Fuzzy Logic: Mathematical Tools for Approximate Reasoning

275 Pages·2001·10.55 MB·English

by  Giangiacomo Gerla

FUZZY LOGIC
TRENDS IN LOGIC
Studia Logica Library
VOLUME 11
Managing Editor
Ryszard W6jcicki, Institute of Philosophy and Sociology,
Polish Academy of Sciences, Warsaw, Poland
Editors
Daniele Mundici, Department of Computer Sciences, University of Milan, Italy
Graham Priest, Department of Philosophy, University of Queensland,
Brisbane, Australia
Krister Segerberg, Department of Philosophy, Uppsala University,
Sweden
Alasdair Urquhart, Department of Philosophy, University of Toronto, Canada
Heinrich Wansing, Institute of Philosophy, Dresden University of Technology,
Germany
Assistant Editor
Jacek Malinowski, Box 61, UPT 00-953, Warszawa 37, Poland
SCOPE OF THE SERIES
Trends in Logic is a bookseries covering essentially the same area as the journal
Studia Logica - that is, contemporary formal logic and its applications and
relations to other disciplines. These include artificial intelligence, informatics,
cognitive science, philosophy of science, and the philosophy of language.
However, this list is not exhaustive, moreover, the range of applications, com
parisons and sources of inspiration is open and evolves over time.
The titles published in this series are listed at the end of this volume.
GIANGIACOMO GERLA
Department of Mathematics and Computer Sciences,
University of Salerno, Italy
FUZZY LOGIC
Mathematical Tools for Approximate Reasoning
Springer-Science+Business Media, B.Y.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5694-8 ISBN 978-94-015-9660-2 (eBook)
DOI 10.1007/978-94-015-9660-2
Printed on acidjree paper
All Rights Reserved
© 2001 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2001.
Softcover reprint of the hardcover 1st edition 2001
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To my wife Loredana,
to my daughters Brunella and Francesca
CONTENTS
CONTENTS ................................................... VII
PREFACE .................................................... XI
CHAPTER 1. Abstract logic in a lattice
1 Introduction............................................. 1
2 Lattices, Boolean algebras, triangular norms. . . . . . . . . . . . . . . . . . . . 1
3 Closure operators and closure systems. . . . . . . . . . . . . . . . . . . . . . . . . 4
4 A Galois connection between operators and classes .. . . . . . . . . . . . . 6
5 Abstract logic in a lattice .................................. 8
6 Continuity for abstract logics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
7 Step-by-step deduction systems ............................. 12
8 Logical compactness .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
9 Product of two abstract deduction systems . . . . . . . . . . . . . . . . . . . .. 15
10 Duality principle for ordered sets ............................ 16
CHAPTER 2. Abstract fuzzy logic
1 Fuzzy subsets for vagueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19
2 Basic notions ...................... . . . . . . . . . . . . . . . . . . . . .. 21
3 Closed and open cuts. . . . . . . . . . . . . .. ....................... 24
4 Fuzzy subsets and continuous chains. . . . . . . . . . . . . . . . . . . . . . . . .. 26
5 Abstract fuzzy logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28
6 Compactness and continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 30
7 Logical compactness ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8 Ultraproduct ofa family of fuzzy models ..................... , 33
9 Fuzzy logic is not monotone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
10 Abstract similarity logic ................................... , 37
11 Any fuzzy logic is equivalent to a crisp logic. . . . . . . . . . . . . . . . . . .. 39
CHAPTER 3. Extending an abstract crisp logic
1 An extension principle for closure operators. . . . . . . . . . . . . . . . . . .. 45
2 An extension principle for closure systems. . . . . . . . . . . . . . . . . . . .. 48
3 Canonical extensions and continuous deformations ............. , 50
4 Dualizing the extension principle. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52
5 Extension of a compact closure operator. . . . . . . . . . . . . . . . . . . . . .. 53
6 Extension of a crisp logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7 Characterizations of the canonical extensions .................. 58
8 Degree of inconsistency of a canonical extension. . . . . . . . . . . . . . .. 61
9 Canonical similarity logic ................................. 64
10 Fuzzy metalogic, facts and preferences. . . . . . . . . . . . . . . . . . . . . . .. 66
CHAPTER 4. Approximate reasoning
1 The heap paradox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
VII
VIII CONTENTS
2 Fuzzy inference rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71
3 Fuzzy Hilbert logic and homomorphisms . . . . . . . . . . . . . . . . . . . . .. 76
4 Degree of consistency and non-monotonicity .. . . . . . . . . . . . . . . . . . 78
5 Step-by-step deduction and continuity. . . . . . . . . . . . . . . . . . . . . . . . . 79
6 Building up fuzzy Hilbert systems by inequalities ... . . . . . . . . . . .. 81
7 Any fuzzy Hilbert system is equivalent to a crisp system ......... 83
8 Bald men, Lukasiewicz conjunction and induction principle. . . . . . .. 86
CHAPTER 5. Logic as managment of constraints on the truth values
I Heap paradox by negative information . . . . . . . . . . . . . . . . . . . . . . . . 89
2 Constraints on the truth values .............................. 91
3 Examples: Zadeh logic, Boolean logic, Probability logic .......... 94
4 Hilbert systems for constraints .... . . . . . . . . . . . . . . . . .. . . . . . . . . . 96
5 Fuzzy logics with a negation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6 Refutation procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100
7 Equivalence to a crisp logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 102
8 Tableaux method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 104
CHAPTER 6. Canonical extension of a crisp Hilbert logic
