UNIVERSITY OF OSTRAVA FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS

APPROXIMATING FORMULAS IN THE FRAME OF BASIC

Ph.D. THESIS

AUTHOR: Martina Daˇnkov´a SUPERVISOR: Irina Perfilieva

2005 OSTRAVSKA´ UNIVERZITA V OSTRAVEˇ PRˇ´IRODOVEDECKˇ A´ FAKULTA KATEDRA MATEMATIKY

APROXIMACNˇ ´I FORMULE V RAMCI´ BASIC LOGIKY.

DOKTORSKA´ DISERTACNˇ ´I PRACE´

AUTOR: Martina Daˇnkov´a VEDOUC´I PRACE:´ Irina Perfilieva

2005 Prohlaˇsuji, ˇze pˇredloˇzen´apr´ace je m´ym p˚uvodn´ım autorsk´ym d´ılem, kter´ejsem vypracoval samostatnˇe. Veˇskerou literaturu a dalˇs´ızdroje, z nichˇzjsem pˇri zpra- cov´an´ıˇcerpal, v pr´aci ˇr´adnˇecituji a jsou uvedeny v seznamu pouˇzit´eliteratury.

Ostrava...... (podpis) Beru na vˇedom´ı, ˇze tato doktorsk´adisertaˇcn´ıpr´ace je majetkem Ostravsk´euniverzity (autorsk´yz´akon C.ˇ 121/2000 Sb., §60 odst. 1), bez jej´ıho souhlasu nesm´ıb´yt nic z obsahu pr´ace publikov´ano. Souhlas´ım s prezenˇcn´ım zpˇr´ıstupnˇen´ım sv´epr´ace v Univerzitn´ıknihovnˇeOstravsk´e univerzity.

Ostrava...... (podpis) “It has long been an axiom of mine that the little things are infinitely the most important.” Sherlock Holmes

5 Summary

An interesting class of decision making systems is that which is based on lin- guistic , i.e. finite sets of IF-THEN rules describing a process, situation, environment etc. These rules often include natural language expressions character- ized by a position on an ordered scale, usually on a real interval. As typical examples of such expressions, let us light, loud, approximately 5, smaller then 7, warmer then 78 etc. The main task is to derive a conclusion from an observation on the basis of linguistic by means of some feasible . This thesis proposes a methodology for dealing with the above-described situa- tion and studies its properties. The background for it is fuzzy predicate calculus. Considered fuzzy IF-THEN rules are understood in two different ways, i.e. as lin- guistically expressed logical conjunctions or implications. We strictly distinguish the syntactic from the semantic level of study. First, we introduce a generalized notion of normal forms aimed at formalizing linguistically expressed collection of fuzzy IF-THEN rules and prove their represen- tation abilities for a special class of formulas. Special inference rules determining conclusion from the observation are proposed as well. Moreover, the properties of such obtained conclusion are studied. Later on, we characterize this special class of formulas from the functional point of view. This leads to the general result relating to an approximations by functions representing normal forms. Finally, we apply all these result and present the complete automated learning method which conceives the rule-base composed of a of IF-THEN rules from data assemble from a real process or environment.

Keywords: Normal forms, Approximate inference, IF-THEN rules, Fuzzy logic, Fuzzy relations, Approximation, Fuzzy Systems, Genetic Algorithms.

