otherwise,

7.3. Minimal Fuzzy Recognizers

351

V'x - IA, y - IA E QdA, a E X. Define QdA : QA ~ [0,1] by l A) _ t(so) - { 0

LdA(x

if x-'A = A

otherwise,

V'x- 'A E QA . Define TdA : QA ~ [0,1] by TdA(x-'A)

_

CA

0

if x E A

otherwise .

Set MdA = (QdA, X, NdA, tdA, TdA) . Next we show that MdA is a complete accessible fuzzy deterministic recognizer such that L(MdA) = A. Clearly QA(A) = t(so) > 0 and LdA(x- 'A) = 0 if A :?~ x- 'A . Thus MdA has a unique initial state. Let x- 'A, y-'A, u- 'A, v-l A E QdA, a, b E X. Let (x- 'A, a, y- 'A) _ (u -'A, b, v-'A) . Then x- 'A = u-'A, y- 'A = v-'A, and a = b. Now (xa) -'A = a-'(x- 'A) = b- '(u -'A) = (ub) -'A . Hence (xa) - 'A = y-'A if and only if (ua) -l A = v - 'A . This shows that PdA is well defined . Let x -I A, y - IA, z-IA E QdA and ltdA(x-'A, a, y- 'A) > 0 and PdA(x-' A, a, z-'A) > 0 .

Then y- 'A = (xa) - 'A = z-'A . Hence MdA is deterministic . By Theorem 7.3.9, QdA is a finite set . Clearly MdA is a complete accessible fuzzy recognizer. Let x E A. Then TdA(x-'A) = CA > 0. Let x = ala2 . . . an, a2 E X for all i . Now ltdA(A, x, x -' A)

>_ = = >

ladA(A, al, al'A) A ttdA(al 1A , a2, (ala2) -'A)A PdA((a,a2)`A, a2, (a,a2a3)`A) A . . . nltdA((ala2 . . . an_l) -l A,an,(ala2 . . . an) - 'A) mAAmAA . . .AmA mA 0.

Thus LdA(A) A ltdA(A, x, x -1 A) A TdA(x -1 A) > 0. This implies that x E L(MdA) . Now let x E L(MdA) . There exists z- 'A E QdA such that LdA(A) A ltdA(A, x, z- I A) ATdA(z -'A) > 0 . Thus LdA(A) > 0, ttdA(A, x, z- 'A) > 0, and TdA(z-'A) > 0. Now ltdA(A, x, z- 'A) > 0 implies that x - IA = z-IA by the definition of ltdA . Hence TdA(x-'A) = TdA(z -'A) > 0 and so x E A. Consequently, A = L(MdA) . 0 Theorem 7.3.11 Let A C X* be recognizable . Suppose Md = (Qd, X, ft,

t, T) is a complete accessible deterministic fuzzy recognizer with behavior A. Let MdA be the complete accessible deterministic fuzzy recognizer as constructed in the proof of Theorem 7.3.10. Then 3 a function f : Qd ~ QdA such that

© 2002 by Chapman & Hall/CRC

352

7. More on Fuzzy Languages (1)

(2) {X - ' A (3) (4)

f (so) = A ; f -' (FdA) = Fd, where Fd = {q E Qd I T(q) > 0} E QdA I TdA(x - 'A) > 0} ; ltdA(f (q), a, f (p)) >_ ft (q, a, p) for all p, q E Qd, a E X ; f is surjective .

and

FdA =

Proof. Define f : Qd ~ QdA by b'q E Qd, f (q) = q -1 o T . By Theorem 7.3.9(3), f maps Qd into QdA . (1) f (s o) = so' o T = A by Theorem 7.3.9(4) . (2) Let q E Fd . Then T(q) > 0. There exists t E A such that q-' o T = t - 'A by Theorem 7.3 .9(3) . Now t E A = L(MdA ) . Hence wdA(A, t, y 'A) A TdA(y 'A) > 0 for some y-1 A . This implies that wdA(A, t, y - 1 A) > 0 and TdA(y - 'A) > 0. Now wdA(A, t, y - 'A) > 0 implies that t -1 A = y-1A and TdA(y 'A) > 0 implies that y-1 A E FdA . Hence f (q) = q-' o T = t - 'A = y - 'A E FdA, i .e ., q E f - '(FdA) . Suppose q E f - '(FdA) . Then TdA(q - ' o T) = TdA(f (q)) > 0. Now q - ' o T = t- 'A for some t E X* . Thus TdA(t - 'A) > 0 and by the definition of TdA it follows that t E A . Since t E A, A E t - 'A = q -1 o T . Hence /t* (q, A, r) A T (r) > 0 for some r E Qd . This implies that q = r . Thus T(q) > 0 and so q E Fd . Consequently f _' (FdA) = Fd . (3) Let q, p E Qd . Now f (q) = q -1 o T and f (q) = p -1 o T . By Theorem 7.3.9, there exists x, y E X* such that q -1 oT = x - 'A and p-' oT = y-'A . Let a E X. Suppose lltdA(f (q), a, f (p)) = 0. Then ltdA(x-'A, a, y - 1 A) = 0 and so (xa) - 'A z,4 y -1 A . We claim that ft (q, a, p) = 0. Suppose ft(q, a, p) > 0. Either there exists t E (xa) - 'A such that t y - 'A or there exists t E y - 'A such that t ~ (xa) - 'A . First suppose that there exists t E (xa) - 'A such that t ~ y - 'A. Since xat E A, /t* (so, xat, r) A T (r) > 0 for some r E Qd . Thus ft* (so, xat, r) > 0. This implies that there exists q', p' E Qd such that /t* (so, x, q') A N (q', a, p') A /t* (p', t, r) > 0 . Since Md is deterministic, q' = q (note this follows from the proof of Theorem 7.3.9 by the choice of x) . Thus ft (q, a, p') > 0 and since Md is deterministic, it follows that p = p' . Hence ft * (p, t, r) > 0 . By the choice of y (as in the proof of Theorem 7.3.9), ft* (so , y, p) > 0. It now follows that ft* (so , y, p) A y* (p, t, r) A T(r) > 0 and so t E y - 1 A, a contradiction. Now suppose there exists t E y - 'A such that t ~ (xa) -'A . Then xat ~ A . Now yt E A . Thus /t* (so, yt, r) A T (r) > 0 for some r E Qd . Thus there exists p" E Qd such that ft * (so, y, p") A ft * (p", t, r) > 0. From this it follows that p = p" . Thus /t* (p, t, r) > 0. Hence y,* (so, x, q) A ft (q, a, p) A /t* (p, t, r) A T (r) > 0 and so xat E A, a contradiction. Hence ft (q, a, p) = 0 . From the definition of /tdA, it now follows that lltdA(f (q), a, f (p)) > ft (q, a, p) . (4) Let x- ' A E QdA . By Theorem 7.3 .9, x-' A = qx' o T . Thus f is surjective. We regard the recognizer MdA as being a minimal complete recognizer of the recognizable subset A, where the term "minimal" refers to © 2002 by Chapman & Hall/CRC

7.4 . Fuzzy Recognizers and Recognizable Sets

353

the properties described in Theorem 7.3 .11 ; in particular, (4) implies that IQdAI <_ IQdj

7.4

Fuzzy Recognizers and Recognizable Sets

In the next five sections, we present the work of [118] . In [118], the definitions of a fuzzy recognizer, Definition 7.4 .1, the recognition of a word, Definition 7.6 .9, accessibility, Definition 7.7.1, and so on are equivalent to the ones given above, i .e., Definition 7.2 .8, Definition 7.2 .10, and Definition 7.2 .16, respectively. However the approach differs and additional results are obtained. Hence we use notation that is compatible with that of [118]. The idea of a reversal, inverse image, accessible part, coaccessible part, and trim part regarding fuzzy recognizers are introduced and their properties are discussed . We characterize the words recognized by a fuzzy recognizer and prove fuzzy recognizability of several crisp sets. We prove that if a set of words is recognized by a fuzzy recognizer, then it is also recognized by its accessible and coaccessible parts . We prove similar results for a complete fuzzy recognizer. We also give a procedure to construct a complete fuzzy recognizer from a given fuzzy recognizer with the property that they recognize the same subset . Let M = (Q, X, /t) denote a fuzzy finite state machine and f : (X')* X* be a monoid homomorphism, where X' is a nonempty set. If f (X') C X U {A}, then f is called a fine homomorphism. Definition 7.4.1 A fuzzy X-recognizes of a fuzzy finite state machine M = (Q, X, /t) is a triple .M = (M, t, T), where t and T are fuzzy subsets of Q. t is called a fuzzy set of initial states and T is called a fuzzy set of final states . A fuzzy X-recognizes will be called a recognizer when the set of input symbols X is understood. Let M = (Q, X, /t) and M' = (Q', X', /t') be fuzzy finite state machines . Define ft x~ :(QxQ')x(XxX')x(QxQ')~[0,1] as follows :

w x ,a ((p, p), (a, a), (q, q)) = ft (p, a, q) A ,a (p, a, q) .

Recall that the fuzzy finite state machine Mx M' = (Q x Q', X x X', ft x /t') is called the direct product of M and M' . Let M = (Q, X, /t) and M' = (Q', X', /t') be fuzzy finite state machines . Define ,tn,a :(QxQ')xXx(QxQ')----> [0,1] © 2002 by Chapman & Hall/CRC

354

7.

More

on Fuzzy Languages

as follows : (w n w) ((p, p), a, (q, q))

= ft (p, a, q)

n la'(p, a, q) .

Recall that the fuzzy finite state machine M n M' = (Q is called the restricted direct product of M and M' . Definition 7.4.2 Let M state machines such that

and M' = 01 . Define

= (Q, X, /t)

Q n Q'

= (Q', X, /t')

x Q', X, /t

n /t')

be fuzzy finite

/tu/t' :(QUQ')XXX(QUQ'),[0,1]

as follows:

(w

U w) (p, a, q)

=

The fuzzy finite state machine and M' .

join of M

f0, a, q)

if p, q E Q

0

otherwise.

w' (p, a, q)

if p, q E Q'

M V M' = (Q U Q', X, /t U /t') is

called the

We recall the definition of a sub-fuzzy finite state machine . Let M = finite state machine . A fuzzy finite state machine called a sub-fuzzy finite state machine of M, written

(Q, X, /t) be a fuzzy MS = (Q s , Xs , ft') is Ms CM,if

(i) QS C Q, XS C X, (n) its = ItIQ .XX.XQ. .

and

Note again that the definition differs from that of a submachine is not required.

7.5

of

a sub-fuzzy finite state machine of M Definition 6.7.7, in that X s = X

of M,

Operations on (Fuzzy) Subsets

Next, we introduce some notation that will simplify the proofs of some of the theorems . Let A C_ X*, B C_ X*, and a, b E X* . Then N1. AB -1 = {x E X* xb E A, for some b E B} . N2. A-1 B = {x E X* ax E B, for some a E A} . N3. Ab-1 = {x E X* I xb E A} . N4. a -1 B = {x E X* I ax E B} . N5. AB = {xy I x E A, y E B} . N6. Ab = {xb I x E A} . N7. aB = {ay I y E B} . Clearly, AB -1 = UbEBAb -1 , A-1 B = UaEAa -1 B, and AB = UbEBAb = UaEAaB .

Let

s : Q ~ [0,1], -y : Q ~ [0,1], x E X*, p E Q, and A C X* .

© 2002 by Chapman & Hall/CRC

Then

7.5. Operations on (Fuzzy) Subsets

355

N8. p * x : Q ----> [0,1] is defined by b'q E Q, p * x (q) = ft * (p, x, q) .

N9. p * A : Q ----> [0,1] is defined by b'q E Q, p * A(q) = V {ft*(p, y, q)

I y

E A} .

N10 . 6 * x : Q ----> [0,1] is defined by b'q E Q, 6 * x(q) = V{6(p) n ft* (p, x, q)

I p

E Q} .

N11 . 6 * A : Q ----> [0,1] is defined by b'q E Q, 6 * A(q) = V{6(p) A w* (p, x, q) I p E Q, x E A} .

N12 . 6 * x-1 : Q ----> [0,1] is defined by b'q E Q, 6*

x-1(q)

= V{6(p)

n ft * (q, x, p) I p

E Q} .

N13 . 6 * A-1 : Q ----> [0,1] is defined by b'q E Q, 6 * A-1 (q) = V{6(p) A w*(q, x,p)

Ip

E Q, x E A} .

Clearly p*A= U,EAP*Y, 6*A= UyEA6*y, and 6*A-1 = UvEA6*Y-1 . The following results are easily proven . Theorem 7.5.1 Let 6 : Q ----> [0,1], x, y E X*, and A, B C X* . Then (1) (6 * x) * y = 6 * (xy) ; (2) (6*A)*y=6*(Ay) ; (3) (6 * x) * B = 6 * (xB) ; (4) (6*A)*B=6*(AB) ; ; (5) (6 * x-1) * y -1 = 6 * (yx)-1 A-1) y-1 * = 6 * (yA)-1 ; (6) (6 * x-1) -1 * B = 6 * (Bx)-1 ; (7) (6 * A-1) * B-1 = 6 * (BA)-1 . (8) (6 *

Theorem 7.5.2 Let 6, 61, 62 be fuzzy subsets of Q and let x, y E X* . Then the following properties hold: (1) 6*A=6; (2) (61U62)*x=(61* x)U(6 2 *x) ; (3) (61n 62)*x=( 6 1* x)n(62*x) ; (4) ( 6 1 U 62) * x -1 = ( 6 1 * x -1 ) U (62 * x (5) (61 n 62) * x -1 = (61 * x -1 ) n (62 * x (6) x0 * x = x0, where x0 is the characteristic function of 01 in Q; (7) xQ * x = UpEQp * x, where xQ is the characteristic function of Q. 0 © 2002 by Chapman & Hall/CRC

356

7. More on Fuzzy Languages Let S : Q ----> [0,1], 'Y : Q ----> [0,1] . N14 . Define 6#-y : X* ----> [0,1] by b'x E X*, 6#y(x) = V {6(p) A y(q) A ft* (p, x, q) I p, q E Q}. N15 . S-1 o/-, ,y = {x E X* I S * y(x) > 0} . When ft is understood, we sometimes write S -1 o 'Y for S -1

oN 'Y .

Theorem 7.5.3 Let S : Q ---> [0,1] and x E X* . Then S * XQ(x) > 0 if and only if 3q E Q such that S * x(q) > 0. Proof. 6 * XQ (x) > 0 ~ V{6(p) A XQ(q) A ft*(p, x, q) I p, q E Q} > 0 ~ V{6(p) A w* (p, x, q) I p, q E Q} > 0 ~ V{6(p) A w* (p, x, q) I p E Q} > 0 for somegEQ~6*x(q)>0. Theorem 7.5.4 Let S and 'Y be fuzzy subsets of Q, q E Q, A, B C_ X*, and x, y E X* . Then the following properties hold: -1 ; (1/ b-1 o ('y * A-1) _ (b -1 o'y)A (2) b -1 o ('y * x-1) _ (b -1 o y)x -1 i (3) (6 A) -t o -y A -'(6-1 o -y) ; (4) (b * x) -1 o r}' = x -1 (b -1 o r}') ; (5) (q * A) -1 o 'y = A -1 (q -1 o ry) ; (6)

(q * x) -1

o 'y = x-1 (q -1 o -y) .

Proof. We prove (1), (3), and (5) . We ask the reader to verify (2), (4), and (6) . (1) z E S -1 o ('Y* A -1 ) ~ S#('Y* A -1 )(z) > 0 ~ 6(po) n ('y* A -1 )(qo) n ft * (po, z, qo) > 0 for some po, qo E Q ~ 6(po) n -y (to) n ft * (qo, xo, to) n ft * (po, z, qo) > 0 for some po, qo, to E Q, xo E A ~ S(po)n'y(to)n[ft * (po, z, qo) Aft* (qo, xo, to)] > 0 for some po, qo, to E Q, xo E A ~ 6(po) n 'y(to) n /t* (po, zxo, to) > 0 for some po, to E Q, xo E A ~ zxo E S -1 o 'Y for some xo E A ~ z E (S-1 o y)A -1 . (3) z E (S * A) -1 o y ~ ((6 * A)-1#-y)(z) > 0 ~ (S * A) (po) A y(go)A ft * (po, z, qo) > 0 for some po, qo E Q ~ 6(to) n [t* (to, xo,po) n'y(go) n 6 t o ) n 'y(go) n ft * (po, z, qo) > 0 for some po, qo, to E Q, xo E A o y for some y* (t o , xoz, qo ) > 0 for some qo, t o E Q, xo E A ~ xoz E S xo E A ~ z E A -1 (S -1 o y) . (5) y E (q * A) -1 o y ~ (q * A) -'#y(y) > 0 ~ q * A(po) A y(qo) A ft * (po, y, qo) > 0 for some po, qo E Q ~ ft* (q, x,po) A y(go) n ft * (po, y, qo) > 0 for some po, qo E Q, y E A ~ 'y(qo) n [ft* (q, x,Po) n ft* (po, y, qo)] > 0 for some po, qo E Q, y E A y(qo) A /t* (q, xy, qo ) > 0 for some qo E Q, yEA~?yEA -1 (q -1 oy) . N16 . Let 61 : Q ----> [0,1] and 62 : Q' ----> [0,1] . Then define 61 n62 QxQ'-[0,1] by d(q,q')EQxQ', (b1 n62) (q, q) = 61 (q) n 62 (q) . © 2002 by Chapman & Hall/CRC

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357

N17. Let 6 1 : Q ~ [0,1], 62 : Q' ~ [0,1] be such that Q n Q' _ 0 . Then define 6 1 U 62 : Q U Q' ~ [0,1] by b'q E Q, 61(q) (61 U 62) (q) - { 62(q)

if q E Q . if q E Q

Theorem 7.5 .5 Let M = (Q, X, /t) and M' = (Q', X, /t') be fuzzy finite state machines. Let 61,ryi be fuzzy subsets of Q and 62, 'Y2 be fuzzy subsets of Q' . Then the following properties hold. (1) (61 U 62) -1 0 NUN' ('Y1 U -Y2) = (6 1 1 o N 'Yi) U (62 1 Off' 'Y2) ; (2) (61 n 62) -1 0 /"n/a , (-yj n -Y2) = (6 1 1 of" 'Yl) n (62 1 0/1' -Y2)Proof. (1) Let x E (61U62)-10('YJU'Y2) . Then (61U62)*(-YJU-Y2)(x) > 0. Thus (61U62)(p)A(ylUy2)(q)A(ftUft')*(p,x,q) > 0 for somep,q E QUQ' . It follows that both p, q belong to either Q or Q' . For, if p E Q and q E Q'and vice versa, then by definition (ft U ft') * (p, x, q) = 0 and consequently (61 U 62) (p) A (-yj U'Y2)(q) A (/t U ft) *(p, x, q) = 0, a contradiction. If p, q E Q, then clearly 61 (p) A yl (q) A ft* (p, x, q) = (61 U 62) (p) A ('Y1 U '2) (q) A (/t U /t')* (p, x, q) > 0. Therefore, 61 #yl (x) > 0 and hence x E 61 1 o ,yl . Similarly, if p, q E Q' , then x E 62 1 0 'Y2 * On the other hand, let x E 6 -1 1 o yl . Then 61 (p) Ayl (q) A ft (p, x, q) > 0 for some p, q E Q, i.e., (61 U 62) (p) n (-yj U 'f2) (q) n (ft U ft') * (p, x, q) > 0. A similar situation holds if x E 62 1 0 y2 . (2) x E (6 1 n 62 ) -1 0 ('Y, n -Y2) (6 1 n 62 )#(-y, n y2)(x) > 0 (6i n 62)(pl,p2) A (yl n'Y2)(gl,q2) A (ft U ft) *((PI,p2),x, (gl,q2)) > 0 for some (PI, P2), (gl,q2) E Q x Q' ~ [61(pl) n'f1(gl) n[t* (pl, x,ql)]n [62(p2) A 'Y2 (q2) n [t * (P2, x, q2)] > 0 for some pl, ql E Q, p2, q2 E Q' ~ [61#'f1(x)] n [62# -Y2 (X)1 > 0 ~ [61#'Yj(x)] > 0 and [6 2#-Y2 (X)1 > 0 ~ x E 61 1 o yl and XE6210-Y2~XE(611o'yl)n(621o-Y2) . Theorem 7.5 .6 Let M = (Q, X, ft) and M' = (Q', X, ft') be fuzzy finite state machines. Let 61, y 1 be fuzzy subsets of Q and 62, 'Y2 be fuzzy subsets of Q' . Then (61 n 62) -1 0wxw' (-yj n -Y2) = (611 0w 'YJ x (62 1 0w' 'Y2) Proof. (x, y) E (61 n 6 2) -1 o (-yj n -Y2) <#~ (61 n62) * ('Y, n-Y2) (x, y) > 0 <#~ (61 n 62)(pl,p2) A (yl n'Y2)(gl,q2) A (ft x ft)*((PI,p2), (x, y), (gl,q2)) > 0 for some (p1, p2), (gl,g2) E Q x Q' <#~ [61(pl) n'f1(gi) n [t*(pl,x,gl)]n [62(p2) A -Y2 (q2) A ft *(P2, y, q2)] > 0 for some pl, ql E Q, p2, q2 E Q' <#~ [61#'f1(x)] n [6_ 2#'Y_ (y)] > 0 <#~ x E 61 1 o '1'1 and y E 62 1 0 'Y2 <#~ (x, y) E x (6 1 1 0'Y1) (62 1 0-Y2)The proof of the following theorem is similar to the proof of Theorem 7.5 .6 . Theorem 7.5 .7 Let 61 : Q ----> [0,1], 62 : Q' [0,1] and x E X* . Then (61n62)*x=(61* x)n(62*x) ; (1) (2) (61n 62)*x -1= (6 1*x -1 )n (62*x -1) .~

© 2002 by Chapman & Hall/CRC

358

7 .6

7. More on Fuzzy Languages

Construction of Recognizers and Recognizable Sets

Definition 7.6 .1 Let M = (M, t, T) and M' = (M', t, T) be fuzzy Xrecognizers of M = (Q, X, /t) and M' = (Q', X, /t'), respectively . Then the fuzzy X-recognizer .M n.M' = (M n M, t n t', T n T') is called the restricted direct product of M and M' . Definition 7.6 .2 Let M = (M,t,T) be a fuzzy X-recognizer of a fuzzy finite state machine M = (Q, X, /t), and M' = (M', t', T') be a fuzzy X'recognizer of a fuzzy finite state machine M' = (Q ', X', /t') . Then the fuzzy X x X'-recognizer .M x .M' = (M x M, t n t', T n T') is called the direct product of M and M' . Definition 7.6 .3 Let M = (M, t, T) and M' = (M', t, T) be fuzzy Xrecognizers of M = (Q, X, /t) and M' = (Q', X, /t'), respectively, where Q n Q' = 01 . Then the fuzzy X-recognizer .M U AT = (M U M, t U t', T U T') is called the join of M and M' . Definition 7.6 .4 The mapping p : X* ~ X* is called a reversal of X* if the following conditions hold : (1) p(A) = A, (2) p(a) = a, (3) P(xa) = P(a)P(x), and (4) P(P(x)) = X, VaEX and xEX* .

It follows by induction that p(xy) = P(y)P(x) b'x, y E X* . Let p be a reversal of X* . Define the fuzzy subset ftp :QxX xQ----> [0,1] by ftP (p, a, q) = ft(q, p(a), p) b'a E X, dp, q E Q. Definition 7.6 .5 If M = (Q, X, ft) is a fuzzy finite state machine, then the fuzzy finite state machine MP = (Q, X, ftP) is called the reversal of M. Theorem 7.6 .6 Let MP = (Q, X, pP) be the reversal of M = (Q, X, p) . Then (wP)*(p,xy,q)

= w* (q,P(y)P(x),p)

Vx, y E X* . 0 Let A C X* and p : X* ----> X* be the reversal mapping. Let AP = {p(x)

© 2002 by Chapman & Hall/CRC

I x E Al .

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359

Theorem 7.6.7 Let S and 'y be fuzzy subsets of Q. Then (S -1 o ,y)P = ,Y -1 06. Proof. p(x) E (6 -1 o y)P x E 6 -1 o y ~ 6# y (x) > 0 ~ 6(p) n y ( q) n p,* (p, x, q) > 0 for some p, q E Q S(p) A y(q) A ([tP) * (q, p(x),p) > 0 for some p, q E Q -y (q) A 6(p) A ([tP ) * (q, p(x), p) > 0 for some p, q E Q p(x) E 'Y-1 O S . 'Y#b(p(x)) > 0 Definition 7.6.8 Let .M = (M, t, T) be a fuzzy X-recognizer. Let p : X* ~ X* be the reversal mapping. Then the fuzzy X-recognizer .MP = (MP, T, t) of MP is called the reversal of M.

Definition 7.6.9 Let .M = (M, t, T) be a fuzzy X-recognizer . Then x E X is said to be recognized by .M if t#T(x) > 0.

Let B(M) denote the set of all words that are recognized by .M. B(M) is called the behavior of .M . Theorem 7.6.10 Let M be a fuzzy recognizer and x E X* . Then the following (1) (2) (3)

conditions are equivalent . x E B( .M); x E t -1 o T; t#T(x) > 0; (4) 3po E Q such that t n T * x-1(Po) > 0; (5) 3qo E Q such that t * x n T(qo) > 0.

Proof. Clearly (1), (2), and (3) are equivalent . (3)x(4) : t#T(x) > 0 ~ V{t(p) AT(q) A w*(p, x, q) I p, q E Q} > 0 3po, qo E Q such that t(po) n T(qo) n [t * (po, x, qo) > 0 ~ 3po, qo E Q such that t(po) > 0 and T(qo) A /t* (po, x, qo) > 0 ~ Elpo E Q such that t(po) > 0 and V {T (q) A /t* (po, x, q) I q E Q} > 0 ~ 3po E Q such that t(po) > 0 and T * x-, (PO) > 0 ~? 3po E Q such that t n (T * x-1) (PO) > 0. (3)x(5) : This can be proved by interchanging the roles of po and qo and using the definition of t * x. Definition 7.6.11 A subset A of X* is called X-recognizable, if there exists an X-recognizer .M such that B(M) = A.

Theorem 7.6.12 Let X be a nonempty finite set. Then the following sets are X-recognizable: (1) {a}, where a E X. (2) A. (3) A*, where A C X. (4) X. 0

Theorems 7.5 .5, 7.6.7, and 7.6 .10 immediately lead to the following theorem. © 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

36 0

Theorem 7.6.13 Let M and .M' be fuzzy recognizers . Then the following assertions hold. (1) B( .M U .A4) = B( .M)UB(.M') . (2) B( .M n .A41) = B( .M)nB(.M') . (3) B( .MP) =B(.M)P .

Proof. (3) We have B(M)P = (t -1 o T)P = T -l o t = B( .MP) . The following corollary is immediate from Theorem 7.6 .13. Corollary 7.6.14 Let A and B be recognizable subsets of X* . Then the following sets are recognizable . (1) A U B . (2) A n B . (3) AP . Corollary 7.6.15 A subset A of X* is recognizable if and only if AP is recognizable .

Proof. (AP)P =A . m In view of Theorems 7.5 .6 and 7.6 .10, we obtain the following theorem. Theorem 7.6.16 Let M be a X-recognizer and .M' be a X'-recognizer. Then B(.M x .M') = B(M) x B(.M') . m Corollary 7.6.17 If A is X-recognizable and B is X'-recognizable, then A x B is X x X'-recognizable. m Theorem 7.6.18 Let A be X-recognizable and B be any subset if X* . Then the following sets are X-recognizable . (1) AB -1 . (2) Ax-' . (3) B-1 A. (4) X -1 A.

Proof. We prove (1) and (3) only. Since A is X-recognizable, there exists a fuzzy recognizer .M = (M, t, T) of a fuzzy finite state machine M = (Q, X, /t) such that B(M) = A. (1) Consider a fuzzy recognizer .M' = (M, t,T*B -1 ) . Then by Theorem 7.5.4(1), t-1 o (T * B-1 ) = (t - 'o T)B-1 = B(A4)B -' = AB -1 . Thus AB -1 is recognizable . (3) Consider the fuzzy recognizer .M' = (M, t, B-1 * T) . Then by Theorem 7.5.4(3), (t * B) -1 o T = B-1 (t -1 o T) = B -1 B(M) = B- 'A . Hence B-'A is recognizable . Let X and X' be sets and let f : (X')* X* be a fine homomorphism. If M = (Q, X, /t) is a fuzzy finite state machine, then define /t' : Q x X' x Q ----> [0, 1] by fft'(p,a,q)=ft(p,f(a),q)

© 2002 by Chapman & Hall/CRC

7.6. Construction of Recognizers and Recognizable Sets

361

b'p, q E Q, a' E X' . This defines a fuzzy finite state machine M' _ (Q, X, /t') and a fuzzy X'-recognizer .M' = (M', t, T) . Definition 7.6.19 The fuzzy X'-recognizer .M' defined immediately above is called the inverse image of M and is denoted by f-1 (.M) .

Theorem 7.6.20 Let S : Q ~ [0,1] and y : Q ~ [0,1] . Then S-1 f-1(S-1 o w 'Y) .

oN , ,y =

Proof. x E S-1 oN,, y ~ 6#-y(x) > 0 ~ S(p) n -y(q) n ft~(p, x, q) > 0 for some p, q E Q ~ S(p) A y(q) A /t* (p, f (x), q) > 0 for some p, q E Q f (X) E 6-1 0/"Y. The following corollaries are easily shown to hold. Corollary ?.6 .21 B(f -1 (.M)) = f-1 (B( .M)) . m Corollary 7.6 .22 If A is X-recognizable, then f-1 (A) is X'-recognizable. Let .M = (M, t, T) be a fuzzy recognizer of a fuzzy finite state machine M = (Q, X, p) and let a E X. Consider Q' = Q U {t'}, where t' ~ Q. Define fl a :Q'XXXQ'~[0,1]by ft(p, b, q)

f'a(p, b, q) =

1 0

if p, q E Q, if p E Q, T(p)>O, b = a and q = t', otherwise .

Clearly, Ma = (Q', X, f a) is a fuzzy finite state machine. Define to [0,1] as follows : __

to(p) - { 0

p))

if p E Q, otherwise .

Let Ta be the characteristic function of {t'} . Then Ma = (Ma, ta,Ta) is the fuzzy recognizer of Ma. Let X be a nonempty finite set, A C X*, and u E X. Recall that An = {xu I

x E A} .

Theorem 7.6.23 Let Ma = (Ma , ta ,Ta) be a fuzzy recognizer of Ma. Then to 1 0Ma Ta = t 1 0A T)a .

Proof. Let x

E to 1 0Aa Ta .

Then (ta#Ta) (x) > 0 . Thus

ta(po) ATa(go)

for some po, q'

E Q' .

n fta(P' , x, qo) > 0

Hence q' = t' and p'

© 2002 by Chapman & Hall/CRC

E

Q.

7. More on Fuzzy Languages

36 2

Now fta(po,x,go) > 0 is possible only if either T(p0 ) > 0 and x = a or there exists y E X* such that /ta(p'o , ya, t') > 0. We consider these two cases . Suppose T(p0) > 0 and x = a. Now t(po) > 0, T(po), and /t*(po,A,po) > 0. Hence t(po) n T(po) n [t* (po, A, po) > 0, i.e., A E L -1 Of,, T. Therefore, x = Aa E (L -1 Off, T) a. Suppose now that y E X* such that ya(po, ya, t') > 0. Then there exists t'o E Q' such that laa(po, y, to) Alta(to, a, qo) > 0. This is true only when T (to) > 0, i.e., to E Q . Thus t(po) AT(to) Aft*(po, y, to) > 0 . Therefore, L#T(y) > 0. Hence x = ya E (L -1 OA, T)a . Thus tat OA, Ta C(6- 1

oA T)a.

We now show inclusion in the other direction . Let x = ya E (t -1 OA T)a . Then y E (L -1 ON, T) . Therefore, t(po) A T(qo) A ft*(po, y, qo) > 0 for some lta (go, a, t') = 1 po, qo E Q . Thus t(po) > 0 and T(qo) > 0 . Now T(qo) > 0 and fta(POI yI qo) = [t * (po, y, qo) > 0 . Hence fta(po, ya, t~) = V{[ta (po, y, r~) A lta(r', a, t') I r' E Q'} > lta(po, y, qo) n fta(go, a, t~) > 0 . Therefore, t(po) A ' Ta(t PO ) AN'a(po, ya, t') > 0 and hence ta(po) ATa(t') Alta(po, ya, t') > 0 since E Q, l.e., x = ya E La 1 OV-a Ta . Thus (t -1 ON, T)a C La 1 ON,a Ta' Hence tat O V- a Ta - (L-1 OAT)a . Corollary 7.6 .24 B( .M a) = B(A4)a .

m

Corollary 7.6 .25 If a subset A of X* is recognizable and u is recognizable. m

7 .7

E

X, then An

Accessible and Coaccessible Recognizers

Definition 7.7 .1 Let M = (M,t,T) be a fuzzy X-recognizer of a fuzzy finite state machine M = (Q, X, ft) and R = {q E Q I * X* (q) > 01 . If Q = R, then M is called accessible. Let .M = (M, t, T) be a fuzzy recognizer of M = (Q, X, lt) . Let R = {q E X* (q) > 01. Consider to = wI RXXXR, La = tIR, and Ta = TI R . Then Ma = (R, X, ta) is a fuzzy finite state machine and .Ma = (Ma, 6a, Ta) is a fuzzy recognizer of Ma . Q

I

T) be a fuzzy X-recognizer of M. Then .Ma = (Ma' ta,Ta) is called the accessible part of M.

Definition 7.7 .2 Let M = (M, t,

Definition 7.7 .3 Let M = (M,6,T) be a fuzzy X-recognizer of a fuzzy finite state machine M = (Q, X, ft) . Let S = {p E Q I T * (X*) -'(p) > 01 . If Q = S, then M is called coaccessible. Consider ftb = wI sXXXs' tb = t1 s, and Tb = TI s. Then Mb = (S, X, ftb) is a fuzzy finite state machine and Mb = (Mb, tb, Tb) is a fuzzy recognizer of Mb . © 2002 by Chapman & Hall/CRC

7.7. Accessible and Coaccessible Recognizers

363

Definition 7.7.4 Let .M b = (Mb, Lb , Tb )

M = (M, t, T) be a fuzzy X-recognizer of is called the coaccessible part of M.

M.

Then

If t(q) > 0, then q E R. Thus whenever t is crisp, so is ta and La = t as sets. If T(q) > 0, then q E S. Hence whenever T is crisp, so is Ta and Ta = T as sets. Theorem 7.7.5 Let .M be a fuzzy recognizer. Then the following properties hold : (1) L-1

O T = (La)-1 O Ta ; L-1 O (2) T = (Lb)-1 O Tb.

Proof. (1) Let x E

L-1 O T .

Then

L(po) AT(go)

L#T(x)

> 0. Therefore,

n [t* (po, x, qo)

>

0

for some po, q0 E Q. Clearly, q0 E R since L(po) A ft * (po, x, L(po) > 0 and y* (po, A,po) = 1 > 0 and so po E R. Hence Ta(go)

But then (La)-1

La#Ta(x) O Ta .

qo) >

0. Also,

n La(po) n [ta*(po' x, qo) > 0.

> 0 and thus x E

(La)-1

O Ta .

Therefore,

L-1 O T

C_

We now show inclusion in the other direction . Let x E (La)-1 OT a . Then a La #T (x) > 0, i .e., La (p0) AT a (go) All' a* (p0,x,g0) > 0 for some p0, go E R. Therefore, L° (p0 ) > 0, Ta (qo) > 0, and ta* (po, x, qo) > 0. Thus L(p0) = La (p0),T(g0) = Ta(g0)

and * (po,x,go) = ta*(po,x,go) .

Therefore, L(p0) AT(go)

Hence L#T(x) > 0 and so x E L-1 quently, L-1 O T = (La)-1 O Ta . (2) The proof is similar to (1) .

At* O T.

0 x,qo)

Thus

> 0.

(L a) -1 O Ta

C_

L-1 O T.

Conse-

Corollary 7.7.6 Let .M be a fuzzy recognizer. Then (1) B(.M) = B(.Ma) ; (2) B(.M) = B(.Mb) .

Theorem 7.7.7 Let .M be an X-fuzzy recognizer. Then if and only if .MP is accessible .

© 2002 by Chapman & Hall/CRC

.M

is coaccessible

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36 4

Proof. .M is a coaccessible fuzzy recognizer ~ Q = {p E Q I T (X*) -1 (p) > 0} ~? b'p E Q, 3xo E X*, and qo E Q such that T(qo) A b'p E Q, 3xo namely p(xo) E X* and qo E Q such that ft * (po, xo, qo) > 0 n T(qo) [t P* (po, p(xo), qo) > 0 ~ Q = {p E Q I T * X* (p) > 0} ~ .MP is an accessible fuzzy recognizer. Definition 7.7.8 A fuzzy X-recognizer .M is called trim if it is both an accessible and coaccessible fuzzy X-recognizer.

Theorem 7.7.9 Let M = (M, t, T) be the fuzzy recognizer of the fuzzy

finite state machine M = (Q, X, /t) and let W = {p E Q I t * X* n T (X*) -1 (p) > 0} . Then .M is trim if and only if Q = W. m

Consider pt = ftjw .x .w, tt = t1 w, and Tt = Tjw . Then Mt = (W X, [tt) is a fuzzy finite state machine and Mt = (M t , tt, Tt) is a fuzzy X-recognizer . Definition 7.7.10 Let .M be a fuzzy X-recognizer. Then .M t = (M t , tt ,Tt) is called the trim part of M.

The following theorem is immediate from Corollary 7.7 .6 . Theorem 7.7.11 If M = (M, t, T) is a fuzzy recognizer, then B(M) _ B(.Mt) . .

Corollary 7.7.12 If .M is trim, then ( .Ma) b = .Mt = (.Mb)a . 0 If .M is an accessible (coaccessible, trim) fuzzy X-recognizer, then Ma = M .Mb = M, Mt = M, respectively) . (

7 .8

Complete Fuzzy Machines

Recall that a fuzzy finite state machine M = (Q, X, ft) is called complete if for all (p, u) E Q x X, there exists q E Q such that ft(p, u, q) > 0. As in Definition 7.2 .13, if M = (Q, X, ft) is a complete fuzzy finite state machine, then the fuzzy X-recognizer .M = (M, t, T) of .M is called complete. Theorem 7.8.1 If M is a complete fuzzy recognizer, then so is .M a . Proof. Let (pa, u) E R x X. Then (pa, u) E Q x X. Since .M is complete, there exists q E Q such that ft (pa, u, q) > 0. Now pa E R ==> t * X* (pa) > 0 . Therefore, 3qo E Q and xo E X* such that t(qo) A /t* (qo, xo , pa) > 0 . This implies that t(qo) > 0 and lt* (qo, xo, pa) > 0. Hence lt* (qo, xo , pa) A ft(pa ' u, q) > 0. Thus /t* (qo, xou, q) = V {ft* (qo, xo' t) nft(t, u, q) I t E Q} > 0 . Hence t(qo) A ft*(go,xou,q) > 0, i.e ., t * X* (q) > 0 and so q E R. Since ,ta(pa , u q) = t(pa u q) Ma is complete. The following definition of completion and the following construction of a completion differs somewhat from that in Section 7.2 . © 2002 by Chapman & Hall/CRC

7.8 . Complete Fuzzy Machines

365

Definition 7.8.2 Let M = (Q, X, /t) be a fuzzy finite state machine. A fuzzy finite state machine Mc = (Q c , Xc, ftc) is called a completion of M, if the following conditions hold: (1) Mc is a complete fuzzy finite machine, and (2) M is a subfuzzy finite state machine of Mc . Let M = (Q, X, ft) be a fuzzy finite state machine that is incomplete. Consider M' = (Q', X, /t'), where Q' = Q U {z}, z ~ Q and

f~ (p, u, q) =

ft(p, u, q) 1

0

if p, q E Q and ft(p, u, q) :A0, if either ft(p, u, r) = 0 b'r E Q and q = z or p = q = z,

otherwise.

Then M' = (Q', X, ft') is called the smallest completion of M. Definition 7.8.3 Let M = (M,t,T) be a fuzzy X-recognizer of a fuzzy finite state machine M = (Q, X, /t) and Mc = (Q c, X, /tc) be a completion of M. Then the fuzzy X-recognizer .Mc = (Mc, tc, Tc) of Mc is called the completion of M, where tc : Qc ~ [0,1] and Tc : Qc ~ [0,1] are such that t(q) 0

if q E Q if q Q.

T(q) 0

if q E Q if q Q.

If Mc is the smallest completion of M, then .Mc is called the smallest completion of .M .

Theorem 7.8.4 Let .M = (M, t, T) be a fuzzy recognizer and .Mc is the smallest completion of .M . Then B(A4) = B( .Mc) . 0

Theorem 7.8.5 If M is an accessible fuzzy recognizer, then so is Mc . Proof. Let .M be accessible. Then Q = {q E Q I t * X*(q) > 0} . Let qc E Qc . If qc E Q, then the desired result holds . Let qc = z. Since M is incomplete, there exists (po, no) E Q x X such that ft(po, no, t) = 0 for all t E Q. But then (ftc) (po, no, z) = 1 > 0. Since M is accessible and po E Q, there exists r E Q and yo E X* such that t(r) A /t* (r, yo, po) > 0. Now (ftc)* (r, youo, z) = V{(ft c)* (r, yo, s) n (ftc) (s, uo, z) I s E Q} > * (ft c ) (r, yo, po) n (ftc) (po, no, z) = (ftc) * (r, yo,po) = ft * (r, yo, po) > 0. Thus t(r) A (ftc) * (r, youo, z) > 0, i.e., t * X*(qc) > 0. Hence Mc is accessible. m Let .M = (M, t, T) be a fuzzy recognizer of a fuzzy finite state machine M = (Q, X, ft) . Recall that P(Q) is the power set of Q . © 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

36 6

Define p,^' : P(Q) x X x P(Q) ----> [0,1] by p - (P, u, R

-

1

1

V{ft(p, u, r)

P(Q) ----> [0 , 1] by

Ip

E P, r E

0{L(p) I p E

P}

R}

if P4 QlorR~01 otherwise,

otherwise,

and T^' : P(Q) ----> [0,1] by T-(P)

_

0{T(p) I p E P}

if P otherwise.

Then M^' = (P(Q), X, /t -) is a fuzzy finite state machine and .M^' _ (M^', t- , T- ) is a complete fuzzy X-recognizer of M^' = (P(Q), X, ft-) . Theorem 7.8.6 Let M = (M, t, T) be a fuzzy recognizer of a state machine M = (Q, X, /t) . Then B(M) = B( .M-) .

fuzzy finite

Proof. Let x

E B( .M - ) . Now x E B(M-) =~> x E t- oA- TL-#T- ( x) > 0 =~- t- (P) AT - (R) A /t - (P, x, R) > 0 for some P, R E P(Q) .

Clearly, by the definition of t- and T- , both P and R are nonempty. Thus [A{L(p) I p E P}] n [n{T(r) I r E R}] n [V{,t* (p, x, r) I p E P, r E R}] > 0 for some P, R E P(Q) and so t(p) > 0 Vp E P, T(r) > 0 Vr E R and there 3t o E P, ro E R such that [t* (to, x, ro) > 0 for some P, R E P(Q) . Hence t(to) > 0, T(ro) > 0, and y,* (to, x, ro) > 0 for some to, ro E Q . Thus t(to) A T(ro) A ft * (to, x, ro) > 0 for some to, ro E Q. This implies that t#T(x) > 0 and so x E t oN T . Hence x E B(M) . Thus B(M- ) C B(M) . Conversely, suppose x E B( .M) . Then x E t oN, T and so t * T(x) > 0 . This implies that t (p) A T (q) A ft * (p, x, q) > 0 for some p, q E Q. Choose P = {p} and R = {q} . It then follows that B(M) C B( .M - ) . m

Theorem 7.8.7 Every fuzzy X-recognizable subset A of X* is the behavior of a complete fuzzy recognizer. m

7 .9

Fuzzy Languages on a Free Monoid

The study of fuzzy grammars, the rules of fuzzy syntaxes, and the recognition ability of fuzzy automata extends the application area of fuzzy set theory. One goal is to reduce the difference between formal languages and natural languages . In this section, we examine fuzzy regular languages, adjunctive languages, and dense languages . We present their algebraic properties. The results are from [217] . In this and the next section, we let X denote a finite alphabet with at least one element . Recall that,F'P(X) denotes the set of all fuzzy subsets of X. We use the superscript T to denote the transpose of a matrix. © 2002 by Chapman & Hall/CRC

7.9 . Fuzzy Languages on a Free Monoid

367

Definition 7.9.1 [160] A finite fuzzy automaton on an alphabet X is a 5-tuple M = (Q, Y, {Tn I u E Xj, ao, al ) such that (1) Q = {ql, q2, . . . , qn} is the set of states, (2) Y = {yl, y2, . . . , yn J C Q is the set of output symbols, (3) {Tn u E X} is the set of fuzzy transition matrices, where Sgiq, : X ----> [0,1] and T _ [S qiqj (u)], qZ, qj E Q, i, j = 1, 2, . . . , n, (4) Qo = [ i l i2 . . . i n ] , ik E [0,1] for k = 1, 2. . . . , n, (5) CI = [ ji

j2

.. .

in

]T'

jk E [ 0 , 1 ] for k = 1, 2. . . . , n.

ao determines the fuzzy subset of initial states and al determines the fuzzy subset of final subsets . Let M = (Q, Y, {T I u E Xf, ao, al ) be a finite fuzzy automaton. Define S : Q x X x Q ~ [0,1] by b(qj, u, qj) = bqiqj (u)

for i, j = 1, 2, . . . , n and Vu E X. Then S is a fuzzy transition function. Let S* be defined as usual. Then S* (q, A, q') = 1 if q = q' and 0 otherwise. Also for all x = ulu2 . . . uk E X*, x :?~ A, 6* (q, x, q') or

=

V {b(q, ul, qi) A S(ql , u2, q2) A . . . A S(qk-1, uk, q~) I qi ~ q2 ~ . . . , qk-i E Q}

0 , TX = Tnl 0 Tn2 0 . . . Tn,

where o is the sup-min composition of fuzzy matrices. Definition 7.9.2 [160] Any member of ,F'P(X*) is called a fuzzy language on the free monoid X* . For all u E X, let x n denote the characteristic function of {u} in X* . Then xn is called the basic fuzzy language generated by u. Let E = {x,, I u E

X} U {A} .

Definition 7.9.3 [160] Let U C_ ,F'P(X*) be such that the following conditions hold: (1) Vc E[0,1], ft EU==> cnyEU,

(2)Ni,N2EU ==> /t l UN 2 EU (3) fti, P2 E U =~> fti o y 2 E U, where (Pi 0 ft2)(x) = V {fti(u) n [t 2(V) I uv = x}, b'x E X*, (4) ft EU==> ,t E U, where ft' is the Kleene's closure of ft, i .e ., = [toUftUft2 U . . .

where [t o (x) = 0 V'x E X* . Then U is called a closed family of fuzzy languages on X* .

© 2002 by Chapman & Hall/CRC

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368

Let .F = {U I U is a closed family of fuzzy languages on X*} .

Clearly,,F'P(X*) E T Definition 7.9.4 [160] Let FR(X*)

= nEC MC .-M .

FR(X*) is called the family of fuzzy regular languages on X* . If f E FR(X*), then ft is called a fuzzy regular language on X* . Clearly, FR(X*) is a subalgebra of ,F'P(X*) generated by E with the four operations in Definition 7.9 .3 . Definition 7.9.5 [160] Let M = (Q, Y, Tn, ao, al ) be a finite fuzzy automaton on X. Define fm : X* ~ [0,1] by Vu E X*, fm(u) = ao o Tn o al . Then fm is called the fuzzy language determined by the fuzzy automaton M.

Example 7.9.6 Let Q = {s, S, F}, X = {a, b}, and S : Q x X x Q ~ [0,1] be defined as follows:

6(s, a, S) S(s, b, s) S(S, b, S) S(S, b, F)

_ _ _ _

.9 .5 .9 .9

and S(q, x, q~) = 0 for any other (q, x, q~) E Q x X x Q. Let ao = (1, 0, 0) and al = (0, 0,1)T. Then

Ta

_

s S F

0 0 0

s

.9 0 0

S

0 0 0

F Tb

_

s S F

.5 0 0

s

0 .9 0

S

0 .9 0

F

Let M = (Q, Y, ~Ta, Tb}, ao, al ), where Y = Q . Then fm (bab)=ao oTboTa oTboal =[ 1

0

0 0 0

0 ]

.5 0 0

.5 0 0

0 0 1

= .5

and fm(ab)=ao oTa oTb oa l =[ 1

0

0 ]

0 0 0

.9 0 0

.9 0 0

0 0 1

= .9 .