I Extending a crisp deduction Hilbert system .................... 109
2 Controlling the inconsistency ............................... 111
3 Necessity logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113
4 A simple example of non-monotone fuzzy logic. . . . . . . . . . . . . . . .. 116
5 Fuzzy filters and fuzzy subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . .. 118
6 Necessity measures as fuzzy theories ......................... 119
7 Fuzzy Hilbert systems and fuzzy subalgebras ................... 121
8 Extensions by continuous triangular norms. . . . . . . . . . . . . . . . . . . .. 124
CHAPTER 7. Graded consequence relations
1 Graded information with graded deductive tools ................ 129
2 Stratified fuzzy closure operators. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 129
3 Stratified fuzzy closure systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 132
4 A characterization of stratified closure systems. . . . . . . . . . . . . . . ... 135
5 A characterization of stratified operators. . . . . . . . . . . . . . . . . . . . . .. 138
6 Stratified deduction systems ............................... . 140
7 Sequents and consequence relations .......................... 142
8 Graded consequences and sequent calculus . . . . . . . . . . . . . . . . . . . .. 144
9 Finite sequent calculus and compact graded consequences. . . . . . . .. 146
10 Graded consequences and stratified operators. . . . . . . . . . . . . . . . . .. 148
CHAPTER 8. Truth-functional logic and fuzzy logic
1 Truth-functional fuzzy semantics. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 151
2 The main properties ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 152
3 Two discontinuous truth-functional semantics .................. 156
4 Any continuous truth-functional semantics is axiomatizable ....... 158
5 Any axiomatizable truth-functional semantics is continuous ....... 160
6 Zadeh (continuous) logic ................................... 163
CONTENTS IX
7 Lukasiewicz (continuous) logic .............................. 165
8 Comparing truth-functional logic with fuzzy logic ............... 168
CHAPTER 9. Probabilistic fuzzy logics
1 Vagueness and uncertainty ................................. 171
2 Logic of super-additive measures. . . . . . . . . . . . .. . . . . . . . . . . . . . .. 172
3 Completeness theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 176
4 Logic of upper-lower probabilities . . . . . . . . . . . . . . . . . . . . . . . . . .. 177
5 Probability logic: semantics .... ............................ 180
6 Probability logic: Hilbert system ............................ 184
7 Completeness theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 187
8 Refutations in probability logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 189
9 Two remarks: probability of formulas, subjective probability . . . . .. 191
10 Belief logic and Boolean logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 194
11 Qualitative probability logics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 196
CHAPTER 10. Fuzzy control and approximate reasoning
1 Information by words versus information by numbers ............ 199
2 Control by triangular norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 200
3 Programs and Herbrand models. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 204
4 Fuzzy programs and fuzzy Herbrand models . . . . . . . . . . . . . . . . . . . 205
5 Logic approach to fuzzy control ............................. 206
6 The logical interpretation suggests new tools . . . . . . . . . . . . . . . . . .. 208
7 Control by implication and negative information . . . . . . . . . . . . . . .. 210
8 Control by similarity and prototypes .......................... 214
9 Logic interpretation of defuzzification: an open question ......... 216
10 The predicate MAMD and some observations ................... 219
CHAPTER 11. Effectiveness in fuzzy logic
1 Introduction ............................................. 221
2 Recursively enumerable fuzzy sets ........................... 221
3 Decidability and fuzy computability . . . . . . . . . . . . . . . . . . . . . . . . .. 225
4 Enumerability by discrete topology ........................... 228
5 Kleene hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 231
6 GOdel numbering and Church Thesis. . . . . . . . . . . . . . . . . . . . . . . . .. 234
7 Reducibility and Universal Machines ......................... 236
8 Effective abstract fuzzy logic ................................ 238
9 Fuzzy logic = enumeration fuzzy closure operator .. . . . . . . . . . . . .. 240
10 Creative fuzzy sets and Godel theorems ....................... 246
11 Sharpened and shaded versions: limitative theorems .............. 248
REFERENCES ................................................ 251
INDEX ....................................................... 261
LIST OF SYMBOLS ............................................ 267
Preface
Fuzzy logic in narrow sense is a promising new chapter of formal logic whose
basic ideas were formulated by Lotfi Zadeh (see Zadeh [1975]a). The aim of this
theory is to formalize the "approximate reasoning" we use in everyday life, the
object of investigation being the human aptitude to manage vague properties (as,
for example, "beautiful", "small", "plausible", "believable", etc.) that by their own
nature can be satisfied to a degree different from 0 (false) and I (true). It is worth
noting that the traditional deductive framework in many-valued logic is different
from the one adopted in this book for fuzzy logic: in the former logics one always
uses a "crisp" deduction apparatus, producing crisp sets of formulas, the formulas
that are considered logically valid. By contrast, fuzzy logical deductive machinery
is devised to produce a fuzzy set of formulas (the theorems) from a fuzzy set of
formulas (the hypotheses). Approximate reasoning has generated a very interesting
literature in recent years. However, in spite of several basic results, in our opinion,
we are still far from a satisfactory setting of this very hard and mysterious subject.
