Mathematical Principles of Fuzzy Logic

Mathematical Principles of Fuzzy Logic

THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE

MATHEMATICAL PRINCIPLES OF FUZZY LOGIC VILEM N O V K University of Ostrava Institute for Research and Applications of Fuzzy Modeling Brafova 7, 701 03 Ostrava 1, Czech Republic IRINA PERFILIEVA Moscow State Academy of Instrument Making Stromynka 20, 107846 Moscow, Russia and University of Ostrava Institute for Research and Applications of Fuzzy Modeling Brafova 7, 701 03 Ostrava 1, Czech Republic JIRI MOCKOR University of Ostrava Institute for Research and Applications of Fuzzy Modeling Brafova 7, 701 03 Ostrava 1, Czech Republic... " Springer-Science+Business Media, LLC

Library of Congress Cataloging-in-Publication Data Novak, Vilem. Mathematical principles of fuzzy logic I Vilem Novak, Irina Perfilieva and Jiri Mockor. p. cm. -- (The Kluwer international series in engineering and computer science ; secs 517) Includes bibliographical references and index. ISBN 978-1-4613-7377-3 ISBN 978-1-4615-5217-8 (ebook) DOI 10.1007/978-1-4615-5217-8 1. Fuzzy logic. 1. Perfilieva, Irina. 1953-. II. Mockof, Jin. III. Title. IV. Series. QA9.64.N68 1999 511.3--dc21 99-37210 CIP Copyright 1999 by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 1999 Softcover reprint ofthe hardcover Ist edition 1999 AII rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Springer-8cience+Business Media, LLC. Printed an acid-free paper.

To David and Martin, to Vitalij, to Sona, Marek and Katerina, with love

Contents Preface xi 1. FUZZY LOGIC: WHAT, WHY, FOR WHICH? 1 1.1 Vagueness and Uncertainty 2 1.2 Vagueness and Fuzzy Sets 6 1.3 What Is Fuzzy Logic 9 1.4 Outline of the Agenda of Fuzzy Logic 12 2. ALGEBRAIC STRUCTURES FOR LOGICAL CALCULI 15 2.1 Algebras for Logics 16 2.1.1 Boolean algebras 16 2.1.2 Residuated lattices and MV-algebras 23 2.2 Filters and Representation Theorems 35 2.3 Elements of the theory of t-norms 41 2.4 Introduction to Topos Theory 52 2.4.1 T opos theory 52 2.4.2 Lattice of subobjects in topos 56 3. LOGICAL CALCULI AND MODEL THEORY 61 3.1 Classical Logic 61 3.1.1 Propositional logic 62 3.1.2 Predicate logic 67 3.1.3 Many-sorted predicate logic 73 3.2 Classical Model Theory 74 3.3 Formal Logical Systems 77 3.4 Model Theory in Categories 83 4. FUZZY LOGIC IN NARROW SENSE 95 4.1 Graded Formal Logical Systems 97 4.2 Truth Values 106 4.2.1 Intuition on truth values and operations with them 106 4.2.2 Truth values as algebras 111 4.2.3 Why Lukasiewicz algebra 112 4.2.4 Enriching structure of truth values 113 4.3 Predicate Fuzzy Logic of First-Order 114