It follows that fm (b mabn ) = .5 if m > 0 and fm (b-abn ) = .9 if m = 0. We see that fm = A, where A is the fuzzy language of Example 7.1 .11.

© 2002 by Chapman & Hall/CRC

7.9 . Fuzzy Languages on a Free Monoid

369

From [160], /t E FR(X*) if and only if there exists a finite fuzzy automaton M such that fm = /t . Definition 7.9.7 [160] We call PL a main congruence if b'x, y E X*, x - y(PL) if and only if Vu, v E X*, uxv E L ~ uyv E L . Example 7.9.8 Let X = {a, b} and L = Jan I n = 0,1 . . . . } . Let x, y E

X* . Then Vu, v E X*, uxv E L if and only if u = az, x = ai, and v = ak for some i, j, k E hY U {0}. Thus x - y(PL ) if and only if either x E L, y E L or x ~ L, y ~ L . Hence the equivalence classes corresponding to - are L and X*\L . Thus the index of PL is 2.

Example 7.9.9 Let X = {a, b} and L = {b'abn I m = 0,1 . . . . ; n =

1. . . . } . Let x, y E X* . Then Vu, v E X*, uxv E L if and only if either u= b', x=b~abk,v=bl (not both k=0,1=0) oru=b'abi,x=bk ,v=b l (j, k, l not all 0) or u = b2, x = bi, v = bkabi (1 z,4 0) for i, j, k, l E hY U {0}. Thus x - y (PL ) if and only if either x, y E L U {b le a k = 0,1, 1 . . . . I or { bk I k = 0,1 . . . . } or x, y E X*\(L U {bka I k = 0,1. . . . } U {bk X' y E .}) . Thus the index of PL is 3. k=0,1

Proposition 7.9 .10 [218] An ordinary language L C_ X* is regular if and only if the index of

PL

is finite . 0

Proposition 7.9 .11 [218] An ordinary language L C_ X* is adjunctive if and only if b'x, y E X*, x - y(PL) =~' x = y . 0

Example 7.9.12 Let X = {a, b} and L = fanbn I n = 0,1. . . . } . Let

x, y E X* . Then Vu, v E X*, uxv E L if and only if either u = anbn , x = V, v = bn -Z-j or u = a2 , x = a n-Z b', v = bn- j where n, i, j E hY U {0} . Thus x - y(PL) implies x = y . Hence L is adjunctive . The index of PL is not finite and L is not regular.

Definition 7.9.13 Let /t E .PP(X*). Then FN is called a fuzzy main congruence with respect to /t on X* if

b'x, y E X*, x - y(FN,) ~ Vu, v E X*, y,(uxv) = y,(uyv) .

Proposition 7.9 .14 Let x, y E X* . Then x - y(FN ) if and only if b'c E [0,1], x - y(PN ), where y c = {x E X* I y(x) > c } . Proof. Suppose that x - y(FN ) . Then Vu,v E X*, y(uxv) = y(uyv) . Thus b'c E [0,1], y(uxv) > c if and only if y(uyv) > c. Hence Vu, v E X*, b'c E [0,1], uxv E /t c if and only if uyv E 1t c . This implies that x - y(PI") . Conversely, suppose that b'c E [0,1], x - y(PN ) . Then Vu, v E *, y(uxv) > c r y(uyv) > c b'c E [0,1] . Thus x - y(FN ) . 0 Proposition 7.9 .15 Let x, y E X* . Then x - y(FN ) if and only if b'c E [0,1], x - y(PN +), where y c+ _ {x E X* I y(x) > c }. © 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

370

Proof. Suppose that b'c E [0,1], x - y(PN ) . Then Vu, v E X*, p,(uxv) > c ~ p,(uyv) > c. Let p,(uxv) = d. Now for any E > 0, we have p,(uxv) > d - E . Therefore, p,(uyv) > d - E for all E > 0. Thus p,(uyv) > d = p,(uxv) . Similarly, p,(uxv) > p,(uyv) . Hence, x - y(FN,) . The converse is easily shown . m Corollary 7.9.16

7 .10

N = OCEfo,11Pw = OCE[o,l]Pw'+ .

F

Algebraic Character and Properties of Fuzzy Regular Languages

Proposition 7.10 .1 Let ft E .FP(X*) . Then ft E FR(X*) if and only if the index of FN is finite . Proof. Suppose that ft E FR(X*) . Then there exists a finite fuzzy automaton M = (Q, Y, T., ao, al ) such that fm = ft . Clearly, fm = ao o T o al . Define a relation p on X* as follows: b'x, y E X*, x - y(p) ~ TX = T. . Clearly, p is an equivalence relation . Let x, y E X* . Then x - y(P)

TX = TV

Vu, vEX*,T,, oTx oT,=T,, oT.oT, du,v E X* ,Tuxv = Tvyv Vu, v E X*, ao o Tuxv o al = ao o Tuyv o al VU, v E X* ' p(uxv) = p (uyv) x - y(FN,) .

Hence p C FN . Since X and Q are finite, {Tu, I w E X*} is a finite set . Therefore, since the index of p is finite, the index of FN is also finite . Conversely, suppose that the index of FN is finite . Let [A], [xl], . . . , [x .] be the distinct congruence classes of FN , where x2 E X*, i = 1, 2, . . . , m. Let M = (Q, Y, {Tu I u E Xj, ao, al) be a finite fuzzy automaton, where .]}, Q = Y = {[A], [xl], . . . , [x

the fuzzy transition matrix is defined as a[w] o Tu = a[wu], and a[A] = [10 . . . 0],aIx il =[0 . . . 010 . . . 0],i=1,2, . . .,Tn . Set ao = a[A], a l = (ftQA]), ft Qxl]), . . . , ft([X.])) T . . Also Since V'x E X*, x E [x], p(x) = p&]) f (x) = ao o Tx

o 01

= a[A] o Tx o al

= Cr[x]

o al = ft([x]) .

Thus b'x E X*, ft(x) = fm(x) . Hence ft = fm and so p, E FR(X*) . © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 371

Proposition 7.10 .2 Let ft E .FP(X*). Then ft E FR(X*) if and only if Vc E [0,1], /tc is regular, and I Im(y) I < oo. Proof. Suppose that ft E FR(X*) . Then the index of FN is finite. By Corollary 7.9 .16, Pt,, D FN , Vc E [0,1] . Let c E [0,1] . Hence the index of PN is finite and so p c is regular . We may assume that the congruence classes of FN are [XI], [x2], . . . , [x,] since the index of F, is finite . Clearly, Vi, i = 1, 2, . . . , m, Vu, v E X*, if u, v E [x2], then y(u) = ft(v). Therefore, IM(ft) = {ft(xl), ft(x2), - . . , ft(xm)I and so I Im(ft) I < oo. Conversely, suppose that Vc E [0,1], ftc is regular, and IIm(ft)I < oo . Then the characteristic function Xw of ftc is a fuzzy regular language. By the resolution theorem of fuzzy sets (see [160]), ft = Uc.E[o,l] c X w , . Let IM(ft) = {Cl, c2, . . . , ck} . Then ft = c1Xw , , U c2X w 2 U . . . U ckXw'k . Hence by Definition 7.9 .4, yE FR(X*) . m Proposition 7.10 .3 (FR(X*), n, U, -) is a de Morgan algebra, where denotes complement of fuzzy subsets.

Proof. Since FR(X*) C .FP(X*) and (.F'P(X*), U, n, - ) is a de Morgan algebra, it suffices to show that (FR(X*), U, n, - ) is a subalgebra of (.FP(X*), U, rl, - ) . By Definition 7.9 .4, it follows that Vtt,v E FR(X*), ft U v E FR(X*) . Also, since y,,v E FR(X*), Vc E [0,1], ft c , V c are regular, and I Im(ft) I and I Im(v) I are finite. Thus Vc E [0,1], (ft n v) c = ftc rl vc is regular, and I Im(y, n v) I < I Im(y,) I + I Im(v) I is finite . Hence by Proposition 7.10.3, ft rl v E FR(X*) . Let x, y E X* . Then x--y(FN)

b'u, v E X*, y(uxv) = y(uyv) b'u, v E X*,1 - y,(uxv) = 1 - y,(uyv) Vu, v E X*, !(uxv) = -(uyv) x - y(Fw) .

Thus since the index of FN is finite, the index of FA is finite . Hence, FR(X*) . m

Proposition 7.10 .4 Let y,v E FR(X*) . Then (1) /t, o v E FR(X*), (2) v - ly, (or y,v -1 ) E FR(X*), where v -ift (or yv -1 ) is called the fuzzy left (right, respectively) quotient of ft with respect to v and is defined as follows: (V -I [t)(y)

=V{ft(xy) n v(x) I x E E*),

(ftv-1) (y) = V{ft(yx) nv(x) I x E E* },

b'y E X*, b'y E X* .

Proof. (1) Let c E [0,1] . Since (ft o v) c = y cvc and ft and v are fuzzy regular, y c and vc are regular . Also since (yov) c = y cvc is an adjoin of two © 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

37 2

regular languages, (ft o v), is regular . Now the degrees of membership of elements of pov are obtained from the degrees of membership of elements of /t and v with the operations max and min. Thus Im(y o v) C Im(y) U Im(v) and so Im(y o v) is finite . Hence /t o v E FR(X*) . (2) For all s, t E X*, s - t(FN,) Vu, v E X* ' y(usv) = y (utv)

Vu, v, x E X*, y,(xusv) = y,(xutv)

Vu, v, x E X*, y(xusv) A v(x) = y(xutv) A v(x)

Vu, v E X * , V XE x=(y(xusv) A v(x)) = VxEx* (y(xutv) A v(x)) Vu, v E X*, (v -l y)(usv) = (v -l p)(utv) s - t(F-,I,) .

Thus, FN C F -i /, . Since /t E FR(X*), the index of FN is finite and so the index of F-i N is also finite . Consequently, v -l y E FR(X*) . m Note that v in Proposition 7.10 .4(2) may be any fuzzy language. Proposition 7.10 .5 Let

f be a homomorphism from X* onto Xl, where Xl is a nonempty set. Then (1) ,t E FR(X*) f(w) E FR(X*) ;

(2) v E FR(X*)

f-1 (v)

E FR(X*) .

Proof. (1) Let y E FR(X*) . Let c E [0,1] and xl E Xl . Now xl E if and only if (f (ft)) (xl ) > c if and only if V{ft(x) I f (x) = xl } > c if and only if 3x', f(x') = xl and ft(x) > c (since I Im(y,) I < oo) if and only if f(x') = xl , x' E /tc if and only if xl E f(ft c ) . Hence (f (ft)), = f(ft c ) . Since /tc is regular, f (ftc ) is regular and so (f (ft)), is regular . Since Im(y,) is finite, it follows that Im(f (ft)) C Im(y,) and so Im(f (ft)) is finite . Hence f(ft) E FR(X*) . (2) Let v E FR(X*) and x, y E X* . Then (f (ft)),

x-y(Ff-,( ))

(f-1(v))(uxv) = (f-1(v))(uyv) Vu,v E X* v(f (uxv)) = v(f (uyv)) vu, v E X* v(f(u)f(x)f(v)) = v(f(u)f(y)f(v)) vu,v E X* v(ul f (x)vl) = v(ulf(y)vi) vul, vi E X1 f(x) -- f(y) (F,) .

Consequently, the index of Ff-i ( ) in X* equals the index of F in X* . Since the index of F is finite, the index of Ff-,( ) is finite and so f-1 (v) E FR(X*). m Definition 7.10 .6 Let y, E .FP(X*) and let h(ft) = V{ft(u)I u E X*}. (1) Then h(y) is called the height of ft . (2) If there exists no E X* such that h(ft) = ft(uo), then no is called a saddle point of ft . © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 373 Proposition 7.10 .7 Let /t E FR(X*) . Then there exists n E N such that /t has a saddle point with length less than or equal to n - 1 . Proof. Since /t E FR(X*), there exists a finite fuzzy automaton M = (Q, Y, Tn , ao, a l ) such that p = fm = ao o T o al, where IQ I = n, ao = ]T c2 d ] I cl cn , and Q I = [ d l 2 . .. do . Clearly, .. . h(ft)

= VxEX*w(x)

=

V

~VxEU1"-1X1N'(x), VxEX*\U11-1Xbp,(x)},

where Xo = {A} . We now show that V XEU 1k-oX k[t(x)

>

VxEX*\Uk-oxklt(x)

.

Let v E X* \ Uk-o X k be such that wl >_ n. Let v = vI V2 v2 E X, i = 1, . . . , m. Then ft(v)

=

= =

where Tvk

. . .V m ,

m >_ n,

Qo o T,t, o Ql QooT2,1oT12o . . .oT2,,,zoal m VI
~7

k = 1, 2, . . . , m . Consider the term d = c 2 n r~~) n r~2~2 n . . . n r(m)

lz,n

A dz,n

The number of different elements in the indices of d is less than or equal to n. Since m > n, there exists p, q with p < q and ip = i q . Now d
c2nr2~2)n . .

.nr~P)12'Ar"9+)

n . . .nr (m-) 1Z,n Ad2m .

If the number m+p-q+1 of indices i, i1, . . . , iP, iq+1, . . . , im is still greater than n, then we repeat the above process and eliminate a part of r until the number of indices is not greater than n . Thus we may assume that m+p-q+l
< -

VI
i

e

i9+1, . . .,

-

+1

TvP+1

Let v' = vlv2 . . . VpVq+ l . . . Vm. Clearly, I V' I = m + p - q < n - 1 . Furthermore, d < ao oTv / oa l = ft(v) < VxeU;k-oxkft(x) . Since d was an arbitrary term in ft(v), it follows that p(v) < VxEUk-oxkft(x) . Thus V xEX* \Uk-oX k[t(x) < VXEU1k-oxklt(x) .

This implies that h(p) = VxEx*tt(x) = VXEU ;k-oxkft(x) . Since Uk=OX k is a finite set, there exists no E X* such that I n o l < n - 1 and h(p) = ft(n o ) .

© 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

37 4

Example 7.10 .8 Let A be the fuzzy language of Example 7.1 .11. Then

is regular, h(A) = .9, and ab is a saddle point of A with length not greater than n - 1, where n = 3. If ftE FR(X*), then there exists a number n such that for any u E X*, u = xyw with Ixyl < n, Iyj > 1, and Vi > 0, ft(xyjw) > ft(u) provided Jul > n.

Proposition 7.10 .9 (Action lemma of the fuzzy pump

Proof. As the proof of Proposition 7.10.7, let ft

u E X* and let u

= VI V2 . . .

m > n. Now

V~ ,

It( U ) =v 1
and n

= IQ I .

Let

nrl)n . . .nr) 221 2,n-lien nd 2~n)'

Since m and n are finite, there exists a term that ft(u) = d =

= fm

d

in the above equality such

ci n r~~) n . . . n r~n') li,n n di,n

Since the number of indices i, il, . . . , i n is greater than n, there exist, as in the proof of Proposition 7.10.7, p, q with p < q, i p = i q , and r(P) r(q+1) r(m) c2 A r~l) > 221 A . . . A iP-12y n 2giq+1 n . . . /~ 2--12m /~ d2m -

ft(u) )

where m + p - q < n. Furthermore, Vi
Ar1,n-1i,n n di ,n -

w (u)

n

1'(i)

n

n

rip)1ip

n

r2991+l

n. . .

That is, ft(vi . . . Vpvq . . . v m ) > ft(u) . Let x = v l . . . vp , y = Vp+1 . w=vq+1 . . . Vm . Clearly, Ixyl < n, Iyj > 1, and ft(xy °w) > y,(u) . Also, d(i)

=

. . Vq ,

and

ci A r~ l) n . . . . . . n r(p ) li P n r(p+1) r(p+1) n A r(q) n. . . A 2 i n . . . A r(q) 2P i p+ 1 2q - 1 2 q 2q-12q P p+1 (

(P+p+1 1) , .. (q) .eTZ q 1,ig Ti p i

nr~q+ l ) n . . . n r(m) n din 2giq+1 inn-12m

j -times

where (i) stands for the set of all indices in the above formula . Thus 101 . . . vp(vp+i . . . Vq) j Vq+i . . . V.) = V{ d(i) I 1 < (i) < n} > ft(u), i.e., tt(xyjw) > ft(u), b'j > 1 . Consequently, b'j > 0, ft(xyjw) > ft(u) . 0 © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 375 Corollary 7.10 .10 Let ft E FR(X*) . Then V'c E [0,1] there exists a pos-

itive integer n such that for any u E /t c, u = xyw with lxyl < n, lyl >_ 1, and b'j > 0, xy3 w E /t c provided Jul > n . m

If A is an ordinary regular language, then its characteristic function xA E FR(X*) and Corollary 7.10.10 becomes the ordinary action lemma of the pump (see [122]) . Hence Proposition 7.10.9 is a generalization of the ordinary action lemma of the pump and for this reason is called the action lemma of the fuzzy pump. We now consider fuzzy adjunctive languages . Definition 7.10 .11 Let A E .FP(X*) . If V'x, y E X*, x - y(Fa) =~, x = y, then A is called a fuzzy adjunctive language .

Example 7.10 .12 Let X = {a, b} and L = {anbn l n = 0,1. . . . } . Define A : X* ~ [0,1] by A(azb z ) _ .9 for i = 0,1, . . . , n, A(az b z ) = .5 for i =

n + l, n + 2, . . . , and A(x) = 0 if x E X*\L . By Example 7.9 .12, L is adjunctive . Let x, y E X* . Then Vu, v E X*, A(uxv) = A(uyv) implies uxv, uyv E L or uxv, uyv E X*\L . Hence it follows that x - y(Fa) implies x = y . Thus A is a fuzzy adjunctive language .

Proposition 7.10 .13 The following statements are equivalent: (1) A E .FP(X*) is a fuzzy adjunctive language. (2) Fa is the identity relation. (3) b'x E X*, [x]F, _ {x} . (4) b'x, y E X*, if x y, then there exist u, v E X* such that A(uxv) A(uyv) . m

Proposition 7.10 .14 Let A E .FP(X*) . Then A is a fuzzy adjunctive lan-

guage if there exist c E [0,1] such that A, is an ordinary adjunctive language.

Proposition 7.10 .15 Let A E .FP(X*) . Then A is a fuzzy adjunctive language ~ V'x, y E X*, {x = y(ncE(o,i) Pa,) =~> x = y} . 0

Lemma 7.10 .16 Let f be a homomorphism from X* onto Xl . Then the following properties hold: (1) dw E .FP(X*), (f(w))c = f(w)c, do E [0,1] . (2) Vv E .F'P(X*), (f -1 (v)) , = f-1 (v),, do E [0,1] . m Proposition 7.10 .17 Let f be a homomorphism from X* onto Xl . Let A E .FP(X*) be a fuzzy adjunctive language . Then f (A) E .FP(Xl) is fuzzy adjunctive .

© 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

37 6

Proof. Since A is fuzzy adjunctive, b'x, y E X*, x - y(nc Pa,) =~, x = y . That is, Vc E (0, 1), Vu, v E X*, uxv E A, ~ uyv E A, implies that x = y . Let f(x), f(y) E f(X*) = Xi and f(x) - f (y)(ncP(f(a) >) . Then Vc E (0, 1), Vu, v E X*, f(u)f (x) f(v) E (f (A)), ~ f(u)f(y) f(v) E (f (A)), . This implies that f(uxv) E (f (A)) c ~~ f (uyv) E (f (A)) c, b'c, Vu, v E X*, which in turn implies that f (uxv) E f(A c) ~ f(uyv) E f (A,), b'c, Vu, v E X* . Hence uxv E A, ~ uyv E A,, Vc,Vu,v E X* . Thus x - y(Pa ,),Vc and so x - y (n,Pa, ) . Hence x = y and so f (x) = f (y) . Therefore, f(A) is fuzzy adjunctive. Proposition 7.10 .18 Let

f : X* ~ Xl be an isomorphism. Suppose that v E .FP(Xl) is a fuzzy adjunctive language . Then f -1 (v) E .FP(X*) is fuzzy adjunctive .

Proof. Let x, y E X* and x - y(n c P(f-, Then uxv E (f -1 (v)) , v X* and so uxv E E f -1 (v c ) ~? uyv E f-1 (v,) (f -1(v))C b'c, Vu, v X* . Thus f (uxv) v, (uyv) b'c, Vu, E E ~ f E v. b'c, Vu, v E X* . Hence f(x) - f(y) (P,,) Vc and so f(x) - f(y) (n c p,) . Thus f (x) = f(y). Since f is one-one, x = y. Therefore, f-1 (v) is fuzzy adjunctive. uyv E

Proposition 7.10 .19 Let A E .FP(X*) . If there exists v E .FP(X*) such that v 1A is adjunctive, then A is fuzzy adjunctive.

Proof. Since v-1 A is adjunctive, Vs,t E X*, s - t(F -la) =~> s = t. Suppose that s - t(Fa) . Then Vu, v E X*, A(usv) = A(utv) . Thus A(xusv) = A(xutv) V'x, u, v E X* . This implies that A(xusv) A v(x) _ A(xutv) A v(x) b'x, u, v E X* and so V{A(xusv) A V(X) I x E X*} = V{A(xuyv) A V(X) I x E X* } Vu,v E X* . Thus (v-1 A)(usv) = (v-1A)(utv), Vu,v E X* and so s t(F -la) . Hence s = t. Therefore, A is fuzzy adjunctive. m

Proposition 7.10 .20 Let A E .FP({z}*) . Then A is fuzzy adjunctive if and only if A is not fuzzy regular.

Proof. Suppose A is fuzzy adjunctive. If A is fuzzy regular, then the index of Fa is finite . This contradicts the fact that A is fuzzy adjunctive and z* is divided into infinite classes that contain only one element by Fa . Conversely, if A is not fuzzy adjunctive, then there exist i, j with i :?~ j and z2 - zj(Fa) . Now {z}* is divided into at most j classes, [A], [z], [z2], . . . ,

[zj-1] .

Hence the index of Fa is finite, i.e., A is fuzzy regular . m © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 377 Proposition 7.10 .21 Let A E F({z}*). If there exists c E [0,1] such that (1) b'm > 1, 3n E N, A(zm+n) > c , (2)Vm>1,31EN,A(z'+l ) 1, b't E N, 3s > t, (A(zs+Z) < c 1,2, . . . ,m), then A is a fuzzy adjunctive language.

or A(zs +Z) > c , Vi =

Proof. It suffices to show that b'r > 0, k >_ 1, zr * zr+k (Fa) . In fact, if k + 1 = m >_ 1, then sets >_ 1 when r = 0 and sets >_ r when r > 1 . From (3), it follows that Vi = 1, 2, . . . , m, A(z s +Z) < c or A(z s +1) >_ c_ Suppose that A(zs+Z) < c , i = 1, 2, . . . , k + 1 . From (1), there exists ml, the smallest integer such that A(z s +k+l+ml) > c . Thus A(z s-r+l z

r zmi

) < c,

A(zs-r+l z r+k z mi ) > c,

i.e ., there exist z' - '+ I, zml E z* such that A(zs-r+lzrzmi)

7~ A(zs_r+lzr+kzml) .

Hence zr * zr+k(Fa) . Suppose that A(z s +z) > c , i = 1, 2, . . . , k + 1 . From (2), there exists s +l+m2) < c . Similarly, it follows m2, the smallest integer such that A(z +k easily that zr * zr+k(Fa) . Definition 7.10 .22 Let A E .FP(X*) and c E [0,1] . Then A is called a c-discrete language if b'x, y E X*, x :?~ y, A(x) >_ c and A(y) > c implies that lxl :A lyl .

Example 7.10 .23 Let X = {a, b} and L = fanbn I n = 0,1. . . . } . Define l

the fuzzy subset A of X*by A(x) = 0 if x E X*\L and A(x) > 0 if x E L. The A is c-discrete for c E (0,1] since no two distinct elements of L have the same length.

Lemma 7.10 .24 Let JXl > 2 and A E .FP(X*) . Then A is a fuzzy adjunc-

tive language if and only if Vu,v E X*, Jul = lvl and u - v(Fa) implies that u = v. 0

Proposition 7.10 .25 Let A E .FP(X*) . Suppose that A is c-discrete and b'w E X*, there exist u, v E X* such that A(uwv) >_ c . Then A is a fuzzy adjunctive language . Conversely, if A is a fuzzy adjunctive language, then b'w E X* there exist u, v E X* such that A(uwv) > 0. Proof. If u - v(Fa) and Jul = lvl , then by the hypothesis there exist x, y E X* such that A(xuy) > c . Since u - v(Fa), A(xuy) = A(xvy) and so A(xvy) > c . Now, since l xuy l = l xvy l and A is c-discrete, it follows that xuy = xvy and so u = v. Hence A is fuzzy adjunctive. © 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

37 8

Conversely, if there exists w E X* such that Vu, v E X*, A(uwv) = 0, then Vu, v E X*, A(uwv) = A(uw 2 v) = 0, i.e., w - W2 (F,\), which contradicts the fact that A is a fuzzy adjunctive language. m Proposition 7.10 .26 If A E .FP(X*) is a fuzzy adjunctive language, then b'w E X*, IA' I = oo, where A' = {x I x E X*wX*, A(x) > 0} . Proof. Since A is fuzzy adjunctive, b'w E X*, IA'l z,4 0. Suppose that IA'l < oo . Let u E A' be such that Jul = V{lvI I v E A'J. Clearly, I AwnI = 0. This contradicts the fact that A is a fuzzy adjunctive language. Proposition 7.10 .27 Let I X I > 2 . Then A is an 0-discrete fuzzy adjunctive language if and only if Supp(A) is an ordinary discrete adjunctive language.

Proof. If Supp(A) is an ordinary discrete adjunctive language, then A is clearly a 0-discrete fuzzy adjunctive language. Conversely, suppose that A is a 0-discrete fuzzy adjunctive language and Supp(A) is not adjunctive. By Lemma 7.10.24, there exist x, y E X* such that x :?~ y, Ixl = lyl, and x - y(Psupp(a) ) . Hence Vu,v E X*, uxv E Supp(A) ~ uyv E Supp(A) . Consequently, Vu, v E X*, A(uxv) > 0 A(uyv) > 0. Since A is a fuzzy adjunctive language, by Proposition 7.10.25, it follows that b'x E X* there exist no, vo E X* with A(uoxvo) > 0. Thus A(uoyvo) > 0 . Since luoxvol = luoyvol and A is 0-discrete, uoxvo = uoyvo and so x = y, a contradiction . Hence Supp(A) must be an adjunctive language. If V'x, y E X* with x :?~ y, x, y E Supp(A), then clearly A(x) > 0 and A(y) > 0. Since A is 0-discrete, Ixl :?~ lyl . Therefore, Supp(A) is discrete . m We now consider fuzzy dense languages . Definition ?.10 .28 Let y E .FP(X*) and w E X* . Let yW =

{x I x E X*wX*, y(x) > c},

c E [0,1] .

If b'w E X*, Ip~ 0, then /t is called a c-dense language . In particular, a 0-dense language is called a fuzzy dense language and we write , Itw

po =

Example 7.10 .29 Let X = {a}. Define the fuzzy subset y of X* by la(an) = n+l

, n = 0, 1, 2, . . . , where ao = A. Then tt is a 0-dense language . Now /t

is not c-dense for any c E (0,1] since there exists n such that n+il < c and for w = an and x E X*anX*, y(x) < n+il < c .

Proposition 7.10 .30 Let w E .FP(X*). Then the following statements are equivalent. (1) w is a fuzzy dense language. (2)VwEX*, lw w I =oo . (3) There exists a fuzzy adjunctive language A with w D A.

© 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 379 Proof. (1) ==> (2) Immediate from Proposition 7 .10 .26 . (2) ==> (3) Define an ordering relation on X * as follows : if Ix I < ly l, then x < y ; if IxI = IyI , then x < y means that x, y is in the lexicographic ordering of elements of X. Now X* = {A<w1<w2< . . .<wn< . . .} . Since Iww I = oo, b'w E X*, ISupp(w) _ and the number of elements in {[x] W} is greater than 1 . Let x1, x2, . . . , x, . be the representative elements from these classes. Define A E .PP (X*) as follows : if x = xi for some i if x :?~ xiforalli . Clearly, A C w . Now b'x, y E X*, x z,4 y, if A(x) > 0, A(y) > 0, then IxJ z,4 and so A is 0-discrete . Since Iwwl I = oo, there exists an element ulwlvl IyI with the shortest length in wwl . We select u l wl v l as the representative element . Then A(u l w l v l ) = w(u l wl v l ) > 0. Similarly, since Iww2 1 = oo, we can choose u2w2v2 E ww2 such that `(u2w2v2) = w(u2w2v2) > 0 and Iww~ I = oo, there exists u2w2v2 E wwa I u1w1v1 I 0 and ui_1wi_1vi_1 I (1) Now w D A implies that I ww I > IA w I b'w E X* . Since A is fuzzy adjunctive, IAw I :?~ 0 b'w E X* by Proposition 7 .10.25 . Thus ww I :?~ 0 b'w E X* . Consequently, w is a fuzzy dense language. m Proposition 7 .10 .31 Let /t E .FP(X*) .

fuzzy dense language.

Then either /t or 7 must be a

Proof. If b'w E X*, IftwI = oo, then /t is fuzzy dense . Suppose that there exists w E X* such that Iftwl < oo . Then l{x I x E X*wX*, y(x) > 0}I < oo . Thus {x I x E X*wuX*, y(x) > 0}I < oo Vu E X* and so {x I x E X * wuX * ,ft(x) = 0}I = oo Vu E X* . This implies that I {x I x E X*wuX*, T(x) = 1 > 0}I = oo .Hence (N)w'I = oo Vu E X* . Since X*wuX* C_ X*uX*, VU E X*, I (N)wul < I (p)ul = oo . Therefore, p is fuzzy dense . Proposition 7 .10 .32 Let w = / t U v and c E [0,1] . Then w is c-dense if and only if either /t or v is c-dense. Proof. Suppose that w is c-dense and v is not c-dense . exists wo E X* such that I {x I x E X*WO X*, v(x) > c} I <

© 2002 by Chapman & Hall/CRC

Then there

38 0

7. More

on Fuzzy Languages

This implies that Vu E X*, I{x I x E X*wo uX*, v(x) > c c oo . Since w is c-dense, b'w E X*, J {x I x E X*wX*, w(x) > c }I = oo, i.e., J {x I x E X*wX*, (ltUv)(x) > c}I = oo Vw E X* . This is equivalent to I{x I x E X*wX*, y(x) V v(x) > c}I = oo Vw E X* . Consequently, J {x I x E X*wouX*, y(x) V v(x) > c}I = oo Vu E X*. From this, it follows that J {x I x E X*w ouX*, y(x) > c}I = oo Vu E X* . Hence VU E X* ' I tt'o' I = oo and so I pu = oo Vu E X* . Thus tt is c-dense . Conversely, suppose that /t is c-dense . Now b'w E X*, I {x

I

x E X*wX*, ft(x)

> 0}I =

and so I {x I x E X*wX*, y(x) V v(x) > c}I = oo Vw E X* . This implies that I{x I x E X*wX*, w(x) > c} I = oo Vw E X* . Hence w is c-dense . m Proposition 7.10 .33 Let w

E .FP(X*) . Then w is c-dense if and only if is c-dense, where wl x*,,x* wl x*,,x* denotes the restriction of w to X*wX*, wEX* .

Proof. Suppose that w is c-dense . Then b'w, u E X*, I {x I x E X*wuX*, w(x) > c}I :?~ 0. Thus there exists x E X*wuX* C_ X*wx* rl X*uX* such that w(x) > c. Hence Vu E X*, there exists x E X*uX* such that wlx*,,x* (x) > c. This implies that Vu E X*, {x I

x E X*uX*,wlx .,x " (x) > c} I :?~

0.

Hence wlx*,,x* is c-dense b'w E X* . Conversely, suppose that b'w E X*, wl x*,,x* is c-dense . Then b'w X*, I{x I x E X*wX*, wlx*,,x*(x) > c }I :?~ 0. Thus b'w E X*, I{x I x X*wX*, w(x) > c} I :?~ 0. Hence w is c-dense .

E E

Proposition 7 .10 .34 Let I X I > 2, w = /t U v, and w be a fuzzy adjunctive language . Then one of the following statements hold. (1) /t or v is a fuzzy adjunctive language . (2) /t and v are fuzzy dense languages.

Proof. Suppose that /t, v are not fuzzy adjunctive and v is not fuzzy dense . Then there exists w E X* such that J {x I

x E X*wX*, v(x)

> 0 }I <

Since /t is not fuzzy adjunctive, there exist u, v E X* such that u :?~ v, Jul = IvJ, and u - v(FN) . Now Iv- 1 < oo and so there exists u such that Jul > V{IzI I z E vw} . Since uw -- vw(FN ), Vx,y E X*, lt(xuwy) = © 2002 by Chapman & Hall/CRC

7.10. Algebraic Character and Properties of Fuzzy Regular Languages 381 ft(xvwy), and Ixuwyl = lxvwyl > Jul . Thus xuwy, xuvy ~ v' and so v(xuwy) = v(xvwy) = 0. Now V'x, y E X*, co(xuwy)

= = = = =

ft(xuwy) V v(xuwy) ft(xuwy) ft(xvwy) ft(xvwy) V v(xvwy) W(xvwy) .

Hence uw - vw(F,) . Since w is a fuzzy adjunctive language, uw = vw, which contradicts the fact that u :?~ v. Thus v is fuzzy dense. Similarly, it can be shown that ft is fuzzy dense. m Proposition 7.10 .35 Let A be a fuzzy adjunctive Then AIx*wX* is a fuzzy adjunctive language .

language and w

E X*.

Proof. Clearly, A = Alx*wx* U Al x*wx* . Let x E X* . Then SI X*wx*

(x)

_ _

A(x) 0 A(x) 0

x E X*wX* x X*WX* x X*wX* x E X*wX* .

Clearly, (AI x*wX*)' is the empty set and so AI x*wX* is not fuzzy dense. By Proposition 7.10 .34, Alx*wX* is a fuzzy adjunctive language. Proposition 7.10 .36 Let I X I > 2, A be a 0-discrete fuzzy adjunction guage and A = ft U v . Then ft or v is a fuzzy adjunctive language .

lan-

Proof. If ft and v are not fuzzy adjunctive, then there exist xl, x2, yl, y2 X* such that xl :A x2, Y1 54 y2, Ixl I = Ix2I , Iyl I = IY2I , and x, -- X2 (Ft,), E = Y1 y2(F ) . Clearly, x 1 yl - x2y1(FN ) and x 1 y l - xly2(F ) . Since A is fuzzy adjunctive, there exist u, v E X* such that A(uxlylv) = y(uxlylv) V v(uxlylv) > 0 by Proposition 7.10.25 . If y(uxlylv) > v(uxlylv), then A(uxlylv) = ft(uxlylv) = ft(ux2ylv) > 0 and furthermore A(ux2ylv) = ft(ux2ylv) V v(ux2ylv) > 0. Since A is 0discrete, IUXIYIVI = lux2y1vl implies that uxlyly = ux2ylv, which contradicts that xl :A x2 . Similarly, if y(uxlyl v) <_ v(uxlyl v), then we can show that yl = y2, a contradiction . Hence ft or v is a fuzzy adjunctive language . Proposition 7.10 .37

Every 0-discrete fuzzy adjunctive language may be written as the disjoint union of two 0-discrete fuzzy adjunctive languages .

© 2002 by Chapman & Hall/CRC

382

7. More on Fuzzy Languages

Proof. Let w be a 0-discrete fuzzy adjunctive language. Let L C_ X* be an adjunctive language and let Bl

= Supp(w)

n

X*LX*,

B2

= Supp(w)

n

X*LX* .

Then B,UB2 = Supp(w), B 1 nB 2 = 01, and w = WIB,UWIB,, WIB l nwlB, = 0 . Now since w is 0-discrete, w I B I and W I B2 are also 0-discrete. Since L is an adjunctive language, b'w E X* there exist u, v E X* such that uwv E L . Also since w is fuzzy adjunctive and uwv E X*, there exist s, t E X* with w(suwvt) > 0. This implies that x = suwvt E Supp(w) r1X*LX* = B1 . Thus b'w E X*, I (WJ Bi ) w J 7~ 0 . Hence WJ B, is fuzzy dense. By Proposition 7.10.25, W J B, is fuzzy dense. _ Similarly, w 1 B 2 is also fuzzy dense since the adjunctivity of L implies that L is adjunctive. m Proposition 7.10 .38

Every fuzzy dense language may be written as the disjoint union of a fuzzy adjunctive language and an 0-discrete fuzzy adjunctive language .

Proof. Let w be a fuzzy dense language. First suppose that w is not fuzzy adjunctive. Suppose that IXI > 2. We consider the case IXI = 1 later. Let X* _ {ul, u2, . . . , un, . . . } . Since w is fuzzy dense, Vu 2 , I {x I x E X*u i X*, w(x) > 0}I = oo. Let x2 = s j ui t 2 E w n `, i = 1, 2. . . . be such that Ix1I
. . .
Let B = {XI, x2, . . . , xn, . WIB is an 0-discrete fuzzy

,

. . } . By Definition 7.10.22 and Proposition 7.10.25, adjunctive language . Next, we show that wIa is fuzzy adjunctive. Suppose that wIa is not fuzzy adjunctive. Since w is not fuzzy adjunctive, there exist V1, V2 such that vl :?~ v2, Iv1I = Iv2I, and vl - v2 (F,) . Also, since w17T is not fuzzy adjunctive, there exist W1, W2 such that wl :A w2, Iw1I = Iw2I , and w l - W2 (F, 17,) . Clearly, v l w l - vlw2(F,, B ) . Since WIB is fuzzy adjunctive, v 1 wl * vlw2(FWIB) . Thus there exist s,t E X* such that wJB(svlwlt) :A WIB(svlw2t) . Now either wJB(svlwlt) > WIB(svlw2t) or WIB(svlwlt) < WIB(svlw2t) . To be specific, suppose that wJB(svlwlt) > WIB(svlw2t) . Clearly, svlwlt E B . If WIB(svlw2t) > 0, then w(svlwlt) > w(svlw2t) > 0, i.e., v1wl - vlw2(F,) . If wI B(svlw2t) = 0 and svlwl t E B, then w(svlw 2 t) = 0 and w(sv 1 w 1 t) > w(sv l w2 t), i .e ., v 1 w l * v l w2 (F,) . If wI B(svlw2t) = 0 and svl wl t ~ B, then svlwlt E B . From v1wl - vlw2(F,1B) and svlwlt ~ B we have that w(sv l w2 t) = WI B (sv l w 2 t) = wlB(sv l w l t) = 0. Hence v 1 w l * vlw2(F,) . Therefore, it always holds that v1wl * vlw 2 (F,), i.e., w(sv l w l t) > w(svlw2t) .

© 2002 by Chapman & Hall/CRC

7.11 . Deterministic Acceptors of Regular Fuzzy Languages

383

However, vl - v2 (F,), v l w l - v 2 w 1 (F,), v l w2 (F,), and furthermore, w(sv 2 w l t) > w(sv2 w 2 t) . Since sv l w l t 7~ sv 2 w lt, I svl w ltI = I sv2witI , svlwlt E B, and by the definition of B, sv2 wl t ~ B, i .e ., sv2 w lt E B . Consequently, w(sv 2 w l t) =WI T(sv2 w lt) > w(sv2 w2t)

>

WIT(sv2w2t),

wl * w 2 (F,1 B), a contradiction . Hence w1 77 is a fuzzy adjunctive language, w = w I B U w I T, and w I B n w 1a = xO . Suppose that IXI = 1. Since w is not fuzzy adjunctive, by Definition 7.9.7, w is a fuzzy regular language . If wIa is not fuzzy adjunctive, then wIa is fuzzy regular . Since w = WIa U WIB, w, = (w I B), U (WIT) , VC E [0,1] . Clearly, w, and (wIa), are regular languages b'c E [0,1] . Since (WI B) c n (w I T)c = Ql b'c E [0,1], (WIB), is a regular language. Also since w is fuzzy regular, I {w(x) I x E X * } I < oo and { (W I B) (x) I x E X * } I < oo. By Proposition 7.10 .2, W I B is a fuzzy regular language . This contradicts that WIB is a fuzzy adjunctive language . Hence wIa is also a fuzzy adjunctive language . Suppose that w is a fuzzy adjunctive language. Now w = wIB U WIT, where W IB is a 0-discrete fuzzy adjunctive language. By Proposition 7.10.15, WI B = WI B, U WIB2 and WIBI n WIB2 = x0, where wIB. and WIB2 are two 0-discrete fuzzy adjunctive languages . Thus i.e .,

w=WI B,

UWIB 2 UWIB .

If WIB2 UwlB is a fuzzy adjunctive language, then the result is true. Suppose that wIB2 U WIBI is not fuzzy adjunctive. Then WIB2 is fuzzy adjunctive. By Proposition 7.10.30, w I B2 U w I a is a fuzzy dense language. From the preceding arguments, it follows that wIa is a fuzzy adjunctive language. Since w = W I B U WIT, the desired result follows . m

7.11

Deterministic Acceptors of Regular Fuzzy Languages

The results in this section are from [228] . It is well known that there is a oneto-one relationship between finite automata, regular (type-3) languages, and regular expressions . This relationship demonstrates the use of regular expressions for describing deterministic finite fuzzy automata and regular fuzzy languages . It introduces a normal form for the production of a regular fuzzy grammar when the max-min rule is used. Specific fuzzy system models based on fuzzy set theory include a description of decision making in a fuzzy environment [14], finite fuzzy automata as learning systems [250], and fuzzy grammars and languages [122, 123]. © 2002 by Chapman & Hall/CRC

38 4

7. More on Fuzzy Languages

Fuzzy automata, grammars, and languages lead to a better understanding of nondeterministic algorithms and of pattern recognition tasks using syntactic pattern recognition techniques. In this section, an algorithm is developed for constructing a deterministic finite automaton that classifies the strings of a language with a regular fuzzy grammar . The derivations of the grammar are governed by the maxmin rule [122, 123] . An equivalent unambiguous regular fuzzy grammar with productions in a normal form is developed from an extension of this algorithm . Definition 7.11 .1 A regular fuzzy grammar is a four-tuple G = (N, T, S, P), where N is a finite set of nonterminals, T is a finite set of terminals, S E N is the starting symbol, P is a finite set of productions, N n T = 0, and the elements in P are of the form A aET, 0
~ aB or A -

k

k

->

a, A, B E N,

Definition 7.11 .2 A finite quasi-fuzzy automaton (fgfa) is a six-tuple

M = (Q, X, Y, S, w, qo), where Q is a finite set of states, X is a finite input alphabet, Y is a finite output alphabet, S : Q x X x [0,1] ----> Q is the fuzzy state transition map, w : Q ----> Y is the output map, and qo E Q is the starting state.

Definition 7 .11.2 differs from the fuzzy machines defined previously in two important ways. First, the interval [0,1] determines the third component of the ordered triple appearing in S's domain rather than containing the image of S. Second the output map is crisp and has Q as its domain rather than Q x X. A regular fuzzy grammar reduces to a conventional grammar when production weights are all equal to 1. Similarly, a finite quasi-fuzzy automaton reduces to a conventional finite-state Moore machine by restricting the transition weights to the value 1. A fuzzy subset of T* is called a fuzzy language in the alphabet T. Given a regular fuzzy grammar G, the membership grade of a string x of T* in the regular fuzzy language L(G) is the maximum value of any derivation of x, where the value of a specific derivation of x is equal to the minimum weight of the productions used. From the max-min rule for a fuzzy language, every string of T* has its highest computable membership grade. It is known [123] that given a regular fuzzy grammar G, a corresponding finite quasi-fuzzy automaton can be constructed that "accepts" the language L(G). The following theorem describes the construction of a deterministic nonfuzzy, finite automaton that computes the membership function of LG) . Unless otherwise specified, we use the symbol X to denote both the automaton input alphabet and the language terminals, i.e., X = T.

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7.11 . Deterministic Acceptors of Regular Fuzzy Languages

38 5

Theorem 7.11 .3 Let G be a regular fuzzy grammar. Then there exists a deterministic Moore sequential machine dfa, with output alphabet Y C_ {c c is a production weight} U {0}, which computes the membership function y: X* ~ [0,1] of the language L(G) .

Proof. We give a five-step algorithm for constructing the dfa. Step 1 : Given the regular fuzzy grammar, obtain the corresponding fqfa. The fqfa is obtained in the same way that a nonfuzzy finite automaton is obtained from a nonfuzzy regular grammar [22], with the exception that a production weight is assigned to the corresponding transition [123] . Step 2 : Obtain the set W of possible nonzero membership grades of strings in the language L(G) . W is taken to be the finite set of distinct production weights or, equivalently, the weights of the transitions of the fqfa. (The reasoning is as follows : (a) the max and min operations do not introduce a weight not al ready assigned to some production and (b) each production weight initially may be the membership grade of some string (or strings) in the language L(G).) Step 3 : For all c E W obtain the regular expression P(c) describing those x E X* such that y(x) > c. (It is known that given a regular fuzzy language and a threshold c, 0 < c < 1, the nonfuzzy "threshold language" L(c) = {x I x E X*, ft(x) > c}

is regular .) The regular expression P(c) can be found in the following method . Examine the transition diagram of the fqfa and retain without weight only those transitions whose weight is equal to or greater than c. This yields a nondeterministic nonfuzzy machine that recognizes the language L(c) and from which the regular expression P(c) can be obtained directly by standard techniques for nondeterministic transition graphs or by conversion to a deterministic finite automaton and solution of the descriptive equations [22] . Step 4: For all c E W obtain the regular expression F(c) describing those strings x of X* such that y(x) = c. (It is known that if F'(cl) and F'(c2) are regular expressions, then so are the Boolean functions of F'(c l ) and F(C2) . Specifically, if F'(cl) and F(C2) define two threshold languages, then the new regular expression F(c2) = F(C2) n F'(cl)

defines the regular language consisting of those strings that are in L(c2) and not in L(cl) .) Consider the finite set W of possible nonzero membership grades in L(G) . Let cl, c2 E W be such that cl > c2 . Then the regular expression © 2002 by Chapman & Hall/CRC

7. More on Fuzzy Languages

38 6

F(c2) = F(C2) n F'(cl)

defines the set of strings {x I

x E X*,x E

L(c2),x

E L(cl)}

_ _

{x I

x E X*,

p(x) > c2, clI 10) < {x I x E X*, 1t(x) = c2}

F(c2) is an equivalence class of the equivalence relation E on X* defined by (x, y) E E if and only if 1t(x) = ft(y) . This procedure, applied to all pairs of

adjacent membership grades in W beginning with the lowest value, yields the disjoint regular expressions defining the dfa . Step 5: Use the regular expressions F(c) to obtain the state transition diagram of the dfa, where c E W. m The procedure for obtaining a state transition diagram by taking derivations of a regular expression is discussed in [22] . Another method of decomposing a fuzzy grammar into nonfuzzy grammars using the concept of level set can be found in [258] .

Corollary 7.11 .4 Let Gl be a regular fuzzy grammar. Then there is an equivalent unambiguous regular fuzzy grammar G2 in which productions have the form A - aB or A -c -> a, where A, B E N, a E T, and 0 < c < 1 .

Construct the dfa as described in proof of Theorem 7.11 .3 . Then there is in P a production A - I --> aB (or A ~ aB, with weight 1 understood) for each transition S(qA, a) = qB . If w(qB) = c :?~ 0, then there is in P a production A -c -> a for each transition S(gA,a) = qB . The starting symbol S of G2 corresponds to the starting state qs of the dfa. Suppose the terminal string x = ab . . . de causes the dfa started in state qs to halt in a state with output k. Then there is a derivation Proof.

S----> aA----> abB----> -------> ab . . .dD__c___> ab . . .de in G2 . Conversely, such a derivation in G2 yields a string that causes the dfa to terminate in a state with output c. G2 is unambiguous since it is obtained from a deterministic finite automaton. 0 The following example illustrates Theorem 7.11.3 and Corollary 7.11 .4 . If x, y E T* in the following example, then we use the notation x + y to denote {x, y} . © 2002 by Chapman & Hall/CRC

7.11 . Deterministic Acceptors of Regular Fuzzy Languages

Example 7.11 .5 (216) Consider the regular fuzzy grammar

387 GI = (N, T,

S, P), where N = {S, A, B}, T = {a, b}, and the productions are as follows: 0 .3

S-

0 .3

S-

-> bS

Bib.

0.5 S0.2 S- -> bA

S~aB 0.5 A -> b

Step 1 : The corresponding fuzzy machine is shown in Figure 7.1 . Step 2: W = {0 .7, 0.5, 0.4, 0.3, 0.2} . Step 3: c = 0.2, c = 0.3, c = 0.4, c = 0.5, c = 0.7,

F'(0 .2) F'(0 .3) F' (0 .4) F' (0 .5) F' (0 .7)

= (a + b)*(ab + bb) = (a + b)*ab = ab = ab = 01 .