6 Anotace

V´yznamnou tˇr´ıdu rozhodovac´ıch syst´em˚utvoˇr´ısyst´emy zaloˇzen´ena jazykov´em popisu, tj. na koneˇcn´emnoˇzinˇepravidel typu JESTLIZE-PAKˇ popisuj´ıc´ıch proces, situaci, syst´emov´eprostˇred´ıap. Tato pravidla ˇcasto obsahuj´ıv´yrazy pˇrirozen´eho jazyka charakteristick´esvou pozic´ına uspoˇr´adan´eˇsk´ale, vˇetˇsinou na re´aln´em inter- valu. Typick´ym pˇr´ıkladem jsou v´yrazy jako lehk´y, hlasit´y, pˇribliˇznˇe5, m´enˇeneˇz7, teplejˇs´ıneˇz78 oC atd. Z´akladn´ım ´ukolem je odvodit z´avˇer z urˇcit´eho pozorov´an´ı na z´akladˇedan´eho jazykov´eho popisu prostˇrednictv´ım vhodn´einference. C´ılem t´eto disertace je navrhnout urˇcitou metodologii pro ˇreˇsen´ı situac´ı uve- den´ych v´yˇsea prozkoumat jej´ıvlastnosti. Tato metodologie je zaloˇzena na kalkulu predik´atov´efuzzy logiky. Uvaˇzovan´apravidla typu JESTLIZE-PAKˇ ch´apeme dvˇema odliˇsn´ymi zp˚usoby - bud’ jako jazykovˇevyj´adˇren´elogick´ekonjunkce, nebo implikace. V cel´epr´aci pˇr´ısnˇerozliˇsujeme syntaktickou a s´emantickou str´anku probl´emu. K tomuto ´uˇcelu nejprve zavedeme zobecnˇen´ypojem norm´aln´ıformy vhodn´ypro formalizaci jazykovˇevyj´adˇren´ych soubor˚upravidel typu JESTLIZE-PAKˇ a uk´aˇzeme, nakolik umoˇzˇnuje reprezentovat formule z urˇcit´etˇr´ıdy. Navrhneme rovnˇeˇzspeci´aln´ı odvozovac´ıpravidla pro stanoven´ız´avˇer˚uz pozorov´an´ıa prozkoum´ame vlastnosti takto z´ıskan´ych z´avˇer˚u. V dalˇs´ıˇc´asti pod´ame funkcion´aln´ıcharakterizaci uveden´etˇr´ıdy formul´ı, jeˇzvede k obecn´emu v´ysledku o aproximac´ıch funkcemi reprezentuj´ıc´ımi norm´aln´ıformy. Na z´avˇer provedeme aplikaci tˇechto v´ysledk˚ua pˇredvedeme metodu plnˇeauto- matick´eho uˇcen´ı, jeˇzvytv´aˇr´ıb´azi pravidel sloˇzenou z pravidel typu JESTLIZE-PAKˇ podle dat z´ıskan´ych z re´aln´ych proces˚uˇci prostˇred´ı.

Kl´ıˇcov´aslova: Norm´aln´ı formy, Pˇribliˇzn´adedukce, JESTLIZE-PAKˇ pravidla, Fuzzy logika, Fuzzy relace, Aproximace, Fuzzy syst´emy, Genetick´ealgoritmy.

7 Preface

Fuzzy are intend to handle information affected by inexactness or in the other words incorporate the vague phenomenon. Enormous number of applications in expert systems verify the role of fuzzy logic as an important tool compiling expert knowledge into a rule-base approximately describing a real system in a natural way. This is done using so called fuzzy IF-THEN rules. Formalizing these rules into the language of logic leads to formulas having special form commensurate with interpretation of available knowledge. The first type of fuzzy rules reflects a functional dependency, e.g. monotone dependency can be read as follows

”X is small” AND ”Y is small” OR ”X is medium” AND ”Y is medium” OR ”X is big” AND ”Y is big”

Such interpretation can be naturally formalized as a disjunctive normal form. A different type of fuzzy rules is that which forms conjunctive normal form from the logical point of view. This type of fuzzy rules can be regarded as summarizing an expert knowledge. Having on mind the same dependency as above, we can read it as

IF ”X is small” THEN ”Y is small” AND IF ”X is medium” THEN ”Y is medium” AND IF ”X is big” THEN ”Y is big”

