Goguen Categories: A Categorical Approach to L-fuzzy Relations

217 Pages·2007·2.95 MB·English

by  Michael Winter

GOGUEN CATEGORIES
TRENDS IN LOGIC
Studia Logica Library
VOLUME 25
Managing Editor
Ryszard Wójcicki, Institute of Philosophy and Sociology,
Polish Academy of Sciences, Warsaw, Poland
Editors
Vincent F. Hendricks, Department of Philosophy and Science Studies,
Roskilde University,
Denmark
Daniele Mundici, Department of Mathematics “Ulisse Dini”,
University of Florence, Italy
Ewa Orłowska, National Institute of Telecommunications,
Warsaw, Poland
Krister Segerberg, Department of Philosophy, Uppsala University,
Sweden
Heinrich Wansing, Institute of Philosophy, Dresden University of Technology,
Germany
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.
Volume Editor
Ewa Orłowska
The titles published in this series are listed at the end of this volume.
GOGUEN CATEGORIES
A Categorical Approach to L-fuzzy Relations
by
MICHAEL WINTER
Brock University,
St. Catharines,
ON, Canada
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-6163-9 (HB)
ISBN 978-1-4020-6164-6 (e-book)
Published by Springe,r
P.O. Box 17, 3300 AADordrecht, The Netherlands.
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© 2007 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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Tomyfamily
Contents
INTRODUCTION ix
1. SETS, RELATIONS, AND FUNCTIONS 1
2. LATTICES 5
2.1 Galois correspondences and residuated operations 13
2.2 Distributive lattices 17
2.3 Brouwerian lattices 18
2.4 Boolean algebras 20
2.5 Special elements 22
2.6 Fixed points 23
2.7 The complete Brouwerian lattice of antimorphisms 25
2.8 Filters 31
2.9 Lattice-ordered semigroups 40
3. L-FUZZY RELATIONS 43
3.1 Basic operations and properties 43
3.2 Crispness 48
3.3 Operations derived from lattice-ordered semigroups 52
4. CATEGORIES OF RELATIONS 55
4.1 Categories 55
4.2 Allegories 57
4.3 Distributive allegories 63
4.4 Division allegories 65
4.5 Dedekind categories 68
4.6 Relational constructions in Dedekind categories 74
4.7 The Dedekind category of antimorphisms 75
4.8 Scalars and crispness in Dedekind categories 79
4.9 Schro¨der categories 85
4.10 Formal languages of relational categories 86
vii
viii GOGUENCATEGORIES
5. CATEGORIES OF L-FUZZY RELATIONS 93
5.1 Arrow categories 94
5.2 The arrow category of antimorphisms 112
5.3 Arrow categories with cuts 122
5.4 The arrow category with cuts of antimorphisms 126
5.5 Goguen categories 127
5.6 The Goguen category of antimorphisms 131
5.7 Representation of Goguen categories 134
5.8 Boolean-based Goguen categories 139
5.9 Equations in Goguen categories 142
5.10 Operations derived from lattice-ordered semigroups 150
6. FUZZY CONTROLLERS IN GOGUEN CATEGORIES 169
6.1 The Mamdani approach to fuzzy controllers 169
6.2 Linguistic entities and variables 170
6.3 Fuzzification 176
6.4 The rule base 176
6.5 Decision module 179
6.6 Defuzzification 179
6.7 Proving properties of a controller 182
6.8 Discussion of the approach 194
INDEX 197
SYMBOLS 201
REFERENCES 203
INTRODUCTION
In a wide variety of problems one has to treat uncertain or incomplete in-
formation. Some kind of exact science is needed to describe and understand
existing methods, and to develop new attempts. Especially in applications of
computer science, this is a fundamental problem. To handle such information,
Zadeh [44], and simultaneously Klaua [22, 23], introduced the concept of fuzzy
sets and relations. In contrast to usual sets, fuzzy sets are characterized by a
membership relation taking its values from the unit interval [0,1] of the real
numbers. After its introduction in 1965 the theory of fuzzy sets and relations
was ranked to be some exotic field of research. The success during the past
years even with consumer products involving fuzzy methods causes a rapidly
growinginterestofengineersandcomputerscientistsinthisfield.Nevertheless,
Goguen [12] generalized this concept in 1967 to L-fuzzy sets and relations for
an arbitrary complete Brouwerian lattice L instead of the unit interval [0,1] of
the real numbers. He described one of his motivating examples as follows:
A housewife faces a fairly typical optimization problem in her grocery shop-
ping: she must select among all possible grocery bundles one that meets as