Step 4: F(0 .2) F(0 .3) F(0 .4) F(0 .5) F(0 .7)

= F'(0 .2) n F'(0.3) = (a+ b) (a+ b)*ab = Ql = ab = 01 .

= (a + b)*bb

Step 5: The deterministic classifier of strings in the regular fuzzy language L(G) is shown in Figure 7.2. Using the method of Corollary 7.11.4, the productions of the equivalent grammar G2 are as follows: S~aA~bB

0.5

A~aC~bD

A

-> b

B~aC~bE

Bib

a/0 .5

Figure 7.1 1 : fqfa obtained from Gl

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7. More on Fuzzy Languages

388

C

aC

bF

D~aC~bE

Cab

E~aC~bE

Deb 0.2 E b

F

Fib.

aC

bE

b

Figure 7.2 2 : dfa obtained from G1

Example 7.11 .6 [28] Consider the string ab . Using G1, S-

0 .5

->

0 .5 ->

aA-

0 .7 ->

abandS-

aB

0.4

->

ab .

We have that ft (ab) = (0 .5 A O .5) V (0 .7/x, 0 .4) = 0 .5 . Using G2, we have that 0.5 S~aA -> ab.

7 .12

Exercises

1. Let .M = (M, t, T) be a fuzzy recognizer of a fuzzy finite state machine M = (Q, X, [t) . Let A = L(M) . If q E Q and t(q) n ft(p, x, q) > 0 for some p E Q and x E X*, then show that x-1 A = X{e} o T, where X{e} is the characteristic function of {q} . 'Figure 7 .1 is from [228], reprinted with permission by Copyright 1974 IEEE . FFggure 7 .2 is from [228], reprinted with permission by Copyright 1974 IEEE .

© 2002 by Chapman & Hall/CRC

7.12. Exercises

389

2. (121) Let .M = (M, t, T) be a fuzzy X-recognizer of M = (Q, X, /t) and let .M' = (M', t', T') be a fuzzy X'-recognizer of M' = (Q', X', ft') . Let f : Q ----> Q' and g : X ----> X' . Then (f, g) is called a homomorphism of .M into AT if b'p, q E Q and Vu E X, (1) ft(p, u, q) <_ w'(f (p), g(u), f(q)), (2) t(q) <_ t'(f (p)), and (3) T(p) <_ T'(f (p)) . If (f, g) is a homomorphism of .M into .M' prove that b'p, q E Q and x E X* , ft * (p, x, q) < ft* (f(p),9(ui) . . .g(u,),f(q)), where x = ul . . .un and u2 E X, i = 1, 2, . . . , n. 3. (121) Suppose that (f, g) is a homomorphism of .M into .M' and (h, k) is a homomorphism of .M' into .M" . Prove that (h o f, k o g) is a homomorphism of .M into .M" . 4. (121) Suppose that M = (M,t,T) is a complete accessible fuzzy Xrecognizer of the ffsm M = (Q, X, ft) . Let A = L(M) . Show that there exists a homomorphism of MA into .M, where MA is the fuzzy X-recognizer of MA . Show that if Supp(t) z,4 0, then MA is a homomorphic image of .M . 5. (121) Let M = (M, t, T) be a complete accessible fuzzy X-recognizer of a ffsm M = (Q, X, [t) . Then .M is called reduced if ql 1 oT = q2 1 OT implies ql = q2 . Let A = L(M) . Prove that .M is reduced if and only if f is injective, where (f, g) : .M ----> MA is a homomorphism and g is the identity map . If, in addition, Supp(t) z,4 0, prove that .M is reduced if and only if .M = .M A. 6. Prove Theorems 7.5.1 and 7.5 .2. 7. Prove (2), (4), and (6) of Theorem 7.5 .4. 8. Prove Theorem 7.6 .6. 9. Prove (2) and (4) of Theorem 7.6.18. 10. Prove (2) of Theorem 7.7.5. 11. Prove Theorem 7.8 .4. 12. Prove that x - y(FL) implies x = y in Example 7.10.12 . 13. Compare the fuzzy state transition map S : Q x X x [0,1] ----> Q in Definition 7 .11 .2 with a function ft : Q x X x Q ----> [0,1] . Show for example that S(q, a, 2) = ql and S(q, a, 2) = q2 is not possible, while ft(q, a, ql) = z and ft(q, a, q2) = z is possible. Show also that S(q, a, 2) = ql and S(q, a, 4) = ql is possible, while ft(q, a, qi) = 2 and ft(q, a, ql) = 4 is not possible.

© 2002 by Chapman & Hall/CRC

Chapter 8 Minimization of Fuzzy Automata 8 .1

Equivalence, Reduction, and Minimization of Finite Fuzzy Automata

The problem of equivalence, reduction, and minimization is completely solved for deterministic automata. For stochastic automata, the problem is studied in detail in [29, 58, 168] . The case for some types of fuzzy automata is given in Chapters 2 and 3. In this and the following three sections, we present an algebraic approach concerning the minimization of fuzzy automata as developed in [231] . The theoretical foundation [169] of the algorithm of Even [58] and the analogies between some aspects of the theory of rings (resp., modules) and the theory of semirings (resp., semimodules) indicate the way for determining a general solution in the case of fuzzy automata. Using the notion of a noetherian semimodule, we present an algorithm for the equivalence of some kinds of fuzzy automata. We recall the definitions of semiring and semimodule [27, 56, 59] and some ideas from the theory of fuzzy automata from Chapters 2 and 3 and [262] . Let S be a set and let *1 and *2 be two binary operations on S, i.e., *Z : S x S ~ S, i = 1, 2 . In this chapter, we call the triple (S, *1, *2) a (commutative) semiring if (S, *1) and (S, *2) are (commutative) semigroups with identity and *2 is distributive over *1, i .e ., b'a, b, c E S, a *2 (b *1 c) (b *1 c) *2 a

_ =

(a *2 b) (b *2 a)

*1 *1

(a *2 c) (c *2 a) .

Let E be a set, S be a semiring and let *'1 : E x E ~ E and *'2 S x E ----> E. The triple (E, *i, *z) is called a left semimodule over S if © 2002 by Chapman & Hall/CRC

391

392

8. Minimization of Fuzzy Automata

for all a, b E S and x, y E E the following conditions hold: (SM.1) (E, *''I ) is a commutative semigroup with identity ; (SM.2) a *2 (x *' y) (a * 1 b) *2 x

_ =

(a *2 x) *' (a *2 y) (a *2 x) *' (b *2 x)

(SM.3) a *2 (b *2 x) = (a *2 b) *2 x .

A function h : (E, *i, *z) ----> (E, *i, *z) is called a morphism of semimodules if the following properties hold: b'a E S, b'x, y E E, h(x *' y) = h(x) *i h(y) and h(a *'2 x) = a *z h(x) .

We sometimes either use - for the operation *'2 or suppress it all together and we replace *'i with +. We also simply call E a semimodule over S or an S-semimodule. The notion of a right semimodule is defined similarly. If S is a commutative semiring, we call E a semimodule. Let M be an S-semimodule. A set X C_ M is called a system of generators for M if X generates M, i.e ., if every element of M is expressible in the form E', aix2, a2 E S, x2 E X, i = 1, 2, . . . , n for some n E N. A quasi-base is a minimal system of generators for M. If the quasi-base is finite, the dimension of M (denoted dim M) is the cardinality of X. Let X be a set, not necessarily finite, and let S be a semiring. Let VX =

XEX

ax - x, ax

E

S, x

E X,

where ax = 0 except for a finite number of elements x E X. Then it follows that VX is an S-semimodule, called a free semimodule . It also follows that the set X is a minimal system of generators for VX. Proposition 8 .1 .1 Let M be a semimodule over S. Then the following conditions are equivalent . (1) Every increasing sequence of sub-S-semimodules of M, i.e., Ml C_ M2 C_ . . . C_ Mk C . . . , such that MZ :?~ MZ_1, is finite. (2) For every sub-S-semimodule of M, there exists a finite minimal system of generators . (3) Every nonempty set G of sub-S-semimodules of M contains a maximal element.

A semimodule that satisfies any of the properties of Proposition 8.1.1 is called noetherian. Proposition 8 .1 .2 If X is a nonempty finite set, then the free semimodule VX is noetherian.

0

© 2002 by Chapman & Hall/CRC

8.1 . Equivalence, Reduction, and Minimization ofFinite Fuzzy Automata393 (1) Let I = [0,1] . Consider the binary operations *1 = max and *2 = min on I. Then the triple (I, *1, *2) = ([0,1], max, min) is a commutative semiring . (2) Let L be a distributive lattice. Then the triple (L, max, min) is a commutative semiring . (3) Let X be a finite set and VX be the free semimodule generated by X over the semiring from (1). The operations on the free semimodule are as follows : Example 8.1 .3

=+ :VXXVX~VX, Lax x+Lbx x=L(ax Vbx)-x, XEX

XVX ----> VX,

=- : [0,1]

XEX

XEX

XEX

XEX

If X is finite, then VX is a noetherian semimodule (see Proposition 8.1 .2) . fuzzy automaton is a quadruple A = are finite sets and /t : Q x X x Y x Q ~ [0,1] .

Definition 8.1 .4 A

where

Q, X, Y

(Q, X, Y, /t),

(The definition of a fuzzy automaton in Definition 8.1 .4 is the same as that of a max-min sequential-like machine in Definition 2.9.1 . We use the differing terminology partially to be consistent with the original papers, but also due to the different approach.) As usual Q is the set of states, X is the input alphabet, and Y is the output alphabet for the fuzzy automaton A. Let y(qj , x2 , yr, qk) = a~ E [0,1] . If the interval [0,1] is replaced by the distributive lattice L (see Example 8.1 .3(2)), we obtain the more general notion of L-automaton, closely related to fuzzy automaton. Every fuzzy automaton A defines the free semimodules V(Q x X) and V(Y x Q) over the semiring [0,1] . Define the function ~ :V(QXX)----> by Vqj E

Q,

Vx 2 E X, r, k

The

V(YXQ)

a; (yr, qk)

array MN = [a~] describes the fuzzy automaton . the words x = xl - X2 2. . . . x1, E X*, y = Y1 ' y2 . . . ' Yp E Y' and the matrices M(x 2 , yr) = [mjk(x2, yr)], where mj k(x 2 , yr) = a~. Denote the max-min product of matrices by o . Then we obtain the expression corresponding

Consider

M(x, y)

=

M(xl, yl) 0 M(x2, y2) 0 . . . 0 M(xP, yP)-

© 2002 by Chapman & Hall/CRC

394

8. Minimization of Fuzzy Automata

If P = P(A,A) is a column matrix with IQ I rows and whose elements are equal to 1, let P(x, y) = M(x, y) o P. Let A = (Q, X, Y, p) be a fuzzy automaton . We call (Q, E) the fuzzy set of initial states, i .e., E is a fuzzy subset of Q. Let q E Q and a E [0,1] . Define Eq : Q [0,1] and Eq : Q ~ [0,1] by dq' E Q, 1

Eq(q) = ~ 0

if q=q' if q4 q'

0 o , Eq(q) = ~ aE[01]

if q=q' if qz,4 q'

with the condition that EgEQ Eq(q) 0 for Eq . The fuzzy automaton A, denoted in this case (A, Eq) (resp. (A, E)), is called initial (resp . weakly initial) . For the fuzzy automaton A, define SE (x, y) = E o P(x, y) . An entry of SE (x, y) indicates the maximal degree of membership for the input word x and the output word y, where (Q, E) is fixed. Let A = (X, Q, Y, y) and A' = (X', Q', Y', y') be fuzzy automata and let E and E' be fuzzy subsets of Q and Q' respectively. Definition 8.1.5 The initial automata (A, E) and (A', E') are said to be equivalent, written (A, E) - (A', E'), if SE (x, y)A = SE' (x, y)A' for all

and yEY* . Let A = A' = (X, Q, Y, /t) . If (A, E) - (A', E'), then E and E' are said to be equivalent on Q, written E - E' . (2) If (A, Eq) - (A', E`), then the states q E Q and q' E Q' are said to be equivalent, written q - q' . (3) A = (X, Q, Y, ft) is said to be equivalently embedded into A' _ (X', Q', Y', /t'), written A ;~ A', if for each q E Q, there exists an equivalent state q' E Q' of A' . (4) A is said to be weakly equivalently embedded into A', written A ;~ A', if for all E : Q ~ [0,1] there exists E' : Q' ~ [0,1] such that (A, E) - (A', E') . (5) A and A' are said to be equivalent, written A - A', if A ;~ A' and xEX* (1)

A'~ A. (6) A and A' are called weakly equivalent, written A s: A' , if A ;~ A' and A' ;~ A . Definition 8.1.6 Let A be a fuzzy automaton. (1) A is said to be in reduced form if for all q, q'

E

Q, q - q' implies

q=q' . (2) A' is called a reduct of A if A' is in reduced form and is equivalent to A . (3) A is said to be in minimal form if for each Eq y (i <_ QI ), there does not exist Eqy (i <_ IQ 1) such that (A, Eq y ) - (A', Eq y ) . (4) A' is called a minimal of A if it is in minimal form and A ; A'.

© 2002 by Chapman & Hall/CRC

8.1 . Equivalence, Reduction, and Minimization ofFinite Fuzzy Automata395

Example 8.1.7 Let X = {u} and Y = {1} . Let Q1 {sl, s 2 } . Define ft, : Q1 x X x Y x Q1 [0,1] and P2 [0,1]

as in Example 2.10.2 . Let El 2

Ml (u,1) = 2

1

1 3

2

3

i

i

3

2 3

and

0

: Q1 ----> [0,1]

M2 (u,

l) _

for n = 2,3= , . . . . For n

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

SE ' (u,1)A,

_

M2 (U

[

2 3

and E2

. In fact, Ml(un, in) _

2,4,6. . . . 1

0

0

n ' in) _ ~ 1

2

2

,

M2(un ,

1n ) =

2 0

0 1

2

~ . We also have that 3

° ~ 11

0

s n z) V (E l (q2) n 3)

(E l (gi)

: Q2 ---->

Y x

1

E, (q2) ] °

[ E1(gi)

i 2 0

0

and Q2 = Q2 [0,1] . Then

= {qi, q2} : Q2 x X x

(E l (gi)

n 3) ] ° [ 1

n z) V (El (q2) n 3) V (Ei(gi) n s) n z) V (El (q2) n 3)

(E l (gi) (E l (gi)

and SE2 (u,l)A 2

=

_ _

~ E2(si)

~

(E2(s2)

(E2 (82)

A

0

E2(s2) ~ 0

n 3)

3)

V

(E2(sl) (E2(sl)

0

0 n

2)

A 2) .

]

°

1

[

1 1

In fact, SE l (un,1n)A,

_

, (E1(q ) (Ei(gi)

n 2) v n z) V

n 2) v (EI(q,) n 3) v (Ei(g2) n z), (Ei(q2)

(Ei(q2)

A

3)

for n=2,3,4, . . . . For n=2,4,6, . . . , SE2 (un,1n)Az

=

(E2(sl)

A z) V

(E2(s2)

A z),

=

(E2(82)

A z) V

(E2(sl)

A 2) .

and for n = 3,5,7, . . . , SEZ (un,1n)Az

We see that if E, (q2) > 3 < E2 (82) and El(gi) > z < E2 (81), then (Al, El) ti (A2, E2). We also have that for appropriate choices of rh and 972 in Example 2.10 .2 and of El and E2 here, rh (un, 1n) = SEl (Un' in )A, and rIz (Un, 1n) _ SEZ (U n' in )A2 b'n E ICY. © 2002 by Chapman & Hall/CRC

396

8. Minimization of Fuzzy Automata

Example 8.1 .8 Let X = {u} and Y = {0,11. Define ft : Q x Xx Y x Q [0,1] as in Example 2.10.3, i.e ., ft(gi, u,1, q2) = 2 and tt(g2, u, 0, qi) = 23 with ft of any other element equal to 0 . Let E : Q ~ [0,1] . Then M(u,1) = 0

M(uu,10) = I 0 0 ~ ,

1010) =

i

0

M(uuu, 010)

0

~

, M(uuu,101) =

0 ] , . . . , and M(u, 0) _

=

0

I

,

0 ] ,

02

3

and M(uuuu, 0101)

_

1

0

0

], M(uuuu,

M(uu, 01) =

0

0

2

] ,

0 ] , . . . . It follows

that r' (X, y) = SE (x, y) A V'x E X* and b'y E Y* . Now SE91(u'1)=[

1

0 ] °

1 0

0 1 0 1 1 1

2

and

Hence it is not the case that qi - q2 . Thus A is in reduced fo 8 .2

Equivalence of Fuzzy Automata :

An Al-

gebraic Approach Let A = (Q, X, Y, /t) be a fuzzy automaton. Define a function t : V (X Y*) ----> VQ as follows: t(A, A) = 1: g, qEQ

t(x,y) -

EgjEQpj(xly)gj 0

if Ix1 = lyl if Ix1 z"~ Iy1,

where pj : X* x Y* ----> [0,1] . It follows that t is a morphism of semimodules. We denote its corresponding matrix by Mt . Construct the sequence Eo C El C . . . C_ E of subsets of E = X* x Y* as follows: Eo =EZ

Ei-i U {(x, y) I x E X*, y E Y* , Ix1 = Iy1 = i} .

© 2002 by Chapman & Hall/CRC

8.2 . Equivalence of Fuzzy Automata: An Algebraic Approach

397

Proposition 8.2 .1 The following assertions hold: (1) VEZ is a sub-semimodule of VEZ+1 for i = 0,1, . . . . (2) tVE2 is a sub-semimodule of tVE2+1 for i = 0,1, . . . . (3) If tV EZ = tV EZ+1, then tV EZ = tV EZ+ k for k = 0,1, . . . . (4) tVEk = tVEk + 1 for some k E N. Proof. (1) Since EZ is a set of generators (quasi-basis) for the semimodule VEZ, i = 0,1, . . . , and Eo C El C . . . C_ E, we have VEo C VEl C

. . .CVE . (2) The result follows here since t is a morphism. (3) See [176] . (4) The result follows here since VQ is a semimodule that is noetherian and VtE C VQ . m

Theorem 8.2.2 Let (A, E) and (A', E') be weakly initial fuzzy automata. Then (A, E) - (A', E') if and only if E o t = E' o t' .

Proof. If

IxI = I yI

for (x, y) E X* x Y*, then by Definition 8.1 .5, SE (x

and since SE(x, y)

2Y) A

I

= SE' (x, Y) A'

= E o (M (x, y) o P),

we have that

E o (M(x, y) o P) = E' o (M' (x, y) o P) .

This expression is equivalent to E o t(x, y) = E' o t' (x, y) for each (x, y) E X* x Y* such that Ixl = lyl . If Ixl :A jyj , then by the definition of t, t(x, y) = t'(x, y) = 0, i.e ., E o t = E' o t' . Conversely, suppose that E o t = E' o t' . Clearly, E o t(x, y) = E' o t' (x, y) for all (x, y) E X* x Y* . Hence E o t(x, y) = E' o t'(x, y) . However,

Mt(x, y) =

0

(x, y) o P

jjyj if ixi = yj

It follows that E o (M(x, y) o P) = E' o (M'(x, y) o P), i.e ., SE (x, y) A SE'(x, y) A' for all (x, y) E X* x Y* such that IxI = Iyl . A similar result is given in Chapters 2 and 3.

=

Corollary 8.2 .3 Let A be a fuzzy automaton. Then E - E' if and only if SE (x, y)A = SE' (x, y)A for all (x, y) E X* x Y* such that Ixl = lyl < n - 1. Proof. If (A, E) ti (A, E'), then SE(x, y)A = SE' (x, y)A for Ixl = jyj 0,1, . . . . Hence Ixl = lyl < n. If SE (x, y)A = SE' (x, y)A for all (x, y) E X* x Y* such that Ixl = Iyl < n-1, then by Proposition 8.2.1(4), it follows that E o (M(x, y) o P) = E' o (M(x, y) o P) . Thus (A, E) - (A, E') . m This is the fuzzy interpretation of the well-known Carlyle theorem [29] for equivalence of stochastic automata . © 2002 by Chapman & Hall/CRC

8. Minimization of Fuzzy Automata

398

Corollary 8.2.4 Let A be a fuzzy automaton. Then the following statements are equivalent:

(2) e0Mt =e'0Mt .0

Corollary 8.2.5 Let A be a fuzzy automaton. Then the following statements are equivalent: (1~ (2)

El q-a

-

e'1 . qj

The i-th and j-th rows in the matrix Mt are identical.

Proof. (1)x(2) Suppose that el . - e' 1 . By Corollary 8 .2 .4, el, o Mt eq, o Mt . However, by the construction of eq,, the i-th and j-th rows in Mt are identical . q-~

qj

q-~

(2)x(1) The proof is straightforward.

Lemma 8.2.6 If (A, e) - (A, e'), then dim(Im(t)) = dim (Im(t')) .

Proof. By Theorem 8 .2 .2, e o t - e' o t' ~ e o Mt - e' o Mt , . Thus it follows that dim(Im(t)) = dim(Im(t')) .

Theorem 8.2.7 Let A and A' be fuzzy automata. If e is given, then the problem of finding e', if it exists, such that (A, e) - (A', e'), is algorithmically decidable. m

© 2002 by Chapman & Hall/CRC

8.2. Equivalence of Fuzzy Automata: An Algebraic Approach

399

For a proof of Theorem 8.2 .7, see the algorithm in Figure 8.1 .

PRINT: NO EXISTS e" SUCH THAT (A, e) IS EQUIVALENT TO (A",6")

Figure 8.1 1 It is stated in [231] that the computing program is not easy to realize since useful standard programs are missing . 'Figure 8. 1 is reprinted from [231] with permission by Academic Press.

© 2002 by Chapman & Hall/CRC

8. Minimization of Fuzzy Automata

400

8.3

Reduction and Minimization of Fuzzy Automata

The reduction and minimization of fuzzy automata are a consequence of the theory of equivalence of fuzzy automata. They are of importance in applications. In this section, we give a completion of the ideas of Santos, which were presented in Chapters 2 and 3. The next result is closely connected with the problem of reduction of fuzzy automata.

Theorem 8.3.1 Let Mt be the matrix associated with the fuzzy automaton A. If Mt contains two identical rows, then there exist fuzzy automata A' and A", with IQI - 1 states each, such that A A' and A A".

-

-

Proof. Suppose that the rows corresponding to the states qi and qj are identical in Mt. Let Q' = Q\{qi) and Q" = Q\{qj). Then the corresponding matrix Mt, (resp. Mt,,) for the fuzzy automaton A' (resp. A") is obtained from Mt by eliminating the i-th (resp. j-th) row. We show that A A' (resp. A A").

-

-

The equivalent state to q E Q, qi # q # qj is q E Q' (resp. q E Q") and vice versa since E: o Mt = E: o Mtr (resp. E: o Mt = o M p ) . The equivalent state to q = qi, qj E Q, respectively, is the state qi E Q' (resp. qj E Q"). The state equivalent to qi E Q' (resp. qj E Q") is qi E Q (resp. qj E Q). The proof of these conditions is a consequence of the definition of E:, of the construction of Mt.

Corollary 8.3.2 For every fuzzy automaton, there exists a reduced fuzzy automaton. All reduced fuzzy automata associated to a given fuzzy automaton have sets of states of the same cardinality. rn

Theorem 8.3.3 For finite fuzzy automata, the relation of equivalence is decidable. rn

The block-scheme (Figure 8.2) of the algorithm proving the equivalence

© 2002 by Chapman & Hall/CRC

8.3. Reduction and Minimization of Fuzzy Automata of two fuzzy automata A and A' is in fact the proof of Theorem 8.3.3.

Figure 8.22

2 Figure

8 .2 is reprinted from [231] with permission by Academic Press .

© 2002 by Chapman & Hall/CRC

401

402

8. Minimization of Fuzzy Automata

PRINT : A" IS NOT EQUIVALENTLY EMBEDDED INTO

A"

Figure 8 .2 (Continued) The following result pertains to the existence and explicit construction of a minimal fuzzy automaton of a given fuzzy automaton . q

Theorem 8 .3.4 Let A = (Q, X, Y, p) be a fuzzy automaton . If E l n En and E0 n contains a fuzzy _ _ 1 E [0,1] as a component, then there exists _ automaton A = (Q, X, Y, y), with ~ Q1 - 1 states such that A ; A. Proof. Suppose =

El

E2

. . .

E .-1

0

Ern+l

. . .

satisfies the condition of the theorem and = [ 0 © 2002 by Chapman & Hall/CRC

0

. ..

0

1

0

. ..

0 1

En

8.3. Reduction and Minimization of Fuzzy Automata

403

. We construct the fuzzy automaton A = (Q, X, Y, 7) _ as follows: Let Q = Q\{qr } and define 7 : V(Q x X) V(Y x Q) as follows : is equivalent to Eq_

n

g(gj, xi) = Lr -a" (y,, qk), r, k

where

at = a~ V (EO M A a~) . We

now show that the states qj, j z,4 m,

with the same indices in A and A are equivalent . For the words with length l = 1, we have

az7. = Vk

farkl = V { V/ O f a rkl k ml ij ml a~

V O

k m

(EO k

= V f a rkl n arm ii ) f kl ij

= ar . ij

In the last equality, recall that Eq~ contains 1 E [0,1] as a component, i.e ., VkO .(Eo A j) j since

a =a Vk

.(E00

rm

n arm))

f0 - (Vk m LE k )

n arm aaj - arm azj .

Suppose the states qj , j :?~ m, are 1-equivalent, i.e ., for every x E X*, y E Y* such that IxI = I yI = l, the following holds: pj (x, y)a = Pi (x, y) A . By the hypothesis, VkO m (EO A pk (x, y)) = pm (x, y) . Thus we have Pi (xix, yry)

n pk (x, 2J))

=

VkOm (aj

=

Pj(xix,yry)

=

(VkOm(aj APk(x,Y)))V O (arj n (VkOm{(E APk(x,y))})) Vk(aj APk(x,y))

That is, the states with the same indices for automata A and A are (l + 1)equivalent and thus equivalent . For each El

E2

.. .

Em-1

Em+1

...

En

for A, there exists an equivalent E =

E2

E1

. ..

Em-1

0

Em+1

. ..

En

for the automaton A. For a given E = (El, E2, . . . , En ) for the automaton A, the corresponding equivalent 7 = (71, 72, . . . , Vin) for A is defined by the equation 7i = Ei V (Em A E°), i z,4 m, where E° is the i-th component of the vector E ° . Thus 9-

SE(x, y)A

=

E

o P(x, y)

vi (Ei n pi (x, y))

(E° (Viom{(Ei APi(x, y))}) V (Em A APi(x, y))) Viom{(Ei V (Em n E9)) n pi (x, y)} V iom{ (Ei n pi (x, Y)) SE(x,y)a-

© 2002 by Chapman & Hall/CRC

404

8. Minimization of Fuzzy Automata

8 .4

Minimal Fuzzy Finite State Automata

Fuzzy finite automata are used to design complex systems. For example, they are useful for a knowledge-based system designer since a knowledgebased system should solve a problem from fuzzy knowledge and should also provide the user with reasons for arriving at certain conclusions . A design tool is more valuable if there exist guidelines to assist the designer to come up with the best possible design. One of the major criteria for a best design is that it be minimal. In this section, we show that a fuzzy finite automaton Ml has an equivalent minimal fuzzy finite automaton M. We also show that M can be chosen so that if M2 is equivalent to Ml, then M is a homomorphic image of M2 . In Sections 8.1-8 .4, we considered fuzzy automata (Q, X, Y, /t), where /t : Q x X x Y x Q ~ [0,1] . Hence /t served both as the state transition function and the output function. In this section, we consider fuzzy automata such that the state transition function and the output function are distinct. We ask the reader to compare the two approaches in the Exercises . A fuzzy finite automaton (ffa) is a five-tuple M = (Q, X, Y, /t, w), where Q, X, and Y are finite nonempty sets, /t is a fuzzy subset of Q x X x Q, and w is a fuzzy subset of Q x X x Y such that the following conditions hold: (1) for all p, q E Q if there exists a E X such that ft(p, a, q) > 0, then there exists b E Y such that w (p, a, b) > 0, (2) for all p E Q, for all a E X if there exists b E Y such that w (p, a, b) > 0, then there exists q E Q such that ft(p, a, q) > 0, (3) for all p E Q, a E X, V{ft(p, a, q) I q E Q} > V{w(p, a, b) I b E Y} . The elements of Q are called states, and the elements of X and Y are called input and output symbols, respectively. /t is called the fuzzy transition function and w is called the fuzzy output function . The approach in this section differs from that of the previous sections in that the fuzzy transition function and fuzzy output functions are distinct in this section . Definition 8.4.1 Let M = (Q, X, Y, /t, w) be a ffa.

(1) Define the fuzzy subset p* of Q x X* x Q as follows: It * (p, A, q)

1 ifp=q

w* (p, xa, q) = V{w(p^ r) A w* (r, x, q) I r E Q}, for allp,gEQ,xEX*, and aEX. (2) Define the fuzzy subsets w* of Q x X* x Y* as follows: _ 1 ifx=y=A w* (p, x, y) - ~ 0 if either x :?~ A and y = A or x = A and y :?~ A ,

© 2002 by Chapman & Hall/CRC

8.4 . Minimal Fuzzy Finite State Automata

40 5

w* (p, xa, yb) = V{w(p, a, b) A ft(p, a, r) A w* (r, x, y) I r E Q}, for allpEQ,xEX*, aEX,yEY*, and bEY.

Let M = (Q, X, Y, y, w) be a ffa . Let p, q E Q, a E X, b E Y. Then * ft (p, a, q) = ft* (p, Aa, q) = V {ft(p, a, r) n ft * (r, A, q) I r E Q} = ft(p, a, q) n * * ft (q, A, q) (since ft (r, A, q) = 0 if r :?~ q) = ft(p, a, q) . Also w* (p, a, b) = w* (p, Aa, Ab) = V{w (p, a, b) Aft(p, a, r) A w* (r, A, A) I r E Q} = V{w(p, a, b) n ft(p, a, r) I r E Q} (since w* (r, A, A) = 1) = w(p, a, b) A (V{ft(p, a, r) I r E Q}) = W(P, a, b), where the latter equality holds by condition (3) . We have thus proved the following result.

Theorem 8.4.2 Let M = (Q, X, Y, ft, w) be a ffa. Then (1) ft =w*IQXXXQ, (2)w

= w* IQXXXY~

Lemma 8.4.3 Let M = (Q, X, Y, ft, w) be a ffa. Then for all p, q E Q, x,uEX*,

* ft (p, xu, q) = V{ft* (P, x, r)

n ft * (r,

u, q)

I

r E Q} . .

Lemma 8.4.4 Let M = (Q, X, Y, ft, w) be a ffa. For all p E Q, x E X*, y E Y*, of Ix1 z,4 Iyl, then w* (p, x, y) = 0.

Proof. Let p E Q, x E X*, y E Y*, and Ix1 :?~ Iyl . Suppose Ix1 > Iy1 and let I y1 = n. We prove the result by induction on n. If n = 0, then y = A and x :?~ A. Hence by definition w* (p, x, y) = 0. Suppose the result is true for all u E X*, v E Y* such that Jul > Iv1 and I v1 = n - 1 . Suppose n >_ 1. Write x = ua, y = vb where u E X*, a E X, v E Y*, and b E Y. Then Jul > Iv1 and Iv1 = n - 1. Now by the induction hypothesis, for all r E Q, w* (r, u, v) = 0. Thus w* (p, x, y) = w* (p, ua, vb) = V{w(p, a, b) A ft(p, a, r)A w*(r,u,v)I r E Q} = 0. Hence for all p E Q, x E X*, y E Y*, if Ix1 > Iyl, then w* (p, x, y) = 0. Similarly, by induction, we can show that for all p E Q, x E X*, y E Y*, if Ix1 < Iyl, then w*(p, x, y) = 0. m Theorem 8.4.5 Let M = (Q, X, Y, ft, w) be a ffa. Then statements 1(a)

and 2(a) are equivalent as are 1(b) and 2(b) . (1) (a) for all p, q E Q if there exists a E X such that ft(p, a, q) > 0, then there exists b E Y such that w(p, a, b) > 0; (b) for all p E Q, for all a E X if there exists b E Y such that w (p, a, b) > 0, then there exists q E Q such that ft(P, a, q) > 0. (2) (a) for all p, q E Q if there exists x E X* such that ft * (p, x, q) > 0, then there exists y E Y* such that w* (p, x, y) > 0; (b) for all p E Q, for all x E X* if there exists y E Y* such that w* (p, x, y) > 0, then there exists q E Q such that ft * (p, x, q) > 0.

© 2002 by Chapman & Hall/CRC

406

8. Minimization of Fuzzy Automata

Proof. (1)x(2) : (a) Let p, q E Q, and x E X* be such that p* (p, x, q) > 0. Let l xl =n . If n = 0, then x = A and so p = q . Also A E Y* and w* (p, A, A) > 0. Suppose the result is true for all u E X* such that Jul = n1. Let x = ua, where a E X and u E X*. Then Jul = n-1 . Now p* (p, x, q) = np* (r, u, q) l r E Q} > O implies that p* (p, a, r) A p* (p, ua, q) = V{p* (p, a, r) (r, u, q) > 0 for some r E Q. Hence p(p, a, r) = p* (p, a, r) > 0 and p* /t* (r, u, q) > 0 . Hence by 1(a), there exists b E Y such that w(p, a, b) > 0 . Also by the induction hypothesis, there exists v E Y* such that w* (r, u, v) > 0. Let y = vb . Then w* (p, x, y) = w* (p, ua, vb) >_ w (p, a, b) A p(p, a, r) A w* (r, u, v) > 0. The result now follows by induction . (b) Let p E Q, x E X*, and suppose there exists y E Y* such that w*(p,x,y) > 0. Then by Lemma 8 .4 .3, lxl = lyl = n, say. If n = 0, then x = y = A. Also p* (p, A,p) > 0. If n = 1, then x E X and y E Y and w(p, x, y) = w* (p, x, y) > 0. Thus by 1(b), there exists r E Q such that p(p, x, r) > 0. Hence p* (p, x, r) = p(p, x, r) > 0. Thus the result is true if n = 0 or n = 1 . Suppose the result is true for all u E X*, v E Y* such that Jul = lvl = n - 1. Suppose n > 1 . Let x = ua, y = vb, where a E X, bEY,uEX*,andvEY* . Then Jul =lvl =n-1 .Noww*(p,x,y)= w* (p, ua, vb) > 0 implies w(p, a, b) > 0 and there exists r E Q such that ft (p, a, r) > 0 and w* (p, u, v) > 0. By the induction hypothesis, there exists q E Q such that p* (r, u, q) > 0 . Hence p* (p, x, q) = p* (p, ua, q) > ft (p, a, r) A p* (r, u, q) > 0. (2)x(1): Straightforward. m

Definition 8.4.6 Let MZ =

(QZ, X, Y, pi, w 2) be a ffa, i = 1, 2. Let qZ E QZ, i = 1, 2 . Then q l and q2 are called equivalent, written ql =Q 1 Q2 q2,

if and only if for all x E X*, y E Y* (wl (ql, x, y) > 0 if and only if w2(g2,x,y) > 0) . Ml and M2 are said to be equivalent, written Ml ; M2, if for all ql E Ql there exists q2 E Q2 such that ql =Q,Q2 q2 and for all q2 E Q2 there exists ql E Q1 such that q2 ~: QIQ2 ql . If Ml = M2 = M, say, then we denote the relation JQ~Q2 by Clearly, JQ, is an equivalence relation .

~Ql .

Lemma 8.4.7 Let M = (Q, X, Y, p, w) be a ffa. Let p, q E Q, x E X*, y E Y*, a E X, b E Y. Suppose p(q, a, p) > 0, w* (p, x, y) > 0, and w(q, a, b) > 0 . Then w* (q, xa, yb) > 0.

Proof. Now w* (q, xa, yb) > w(q, a, b) A p(q, a, p) A w* (p, x, y) > 0. 0 Remark 8.4.8 Let M = (Q, X, Y, p, w) be a ffa. Let p, q E Q, x E X*, y E Y*, a E X, b E Y. We assume for the rest of this section that w* (q, xa, yb) > 0, p(q, a, p) > 0, and w (q, a, b) > 0 implies w* (p, x, y) > 0 .

Proposition 8 .4.9 Let MZ = (QZ, X, Y, pi, w2) be a ffa i = 1, 2 . Let gj,p2 E QZ, i = 1,2 . Suppose ql ~:QIQ2 q2 and there exists a E X such that ft, (gl,a,pl) > 0 and p2(g2,a,p2) > 0 . Then pl ^QIQ2 p2© 2002 by Chapman & Hall/CRC

8.4 . Minimal Fuzzy Finite State Automata

40 7

Proof. Let x E X*, y E Y* . Suppose w*1 (p l , x, y) > 0. By hypothesis, there exists a E X such that ft, (gl,a,pl) > 0 and p2(g2,a,p2) > 0. Since Ml is a ffa, there exists b E Y such that wI(gl, a, b) > 0. Thus by Lemma 8.4 .7, wl (ql, xa, yb) > 0. Hence w2 (q2 , xa, yb) > 0 and w 2 (q2 , a, b) > 0 since ql ~Ql Q2 q2 . Now w2* (q2, xa, yb) > 0, p2 (g2, a, p2) > 0, and W2 (q2, a, b) > 0. Hence by Remark 8.4 .8, w2 (p2, x, y) > 0. Similarly we can show that w2 (p2, x, y) > 0 implies w*1 (pl, x, y) > 0. Thus pl ~QlQ2 p2 . 0 Remark 8.4.10 Let MZ =

(QZ, X, Y, pi, w2) be ffa, i = 1, 2, 3. Let qZ, pi E i = 1, 2,3. (1) Suppose ql ^QIQ2 q2 and q2 ~Q2Q3 q3 . Then for all x E X*, y E Y*, w* (ql, x, y) > 0 if and only if w2 (q2, x, y) > 0 if and only if w3 (q3, x, y) > 0. Hence ql ~QIQ3 q3(2) Let pl ^Q, ql and ql ^QIQ2 q2 . Then for all x E X*, y E Y*, wl (pl, x, y) > 0 if and only if wl (ql, x, y) > 0 if and only if w2 (q2, x, y) > 0. Hence pl ^Ql Q2 q2 . (3) Let pl ^QIQ2 q2 and ql ^QIQ2 q2 . Then for all x E X*, y E Y*, wl (pl, x, y) > 0 if and only if w2 (q2, x, y) > 0 if and only if wl (ql, x, y) > 0. Hence pl ,~Ql ql . QZ,

From Remark 8.4 .10, it follows that we can use the same symbol ,~ for states to be equivalent, whether states are in the same set or different sets. (QZ, X, Y, pi, w2 ) be ffa, i = 1, 2. Let f : Q1 be a function . f is called a homomorphism of Ml into M2 if (1) for all ql E Q1, a E X, b E Y, wl (ql, a, b) > 0 if and only if w2 (f (ql), a, b) > 0, (2) (a) for all pl, ql E Q1, a E X if ft, (gl, a, pl) > 0, then

Definition 8.4.11 Let MZ = Q2

p2 (f(gl),a,f(pl))

> 0,

(b) for all pl, ql E Q1, a E X if p2 (f (ql), a, f(p l )) > 0, then there exists rl E Q1 such that ft, (ql, a, rl) > 0 and f(pl) = f (rl) . M2

phism

is called a homomorphic image of Ml , if there exists a homomorMl ~ M2 such that f is onto.

f :

Definition 8.4.12 Let M = (Q, X, Y, p, w) be a ffa. M is said to be minimal if for all p, q E Q, p = q implies p = q. Theorem 8.4.13 Let Ml =

(Q, X, Y, p l , wl) be a ffa. Then there exists a minimal ffa M such that Ml ~ M. Furthermore, M can be chosen so that if M2 is any ffa such that M2 ~ Ml , then M is a homomorphic image of M2 .

© 2002 by Chapman & Hall/CRC

408

8. Minimization of Fuzzy Automata

Proof. Set Q = {[q] I q E Q, I, where [q] is the equivalence class containing q and is induced by the equivalence relation ;Q1 . Define /t QxXxQ~[0,1]andw :QxXxY~[0,1]by w([q], a, [p]) = V{wl(s, a, t) I s, t E Q1, s ~ q, t ~ p}, and w([q], a, b) =V{wl(s, a, b)

I s E Q1, s ~ q}

for all [q], [p] E Q, a E X, and b E Y. Let ([q], a, [p]) = ([q], a, [p]) . Then q ,~ q' and p ,~ p' . Hence V{ftl(s, a, t) s, t E Q1, s ,~ q, t ,~ p} = V{ftl(s', a, t') I s', t' E Q1, s' ,~ q', t ,~ p'} since is an equivalence relation on Q1 . Thus /t is well defined. Similarly w is well defined . Let [q], [p] E Q. Suppose that there exists a E X such that /t([q], a, [p]) > 0. Then there exists s, t E Q1, s ; q, t s: p such that ft, (s, a, t) > 0. Since Ml is a ffa, there exists b E Y such that wl (s, a, b) > 0. Hence w([q], a, b) >_ wl (s, a, b) > 0. Now suppose [q] E Q, a E X, and there exists b E Y such that w([q], a, b) > 0. This implies wl (s, a, b) > 0 for some s E Q1, s ; q. Since Ml is a ffa, there exists t E Q1 such that ft, (s, a, t) > 0 . Now ft([q], a, [t]) > ft, (s, a, t) > 0. Let [q] E Q, a E X. Since ; is an equivalence relation on Q1, we can write Q1 = Uk 1 [pi], where Q = [PI 1, [p2], . . . , [pk]} and the [pi] are distinct . Now V{w([q], a, [p]) I [p] E Q} =Vklw([q], a, [p2]) =Vk1(V LN1(s,a,t) . Also, V{w([q], a, b) I b E Y} = V{V{wl(s, a, b) I s ~ q} I s ~ q, t ~ pi} b E Y} = V{V{wl (s, a, b) I b E Y} I s ~ q} < V{V{wl (s, a, r) I r E Q 1 }

I[p2]j) s ~ q} = V(V {wl (s, a, r) I s ~ q, r E Uk = Vk 1(V LN'1(s, a, r) s ~ q, r E [pi]}) = Vk1(V{ftl(s, a, r) I s ~ q, r ~ pi}) = V{w([q], a, [p]) [p] E Q} . Consequently, M = (Q, X, Y, ft, w) is a ffa. First we show that for all [q] E Q, x E X*, y E Y*, w* ([q], x, y) > 0 if and only if wl (q, x, y) > 0. Let [q] E Q, x E X*, y E Y*, and wl (q, x, y) > 0 . Then IxJ = Iyj = n, say. If n = 0, then x = y = A. Thus w*([q], x, y) = w* ([q], A, A) = 1 > 0. If n = 1, then x E X and y E Y. Hence wl (q, x, y) = wi(q,x,y) > 0 . Hence w * ([q], x, y) = w([q],x,y) = V{wl(s,x,y) I s E Q1, s ~ q} >_ wl (q, x, y) > 0. Thus if n = 0 or n = 1, then wl(q, x, y) > 0 implies w* ([q], x, y) > 0. Suppose the result is true for all u E X*, v E Y* such that Jul = Iv1 = n - 1 . Let n > 1 . Now x = ua, y = vb for some aEX,bEY,uEX*,vEY* . Then Jul=lvl=n-1.Thuswl(q,x,y)= wl (q, ua, vb) = V{wl (q, a, b) A ft, (q, a, r) A wl (r, u, v) r E Q1 } > 0 implies that there exists r E Q such that wl(q, a, b) > 0, ft, (q, a, r) > 0, w* (r, u, v) > 0. By the n = 1 case and the induction hypothesis, w([q], a, b) > 0 and w* ([r], u, v) > 0. Also ft([q], a, [r]) > ft, (q,a,r) > 0 . Hence w* ([q], x, y) = w*([q],ua,vb) > w([q], a, b) /alt([q], a, [r]) Aw*([r], u, v) > 0. The result now follows by induction . Now suppose w* ([q], x, y) > 0 for some [q] E Q, x E X*, y E Y* . Then Ix1 = Iy1 = n, say. If n = 0, then x = y = A and so wi(q,x,y) = 1 > 0 . If n = 1, then x E X and y E Y and so w([q], x, y) = w*([q], x, y) > 0 .

© 2002 by Chapman & Hall/CRC

8.4. Minimal Fuzzy Finite State Automata

409

Thus there exists s E Q1, s ; q such that w l (s, x, y) > 0. Since s s: q and w l (s, X, y) > 0, we must have w l (q, x, y) > 0, i.e., wl (q, x, y) > 0. Thus the result is true for n = 0 and n = 1 . Make the induction hypothesis that the result is true for all u E X*, v E Y* such that Jul = lvl = n - 1. Let n > 1 . Now x = ua, y = vb for some a E X, b E Y, u E X*, v E Y* . Hence Jul = lvl = n - 1 . Thus w*([q],x,y) = w*([q],ua,vb) = V{w([q],a,b) A ft([q], a, [r]) Aw*([r], u, v) l [r] E Q} > 0. This implies that there exists [r] E Q such that w([q], a, b) > 0, ft ([q], a, [r]) > 0, w*([r], u, v) > 0. Now ft([q], a, [r]) > 0 implies ft, (s, a, t) > 0 for some s, t E Q1, s s: q, t s: r. Since Ml is a ffa, there exists d E Y such that wl (s, a, d) > 0 . Thus wl (q, a, d) > 0 since s ; q . This implies that there exists t' E Q1 such that ft, (q, a, t') > 0 since Ml is a ffa . Now since s ; q, ft, (s, a, t) > 0 and ft, (q, a, t') > 0, we have that t ,~ t' by Proposition 8.4.9 . Thus t' ,~ r. Now w ([q], a, b) > 0 implies w l (q, a, b) > 0 by the n = 1 case and w* Qr], u, v) > 0 implies wl (r, u, v) > 0 by the induction hypothesis. Since r ; t', w*1 (t', u, v) > 0. Hence wl (q, x, y) = wl (q, ua, vb) >_ wl (q, a, b) A ft, (q, a, t') A wl (t', u, v) > 0. This proves our claim . Let [q], [p] E Q and [q] ,~ [p] . Let x E X* and y E Y* . Now wl (q, x, y) > 0 if and only if w* ([q], x, y) > 0 if and only if w*([p], x, y) > 0 (since [q] ; [p]) if and only if wl (p, x, y) > 0. Thus q ; p and so [q] = [p] . Hence M is minimal. Since for all q E Q1, x E X*, y E Y*, w*1(q,x,y) > 0 if and only if w* ([q], x, y) > 0, q ~ [q] . Thus Ml ,~ M. Let M2 = (Q2, X, Y, ft2, w2) be a ffa such that M2 ; MI . Define f Q2 ~ Q as follows : for all q2 E Q2 there exists ql E Q1 such that q2 ~ ql . Define f (q2) = [ql] . Suppose q2, p2 E Q2 and q2 = p2 . Then there exists gl,pl E Ql such that q2 ,~ ql and p2 ,~ pl . Thus by Remark 8.4 .10, ql = pl and so [ql] = [pl] . Hence f is well defined . Now for all [ql] E Q, ql = [ql] . There exists q2 E Q2 such that q2 ,~ ql . Now f (q2) = [ql] . Hence f is onto. We now show that f is a homomorphism . (1) Let q2 E Q2, a E X, b E Y. Suppose w2 (q2, a, b) > 0. Since M2 ,: Ml, there exists ql E Q1 such that q2 ; ql . Then wI(gl, a, b) > 0 . Thus w (f (q2), a, b) = w([ql] , a, b) > wl (ql, a, b) > 0. Now suppose w (f (q2), a, b) > 0. Let f (q2) = [ql] . Then q2 = ql . Since w([ql], a, b) = w (f (q2 ), a, b) > 0, w l (s, a, b) > 0 for some s E Q1, s ~ ql . Since s ~ ql, w l (ql, a, b) > 0 . This implies that w2 (q2, a, b) > 0 since q2 ,~ ql . Hence w2 (q2, a, b) > 0 if and only if w(f (q2), a, b) > 0 for all q2 E Q2, a E X, b E Y. (2) First suppose ft2 (q2, a, p2) > 0 for some q2, p2 E Q2 and a E X. There exists gl,pl E Q1 such that q2 s: ql and p2 s: pl . Thus f (q2) = [ql] and f (p2) = [PI I . Since ft2(g2,a,p2) > 0 and M2 is a ffa, there exists b E Y such that w 2 (q2 , a, b) > 0. This implies that w l (ql, a, b) > 0 since q2 ~ ql . Since Ml is a ffa, there exists tl E Q1 such that ft, (ql, a, ti) > 0 . Since q2 ^ ql, ft, (gl, a, tl) > 0 and ft2 (q2, a, p2) > 0, we have that tl ~ p2 by Proposition 8.4 .9. By Remark 8 .4.10, tl ~ pl. Hence ft(f (q2), a, f (p2)) = [t([gl], a, [pl]) > ft, (ql, a, tl) > 0. Now suppose ft(f (q2), a, f (p2)) > 0 for some q2 , p2 E Q2 and a E X. There exists gl,pl E Q1 such that q2 ; ql and © 2002 by Chapman & Hall/CRC

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8. Minimization of Fuzzy Automata

p2 ^ P1, f (q2) = [q,] and f (p2) = [p,] . Thus ft([q,], a, [p,]) > 0. This implies /t l (s l , a, tl) > 0 for some sl, tl E Q1, s, s: ql, and tl s: p l . Since M, is a ffa, there exists b E Y such that w, (sl, a, b) > 0 . This implies w, (ql, a, b) > 0 and so W2 (q2, a, b) > 0 since q2 ; ql . Since M2 is a ffa, there exists r2 E Q2 such that fI2(g2, a, r2) > 0. Finally we show that f (r2) = f (P2) . Since s, q, and q2 ,~ ql, s, ,'= q2 by Remark 8.4.10. Hence by Proposition 8.4.9, tl ,: r2 . Once again by Remark 8.4 .10, p, ~ r2 . Hence f (r 2 ) = [PI ] = f (p2) .