Generally, we may speak about the same system or process but we describe it in

8 two different ways with distinct functional realization. On the other side, we expect that both descriptions are at least partially equivalent. The aim of this thesis is to propose generalized normal forms suitable to formalize a collection of fuzzy IF-THEN rules, which satisfy the natural requirement for partial equivalence and further investigate their relationship. Furthermore, the significant task comprises of introducing a new automated learning mechanism able to find the normal forms on the basis of an available information. The results together with the whole methodology from this thesis will be used in the software system LFLC 2000 developed on Institute for Research and Applications of Fuzzy Modeling. This software system allows to design, test and apply linguistic descriptions, and additionally, it is endowed with automated learning procedures. It already proved itself to be useful in practical applications. I want to express my gratitude to my supervisor Prof. Irina Perfilieva for her support, valuable comments and permanent encouragement which made it possible to finish this thesis. Moreover, I would like to thank my boss Prof. Vil´em Nov´ak for working conditions he created in Institute for Research and Applications of Fuzzy Modeling. The warm thanks goes to Libor Bˇehounek and Anton´ın Dvoˇr´ak for the watchful reading throughout this thesis. I would also like to thank colleagues from our Institute making the friendly atmosphere. Last but not least, I thank my parents for warm atmosphere which gave me strength in difficult times.

Ostrava, May 2005 Martina Daˇnkov´a

9 Table of Contents

Summary 6

Anotace 7

Preface 8

Table of Contents 10

List of Figures 13

List of Symbols 15

1 Introduction 16 1.1 Fuzzy Logic in the modern scientific world ...... 16 1.2 NormalformsandfuzzyIF-THENrules ...... 17 1.3 GeneralizationofNormalforms ...... 18 1.4 Theoutlineofthiswork ...... 20

2 Preliminaries 22 2.1 Fuzzysetsandfuzzyrelations ...... 22 2.1.1 Basicnotions ...... 22 2.1.2 T-norms and their residua as logical operations ...... 23 2.1.3 Extensional fuzzy relations ...... 28 2.2 Basicfuzzypredicatelogic ...... 34 2.2.1 Algebraic structures for BL and BL∀ ...... 35 2.2.2 Syntaxandsemantics...... 37 2.2.3 Provable propositional and predicate tautologies ...... 41 2.2.4 Normal Forms in BL∀ ...... 43

10 3 Logical approximation 48 3.1 Normal Forms for Extensional Formulas ...... 50 3.1.1 Characterization of extensional formulas ...... 52 3.1.2 Logical Approximation by Finite Normal Forms ...... 56 3.2 Logical approximation in approximate inference ...... 65 3.2.1 Approximate based on Normal Forms ...... 67 3.2.2 Rules of inference and the theory of fuzzy control ...... 71 3.2.3 Properties of approximate inferences ...... 73 3.2.4 Consequences to fuzzy control ...... 76

4 Approximations over linearly ordered BL-algebras 83 4.1 Normal Forms for extensional fuzzy relations ...... 84 4.1.1 Approximating abilities of Normal Forms ...... 85 4.1.2 Representation of extensional fuzzy relations ...... 87 4.1.3 Illustrations ...... 88 4.2 Functions associated with formulas of BL-logic ...... 88 4.2.1 Normal Forms for functions represented byLukasiewicz formulas 91 4.2.2 Functions associated with Product (Goguen) formulas . . . . . 94 4.2.3 Functions associated with G¨odel (Minimum) formulas . . . . . 100 4.2.4 Normal forms for functions associated with formulas of BL-logic106 4.3 Characterization of extensional fuzzy relations ...... 108 4.3.1 Extensionality w.r.t. Reflexivity means Lipschitz Continuity . 110 4.3.2 From Lipschitz Continuity to Extensionality w.r.t. Reflexivity 114 4.3.3 Notes on ∧-extensionality ...... 124 4.3.4 Approximation Theorem ...... 125

5 Genetic Algorithm + Logical Approximation = Optimized Rule-Base 128 5.1 IntroductiontoGeneticAlgorithms ...... 129 5.1.1 Basicprinciples ...... 130 5.1.2 Observations ...... 133 5.1.3 Simple genetic algorithm ...... 134 5.2 Genetic algorithms in logical approximation ...... 136 5.3 LearningfuzzyIF-THENrules...... 141