Vlll MATHEMATICAL PRINCIPLES OF FUZZY LOGIC 4.3.1 Syntax and semantics 4.3.2 Basic properties of fuzzy theories 4.3.3 Consistency of fuzzy theories 4.3.4 Extension of fuzzy theories 4.3.5 Henkin fuzzy theories 4.3.6 Complete fuzzy theories 4.3.7 Lindenbaum algebra of formulas and its properties 4.3.8 Completeness theorems 4.3.9 Sorites fuzzy theories 4.3.10 Partial inconsistency and the consistency threshold 4.3.11 Additional connectives 4.3.12 Disposing irrational logical constants 4.4 Fuzzy Theories with Equality and Open Fuzzy Theories 4.4.1 Fuzzy theories with equality 4.4.2 Consistency theorem in fuzzy logic 4.4.3 Herbrand theorem in fuzzy logic 4.5 Model Theory in Fuzzy Logic 4.5.1 Basic concepts of fuzzy model theory 4.5.2 Chains of models 4.5.3 Ultraproduct theorem 4.6 Recursive Properties of Fuzzy Theories 115 120 129 132 139 140 142 147 150 151 153 154 158 159 160 166 168 169 171 172 176 5. FUNCTIONAL SYSTEMS IN FUZZY LOGIC THEORIES 179 5.1 Fuzzy Logic Functions and Their Representation by Formulas 181 5.1.1 Formulas and their relation to fuzzy logic functions 181 5.1.2 Piecewise linear functions and their representation by formulas 185 5.2 Normal Forms for FL-Functions and Formulas of Propositional Fuzzy Logic 189 5.2.1 Normal forms for L-valued FL-functions 190 5.2.2 Normal forms for functions represented by formulas 195 5.3 FL-Relations and Their Connection with Formulas of Predicate Fuzzy Logic 201 5.3.1 FL-Relations and Their Representation by Formulas 201 5.3.2 Normal Forms and Approximation Theorems 204 5.3.3 Representation of FL-relations and Consistency of Fuzzy Theories 208 5.4 Approximation of Continuous Functions by Fuzzy Logic Normal Forms 210 5.4.1 Approximation of continuous functions through FL- relations 210 5.4.2 Approximation of continuous functions determined by a defuzzification operation 213 5.5 Representation of Continuous Functions by the Conjunctive Normal Form 218 6. FUZZY LOGIC IN BROADER SENSE 223 6.1 Partial Formalization of Natural Language 224 6.1.1 Evaluating and simple conditional linguistic syntagms 224 6.1.2 Translation of evaluating syntagms and predications into fuzzy logic 227 6.1.3 The meaning of simple evaluating syntagms 231 6.2 Formal Scheme of FLb 239 6.3 Special Theories in FLb 242 6.3.1 Independence of formulas 243 6.3.2 Deduction on simple linguistic descriptions 247 6.3.3 Fuzzy approximation based on simple linguistic descriptions 249

Contents IX 7. TOPOl AND CATEGORIES OF FUZZY SETS 7.1 Category of O-sets as generalization of fuzzy sets 7.2 Category of O-fuzzy sets 7.2.1 O-fuzzy sets over MV-algebras 7.3 Interpretation of formulas in the category CO - Set 8. FEW HISTORICAL AND CONCLUDING REMARKS References Index 259 261 275 275 286 299 305 315

Preface This book is an attempt to provide a systematic course of the formal theory of fuzzy logic. We made a lot of effort to be precise, but at the same time to explain the motivation and interpretation of all the results and, if possible, to accompany the theory by examples. There are a lot of other books on various aspects of fuzzy logic. Our book is more specific from the point of view of several aspects. First, it is based on logical formalism demonstrating that fuzzy logic is a well developed logical theory. Second, it includes the theory of functional systems in fuzzy logic, which provide explanation of what, and how can be represented by formulas of fuzzy logic calculi. Third, except for the generalization of the classical way of interpretation within the environment of fuzzy sets constructed over classical sets, it also presents much more general interpretation of fuzzy logic within the environment of other proper categories of fuzzy sets stemming either from the topos theory, or even generalizing the latter. Last but not least, the leading philosophical point of view is presentation of fuzzy logic especially as the theory of vagueness as well as the theory of the common-sense human reasoning, which is based on the use of natural language, the distinguished feature of which is vagueness of its semantics. We expect the book to be read by people interested in fuzzy logic and related areas, and also by logicians, mathematicians and computer scientists interested in mathematical aspects of fuzzy logic. It can be used in special courses of fuzzy logic, artificial and computational intelligence, in master and post-gradual university studies, in advanced courses on various applications including fuzzy control, decision-making and others. The book is divided into eight chapters. The first chapter is introductory and it provides motivation for the development of fuzzy logic, describes its structure and outlines its potential for applications. Fuzzy logic is there divided into that in narrow sense (FLn), which is a special many-valued logic aiming at description of the vagueness phenomenon and that in broader sense (FLb), whose aim is to provide a formal theory for modeling of natural human deduction based on the use of natural language. It is argued that characterization of the vagueness phenomenon is fundamental for further development of fuzzy logic as well as its applications.