8 .5

Behavior, Reduction, and Minimization of Finite L-Automata

In the remainder of the chapter, we consider [174] . Issues concerning the behavior, equivalence, reduction, and minimization have been completely studied for deterministic, non-deterministic, and stochastic automata [220] . The approach given in Chapter 2 and Sections 8.1-8.3 Q203, 231] also) for fuzzy automata is useful if it is known how to solve systems of linear equations over the bounded chain ([0,1], max, min). It is established in [203] that if the system is compatible, the solution is among the m-fold variations with repetitions over n elements . Since the number of these variations is Mn' the time complexity function of a search manner algorithm is exponential . Following [170], a polynomial time algorithm for solving systems of linear equations over IL is given. This result is purely algebraic . It allows one to compute the behavior matrix, to study approximate (E-) equivalence, E-reduction, E-minimization, and to prove their algorithmical decidability for each finite L-automaton. Moreover, the concept of computational complexity and algorithmical decidability as described in [220] and the properties of the chain according to [125] are used here. The terminology for automata theory is as in Sections 8.1-8.3.

8 .6

Matrices over a Bounded Chain

We write IL for the bounded chain IL = (L, V, A, 0,1) over the linearly ordered set L with upper and lower bounds 1 and 0, respectively. Let I, J be index sets and K a finite index set. Let MIXJ denote the matrix [mid], where mid E L for each (i, j) E I x J. Then MIXJ is referred to as a matrix over IL. The matrix MIXJ = [mid] is the product of AI X K = [aik ] and BK X J = [bkj], if mid =Vk E K(aik Abk j ) for each i E I and each j E J. Let f : L ---> [0,1] be an injective function such that x - y ===> f (x) f (y) . The distance with respect to f between x and y for x, y E L is © 2002 by Chapman & Hall/CRC

8.6. Matrices over a Bounded Chain

411

defined as follows : df(x,y) = If (x) -f(y)1 .

Let Ix - yI = df(x,y) for x,y E L . In the remainder of the chapter, we assume f to be fixed. Let AIX J = [aij] and B IX J = [bij] be matrices over IL and E E [0,1] be fixed. We say that the i-th row in A is E-close to the k-th row in B if I ais - bks I S E for each s E J. We also say that A and B are E-close, written d(A, B) S E, if I aij - bij I S E for each i E I and each j E J. Proposition 8.6 .1

If A,, X i = [a i] and IV{ai I

i E I}

Bnx l =

-V{bi I

[bi] are E-close, then

i E III <-

E.

Proof. Let V{ai I i E I} = ak and V{bi I i E I} = b,.. Then ak - E S S b,. S a,.+E S ak+E and so ak-E S b,. S ak+E . Thus -E 5 b,.-ak S E and I ak - b, I - E. Hence I V {a i I i E I} - V{ bi I i E III = I ak - b, . I <- E. The result in Proposition 8.6.1 is valid for the transposed matrices AT and BT. bk

If the products below make sense, then for A, x J,

Proposition 8.6 .2

the following statements hold : (1) (2) (3)

Let

B) 5 E B) , E d(A, B) , E d(A,

d(A,

CB) 5 E; BT) , E; d(CAT, CBT) , E. d(CA,

d(AT,

Proof. Let A = [aij], B = [bij], Cp x j = h and u denote, respectively, the vectors h

where

= h(a, j)

B, X J

= (hk(i, j)), kEI ;

hk(i, j) = Cik Aakj

and

[cij], CA = [hij],

u = u(i, j)

= (uk(i, j)),

uk(i, j) = Cik n bkj .

CB =

[uij].

kEI,

Then

hij = Vk(eik Aakj) = Vkhk(i,j), uij = Vk(eik Abkj) =Vkuk(i, .j)-

From I aij

I hk (i, j)

- bij I

S E, it follows that I h

- uI

S

- uk (i, j) I = I Cik n akj - Cik A bkj I =

E

since I eik - Cik I = 0 or E or Iakj - bkjl Cik - bkj or Iakj - eik I .

If Ih k (i, j) - uk (i, j) I = I Cik - bkj I , then Cik / akj = Cik implies Cik - akj and Cik n bkj = bkj implies bkj 5 cik, i.e ., in this case bkj S Cik S akj and I cik - bkj I <- Iakj - bkj I <- E . If hk (i, .j) - uk (i, .j) I = Iakj - cik I the proof is similar. © 2002 by Chapman & Hall/CRC

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By Proposition 8 .6 .1, d(h,u) 5 E implies IVk(hk) -Vk(uk)I S E, which is equivalent to ~h(i, j) - u(i, j) S E and I hij - uij 5 E for each i E I, j E J. The last inequality means d(CA, CB) E. (2) The proof is similar to that of (1) . (3) The proof follows from (1) and (2) . We now see that the relation E-closeness of matrices over IL is invariant .

8.7

Systems of Linear Equivalences over a Bounded Chain

We now review the main results in [170] on computing a solution of a system of linear equations over IL. We assume all matrices are over the bounded chain IL. Let I and J be finite sets with III = m E hY and IJI = n E N. By A o X = B, we mean the system : (ail (aml

n xl ) V (ail n x2) V . . . V (aln n xn)

n xl) V (am2 n x2) V . . . V (amn n xn)

=

=

bl bm

(8 .1)

with coefficients AIXJ = [aij], unknowns XJX{1} = [xj], and constants B, X{1} = [b2] . We assume that bl >, b2 >, --- >, bm . A matrix X° = [x°] is called a (point) solution of (8.1) if A o X° = B holds. The system (8.1) is said to be solvable if it possesses at least one solution; otherwise it is said to be unsolvable. In order to propose a polynomial time algorithm for solving the system (8.1), we introduce the following symbolic matrix 13 = [bij]: bij =

S

E G

if if if

aij < b2, a ij = b2, aij > b2 .

We are interested in determining if there exists xj E L such that aij A xj - b2 . We mark the i-th equation in (8.1) in a marker vector IND if aij A xj = bi holds. Theorem 8.7.1 [170] Consider the system AoX = B. Then the following statements hold . (1) If k is the greatest number of the row with a G-type coefficient in the jth column of 13, then x j = bk implies aij A xj = bi for i = k and for each i > k with aij = bi as well as for each i' < k with ai,j >, b2, = b2 . (2) If the j-th column in 13 does not contain a G-type coefficient and r is the smallest number of the row with E-type coefficient in the j-th column, then xj = bT means aij A xj = bi for each i > r with aij = b2 . (3) If the j-th column in 13 does not contain either a G-type or an E-type coefficient, then aij A xj < bi for each xj E L . m

© 2002 by Chapman & Hall/CRC

8.7. Systems of Linear Equivalences over a Bounded Chain The implementation of Theorem

8 .7 .1

413

gives the following.

Algorithm 8.7 .2 [170] For computing a solution of the system (8.1): 1. Enter the matrices A, B. Form the matrix 13 . Erase IND . j=0. 3. j=j+1. 4. If j>ngoto8 . 5. If the j-th column in 13 does not contain a G-type coefficient, then go to 6. Otherwise xj = bk, put marks in IND for i = k, for each i > k with aij = bi and for each i < k if aij >, bi = bk . Go to 3. 6. If the j-th column does not contain an E-type coefficient then go to 7. Otherwise xj = b,., put marks in IND for i = r and for each i > r with aij = b2 . Go to 3. 7. xj = 1 . Go to 3. 8. If there exists at least one unmarked row in IND then the system is unsolvable . If all rows in IND are marked, then the system is solvable and the point solution is determined in steps 5,6, and 7. Theorem 8.7.3 [170] The following problems are algorithmically decidable in polynomial time for the system (8 .1) : (1) whether the system is solvable or not; (2) computing a point solution if the system is solvable; (3) obtaining the numbers of the contradictory equations if the system is unsolvable . 0 An n-tuple (X,, .Xn ) with Xi C L for i = 1, . . . , n is an interval solution of (8 .1) if each (XI, . . . , xn) with xi E Xi is a point solution of (8 .1) and (XI , . . . , Xn) is maximal with respect to this property. Using Algorithm 8 .7 .2, the interval solutions of (8 .1) can be found . For j, j' E J, define j - j' if bij = bij , for each i E I. Then - is an equivalence relation on J. Let [j] = {j' j' E J and j - j'} . Algorithm 8.7 .4 [170] For obtaining the interval solutions of the system (8 .1), if (8 .1) is solvable : 1. Compute the point solution by Algorithm 8 .7 .2 . Obtain the equivalence classes [j] for J. 3. For each j E J (i) if the j-th column in 13 does not contain a G-type coefficient, then go to 3(iv); (ii) of I [j] I = 1 then Xj = {bk} ; go to 3 for the next j ; (iii) for each j' E [j] form Xj, as follows: X, _ go to 3 for the next j ;

© 2002 by Chapman & Hall/CRC

{bk} bk]

ifj=j if j 7~Y~[0, ;

414

8. Minimization of Fuzzy Automata

(iv) if the j-th column does not contain an E-type coefficient, then go to 3(vii), (v) of I [j] = 1, then Xj = [b,,1] ; go to 3 for the next j; (vi) for each j' E [j] form Xj, according to the rule Xj,

= ~ [b,,1]

if j = if j jjL l ;

go to 3 for the next j ; (vii) if the j-th column contains only S-type coefficients, then Xj = L for each j E [j] ; go to 3 for the next j .

Theorem 8.7.5 [170] The time complexity function of each algorithmical realization for finding all interval solutions of (8 .1) is exponential. 0

The column matrix B IX{1} _ [bi] is called a convex linear combination of Aj = [a2j]IX{i}, j E J, with coefficients x j E L if B = (Al A x l ) V . . . V (A n A xn ), i.e ., bi = VjE j(aij A x j ) for each i E I. The next result follows from Theorem 8.7 .3 and Algorithm 8.7 .2 . Corollary 8.7.6 Let Aj, j

E J, and B be as above. It is algorithmically decidable whether B is a convex linear combination of Aj, j E J. m

.9 .7 .1 .7 1 .3 .2 .5 .6 G S S E G S = S E G .8 and 0 are .5

Example 8.7.7 Consider the system AoX = B, where A = X=

xl x2 x3

, and B =

.8 .7 .5

. It can be seen that 13

.8 .7 is a maximal solution and that .5 minimal solutions. and that

8 .8

.8 .5 0

Finite IL-Automata-Behavior Matrix

An IL-automaton is a quintuple A = (Q, X, Y, M, IL), where (1) Q, X, Y are nonempty sets of states, input letters, and output symbols, respectively; (2) IL = (L, V, A, 0,1) is a bounded chain ; (3) M = {M(x, y) = [mqq' (x, y)] I x E X, y E Y, q, q 1 E Q, mqq ' (x, y) E L} is the transition-output matrices (the stepwise behavior) of A. If Q, X, Y are finite, then A is called finite IL-automaton . We denote by AO the class of all finite IL-automata. The membership degree mqq ' (x, y) determines the stepwise transitionoutput behavior of A as follows : in step k the automaton is in state q and © 2002 by Chapman & Hall/CRC

8.8. Finite IL-Automata-Behavior Matrix

415

receives the input letter x. It outputs letter y in step k and reaches the state q' in step k + 1 . In order to define and consider the complete input-output behavior of A = (Q, X, Y, M, IL) E Ao, we introduce the following notation: As usual, X* (resp. Y*) is the free monoid over X (resp . Y) with the empty word A as the identity ; U = [u2j] is the square unit matrix: u 2j = 0 if i z,4 j; u2j = 1 if i = j; E=[ 1 1 . .. 1 ] is the column matrix with all elements 1 ; QXt1} (X x Y)* = {(x, y) I x E X*, y E Y*, Ixl = y1} . The matrix M(x, y) defines the transition-output behavior of A E AO for (x, y) = (x1 . . . xk, y1 . . . yk) E (X x Y)* if M(x, y)

_

M (x1, yl) o . . .

o M(xk, yk)

for (x, y) :?~ (A, A), if (x, y) = (A, A) .

The input-output behavior of A for (x, y) E (X x Y)* is defined by the matrix T(x, y) T(x, y)

= ~ E (x, y)

oE

if (x, y) (A, A), if (x, y) _ (A, A) .

Every element mqq ~ (x, y) in M(x, y) defines the operation of A under the input word x beginning at state q, outputting the word y and reaching the state q' . Every element t,, (x, y) in T(x, y) defines the operation of A when it receives the word x beginning at state q and outputs the word y. From the matrices T (X1 y) I we shall construct the semi-infinite matrix T of the complete input-output behavior of A and the finite submatrix B of the behavior of A. Suppose that a lexicographic order on (X x Y)* is given . Let T(i) be a finite matrix with columns T(x, y) ordered according to the lexicographic order on (X x Y)* and such that Ix1 = lyl _ i. By definition, T(0) = T(A,A) = E. Let T be the semi-infinite matrix of the complete inputoutput behavior of A with columns T(x, y) indexed according to the above order. Let B(i) be the finite matrix obtained from T(i) by omitting all columns that are convex linear combinations of the previous columns . Let B be the matrix constructed from T by omitting the columns that are convex linear combinations of the previous columns. The matrix B is called the behavior matrix for the IL-automaton A. For arbitrary matrices C', C" we write C' C_ C" if each column of C' is a column in C" . If C' C_ C" and C" C_ C', then we write C' - C" . Clearly, B(i) C_ T(i) C_ T(i + 1) C_ T for each i = 0,1, . . . . Let .M = {mee, (u, v) I u E X, v E Y, q, q' E Q} and .F = {tq (x, y) I (x, y) E (X x Y)*, q E Q} be the sets of the distinct entries for the matrices M(u, v) E M and T, respectively. For any A E Ao, we have ,F C .M U {1}, I A41 = r and ICI = s are finite, and s _ r + 1 . These properties follow from the chain properties and the algebraic operations for matrices over IL. © 2002 by Chapman & Hall/CRC

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8. Minimization of Fuzzy Automata

The following result is proven in Chapter 2 and Sections 8.1-8 .3. For IL = ([0,1], V, A, 0, 1), the generalization to the next result is easy. Theorem 8.8.1 Let A E Ao . Then the following statements hold:

(1) If there exists iE hY such that T(i) - T(i + 1), then T(i) - T(i + l) for all l = 0, 1, 2, . . . . (2) If there exists iE hY such that B(i) = B(i + 1), then B(i) = B(i + l) for all l = 0,1, . . . . (3) There exists an integer k S s1Q1 - 1 such that T(k) - T(k+ 1) and B(k) = B. m

Corollary 8.8.2 The behavior matrix of any AEAO is finite . m For equivalence, reduction, and minimization of automata, the main question is how to compute B. Thus for any column in T, we must determine whether it is a convex linear combination of the previous columns in order to select B from T. Recalling Algorithm 8.7.2, Theorem 8.7.3, and Theorem 8.8 .1(3), we propose the following algorithm for completing the behavior matrix . Algorithm 8 .8.3 Computation of the behavior matrix: 1. Enter M, s, Q1_ 2. Obtain k S s1Q1 - 1 such that T(k) - T(k + 1) . 3. Use Algorithm 8.7.2 to select B(k) from T(k) . 4. B = B(k) .

8 .9

E-Equivalence

Let A = (Q, X, Y, M, IL) E Ao be given. Let E E [0,1] . The states q, q' E Q are called E-equivalent, written qEq' if for all (x, y) E (X x Y) *, tq (x, y) - tq(x, y) I <_ E, written qEq' . In particular, for E = 0 we have tq (x, y) = tq ' (x, y) for all (x, y) E (X x Y) * and the states q, q' are called equivalent, written q - q' . Theorem 8.9.1 Let A E Ao be given. Let q,

q' E Q . and q' are E-equivalent if and only if their corresponding rows in T(k) - T(k + 1) for k S s1Q1 - 1 are E-close; (2) q and q' are equivalent if and only if their corresponding rows in B are equal.

(1)

q

Proof. (1) The proof follows from Theorem 8.8.1. (2) The necessity is clear. Conversely, let bqj = bqlj for each j in B = [b2j] . An arbitrary column t(x, y) in T is a convex linear combination of the columns of B, i.e ., t(x, y) = B o X . For tq (x, y) and tq' (x, y), we have tq (x , y) = V7 (bg7 A x7) = V7 (bq'7 A x7) = tq' (x, y)-

© 2002 by Chapman & Hall/CRC

8.9. E-Equivalence

417

The next algorithm determines whether two states in a given finite ILautomaton are E-equivalent, resp. equivalent. Algorithm 8.9.2 1. Enter E, M, IQ I , s. Construct T(k) for the smallest k S s1Q1 - 1 with T(k) - T(k + 1) . 3. If E :?~ 0 check whether the rows corresponding to q and q' are E-close in T(k) and go to 6. 4. Use Algorithm 8.8.3 to select B from T(k) . 5. Check whether the rows corresponding to q and q' are equal in B. Go to 7. 6. Print whether q E q' . Go to 8. 7. Print whether q - q' . 8. End.

Corollary 8.9 .3 For any A algorithmically decidable. m

E

Ao, the relation E-equivalence of states is

We can define an E-partition over Q for a given A E Ao from the relation E-equivalence of states . A subset QZ C Q is an E-class with center qZ E QZ if giEq holds for each q E QZ and giEq does not hold for each q E Q\QZ . If every state q E Q belongs to exactly one E-class, then the family Ni i S IQI} defines an E-partition over Q . E-equivalence of states is not an equivalence relation, but E-partition as its analogue is a sufficient tool for our purposes. Algorithm 8.9.4 Construction of an E-partition over 1. Enter Q, E. 3.

Q=Q .

Denote by qE Q the element with the smallest index. 4. Form the E-class QZ = {q2 I gEg2, qZ E Q} using Algorithm 8.9 .2.

5.

n Write QZ as the i-th E-class with the center q .

6. Q =_ Q\QZ . 7. If Q = Ql go to 9. 8. Go to 3.

9. End.

For E = 0, we obtain the factor set Q/ - under the equivalence relation equivalence of states . Corollary 8.9 .5 For all A E Ao, the construction of an E-partition of the set of the states Q is algorithmically decidable. m

© 2002 by Chapman & Hall/CRC

418 8 .10

8. Minimization of Fuzzy Automata E-Irreducibility

Let A = (Q, X, Y, M, IL) and A' = (Q', X, Y, M', IL) be finite IL.-automata with the same X, Y, L and f . Let E E [0,1] . The state q E Q is said to be E-equivalent to the state q' E Q', written qEq', if It,, (X, y) - tq, (~, y) E for all (x, y) E (X x Y) * . A is E-equivalent to A', written AEA', if for all q E Q there exists q' E Q' such that qEq' and vice versa . For E = 0, A is called equivalent to A' . Theorem 8.10 .1 Let A = (Q, X, Y, M, IL) and A' = (Q', X, Y, M', IL) be finite IL-automata. The states q E Q and q' E Q' are E-equivalent if and only if I tq (x, y) - tq, (x, y) I <_ E for all (x, y) E X* x Y* with xI = Iyl cIQI+IQ'I - 1 , where c = LF U PI . Proof. Consider the automaton A 0 A' = (Q U Q', X, Y, M 0 M', IL), where Q n Q' _ 01 and M (@ M' = {M" (u, v) = [m"q, (u, v)], u E X, v E Y, q, q' E Q U Q'} is defined as follows : mqq ' (u, v) if q, q' E Q, mqq , (u, v) _ mq q, (u, v) if q, q' E Q', 0 otherwise . By Theorem 8.9 .1, two states in A() A' are E-equivalent if and only if the corresponding rows in T(k) - TA®A, are E-close, where k = cIQI+IQ'I - 1, c = I~F U P I (the number of distinct entries in T and T') . Corollary 8.10 .2 The relation E-equivalence of finite IL-automata is algorithmically decidable in the class Ao . 0

The proof of Corollary 8.10.2 follows from Theorem 8.10.1 and Algorithm 8.9 .2 . An automaton A E Ao is called E-irreducible if for each qZ, qj E Q, i :?~ j, giEqj implies qZ = qj. The next result follows directly from Corollary 8 .9.3. Corollary 8.10 .3 For any A A is E-irreducible or not. m

E Ao, it is algorithmically decidable whether

An automaton A T is an E-reduct of a given automaton A if A T is Eirreducible and AEA, For E = 0, A T is called a reduct of A. Theorem 8.10 .4 For all automaton A automaton A T .

E

Ao, there exists an E-reduced

Proof. The construction of A T = (X, QT, Y, MT , IL) as an E-reduct of A = (X, Q, Y, M, IL) contains the following steps : 1 . Take X, Y, IL for AT the same as for A. 2. Obtain MT from M as follows : if M = {M(u, v) I M(u, v) = [m2j (u, v)], uEX, vEY, i, j < IQI}, then M,. contains the square matrices of order IQTI , © 2002 by Chapman & Hall/CRC

8.11 . Minimization

419

MT = {M'(u, v) I u E X, v E Y{ . Here M(u, v) = [mg,,, .(u, v)], where and qj are the centers of the E-classes and Qj, respectively. EvqZ QZ

mg,,, (u, v) is calculated from the elements of the i-th row of ery element . M(u,v) E M, belonging to the E-class with center qjEQj according to the rule m eiej

(u' v)

- V4TEQj

(mi,(u,

v)) .

The automaton AT is E-irreducible by construction. AEA,

Consequently,

The automaton AT constructed as in the proof of Theorem 8 .10 .4 is called a natural E-reduct of A E Ao. Corollary 8.10 .5 For every A E Ao, there are a finite number of natural reducts. m

It follows from Corollary 8.10.5 that all natural E-reducts can be constructed for any A E Ao . A method for obtaining all E-reducts for A E Ao is as follows : 1. Compute the behavior matrix B for A using Algorithm 8.8 .3. 2. Construct all natural E-reducts AT for A (cf. Theorem 8 .10 .4) . 3. Compute the behavior matrix B,. for each A, 4. For each natural E-reduct AT = (X, QT, Y, MT , IL) obtain all E-reducts {A,, _ (X, QT, Y, M,., IL){,where M, _ {M'(u, v) u E X, v E Y{ and each M(u, v) is a solution of the system M(u, v) o BT = MT (u, v) o BT for M' (u, v) .

We are interested not only in a point solution of these systems but in each solution. According to Algorithm 8.7.4 and Theorem 8.7.5, there is a continuum of E-reducts for A E Ao. The above results allow for the selection of a suitable reduced automaton having behavior E-equivalent with the original one . Using fuzzy acceptors as recognizing systems and the appropriate E-equivalence and E-reduction, feature extraction can be carried out in order to exhibit properties from primary data and to make reduction of the features characterizing a class of objects . 8 .11

Minimization

Let Oq = [CZ]Q,{1} be the column matrix with c2 = 0 if qZ = q. Let l q = [d2]QX{1} be the column matrix with d2 = 1 if qZ =q, d2 = 0 otherwise. The automaton A E AO is said to be state minimal (resp . E-state minimal) if there does not exist q E Q such that 1q o T = Oq o T (resp. l q o TA o T) for some column matrix Oq . © 2002 by Chapman & Hall/CRC

420

8. Minimization of Fuzzy Automata

Theorem 8.11 .1 The automaton A E AO is state minimal (resp. E-state minimal if and only if there does not exist q E Q such that 1q o B = Oq o B (resp. 1q o BEOq o B) for some column matrix Oq. Proof. Since B is a submatrix of T, the first part of the proof follows easily. Conversely, suppose 1 q o B = Oq o B holds, i.e., the q-th row in B is a Oq -convex linear combination of the other rows in B, b qj = Vq'EQ\Iql (bq'j A Cq ') for each j. Then we obtain for the state minimization

Vj(bgj n xj) Vi ((Vq'EQ\{gl(bq'j n Cq')) n xj) Vq'EQ\{q}(Vj(bgj n xj) n Cq') Vq'EQ\Igl(tq'(u,v) ACq') . The E-state minimization follows from Proposition 8.6 .2 and Theorem 8.10.1. tq(u , v )

=

Corollary 8.11 .2 For all A E A, it is algorithmically decidable whether A is state (resp. E-state minimal or not. Proof. The algorithm for the state minimal part consists of the following steps : 1 . Compute the behavior matrix B for A. 2. For all q E Q, solve the system 1q o B = Oq o B for Oq. 3. If there does not exist q E Q with 1q o B = Oq o B, then A is state minimal. For the E-state minimization, we must change the meaning of the matrix in Algorithm 8.7 .4 as follows : 13 13 = (bij),

bij=

S E G

if aZj - E < b2, if aZj -E-bi -aij +E if aij + E > bi,

for suitable E E [0,1] . Then as a result of Algorithm 8.7.4 (or Algorithm 8.7 .2), we obtain A o XEB. Thus, 1q o BEO q o B can be solved for Oq in steps 2, 3 above. In [104], E-equivalence for stochastic automata is examined . The main result is that there is a need to consider E-equivalence only for a subclass SSo of the class of all finite stochastic automata (A E SSo if and only if any column in T is a substochastic linear combination of the columns in B) . The reason for this restriction is that Proposition 8.6 .2 and Theorem 8.9.1 and the related statements are not valid for the stochastic case. The results show that all results given in Sections 8.10, 8.11, 8.12 above are true again only in SSo (for k = IQ I - 1) . For E = 0, we have now obtained the results of Sections 2.6 and Sections 8.1-8.3 as a particular case. © 2002 by Chapman & Hall/CRC

8.12. Exercises

8 .12

421

Exercises

1. Prove Proposition 8.1 .1 . 2. Prove Proposition 8.1 .2 . Let M = (Q, X, Y, S) be a fuzzy automaton (Definition 8.1.5) and let M' = (Q, X, Y, ft, w) be a fuzzy finite state automaton (Section 8.5) . Call M and M' (1) strongly equivalent if V(q, x, y, q') E Q x X xY x Q, S(q, x, y, q') _ (q, x, q~) = w(q, x, y) ; ft (2) equivalent if V(q, x, y, q') E QxXxYxQ, S(q, x, y, q') = ft(q, x, q')A w(q, x, y) ; (3) weakly equivalent if V(q, x, y, q') E Q x X x Y x Q, S(q, x, y, q') > 0 if and only if ft (q, x, q') A w(q, x, y) > 0. 3. Prove (3) of Proposition 8.2 .1 . 4. If M and M' are strongly equivalent, prove that V(q, x, y, q') E Q x X* x Y* x Q, S* (q, x, y, q') = ft* (q, x, q') = w*(q, x, y) .

5. If M and M' are equivalent, prove that V(q, x, y, q') E Q xX* xY* 6 * (q, x, y, q~) = ft * (q, x, q~)

n w* (q, x, y) .

6. If M and M' are weakly equivalent, prove that V(q, x, y, q') E Q x X* x Y* x Q, S* (q, x, y, q') > 0 if and only if ft* (q, x, q') A w* (q, x, y) > 0. 7. Compare minimality requirements of M and M' under various equivalence assumptions . 8. Consider the system AoX = B of Example 8.7 .7. Use the algorithms .8 .8 of Section 8.7 to show that .7 is a maximal solution and .5 .5 0 .8 d 0 are minimal solutions . .5

© 2002 by Chapman & Hall/CRC

Chapter 9 L-Fuzzy Automata, Grammars, and Languages 9 .1

Fuzzy Recognition of Fuzzy Languages

In this chapter, we consider fuzzy automata, grammars, and languages, where the interval [0,1] is replaced by a lattice . We begin our study by considering the work of [94]. We give an application of the theory of fuzzy recognition to the theory of probabilistic automata and discuss the closure properties of some fuzzy language classes corresponding to machine classes. In formal language theory, languages are classified by the complexities of machines that recognize them. Typically, the machines are finite automata, push-down automata, linear bounded automata, and Turing machines . It is also of interest to develop the theory of recognition of fuzzy languages by machines and their classification by the complexities of machines that recognize them. The theory should be a reasonable extension of the ordinary language recognition theory. In ordinary language recognition theory, a machine is said to recognize a language L if and only if for every word in L, the machine decides that it is a member of L and for a word not in L, the machine either decides that it is not a member of L or loops forever . That is, a machine may be said to recognize a language if and only if the machine computes the characteristic function of the language. Thus it is natural to define a machine to recognize a fuzzy language if and only if the machine computes its fuzzy membership function . The question arises concerning the meaning for a machine to have a fuzzy membership function. In ordinary language theory, it is defined that for a given input word a machine computes the characteristic function value at 1 if and only if it takes one of special memory-configurations, such as configurations with a © 2002 by Chapman & Hall/CRC

423

424

9. L-Fuzzy Automata, Grammars, and Languages

final state and configurations with the empty stack for cases where the machine has pushdown stacks . Hence a straightforward extension is to define each fuzzy membership function value to be represented by some memory configuration of the machine. In other words, the memory configuration, which the machine moves into after a sequence of moves for a given word, should be uniquely associated with the membership function value of the word. Even if the machine obtains a configuration uniquely associated with the membership function value of the word, we must have that the machine computes the membership function value of the word. We can be sure that the machine knows the value if it is to be able to answer exactly any question about the value . Furthermore, it is also felt in [94] that it is essential that the values represented by memory configurations of the machine are lattice elements . Thus, we require that the machine should be able to compare the fuzzy membership function value of a given word, which is stored in its memory, with any lattice element that is given to the machine as a question. Such a lattice element will be called a cutpoint . We assume that a cutpoint is represented by an infinite sequence of symbols in a finite alphabet, following the infinite expansions of decimals . For any cutpoint c, the machine having the memory configuration associated with the fuzzy membership function value ft(x) of a given input word x should be able to read, as a subsequent input, the infinite sequence corresponding to the cutpoint c sequentially, and after a finite step of deterministic moves, it can determine which of the following four cases is valid, (1) ft(x) > c, (2) p(x) < c, (3) p(x) = c, and (4) p(x) and c are incomparable.

9 .2

Fuzzy Languages

Let X be a finite set of symbols called an alphabet . Let L be a lattice with minimum element 0. An L-fuzzy language over X is defined to be a function from X* into L. At times, we omit the symbol L if its presence is clear . If /t is an L-fuzzy language over X and x E X*, then p(x) represents the membership grade for x to be in the L-fuzzy language . We consider languages associated with an L-fuzzy language /t and a cutpoint c, namely LG(w,

C) =

{x E X* I w(x) > C},

LGE(w,C) = {x E X* I w(x) > C} . We call such languages cutpoint languages for /t and c. Let A be a finite alphabet . Let 0°° be the set of all infinite sequences of symbols in A extending infinitely to the right. A one-to-one function r from L into 0°° is called a representation of L over A if there exists © 2002 by Chapman & Hall/CRC

9.2. Fuzzy Languages

425

a function d : r(L) x r(L) , N U {0} that assigns to any distinct two elements r(l) and r(m) in r(L) a positive integer d(r(l), r(m)) such that d(r(l), r(m)) = d(r(m), r(l)), d(r(l), r(l)) = 0 and conditions (1) and (2) below are satisfied . (1) Let l, m E L, l :?~ m. Then for the prefix wl of r(1) and the prefix w, of r(m) of length d(r(l), r(m)), respectively, and for any a',,3' E A-, either condition (a) or (b) holds: (a) Either w, a' or w,,3' is not in r(L). (b) Both w, a' and w,,3' are in r(L), and r -1 (wia') > r -1 (wr ,3~) if l > m r -1 (wia' ) < r -1 (w ,/3') if l < m and r -1 (wia') and r-1 (w ,,3') are incomparable if l and m are incomparable. (2) For any l, m, andninLwith 1>m>n, d(r(1), r(n)) S d(r(1), r(m)) A d(r(m), r(n)) . For a,,3 E r(L), d(a,l3) is called the D-length of a and,3 . For l E L, r(l) is called the representation of l with respect to r . A lattice does not always have a representation . However, we consider from now through Section 9.8 only lattices that have a representation over some finite alphabet . Condition (2) means that representations of lattices are restricted to those that have decimal expansions of real numbers . However, there may be many representations for a lattice . Example 9 .2 .1 Let L be a lattice with a finite number of elements . If L has elements 11, . . . , lk, let A = {h, . . . , lk} . If r(l2) = lilil2 . . . for 1 S i 5 k, then r is a representation of L over A .

Example 9.2.2 Let L[0,1] be the set of all real numbers in [0,1] with the usual ordering. A representation rl is given as follows: Let A = {0,

1, 0, 1} . For l E L[0,1], let e(l) be the binary expansion of l not of the form w111 . . . with w E {0,1}*0 . For any rational number l in [0,1], set e(l) = wowlwlwl . . . such that if e(1) = w'owlwlwl . . ., then ~wo~ 5 Iwol

and ~wlI S ~wij . Let e (l) = wo wl wlwl . . . , where wl=ala2 . . . ak for wl = ala2 . . . ak with a2 E {0,1}, i = 1, 2, . . . , k. Then let r, (1) =e(1) if l is irrational, rl (l) =

© 2002 by Chapman & Hall/CRC

e (l) if 1 is rational .

42 6

9. L-Fuzzy Automata, Grammars, and Languages

Example 9 .2 .3 Let L[o

1] u

denote the set of all rational numbers in [0,1]

with the usual ordering. Let A = {0, 1, 0, 1} . Define a representation r2 : L[o 1] u

--->

A' as follows: r2(0)

= 000,-

. . , r2(1) =111, . . . , and for l

such that e(l) = w000 . . . with w E {0,1}*1, define r2 (l) = w 000 . . . and for other l in L[o,l],, define r2(l) = e(l) .

9.3

Fuzzy Recognition by Machines

Machines treated in Sections 9 .3-9 .8 may be finite automata, pushdown automata, linear bounded automata, and Turing machines . These machines can be represented in the following manner . A machine has an input terminal that reads input symbols and A sequentially, a memory storing and processing device, and an output terminal . Formally, a machine is defined to be an 8-tuple M = (1b, I', '@, 0, S, S2, K, -yo) , where -D is a finite set of input symbols ; I' is a finite set of memory-configuration symbols ; T is a finite set of output symbols ; S2 is a finite set of partial functions {w2 } from I'* into I'* ; 0 is a partial function from I'* into IF', for some n > 1 ; (For a memory configuration 'Y E I'*, 0(-y) designates the instantaneously accessible information of 'Y by M .) ; S is a partial function from (,b U {A}) x I+n into P(Q) ;

is a partial function from I+n to @ U {A} ; -yo is an element in I'*, called the initial memory configuration . A memory configuration ~ is said to be derived from a memory configuration 'Y by u E -D U {A}, denoted by 'Y ===> ~~, if and only if there exists K

n

p = 0(y) and cot E S(u, p) such that w2(y) = ~ . Let w E V. A memory configuration ~ is said to be derived from a memory configuration 'Y by w, written y ==~> ~~, if and only if there exist ul, u2, . . . , ul with u2 in b U {A} w

such that w = ulu2 . . . ul, and there exist yo,'yl . . . . . . l with y2 E I'* such that yo = y, yi = , and y 2 ===> y2+1 for all 0 S i S l - 1 . (-y ===> -y is

valid for all y E I'* .) Given an input word w E V and the initial memory configuration -yo first, M reads input symbols or A sequentially along w, changes memory configurations step by step possibly in a nondeterministic way and reaches y such that -yo ==~> y, and emits the output K(0('Y)) .

Clearly, a machine M = ('D, I', '@, 0, S, S2, K, y o ) can be restricted to a specified family of automata such as Turing machines, pushdown automata, and finite automata for appropriate choices of I', 0, S, S2, K, and 'Yo . A machine M = (,D, I', IQ, 0, S, S2, K, y o ) is called a deterministic machine if for any memory configuration y such that -yo ===> y for some x

x E V, if S(A, 0(y)) :?~ 0, then S(A, 0(y)) contains at most one element

© 2002 by Chapman & Hall/CRC

9.3. Fuzzy Recognition by Machines

427

and S(u, 0(y)) = Ql for any u E -D, and if S(A, 0(y)) = 0, then for any u E -D, S(u, 0(y)) contains at most one element . Let L be a lattice with a minimum element 0 and tt : X* ---+ L be an L-fuzzy language over an alphabet X. Let r be a representation of L over an alphabet A. A machine M = (1b, I','@, 0, S, S2, K, -YO) is said to fuzzy recognize ft with r if and only if the following conditions hold: (1) -D = X U A U {t{, where t is an element not in X U A. (3) There exists a partial function v from I'* into L such that the conditions (i)-(iv) hold: (i) Let Sx = {y I yo ~ y{ for all x E X* . Then for all x E X* such that Sx :?' 01, Sx C Dom(v) and v x = V{v(y) I y E SxJ exists. If Sx = 01, v x is undefined_ (ii) Let y be any memory configuration in Sx . Let My denote the machine A r, "If, 0, S', 0, K,'Y), where S' is the restriction of S over (A U {A{) x I'n . Then My is a deterministic machine . (iii) For any memory configuration y E I'*, if K(0(-Y)) is in 1Q, that is, K(0(-Y)) :?~ A, then for any u E -DU{A{, S(u, 0(y)) is empty . Also, K(0(-Y)) is in T only if yo ===> y for some x E X*and y in PRE-,(L) . (wl is called xty

a prefix of a word or an infinite sequence a if a = w l , 3 for some 3. Let lI be either a set of words or a set of infinite sequences . PREII is the set of all prefixes of elements in II .) (iv) Let y E Dom(v) . For all l E L, there exists a prefix v of r(l) and y' in I' such that y ==~> y' and the following properties hold: v

is K(0('Y')) is K(0('Y')) is K(0('Y')) is K(0('Y'))

> = < ! if

if v(-Y) > l, if v(y) = l, if v(-Y) < l, v(y) and l are incomparable .

Consider xt in X*t, where t indicates the end of the input sequence. Then a machine M moves possibly nondeterministically into some memory configuration y such that v(y) is defined. Let Sx denote the set of all such y's . Consider the maximum value v x of {v(y)jy E Sxf as the value of x computed by M . If Sx is empty, then the value of x cannot be computed by M. A sequence of moves from the initial memory configuration to a memory configuration in Sx is called a value computation for x. If the machine M completes the value computation for x E X*, that is, if Sx :?' 01, then M is required to have the following ability. Let 'Y be any memory configuration in Sx . Then My should be able to compare v(y) with any element l in L if the representation r(l) of l is presented to My as a question. In other words, when r(l) is given to My , My moves deterministically reading input symbols in {A} U A along the infinite sequence r(l) and emits one of >, <, =, and ! following the order of v(y) and 1 in L after reading a finite length of a prefix of r(l), and halts (see (iv)). © 2002 by Chapman & Hall/CRC

42 8

9. L-Fuzzy Automata, Grammars, and Languages

A sequence of moves of M from ,y E Dom(v) to a halting configuration is called an order-comparing computation for r y . Let the fuzzy language ft : X* ---+ L be such that (~) -

vx

0

if vx is defined otherwise .

Then /t is said to be fuzzy recognized by the machine M with the representation r. By condition (iii), if M reads a word in A* that is not any prefix of r(l) for any l E L, then M does not emit any of >, <, =, and! . In other words, if M is given an illegal question, then M does not answer . Let To, TI, T2, and T3 be the classes of Turing machines, linear bounded automata, pushdown automata, and finite automata, respectively. Let DTo , DTI , DT2 , and DT3 be the classes of deterministic Turing machines, deterministic linear bounded automata, deterministic pushdown automata, and deterministic finite automata, respectively. An L-fuzzy language /t is said to be fuzzy recognized by a machine in TZ(DTZ) if /t is fuzzy recognized by a machine in TZ (DTZ) with some representation r, for i = 0, 2, and 3. We say that an L-fuzzy language /t : X* ---+ L is fuzzy recognized by a (deterministic) linear bounded automaton if /t is fuzzy recognized by a (deterministic) Turing machine M with some representation r in the following manner : Let x E X* and l E L. If

for some prefix yI of xt, or 'YO ===> 'Y

for some y in PREry(L), then 1ryj S t Ixl for some constant t, where 'yo is the initial configuration of M. In other words, lengths of memory configurations in M for any x E X* are always less than or equal to some constant times Ixl throughout the value computation and the order-comparing computation of x with any cutpoint in L. Example 9.3.1 Let X = {a, b} and w E X* . Let n a (w) and nb(w) denote the numbers of occurrences of a and b in w, respectively . The fuzzy language PI : X* ---> L[o,I]Q

defined by b'w E X*, PI (w)

= 2 + (2)Ina~w>-nb(w)I+I

is fuzzy recognized by a deterministic pushdown automaton.

© 2002 by Chapman & Hall/CRC

9.3 . Fuzzy Recognition by Machines

429

A pushdown automaton M = (1b, I','@, 0, S, S2, K, -YO) with

'Y2

in Example

9.3 .1 fuzzy recognizes pl, where b = X U A U {t} with X = {a, b} and 0 = {0,1, 0,1}, I' = Q U {zo, a, b,1}, where Q = {qo, ql, q>, q<, q=}, T = {>, <, _}, 0 is a partial function from I'* into I'2 such that 0(gxu) = (q, u)

for all

q E Q, x E {A} U zo{ab}*

and u

E {zo, a, b, l, l},

0 = {Wla,Wlb,Wa,Wb,W-,Wj,Wi

where b' x

E I'*

'Yo = gozo W l (x) = Au, for x E I'* and u E {a, b} W (x) = xu, for x E I'* and u E {a, b} W_ (xu) = x, for x E I'* and u E {a, b} W, (golxu) = golx1, for u E {a, b} Wi(gozo) = Wi(gozol) = glzo 1 W > (qlx) = q>x W< (qlx) = q<x W=(qlx) = q=x .

S : b x I'2 ---+ P(Q) is defined as follows : 6(t, (qo, zo)) 6(u, (qo, zo)) 8(u, (go, u)) 6(a,(go,b)) 6(t, (qo, u)) 6(t, (qo,1)) 8(1, (qj,1)) 6(0, (q1,1)) 6(1, (qj, u))

{Wi}, {Wlv} S(u,(go,l))

_

6(b, (go, a))

{W~}

if u E {a, b}, if, u E {a, b},

{W1}

if u E {a, b},

{ W i} s(0, (qj, u))

if u E {a, b}, if u E {a, b},

S(0, (q j , zo)) 8(u, (qj , zo))

if u E {0 ,1}, if u E {0,1, 0},

6(u, (q1,1)) 6(1, (q1,1))

Let the partial function K(q n , u)

K

from I'2 into T U {A} be defined as follows :

= 97 for 97

Define v : I'* ---+ [0,1]Q by

© 2002 by Chapman & Hall/CRC

E

T and u

E {zo, a, b, l, l} .

43 0

9. L-Fuzzy Automata, Grammars, and Languages

and v(glzolunl)

= + ( 2) n +l , 2

for u E f a, bf and n >, 1 .

Let ft' : X* ---> L[0 , 1] be such that ft', (w) = ft, (w) for w E X*. Then it follows that ft' is fuzzy recognized by a deterministic pushdown automaton with the representation rl of a previous example. 9 .4

Cutpoint Languages

Let ft be an L-fuzzy language over X, where L is a lattice with minimum element 0. Then l E L is called an isolated cutpoint of ft if one of the following three conditions holds: (1) There exist 11 and 12 in L such that 11 < l < 12 and Vx E X* such that ft(x) :?~ l, either ft(x) S 11 or ft(x) >, 12(2) l is a maximum element of L and there exists 11 :?~ l in L such that Vx E X* with [t(X) :?~ l, [t(X) S 11. (3) l = 0 and there exists 1 2 :?~ 0 in L such that Vx E X* with y(x) :?~ 0, Theorem 9.4.1 Let L be a lattice with minimum element 0.

Let ft X* ---+ L be an L-fuzzy language and let l be an isolated cutpoint of ft. Then, for i = 0, 1, 2, 3, if ft is fuzzy recognized by a machine in Ti, then LG E (f, l) and LG (f, l) are recognized by a machine in Ti .

Proof. Suppose that ft is fuzzy recognized by a machine M = (X U A U ftf , I', T, 0, S, S2, K,'YO)

in TZ with a representation r over A. Since l is an isolated cutpoint of ft, either (1), (2), or (3) holds . We only prove the case when (1) holds . The proofs for other cases are similar . Let 11 and 12 in L be such that 11 < l < 12 . Suppose that Vx E X* with ft(x) :?~ l, either ft(x) > 12 or ft(x) S 1 1 . Let dl and d2 be the D-length of r(11) and r(l) and that of r(l) and r(12), respectively. Let d3 = dl V d2 and let w E A* be the prefix of r(l) of length d3 . From the definition of fuzzy recognition, the set L[M, >1] is recognized by a machine in Ti, where L[M, >,] is the set of all words of the form xty with x E X*, and y E A* is such that 'Yo ==~> 'Y and K(0(-Y)) is = or >. We now show that

xty

fx E X*ly(x) > if = fx E X*Ixty E L[M, >] for some y E PRE wA*f . If y(x) >, l, then there exists 'Y E I'* and y E PREfr(1)f C PREwA* such

that 'Yo ~ 'Y and K(0(-y)) is = or >.

© 2002 by Chapman & Hall/CRC

Conversely, suppose that 'Yo ~ 'Y

9.4. Cutpoint Languages

431

and K(0(-Y)) is = or >, where x E X*, y E PREwA*, and ,y E I'* . If y E PREr(l), then clearly ft(x) >, l . Otherwise, there exists l' E L such that y(x) >, l' and for some w' E A* and a E A', r(l') = ya = ww'a. From the definition of D-length, neither l' and 1 1 nor l' and 12 are incomparable, and also neither l' < 1, nor 12 < l' holds . Therefore, 1 1 S l' S 12 . This implies that ft(x) = l or ft(x) >, 12. Thus ft(x) >, l. Clearly, there exists a gsm-mapping G, [96, p.272], such that LGE(lt, l) = G(L[M, >,]

n X*cPREwA*) .