The aim of this book is to furnish some theoretical devices and to sketch a
general framework for fuzzy logic. This is also in accordance with the non
Fregean attitude of the book. Indeed, we aim to give some instruments to define
rough mathematical models of the wonderful human capacity of reasoning with
vague notions and not to propose a unique rigorous formalized (multivalued) logic
able to settle such human activity. Consequently, in the book there is no definitive
choice of the logical connectives and of their associated interpretations. Also, our
constant usage of the set of truth values given by the unit real interval [0, I], is only
due to the purpose of simplifying the treatment of the material and not to the
conviction that different sets of truth-values are useless. In any case, the book is
mainly an exposition of our ideas and research and it does not have any pretension
of completeness. In particular, we do not expose the very important fuzzy logic
related to Lukasiewicz truth-functional logic. Very good books and papers exist on
this argument and we strongly suggest their direct reading (see, for example,
Pavelka [1979]c, Cignoli, D'ottaviano and Mundici [2000], Hajek [1998], Novak,
Perfilieva and Mockor [1999], Turunen [1999]).
Mainly, three tools are proposed and examined:
-the theory of fuzzy closure operators,
-an extension principle for closure operators,
-the theory of recursively enumerable fuzzy subsets.
Indeed, we embrace Tarski's viewpoint, according to which a monotone logic
is a set (of formulas) together with a closure operator (the deduction operator).
Consequently, in Chapters I and 2 the theory of closure operators in a lattice is
exposed and applied to outline an abstract approach to fuzzy logic. In Chapters 3
and 6 an extension principle for classical closure operators is also proposed and
largely used. This principle enables us to extend any crisp logic into a fuzzy logic.
In Chapter 7 such an approach is generalized by showing that it is possible to
associate any chain of crisp logics with a fuzzy logic. In such a way we obtain a
XI
XII PREFACE
very interesting class of fuzzy logics we call "stratijied". Necessity logic, graded
consequence operator theory and similarity logic all belong to this class. Chapter 4
introduces the notion of fuzzy inferential apparatus in a "Hilbert style" by giving a
fuzzy set of logical axioms and a suitable set of fuzzy inference rules. In particular,
we prove that this approach is equivalent to the theory of continuous fuzzy closure
systems (see Theorem 2.6). In Chapter 5 we extend the definition of approximate
reasoning usually proposed in the literature by assuming that a deduction apparatus
is a tool to calculate constraints on the possible truth values of the formulas. Truth
functional multi-valued logic is examined in Chapter 8 where a strong connection
between axiomatizability and continuity of the logical connectives is established
(see Theorems 4.5 and Theorem 5.5). In Chapter 9 we propose fuzzy logics
probabilistic in nature which are strictly related to super-additive probabilities,
upper-lower probabilities, lower envelopes and belief measures. In Chapter lOwe
present a tentative approach to fuzzy control by translating any system of IF
THEN fuzzy rules into a system of fuzzy clauses, i.e., a fuzzy program. This
enables us to unify the treatment based on the triangular norms and the treatment
based on the implications. Finally, in Chapter 11 we extend to fuzzy sets the
fundamental notions of decidability and recursive enumerability. In accordance, a
definition of an effective Hilbert system is proposed and compared with a
definition of enumeration fuzzy operator. This will enable us to get a constructive
version of the just quoted Theorem 2.6 (see Theorem 9.2). Also, the notion of
recursively enumerable fuzzy subset enables us to obtain several interesting
limitative results for fuzzy logic.
Note that three types of fuzzy logics are considered. The first type arises from a
fuzzijication of the metalogic. Necessity logic, similarity logic and graded
consequence theory, are typical examples. Indeed, these logics are obtained by
admitting that the notions of hypothesis, identity between formulas, logical
consequence, can be vague. In this case, the worlds we will describe are "crisp";
vagueness arises from the language and the deductive apparatus we use. The
second type is related to truth-functional multi valued logic. In such a case fuzzy
worlds are considered, i.e., worlds whose elements can have graded properties.
Finally, we define logics, probabilistic in nature, which are related to "belief
measures". Perhaps, we can allocate these logics to the first class. Indeed, in this
case vagueness concerns the metalogic notion of "believable".
We expect the book to be read by people interested in artificial intelligence,
fuzzy control, formal logic, philosophy. The book is almost completely self
contained but some familiarity with classical logic is required. Moreover, Chapter
11 assumes some acquaintance with the theory of recursive functions.
We are indebted to Loredana Biacino for the critical reading of the manuscript
and for several ideas and suggestions. We wish to thank my collaborator Ferrante
Formato for stimulating comments and cooperation and Professor Joe Saporito for
checking the English style of the manuscript. Finally, we wish to thank the
anonymous referee for spending many hours reading preliminary versions, and
writing long and interesting reports.
Napoli, February 25, 2000.