well as several criteria of optimality, such as cost, nutritional value, quality,
and variety. The partial ordering of the bundles is an intrinsic quality of this
problem. (Goguen [12] 1967)
It seems to be unnatural – comparing apples to oranges – to describe the
criteria of optimality by a linear ordering as the unit interval. Why should the
nutritional value of a given product be described by 0.6 (instead of 0.65, or
any other value from [0,1]), and why should a product with a high nutritional
valuebebetterthanaproductwithhighqualitysincethosecriteriaareusually
incomparable?
Thisobservationhasledtothetheoryofmultiobjective ormulticriteria opti-
mization problems (cf. [13]). Instead of combining several criteria into a single
number, and choosing the highest value, the concept of Pareto optimailty is
used. In this approach the elements that are not dominated are taken for fur-
ther considerations. Here an element x is said to dominate an element y if the
valuex foreachobjectiveiisgreaterthanorequaltothecorrespondingvalue
i
y of y. Traditional techniques of optimization and search have been applied
i
ix
x GOGUENCATEGORIES
in this area. Recently, even genetic algorithms have been used to search for
multicriteria optima (e.g., [30, 31]).
Oneimportantnotionwithinfuzzytheoryis0-1crispness.A0-1crispsetor
relation is described by the property that their characteristic function supplies
either the least element 0 or the greatest element 1 of the unit interval [0,1] or
more general the complete Brouwerian lattice L. The class of 0-1 crisp fuzzy
sets or relations may be seen as the subclass of regular sets or relations within
thefuzzyworld.Especiallyinapplications,thisnotionisfundamental.Wewant
to demonstrate this by considering two examples.
Infuzzydecisiontheorythebasicproblemistoselectaspecificelementfrom
a fuzzy set of alternatives. Therefore, several cuts are used [9, 24]. Basically,
an α-cut of a fuzzy set M is a set N such that an element x is in N if, and
only if, x is in M with a degree ≥α. Analogously, an α-cut of a fuzzy relation
R is a crisp relation S such that a pair of elements is related in S if, and only
if, they are related in R with a degree ≥ α. Some variants of this notion may
also be used. By definition, these cut operations are strongly connected to the
notion of crispness. In particular, using the notion of crispness, one may define
cut operations, and a cut operation naturally implies a notion of crispness.
Another example might be the development of a fuzzy controller. Usually
the output of the controller has to be a 0-1 crisp value since it is used to
control some nonfuzzy physical or software system. Therefore, a procedure,
called defuzzification, is applied to transform the fuzzy output into some 0-1
crisp value. This list of examples may be continued. The bottom line is that a
convenient theory for L-fuzzy relations should be able to express the notion of
crispness.
Today, fuzzy theory as well as its application is usually formulated as a
variation of set theory or some kind of many-valued logic (e.g., cf. [2, 14, 26]).
Althoughmanyalgebraiclawsaredeveloped,theseformalizationsarenotalge-
braic themselves. But an algebraic description would have several advantages.
Applicationsoffuzzytheorymaybedescribedbysimpletermsinthislanguage.
In this way, we get in some sense a denotational semantics of the application,
and,hence,amathematicaltheorytoreasonaboutnotionsascorrectness.One
may prove such properties using the calculus of the algebraic theory, which is
quite often more or less equational. Furthermore, this denotational semantics
may be used to get a prototype of the application.
On the other hand, the calculus of binary relations has been investigated
since the middle of the nineteenth century as an algebraic theory for logic and
set theory [36, 37]. A first adequate development of such algebras was given
by de Morgan and Peirce. Their work has been taken up and systematically
extendedbySchro¨derin[34].Morethan40yearslater,Tarskistartedwiththe
exhaustivestudyofrelationalgebras[35],andmoregenerally,Booleanalgebras
with operators [17].
The papers above deal with relational algebras presented in their classical
form. Elements of such algebras might be called quadratic or homogeneous;
relations over a fixed universe. Usually a relation acts between two different