Since classes of recursively enumerable sets, context-free languages, and regular sets are closed under a gsm-mapping operation, respectively, it holds for i = 0, 2, and 3 that LGE(lt, l) is recognized by a machine in Ti . Suppose that i = 1 . Then machine M is a Turing machine such that for some constant t, I _ t I x I for any x E X* and for any memory configu ration ,y such that 'Yo ===> 'Y for some y E PRE{r(l) 11 E L} . A machine xty

M' is a modification of M as follows, M' moves reading xt in the same way as M moves reading xt for x E X* . After reading xt, M' continues to

read A and changes sequentially memory configurations in the same way as M reads some y E PREwA* . In order to move in this way, M' has an autonomous finite state machine as a submachine that generates any y E PREwA* nondeterministically. Clearly, M' E T1 and thus the set L(M') of all words xt (x E X*) for which M' emits > or = is recognized by a machine in T1 . It follows that L(M) = {xt I x E X*, y E PREwA* and xty E L(M, >)}. Therefore, L(M') = LGE(lt, l) t. Hence LGE(ft, 1) is recognized by a machine in T1 . The proof for LG(lt, l) is similar . m Corollary 9.4 .2 Let L be a finite lattice and let ft : X * ~ L be an L-fuzzy language . Then, for i = 0, 1, 2, 3, if ft is fuzzy recognized by a machine in Ti, then for any l E L, LGE(lt, l) and LG(lt, l) are recognized by a machine in Ti . Proof. Suppose that ft is fuzzy recognized by a machine M = (X U A U {t}, I',,@, 0, S, S2, K,'Yo)

in Ti. Let L = {11,12 . . . . , i s } . Then there exists wj E A* such that r (1j) E wj A', but r (1j) ~ wk A' for 1 <_ j < k <_ s . Let L[M, >] be the set of all words of the form xty with x E X* and y E A* such that 'Yo ==> ,y and xty

is = or > . Then it follows from the definition of fuzzy recognition that L[M, >] is recognized by a machine in Ti . Clearly, it follows that K(0(-Y))

{x

E

X* I y(x) > lj } = {x

© 2002 by Chapman & Hall/CRC

E X*

I xty

E

L[M, >] for some y

E

PREwj A

432

9. L-Fuzzy Automata, Grammars, and Languages

for j = 1, . . . , s such that lj is not the minimum element of L. Moreover, if lj is the minimum element of L, then {x E X* I ft(x) > l j } = X* is clearly recognized by a machine in Ti. The remainder of the proof is the same as in the proof of Theorem 9.4 .1 . Theorem 9.4.3 Let L be a lattice with minimum element 0 that has a representation ro over D0 . Let /t : X* ~ L be an L-fuzzy language such that ft(X*) is finite . Then for i = 0, 1, 2, 3, if LGE(lt, ft(x)) is recognized by a machine in TZ b'x E X*, then /t is fuzzy recognized by a machine in Ti. Proof. Suppose that ft (X*) _ {h, 12, . . . , i s } . Let Mj be a machine recognizing LGE(/t, lj) for j = 1, . . . , s. A machine M that fuzzy recognizes /t with a representation r over A is given as follows : Let A = Do U A l U 02, where Di = {ll, 1 2 . . . . . ls} such that 1' is a new symbol corresponding uniquely to l2 for i = 1, . . . , s and A2 = P(Al) x P(Al) . Define r as follows : r(lj ) r(l)

hro(lj ) (A,, Bi)r0(l)

=

=

for j = 1, . . . , s if l ~ y,(X*),

where Al ={h I lj >11andBI={h I l j lk, (3) li < lk, and (4) lj and lk are incomparable, and then halts . Reading (Al, BI), M(~j) emits, respectively, >, <, and ! according to the _ cases (1) h E A, (2) h E B1, (3) h ~ Al U B1, and then halts . Let Fj _ {yT '}'T E Fj I and let v('YT) = lj for all %3r E I'j and for all j = 1, . . . , s . If ft(X) = lk, then Sx = Uii < b I'j . Therefore, i

it W = V(W) = V{V(-r) I

'r

E Sx}

Thus /t is fuzzy recognized by M. The following corollaries follow from Theorems 9.4.1, 9.4.3, and Corollary 9.4 .2. © 2002 by Chapman & Hall/CRC

9.4. Cutpoint Languages

433

Corollary 9.4 .4 Let L be a totally ordered set with a minimum element

and that has some representation or let L be a finite lattice. If /t is an L-fuzzy language over some alphabet X such that ft(X*) is finite, then for i = 0, 1, 2, 3, /t is fuzzy recognized by a machine in TZ if and only if b'x E X*, LGE(lt, ft (X)) is recognized by a machine in Ti . m

Corollary 9.4 .5 Let L be a language over X and let XL : X* ----> B be the characteristic function of L, where B is the Boolean lattice with two elements . For i = 0, 1, 2, 3, L is recognized by a machine in TZ if and only if XL is fuzzy recognized by a machine in Ti . m Corollary 9 .4.5 shows that the representation concept for fuzzy languages introduced in this section is a fairly good extension of the one for ordinary languages . Example 9.4 .6 Let L be a lattice with a minimum element 0 and a maxi-

mum element 1. An L-fuzzy context-free grammar is defined to be a quadruple G = (N, T, P, S), where NUT is a finite set of symbols with Nr1T = 0, T is the set of terminal symbols, N is the set of nonterminal symbols, S E N, and P is a finite set of production rules of the form

with A E N, x E (N U T)*, and l E L. For any y, z E (N U T)*, we write i y ----> z if there exists u, v E (N U T) * and A z = uxv. We write

z is in P such that y = uAv and

i

y~y * y

0 z *

y

m z *

for all y, z E (N U T)*, and

if and only if there exists a sequence of elements y0, yl, . . . , yn E (N U T) * i. such that y0 = y, yn = z, y2_1 4 yZ for i = 1, . . . , n and AZ l l2 = m. For any x E T*, let lx = V{l E L I S ='~> x} . The L-fuzzy language /t defined by ft(x) = lx , for all x E T*

© 2002 by Chapman & Hall/CRC

43 4

9. L-Fuzzy Automata, Grammars, and Languages

is said to be generated by G. An L-fuzzy language generated by some Lfuzzy context-free grammar is called an L-fuzzy context-free language. (L_ fuzzy phrase structure, L-fuzzy context-sensitive, L-fuzzy regular grammars and languages are similarly defined, respectively .) L[0,1]-fuzzy context-free languages were studied in Chapter 4. From Proposition 18 in [261] and Theorem 9.4 .3, it follows that any L[0,1] fuzzy context-free language is fuzzy recognized by a pushdown automaton.

Example 9.4.7 Let X = {a, b, c} and let y2 be the L[0 1]-fuzzy language over X defined by Vi, j, k E hY U {0} It2(a'. h'ck) _

(1)~i-j I+1 + (1)h-k1+1 2

2

and It2(w) = 0 if w ~ a*b*c* . Then 1 is an isolated cutpoint of ft2, and LGE(It2 ,1) = {a'b'c2 I i E NU{0}} is not recognized by any pushdown automaton. Thus by Theorem 9.4 .1, ft2 is not fuzzy recognized by any pushdown automaton. It follows that N2 is fuzzy recognized by some deterministic linear bounded automaton with the representation rl of Example 9.2.2.

We now consider regular representations . The following theorem is clear from the proof of Theorem 9 .4 .1 . Theorem 9.4.8 Let L be a lattice with a minimum element. Suppose that

the L-fuzzy language ft is fuzzy recognized by a machine in TZ(DTZ) with a representation r and that r(l), l E L, is generated by an autonomous finite automaton sequentially . Then LG(lt, l) and LGE(lt, l) are recognized by a machine in TZ(DTZ), where i = 0, 1, 2,3. 0

We next introduce the concept of a regular representation . Let L be a lattice with a minimum element . A representation r of L is said to be regular if for all l E L, r(l) is generated sequentially by some autonomous finite automaton . The following corollary follows from Theorem 9.4 .8 . Corollary 9.4.9 Let L be a lattice with a minimum element. For i =

1, 2, 3, if an L-fuzzy language ft is fuzzy recognized by a machine in TZ(DTZ) with a regular representation, then for all l E L, LG(lt, 1) and LGE(ft, 1) are recognized by a machine in TZ(DTZ) . 0

The representation of r2 of L[0,1], shown in the lar . Thus the L-fuzzy language ft, of Example 9.3 .1 LG(lt, l) and LGE(lt, l) are context-free languages . of restricted type have regular representations . For © 2002 by Chapman & Hall/CRC

Example 9.2 .3 is reguand for any l E Lfo,ll,, Clearly, only lattices example, L[o,1] cannot

435 9.5. Fuzzy Languages not Fuzzy Recognized by Machines in DT2 have a regular representation . However, fuzzy recognition of a fuzzy language with a regular representation is of interest since, by Corollary 9.4.9, it gives a property independent of cutpoints of the fuzzy language with respect to recognition of its cutpoint languages .

9 .5

Fuzzy Languages not Fuzzy Recognized by Machines in DT2

In view of Theorem 9.4.1 and Corollary 9.4.2, L-fuzzy languages not fuzzy recognized by a machine in TZ for i = 0, 1, 2,3 are easily found. However, Theorem 9.4 .1 and Corollary 9.4 .2 cannot be used to find an L-fuzzy language whose membership function-values distribute densely over L and that is not fuzzy recognized by a machine in TZ for i = 0, 1, 2. Hence it would be of interest to find such a language. Although we do not have such an example, we do have the following example . Example 9.5.1 Let X = {0,1} and let over X such that for a2 E X, i = 1, 2, . . . ,

la3 be an L[0,1],-fuzzy language k,

Ps(ala2 . . . ak) = a12 -1 + a22 -2 + . . . + ak2-k (binary expansion, and P3 (A) = 0 . We now show that la3 is not fuzzy recognized by any deterministic pushdown automaton. Suppose that la3 is fuzzy recognized by a deterministic pushdown automaton

with a representation r over A . Let

Ll =

{xty

I

'Yo

ty'Y and K(B('Y)) _ (_)}

Then Ll is a context-free language included in X*t0* . Let L2 = L1 n 0*1*t0* . Then L2 is also a context-free language . Since M is deterministic, for any xty E L1, there exists a E A' such that r(ft3(x)) = ya . Due to the pumping lemma of the theory of context-free languages, there exists a constant k such that if ~zj >_ k and z E L2 , then z = uvwxy such that vx :?~ A, Ivwxl <_ k, and for all i, uv'wx'y E L2. Let m >_ k and let zP be an element in L2 of the form OP1mtgp , gP E A*, for any p E N U {0} . Then zP = u pvp wp xp yp , where vp xp :?~ A, vp wp xp l <_ k, and for all i E hY U {0}, up vpwp xPyp E L2 . Since M is deterministic and halts immediately after it emits (_), there does not exist x E X* and y, y' E A* with y :?~ y' such

I

© 2002 by Chapman & Hall/CRC

43 6

9. L-Fuzzy Automata, Grammars, and Languages

that both xty, xty' E L2 . Since ft3(x) 7~ ft3(x') for x and x' in 0*lm with x z,4 x', there do not exist x and x' in O*lm with x :?~ x' and y E A* such that xty, x'ty E L2 . Thus for all p, neither up nor yP contains t . Since t cannot occur in either vp or xp, wp contains t for all p. Hence vp :?~ A and xp :?~ A for all p > 0 . Also b' p > 0, we can write vp = Pp for some sp > 1 and wp = 11PtWp for some tp > 0 and Wp E A* . Since l vpwpxpl <_ k for all p, there exist non-negative integers p and q, and W and A :?~ x E A* such that p < q, Wp = Wq = W sp = sq = s, and xp = xq = x. Thus z

P

z4

and for all i

= =

Opl,twxyp Og1'tWxyq

E N U {0}

Opl m- sls ztWxzyp Ogj m- s1sitWXZyq There exist ao, ctrl

E

E L2 E

L2 .

A' such that

r-1 (WYPC'O) = ft3(oplm-S) < P3(Op l m) = r-1(WxyPCti1) . Let do be the D-length of Wyp ao and Wxypcxl, and let j > do . Then there exists a2 E A' such that r-1(WXZypC'2) = ft3(Op l m-S1Sj ) > ft3(Op1m) . Hence the D-length dl of Wyp ao and Wxiypa2 is less than or equal to do . Since j > do, j > dl . However, there exists a3 E A' such that r-1(WxjygC'3) = ft3(Op l m-S1Sj ) < ft3(Op l m-S ) = r -1 (WyPaO)l which contradicts the definition of a representation. Thus la3 cannot be fuzzy recognized by any deterministic pushdown automaton.

It is well known in the theory of probabilistic automata [168] that LGE(lt, l) and LG(lt, l) are regular sets for any l E L[0,1], . Thus it is not true in general that an L-fuzzy language is fuzzy recognized by a machine in T3 (even in DT3) even if all its cutpoint languages are recognized by machines in T3. Consequently, the converse of Corollary 9.4 .9 does not hold .

9 .6

Rational Probabilistic Events

In this section, it is shown that any rational probabilistic event is fuzzy recognized by a deterministic linear bounded automaton with a regular representation . A rational probabilistic automaton A with n states over a finite alphabet X is a triple A = (7r, {A(u) l u E Xj, 97), where 7r is a 1 x n 7r2 matrix [ 7r 1 . . . 7rn ] with 7r2 E [0,1]Q for i = 0, . . . , n such that © 2002 by Chapman & Hall/CRC

9.6. Rational Probabilistic Events

437

1 7r2 = 1, for all u E X, A(u) is an n x n stochastic matrix such that all components of A(u) are in [0,1]Q and rl is an n x 1 matrix such that all components of rl are in [0,1]Q . An L-fuzzy language p : X* ----> L[0,1 ], is said to be realized or accepted by a rational probabilistic automaton A if p(A) = 7rrl and for any m E hY and u2 E X for i = 1, . . . , m

p(UIu2 . . .

urn)

= 7rA(u 1 )A(u2) . . . A(ur )9J .

An L-fuzzy language p is called a rational probabilistic event if p is realized by some rational probabilistic automaton. The reader is referred to [168] for properties concerning probabilistic automata . Theorem 9.6.1 Every rational probabilistic event is fuzzy recognized by a machine in DT, with a regular representation .

Proof. Suppose that p : X* ----> L[0,1 ], is a rational probabilistic event . Assume that p is realized by a rational probabilistic automaton A = (7r, {A(u) In E Xj, rl) ,

with n states . Let all components of 7r, A(u)'s (u E X), and rl be represented in the form h/k for h, k E hY such that k is common to all of them. For any word x = ulu2 . . . Um of length m >_ 1, let 7r(x) = (7r1(x), 7r2(x), . . . ,7r .(x)) be 7rA(u1)A(u2) . . . A (u ,) and let 7r (A) = 7r . Then for any word x of length m, 7r2(x) is represented in the form h2(x)/k-+1 with 0 <_ h2(x) <_ km+1, for i = 1, . . . , n, and p(x) is represented in the form w(x)/km+ 2 with 0 < w(x) < km+2 . We denote the binary expansion of a nonnegative integer j by b(j) and the length of b(j) by b(j) j . Then we can construct a Turing machine z such that when z reads an input word x E X*, z gives the output string T(x) = b(h1(x))#b(h2(x))# . . . #b(hn(x))#b(w(x))#b(klx1+2)

on its tape using at most d I x l working spaces, where d is a constant independent of x. Clearly, T(xu) with u E X can be computed by a Turing machine from T(x) at most a constant multiple of Ib(klx'l+1)I spaces and that for any non-negative integer s, Ib(ks)I <_ d's for some constant d' . Thus by induction on the length of an input word, it follows that there exists a Turing machine z that computes T(x) for x E X* using at most d I xl spaces . Subsequent moves of z are defined as follows : If z with a memory configuration T(x) reads t as its next input symbol, then z transforms T(x) into b(s(x))#b(s'(x)) on its tape, where s(x)/s'(x) = w(x)/klx1+ 2 and s(x) and s(x) are relatively prime. (If w(x) = 0, z prints 0 and if w(x) = k1x1+2, z prints 1 on its tape.) Since division by a binary number not greater than b(klxl+2) can be done with at most a constant multiple of Ib(klxl+2) I spaces, z can give b(s(x))#b(s'(x)) on its tape using d I xl spaces after it reads xt . © 2002 by Chapman & Hall/CRC

43 8

9. L-Fuzzy Automata, Grammars, and Languages

0'°° We now define a representation r : L[o 1] Q as follows, where A' = {0,1} x {0,1, #} : Define Tl : A {0,1, #}°° and T2 : 0'°° ----> {0,1}°° by TZ(ul, u2) = u2 for (UI, u2) E A' and Ti(x) = TZ(ul)TZ(u2)TZ(u3) . . .

for x = ulu2u3 . . . with uj E X, j > 1, for i = 1, 2. For l E L[ 0,1], with 0 :?~ l 1, let l = s1s', where s and s' are relatively prime positive integers . Then r(l) is the element in 0'°° such that T I (r(l)) = b(s)#b(s')### . . . and T2(r(l)) = e(l), where e(l) was given in Example 9 .2 .2 . Then r(0) = (0 , 0) (#, 0) (#, 0) (#, 0) . . . and r(1) = (1,1) (#, 1) (#, 1) (#,1) . . . . Clearly, r is a representation since the D-length of r(l) and r(m) with l :?~ m may be defined as the positive integer k such that T2(r(l)) and T2(r(m)) differ first at the kth digit . It follows easily that r is a regular representation . If z with b(s(x))#b(s'(x)) on its tape is given a representation r(l) of l E L[0,1]u as an input succeeding xt for x E X*, z computes the binary expansion of s(x)Is'(x), digit-by-digit, and compares it with T2 (r(l)) . In parallel with this computation, z compares sequentially b(s(x))#b(s'(x)) with TI(r(l)) . Then either z emits = and halts if b(s(x))#b(s'(x)) = b(s)#b(s'), where TI(r(l)) = b(s)#b(s')### . . . , or z emits > or < and halts according to the comparison of the first distinguished digit of T2(r(l)) and the binary expansion of s(x)ls'(x) . Since the digit-by-digit generation of the binary expansion of s(x)ls'(x) can be made using at most a constant multiple of lb(s'(x))l spaces, z can do the above value comparing computation for x E X* using at most d I xI spaces. Thus p is fuzzy recognized by z in DTI with the regular representation r. The following result, which was proved in [230], follows directly from Theorem 9.6 .1 . Corollary 9.6 .2 Let p : X* ----> L[0,1], be a rational probabilistic event. Then for all l E L[O,I]Q , LG(lt, l) and LGE(N, l) are recognized by a machine in DTI . m 9 .7

Recursive Fuzzy Languages

The relation between deterministic machines and nondeterministic machines with respect to the fuzzy recognizability of fuzzy languages differs somewhat from that of ordinary languages . We show that in the fuzzy recognition of fuzzy languages, nondeterministic Turing machines are more powerful than deterministic Turing machines. Let {t0, t1, t2 . . . . } be an enumeration of deterministic Turing machines . Let L3 be a lattice with three elements 0, c, and 1 such that 0 < c < 1 . Let X = {u} . Let la4 and lay be L3-fuzzy languages over X defined as follows : For n E hY U {0}, ft4(un)

-

1 0

if to with the blank tape eventually halts otherwise

© 2002 by Chapman & Hall/CRC

9.8. Closure Properties la5(Un)

-

1 c

439 if to with the blank tape eventually halts otherwise .

Lemma 9.7.1 Let la4 and [t5 be defined as above . Then la4 is fuzzy recognized by a deterministic Turing machine. [t5 is fuzzy recognized by a nondeterministic Turing machine, but it is not fuzzy recognized by a deterministic Turing machine .

Proof. Clearly, there exists a deterministic Turing machine that fuzzy recognizes 1a4 . Since LGE(/t5 ,1) and LGE(N5, c) are recursively enumerable languages, it follows from Theorem 9.4.3 that [t5 is fuzzy recognized by a Turing machine. Suppose that [t5 is fuzzy recognized by a deterministic Turing machine. Then it follows that the language {un I lt 5 (un) = c} is recursively enumerable. Thus the halting problem of Turing machines is solvable, which is a contradiction . Hence [t5 is not fuzzy recognized by a deterministic Turing machine. Let L(Tj) and L(DTj) be the sets of L-fuzzy languages fuzzy recognized by a machine in Ti and DTi, respectively, for i = 0, 1, 2,3 . Theorem 9.7.2 (1) L(To) D L(DTO ) . (2) L(T2) L(DT2) . (3) L(T3) = L(DT3) . Proof. (1) is immediate from Lemma 9.7.1. (2) follows from Corollary 9.4 .4. (3) follows easily. It is not known whether or not L(TI ) D L(DTI ) . Considering Lemma 9.7 .1, it seems reasonable to define recursive Lfuzzy languages as follows: An L-fuzzy language ft over X is recursive if and only if ft is fuzzy recognized by some machine M = (X U A U {t}, I', T, 0, S, S2, K, 'o)

in DTo with some representation r over A with the condition that Sx zA 01 for any x E X*, where Sx = {,y E I'* I 'Yo ==> -Y} . xt Clearly, any L-fuzzy language in L(T3) is recursive . The proof of the following proposition follows easily. Proposition 9.7 .3 An L-fuzzy language in L(DT2) U L(TI ) is a recursive fuzzy language . 9 .8

Closure Properties

We now consider closure properties of the classes of fuzzy languages corresponding to machine classes Ti 's under fuzzy set operations. We show © 2002 by Chapman & Hall/CRC

44 0

9. L-Fuzzy Automata, Grammars, and Languages

that these properties are the extensions of certain closure properties of ordinary languages such as regular sets, context-free languages, contextsensitive languages, and recursively enumerable sets. We consider L[01] fuzzy languages . Theorem 9.8.1 Let

/t and v be L[01 ]-fuzzy languages over X. Let i E {0,1, 2,3f . If /t and v are fuzzy recognized by a machine in Ti, then /t U v is fuzzy recognized by a machine in Ti .

Proof. Suppose that /t and v is fuzzy recognized by a machine Ml with a representation r l : L[ 0,1] ~ Di° and a machine M2 with a representation r 2 : L[0,1] ~ A", respectively. Let A = O1 x 02 . For j = 1, 2 define Tj : 0°° ~ 0~ as follows : Tj((al,a2)) = aj with aj E Aj and for y = bjb2b3 . . . with bk E A, k >_ 1, T,j (y) = Tj (bl) Tj (b2) Tj (b3) . . . . Let r : L[01] ~ A' be such that for any l E L[0,1], T1(r(l)) = r1(l) and T2(r(l)) = r2(l) . Then r is a representation of L[o 1] over A since the Dlength d(r(l), r(m)) of r(l) and r(m) for any distinct l, m E L[0,1] can be defined as d(r(1),r(m)) = d(r1(1),r1(m)) . A machine M that fuzzy recognizes ft U v with the representation r is given as follows : M has Ml and M2 as submachines . For any word xt with x E X*, M first reads A, chooses nondeterministically Ml or M2, and simulates the chosen machine hereafter reading xt . If M reads through xt simulating Mj , j = 1, 2, and if M is given r(l), l E L[0,1 ], then M moves as Mj does if it is given Tj (r(l)) = rj (l) . Clearly, M fuzzy recognizes ft U v . Theorem 9.8.2 Let i E {0,1, 3} . Let

/t and v be L[0,1] -fuzzy languages over X. If /t and v are fuzzy recognized by a machine in Ti, then /t n v is fuzzy recognized by a machine in Ti .

Proof. We prove the result for i = 1. The proofs for the i = 0 and 3 cases are similar . Suppose that /t and v are fuzzy recognized by a machine Ml in Tl with a representation rl : L[0,1] ~ A' and a machine M2 in Tl with a representation r2 : L[0,1] ----> 0~ . Let 0 = Ol x 02 . Define a representation r : L[o 1] ~ 0°° as defined in the proof of Theorem 9.8.1 . We determine a machine M in Tl that fuzzy recognizes /t n v with the representation r with M having Ml and M2 as submachines. When M is given xt with x E X* as an input word, submachines Ml and M2 of M move in parallel reading xt, and M reaches the configuration 'Y corresponding to the pair of configurations 'Yl of Ml and 'Y2 of M2, which they reach after reading xt, respectively. For this computation of M, at most a constant multiple of Ixl spaces is necessary. If M with the configuration 'Y is given r(l) with l E L[0,1] as a subsequent input, then M makes Ml with the configuration y1 read T1(r(l)) = r1(l) and in parallel makes M2 with the configuration 'Y2 read T2(r(l)) = r2(l) . When one of M1 or M2 emits an output, i.e., one of >, <, and =, M remembers it in a state and then continues to make another submachine move until it emits an output . If © 2002 by Chapman & Hall/CRC

9.9 . Fuzzy Grammars and Recursively Enumerable Fuzzy Languages 441

both Ml and M2 emit >, then M emits > . If one of them emits = and another emits > or =, then M emits =. If either one of them emits <, then M emits <. M halts immediately after M emits an output . Since Ml and M2 are machines in Tl , M needs at most a constant multiple of Ixl spaces in order to do the above value-comparing computation for x E X* . Thus M in Tl fuzzy recognizes /t n v. Since for two context-free languages L1 and L2, Ll nL2 is not necessarily a context-free language, it follows from Corollary 9.4 .4 that for two L-fuzzy languages /t and v each of which is fuzzy recognized by a machine in T2, tt rl v is not necessarily fuzzy recognized by a machine in T2 . However, a similar proof to that of Theorem 9.8 .2 shows that if /t is fuzzy recognized by a machine in T2 and v is fuzzy recognized by a machine in T3, then /t n v is fuzzy recognized by a machine in T2 .

9 .9

Fuzzy Grammars and Recursively Enumerable Fuzzy Languages

The purpose of this and the next section is to prove that an L-fuzzy language is generated by an L-fuzzy grammar if and only if it is recursively enumerable, that is, its set of L-fuzzy points is recursively enumerable . As an immediate consequence, we give simple proofs that the union, the intersection, the concatenation of two generated L-fuzzy languages is a generated L-fuzzy language. The results are from [73] . L-fuzzy grammars and generated L-fuzzy languages were introduced in [122] in order to examine ambiguous languages . In the classical case, a language is generated by a grammar if and only if it is recursively enumerable, [96, p. 150] . The question thus arises if such a result holds for generated L-fuzzy languages also. This leads to the question of a suitable definition of recursive enumerability for L-fuzzy subsets . In [17], an L-fuzzy subset whose set of fuzzy points is recursively enumerable is called recursively enumerable . There are several arguments that justify such a definition . For example, it is in accordance with the concept of computability for L-maps given in [200, 210] . Also, in [72], it was shown that if L is finite, an L-fuzzy subset is recursively enumerable if and only if it is the domain of an L-map computable via a Turing L-machine . In this and the next section, we prove that if L is finite, then an Lfuzzy language is generated by an L-fuzzy grammar if and only if it is a recursively enumerable L-fuzzy subset. Recursive enumerability is a very manageable concept . Consequently, this also provides a simple tool to in vestigate the properties of generated L-fuzzy languages . As stated earlier, we give a simple proof that the union, intersection, and the concatenation of two generated L-fuzzy languages is also a generated L-fuzzy language. © 2002 by Chapman & Hall/CRC

442

9. L-Fuzzy Automata, Grammars, and Languages

This allows the results given in [17, 70, 71] about the relationship between fuzziness and decidability to be extended to L-fuzzy languages . We call an L-fuzzy subset /t crisp if for every x E X, either y(x) = 1 or y(x) = 0. We identify the classical subsets of X with the crisp L-fuzzy subsets via characteristic functions . If c E L and c 0, then the c-cut of /t is the set /tc = {x I y(x) > c}. Let x E X and c E L. Then the L-fuzzy subset x, of X is defined by x,(y) = c if x = y and x,(y) = 0 otherwise . An L-fuzzy point x, is said to belong to /t provided that y(x) >_ c. We denote by P(X, L) and by P(p) the set of the L-fuzzy points of X and the set of the L-fuzzy points of ft, respectively. An L-fuzzy grammar is a 4-tuple G = (N, T, P, S) with N and T finite sets, the nonterminal and terminal symbols, respectively, such that N n T = 0, S E N (the initial symbol), and P a finite set of L-fuzzy productions, i.e., elements of the form x -c~ y with c E L, c :?~ 0, and X' Y E (NUT)* . We say that an L-fuzzy grammar G = (N, T, P, S) is in normal form provided that if x -c~ y is an L-fuzzy production of G, then either x, y E N* or x E N and y E T. In an L-fuzzy production x C ----> y, c represents the membership degree of the rewriting rule x ~ y . If c = 1, we write x ~ y for x -c~ y. If wxw' and wyw' are elements of (N U T)* and x -c~ y belongs to P, then we say that wyw' is directly derivable from wxw' with degree c in the L-fuzzy grammar G. If w and w' are in (N U T)*, a derivation chain in G from w to w' is a pair of words (w1 . . . WP, cl . . . cp _ 1) such that w1 = w, wp = w', and w2+ 1 is directly derivable from w2 with degree c2 . A derivation of w is a derivation chain from S to w. The element c 1 A c2 A . . . A cp_ 1 is called the degree of the derivation. An L-fuzzy language /t : T* ~ L is generated by the L-fuzzy grammar G provided that, for every w E T*, /t(w) = V{c E L I c is the degree of a derivation of w}. It follows that an L-fuzzy grammar G utilizes only a finite subset X of elements of L and that the sublattice L' generated by X is finite even in the case that L is infinite. This means that every generated L-fuzzy language is a generated L'-fuzzy language, where L' is finite. As a consequence, in the next section, we assume that the lattice L is finite. 9 .10

Recursively Enumerable L-Subsets

An effective codification of T*, P(T*, L), and L is possible since T and L are finite . Then concepts such as a partial recursive function from T* into L and such as recursive enumerability for subsets of T* and P(T*, L) are defined, [182] . An L-fuzzy subset /t is said to be recursively enumerable if its set of points P(p) is a recursively enumerable subset of P(T*, L) and /t is called decidable if it is a recursive function from T* into L . An L-fuzzy subset ft is called a projection of a decidable L-fuzzy relation if there exists a finite set B and a decidable L-fuzzy subset v of T* x B* such that © 2002 by Chapman & Hall/CRC

9.10. Recursively Enumerable L-Subsets

44 3

y(x) = V{v(x, y) I y E B*} . The following proposition is proved in [17] .

Proposition 9.10 .1 Let /t be an L-fuzzy subset of T* . Then the following statements are equivalent . (1) /t is recursively enumerable. (2) Every cut of /t is recursively enumerable. (3) y(x) is a projection of a decidable L-fuzzy relation. (4) y(x) =1im v(x, n) with v recursive and increasing with respect to n.

Proof. (1)x(2) : Immediate. (2)x(3) : Suppose that IBI = 1. Then we can identify B* with N. It follows that y(x) = V{c E L\{0} I x E ttj . Since /t, is recursively enumerable, a partial recursive function v, exists whose domain is /t c. Let v : T* x hY x L ----> L be defined as follows : v(x, n, c)

c 0

if v,(x) is convergent in less than n steps, otherwise .

Clearly, v is recursive and y(x) = V{v(x, n, c) I n E N, c E L} . By identifying hY x L with N, and therefore with B*, via a suitable codification, we obtain (3) . (3)x(4) : Since we can identify B* with N, we can assume that y(x) _ V{v(x, n) I n E N} with v recursive . Set v'(x, n) = v(x,1) V v(x, 2) V . . . V v(x,n) . Then y(x) = limv'(x,n) and v' is recursive and increasing with respect to n . (4)x(1) : It suffices to note that x, E P(p) if and only if ft(x) > c if and only if there exists n E hY such that v(x, n) >_ c. Thus P(p) is recursively enumerable. Proposition 9.10 .2 The set of recursively enumerable L-fuzzy subsets of

T* is a lattice, the minimal lattice containing the recursively enumerable subsets and the L-fuzzy subsets that are constant maps . Namely, an Lfuzzy subset /t : T* ~ L is recursively enumerable if and only if /t admits a decomposition /t = (c l A XI) V . . . V (cn A xn) with c2 E L and where XZ is the characteristic function of a recursive enumerable subset, i = 1, . . . , n.

Proof. Let /t and v be recursively enumerable . Then by Proposition 9.10.1(4), y, (x) = limlt'(x,n) and v(x) = limv'(x,n) with /t' and v' recursive and increasing with respect to n. Then y(x) V v(x) = limlt'(x,n) V v'(x, n) and ft (x) A v(x) = lim ft'(x, n) A v'(x, n) . Since /t' V v' and /t' A v' are recursive and increasing with respect to n, /t V v and /t A v are recursive enumerable This proves that the set of recursively enumerable L-fuzzy subsets forms a lattice . Let /t be a recursively enumerable L-fuzzy subset. Set if

[t(X) x ~(x) _ { 10 otherwise . © 2002 by Chapman & Hall/CRC

>c

444

9. L-Fuzzy Automata, Grammars, and Languages

Since /t c is recursively enumerable, Xc is the characteristic function of a recursively enumerable subset . Clearly, ft = V{c A X c c E L\{0}} . Thus /t is generated by the constants c and the recursively enumerable crisp L-fuzzy subsets xc . We now prove our main result, i.e., an L-fuzzy language /t : T* ----> L is generated by an L-fuzzy grammar if and only if it is recursively enumerable. However, first we prove the following lemma. Lemma 9.10 .3 If ft : T* ~ L and v : T* ~ L are L-fuzzy languages generated by normal form L fuzzy grammars, then /tUv is an L fuzzy language generated by a normal form L fuzzy grammar . Proof. Suppose that y and v are generated by the normal form Lfuzzy grammar G' = (N', T, P', S') and G" = (N", T, P", S"), respectively . Without loss of generality, we assume that N'nN" = 0. Let G be the normal form L-fuzzy grammar (N' U N" U {S}, T, P, S), where S is a new symbol not in N' U N", and P contains S ~ S', S ~ S", and all production in P' and P" . Let A : T* ~ L be the L-fuzzy language generated by G . Then A = /t U v . In a sense, the derivations of G consist of the derivation of G' and the derivations of G". That is, if (w1 . . . wp , ul . . . Up -,) is a derivation of G' (of G") with w 1 equal to S' (to S"), then (Sw1 . . . wp, lul . . . up-1) is a derivation of G' (of G") . Furthermore, since N' and N" are disjoint, every derivation of G can be obtained either from a derivation of G' or from a derivation of G" as above. Thus it follows that A = ft U v. Theorem 9.10 .4 Let /t be an L fuzzy language of T* . Then the following statements are equivalent . (1) /t is generated. (2) /t is recursively enumerable. (3) /t is generated by a normal form grammar . Proof. (1)x(2) : Suppose that /t is generated by the L-fuzzy grammar G. Let P1,P2, . . . be an effective enumeration of the derivation of G . Moreover, let v be defined by v(w, n)

the degree of Pn 0

if Pn is a derivation of w otherwise .

Then it follows that /t is the projection of v, and thus, /t is recursively enumerable by Proposition 9 .10 .1 . (2)x(3) : Suppose that /t is recursively enumerable . Then by Proposition 9.10.2, /t = (c1 A x1) V . . . V (cn A xn), where XZ is the characteristic function of recursively enumerable subset Si of T*, i = 1, . . . , n. Let GZ be a classical grammar in normal form producing Si and let GZ be the normal form L-fuzzy grammar obtained from GZ by substituting each production x ----> y with x _c~ y. It follows that GZ produces the L-fuzzy language c2 AXi . © 2002 by Chapman & Hall/CRC

9.10. Recursively Enumerable L-Subsets

44 5

From Lemma 9.10.3, it follows that ft is generated by a normal form L-fuzzy language . (3)==>(1) : Immediate . We now consider some closure properties for generated L-fuzzy languages . Recall that the concatenation of two L-fuzzy languages ft and v is the L-fuzzy language A defined by b'w E T*, A(w) = V{ft(x) A v(y) w = xy} . Moreover, the Kleene closure /t°° of ft is such that ft'(w) _ V{ft(xl) A . . . A ft(xe) I w = xi . . . xq , q E N} . Corollary 9.10 .5 The class of all generated L-fuzzy languages is a lattice.

In particular, it is the minimal lattice containing the generated (classical) language and the L fuzzy languages that are constant functions. Furthermore, if ft and v are generated languages, then the concatenation of ft and v and the Kleene closure of ft are generated.

Proof. The first part of the corollary follows from Proposition 9.10.2 . Suppose that ft(x) = Vnft'(x, n) and v(x) = Vn v'(x, n) with ft' and v' recursive. Then A(w) = V{ft'(x, n) A v'(x, m) I w = xy, n, m E N} . Since it is possible to codify the set, { (x, y, n, m) I w = xy, n, m E N}, A is a projection of a decidable relation. Thus A is recursively enumerable and therefore generated . Likewise, it is possible to express the Kleene closure of ft by the formula ft-(w) = V{ft'(xl, nl) n . . . n ft'(xq, ne ) I w = xi . . . xq , ni, . . . , nq E N} . Since it is possible to codify the set {(x l . . . x q , q, nl , . . . , nq ) I xl . . . xq = w, q, nl, . . . , nq E N}, ft' is recursively enumerable and therefore generated . Theorem 9.10.4 allows for the transfer of results on the relationship among imprecision, decidability, and recursive enumerability given in [17, 70, 71] to L-fuzzy languages . Namely, we assume that L is a finite sublattice of the interval [0,1] containing -1 . We call an L-fuzzy language infinite indeterminate (almost-everywhere indeterminate) provided that the set {x E T* I s(x) = 2 } is infinite (cofinite) . If A and A' are L-fuzzy languages, we say that A' is a sharpened version of A or that A is a shaded version of A' if A(x)

>2

==> A(x) > A(x) and A(x)

< 2 ==> A(x) < A(x) .

Hence we conclude the following results from Theorem 9.10.4: 1. A generated infinitely indeterminate L-fuzzy language exists with no decidable sharpened version, [17, Proposition 5.1] . That is, it is not possible to obtain decidability by using the indeterminateness of a generated fuzzy language, in general . 2. An infinitely indeterminate L-fuzzy language exists with no generated shaded versions, [71, Proposition 4.4] . This provides an example of a fuzzy language that is not generated in a strong case. © 2002 by Chapman & Hall/CRC

446

9. L-Fuzzy Automata, Grammars, and Languages

3. A generated classical language exists whose unique decidable shaded versions are the L-fuzzy languages infinitely indeterminate, [17, Proposition 5.2] . Consequently, it is not possible to shade a generated classical language in order to obtain decidability . 4. A classical language exists whose unique generated shaded versions are the L-fuzzy languages almost everywhere indeterminate, [71, Proposition 5.2] . Therefore, it is not always possible to obtain generated languages by shading a classical language.

9 .11

Various Kinds of Automata with Weights

Various types of automata such as fuzzy automata, max-product automata, and integer-valued generalized automata [233, 234] have been introduced as a generalization of well-known deterministic automata, nondeterministic automata, and probabilistic automata . These automata have the common property that they have "weights" associated with the state transitions as well as initial and final distributions . Clearly, probabilistic automata can be considered as automata with "weights." The operations of max and min and product have been used with these automata . By using addition and multiplication as the operations and the probabilities as the weights, probabilistic automata can be defined. Moreover, integer-valued generalized automata have integer weights and addition and multiplication as operations. We now present the results of [148] in order to present a general formulation of automata with weights by extracting the basic properties common to the existing automata and by incorporating the appropriate algebraic systems with automata systems and by performing the operations of the algebraic systems to the state transition functions and initial and final distribution functions of the pseudoautomata defined later. We continue the theme of using a lattice L rather than [0,1] . The concept of L-fuzzy relations enables us to define fuzzy automata, l-semigroup automata, lattice automata, dual lattice automata, max-product automata, and so on. Moreover, L-fuzzy relations are important in formulating other kinds of automata with weights such as semiring automata, ring automata, integer-valued generalized automata, and field automata . Definition 9.11 .1 Let X be a set and L be a lattice . An L-fuzzy relation on X is a function /t from X x X into L, i.e., ft :XxX----> L.

(9 .1)

We let V and A denote the operations of supremum and infimum on L, respectively. In the remainder of the chapter, the structure of the membership space L is assumed to be a complete lattice ordered semigroup and a complete distributive lattice because of the concept of composition of L-fuzzy relations defined below. © 2002 by Chapman & Hall/CRC

as =lla2 also for (L, follows as the there Various L iIf *pl, If *Eordered V, follows all iscomposition to is aL (VZEZyi) satisfies semigroup lan(X1, the L I,complete P2, *) the axisare the is 9an complete Eaoperations Kinds aThe semigroup (or complete two L, index associativity Xn+1) complete I, w2(x, semigroup =VZEZ(x* lan the product distributive Let elements of operation are of set, distributive following Automata z) lattice z) ft, L-fuzzy lattice L-fuzzy V= operation =A{wj lattice and and V{wl(X, V{wj(X, yi) 0of X2, Lla1(XI,X2) that (l-semigroup ordered *laws and ft, composition relations distributive N2 relations (L, Ais(X, with replaced and are in 1V, is0*x lattice x, are be 1, 1*x y) ,y)in Xn LA), asemigroup *VL-fuzzy A Weights dual N2 as an semigroup w2(y, Ew2(y, *can then A X}, follows P2(X2,X3) = on l-semigroup by laws, in similar of for be z) X, z) 0,relations x, AL-fuzzy adefined short) I(or I in *yythen denoted distributive yyFor with result EE L*=X}, El-semigroup X} XI' =L and all VZEZ(xj itrelations, by identity on (L, *holds = isx, by replacing lan(Xn,Xn+1) is(L, V, X aPIP2, y, denoted lattice, complete xi, *), *V, for Then y) (L, under yi we *) then (9isV V, Ecan such the the deby L, *), L

9.11 .

447

If and lattice L where

.

.

x

and

.

(VZEZxi)

If becomes

.

Definition .11 .2 composition : fined (1) then

.

(9 .2)

wIw2(X, where (2)

. (9 .3)

wIw2(X, Since different A

: .

It

(9 .4)

Due write It

where (9 .4) . If that

.. .

V I

. ..

"..

.. .

(9 .5) .

XV0 x*0 XV1 x*1

© 2002 by Chapman & Hall/CRC

= = = =

.3)-

(9 .6)

448

9. L-Fuzzy Automata, Grammars, and Languages

then they are called a zero and an identity of L, respectively. For example, let L be ([0,1], V, .), where the operation - is ordinary multiplication. Then L is an l-semigroup with zero 0 and identity 1. Moreover, let the Cartesian product of [0,1] be written as [0,1] 2 and the operations V and - be defined as (a, b) V (c, d) _ ((a V c), (b V d)), (a, b) - (c, d) _ (a . c, b . d), for each (a, b), (c, d) E [0,1]2 . Then L = ([0,1]2, V' .) is an l-semigroup with zero (0, 0) and identity (1,1) . For the l-semigroup L with zero 0 and identity 1, define the identity relation t by V'x, y E L, 1 t(x~y) =~ 0

ifx=y if xz,4 y.

(9.7)

Then we have qt = frt = ft,

(9.8)

for each L-fuzzy relation ft . Clearly, every L-fuzzy relation ft over X is representable by a matrix if X is a finite set. Let X = {xl, x2, . . . , x } . Then ft is represented by the n x n matrix, We now define what is called a pseudoautomaton in [148] in order to derive various kinds of automata with weights . Definition 9.11 .3 A pseudoautomaton is a 6-tuple A= (Q, X, W, ft, 7r~ 97),

where (1)

(9 .9)

Q is a finite set of states, (2) X is a finite set of input symbols, (3) W is a weighting space, (4) ft is a weighting function such as

ft :

Q

x X x

Q ---->

W

(9.10)

and is called a state transition function. The value ft(q, a, q') of (q, a, q') E Q x X x Q represents the weight of transition from state q to state q' when the input symbol is a, (5) 7r is an initial distribution function, where 7r :Q----> W (6)

97

(9.11)

is a final distribution function, where 97 :Q----> W.

© 2002 by Chapman & Hall/CRC

(9 .12)

9.11 . Various Kinds of Automata with Weights

449

We do not consider time and outputs for the sake of simplicity. If we consider time, the state transition function /t, the initial distribution function 7r, and the final distribution function rl are given as ft : QxXxQxT~W TAW 9 :QxT~W where T is a subset of Let Y be a set of output symbols . QxXxYxTinto W.

Then the output function maps

Definition 9.11 .4 A weighted-automaton or simply automaton A* is a 6-tuple A* = (Q, X, W ft*, 7r, 9),

(9 .13)

where Q, X, W 7r, and 9 are given in Definition 9 .11 .3 and ft* : QXX*XQ----> W.

(9 .14)

We now derive various kinds of automata with weights by introducing binary operations on the weighting space W of the pseudoautomata and by giving the extension rules for obtaining /t* from /t.

L-Semigroup Automata (la) . Here the weighting space is a complete lattice ordered semigroup (lsemigroup for short) L = (L, V, *) with identity 1 and zero 0, where the operation * is the semigroup operation in L . The state transition function /t, the initial distribution function 7r, and the final distribution function 97 are given by replacing W in (9.10), (9.11), and (9.12) by L as follows : ft : QxXxQ----> L

(9 .15)

L

(9 .16)

97 : Q ----> L.

(9 .17)

(1b). Using the concept of composition of L-fuzzy relations (9.2), the state transition function /t* for input strings in X* is obtained recursively as follows : © 2002 by Chapman & Hall/CRC

450

9. L-Fuzzy Automata, Grammars, and Languages For A,xEX*, and aEX, 1 0

~* (q, A, q) _

if q = q~ if gzAq,

w* (q, xa, q) = v{w* (q, x, q') * w(q', a, q) I q' E Q},

(9.18) (9.19)

where q, q' E Q, and 1 and 0 are identity and zero of L, respectively. Suppose that the automaton starts from a certain initial state, say, qo . Then the initial distribution function 7r is concentrated at qo, i.e ., 7r (q)

_

1 ~ 0

if q = qo if q qo .

(9.20)

Let F C_ Q be a set of final states . Then the final distribution function 97 is defined as 1

ifgEF

97 (q)-~ 0 ifq~F

(9.21)

Given the expression (9.19) and the initial distribution 7r and the final distribution rl, the weight, denoted by w(x), of the string x of the automata is defined by w(x) = V{7r(q) * w*(q, x, q) * ?l(q) I q, q E Q},

(9.22)

where x E X* . Since there exists an order relation >_ in the l-semigroup L = (L, V, *), the language L(A, c) accepted by the l-semigroup automaton A with parameter c can be defined by L(A, c) = {x E X* I w(x) > c},

(9.23)

where c is called a threshold (or cutpoint) and is a member of the weighting space L .

Max-Product Automata

(2a) Let the weighting space be L' = ([0,1], V, -) in the l-semigroup automaton of (la, b), where the operation - represents ordinary multiplication. Then L' is an l-semigroup with identity 1 and zero 0. Moreover, /t, 7r, and rl are obtained by replacing L in (9.15)-(9 .17) by [0,1], i.e., ft :

QXX XQ---> [0,1],

7r : Q ---> [0,1],

© 2002 by Chapman & Hall/CRC

(9.24) (9.25)

9.11 . Various Kinds of Automata with Weights 77 : Q ----> [0,1] .

451 (9 .26)

(2b) /t* and w are obtained by replacing * by - in (9.18), (9.19), and (9.22) . * [t (q, A, q) _ q..

1

0

if q = q~ if gzAq,

(9 .27)

) - ft (q", a, q) I q" E Q},

(9 .28)

w(x) = V{7r (q) - [t * (q, x, q) - 97 (q) I q, q' E Q}.

(9 .29)

w* (q, xa, q) = V{w* (q, x,

Clearly, a max-product automaton is a special case of the l-semigroup automaton of (1a,b) and is also a special case of the semiring automata of (14a,b), defined later.

Lattice Automata (3a) A complete distributive lattice L = (L, V, A) is the weighting space. Moreover, /t, 7r, and rl are given from (9.10)-(9 .12) by ft : QxXxQ----> L,

(9 .30)

7r :Q----> L,

(9 .31)

97 : Q ----> L.

(9 .32)

(3b) By using the concept of composition of L-fuzzy relations (9.3), /t* and w are obtained as follows : * [t (q, A, q') _

1 0

if q = q~ if gzAq,

(9 .33)

w* (q, xa, q) = V{w* (q, x, q") n w(q', a, q) I q' E Q},

(9 .34)

w(x) = V {7r(q) A w* (q, x, q) A ?7(q) I q' E Q},

(9 .35)

where 1 and 0 are the maximal and minimal elements of the complete distributive lattice L, respectively. Expressions (9.34) and (9 .35) are obtained by replacing * by A from (9.19) and (9 .22) . A complete distributive lattice is a special case of a complete lattice ordered semigroup . Likewise, a lattice automaton is considered to be a special case of an l-semigroup automaton . The operations V and A are dual in a complete distributive lattice L = (L, V, A) . Thus the dual automata of lattice automata can be formulated in the following manner. © 2002 by Chapman & Hall/CRC

452

9. L-Fuzzy Automata, Grammars, and Languages

Dual Lattice Automata (4a) This is the same as (3a) . (4b) Using the concept of composition w are given as follows : * [t (q, A, q~) _

w* (q, xa,

q) =

w(X)

A{w* (q,

0 1

of

L-fuzzy relations (9 .4), /t* and

if q = if q7~ q~ q,

(9.36)

" x, q ) V w(q', a, q) I q' E Q},

= n{7r(q) n w*(q, x, q) v ?7(q)

I q' E Q} .

(9.37) (9.38)

Given a certain initial state qo and a final state set F, 7r and 97 of the lattice automata of (3a,b) are given as follows in the same manner as (9.20) and (9 .21) . _ 1 7(q) - { 0

if q=qo if q qo .

(9 .39)

_ 1 - { 0

ifgEF if q ~ F

(9 .40)

97 (q)

However, follows :

7r

and

97 of

the dual lattice automata

of

(4a,b) are defined as

0 7r (q) = ~ 1

if q = qo if q qo .

(9.41)

0 97 (q)-~ 1

ifgEF ifq~F.

(9.42)

(Pessimistic) Fuzzy Automata [250, 143, 212, 31, 65, 140, 193, 259] (5a) If the weighting space, J = ([0,1], V, A), is adopted, then J is a complete distributive lattice under the operations V (max) and A (min) . Furthermore, 7r, 7r, and 97 are as follows . ft :

© 2002 by Chapman & Hall/CRC

QxX xQ----> [0,1],

(9.43)

7r : Q ----> [0,1],

(9.44)

97 : Q ----> [0,1] .

(9.45)

9.11 . Various Kinds of Automata with Weights (5b)

/t*

453

and w are defined as follows : * [t (q, A, q)

w* (q, xa,

i

q) = V{(ft* (q, x, q")

w(x) = V{(7r(q)

if if

1

0

q=q' gzAq',

n w(q', a, q)) I

n w* (q, x, q) n 97(q)) I

(9 .46) q' E Q},

(9 .47)

q, q E Q} .