xii MATHEMATICAL PRINCIPLES OF FUZZY LOGIC The second chapter is an overview of the basic algebraic concepts necessary for characterization of the structure of truth values and for understanding to the subsequent chapters. The third chapter briefly reminds basic concepts of classical logic, and also the notation which is employed further. The main chapters of the book are fourth to seventh. The fourth chapter contains explanation of fuzzy logic in narrow sense. We confine ourselves mainly to fuzzy logic based on the Lukasiewicz algebra of the truth values. Many reasons have been given arguing that this requirement is a necessary consequence of the assumptions, which seem to be natural and convincing. Among them, essential are continuity of the connectives (this follows from the conviction that vagueness phenomenon requires continuity) and completeness (balance between syntax and semantics). We prove lemmas and theorems characterizing behaviour of FLn, including deduction, contradiction, and others. The main result is the completeness theorem. Besides this, we also outline fundamentals of the model theory and discuss questions concerning computability. The fifth chapter describes functional systems related to propositional and predicate calculi of fuzzy logic. A notable role here is played by the well known McNaughton theorem. We provide a constructive proof of it and then, deduce a canonical representation of formulas of propositional fuzzy logic as well as some special representation of formulas of predicate fuzzy logic. A special attention has been paid to the generalization of disjunctive and conjunctive normal forms into fuzzy logic. Among the main properties of them, the ability to represent approximately continuous functions including evaluation of the quality of the approximation has been investigated. The sixth chapter extends the previous theory to obtain a theory of parts of natural language to constitute fuzzy logic in broader sense. We present a formalization of the concepts of intension and extension, evaluating linguistic syntagms and linguistic description (a set of IF-THEN rules), and demonstrate some theorems characterizing their behaviour. The seventh chapter is more abstract and deals with fuzzy sets and fuzzy logic within category theory. First, we present properties of three possible categories of fuzzy sets. The subsequent sections provide the reader with a general picture of fuzzy logic interpreted in top os and its place in topos logic. We tried to show two possible approaches. The first one is based on top os theory and Heyting algebra structures defined on the set of subobjects. The second one is based on the MV -algebra structures defined on these sets. In this book, we present some of the results and consequences, which could be useful from the point of view of fuzzy set theory. In particular, we tried to show explicitly similarities between the obtained categories and topoi, and to show how the internal logic in these categories could be directly developed. The book is concluded by a short, eighth chapter giving a brief overview of the history of fuzzy logic and outlining some of its actual problems to be solved in the future. Sections 2.4, 3.4 and Chapter 7 have been written by J. Mockor. Chapter 5 has been written by 1. Perfilieva and Chapters 4 and 6 by V. Novak. The rest has has been written by by the latter two authors together.

PREFACE xiii This book has been prepared mostly in the Institute for Research and Applications of Fuzzy Modeling of the University of Ostrava, Czech Republic, which has been established on the basis of the project VS 96037 of the Ministry of Education, Youth and Physical Training of the Czech Republic. Some parts have been prepared also in the Moscow State Academy of Instrument-Making and Informatics, Russia, and Institute of the Theory of Information and Automation of the Academy of Sciences of the Czech Republic. We wish to express our thanks to all who helped us in the preparation of this book. On the first place, we thank to people whose work has been for us the main source of knowledge and inspiration, namely P. Hajek from UI AS OR, U. Hahle from the University of Wuppertal, D. Mundici form the University of Milano and A. dinola from the University of Salerno. Further, we thank to the people who have read the manuscript and helped us to improve it on many places, namely to R. Belohlavek and A. Dvorak from IRAFM, University of Ostrava, and to S. Gottwald from the University of Leipzig. We want also to thank to our Chinese friends, M. Ying and G. Chen from Beijing Tschingua University, and G. Wang from ShaanXi Normal University for their hospitality, which helped to write some parts of this book during the stay of the first two authors in China. Last but not least, we want to thank to Prof. T. Kreck, director of the Institut fur Mathematik in Oberwolfach, and also to other workers of it for providing us unexceptionable working conditions in which significant portions of the book have been written. Ostrava, May 1999 Vilem Novak, Irina Perfilieva, Jiif Mockof