11 6 Conclusions 147

List of Author’s Publications 149

Bibliography 150

Index 158

12 List of Figures

2.1 Extensionality of a fuzzy relation w.r.t. different binary fuzzy relations 30 2.2 Illustration of the relations from Example 2.1.4...... 33

3.1 Fuzzy sets associated with the linguistic expressions from Example 3.2.1...... 70 3.2 A fuzzy control system based on DNF or CNF...... 71

4.1 Approximation of f(x) = sin(x) (black line) on [0, 1] by functions

represented by fDNF,5 (violet line) and fCNF,5 (blueline)...... 89 4.2Lukasiewicz implication expressed using (4.2.6) in c = [0.5, 0.5]. . . . 93 3 ¯ 4.3Lukasiewicz implication approximated by i=1 fT,ci (x), where ci = [c ,c ], i = 1, 2, 3, and c ’s are from {0.25, 0.5, 0.8}...... 94 i i i W 4.4 Monomial domains specified by →⊙...... 95 3 ¯ 4.5 Product implication approximated by i=1 fT,ci (x) in the form (4.2.13), where c = [c ,c ], i = 1, 2, 3, and c ’s are from {0.25, 0.5, 0.8}. . . . . 100 i i i i W 4.6 Illustration of a combination r1(x1,c1) ⊗ r2(x2,c2) of the type (4.3.6). 115 4.7 Relation from Figure 4.6 for ⊙...... 116 4.8 Illustration of the relation of the type (4.3.6) making FL-function F given by (4.3.20) extensional ...... 123 4.9 Illustration of the relation making FL-function F (x, y)= xy extensional124

5.1 Result of the approximation of FL-function f(x) = (1+e−x2 cos(36x))/2

by fDNF,k(x)usinggeneticalgorithmLAA...... 139 5.2 Application of LAA to f(x, y)= xy with ε = 0.1 and MaxNumber = 3.140 5.3 Normal form based Genetic fuzzy system...... 144

13 5.4 Application of AFL to the input data (blue dots) assumingLuka siewicz t-norm...... 145 5.5 Application of AFL to the input data (blue dots) assuming the t-norm with the additive generator g(x)=(1 − x)2...... 146

14 List of Symbols

∗, t-norm, 24 ¬, negation, 37 J, language of BL∀, 37 ∨, disjunction, 37 J(T ), language of T over BL∀, 38 ∧, conjunction, 37

Nk, set of k nodes, 84 →∗, residual operation, 25, 26 T , theory over BL∀, 38 ⋆, binary operation on L, 28 V n(A), the set of all variations of order ⊢, provability sign, 39 n from A, 84 ∨, supremum, 35 CNF, conjunctive normal form, 44 ∧, infimum, 24

CNFϕ,k, CNF for formula ϕ and k con- de, extended standard metric, 111

stant symbols, 51 ds, standard metric, 34

DNF, disjunctive normal form, 43 fCNF,k, CNF for L-fuzzy relation f and

DNFϕ,k, DNF for formula ϕ and k con- k nodes, 84

stant symbols, 51 fDNF,k, DNF for L-fuzzy relation f and ∃, existential quantifier, 37 k nodes, 84 ∀, universal quantifier, 37 BL∀, basic predicate logic, 34 ⊂, fuzzy subset of, 23 ∼ BL, basic propositional logic, 34 ↔∗, biresidual operation, 25, 26 N, set of natural numbers, 34 GR, generalization rule, 38 N0, set of natural numbers with 0, 34 R, set of real numbers, 34 MP, modus ponens, 38 R+, set of positive real numbers, 34 Z, set of integers, 34 L, BL-algebra, 35 M, L-structure, 39 ⊙, Product t-norm, 24 ⊗,Lukasiewicz t-norm, 24 △, Baaz delta, 90 &&, strong conjunction, 37 ≡, equivalence, 37