(9 .48)

Since J = ([0,1], max, min) is a complete distributive lattice, a fuzzy automaton is a special case of a lattice automaton (3a,b). Therefore, /t* and w of (9.46) - (9 .48) are obtained from (9.33) - (9 .35) by replacing V with max, A by min .

Optimistic Fuzzy Automata [250,143,212,140] (6a) This is the same as (5a) . (6b) p* and w are defined as follows : w* (q, A, q) = {

if if

1

q=q' gzAq~,

" E Q}, w* (q, xa, q') = A{(w* (q, x, q") V w(q", a, q')) I q w(X) = A{(7r(q) V w* (q, x, q) V 97(q)) I q, q E Q} .

(9 .49) (9 .50) (9 .51)

Optimistic fuzzy automata are special cases of dual lattice automata of (4a,b) . Given an initial state qo and a final state F, 7r and 97 of the fuzzy automata of (5a,b) are obtained from (9.39) and (9 .40) . However, 7r and 9 of an optimistic fuzzy automata are obtained from (9 .41) and (9.42) [143] .

Mixed Fuzzy Automata [212] (7a) This is the same as (5a) . (7b) /t* and w are given by using the concept of convex combination of fuzzy subsets, i.e., w* (q, x,

q) = awPF(q, x, q) + bw*OF(q, a, q),

(9 .52)

w(x) = awPF(x)

(9 .53)

+

bwO F (x),

where PP F and It*0F are state transition functions defined by the pessimistic fuzzy automata of 9 .11 and the optimistic fuzzy automata of (6a,b), respectively. This is the same for WPF and WOF . Also a, b are nonnegative real numbers such that a + b = 1. © 2002 by Chapman & Hall/CRC

454

9. L-Fuzzy Automata, Grammars, and Languages

Composite Fuzzy Automata [250, 65] (8a) This is the same as (5a) . (8b) /t* is obtained by operating between p. This is the same for WPF and WOF .

PPF

and

7r*OF

with probability

Nondeterministic Automata [220] (9a) J' = ({0,1}, max, min} is adopted as the weighting space. Clearly, J' forms a distributive lattice (more precisely, a Boolean lattice) and /t, 7r, and 97 are given as follows : ft : QXXXQ----> {0,1},

(9.54)

7r :

Q ----> {0,1},

(9.55)

97 :

Q ----> {0,1} .

(9.56)

(9b) This is the same as (5b) . Nondeterministic automata are special cases of the fuzzy automata of (5a,b) (or l-semigroup automata) .

Deterministic Automata [220] (10a) This is the same as (9a) plus the additional constraints that there exists unique q' E Q such that 7t(q, a, q') = 1 for each q E Q and a E X and 7r(q, a, q") = 0 for q" :?~ q, and there exists unique q' E Q such that 7r(q') = 1 and 7r(q") = 0 for q" :?~ q' . As for 97, let F be a set of final states. Then 97(q)-~

1 0

ifgEF ifq~F.

(10b) This is the same as (9b) . Clearly, deterministic automata are special cases of the nondeterministic automata of (9a,b) and also of the probabilistic automata of (18a,b) defined later.

Boolean Automata (11a) Here the weighting space is a complete Boolean lattice B = (B, V, A), where the operations V and A are supremum and infimum in B. Clearly, a Boolean lattice is a special case of a distributive lattice . Then 7r, 7r, and 97 are as follows : ft : QxXxQ----> B,

© 2002 by Chapman & Hall/CRC

( 9.57)

9.11 . Various Kinds of Automata with Weights

(9 .58)

7r :

97 :

455

Q ----> B.

(9 .59)

(11b) This is the same as (3b) . Boolean automata and dual Boolean automata, defined next, are special cases of the lattice automata of 9 .11 and the dual lattice automata 9 .11, respectively. In [149], a B-fuzzy grammar is defined to be a 7-tuple (N, T, P, S, J, /t, B), where N is the nonterminal alphabet, T is a terminal alphabet, S E N is an initial symbol, P is a finite set of productions, J is a set of production labels, and /t : J ~ B. We summarize some of the results of [149] in the Exercises .

Dual Boolean Automata (12a) This is the same as (11a) . (12b) This is the same as (4b) .

Mixed Boolean Automata (13a) This is the same as (11a) . (13b) Using the concept of a convex combination of B-fuzzy sets [25], and w are defined as follows : (a A [t * w = (a

)

V (a

/t*

n PDB)(9 .60)

AWB) V (a AWDB),

(9 .61)

where [ta and It*DB are state transition functions that are defined in the Boolean automata of (11a,b) and the dual Boolean automata of (12a,b), respectively, and a, a E B, where a is the complement of a.

Semiring Automata Recall that a set R with the operations of addition + and multiplication x is called a semiring if the following three conditions are satisfied: (1) + is associative and commutative ; (2) x is associative ; (3) x distributes over +, i .e., ax(b+c)=axb+ axc, (b+c) xa=bxa+cxa,

for all a, b, c E R. The semiring R is called a semiring with identity 1 and zero 0 if 1 is the identity under x and 0 is identity under + in R. © 2002 by Chapman & Hall/CRC

456

9. L-Fuzzy Automata, Grammars, and Languages

For example, let R = ([0, oo), +, -) with ordinary addition + and ordinary product . . Then [0, oo) is a semiring with identity 1 and zero 0. Similarly, the set of natural numbers containing 0 is also a semiring with identity and zero under + and .. Also, R = ([0,1], V, -) is a semiring with identity and zero. Note that this R is also an l-semigroup . In general, l-semigroups and complete distributive lattices are special cases of semirings with identity and zero. (14a) The weighting space is a semiring R = (R,+, x) with identity 1 and zero 0. Moreover /t, 7r, and rl are given by: ft :

xX x

Q

R,

Q T

(9.62) (9.63)

71 :Q----> R.

(9.64)

(14b) /t* and w are given as follows : [t* (q, A, q) = [t* (q, xa,

w (x)

q)

~

1 0

if if

= 1: {w* (q, x, q qii EQ

= 1: {'r(q) x q,q'EQ

"

q = q' q zA q~,

(9.65)

) x w (q', a, q)},

(9.66)

lr* (q,x,q) x 9 7(q) .

(9.67)

As a special case of semiring automata, there exist l-semigroup automata of (1a,b), mar-product automata of (2a,b), lattice automata of (3a,b), fuzzy automata of (5a,b), nondeterministic automata of (9a,b), Boolean automata of (12a,b), and so on.

Plus-Weighted Automata [188,235] (15a) The weighting space is R = ([0, oo), +, .), where + and - are ordinary addition and multiplication. Clearly, R = ([0, oo), +, -) is a semiring with identity 1 and zero 0. Then /t, 7r, and rl are defined from (9.62)-(9 .64) as follows : ft :

QXX XQ----> [0,

7r :

© 2002 by Chapman & Hall/CRC

Q ----> [0,

(9.68) (9.69)

9.11 . Various Kinds of Automata with Weights 'q : Q ----> [0,

457 (9 .70)

00)

(15b) /t* and w are defined by letting + be ordinary addition and replacing x by - in (9.65)-(9 .67) . 7r* (q, A, q)

w* (q, xa, q')

=

= E

qii EQ

w(x) = 1:

1

~

0

if if

q=q' qz,4 q~,

" {w* (q, x, q ) - 7r(q", a, q')},

{7 r (q) - 7r * (q, x, q) - 97(q)}

q,q'EQ

(9 .71)

(9 .72)

(9 .73)

Weighted automata are special cases of the semiring automata of (14a,

Max-Weighted Automata [188] (16a) The weighting space is R = ([0, oo), max, .), with ordinary multiplication . . Clearly, R is a semiring with unity and zero. Here, /t, 7r, and 97 are given the same way as (9 .68)-(9 .70). (16b) 7r* and w are defined as follows : 7r * (q, A, q) _

7r * (q, xa, q) =

w(x) =

m

Q

1

0

if if

q= q~ gzAq,

{w* (q, x, q') - 7r(q", a, q)},

Q{7r(q) - 7r * (q, x, q) - 97(q)} qm E

(9 .74) (9 .75) (9 .76)

Max-weighted automata are special cases of the semiring automata of (14a,b) . Max-product automata of (2a,b) can be obtained from max-weighted automata by replacing [0, oo) by [0,1] .

Natural Numbered Automata (17a) The weighting space is hY U {0} with ordinary addition and multiplication . Moreover, /t, 7r, and 97 are given as follows : ft :

© 2002 by Chapman & Hall/CRC

QXXXQ---->

NU{0},

(9 .77)

458

9. L-Fuzzy Automata, Grammars, and Languages 7r :

Q ----> N U {0},

(9.78)

97 :

Q ----> N U {0} .

(9.79)

(17b) This is the same as (15b) . Natural numbered automata are special cases of the weighted automata of (15a, b) . Max-natural numbered automata can be easily defined in a manner similar to a max-weighted automata of (16a,b) .

Probabilistic Automata [220,168] (18a) Here the weighting space is ([0,1],+, -) . Then /t,

7r,

and

ft : QXX XQ~ [0,1],

For

97,

a,

are (9.80)

7r :

Q ----> [0,1],

(9.81)

97 :

Q ----> [0,1] .

(9.82)

In addition, the following constraints of /t, gEQandaEX, Eq'EQ P(q,

97

q~) = 1,

let F be a final state set . Then 1 97 (q)-~ 0

7r,

and

97

are assumed . For all

1. Eq' EQ 7r (q) =

ifgEF ifq~F

(9.83)

(9.84)

(18b) This is the same as (15b) . There exists another definition of 97 different from (9.84), [233] . That is, in the same way as /t and 7r of (9.83), we have qEQ

97(q)

=1.

(9.85)

Generalized Probabilistic Automata [233,164] (19a) This is the same as (18a) with the assumption that the image of 97 not the unit interval [0,1], but the set of real numbers (-oo, oo), i.e., 97 :

by

Q ----> (- 00, oo) .

is

(9.86)

(19b) This is the same as (18b) . The language accepted by generalized probabilistic automata is defined L(A, c) = {x E X* I w(x) > c},

where c E (-

(9.87)

) . As for the probabilistic automata of (18a,b), c E [0,1] .

© 2002 by Chapman & Hall/CRC

9.11 . Various Kinds of Automata with Weights

459

Rational Probabilistic Automata [234] (20a) This is the same as (18a) with the assumption that the values 7r(q, a, q') and 7r(q) are rational numbers in [0,1] . (20b) This is the same as (18b) .

Ring Automata (21a) The weighting space is a ring [125] with identity R = (R,+, .) . Here 7r, 7r, and 97 are ft : QxXxQ----> R,

(9 .88)

R,

(9 .89)

97 : Q ----> R.

(9 .90)

(21b) This is the same as (14b) . Ring automata are special cases of the semiring automata of (14a,b) . The weighted automata of (15a,b) and the max-weighted automata of (16a,b), which are special cases of the semiring automata of (14a,b), are not special cases of ring automata .

Integer-Valued Generalized Automata [233,234] (22a) The weighting space is 7L = (7L, +, .), where 7L is a set of integers and the operations + and - are ordinary addition and product, respectively. Clearly, 7L is a ring with identity. Here /t, 7r, and 97 are ft : QXX XQ----> 7L,

(9 .91) (9 .92) (9 .93)

(22b) This is the same as (15b) . Integer-valued generalized automata are a special case of the ring automata of (21a,b) .

Field Automata (23a) Let the weighting space be a field F = (F,+, .), [125] . Then /t, 97 are ft : QxXxQ----> F,

© 2002 by Chapman & Hall/CRC

7r,

and

(9 .94)

460

9 . L-Fuzzy Automata, Grammars, and Languages 7r : Q ---->

F,

(9.95)

9 :

F.

(9.96)

Q ---->

(23b) This is the same as (21b) . Clearly, field automata are a special case of ring automata of (21a,b) . An integer-valued generalized automata ((22a,b)), which is a special case of ring automata, is not a special case of field automata .

(Real-Valued) Generalized Automata [233, 234, 235] (24a) The weighting space is F = ((-oo, oo), +, .), where (-oo, oo) is a set of real numbers, and + and - are ordinary addition and multiplication . Here 7r, 7r, and 97 are ft

: QXX XQ----> (

(-oo, oo),

(9.98)

Q ~ (-00, 00) .

(9.99)

7r : Q ~

97 :

(9.97)

(24b) This is the same as (15b) . Real-valued generalized automata are a special case of the field automata of (23a,b) .

Rational Automata (25a) The weighting space is Q = (Q, +, .), where Q is a set of rational numbers, and + and - are ordinary addition and multiplication . Here /t, 7r, and 97 are ft

: QXXXQ----> Q

(9.100)

7r :Q~Q,

(9.101)

97 :Q~Q .

(9.102)

(25b) This is the same as (15b) . Rational automata are a special case of the real-valued generalized automata of (24a,b) and also of field automata of (23a,b) . We have presented various kinds of automata with weights . Some of these automata are lacking of physical images . However, for example, from the fact that the classes of languages defined by rational probabilistic automata of (20a,b) and integer-valued generalized automata of (22a,b) are © 2002 by Chapman & Hall/CRC

9.12. Exercises

461

equal, various problems concerning rational probabilistic automata can be solved by investigating the properties of integer-valued generalized automata [233,234] . Consequently, automata with weights are important in investigating properties of well-known automata such as deterministic automata and probabilistic automata . Furthermore, they are useful models of learning systems, gaming, and pattern recognition as in the case of fuzzy automata [250, 31, 65] . The set of complex numbers forms a field. Hence as a special case of field automata of (23a,b), we can define complex numbered automata. We cannot, however, define a language accepted by these automata in the same way as (9 .23 because of the fact that there does not exist an order relation >_ on the set of complex numbers . However, using the concept of a mapping of the set of complex numbers into a certain algebraic system with ordering, say, by transforming the complex number z to the absolute value ~zj, we can define a language by a complex numbered automata A as L(A,z) = {x E X* I Iw(x)l >_ ~z1}. Along these lines, the concept of a valuation of a field can be considered. A valuation of a field F is a function v of F\{0} into an additive abelian totally ordered group such that for all x, y E F\{0}, v(xy) = v(x) + v(y) and v(x + y) > v(x) A v(y) . If we do not use this concept of mapping, we would have to restrict our choice of a ring or a field to those with an ordering as weighting spaces, [18, 183] . 9 .12

Exercises

1. For Theorem 9.4.1, prove the cases, where (2) or (3) holds . 2. Give the proof of Theorem 9.8.2 for i = 0 and 3. 3. Show that if a L[o 1]-fuzzy language ft is fuzzy recognized by a machine in T2 and a L[o 1]-fuzzy language v is fuzzy recognized by a machine in T2, then ft n v is fuzzy recognized by a machine in T2 . 4. If P1, P2, P3 are L-fuzzy relations on a set X, prove that (It, P2)P3 = Itl(1t2ft3) 5. (149 Show that type 2 (context-free B-fuzzy grammars can generate type 1 (context-sensitive) languages although type 2 fuzzy grammars cannot generate type 1 languages [145] . 6. (149 Show that the generative power of type 3 (regular) B-fuzzy grammars is equal to that of the ordinary type 3 grammars.

© 2002 by Chapman & Hall/CRC

Chapter 10

Applications 10 .1

A Formulation of Fuzzy Automata and Its Application as a Model of Learning Systems

In [26], a formulation of a class of stochastic automata on the basis of Mealy's [138] formulation of finite automata has been carried out . In [68], a stochastic automaton as a model of a learning system operating in an unknown environment was employed. The formulation of fuzzy automata is similar to that of the stochastic automata proposed in [26] and the fuzzy class of systems described in [256]. The advantage of using fuzzy set concepts in engineering systems has been discussed in [255, 256] . In [212], a general formulation has been given to cover both fuzzy and stochastic automata. The next five sections are based on [250] and deal with a specific formulation of fuzzy automata and their engineering applications .

10 .2

Formulation of Fuzzy Automata

Definition

10 .2 .1 A (finite) fuzzy automaton is a quintuple (Q, X, Y,

It, w), where (1) Q nonempty finite (2) X nonempty finite (3) Y nonempty finite (4) /t is a fuzzy subset (5) w is a fuzzy subset

set (the set of internal states, set (the set of input states, set (the set of output states, of Q x X x Q, i .e., /t : Q x X x Q of Q x X x Y, i . e., w : Q x X x Y

[0,1], [0,1] .

In Definition 10.2.1, /t is called the fuzzy transition function and w the fuzzy output function . Recall that in Section 8.4, a fuzzy finite automaton was defined as in Definition 10.2.1, but with some added conditions . © 2002 by Chapman & Hall/CRC

463

464

10. Applications

It is at times convenient to use the notation PA for a fuzzy subset of a set S, where A is thought of as a fuzzy set and PA gives the grade membership of elements of S in A. At times, A may be merely a description of a fuzzy subset ft of S. Let Q = {ql, q2, . . . , qn }, X = {x1, x2, - . . , XP}, and Y = 01, Y2, - . . , Y,1Then PA (gj, xj , q,) is the grade of transition (class A) from state qZ to state qn, with input xj or q(k) = qZ to qm or q(k + 1) = qm,

(10.1)

where the input is xj or x(k) = xj and where k denotes the discrete time element . Hence we write PA (g2, xj, qm) = ft {q(k) = qi, x (k) = xj, q (k + 1) = qm} .

In order to decide the existence of the transition, a pair of thresholds

c, d may be introduced, where 0 < d < c < 1 . This leads to a three level

logic such as (1) x E A, or "true" if PA(x) > c, (2) x ~ A, or "false" if PA(x) < d, (3) x has an indeterminate status relative to A, or "undetermined" if d < PA(x) < c,

where A is a fuzzy set defined as the transition between states for a particular input ; x denotes the 3-tuple (qj, xj , q,) as defined in (10 .1), where qi, qm E Q; xj E X, i .e., PA(x) = PA(gl, xj, qm) . The function may be dependent or independent of k, the number of steps . If ft is independent of k, ft is called a stationary fuzzy transition function. As we will see, nonstationary fuzzy transition functions are used to demonstrate learning behavior of a fuzzy automaton . For now, let ft be independent of k with fuzzy transition matrix Txj for all xj E X. The Txj are of the following form: xj

q +1 gi,g2,--q,n_ . q .

PA (gi, xj, qm)

© 2002 by Chapman & Hall/CRC

10.2. Formulation of Fuzzy Automata The entries of Txj are

PA (gj, x j , q, ) .

465

The fuzzy transition table is as follows :

where [A] denotes the fuzzy matrix whose ith row and jth column is PA (gj, a, qj), for all qZ, qj E Q, and [AO] denotes the fuzzy matrix formed similarly by WA(gj, a, yj) for all qj E Q, yj E Y. The input sequence transition matrix for a particular n-input tape sequence is defined by an n-ary fuzzy relation in the product space Tl x T2 x . . . x Tn . The fuzzy transition function is as follows : Let Xj (k) be an in put sequence of length j, i.e., x1(k), x 2 (k + 1), . . . , x j (k + j - 1) . Then PA (gi, X, (k), qs) = PA (gi, xj , X0 . . . . , x t , qs) is the grade of transition (class A) from state qj or q (k) = qj to qs or q(k+p) = qs when the input sequence is x (k) = xj , x (k + 1) = xo , . . . , x (k + p - 1) = x t . For identical inputs x s in a sequence of length j, Xj(k) is denoted by xs(k) . The composition of two fuzzy relations NA and NB on a set S is denoted by PB o NA and is the fuzzy relation in S given by either ( 1 ) PBoA(x, y) = V {PA (X, v) n PB(v, y) I v E S} or ( 2 ) PBoA(x, y) = A{PA(x, v) V PB(v, y) I v E S} .

Based on each definition, we have a particular kind of fuzzy automata. When these two kinds ofautomata operate together, they act as a composite automaton similar to the structure of a zero-sum two-person game. We illustrate this in a later section, where a learning model is proposed . The above concepts have been studied previously. However, we have introduced new notation. Thus we concentrate on developing automata based on both definitions using the new notation . The composition given in (1) is often called the pessimistic case while that of (2) is often called the optimistic case. Now PA (qj, X2 (k), qT) = V{PA(gh xj, q) A PA (q, xo, qT) I q E Q} .

In general, wA(qj, Xj (k), q .)

=

wA(qj, Xj-j(k)xj (k + j -1), qm ) [t * (th, xl(k)x2(k+ 1) . . . xj (k+,l - 1),qm) V{[Wgi,xi,qT l ) n ItA(grl,x2,Yrz) A . . . A NA(grj_1, xj, elm) I q,-i E Q, i = 1, 2, . . . , j -

© 2002 by Chapman & Hall/CRC

1}.

466

10. Applications

Example 10.2 .2

Let

Txl

q(k)

be given by the following table:

I

q(k + ql

1)

q2

q3

q4

U11

u12

u13

u14

q2

u21

u22

u23

u24

q3

u31

u32

u33

u34

u41

u42

u43

u44

ql

q4 Then

PA W31 xlI q2) = u32, fta(g3, X21, q2)

=

A [tA(qj, xl, q2)) I qj E Q, j=1,2,3,4} (u31 A u12) V (u32 A u22) V (u33 A u32) V

VWtA(g3, xl, qj)

_

(u34

fta(g3, X31, q2)

=

n u42),

n U11 n U12), (U31 n u12 n U22), (U31 n u13 n u32), n u14 n U42), (U32 n u21 n U12), (U32 n u22 n u22), n u23 n U32), (U32 n u24 n U42), (U33 n u31 n u12), n u32 n U22), (U33 n u33 n U321, (U33 n u34 n u42), n u41 n U12), (U34 n u42 n U22), (U34 n u43 n u32), n u44 n u42)I-

V~(u31 (u31 (u32 (u33 (u34 (u34

Note, fta(g3' xi, g2)

Vex

{[t* (q3, xi, qj)

j=1,2,3,4}

2 (ltA(g3, X1, qx) 2 (PA(g3, X1, q3)

n

f~A(g~ ~ xl, q2)

n u12) V n u32) V

I

qj

(PA(g3, X21, q2)

(ltA(g3, X21, q4)

E Q,

n u22)V n u42) .

This last relation serves as an iteration scheme for a particular sequence of input tape . Similar results for the min-max relation (2) hold.

10 .3

Special Cases of P .1zzy Automata

We first consider deterministic automata, where the set A is no longer a fuzzy set . A is an ordinary set where /t takes two values, 0 and 1. Hence, the entries for each row of matrices [A], [B], [C], . . . , [Aj], [Bj], . . . will have only one 1 and the rest 0. Thus the skeleton matrix [D] of a deterministic automaton will be [D] = [A] + [B] + [C] + . . . © 2002 by Chapman & Hall/CRC

+ [H],

10.3. Special Cases of Fuzzy Automata

467

which is a reduction similar to that in the stochastic automata formulated in [26]. For a two-step transition chain, pA(gl, xy, qT) is determined by finding the maximum of the minimum of pairs of values of four types (0, 0), (0, 1), (1, 0), (1,1) . The total number of paths of length 2 from state qi to state qT is equal to di, =

X,YEX

For a path of length 3 to exist from dIT

-_

ftA(gi xy, gT) . qi

to

Ex,y,zEX

qT,

ftA(gl ,Xyz,gT)

It consists of terms such as (1,1,1), (1, 0, 1), . . . , (0, 0, 0) ; of course, only (1,1,1) defines the existence of the path. The following definition of a nondeterministic automaton has been used in [76]. A nondeterministic automaton is a quintuple A = (Q, X, S, So, F), where Q is a nonempty finite set of objects (internal states), X is a nonempty finite set of objects (input states), S is a function from Q x X into P(Q), and So, F are subsets of Q and So is nonempty. To fit this model to the fuzzy automata formulation, the set A is no longer a fuzzy set . It is an ordinary set, where /t takes only two values, 0 and 1 . The entries for each row of the matrices [A], [B], [C] . . . . are either 1 or 0. There is no restriction on the number of 1's in each row of the matrices. If all the entries of a particular row are zero, then the transition is from the state to the empty set 0. If a particular row has more than one 1, then the transition function may map the succeeding state into any one of a number of possible states. For example, if TX1 is given as 1 0

TX1 -

1

1 0

1

0 0

1

then the state transition is determined as follows : q(k) q1 q2 q3

I

x(k) q(k

+ 1)

{qi, q2}

{gi, q2, q3}

It is felt in [250] that the transition function /t is so general that extra constraints may be incorporated . A special restriction is that the row sum of all transition matrices be equal to 1 similar to that of the stochastic automata . That is, [A], [B], [C] . . . . have exactly the same structure as © 2002 by Chapman & Hall/CRC

468

10. Applications

that of stochastic matrices. This type of automata is called normalized fuzzy automata. There are many properties that can be examined for fuzzy automata similar to those for deterministic automata and stochastic automata [249].

10 .4

Fuzzy Automata as Models of Learning Systems

A basic learning system is given in Figure 10.1. Y

Unknown Environment

X

"Student" Decision Maker

Learning Section

Performance Evaluator "Teacher" "Learning System" Figure 10 .1 1 : Basic Learning Model 'Figure 10 . 1 is from [250], reprinted with permission by Copyright

© 2002 by Chapman & Hall/CRC

1969 IEEE .

10.4 . Fuzzy Automata as Models of Learning Systems

469

The proposed model represents a nonsupervised learning system if a proper performance evaluator can be selected [161, 162] . The learning section primarily consists of a composite fuzzy automaton . The performance evaluator serves as an unreliable "teacher" who tries to teach the "student" (the learning section and the decision maker) to make correct decisions . The decision executed by the decision maker is deterministic . Since online operations are required, the decision will be based on the maximum grade of membership . That is, for S2 = {w2 I i = 1, 2, . . . , r}, the set of all pattern classes, it is decided that x is from h2, or x - w 2 if

ft, (x) = V{ft, (x) I wj E Q, .i = 1, . . . , r}, where w2 E S2, i = 1, 2, . . . , r. If the decision maker is allowed to stay undecided and defer its decision, especially at the beginning of the first few learning steps, then it will be decided that x - w2 if

ft, (x) = V {ft, (x) I wj E Q, .i = 1, . . . , r} > c and be undecided if

It, (x) = V{ft, (x) I wj E Q, .l = 1, . . . , r} < c, for some predetermined c, 0 < c <_ 1 . Under this condition, the decision maker is allowed to use a pure random strategy for making decisions until it can make a decision with a certain degree of confidence . For an n-state fuzzy automaton, let ~i (k) be the grade of membership for the automaton to be at state qi in step k. The entries of the fuzzy transition matrix are denoted by

ft'i (k) = ft{q(k) = qZ, x(k) = xi, q(k + 1) = qi}. Then for input x(k) = xl, we have by Definition 10 .2 .1 that for j =

1,2, . . . ,n,

w i (k + 1)

=

V{(w .(k) A ftmi (k)) I m =1, 2, . . . , n ; m z,4 j}

Using a similar approach as given in [68], the learning behavior is reflected by having nonstationary fuzzy transition matrices with a convergent property. In order to simplify the problem, let

It'i (k) It~i (k)

_ -

for all m ~ j ft~i (k - 1), Ci Itji (k - 1) + ( 1 - ci)di,

where 0 < ci < 1, 0 < di < 1, j = 1, 2, . . . , n . The structure of the learning

© 2002 by Chapman & Hall/CRC

470

10. Applications

section is illustrated in Figure 10.2.

n (k)

(k+1)=Vmn [AR . (k)

I

(k)

h

N~i (k)

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

I

g i (k+1)= ANIIw m (k),

N m ;(k )I

-------------------------------------------------

Composite Fuzzy Automaton

2

Figure 10.2 : Learning Section The composite automaton switched by p = z constitutes the major part of the learning section . That is, with p = z, the composite fuzzy automaton operates between (1) w,(k+1)=V{(w.(k)Awm;(k)) I m=1,2, . . .n} and w, (k + 1) = n{(wm(k)V ft' (k)) I m= 1,2 . . . . n} . (2) The fuzzy automaton starts with no a priori _ information P_, (0) = 0 or 1 for all _ j, or with a priori information (0) = Aj (0) for some Aj and for all ~, j ; 0 <_ Aj (0) <_ 1. The convergence of the above algorithm can be shown as follows : Aj (k) ----> Aj as k ----> oo for some Aj .

Hence lttj (k) ----> Aj as k ----> oo. The algorithm is said to converge if ~j (k) Aj as k ----> oo . Thus as k ----> oo, we must show that V{(k) n Aj) I rn = 1, 2, . . . , n} n{(k) V ~j) I rn = 1, 2, . . . , n}

(10.2)

so that we have the same output Pj (k + 1) . It follows that n{wm(k) I m =1, 2, . . . , n} < a ;

< v{wm(k)

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

(10.3)

must hold in order for (10 .2) to hold. However, ~j (k) ~ ~j as k ~ oo . Therefore, (10.3) holds as k ~ oo . Now Pj (k) ~ Aj can be shown in the following manner . There exists at least one k such that

2

w,(k+1)=V{(wm(k)A~j) I rn=1,2, . . .,nf Figure 10 .2 is from [250], reprinted with permission by Copyright 1969 IEEE .

© 2002 by Chapman & Hall/CRC

10.5. Applications and Simulation Results

471

and ~j (k + 2) = A{(P .(k + 1) V ~j) I m = 1, 2, . . . , n},

for large k.

Suppose that ~j > V {fin, (k) I m = 1, 2, . . . , n} . For other values of _Aj, we have Pj (k + 1) = ~j . Then ~j(k + 1) = V{~m(k) I m = 1, 2, . . . , n} .

a;

However, ~,(k + 2) = n{(w1(k + 1) V a;), . . . , (V{wm(k) V (~n(k + 1) V ~j )} . Thus ~j (k + 2) = ~j . Hence

1,2 . . . . , n}), . . . ,

m =

Pj (k) ~ Aj as k 10 .5

Applications and Simulation Results

The learning model proposed above has been applied to engineering problems . We consider its applications to pattern classification and control systems . Let Z(k) be the instantaneous performance evaluation at the kth step of learning [162]. The performance evaluator M(Z, k) must be such that it is bounded above, i.e., 3 a real number T such that 0<M(Z,k)
k=1,2, . . .

and it must converge, i.e., 3 a real number M such that lim M(Z, k) = M > 0.

k~oo

The goal of the system is to maximize or minimize M(Z, k) . Hence M(Z, k) is an estimator of Z(k) at the kth step of learning . For example, let M(Z, k) = ;_7- ) 'Z (i).

Then M(Z, k)

k- , k [kll Ek1 Z(2)] + Zk k k 1 M(Z,k-1)+ , Z

k

k=1,2, . . . .

In this example, M(Z, k) is a sample average of Z(k) estimated recursively at each step of learning . We now give an application to pattern classification. © 2002 by Chapman & Hall/CRC

472

10. Applications

A pattern recognition system with nonsupervised learning is given in Figure 10.3. X'

"Student" 'X

Feature Extractor or Receptor

X

"Learning System"

Learning Pattern Classifier Figure 10 .3 3 : Learning Pattern Classifier

The role of the input and output is explained in what follows . The pattern classifier receives a new sample from the unknown environment during each time interval. After the new sample is processed through the receptor, 3 Figure

10 .3 is from [250], reprinted with permission by Copyright 1969 IEEE .

© 2002 by Chapman & Hall/CRC

10.5. Applications and Simulation Results

473

the output is fed to both the decision maker for classification and the performance evaluator for performance evaluation . The performance criterion of the system has to be selected so that its maximization or minimization reflects the clustering properties of the pattern classes, i.e., the unknown environment . Because of the natural distribution of the samples, the performance criterion can be incorporated into the system to serve as a teacher of the learning pattern recognizer. Concerning the problem of pattern classification, the nonsupervised learning model is formulated as follows . It is assumed that the classifier (the decision maker has available sets of discriminant functions character ized by sets of parameters. The system adapts itself to the best solution with a proper specification of the performance evaluation and without any external supervision . The best solution denotes the set of discriminant functions that gives the minimum misrecognition among the sets of discriminant functions for the given set of training samples . The criterion used in this particular case is based upon the sample averages and the average deviations from the sample averages of the training patterns generated by each set of discriminant functions . The best set of discriminant functions must give the maximum total distance between its sample averages and the minimum total sample deviation (average of the squared deviation from the sample averages . Let S2 = {Wi I i = 1, 2, . . . , r} be the set of pattern classes. Let X1, X2 . . . . be the sequence of incoming samples from the unknown environment in Rd . Suppose n sets of discriminant functions are given a priori . The decision maker may assume the structure as shown in Figure 10.4.

Figure 10.4 4 : Multiclass Pattern Classifier Then Kkl 4 Figure

I W1[Ei-1 SkiI

-W2[Ei>jl 1=1,2, . . . ,n,

- M%j)111

10 .4 is from [250], reprinted with permission by Copyright 1969 IEEE .

© 2002 by Chapman & Hall/CRC

474

10. Applications

where Kk is the performance evaluation for the lth set of discriminant functions at the kth step of learning, Sk i is the sample deviation for the ith class of the lth set of discriminant functions at the kth step of learning, Mk a is the sample average for the ith class of the lth set of discriminant functions at the kth step of learning, and 0<

wl,w2 <

Let the nk,i denote the number of samples belonging to wi up to the kth step of learning. The values of the K%, Ski , and Mk i are estimated recursively as follows . For the lth set of discriminant functions, if i=1,2, . . . ,r,

Xk+1 - Wi,

then =

Mk+l,i

Sk+l i

=

nk,i nk,i

1 nk,i

+1

Mk,i

(Xi -

+1

+

Xk+1 nk,i

,

+ 1

Mk+l,i)/(X7

- Mk+l,i) .

From the above two equations, we obtain Sk+l,i

=

n k,i Mk,i

Ski +1

+

n k,i (nk,i

+ 1)

2

(Xk+1 - Mki)~(Xk+1 - Mki)

Also, Mk +l,j Sk+1,S

= =

Mki Ski

for j :?~ i for j zA i.

The minimum Kk, l = 1, 2. . . . , n, serves to indicate the best solution among the n sets of discriminant functions . The patterns used in the computer simulations are the characters A, B, and C. The four features xl, x2, x3, and x4 are extracted from each character . The extracted features correspond to the number of distinct intersections of the sample character with lines al , a2 , a3, and a4 as shown in © 2002 by Chapman & Hall/CRC

10.5 . Applications and Simulation Results

475

Figure 10 .5 .

Sample Feature : [x 1 , x2 , x 3 , x 4 ] = [32321 Figure 10.5 5 : Description of Feature Extraction

Specific computer simulations are conducted for (1) two equal numbers of training characters from A and B using one hyperplane and (2) three equal numbers of training characters from A, B, and C using three hyperplanes . The estimation of the n membership functions ~j(k) at the kth step of learning is as follows :

_

3

Aj(k)=1-Kk

c>V{Kk

.j=1,2, . . ., n ; k=1,2, . . . ;

I j=1,2, . . . ,n; k=1,2, . . .}

with

lim Aj (k) = Aj ,

k~oo

0 <_ Aj < 1 .

5 Figure 10 .5 is from [250], reprinted with permission by Copyright 1969 IEEE .

© 2002 by Chapman & Hall/CRC

476

10. Applications

The details of the computer flow diagram are shown in Figure 10.6.

Selector

Performance Evaluation K

mation of

Feature Extracto or Rece to

Decisi n Maxim m

XE 0) .

Learning Algorithm Figure 10.6 6 : Computer Flow Diagram for Character Learning

Ten sets of predetermined hyperplanes are used in both examples with 60 samples from each class . The incoming samples are introduced to the system in a random fashion. The results of the learning curves, /t j (lc) versus lc for each example, are given in [250, Figures 7 and 8, p. 220] . 6 Figure

10 .6 is from [250], reprinted with permission by Copyright 1969 IEEE .

© 2002 by Chapman & Hall/CRC

10.5. Applications and Simulation Results

477

We now give an application to control systems . The learning controller presented here is similar to that presented in [161] . We give a brief explanation of it. A time-discrete plant may be described as x(k + 1) = Ok+1 [x(k), u(k + 1)],

where x(k), x(k + 1) E Qx = {xi I i = 1, 2, . . . p; p <

and u(k) E SZ = {ui I i = 1, 2, . . . 'P; p < oo}.

Here x(k) is the observed response of the plant at the (k+1)th instant when the control action u(k+ 1) is applied . It is assumed that Ok is unknown for k = 1, 2, . . . . The instantaneous performance evaluation of a control action u(k) is given by Z(k + 1) = w(x(k), u(k + 1), x(k + 1)) with the set {Z(k) I k = 1, 2. . . . } bounded above, say 0
The goal of the control is to minimize Mk + 1(ZIu(k),x(k),u(k + 1)) and the sample average of Z. The sample average for each control policy is estimated as follows . Let u(k+1) = ul be applied after observing u(k) = uj and x(k) = xi . Then Mk+1[ZIuj'xZ'ui]

n 1 n+1Mk[Zluj,xi,ul]+ n 1 Z(k+1)

Mk+1 [Zluj , xi, Uh] = Mk [Zl uj , xi, Uhl, h = 1,2, . . . gy p, h :?~ l,

where n = n(j, i, l) denotes the number of occurrences of u(k) = uj, x(k) = xi, and u(k + 1) = ui . [ZITj , xi, ui] . lak+i [ui uj, xi] = 1 - Mk+i

(10.4)

By equation (10.4), the system is able to associate the control action with the maximum grade membership with the minimum sample average of Z. © 2002 by Chapman & Hall/CRC

478

10. Applications

Figure 10.7 illustrates the learning controller.

"Student"

~

(Decision Maker) Max

i j (k+l )

j=1,2, . . .,n

(Learning Section)

i

(k+l)= [u [u /u(k),x(k) ] ~j j=1,2, . . .,n ...... ...... . ....... ..L... ...... . . .... . . ...... .

"Teacher"

Sample Average Evaluator

M+1 (Z/u(k),x(k) ,u(k+1) )

g

"Learning Controller" Figure 10.7 7: Proposed Learning Controller

An application of a class of fuzzy automata as a model of learning systems was proposed above. A nonsupervised learning algorithm for a fuzzy automaton was presented, together with its application to pattern recognition and automatic control problems . Computer simulations of character recognition by the authors of [250] showed satisfactory results. The following is a pertinent result. 7 Figure

10 .7 is from [250], reprinted with permission by Copyright 1969 IEEE .

© 2002 by Chapman & Hall/CRC

10.6. Properties of Fuzzy Automata

479

Theorem 10.5.1 A necessary and sufficient condition for A{{uj2 V uk

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

= =

V{{uj2 A uk i=1,2, . . . ,n} j =1,2, . . .,s} uk

is that A{V{uj2

n} I

s}

< <

uk V{A{uj2

n} I j = 1, 2, . . . , s} .

Proof. We first prove the necessity. We may assume without loss of generality that (1) ujl = VZ{ujzj > ujn = AZ{uj2}, (2) UP I = Vj{Jajl} >- aqn = njf{ujn}, (3) umn = Vj L ujn} , all = nj L ujl} .

Then

Fl F2

= =

= =

n{{ujj V uk I (UII V uk) A . V{{uj2 A uk I (uln A ak) V .

i = 1, 2, . . . , n} I j = 1, 2, . . . A upi A . . . A (us l V uk) i = 1, 2, . . . , n} I j = 1, 2, . . . V aqn V . . . V (u sn A uk) .

Assuming the value of uk at different regions, one can show that if Fl F2 = uk, then all < uk < It mnThe sufficiency is immediate.

10 .6

Properties of Fuzzy Automata

We explore some properties similar to those of deterministic automata and probabilistic automata. We consider only ergodic, stationary, periodic, and aperiodic fuzzy transition matrices. The numerical illustrations are given to illustrate a specific property of fuzzy automata and also to furnish computational examples of the concepts discussed above. The computational procedure of Section 10.2 is followed . After a certain number of iterations of identical inputs, it is possible for the overall fuzzy transition matrix to remain the same. For example, let [Ti,]~l> =

0.9 0.2 0.8

0.5 0.4 0.1

0.3

0.95 0.25

Then 2 [TZl ]( ) =

© 2002 by Chapman & Hall/CRC

0.9 0.8 0.8

0.5 0.4 0.5

0.5 0.4 0.3

480

10. Applications [TZ 1 ]( 3 > =

[TZ 1 ](4) _

0 .9 0 .8 0 .8

0 .5 0 .5 0 .5

0 .5 0 .5 0 .5

0 .9 0 .8 0 .8

0 .5 0 .5 0 .5

0 .5

0.4 0 .5

= [TZ1](5)

[TZ 1 ](n)

A stationary fuzzy transition matrix with the property [Ti,](') ----> T as called an ergodic fuzzy transition matrix . Certain stationary fuzzy transition matrices have no ergodic property, but have a periodic property. As an illustration, consider the following examples . n ~ oo is

Example 10.6 .1 [TZ](1) -

0 .3 0 .8

1 .0 0 .1

[T (2) -

0 .8 0 .3

0 .3 0 .8

[T (3) _

0 .3 0 .8

0 .8 0 .3

[TZ] (4) _

[ 0 .8 0 .3

0 .3 0 .8 0'3 0 .8

0 .8 ] 0 .3

0 .3 0 .9 0 .5

0.4

0.7

0 .5 0 .5 0 .8

0.7

0.4 0.7

0.7

0.4 0.7

The matrix keeps oscillating between period of oscillation is 2 .

Example 10.6 .2 [TZ]~ 1 > =

[T]

(2)

=

(3) = [T]

© 2002 by Chapman & Hall/CRC

0 .6 0 .5

0 .1 0 .8

0 .6

0.4

0 .6

0 .6 0 .2 1

0 .6

1

0 .6 0 .6

0.7

1

and [ 0'8 0.3

0 .3 0 .8

.

The

10.7. Fractionally Fuzzy Grammars and Pattern Recognition

[T (9)

_

4 [TZ]( > =

0 .5 0 .7 0 .6

0.6 0.6 0.7

0.7 0.6 0.6

[TZ]( s> =

0 .6 0 .6 0 .7

0.7 0.6 0.6

0.6 0.7 0.6

[Ti] (6)

=

0 .7 0 .6 0 .6

0.6 0.7 0.7

0.6 0.6 0.7

[TZ] (7> =

0 .6 0 .7 0 .6

0.7 0.6 0.7

0.7 0.6 0.6

[TZ] (8) =

0 .6 0 .6 0 .7

0.7 0.7 0.6

0.6 0.7 0.6

[T (6) ;

[T,](10)

= [TZ]( 7 ) ; [TZ]

(11)

481

= [TZ](8) ; . . . .

The matrix in Example 10.6.2 has a period equal to 3. From Examples 10.6.1 and 10.6.2, it can be said that ergodicity has a period equal to 1. A fuzzy transition matrix having a period of length 1 is considered as an aperiodic fuzzy transition matrix .

10 .7

Fractionally Fuzzy Grammars and Pattern Recognition

A type of fuzzy grammar, called a fractionally fuzzy grammar, is presented in this and the next section . It was first introduced in [44] . These grammars are especially suitable for pattern recognition because they are powerful and are easily parsed . Formal language theory has been applied to pattern recognition problems in which the patterns contain most of their information in their structure rather than in their numeric values [69, 223, 250, 226, 225]. In order to increase the generative power of grammars and to make grammars more powerful for the purpose that they become more suited to pattern recognition, the concept of a phrase structured grammar can be extended in several ways. One way is to randomize the use of the production rules. This results © 2002 by Chapman & Hall/CRC

482

10. Applications

in stochastic grammars [223, 67, 109] and fuzzy grammars . Languages produced by fuzzy grammars have shown some promise in dealing with pattern recognition problems, where the underlying concept may be probabilistic or fuzzy [250, 226, 225] . A second way of extending the concept of a grammar is to restrict the use of the productions [106, 190, 191] . This results in programmed grammars and controlled grammars . These grammars can generate all recursively enumerable sets with a context-free core grammar . Programmed grammars have the added advantage that they are easily implemented on a computer. Cursive script recognition experiments [54, 52, 53, 139, 124, 213] in the 1960's and 70's had the major emphasis on recognizing whole words. None used a syntactic approach . A typical method presented in [53] decomposed the words to be recognized in to sequences of strokes that were then combined into letters and into words. In [139], an attempt was made to distinguish words that are similar in appearance such as fell, feel, foul, etc . All of these experiments except the ones in [213] input their data on a graphics device and kept the sequence of the points as a part of the data . In [213], pictures of writing were inputted and hence the sequence information was not available . In the following, we show that the languages generated by the class of type i (Chomsky) fractionally fuzzy grammars properly includes the set of languages generated by type i fuzzy grammars . We also show that the set of languages generated by all type 3 (regular) fractionally fuzzy grammars is not a subset of the set of languages generated by all unrestricted (type 0) fuzzy grammars . We show that context-sensitive fractionally fuzzy grammars are recursive and can be parsed by most methods used for ordinary context-free grammars . We describe a pattern recognition experiment that uses fractionally fuzzy grammars to recognize the script letters i, e, t, and l without the help of the dot on the i or the crossing of the t. We discuss the construction of a fractionally fuzzy grammar based on a training set . A fuzzy grammar FG is a six-tuple FG = (T, N, S, P, J, /t), where T, N, S, P are, respectively, the terminal alphabet, the nonterminal alphabet, the starting symbol, and the set of production rules as with an ordinary grammar . J = {r2 I i = 1,2, . . . , n} is a set of distinct labels for the productions in P, and /t is a fuzzy membership function, /t : J ~ [0,1] . Let V = T U N. Suppose that rule r2 is a ~ 3. Then we write the application of r2 as follows : ycti6 M('i) r}' S, where cti, 3, -y, S E V* . If BZ E V*, i = 0, 1, 2. . . . , m, and S = Bo

F (r) _~ r1

© 2002 by Chapman & Hall/CRC

V(r2) V(r3) Bl _~ 02 _~ 03 . . . r2 r3

F(Tn) r'n

Bm

= X

10.7. Fractionally Fuzzy Grammars and Pattern Recognition

483

is a derivation of x in FG, we write S

=

Bo

"(r1T2+

. .T~n)

r1 r2 r3 . . . r-

8772

-

x)

where we write ft(rir2r3 . . . rm ) for ft(ri) A ft(r2) A . . . A ft(r~ ). The grade of membership of x E T* is given by the function ~ : T* --+ [0,1], ~(x) =V{ft(rlr2r3 . . .r,)},

where the supremum is taken over all derivations of x E L(FG), the language generated by FG . It follows that ordinary grammars, i.e., those with ft(ri) = 1 for all r2 E J, are a special case of fuzzy grammars . Fuzzy grammars can be classified according to the form of the production rules. In Chapter 4, it has been shown that for every context-free fuzzy grammar G, there exists two context-free fuzzy grammars G9 and G, such that L(G) = L(G9) = L(Gc.) and Gg is in Greibach normal form and G, is in Chomsky normal form. Example 10.7.1 Consider the fuzzy grammar FG = (T, N, S, P, J, ft), where T = {a, b}, N = {S, A, B, C}, and P, J, and ft are given as follows: rl r2 r3 r4 r5 r6 r7 r8 r9

: : : : : : : : :

S A B A A A C C A

AB a b aAB aB aC a as B

ft(rl) ft(r2) ft (r3) ft (r4) ft(r5) ft (r6) ft(r7) ft (r8) ft(rs)

= = = = = = = = =

1 1 1 0.9 0.5 0.5 0.5 0.2 0.2

The language generated by this fuzzy grammar consists of strings of the form a'b' with n, m > 0 . The membership of these strings is given as follows:

(a'b') _

1 .9 .5 .2 0

if m=n=1 if m=n :?~ 1 if m = n f 1 if m=nf2 otherwise.

Hence it follows that this grammar generates strings of the form a'b', where In-ml < 2.

As we have previously seen, a fuzzy grammar can be used to generate ordinary languages by the use of thresholds . One such language with threshold c is the set of strings L(FG, c) = {x E L(FG) l ~(x) > c} . © 2002 by Chapman & Hall/CRC

484

10. Applications

There are two other threshold languages defined in [147], namely, the twothreshold language and the equal-threshold language. They are defined as follows : L(FG, cl, c2) = {x E L(FG) I cl < ~(x) < c21

and L(FG, =, c) = {x E L(FG) I ~~(x) = c} .

The language L(FG, c) is most often used to compare the generating power of a fuzzy grammar to that of an ordinary grammar . However, L(FG, c) = L(G), where G is the grammar obtained from the FG by removing all productions whose fuzzy membership is less than or equal to c and then removing the fuzziness from the remaining rules. Hence it is stated in [44] that the use of threshold language seems limited . 10 .8

Fractionally Fuzzy Grammars

The patterns in syntactic pattern recognition are strings over the terminal alphabet . These strings must be parsed in order to find the pattern classes to which they most likely belong . Back tracking is required by many parsing algorithms [3] . That is, after applying some rules, it is discovered that the input string cannot be parsed successfully by this sequence of rules. Rather than starting all over again, it is desirable to reverse the action of one or more of the most recently applied rules in order to try another sequence of productions . It is sufficient with ordinary grammars to keep track of the derivation tree as it is generated with each node being labeled with a symbol from T U N, where T is the set of terminal symbols and N is the set of nonterminal symbols. However, this tree is not sufficient for fuzzy grammars since the fuzzy value at the ith step is the minimum of the value at (i -1)th step and the fuzzy membership of the ith rule. If this minimum was the ith rule's membership, there is no way of knowing the fuzzy value at the (i - 1)th step . Hence the fuzzy value at each step must also be remembered at each node. Consequently, the memory requirements are greatly increased for many practical problems . Another drawback of fuzzy grammars in pattern recognition is that all strings in L(FG) can be classified into a finite number of subsets by their membership in the language. The number of such subsets is limited by the number of productions in the grammar . This is due to the fact that if x E L(FG) with a membership ~(x), then there must be a rule in FG with the membership P(x) since P(x) = A{ft(rjj) I i = 1, 2, . . . , m} for some sequence of rules r21 rig . . . ri,n in P. Thus L(FG, c) for some threshold c is always a language generated by those rules in the grammar with a membership greater than c. © 2002 by Chapman & Hall/CRC

10.8 . Fractionally Fuzzy Grammars

48 5

To overcome these drawbacks, we present a method introduced in [44] of computing the membership of a string x that can be derived by the m sequence production rules, rl r2 . . . r k , of lengths lk, where k = 1, 2, . . . , m. This brings us to the next definition. Definition 10.8.1 A fractionally fuzzy grammar is a 7-tuple FFG =

(N, T, S, P, J, g, h), where N, T, S, P, and J are the nonterminal alphabet, the terminal alphabet, the starting symbol, the set of productions, and a distinct set of labels on the productions as a fuzzy grammar, respectively. The functions g and h map J into hY U {0} such that g(rk) < h(r2) b'r2 E J. A string is generated in the same manner as that by a fuzzy grammar. The membership of the derived string is given by V'x E T*, ~~b 1g(r~)

(x) = V{ E i

~= i h(r~ )

I k=1 2, . . . ,m},

where 0/0 is defined as 0.

Since 0/0 is defined to be 0, it follows that 0 <_ y(x) <_ 1 V'x E T* . Clearly, backtracking over a rule r can now be accomplished by simply subtracting g(r) and h(r) from the respective running totals. We now give an interpretation of Definition 10.8 .1 in a heuristic sense . As each rule r is applied, g(r) and h(r) are added to the respective running totals for the numerator and denominator of the fuzzy membership . Clearly, the fuzzy membership of a string could be any rational number in [0,1]. Moreover the number of fuzzy membership levels is not limited by the number of productions . With a view to pattern recognition, it follows that the amount of impact a rule has on the final membership level is proportional to the value of h(r) . The membership of the string tends to increase if g(r) is approximately equal to h(r). It tends to decrease if g(r) is much less than h(r) or if g(r) is close to 0. Rules for which g(r) and h(r) are both 0 have no effect on the membership . Hence it is possible to divide the rules into three classes. Those that strongly indicate membership in the class, those that strongly indicate membership in another class, and those that serve little purpose in separating the classes, but that are traits between different classes . Example 10.8.2 Consider the fractionally fuzzy grammar that has T = {a, b}, N = {S}, and P, J, g, and h given as follows: rl r2 r3 r4

: : : :

S S S S

ab aSb aS Sb

g(ri) g(r2) g(r3) g(r4)

=1 =1 =0 =0

h(rl) h(r2) h(r3) h(r4)

= = = =

1 1 1 1.

Consider the string a3b5 . The following sequences of labels yield three of the ways the string a3b5 can be derived: r2 , r2, r4 , r4 , rl and r2, r3 , r4 , r4, r4,

© 2002 by Chapman & Hall/CRC

48 6

10 . Applications

rl and r3, r3, r 4 , r4 , r 4 , r 4 , rl yield the string a3b5 with associated values 3 , 2, and 3 V 5 V 5 = 3. 5, 5 5 5 respectively. It follows that ft(a3b5) = 5 5 5 5 Wbm This grammar generates the fuzzy language I n, m E NJ . The membership of the string anbm is given by (anbm)

_ nlxm nV m

This set of strings is the fuzzy set of strings that are "almost" anbn . That is, the first pair of rules generates the set of strings anbn for n > 0, and the second pair of rules allows for variations in the number of a's and the number of Vs. The closer n and m are, the greater the membership of the string. This follows since g(r) = h(r) for the first pair of rules and g(r) < h(r) for the second pair. While the grammar shown in Example 10 .7 .1 could only measure finite differences between m and n, the grammar here measures membership on percentage difference between m and n. This is similar to the way one would judge whether or not the lengths of two lines were the same .

We now compare the relative generative power of fractionally fuzzy grammars to that of fuzzy grammars . Theorem 10.8.3 The set of all languages generated by type i fractionally fuzzy grammars properly contains the set of all languages generated by type i fuzzy grammars, where i = 0, 1, 2, and 3. Proof. From the preceding remarks, it follows that the number of distinct levels of membership in a language generated by a fuzzy grammar is limited by the number of production rules in the fuzzy grammar and this number is finite. Hence it suffices to show that for every fuzzy grammar of type i, there is a fractionally fuzzy grammar of type i which generates the same language, and that there exists a fractionally fuzzy grammar of type i which generates a language with an infinite number of distinct membership levels . Let FG = (N, T, S, P, J, /t) be a fuzzy grammar of type i. We can construct a fractionally fuzzy grammar FFG = (N', T', S', P', J', g, h) as follows: Let c2 be the distinct values of ft(rz) for i = 1, 2, . . . , m . There is no loss in generality in assuming that cl > c2 > . . . > c . . Let gZ and h2 be integers such that c2 = ~, i = 1, 2. . . . , m. For all A E N, define m distinct symbols AZ , i = 1, 2, . . . , m. These symbols together with the new starting symbol S' make up the set of nonterminals N' for the fractionally fuzzy grammar . Let P' initially be the set of rules ro :

S' ----> Si

g(ro) = gz

h(ro) = hi,

i = 1, 2, . . . , m. For each rule rj : cti ~ 3 in P and each c 2 , i = 1, 2, . . . , m, if ft(rj) > c2 , add to these rules the rule r~ : cti2 X3 2 g(r~) = gZ

© 2002 by Chapman & Hall/CRC

h(r~) = hi,

10.8 . Fractionally Fuzzy Grammars

48 7

where a2 and ,3 2 are the strings a and, 3 with each nonterminal A replaced with the associated new nonterminal symbol AZ . The FFG has, in effect, m subgrammars reachable by the ro rules. The ith subgrammar produces only those strings that would have been generated with a membership of at least c2 in L(FG) . In the ith subgrammar, these strings have membership = c 2 . Since their membership in L(FFG) is defined as the supremum of all derivations, it follows that each string has equal membership in both languages . That is, L(FFG) D L(FG) . To show the proper inclusion, consider the following fractionally fuzzy grammar : rl : r2 :

S S

a aS

g(rl) = 1 g(r2) = 0

h(rl) = 1 h(r2) = 1.

This fuzzy grammar generates strings of the form an for n E N. Since the fuzzy membership of the string an is n , the language clearly has an infinite number of distinct membership levels . Since the grammar is regular, it is also context-free, context-sensitive, and a member of the class of type-0 grammars. In order for fractionally fuzzy grammars to be of use in pattern recognition, it must be possible to determine whether a given string is a member of the language. That is, in order to apply fractionally fuzzy grammars to pattern recognition, an algorithm is needed that can compute in bounded time the membership of a string in L(FFG) . The following lemma leads to Theorem 10 .8 .7, which proves that context-sensitive fractionally fuzzy grammars are recursive and such an algorithm exists . Lemma 10.8.4 Consider a context-sensitive fractionally Suppose a derivation contains the sequence . . .---->

O Z ----> OZ+1 ----> . . .---->

fuzzy

grammar.

OZ+k ----> . . .,

where OZ = OZ +k. Then either k <_ nP, where n = ~ V1 and p = length of the string OZ , or OZ +j = OZ+ , for 0 < j < m < nP .

101

is the

Proof. The result clearly follows because of the noncontracting nature of context-sensitive grammars (i.e., O.,,j <_ 10,1 for u <_ v) and since there are exactly nP distinct strings over V of length p. Lemma 10.8.5 Let FFG be a fractionally fuzzy grammar. Suppose that x E L(FFG) is derivable by the sequence

S=0o ----> 01 ----> 02 ----> . . .----> On =x . If 03 = Ok for j < k and 0 <_ j < n, then the membership of x in FFG is at least EM-1 g(rZj

+

Em-k+1 g(ri,n)

E M-1 h(rZj + Em-k+1 h(rjj © 2002 by Chapman & Hall/CRC

V

Em-j g(ri'n) Em=j h(rij

488

10. Applications

Proof. The loop in the derivation sequence from O j to O k can be removed. The first argument of the maximum represents the membership given to x by this shortened derivation. The loop can also be repeated as many times as desired . As the number of times the loop is repeated is increased, the membership of x approaches the second argument . Since there may be other derivations of x, the membership of x in L(FFG) is at least the maximum of these two terms. Lemma 10.8 .5 may be applied repeatedly in a derivation. Hence if a derivation contains a loop nested within a loop, the loops can be considered separately and the membership of the string is the maximum given by the loop-free derivation, the inner loop, and the outer loop. Lemma 10.8.6 Let FFG be a context-sensitive fractionally fuzzy grammar

with IV I = n. Let Ro = {S}. Let Rk be the set of all strings over V of length k that can be directly generated from a string of length less than k. Let R~ be the set of all strings of length k that can be directly generated from a string in R.~_ 1 , j = 1, 2, . . . . Then the set Rk = RkRk . . . Rk contains all strings over V of length k that can be generated by the FFG . Moreover, the derivation needed to generate Rk contains all the simple derivation loops on strings of length k in L(FFG) .

Proof. Suppose that there exists a 0. such that O m ~ Rk, 10 m l = k, and Br is derivable from 02 E Ro . Let 02 02+1 . . . OZ+j = Om be the shortest sequence from 02 to Br for FFG. Since j > n by assumption, it follows by Lemma 10.8 .4 that this sequence must contain a loop and is thus not the shortest sequence. Hence no such Br exists. Suppose that Oj ~ Oj+1 . . . ~ Oj+s = Oj is a simple loop in FFG, where I O j = k. Suppose this loop was not detected by the derivations that generate Rk and that this loop can be detected from 02 E R.0 by the shortest sequence OZ ----> OZ+1 . . . ----> OZ+T =

0j .

By assumption, r + s > n since the loop was not detected by generating Rk . However, the r + s strings 02, O2+,, . , Oj+s_i cannot all be distinct. Since 02, 0 2+1, . . . , OZ+T are distinct by assumption and O j , . . . , Oj+s_i are also distinct by assumption, we have that 02 + = Oj+, for some 0 <_ u < r and 0 < v < s. Thus the derivation sequence Oi ----> Oi+1 . . .---->

Oi+., = Oj+v ----> Oj+v+1 . . .-~ Oj+s = Oj ----> . . .----> Oj+v

also detects the simple loop and is shorter than the original sequence. This contradicts the original assumption. Consequently, no such loop can appear in a derivation. Theorem 10.8.7 If a fractionally fuzzy grammar FFG is a context-sensitive fractionally fuzzy grammar, then it is recursive.

© 2002 by Chapman & Hall/CRC

10.8 . Fractionally Fuzzy Grammars

48 9

Proof. Let IV I = n. For any string x, we need only generate R1, R2 . . . . , Rk , where k = Ix I . Let RZ be the set ofall strings derivable from the starting symbol in i steps . Since i is finite, RZ can be determined . Since it takes nj steps to generate Ri from Ro, Rr contains Ri if m >_ 1 + n + n2 + . . . + ni . Thus in a finite number of steps, all strings of length k can be found and all loops in these derivations can be detected. Hence it can be determined whether or not x E L(FFG) and its membership if it is . The following example provides an interesting property of c-fractionally fuzzy grammars. Example 10.8.8 For a regular fractionally fuzzy grammar FFG, the lan-

guage L(FFG, c) is not necessarily a regular language : Consider the two fractionally fuzzy grammars FFG1 = (T, N, X, P, J, g1, hl) and FFG2 = (T, N, S, P, J, 92, h2), where T = f0,1}, N = f S, Af, and P is given as follows: r1 r2 r3 r4 r5

: : : : :

SOS S~0 S~A A 1A A 1.

Let hl(r2) = h2 (r 2) = 1 for i = 1, 2, 3, 4, 5 and let gl(ri) = 1 for i = 1, 2, 3, 91(ri) = 0 for i = 4, 5. Let 92(ri) = 0 for i = 1, 2, and let 92(ri) = 1 for i = 3, 4, 5. Clearly, both the grammars produce strings of the form Onl ,, where m, n > 0 and m + n > 0. Consider the string 02 13. Now S~OS~OOS~OOA~001A~0011A_""~ 00111 .

2+1 and 3+1 Hence ft, (o213) = 36 = 3+2+1 N2 (0 2 13) = 46 = 3+2+1 . It follows that On1,n the fuzzy membership of is given by the following two equations: n+1 m+n+1

Itl(On1m)=

and

m+l m+n+l

0

ifm>0 of m=0.

Assume that the set L(FFG1, 0.5) =

{OnjmI

n > mf

and the set L(FFG2, 0.5) =

fOnlm

I n < mf

are regular. Since regular sets are closed under intersection, the set L(FFG1, 0.5)

n

L(FFG2, 0.5) = fOnln I n > Of

must be regular. However, it is known that fOnln I n > Of is context-free and not regular. Thus the assumption is false and the desired result holds.

© 2002 by Chapman & Hall/CRC

490

10 .9

10. Applications

A Pattern Recognition Experiment

In [44], an experiment was developed to test the usefulness of fractionally fuzzy grammars in pattern recognition. A pattern space consisting of a set of strings was chosen in order to use script writing . The data were input to a computer on a graphics tablet . This data consisted of strings of points in a 2-dimensional space of the tablet . The data were a sample of seven persons' handwritings . Each person was given a list of 400 seven-letter words and was told approximately how large the person should write . The first three persons wrote all 400 words while the last four wrote the first 100 words . The data was digitized in a continuous mode by the computer whenever the pen was down. Each point collected in this manner was compared to the previously stored point to determine if the distance between them was greater than a given threshold. The threshold was chosen to be about 0 .04 inch. If the distance was not greater than the threshold, the new point was discarded and a new position of the pen was read. If the threshold was exceeded, this point was added to the data and the process was repeated . This procedure resulted in a record of 250 points in the X - Y plane (with zero fill-in for each seven-letter word written. Some examples of words input to the computer are shown in Figure 1 of [44, p. 344] . These points were converted into a string of symbols that would comprise the terminal alphabet . This was accomplished by comparing each adjacent pair of points to see the relative direction traveled by the pen at that point. The directions were then classified into one of eight directions . Each direction was separated by 45 degrees with class 0 centered at 0 degrees (the positive X-direction and the remaining classes being numbered 1 through 7 in a counterclockwise direction . Hence the terminal alphabet consisted of eight octal digits, i.e., T = {0,1, 2, . . . , 7}. Figure 2 of [44, p. 345] shows that quantization of directions introduced some distortion into the data. The individual letters were separated by an operator using an interactive graphics program . These letters then consisted of strings of octal digits whose lengths varied from 10 to about 70 characters in length . The crossing of and the dotting of the were deleted since they did not necessarily follow the basic letter without other letters intervening . Hence in order to keep the computer time down, only four letters were used in the test . The machine was asked to separate the i's, e's, Vs, and the l's without the dots on the and the crossings of the Vs. In view of Example 10.8 .8, it was also decided to use only regular fractionally fuzzy grammars . The grammars listed in Figure 10.8 were generated by cut and try methods

is

is

is

© 2002 by Chapman & Hall/CRC

10.9. A Pattern Recognition Experiment

491

based on the following ideas.

Production Rule S OS

h, 0/0

h.;, 0/0

h, 0/0

h, 0/0

S

1A

14/14

10/14

0/18

0/18

S A

2B OA

12/12 0/1

8/14 0/1

0/18 0/1

0/18 0/1

A

1A

0/0

0/0

0/0

0/0

A A A A A A B B B

2B 2B 3C 4D 5E 6F OB 1B 2B

0/0 0/0 1/1 0/1 0/16 0/16 0/2 0/1 0/0

0/0 0/0 0/3 0/1 8/8 8/8 0/2 0/1 0/0

0/0 0/0 4/4 4/4 0/4 0/4 0/2 0/1 0/0

0/0 0/0 0/3 0/1 4/4 4/4 0/2 0/1 0/0

B B B B B C C C C C D D

3C 3J 4D 5E 6F 3C 4D 5E 6F 7G 4D 5E

1/1 0/2 0/0 0/5 0/16 2/2 5/5 1/1 0/3 0/7 2/2 1/1

0/3 0/2 0/0 5/5 8/8 0/7 0/5 0/3 3/3 6/6 0/7 0/3

4/4 0/2 4/4 3/3 0/4 4/4 4/4 2/2 1/1 0/3 5/5 2/2

0/3 0/2 0/0 5/5 4/4 0/7 0/5 0/3 3/3 6/6 0/7 0/3

D

6F

0/0

0/0

0/0

0/0

© 2002 by Chapman & Hall/CRC

Comments Allows horizontal initial stroke Begining of a letter . Initial membership Ditto Small backtrack in direction (noise) Expected in all up strokes No effect Ditto Ditto Top of a letter not pointed Ditto Top sharply pointed Ditto Noise up on stroke Noise up on stroke Expected on up stroke (no effect Rounded top of a letter Noise sequence started Ditto Neutral top of a letter Pointed top of a letter Rounded top of a letter Ditto Neutral top of a letter Slightly pointed top Pointed top Very open loop of a letter Open loop or a highly slanted letter No effect

492

10. Applications Production Rut e

ge he

9-i h.;,

E 5E 0/1 0/1 E 6F 0/2 0/2 E 7F 0/1 0/1 F OH 0/0 0/0 F 0 0/0 0/0 F 7G 0/2 0/2 F 7 0/2 0/2 F 6F 0/1 0/2 F 6 0/2 0/2 F 5F 0/1 0/1 G OH 0/0 0/0 G 0 0/0 0/0 G 7G 0/2 0/2 G 7 0/2 0/2 G , 6G 0/3 0/3 G 6 0/2 0/2 H OH 0/0 0/0 H 0 0/0 0/0 J 2B 0/0 0/0 Figure 10.88 . The List Fuzzy Grammar for the

Comments 2/2 0/1 Possible loop 2/2 2/2 Expected part of down stroke 0/1 0/1 Noise on down stroke 0/0 0/0 End of a letter (tail) 0/0 0/0 Ditto 2/2 2/2 Down stroke 2/2 2/2 Down stroke . End of a letter 2/2 2/2 Down stroke 2/2 2/2 Down stroke . End of a letter 0/1 0/1 Noise on down stroke 0/0 0/0 No effect (tail) 0/0 0/0 End of a tail 2/2 2/2 Down stroke 2/2 2/2 Ditto 0/3 0/3 Noise 0/2 0/2 Noise or end of a letter 0/0 0/0 Tail of a letter (no effect) 0/0 0/0 Ditto 0/0 0/0 Noise of Production Rules of the Fractionally Experiment. 91 h,

9t h*

Since all the letters under consideration started with a near horizontal, left to right stroke (octal direction 0) and continued in a counterclockwise direction (increasing octal direction) until returning to a near horizontal tail, the same set of production rules was used for all classes. The productions used the nonterminal symbols A, B, . . . , G to represent the highest octal direction so far encountered, A representing 1, B representing 2, and so on. In the ideal case, only higher octal directions and higher nonterminal representations are reachable from any nonterminal symbol . However, to allow for noise in the less curved portions of the letters, the terminal symbol generated was allowed to be one less than the highest so far generated. A change of direction of more than 225 degrees counterclockwise was not allowed since this would never occur in these letters . In order to allow a tail of any length to be affixed to the ideal letter, the nonterminal symbol H was added . The grammar was tested on a training set and was found to accept most of the strings . In order for all strings in the training set to be accepted, minor modifications were made. For example, J was added to the nonterminal alphabet to pick up an unusual noise condition . The fractionally fuzzy membership functions were developed using the following criteria . First, a rule that could not help distinguish one class from another was given the value 0/0 and would then have no effect on the final membership assuming some rule r, for which h(r) :?~ 0, was also applied. 8 Figure

10 .8 is from [44], reprinted with permission from Academic Press .

© 2002 by Chapman & Hall/CRC

10.9. A Pattern Recognition Experiment

493

Second, a rule for which h(r) was small would have little effect on the final membership of any string generated by that rule. Third, any rule for which h(r) was large would have a large effect on the final membership of any string generated by using that rule. Fourth, if rule r was used, the fuzzy membership of the string would be changed in the direction towards the value h(T) by that application of rule r. Hence if h(T) were close to 1, then

the membership of the string would be increased and if h(") were close to 0, the membership of the string would be decreased . Finally, a rule that was used in all strings could be given a membership value that could serve as a starting point from which subtraction could occur by rules with = 0 1(T) and to which addition could occur by rules with = 1. Either the sech(") ond or the third rule in Figure 10.8 must be used in any valid derivation. Some comments are included in Figure 10.8 to give some insight into why the membership functions for that rule were chosen . For example, the rule B ~ 6F is used when a vertical line changes direction abruptly from up to down. This would indicate a sharply pointed crown and the letters i and t are reinforced while e and l are reduced in membership when this rule is used. After adjustment on the training set to allow a threshold of 0.5 or more to indicate "in the class" and less than 0.5 to indicate "not in the class", the grammars were used on a random sampling of 121 letters from the remainder of the patterns . The strings were parsed in a top down (left to right manner by a program written in SNOBOL 4 programming language . The results of this test are summarized in Figure 10.9 . Class E I L T Method 1 : % error 110 16 28 74 Method 2 : % error 10 4 5 27 Figure 10.99 : Results of the Experiment

Two methods of categorizing were tested. The first classified the letter into any class for which the pattern had a fuzzy membership of 0.5 or more. Some letters were not classified while others were classified into more than one class. The method was considered successful if the correct class was included among other classes since a contextual post-processor could be used to find the correct letter . The second method classified the pattern into the class that had the highest fuzzy membership . As expected, the second method had better results, with 90% of the e's, 96% of the i's, 95% of the l's, and 73% of the is correctly classified. The only distinction between a t and an l is the width of the loop. Since many of the is were quite wide, they were incorrectly classified as l's. If the presence of one or more is was detected by the presence of absence of a horizontal line written directly above some portion of the word, most of these incorrect classifications could be corrected by a contextual post-processor such as 9 F' igure

10 .9 is from [44], reprinted with permission from Academic Press .

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described in [54] . The distinction between the e's and l's could have been improved if the data were prescaled to eliminate differences in the average height of the letters generated by the different subjects . Considering the similarities in the four letters tested, the results were considered in [44] to be quite good. 10 .10

General fizzy Acceptors for Syntactic Pattern Recognition

In [62, 63], the syntactic approach to pattern recognition was examined by using formal deterministic and stochastic languages . In [175], fuzzy regular languages for pattern description in relationship with finite fuzzy acceptors were considered, where max-min was used as the composition of L-fuzzy relations. In [98], general fuzzy acceptors were considered using sup-*-composition for binary operations * preserving the properties concerning E-equivalence and which has min as the greatest lower bound . We consider [98] in the next two sections . Let (L, V, A, 0,1) be a complete lattice with upper and lower bound 1 and 0, respectively. Let f : L ~ [0,1] be an injective isotonic map, i .e., a one-to-one function such that u <_ v ==> f (u) <_ f (v) . Here and in the next section, we assume f is fixed. For x = (xl, . . . Ix.), Y = (yl, . . . y.) E Ln (7L E N), the distance IIx - y11 with respect to f between x and y is defined as VZ lIf(XZ)

- f(yZ)I-

As usual, an L-fuzzy subset of a set S is a function from S into L . Let L(S) = {/ t I /t : S ----> L}. Let I, J K be arbitrary nonempty sets. Any binary operation * on L can be used for the construction of the composition of L-fuzzy relations [50] as follows . Let /t E L(I x J), v E L(J x K) . Define the sup-*-composition /t o v by V(i, k) E I x K, ft o v(i, k) = V{lt(i,j) * v(j, k)

I

.j E J} .

Let a E L(I) be an L-fuzzy subset of I . Then an L-fuzzy subset a o ft of J can be defined by Q o /t(i)

= V{a(i) * N-(i,i) I

i E I},

Vi

E J.

If p E L(J), /t o p is defined similarly. Moreover for a' E L(I), a o a' can be defined to be a scalar . In order to consider the complete transition behavior of a general acceptor and how to compute it, the following definition is needed, [19, 91] . A triple (L, *, <), denoted by L, is called a commutative cl-semigroup © 2002 by Chapman & Hall/CRC

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if (L, V, A, 0,1) is a complete lattice with upper and lower bounds 1 and 0, respectively, and (L, *) is a commutative semigroup such that the following

property holds:

a * (VA) = V{a * b I b E A}, VA C L .

The real unit interval equipped with an averaging operator M = *, [141], gives an example of a commutative cl-semigroup as does a complete Heyting algebra with A = *Definition 10.10 .1 A fuzzy acceptor A over G is a six-tuple A = (Q, X,

where Q is a nonempty set of states; X is a nonempty set of input symbols; Q L is the initial state distribution ; -r : Q L is the final state evaluation; .M = {M(x) I Q x Q ----> L I x E X} is the set of transition functions; G is a commutative cl-semigroup .

t, T, .M, G),

(1) (2) (3) (4) (5) (6)

t

:

If * = A and Q and X are finite sets, then this is the same as the definition in [175] . We consider the complete transition behavior of an acceptor and how to compute it. We now extend the stepwise transition behavior of A. Let X be the set of input symbols . Let X l = X and Xi = X x X x . . . x X (j times), j E N. Let S[X] denote the disjoint union of the sets X l , X . . . . . . Then the elements of S[X] are sequences (xii, xi2, . . . , xi-), xii E X, m = 1, 2, . . . . Define a multiplication on S[X] by juxtaposition, i.e., (il ~ xi2 ~ . . . ,

xi-)

(

x

il

I

x72 l . . . , x7J

=

x

(

zl

I

xi2 l . . . ,

xim ~

X

il

~

x72 ~ . . . , x in

-

Then S[X] is a free semigroup with respect to this operation. Let ,F[X] denote S[X]U{A} . Then ,F[X] is a free monoid generated by X with identity A.

Every element of ,F[X] is called a word on X. Every word x E S[X] can be represented by x = xil x i2 . . . where xi2 , . . . , xi,, E X . Then x is said to have length k . Let A = (Q, X, G) be a fuzzy acceptor . For any input word x E S[X], the transition function M(x) is computed by the expression xik

,

Xil

,

t, T, .M,

M(x) = M(xiJ ° M(xi2) 0 . . . o M(xib)

I

where x = xilxi2 . . . E S[X] and where M(A) may be regarded as the identity. The expression represents the complete behavior of A in k consecutive steps . Every element M(x) gq , is the grade of membership if the input word is x E Xk with the beginning state q E Q at instant t and the last state q' E Q at instant t + k . The set of all transition behavior fuzzy relations describes the complete behavior of A and we use the notation xik

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1 0 . Applications

.M*(A) = {M(x) I x E .F[X]I . If x E .F[X], t o M(x) o T is the analytic extension of the stepwise behavior of A . Let L = Q = [a, b] be an interval . Then Vx E M, the transition function M(x) is a fuzzy subset of [a, b] x [a, b] . The initial state distributions and the final state evaluations are fuzzy subsets of [a, b] . Hence fuzzy acceptors may be extended to continuous types . In particular, if a state set Q is a subset of [a, b] such that QI = n for some n E N, then the transition functions can be represented as n x n matrices . The initial state distributions and the final state evaluations also are vectors of dimension n .

10 .11

E-Equivalence by Inputs

Definition 10 .11 .1 Let p, ft' : I x J ~ L, and let E E [0,1] be fixed. Then ft and ft' are called E-close, denoted by ftEft', if IIw(ij) - w'(i,j)II <_ E, V(i, k) E I x J. The following lemma and definition are needed to study the properties of E-closeness . Lemma 10 .11 .2 (95) Let f and g be bounded, real-valued functions on a set X. Then IVXEXf( x) - VxEXg( x)I <_ VXEX f(x) - 9(x)I ,

and InxEXf (x) - nxEXg(x) I < VxEX f(X) - 9(x)

Definition 10 .11 .3 A binary operation * on L is called contractive if I x * y - x/ * y/ I <-I (x, y) - (x" y') I, Vx, x', y, y' E L . Fuzzy acceptors with contractive operations * are called contractive fuzzy acceptors. It follows that min and max are simple contractive operations that are associative . To construct contractive fuzzy acceptors L, (L, *) must be a commutative semigroup . In [141], averaging operators between min and max are summarized, and pictorial representations are given . Averaging operators are defined as follows : An averaging operator is a function M : [0,1] x [0,1] ~ [0,1] such that the following conditions hold Vx, y E [0,1] : (1) x n y < M(x, y ) < x v y, M ~ {n, Vj ; (2) M(x, y) = M(y, x) (commutative) ; (3) M is increasing and continuous ;

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(4) M(o, 0) = 0, M(1,1) =1 . It is known that there exist no associative strictly increasing averaging operators [47] . An associative averaging operator that is not strictly increasing is, for fixed p E [0,1], y p x

M(x, y) = med(x, y, p) =

ifx
(10.5)

This median operator is contractive . It follows that min and max are the least and greatest contractive operations of associative averaging operators, respectively. The concept of fuzzy subsets has been incorporated into the syntactic approach at two levels . The pattern primitives are themselves considered to be labels of fuzzy sets, e.g., such subpatterns as "almost circulars arcs", "gentle", "fair", "sharp" curves are considered. Also, the structural relations among the subpatterns may be fuzzy, so that the formal grammar is fuzzified by the weighted production rules and the grades of memberships of a string are obtained by max-min composition of the grades of the productions used in the derivation . When the pattern primitives are extracted from an image with low quality or a deformed pattern, the min operation in the max-min composition of the grades of production is sensitive to distortion of primitives. In such a case, the above median operator, which is well known to be useful for noise suppression, preserves the grade of membership of primitives better than min operators . Consequently, supmed-composition rather than max-min may work well if the parameter p in (10 .5) is decided appropriately. Proposition 10 .11 .4 Let * be a contractive operation on L . Let /t, /t', v, v' be L-fuzzy relations. If compositions below make sense, then the following properties hold: (1) ftE/t' ==> v o y, o v'Ev o ft' o v' (invariance, (2) ftEft', vEv' ==> /t o vE/t' o v' (coordination) . Proof. Since property (1) can be easily obtained from (2), we only prove (2) . Let the L-fuzzy relations ft and ft' on I x J be E-close, and let the L-fuzzy relations v and v' on J x K be E-close. Then for V (i, k) E I x K, II(wov)(i,k)-(,t'Ov')(i,k)II

=

<_ <_ < © 2002 by Chapman & Hall/CRC

IIVjEJw(i,j)*v(j,k)VjEiit, (i ,j) * vi(j,k)II VjEJIIN'(i,j)*v(.l,k)w'(i, .j) * U'(j, k) II VjEJIIw(iJ) - w'(iJ)IIV I U(j, k) - v (j, k) I I E,

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10. Applications

where the first inequality comes from Lemma 10.11 .2 and the second inequality comes from the assumption that * is contractive . Thus [to vElt' o v' . To obtain an E-partition on X, we define E-equivalence, keeping in mind the definition for E-closeness . Definition 10.11 .5 Let A = (Q, X, t, T, .M, G) be a fuzzy acceptor and E E [0,1] . Then x, y E .F[X] are said to be E-equivalent, written xEy, if the following conditions hold : (1) x, y E Xk for some n E hY, (2) M(x)EM(y) . The following corollary follows from Proposition 10.11 .4(1) . Corollary 10 .11 .6 Let A = (Q, X, t, T, .M, G) be a contractive fuzzy acceptor and let x, y E .F[X] . If xEy, then t o M(x) o TEt o M(y) o T. m Proposition 10 .11 .7 Let A = (Q, X, t, T, .M, G) be a contractive fuzzy acceptor and E E [0,1] . Then the following properties hold: (1) if xEy for x, y E .F[X], then (uxv)E(uyv), VU, v E Xk ' (2) if xEx' and yEy' for x, x', y, y' E .F[X], then xyEx'y' . Proof. (1) Let xEy. Then M(x)EM(y) . By Proposition 10.11 .4(1), M(uXv) = M(u) o M(x) o M(v)EM(u) o M(y) o M(v) = M(uyv) .

Thus (uxv)E(uyv) . (2) The proof uses Proposition 10.11.4(2) and is similar to that of (1) . Note that the relation E-equivalence is reflexive and symmetric, but not transitive. However, an E-partition on X is needed for syntactic pattern recognition. Thus we need the following definition. Definition 10.11 .8 Let A = (Q, X, t, T, .M, G) be a fuzzy acceptor and E E [0,1] be fixed. For every x E X, an E-class [x] with the center x is defined to be [x] = {x' E X I xEx'} . A family of E-classes is called an E-partition of X if it consists of disjoint subsets that cover X. By choosing a suitable operation *, we can compute an E-partition on X through the following Algorithm from Section 8.9 and [174]. Algorithm 10.11 .9 1. Define a contractive binary operation . 2. For every x E X, compute t o M(x) o T. 3. Choose a center with the greatest value t o M(x) o T. 4. Construct the E-class [x] . 5. Replace X with X\ [x] . If X z,4 0, go to step 3. 6. Print all E-classes . © 2002 by Chapman & Hall/CRC

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499

From this algorithm, we compute an E-partition of X. The choice of the center x for an E-class [x] depends on the task under examination and on the user's need . To compute only an E-partition of X, it suffices to choose an operation that is contractive for each argument, i.e ., llx * y - x' * yll <_ Ix - X'11 and ll x * y - x * y'll < lly - y'll for all x_ x', y, y' E L . This yields (1) of Proposition 10 .11 .4 . For example, let * be Yager's t-norm with r > 1 *(x,y) = 1 - ( 1

n

x) T (1-

+(l -Y) T) .

Then l l0 * lOxll <_ 1 and l l0 * /dyll < 1 . This implies that it is contractive in each argument . Hence by using Yager's t-norm, we can obtain an Epartition on X . Proposition 10 .11 .10 Let A = (Q, X, G) be a contractive fuzzy acceptor. Let [v] = {v' E vEv'} and [x] = {x' E X l xEx'} be E-classes with center v E Xk and x E X, respectively . Then vx and xv are centers of E-classes on X k l . t,

Xk

T, .M,

l

+

Proof. The proof follows from Proposition 10 .11 .4(2) . Let X k IE denote the quotient set on X k , k = 1, 2, . . . . As a natural extension of Algorithm 10 .11 .9 by Proposition 10 .11 .7, we have the following algorithm by Proposition 10 .11 .10 . It forms an E-partition on X k for each kEN.

Algorithm 10 .11 .11 1 . Input k E N. 2. For i = 1, compute X/E = X l 1E by using Algorithm 10 .11 .9 . 3. If i > k, go to step 5 . 1 4. For i = i + 1, compute IE by using Proposition 10 .11 .10 . Go to step 3. 5. Print all E-classes of Xk IE . XZ+

10 .12

Fuzzy-State Automata: Their Stability and Fault Tolerance

Special branches of systems theory have achieved a high level of development and sophistication . Attempts to unify these branches have been made, [105, 23, 251, 178] . In the next five sections, we present the work of [36], which introduces geometrical and stability concepts from dynamical system theory into the theory of abstract automata, but without giving up the two main features accounting for the performance of automata or sequential machines, i .e ., discrete state spaces and sequential operation at consecutive discrete values of time . The concept of tolerance space, introduced in [178], is used . This concept is carried over to the state space of abstract automata .

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10. Applications

We first present a brief account of the elementary properties of tolerances and tolerance spaces in Section 10.13. We then introduce in Section 10.14 a special class of finite automata with given tolerance, the class of fuzzy-state automata . Their state transitions exhibit stability properties. We examine the class of fuzzy-state automata with respect to its order structure. In Section 10.15, stability of fuzzy-state automata is studied . The appearance of approximate fixed point and attractors in the state space of fuzzy-state automata and almost periodicity of their state transitions are studied . The application of the previously developed framework is dealt with in Section 10.16. For this, the field of fault-tolerant computing is chosen, i.e., the ability of certain real automata to execute, within a given tolerance, specific programs regardless of failures in their performance or hardware . The goal of [36] was to understand how biological systems work, in particular, how they control malfunctions . With this in mind this capability was simulated on an abstract level. With respect to fault tolerance, von Neumann noted the differences between biological and artificial systems . He states that in a biological system, not every error has to be caught, explained, and corrected ; see [239, p. 71] . 10 .13

Relational Description of Automata

The concept of a tolerance space is due to the authors of [178], [265], and [179], although these authors used different names for tolerance spaces . In [5] and [43], tolerance spaces were used in connection with automata and control theory. Some reasons for studying tolerance spaces are the nature of our perception of space and time, [178], and state space properties of man-made systems like digital computers . Strong similarities between tolerance spaces and topological spaces exist. There are substantial and very important differences, however . A relation on X is called a tolerance relation if it is reflexive and symmetric . A tolerance space (X, T) (or simply X) is a set X with a tolerance relation T on X. If (x, x') E T, we say x is within tolerance of x' and write xTx'; iX = X x X is called the big tolerance and SX = {(x, x) x E X} is called the little tolerance on X. Example 10.13 .1 (1) (N°, -) is a tolerance space, where NO = N U {0} and - is the relation {(m, n) I I m - nI < 1, m, n E N°}. (2) Inj = ({0,1, . . . , n}, i) is called the standard n-simplex. Let p, a C_ X x Y, S C_ X, and T and S be tolerances on X and Y, respectively. Then we introduce the following notation: (2) S - p = {y 13x E S, xpy} and xp = {x} - p. © 2002 by Chapman & Hall/CRC

10.13. Relational Description of Automata

50 1

(3) p - a = {(x, y) I Elx' such that xpx', x'ay} ; (P, p)T = p-1 - T - P, the image of T under p acting on Q x Q. Similarly, p*T = SY U (p, p) T and P*S=P .S .P-1 . (4) P l = 6X; P1 = P; P n = Pn-1 - P. (5) P\a = {(x, y) E P I (x, y) ~ a} ; P = Q x Q\P ; P = P° U P U P-1 . We use S (t) throughout for the little (big) tolerance and omit indicating the underlying tolerance space if it is understood .

Definition 10.13 .2 Let T and a be tolerances on X and Y, respectively. A relation p C X x Y, p :?~ 0, is called a fuzrelation (or fuzzy if p* T C_ a . A complete and univalued fuzrelation is called a fuzmap . If f : (X, T) ----> Y is a set theoretic map, f* T is called the coinduced tolerance on Y. If t : Y ~ X is an inclusion, the induced tolerance OT on Y is called the subspace tolerance of Y and (Y, t*T) (or loosely (Y, T) or Y) is a subspace of X. It follows that f* T is the least tolerance on Y such that f : (X, T) ~ (Y, f *T) is a fuzmap and it is the unique tolerance on Y that has the universal property that for all tolerances a on Z and for all set-theoretic maps g : Y (Z, a), f - g : (X, T) ~ (Z, a) is a fuzmap if and only if g : (Y, f* T) (Z, a) is a fuzmap, [179] . Similarly, if g : Y ----> (X, T) is a set-theoretic map, the induced tolerance on Y, g*T, is the biggest tolerance such that g : (Y, g*T) ~ (X, T) is a fuzmap . It has a universal property dual to that of the coinduced tolerance . Tolerance spaces and fuzmaps form a category denoted by Fuz, [179] . Example 10.13 .3 Let X = Y = N° and T = - = {(m, n) I Im - n1 < 1, m, n E NO } . Let p = {(m, m + 1) I m E NO }. Then (p, p)T =p -' T . p = {(1,1), (1, 2)} U {(m, n) I Im - n1 < 1, m E hY\{1}, n E NJ . (Note that

(0, n) ~ p-1 b'n E N.) Clearly, p*T = {(0, 0)} U (p, p) T and p*T C_ T. It follows that p*T = p . T p-1 = {(0,O),(0,1)1 U {(m, n) I I m - nl < 1, . Let f : N° ----> 7L be such that f (m) = m V'm E NO. Then m E N, n E N°} f*T = SZ U f-1 . T . f = SZ U T. Let t : hY ----> No be such that t(m) = m b'm E N . Then t*T = t - T -1 = {(m, n) I Im - n1 < 1, m E N, n E hY° } _ T\{(0, 0), (0,1)} .

Definition 10.13 .4 Let (XI, TI) and (X2, T2) be tolerance spaces . The union of (XI, TI) and (X2, T2) is defined to be the pair (X1 U X2, T = (L1 * T1)

U (t2*T2)), where tj is the injection tj : Xj ----> X1 U X2, j = 1, 2. Their product is defined to be the pair (X 1 X X2, T1 T2 = n(prj *Tj)), where pry is the projection pry : X1 x X2 ~ Xj , j = 1, 2 .

The tolerance T in Definition 10.13.4 is the least tolerance such that all injections tj are fuzmaps and TIT2 is the biggest tolerance such that all projections pry are fuzmaps, [179] . Definition 10.13 .5 Let x

E X. The tolerance neighborhood of x is defined to be the subspace N(x) = (xT, t*T), where t : xT ----> X is an injection. For A C X, let N(A) = UxEAN(x) .

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If p is a fuzrelation and xp :?~ 0, then N(x)p C N(xp). Let (X, T) and (Y, a) be tolerance spaces and let p C_ X x Y be a fuzrelation. Then p :?~ 01 and p* T C_ a. Let x E X. Then N(xp) = xpQ = Ely' E Y, (x, y') E p and (y', y) E al and N(x)p = {y E Y {y E Y I 3x' E OT, x'py} . Let y E N(x)p. Then there exists x' E OT such that (x', y) E p. Thus (y, x') E p-1 . Now xp z,4 0 so (x, y') E p for some y' E Y. Since (x, x') E T, (x', x) E T . From (y, x') E p-1 , (x', y) E T, and (x, y') E p, we conclude that (y, y') E p-1 Tp C a . Hence (y', y) E a . Thus y E N(x)p . Therefore, N(x)p C N(xp) . Example 10.13 .6 Let X = {x, y, z} and T = {(x, x), (y, y), (z, z), (x, y), (y, x), (y, z), (z, y)} . Then (X, T) is a tolerance space. Now xT = {x, y} .

Let t : {x, y} ----> X be such that t(x) = x and t(y) = y . Then t*T = tT6-1 = {(x, x), (x, Y), (y' Y), (y' x) }-

Definition 10.13 .7 Let g, f : (X, T) ~ (Y, a) be fuzmaps. The f and g are called homotopic, written g - f, if there exists m E N° and a sequence {FZ I i = 0,1, . . . , m} of fuzmaps FZ : (X, T) ~ (Y, a) such that the following conditions hold: (1) f = FO, g = Fm, (2) (FZ,FZ + 1)TCa, i=0,1, . . . ,m-1 .

A fuzmap f : (X, T) ~ (Y, a) is said to be null-homotopic if it is homotopic to a constant map and two tolerance spaces X and Y are homotopy equivalent if there exist fuzmaps f : X ~ Y and g : Y ~ X such that f - g - SX and g - f -_6Y. There is a natural metric on a tolerance space (X, T), namely, the hopmetric d : X x X ~ N° U{oo}, where d(x, x') = A{m 13 a fuzmap w : (N O ,--) ~ (X, T) such that w(0) = x, w(m) = x'} and d(x, x') = oo if there is no such w . Definition 10.13 .8 Let A, B be subspaces of (X, T) . The interior of A, written intA, is defined to be the set, intA = {x E A I d(x,X\A) > 1}, where d(x, A) = nx,EA{d(x, x')} . The boundary of A, written as OA, is

defined to be the set OA = N(A)\intA . The tolerance space X is called T-connected if for all (x, x') E X x X, d(x, x') :?~ oo and is called contractible if it is homotopy equivalent to a point, i.e., to a tolerance space (Y,6) with IYI = 1. The component C(x) of x' in X is the maximal Tconnected subspace of X containing x' .

Disregarding transition probabilities, we represent the state transitions of an automaton and its outputs by its next-state relation A C_ Q x X x Q and its output relation w C_ Q x Y. Q is the nonempty set of states of the automaton, X its input, and Y its output alphabet . The quintuple A, = (Q, X, Y, A, w)

© 2002 by Chapman & Hall/CRC

10.13. Relational Description of Automata

50 3

is called an (abstract) automaton with output. The triple A = (Q, X, A)

is called an (abstract) automaton. Moreover, A is assumed to operate sequentially on a discrete time scale. Definition 10.13 .9 An automaton A is called deterministic if V(q, x) E

Q x X, (q, x, r), (q~, x~, r~) E Q x X x Q, q = q' and x = xt implies r = r~ . It is called complete if V(q, x) E Q x X there exists r E Q such that (q, x, r) E A and finite if IQ I is finite .

In general, A is a nondeterministic automaton . Let x E X. Let S x C_ Q x Q be such that (q, q') E S x if and only if (q, x, q') E A. For computational convenience, the relation A is often given as a family D of binary statetransition relations : D = {Sx I x E X} . Definition 10.13 .10 Let x = xlx2 . . . x n E X*, where x2 E X, i =

1,2 . . . . n. Then the (state) transition relation of Ax under the action of input word x is defined as SA = S, S x = Sxl .6X2 . . . ' .6 X,, , its output relation as wx = 6X , w, and Xl = {x E X* I Ixl = l}, Ixl the length of x .

A conventional function-type description of an automaton appears in many cases not to be adequate when dealing with complex systems . For example, transition modification and memory errors may turn a deterministic automaton into a nondeterministic one. Hence a relational description of automaton has been chosen here. Definition 10.13 .11 An automaton A =

is called a sub _ automaton _ of an automaton A if X = X, Y = Y, Q C_ Q, 0 = A f1 (Q x X x Q), and w = w rl (Q x Y) .

Every subset of Q determines a unique (in general incomplete) subautomaton of A. For example, a special subautomaton of A is given by the successor set of state q E Q, where we recall that this set is defined by R(q) = 17({q}) = {q' I 3x E X* such that q' E q6x} . Recall that a cover on Q with the substitution property is a family S = {QZ I i E I} (I an index set) of nonempty distinct subsets Qj of Q such that UiEIQi = Q and for all x E X and for all Qj E S, there exist Qj E S such that Qj " Sx C Qj . Definition 10. 13 .12 Let A = (Q, X, A) be a complete finite automaton and S a cover on Q with the substitution property . The state-transition relations of the automaton A/S = (X, S, AS) are defined as follows: for all x E X and Qj,Qj E S, QZSSQj if and only if Qj - S x C Qj .

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10. Applications

Definition 10.13 .13

Let A = (Q, X, A) be an automaton and T a tolerance on Q . Then (A, T) is called an automaton with tolerance . The tolerance space (NO, -) is considered as the underlying time set of (A, T) .

Let a be a tolerance on the output alphabet Y of Au expressing indistinguishability of outputs and let w be a set-theoretic function . Then (A, w*a) is an automaton with natural tolerance . As another example, let M be a digital computer whose states are given by the contents of its registers. Two states of M may be defined as being within tolerance if and only if they differ in the contents of a limited number of registers [5].

10 .14

R1zzy-State Automata

In this section, we introduce a class of automata with tolerance whose state transitions exhibit stability properties . Definition 10 .14 .1

An automaton with tolerance (A, T), A = (Q, X, A), with fuzzy state-transition relations is called a fuzzy-state automaton.

We prove in the following lemma that the successors of states within tolerance of a fuzzy-state automaton all stay within tolerance under the action of any input word. We make use of this property when we analyze the effects of temporary and permanent state errors on the performance of fuzzy-state automata . Lemma 10 .14 .2

Let (A, T), A = (Q, X, A), be an automaton with tolerance . (1) Then A is a fuzzy-state automaton if and only if Sx* T C_ T for all XEX . (2) Suppose that A is deterministic and complete. Then (A, T) is a fuzzy-state automaton if and only if T C_ S x *T for all x E X . (3) Both formulas for T hold for all x E X if and only if they hold for all xEX* .