15 Chapter 1

Introduction

1.1 Fuzzy Logic in the modern scientific world

The term fuzzy logic is understood in several different ways. A recent opinion says that the meaning of this term can be formulated as fuzzy set-based methods for approximate reasoning, which is a subtopic of Artificial Intelligence, [66, 23]. This seems to be rather misguided since the term logic is used to denote some formal system or calculus. The confusion between content of fuzzy logic and its applications follows mainly from the absence of communication between the groups of fuzzy logicians and system engineers. Let us quote from H´ajek [28]: “Fuzzy logic is a logic. It has its and and notion of consequence. It is a study of consequence.” This quotation is however non-delimitative and emphasizes only the fact that fuzzy logic should preserve some general attributes. In Zadeh’s sense [65], fuzzy logics grasp many- valued logics aiming at formalization of approximate reasoning. By many, formal calculi of many-valued logic are considered to be the kernel or base for fuzzy logic. An example of such an approach has been presented in [6], where the class of “fuzzy logics” is delimited as weakly implicative logics which are complete with respect to linear semantics. Nevertheless, each such a delimitation brings restrictions leading to infertile hag- gling about dismissed cases. To propose an universal characteristics embracing all “fuzzy” logics and “fuzzy logic” based methods is thus nearly impossible. Espe- cially, when the motivation for a different formalization is grounded by appropriate arguments. An outstanding example is Nov´ak’s graded approach leading to fuzzy

16 logic with evaluated syntax [44], based on Rational Pavelka logic of [50]. Originally, the motivation for doing fuzzy logic was to represent expert knowledge in a unified form, so called IF-THEN rules, which soundly reflects this knowledge and model a physical systems. Involving an inference mechanism consistent with common human reasoning to such a model leads to an application of fuzzy logic in control proposed by L. Zadeh in [63]. Up to now, numerous real-world appli- cations verifying Zadeh’s concept have been realized. The list of the most known applications together with their historical ordering can be found in [47]. In spite of the original attempt to this field, in the last few years fuzzy rules based systems are more often considered as non-fuzzy universal approximators of functions, e.g. in works of B. Kosko [36], L. Wang and J. Mendel [61], Y. Yam, H. T. Nguyen, V. Kreinovich [67], or I. Perfilieva [57]. The word non-fuzzy is used because these approximators do not use an expert knowledge to assemble control laws and consequently the readability of fuzzy rules, as pieces of knowledge formulated using expressions of a small fragment of natural language (see Chapter 6 in [44]), is lost. However, most of such approximation methods deal with fuzzy numbers or sim- ilarities between objects in the background as e.g. in [9, 21] or fuzzy orderings [7]. This fact seems to approve these methods to keep the name fuzzy in the context of fuzzy logic and fuzzy . Nevertheless, it should be pointed out that we cannot avoid the natural language level when communicating with humans and consequently the interpretation of fuzzy IF-THEN rules should be understandable also in the common sense.

1.2 Normal forms and fuzzy IF-THEN rules

As it has been mentioned before, it is possible to implement human reasoning in a form of a sequence of IF-THEN rules used as the basis for control of some physical system. Such control laws (rules) express qualitative characterization of variables using fuzzy predicates and functional dependencies between variables using condi- tional sentences with fuzzy predicates. Basically, two different interpretations of fuzzy IF-THEN rules are distinguished [32, 46, 44, 24]. The first type of interpretation is called fuzzy interpolation [58, 43] or fuzzy approximation [56, 20], which reflects knowledge about some imprecisely known function. The second interpretation can be described as logical inference in the presence of fuzzy information. Here, logical inference means derivation of new

17 facts from some other known ones using formal deduction rules. Due to [46], whenever we understand IF-THEN rules as Max-t-norm rules in fuzzy systems, they are of the first type, and when as Min-t-implication rules, they are of the second type. Moreover, logical inference makes possible to provide inter- polation in a similar manner as Max-t-norm rules, see also [45]. Special formulas, introduced by I. Perfilieva in [44] and called normal forms, aggregate available local information about a fuzzy relation associated with the formula of fuzzy predicate calculus. Thus, these normal forms can be viewed as collections of fuzzy IF-THEN rules and consequently they are related to the field of fuzzy approximation.