Proof. (1) The proof here is immediate. (2) Since T C_ S x *T and Sx1 - S x C_ S, we have that Sx 1 - T - S x C_ Sx 1 Sx . T Sx 1 S x C_ T . If A is a complete fuzzy-state automaton, then T C_ Sx . 6X 1 - T Sx . 6 X 1 C_ Sx - T - Sx1 for all x E X. (3) The desired result follows from the fact that (Sx . 6v)* T = bx*(6v*T) and (Sx - Sy ) * T C Sxx (6v. T) for all x, y E X . Lemma 10.14.2(1) can be simply restated as follows : a fuzzy-state automaton if and only if V'x E X, (q, x, q~) q~ ) q~ ) (q~~, x, .. E A implies (q~, . . E T . © 2002 by Chapman & Hall/CRC

A = (Q, X, A) is E A, (q, q") E T,

10.14. Fuzzy-State Automata

505

Lemma 10.14.2 implies that the transition relations of a fuzzy-state automaton are fuzrelations for all x E X* and thus are metric decreasing, i.e., V'x E X*, q,4 E Q, d(q, 4) >_ d(q*, q*), where q* E q6x and q* E q6 x. This clearly represents a stability property. Let A = (Q, X, A) be a finite automaton . Let F(A) be the set of all fuzzy-state automata with A as first coordinate. Let F(A,T) = {(A, a) E F(A) I a C T, a a tolerance on Q}. Then F(A) forms a complete, distributive lattice with respect to the ordering of automata with tolerance given by (A, a) <_ (A*, T) if and only if A is a subautomaton of A* and a C T. This follows from the next result . Theorem 10.14 .3 Let A = (Q, X, A) be an automaton. Then the follow-

ing assertions hold . (1) The partially ordered set FA of symmetric and reflexive binary relations p on Q with 6 x* p C p for all x E X (ordered by set inclusion) forms a complete, distributive lattice, which is a sublattice of the lattice of all tolerances on Q . (2) FA is closed under the relative product if A is complete . FA is then a lattice ordered monoid if A is complete and deterministic.

P1

Proof. (1) The proof follows from [253] . (2) If A is complete, 6 C_ 6 x '6x 1 , 6x 1 'P1'P2'6x C 6x1'P1'6x'6x1'P2'6x C_ 1 ' P2, P1, P2 E FA . If A is deterministic, 6x - 6 x C 6, 6x 1 - 6 - 6x C 6, and

6x .6=6 -6x=6XVxEX .

It follows from Theorem 10.14.3 that F(A, T) 0 has a unique maximal element . This element is denoted by (A, T*) . It plays an important role later. The unit of F(A) is (A, t) . If A is deterministic, F(A) has zero (A, 6) and so F(A,T) :?~ 01 . Example 10.14 .4 We consider coarsening of a complete, finite fuzzy-state automaton (A, T) . The tolerance T defines a (unique) cover ST on Q with the substitution property as follows: Let QZ C Q. Then QZ E ST if and only if (a) qTq* for all q, q* E QZ and (b) if qTq* for all q* E QZ, then q E QZ .

The coarsening of (A, T), namely A = (AIST ,'r), is a fuzzy-state automaton, where ~r is defined below. In order to construct tolerance T, define p C ST x Q by QZpq if and only if q E QZ . It follows that (Sx, 6x)p C p for all x E X. Define ~r as T = p*T . Clearly, T is reflexive and symmetric and 6x* ~r C_ ~r b'x E X since (6x, 6x)T C 6X` - P - 6x . 6x 1 - T - 6x . 6x 1 - P -1 6x C P - T P -1 . The automaton with tolerance appearing in the next result is a generalization of the concept of a tolerance automaton, introduced first in [5]. Example 10.14 .5 Let (A, T) be an automaton with tolerance such that Pr13A C T. Then (A, T) is called a tolerance automaton. © 2002 by Chapman & Hall/CRC

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10 . Applications

We now prove the following statement : If (A, T) is a finite tolerance automaton, then there exists n E hY with 1 <_ n <_ IQ I -1, such that (A, Tn) is a fuzzy-state tolerance automaton : (a) Since T = S U T, it follows by induction that T' C_ T'+ 1 , m = 1, 2, . . . . Therefore, there exists n E hY U {0} such that Tn = Tn+j, j 0, 1, 2, . . . , since Q x Q is finite . Moreover, since T can be represented by a IQI x IQI Boolean matrix having only ones as diagonal elements, T has a characteristic exponent n < IQI - 1 . (b) Let qT'q~, q* E pr2(gA), and q~* E pr2(g0) . Then q*Tq, q~* Tq~, and so q*T2 +m q'*, i .e ., Sx* Tm C_ T 2 +m V'x E X . Let n be as in part (a) . Then Sx* Tn C_ T2+n =T n for all x E X . Thus (A, n) is a fuzzy-state automaton with pr13A C T C Tn . Any state of a tolerance automaton is in tolerance with its predecessors and any input word x of a tolerance automaton defines a fuzmap

Wx : (N U {0},

i.e.,

a path in

Q.

Q,T ,

') ---->

) The "phase space velocity" c = d(g,,ge of A is less than or

equal to 1 . In this sense, a tolerance automaton has inertia that gives rise to stable behavior . The example closing Section 10 .13 is also an example of a tolerance automaton.

Example 10 .14 .6 The automaton A* given in Table 10 .1 determines 64 automata with tolerance . This is easy to see from the following reasoning : Since there are 4 states, a tolerance can be represented by a 4 x 4-matrix whose entries consist of 0's and 1's. Since a tolerance is reflexive and symmetric only 6 positions of the matrix determine a tolerance . Thus there are 64 = 2 6 possible tolerances.

Table 10 .1 : Next State Relation of A* inputs of A*

A* x1 x2

qo qo q2

q1 qo q2

q2 q1 qo

q3 q1 qo

states of A* next states of A*

F(A*) is shown in Figure 10 .10. (A*, T19) is the only nontrivial tolerance

© 2002 by Chapman & Hall/CRC

10.14. Fuzzy-State Automata

50 7

automaton of F(A*) .

Figure 10 .1010 : Lattice F(A*) The following Boolean matrices (with respect to the ordering of the states given by (qo, ql , q2, q3)) represent the basic tolerances of F(A*) . T2

1 0 0 0

0 1 0 0

1 1 1 0

1 1 0 0

T5

T3

0 0 1 1

0 0 1 1

1 1 0 0

1 1 0 0

1 0 1 0

0 0 0 1

1 1 1 1

1 1 0 0

T7

0 0 1 0

0 0 0 1

1 0 1 0

1 0 0 1

"Figure 10 .10 is from [36], reprinted with permission by Kluwer Acadernic/Plenuin Publishers .

© 2002 by Chapman & Hall/CRC

10. Applications

We now determine ( A ,T * ) . The following procedure determines the maximal symmetric binary relation p, on Q , with the property that (S,, S,)p, g p, g T V x € X . Thus ( A ,p,) = ( A ,T * ) (unit of F ( A ,T ) ) if F ( A ,T ) is nonempty, e.g., if A is deterministic. Define the relation p(k) as follows: Step 1: p(1) .- T . Step 2: For all k 1, qp(k l ) p if and only if V x € X U {A) and V(qf,p f ) E Q x Q it follows from qS,qf , pS,pf that qfp(k)pf. Step 3: If p(k 1) = p(k) go to Step 4, else go to Step 2. Step 4: p, = p(ko),where Lo is the smallest index k with p(k) = p(k+l). Clearly, p, is symmetric since T is symmetric and p, is the maximal relation with the properties stated above since p g T implies that p, g p, . Consequently, a g p, if (S,, S,)a g a V x E X .

>

+

+

Example 10.14.7 Consider Figure 10.10. Let

Then qoSz1qo and qzSxl ql , but (40,ql)$ p ( 1 ) Thus (q0,q2) $ 4 2 ) . Moreover, qlSx1qo and qsSxlqi. Since (qo,qi) $ p ( l ) , (qo,qs) $ p(2). Now (q2,qs) € ~ ( 2since ) qzSxlqi,qsSxlql, qzS,,qo, and qsS,,qo. It follows that ( A ,T * ) $ F ( A * ) and p(2) = p(3) = ~ 2 Hence . ( A * ,T * ) = ( A * ,~ 2 ) . Consider an automaton with tolerance ( A ,T ) . Tolerance T* determines the maximal set of pairs of automata states that are in tolerance T and whose successors under any input sequence are also in tolerance T . Furthermore, if two states are not related by T* , then none of their predecessor pairs contains states within tolerance T* since T* = L\T* is reflexive, symmetric, and 6, *T* C T* V x E X , [253].

10.15

Stable and Almost Stable Behavior of Fuzzy-State Automata

In this section, we consider questions about the stable behavior of finite fuzzy state automata that have counterparts in the theory of topological dynamical systems. With this in mind, we state the following definition.

© 2002 by Chapman & Hall/CRC

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Definition 10.15 .1 Let (A, T), A = (Q, X, A), be an automaton with tolerance, S a subspace of Q, and V C_ X* . The subspace S is said to be

V-stable if its neighborhood N(S) is V-invariant, i.e ., if N(S) .6, C N(S) Vv E V. S is said to be almost V-stable if there is a natural number l such that N(S) f1 UZ-ogbv+' z,4 0

holds for all n E N U {0}, q E S, and v E V. If S is X* -stable and accessible from every state of T C Q, then S is called an attnacton set of T.

Clearly a set S of automata states is almost V-stable if its neighborhood is V-invariant, i.e., if S is V-stable. Stable and almost stable sets of automata states characterize the recurrent state motions of automata with tolerance. The property of attractor sets can be illustrated in the following manner . Let automaton (A, T) be driven by a stationary random input source that generates words such that the probability of any input x E X following an arbitrary word is greater than k, where k > 0. Let A be at time to = 0, say, in T, and p(S, t) be the probability of its state at time t belonging to the neighborhood of an attractor set S of T, if it exists . Then p(S, t > I R(T) I) >_ k~R(T) 1-1 and p(S, t > I R(T) I) = 1 for an autonomous automaton (hence the name "attractor set"). Clearly, Q is an attractor set of any T C_ Q. The set of reset states of an identity-reset automaton with tolerance S is an attractor set of its state set and every state of automaton (A*, TO is almost X-stable with l a = 0, lb = h = ld = 1 (see Figure 10.10). In the following, we assume that the automaton A = (Q, X, A) is complete and deterministic . The following theorem is an adaptation of Poston's "approximate fixed point theorem" to finite automata . Theorem 10.15 .2 Suppose that the state space of a finite, fuzzy-state au-

tomaton is contractible . Then b'x E X*, it contains an x-invariant and x-stable subspace, Px :?' 0, whose elements are mutually within tolerance T (approximated fixed points .

Let automaton A be autonomous and Q be the union of contractible, Sxconnected subspaces, where x E X. Let T be the natural tolerance on Q with respect to an output relation w . Then Px (which is maximal with respect to the properties of Theorem 10. 15 .2) is an attractor set of Q . Now p(Px , t) = 1 for t sufficiently large . Because of unobservable (unmeasurable) differences of outputs, automaton A seems to be caught in a final state. However, this may not be the case if IPx I > 1 . Another example of an attractor set can be determined in the following manner . By Lemma 10 .15.3, for any q E Q, 00R(q) is an attractor set of R(q) if 00R(q) :?~ 01 and if intR(q) is empty, or else strongly connected, and Q\R(q) is X-invariant . Since R(q) is X-invariant, OR is then X*-stable and either q E 00R(q) and so R(q) C_ 00R(q), or else q E intR(q) . However, this implies that 00R(q) is reachable from any state of © 2002 by Chapman & Hall/CRC

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10 . Applications

Clearly, both conditions hold if all R(q) with 00R(q) :?~ 01 are strongly connected, e.g., if (A, T) is a fuzzy-state permutation automaton . R(q) .

Lemma 10.15 .3 Let (A, T) be a fuzzy-state automaton and S be a subspace of its state space. Then the boundary r9S of S is X*-stable if S and Q\S are X-invariant.

Proof. If Ixl = 0 or r9S = 01, then r9S is x-invariant since SA = S . Suppose r9S is x-invariant for all x with xl < n and r0S :?~ 01 . (1) If q E r9S\S, then there is a state q E S such that (q, 46 x) E T for x E Xn . Thus (q6x , _qS xa ) E T, q6x E S, and -qSxa ~ S, i .e. , q6x E r9S for all YEXn+1 . (2) If q E OS n S, then there is a state q ~ S such that (q, q6x) E T for x E Xn . It follows that (4Sa, g6xa) E T for all a E X, i.e., d(4Sa, g6xa) < 1 . Suppose that g6xa E intS. Then 4Sa E S and R(q) n S z,4 0. This is a contradiction since 4S a E S. Hence g6xa E S\intS . Since (A, T) E F(A), r9S is X*-invariant and X*-stable . Lemma 10.15 .4 Let A be a complete, deterministic, connected, and au-

tonomous automaton with tolerance T. Then (A, T) is almost periodic if and only if every state of A is in tolerance with a periodic state of A.

Proof. Let q E Q. Let X = {x} since A is autonomous . The orbit of q under x, i.e., O = {q6x}°_o, is the union of two sets Ot and OP (p = ~QPj), where OP is nonempty and permuted by S x and Ot .67x C OP for s >_ t = Ot . (1) Let qTq, q be periodic. Then q E OP . There exist natural numbers r and s < p such that q6x = q Sx since A is connected and deterministic. Hence q6x+P-S E N(q) and Uz±O

That is, if

q

-s-l

+Z

gbx

n

has period p, then

N(q) :?~ 0,

p+r-s-1. (2) Suppose that

q

q

n = 0, 1, 2, . . . .

has an almost period not greater than

E Q is almost periodic. Then it follows that N(q)

n

UZ-ogbx+2

7~ 0.

Thus N(q) n OP z,4 0. Hence N(OP) = Q if A is almost periodic . Furthermore, N(Op) - Sx C N(O p - Sx) = N(Op) if (A, T) E F(A) for A arbitrary. An automaton (A, T) is said to be almost periodic if every state of A is almost X-stable . The connection between almost periodic and permutation automata can be seen from the next result. Theorem 10.15 .5 The state space of a deterministic, almost periodic fuzzystate automaton is the union of a finite number of neighborhoods of closed stable orbits (cycles .

© 2002 by Chapman & Hall/CRC

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51 1

Proof. The proof follows from part (2) of the proof of Lemma 10 .15.4. The greatest common divisor of the lengths of all proper cycles of a deterministic automaton A is called the period of A, [75] . Let d divide the period of A. Let 7r be a path from state q to state q' and I7rI denote its length. The distance from q to q', written I q, q' I d , is defined as I7rI (modulo d) if q and q' are connected and is defined to be oo otherwise . It follows that this distance is unique . Assume that the period of A is greater than 1. Now tolerances Td on Q may be defined as follows. For all q, q' E Q, gTdq' if and only if q = q' or I q, q' I d = s or d - s. Automata (A, Td are almost periodic. Moreover, (A, Td) is a tolerance automaton and (A, A) is a fuzzy-state automaton if the period of A is even and A is complete. Example 10.15 .6 Let the automaton A be as specified in Table 10 .2. Table 10 .2: Next State Relation of A 0

xl

x2

qo

qi

q2

g3

q4

qi qi

q2 q4

qi qi

go, q2 q2

g3 g3

The lengths of the proper cycles are 2 and 4 . Thus d = 2 . Now Iqo, gsl2 = 3 (mod 2) = 1 and Iqo, g4I2 = 2 (mod 2) = 0. For s = 0, s :A Iqo, 8312 :A d - s . For s = 1, Iqo, 8312 = s. Hence go T2 q3 . For s = 0, ° Iqo, g4I2 = s . Thus qOT 2 q4 . For s = 1, Iqo, g4I2 z,4 s or d - s . It follows that the tolerances T2 and T° are represented by the following Boolean matrices. 1 1 0 1 0

1 1 1 0 1

T12

0 1 1 1 0

1 0 1 1 1

0 1 0 1 1

1 0 1 0 1

0 1 0 1 0

To2

1 0 1 0 1

0 1 0 1 0

1 0 1 0 1

An effective procedure for evaluating the period of any finite automaton is given in [75] . 10 .16

Fault Tolerance of Fuzzy-State Automata

An actual machine occasionally makes errors computing its next state . Consequently, it is unreliable to some extent We now consider this unreliability. It is possible to emphasize the essential aspects of more concrete situations within our abstract framework. A machine exhibits stability in some sense if it overcomes the influence of its errors, i .e., if after some time these influences become "tolerable ." We will show that fuzzy-state automata behave stably in this sense with respect to certain faulty state transitions . © 2002 by Chapman & Hall/CRC

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10. Applications

Let A = (Q, X, A) be a (not necessarily deterministic) finite automaton, T a tolerance on Q, and QP = pr3A . Definition 10.16 .1

A binary relation 0 on Q with pr1O _D QP together with tolerance OTT, where t : 0 C_ Q x Q, is called a state relation of (A,T) . A state relation 0 is called compatible if 0 C_ T, inessential if 0 C_ p,r (cf. Section 10 .14, and consistent if it is fuzzy . We say that 0 changes q if &\S) z,4 0 . 0 is called (p, l)-bounded (by (A, T)) if for all xEX* with 1xl >p, (10.6) and (10 .6) does not hold for (p - 1, l) or for (p, l - 1) . If 0 is (p, l)-bounded, then 0 U 0-1 is also (p, l)-bounded . In general, the bounds of two state relations 0 and 0+ are known, it is usually not difficult to derive bounds for relations such as 0 U 0+ , n + , and 0 - 0+ . For example, 0 U 0+ is (max(p, p+), max(l, l+))-bounded . We present below an algorithm for determining the bound of a state relation. State transition errors may be described by a state relation . We then visualize an element (qj,qj) E 0 as follows . Automaton A goes with some nonzero probability into state qj when it should go into state qZ, due to a permanent or temporary modification in the state transitions of A. Clearly, error e E 0 n S is an improper error . if

o o

Definition 10.16 .2

A state relation (0, t*TT) is called a permanent transition modification (t-modification if it modifies automaton A, i.e., if automaton AO = (Q, X, AO := A - 0) replaces automaton A. Automaton A" is called the modification of A due to 0 and A the reference automaton. The state relation 0 is called a (temporary) error of A if A is not modified by 0 .

Most input errors can be interpreted as state transition errors, i.e., memory errors, [84]. The is also true for errors in the combinational output logic of sequential circuits . Consequently, we concentrate on memory errors. Within the relational framework, we are less interested in the physical causes of errors, but rather in the qualitative aspects of the role errors play in the performance of modifiable systems . In what follows, we assume that the meaning of tolerance T is that a single error e E 0 is tolerable in some appropriate sense if e E T. Examples can be found in [37] . Lemma 10.14.2 states that compatible errors of a fuzzy-state automaton (A, T) are inessential and remain inessential under the action of any input word. The maximal compatible state relation of (A, T) is TQ = (T, TT), and (p,r ,TT) is its maximal inessential state relation if pr1pr _D QP . Since the tangent bundle of the tolerance space (Q, T) is the composite map t TQ C Q x Q -PTA Q, [179], and the tangent space Tj Q to Q at state © 2002 by Chapman & Hall/CRC

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q is the tolerance space TqQ = (tq, TT), the compatible changes of state q determine the tangent space at q in a sense . In [37], such "geometric" properties of errors and t-modifications are studied . There modification tolerance of automata is considered, i.e., with masked and correctable tmodification of an automaton. However, in the following, relational (settheoretic) properties of temporary errors are considered. We now consider bounded state errors . Fault tolerance is an important design parameter . The goal is to design systems that stay operational despite failures and, in fact, can repair themselves in response to their own failure . Intuitively, automaton (A, T) overcomes the influence of error 0 if after some time it takes a state that is, and from then on stays, in tolerance with the correct state . Definition 10.16 .3 An automaton (A, T) is said to T-correct (to correct error 0 if there exists p E hY such that 0 is (p,1)-bounded ((p, 0)-bounded by (A, T) . Correction (self-synchronization) by deterministic finite automata has been studied in [85], [84], [42], [224], and elsewhere . Algorithms have been given for determining which temporary state errors are corrected by a de terministic automaton within a certain amount of time (assuming tacitly that these errors do not alter state transitions .) Correction of input errors has been studied in [252] and [84] . There is a strong connection between the capability of correcting an input error and that of correcting the temporary state errors caused by this input error . Example 10.16 .4 Consider the fuzzy-state automaton (A*, T5) given in Figure 10 .10 . The errors 4'1 and 02 are given by 0 0 0

1 0 0 0

0 0 1 0

0 0 0 0

_

The error 4'1 is (0,1)- and (1, 0) -bounded since E F(A) . The error 02 is (1,1)-bounded .

0 0 1 1 4'1

0 0 0 1

1 0 0 0

1 1 0 0

is compatible and (A*,

T5)

Bounds of an error and error-correcting input words can be determined by the following error graph procedure as long as the next state relation of an automaton is not too large . Error graph. (1) The vertices are given as (l/ab), a, b E Q, if aTl b and as (-/ab) if (a, b) ~ TL , l = 0, 1, 2, . . . , ((0/qq) - (q)) . (2) An oriented i-edge points from vertex (l/ab) to vertex (l'/cd) if and only if the (unordered) state pair {a, b} goes into state pair {c, d} under © 2002 by Chapman & Hall/CRC

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10. Applications

input x2 . The error graph of (A*, T5) is given in Figure 10.11 .

2

Figure 10 .11 11 : Error Graph of (A*,

T5)

Test A. The following algorithm gives a test by which one can use to decide whether or not a given temporary error 0 of automaton (A, T) is (p, l)-bounded and, hence, T-connected by (A, T). Step 1: p := 0; l P := Q I . Step 2: Op := U1 w 1 -p (6,,, 6w) W; Op (1) ~_ OP ; OP (k+1) :_ OP (k)U{(q, q+)

Elx E X, (q, q+) E OP(k) such that g6 x q, q+6x q+) . Step 3: 0P+ := Uk>10P(k) . Step 4: If there exist 1 1 E hY with 1 <_ 1 2 < lP such

lP

= min{l2J and

that

0P C_ Tl t ,

0 is (p, 1 p )-bounded, else l p := lp - 1 . Step 5: p := p + 1 if p <_ IQI2 - 1 go to Step 2, else stop and

set

unbounded.

It follows that 0P, is the minimal binary relation on Q such that (i) 6X)O+ C_ for all x E X*, [253]. Thus for all z = xv Op C 0P ; (ii) (6x, 0P with Ixl =p, C 0P . In order to determine if automaton A recovers from error 0 after a finite amount of time, it must be determined if there is some p, 0 <_ p < IQI 2 - 1, such that 0 is (p, l)-bounded. If 0 is T-corrected by (A, T) for A complete, then it is also T-connected by any deterministic automaton (A, Q) > (A, T) . The precedessors of a single error that is not "Figure 10 .11 is from [36], reprinted with permission by Kluwer Acadernic/Plenuin Publishers

© 2002 by Chapman & Hall/CRC

10.16. Fault Tolerance of Fuzzy-State Automata

51 5

T-corrected by a fuzzy-state automaton (A, T) are also not T-connected by automaton (A, T) (see the comments at the end of Section 10 .14) . The capability of an automaton to recover from errors is a strong restriction on its state transitions and tolerance . Somewhat weaker notions are given in the next definition . Definition 10 .16 .5 Error 0 is 1-corrected (eventually 1-corrected) by automaton (A, T) if for all e E 0 (and all z E X*) there is a word x such that (Sx,6x)e n T l :?~ 0 ((Szx,szx)e n Tl :?4 0) . Example 10 .16 .6 Consider the automaton (A*, TO . The error i is eventually 0-corrected by A* since x = xlxl is a reset word of A* (cf. Figure 10.11 . Test B : In order to decide whether a temporary error 0 is eventually 1-corrected by a fuzzy-state automaton (A, T), the maximal eventually 1corrected set of state pairs 0 of A is constructed by the following procedure:

nT 1

Step 1 : Construct relation 0 L ,= {(q, q) 13x E X* such that (Sx , 6x ) (q, q) ~ 0} . If (A, T) E F(A), then `Y1 C 02 C . . . and O j = Oj+1 for some N ; 0o := 0' .

_ and Sx 1 . 0 + . Sx C 0 + such that C 1 m ; M_1 for all x E X . Since 0o _D m 01 __ . . . , "fit7='fit+1 for some t E hYU{0} := 0i . Then error 0 is eventually 1-corrected if and only if 0 C 0.

Step 2 : Construct relation

The proof of the next result can be obtained by modifying the proof of a corresponding result in [252] . Theorem 10 .16 .7 Let A be a finite deterministic automaton. An error 0 of A is eventually 1-corrected if and only if A is 1-synchronized with respect to 0. That is, A driven by a random source and started either in state q or else in a state q E qO is brought, in the long run, into states within tolerance Tl . However, if A is a fuzzy-state automaton, its correcting input word can be chosen independently of e E 0, and thus 0 is (p, l)-bounded for some l E N . We now present a more general result . Theorem 10 .16 .8 All temporary errors of a complete fuzzy-state automaton (A, T), A = (Q,X,A), are 1-correctable, for some l < IQI - 1, if and only if there is a (generalized) reset x E X* for automaton A such that Qx C C(q), q E Q, where Qx := Q - Sx . Proof. It follows that Qx C_ C(q) if and only if there is some r E NU{0} such that Qx C_ qTT . Thus since Q x C_ C(q), q E Q, we have that for any error 0, T2T C T IQI-1 . (Sx, 6 x)4' C (+ 1 - t - (Sx C T' q x qTT C

© 2002 by Chapman & Hall/CRC

516

10. Applications

Suppose that any error of A is 1-correctable . Let ~Q1 = n and Qr _ {qZ}z, o, 1 <_ m <_ n . There exists x E X* and qZ E gibx, i = 0, 1, such that gxT l gx, gi6x C qZ T C_ gOxTl+ 1 since Sx is fuzzy. Thus Qi C goTi+1 . Suppose now that there is some x E X* such that Q~ C gO T1 +1 , 1 < m < n, where qZ E g Z S x . Let be 1-corrected by E X*. There exists q2 E gj6x, i = 0,m+1, such that go Tlgm+1 . Again for qZ E Q,, gj6xx C (gOTi+1 ) 6x C_gOTl+1 and q, + 1Sxx C qL+1 T and Q;,~+1 C_ go Tl + l . This implies that there exists xEX*,gEQsuch that QXC4T 1 +1 . If Qx is T-connected or if Q is contractible, then there exists q E Q such that Qx C_ C(q) . Hence all errors of A are 1-correctable for some l <_ IQI - 1. For topological spaces, Q is contractible if and only if S = SA is null-homotopic . The following result gives a converse. Corollary 10 .16 .9 If all temporary errors of an automaton A are 1-correctable, l < 2, then (Qx,i*T) is t-connected and (Q, T) is contractible in addition, S x - SA (since then Sx - c, c a constant map, and Qx = qTl+1) .

if,

Moreover, if all errors of an automaton A are 1-correctable, then every error is also eventually 1-corrected and so A is 1-synchronized with respect to every error . Modification tolerance of abstract automata and the problem of finding meaningful tolerance between automata states are studied in [37]. These tolerances should be based on the minimal amount of time needed in order to correct state errors . The approach presented here may also be extended to failure tolerance with respect to spatial propagation of errors in iterative networks .

10 .17

Clinical Monitoring with mata

R1zzy

Auto-

A framework for an intelligent bedside monitor is presented in this section. The material is from [222]. The purpose of the monitor is to derive an abstraction of the current status of a patient by performing fuzzy state transitions on pre-processed input continuously supplied by clinical instrumentation . An implementation called DiaMon-1 has been used for off-line evaluation of data of patients suffering from adult respiratory distress syndrome . Intensive care monitors display more and more information as new devices for on-line sampling of physiological data become available . Hence the clinical staff is faced with the problem of monitoring the monitor . It is difficult for humans to perceive and interpret a large number of timevarying parameters, [38, 86, 238] . If parameters interact in such a way that only certain groupings provide hints for critical conditions, then the situation becomes more complicated. When the meaning of a value depends © 2002 by Chapman & Hall/CRC

10.17. Clinical Monitoring with Fuzzy Automata

517

on the patient history, another complicating factor occurs. Context-specific alarms for which no absolute thresholds can be established provide a good example of this [38, 214]. In [222], a formal framework was presented for the design of monitors that have the following properties: (1) abstraction from objectively observed (quantitative) parameters to (qualitative) stages of a disease, (2) early indication of improvement in or deterioration of the patient's state by providing smooth transitions between stages, and (3) consideration of previous events, i .e., history-based interpretation of data. A state monitor traces the patient's change of state with time, i.e., that records the progress of his illness . A state is considered to be an abstraction of the patient's status that accounts for a specific stage of a disease . Possible paths from one state to the next are provided by the transitions . The transitions depend on input, i .e., events that need to occur or conditions that need to be satisfied for a transition to take place. The input is obtained by processing objectively and preferably automatically acquired data. A state monitor is an abstract model of the medical knowledge in a specific area. Medical decision making is based on knowledge that must consider uncertainty . Thus judgment of the current state of a patient is often a matter of degree [2] . Hence transition from one stage of a disease to another is usually smooth rather than abrupt . The state monitor presented here is based on the concept of a fuzzy automaton rather than on the concept of a conventional one in order to allow for smooth transitions . In the following definition, we use the same terminology to define a different notion of a fuzzy automaton . This definition is confined to this section and hence no confusion should arise. Definition 10.17 .1 A fuzzy automaton is a quadruple A = (Q, qo, X, S), where Q is a finite set of states, qo is a fuzzy subset of Q called the

fuzzy initial state, X is a_finite set of input symbols, and S : Q x X ----> Q is a transition function . Let it be a fuzzy subset of X, t = 0, 1, 2, . . . . Define qt+l : Q ----> [0,1] by for each q E Q, {qt(q') A Zt (X) qt+1 (q) _ { V 0

b(q', x) = q, q' E Q,

E x},

if s-1(q) if 6 -1 (q) _ 0 .

Define S : Q x ,FP(X) ----> .FP(Q) by b(qt, a) = qt+1

(10.7)

where Q={qt I t=0,1, . . .} .

A sequence of fuzzy states is denoted by {gt}t-o and is said to be increasing if qt+1 qt for all t . © 2002 by Chapman & Hall/CRC

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10. Applications

Since monitoring is a continuous process that is terminated on the exhaustion of input data rather than on arrival at a certain state, a set of final states is not defined here. The definition of a fuzzy automaton that appears here differs from the usual ones; see [49,51,212,250] for example. The transition function is crisp rather than fuzzy. Uncertainty expressed in a fuzzy input alone results in a partial transition from one state to another . A general statement about how strongly two states are related is not possible. Except for the fuzzy initial state, the fuzzification of the automaton is completely determined by the extension principle . A fuzzy automaton exhibits properties quite different from those of a crisp one. The current state is a fuzzy subset of the set of states . Consequently, the automaton can perform different (partial) transitions simultaneously and therefore track parallel paths. Another different property is that crisp automata report an error on input not accounted for at the current state while a fuzzy automaton reacts on low or zero membership grades in the fuzzy input with continuously decreasing membership grades in its current state as obtained by (10 .7) . This phenomenon causes a decrease in certainty that seems consistent with all repeated applications of fuzzy set operations. Example 10.17 .2 Let Q = {ql, q2, q3, q4, q5} and X = {a, b, c} . Define S : Q x X ----> Q as follows:

b(qj, a) = q3, b(qj, b) = q4, b(qj, c) = q2, b(q2, a) = q2, b(q2, b) = q4, 6(q2, C) = q2, b(q3, a) = q3, b(q3, b) = q5Define

= q4, = q5, = q4, = q4, b(q5 , a) = q5, b(q5, b) = q5, b(q5, c) = q5,

(gl) = 4, for i = 3,_4, 5. Define io : X ----> [0,1] by_ 1-0 (a) = Let i t = io fort = 1,2, . . . . Then do :

gi(gi) gi (q2) 4-1

(q3)

gi (q4) 4-1

(q5)

Q ~ [0,1] as follows:

b(q3, c) b(q4, a) b(q4, b) b(q4, c)

0, (do

i

(qj)

do

do

z,

(q2) = 4, and do (q,) = 0 LO (b) = 4, and io(c) = 4 .

n io (c)) V (do (q2) n io(a)) V (do (q2) n io (c))

(gi) A io(a)) 1 2 (40(gi) n LO (b)) V (40 (g2) 1 4 (do

0.

© 2002 by Chapman & Hall/CRC

n LO (b))

10.17. Clinical Monitoring with Fuzzy Automata

519

In a similar manner, we obtain

42(gl) 42 (q2) 42 (q3) 42 (q4) 42 (q5)

0, _1

4~ 1 2~ _1 4~ 1 4'

In fact, gt(g2) = q2 (qj) for i = 1, 2, 3, 4, 5, and t = 2, 3, . . . .

Automatically acquired data are generally precise and hence no source of fuzzy input is required by the automaton defined above. Also, if every single parameter value acquired represented an input on its own, (1) the number of possible input symbols needed to be accounted for would be too large for the automaton to handle and (2) the automaton would continuously change its state in order to react to a certain input ; otherwise the input would remain unconsidered and hence become lost. Therefore, a fuzzy automaton by itself is not very appropriate to perform monitoring . The data are therefore pre-processed by a function that abstracts from single input parameters by generating fuzzy events that are passed on to the automaton . Definition 10.17 .3 Let A = (Q, qo, X, S) be a fuzzy automaton, R1 through

Rn be the parameter ranges, where n is the number of parameters observed, P = R1 x . . . x Rn be the parameter value space, and f : P ~ .PP(X) be a function that maps_parameter tuples to fuzzy subsets of the input alphabet of A . Then M = (A, P, f) is called a state monitor.

Note that,F'P(X) specifies the interface between preprocessing of data through f and interpretation of input through A. Thus f can therefore be replaced by any computable method that yields a suitable fuzzy subset . From the definition of S, it follows that other than the state membership values of qo, those of qt can only be introduced through fuzzy input. The following lemma follows from (10 .7) . Lemma 10.17 .4 If a fuzzy automaton A is repeatedly fed with constant fuzzy input i, then the set of fuzzy states it transitions between is finite . m Define the height of qt, written hgt(gt), to be V{qt(q) I q E Q} for t = 0, 1, 2, . . . . The sequence {hgt(gt )1-o , is a decreasing function of t.

This reflects a loss of certainty in the automaton . Even if the input does not change, hgt(gt) can decrease rapidly. Hence in practice, when the monitor is provided input in rapid succession, once the current state is the empty set, it can never recover . As a matter of fact, if the automaton does not contain any feedback loops, i .e., does not provide circular transitions, it will arrive © 2002 by Chapman & Hall/CRC

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10. Applications

at the empty state after at most as many steps as there are states. Rather than leaving the responsibility for providing appropriate feedback loops to the designer of the automaton to overcome this undesirable property, we next define a property that overcomes this inadequacy. Definition 10.17 .5 A fuzzy automation is said to provide a peak hold if b'q , q E Q and b'x E X,

6(q, x) = q implies S(q, x) = q.

(10.8)

The condition in Definition 10.17.5 says that there is a transition for every state on every input that leads to the state . The condition implies that no state can be entered and left on the same input, else S would not be single-valued . The peak hold guarantees that the maximum evidence for a state provided by its predecessors is memorized and held as long as input of ongoing transitions can support it. However, the peak hold may also be sustained by an input other than the one that initially led to that state since a state does not remember its predecessor . Thus the grade of membership can unintentionally remain high. Peak hold has a positive side effect in that the automaton cannot oscillate on constant input [250], a property that would clearly not be acceptable in the clinical setting since stable input should be reflected in stable output . This can be seen from the next result. The fuzzy automaton in Example 10.17.2 provides a peak hold. Theorem 10.17 .6 The fuzzy state of a fuzzy automaton with peak hold always becomes stable after a finite number of repetitions of the same input.

Proof. We show that for t > 1, there is a positive integer r such that qt C qt+1 C . . . C qt+' = qt+T+1 = . . . .

(10.9)

This is accomplished by showing that (1) {qt }t-- o is increasing, i .e., qt C qt+l C . . . and (2) 3r such that S(qt+T, i) = qt+,.. For all subsequent states, (10.9) follows from the fact that S is a function . _ For every fuzzy state qt with t > 0 and every input i, it follows by (10.7) that for every state q, there is a transition that determines its membership value, i.e., b'q E Q, S -1 (q) ~ 0. b'q E Q, 3q' and -T such that 6(q', x) = q and qt (q) = qt -1(q~) Thus, qt (q) < i(x) and so qt (q) A7(x) x) = q then implies

=

qt+1 (q) = qt(q) V (V{qt (q)

qt(q) . Repeated input of i and S(q, A 7(x) I

qt+1 (q) > qt(q), © 2002 by Chapman & Hall/CRC

n a(x) .

b(q, x) = q}),

10.17. Clinical Monitoring with Fuzzy Automata

521

and so qt+l D qt . By Lemma 10. 17.4, there is no infinite sequence of fuzzy states such that qt+1

=

b(qt,

i) and

qt+1

D

{qt}t-o

qt .

Consequently, there exists an r such that qt+T+1 C qt+T for t = 0, 1, 2, . . . . Since {qt}t_o is increasing, qt+T+1 must equal qt+T . m The proof of Theorem 10.17.6 shows that {_gt}t°o does not converge to the empty state . It can take several steps for A to become stable since the fuzzy input can cause a propagation of higher membership grades along a sequence of transitions . The fact that nonfuzzy deterministic automata with peak hold are stable after one step provides another example of how fuzzification yields more general results . The height of the current state is still decreasing even with the peak hold property since high grades of membership cannot be regained once they are lost . A situation where the grade of membership of one state decays while its successor's rises is a source of loss. This behavior does not model the natural decision process correctly because once a decision has been made, it is usually pursued rather uncritically until there is sufficient evidence for another decision to be made. By introducing the idea of a threshold, the state monitor can be modified to adopt this kind of behavior . A state is called active when its grade of membership in the current state exceeds a certain threshold c. An active state is defined to remain active until there is a transition that induces activity of one of its successors, i .e., qt (q) qt+1(q)

=

(10 .7)

c and 6(q, x) = q' and otherwise . if

qt(q)>

x such that it (x) > c

~q~,

(10 .10)

Thus once a state has reached a certain grade of membership, it keeps it until a transition can pass it on to one of its successors. Thus the height of the current state is always greater than c. Therefore, a certain level of uncertainty is thus always being maintained . For c = 1, (10.10) implies that there is always at least one state q such that qt(q) = 1 . This accounts for the fact that the patient is considered to be at least in one state at a time, even if no successor with more evident support could yet be determined. We note that (10.10) has only slight effect on the proof of Theorem 10.17.6. In (1) of the proof of Theorem 10.17.6, {qt}t°o is still increasing since the peak hold also works for qt (q) > c and qt (q) can only drop below c once, namely on the first input . Also, (2) of the proof of Theorem 10.17.6 still holds because Lemma 10.17.4 is not affected. Moreover, (10.10) cannot prevent the automation from oscillation without peak hold even though the height is kept above c. © 2002 by Chapman & Hall/CRC

522

10. Applications

Further discussion of clinical monitoring can be found in [222] . We note that another approach using techniques from fuzzy set theory can be found in [214] . Nonfuzzy approaches can be found in [87, 238]. 10 .18

Systems

R1zzy

In the final section, we consider fuzzy systems for two reasons . First, they resemble fuzzy automata and, second, they have interesting applications to information retrieval [160] . System theory provides a framework for describing general relationships of the empirical world. We are mainly interested in the concept of reachability, observability, stability, and realization . Let Q be the state space, X the input space, and Y the output space. A deterministic dynamic system (time invariant) is a complex S = {Q, Y, S, ,3}, where S : Q x X ~ Q is the dynamics, qt+1 = S(qt, ut ), and,3 : Q ~ Y is the output map . A nondeterministic system is the complex S = {Q, X, Y, S, /3} with the : Q dynamics S : Q x X ~ P(Q), qt+1 E S(qt, ut ), and the output map/3 P(Y) .

This definition can be generalized by considering not only the states, but also the inputs and outputs as being subsets . Definition 10.18 .1

An abstract system is a complex Sa = LQ,X,Y,b,01

such that S : P (Q) x P (X) ----> P (Q), 3 : P (Q) ----> P (Y) .

We next present the application given in [160] of fuzzy systems to information retrieval systems . Example 10.18 .2

We consider an Information Retrieval system defined in terms of its response to a request for information . An IR system reports on the existence and location of information items relating to a request and does not change the knowledge of the user on the subject of the request . If the search criteria are based on the contents of an information item, then it becomes necessary to use content identification such as a set of descriptors attached to each item . In such cases, it is customary to assign descriptors, normally chosen from a controlled list of allowable terms . That is, a request is defined to be recovered from the store . An IR system compares the specification of required items with the descriptions of the stored items, and retrieves, or lists, all the items that correspond in some defined way to that specification . Consequently, the IR system can be characterized by having as inputs a subset of a set D of information items and a subset of a set R of requests . If the system is presented with new documents, the system must process them to obtain descriptions . The same sort of thing must be done

© 2002 by Chapman & Hall/CRC

10.18. Fuzzy Systems

52 3

with the requests . The next step is the comparison. The ultimate response of the system to a request is a partial ordering on the set of information items.

Let Pf (S) be the set of all finite subsets of a set S. Let ~, rl be functions ~ : D Q, rl : R Q, where Q is the set of descriptions. Let A = Pf(D), B = Pf(R), C = Pf(Q), and 'R be the set of partial orderings on Pf(D), i.e., for p E 1Z, M, N E Pf(D), MpN if and only if N is more relevant to a given request than M. We use the following variables to describe the state of the IR system : ql is the set of incoming documents waiting to be processed ; q2 is the document file; q3 is the document description file; q4 is the set of incoming requests waiting to be processed ; q5 is the request currently being processed ; q6 is the request that was just processed ; q7 is a partial ordering on Pf(D) (i .e., in 'R) induced by the request q6 . The input variables are ul = the documents coming to the system; u2 = the requests coming to the system ; and the output variables are yl = q6 is the request that was just processed ; y2 is a subset of the document file that is maximal with respect to q7 . Using the above notation, the input space is X = AxB, the output space is Y = R xA, and the state space is H=AxAxCxBxRxRX R . Let the initial state be given as follows : ql (0) = {di E D i = 1, 2, . . . , m} q2(0) q3(0) q4(0) q5(0) q6 (0) q7(0)

= = = = =

{el, e2, - . . , enJ l ~wl, w2, " . . , wn J with wi = ~( ei), i = 1 , 2. . . . , n {rl . . , ri}, ri E R r'

E 'R. The state equation of the IR system can be written in the form: q(k + 1) = 6(q(k), u (k)), k = 0, 1, 2, . . . , where S : R x X ----> R is the dynamics and the output is y(k) =,3(q(k)), where,3 : R ----> y is the output map . The state equations can be written as follows : ql(k + 1) _

(ql(k~) U ul(k))\~dk+11,

q2(k + 1) = q2(k) U ul(k) © 2002 by Chapman & Hall/CRC

~f

q, (k)

0

524

10. Applications q3(k+

q4(k

{~(dk+1)},

1) _ { g3(kj,U

+ 1) _

~f ql(k)

[q2~~ U u2(k)]\{rk+1}

rk+1,

0 q6 (k q7 (k

+ 1)

+ 1)

if if

q4 (k) q4 (k)

if if

q4(k) 7~ q4 (k) =

0 01

01 0

= q5 (k)

= -r (97 (q5 (k))),

where r is the ordering in A induced by 97 (q5 (k)) . The output equations are as follows : Y1(k) = q6(k)

and Y2(k)

C

q2(k) .

The notation dk +1 in the first equation means that between k and k + 1, dk+ 1 is processed, k = 0, 1, . . . . The same thing holds for rk + 1 . It follows that q7(k + 1) is an ordering in A and y2(k) is the subset of documents that give a "best response" at the request q6(k) . It follows that the IR system is a complex, SIR

= {x, X, Y, 6, l0},

with S and,3 as above. The dynamics S can be extended to H x X* considering sequences of pairs ("documents," "requests") . If U and V are two sets, there is an injection i :(UXV)*~U*XV* such that i((u1,v1)(u2,v2) . . .(un,vn))

= (u1 u2 . . . .un,VIV2 . . . .vn) .

It follows that an element of X* can be considered as a pair of the form ("sequence of documents," "sequence of requests"), where the sequences have the same length. © 2002 by Chapman & Hall/CRC

10.18. Fuzzy Systems

525

The system SIR is reachable, from the state q(0) E R if each state in R can be reached with a suitable input sequence. This means that the input space (i.e., documents and requests) is rich enough to cover each state (i .e., document file, requests, orderings, etc.). The system SIR is observable if, knowing the system's response (documents that are relevant to the request), for each sequence of inputs (requests and documents), the state from which the system is started can be uniquely determined. We are thus led to the concept of a fuzzy system . Definition 10.18 .3 A fuzzy system is a complex Sf = {Q, X, Y, 6"31

with S : ,FP(Q) x ,FP(X) ----> ,F'P(Q), the fuzzy dynamics, and /3 .FP(Q) ----> .FP(Y), the fuzzy output map. The state equation can be written as follows : qt+i = S(qt, ut), ut

E .FP(X), qt,

qt+i

E .FP(Q),

where qt , qt+l are, respectively, the fuzzy states at time t, t + 1 and ut the fuzzy input at time t. The output equation becomes yt = ,3 (qt), qt

E .FP(Q), yt E .FP(Y),

where yt is the fuzzy output at time t. Let (.FP(X))* denote the free monoid generated by .FP(X), i.e., the set of sequences of fuzzy inputs. Then S can be extended to .FP(Q) x (.FP(X))* by defining S : .FP (Q) x ( .FP(X))* - .FP (Q)

as follows : (1) S(a, A) = a, b'a

E FP (Q), (2) 6(a, A*w) = 6(6(a, A*), w), da E .FP(Q), A* E (.FP(X))*, w E

.FP(X) .

If the initial state ao

E .FP(Q) is

fixed, we can define the function:

Sao : (.FP(X))* - .FP(Q)

by VA *

E (.FP(X))*, Sao (A*) = S(ao,A*) .

Hence S(ao, A*) is computed by starting the system in state ao, feeding in the input sequence A* , and determining the final state . With these definitions, we can express two basic concepts of systems theory, namely, reachability and observability. © 2002 by Chapman & Hall/CRC

526

10. Applications

Definition 10.18 .4 The fuzzy system Sf is called reachable from the state ao if Sao is onto, i.e .,

Va E .FP(Q), 3a* E (.FP(X))* such that sao (A*) = a. The image of Sao , Im Sao C ,FP(Q), is called the reachability set of the system Sf from ao .

Composing S ao with, 3 we obtain the fuzzy response function (or behavior) of the system Sf, i.e ., f, = 0 O Sao ,

Thus f'. : (.FP(X))* -

FP(Y)

and so f,. (A*) =,3(b(ao,A*)) VA * E ( .FP(X)) * .

Definition 10.18 .5 A fuzzy system Sf is called observable if the assignment a

F-~

fa is one-to-one .

The interested reader is urged to see [160] for further details and interesting ideas including the concepts of stability and realization . 10 .19

Exercises

1. Let a, aii E II8 for i = 1, 2, . . . , m and j = 1, 2, . . . , n . Prove that (V{A{aj2 i = 1,2, . . . , m} I j = 1,2, . . . , n}) n {a} = V{n{aj2 lea I i = 1,2, . . . , m} I j = 1,2 . . . . , n} and that (A{V{aj2 I i = 1,2 . . . . , m} I j = 1, 2, . . . , n}) V {a} = A{V{aj2 Aa I i = 1, 2, . . . , ml I j = 1, 2, . . . , n}. 2. Prove that the matrix in Example 10.6 .2 has period equal to 3. 3. Prove that y(anbm) = 'A' in Example 10.8 .2. 4. Prove Lemma 10 .11 .2. 5. Let M be the median operator of (10 .5) . Show that M is associative, not strictly increasing, and contractive . 6. Prove (2) of Proposition 10.11.7. 7. Show that (No , -), where No = hY U {0} and ({0,1, . . . , n}, i), of Example 10.13.1 are tolerance spaces. © 2002 by Chapman & Hall/CRC

10.19 . Exercises

527

8 . For f in Definition 10.13 .2, prove that f*T is the least tolerance on Y such that f : (X, T) ~ (Y, f *T) is a fuzmap . Prove also that it is the unique tolerance on Y such that b' tolerances Q on Z and b' set-theoretic maps g : Y ~ (Z, Q), f - g : (X, T) ~ (Z, Q) if and only if g : (Y, f. T) ~ (Z, a) is a fuzmap . 9 . If g : Y ----> (X, T) is a set-theoretic map, prove that g*T is the biggest tolerance on Y such that g : (Y,g*T) ----> (X, T) is a fuzmap . 10 . In Definition 10 .13 .4, prove that T is the least tolerance such that all injections tj are fuzmaps, and TIT2 is the largest tolerance such that all projections pry are fuzmaps . 11 . Prove (1) of Theorem 10 .14 .3 . 1 12 . In Example 10 .14 .6, show that T7 and T9 yield T14 =

and Tg and T9 yield T15 = T14 and T15 .

1 1 1 0

1 1 1 1

1 1 1 0

0 1 0 1

1 1 1

. Then determine T19 from

13 . In Example 10 .15 .6, show that SQ U Sx 1 T°Sx C T° for x E {xl, X2114 . Prove that the condition in Definition 10 .17.5 implies that no state can be entered and left on the same input, else S would not be singlevalued .

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