1.3 Generalization of Normal forms

Originally, the notion of normal form is well known from the . There, the normal form of a formula is understood as its simplification having the form of disjunction or conjunction of elementary terms. Moreover, they are equivalent to initial formulas in the classical sense. Different formal as well as algebraical proofs of this equivalence can be found in almost every book about classical . The situation in fuzzy logic is rather ambiguous. When searching the literature, it is possible to find two main ways of generalization of normal forms into the fuzzy case:

• in a way that the equivalence with an initial formula is fulfilled,

• and the other way can be characterized as preserving a ”partial” equivalence.

The first variant relates to a general characterization of normal forms from classical logic, i.e. that they are mostly based on two logical operations (connectives) which join literals into the unified forms and moreover, they are equivalent with the initial formula or, in other words, they are equal to fuzzy relation associated with this formula. Let us recall the results of P. Cintula and B. Gerla [13], where the normal forms represent formulas of propositional product logic. The representation of functions represented by formulas of propositionalLukasiewicz and G¨odel logics can be found in [27]. Constructive proofs for formulas ofLukasiewicz logic has been introduced

18 by I. Perfilieva in [44] and lately by A. Di Nola and A. Lettieri in [25]. Another known result for formulas over MV-algebras is that of D. Mundici published in [11]. All these methods have a common basis. They arise from functional systems as- sociated with the given fuzzy logic and then transform algebraic formulas associated with the corresponding formulas of the respective formal logical system. The way of transformation and the logical connectives used differentiate these methods. The second attempt is based on the shape of classical normal form for Boolean functions where the classical connectives are replaced by generalized ones. Having such a generalization, we can formulate problems whose solutions clarify the rela- tionship between normal forms and the initial fuzzy relation. Differences between such approaches are determined by the interpretation of inner parts of the classical normal form. In the works of I. Perfilieva [52, 56], inner part of a normal form consists of two parts combined by conjunction: the first part expresses the similarity of input to some (fixed) point, and the second part describes a value of the fuzzy relation in this point. In the full generality, we can understand the first part as a characterization of a local domain by its membership function and the second part as a value of the fuzzy relation provided that its arguments lie inside the respective local domain, see my paper [17]. From the other works of this type, recall a generalization introduced by I.B. T¨urk¸sen in [59], where the inner parts of normal forms remain untouched. Hence, these generalized normal forms are of the same shape as the classical ones but with different functional interpretation. Since they are constructed on the basis of a Boolean function, it is possible only to study their relationship with respect to different interpretations of logical connectives, but they bring nothing new on the syntactical level. And the claim about partial equivalence works just in the case of interpreting the connectives by the operations forming the Heyting algebra. In this thesis, we consider a direct generalization of the second type from the normal forms for the Boolean functions associated with the formulas of classical propositional logic

c1 cn DNFf (x)= x1 ∧ . . . ∧ xn , (1.3.1) f(_x)=1 c′ ′ 1 cn CNFf (x)= x1 ∨ . . . ∨ xn , (1.3.2) f(^x)=0

19 which is equivalent to

c1 cn DNFf (x)= x1 ∧ . . . ∧ xn ∧ f(c1,...,cn), (1.3.3) n (c1,...,c_n)∈{0,1} c1 cn CNFf (x)= x1 ∧ . . . ∧ xn → f(x1,...,cn), (1.3.4) n (c1,...,c^n)∈{0,1} into the fuzzy case, so that we exchange {0, 1}-valued operations by [0, 1]-valued ci ones, the Boolean function by a fuzzy relation and xi by a description of the neigh- bourhood of ci expressed by binary fuzzy relation R for all i = 1,...,n. This generalization, as we will see later, can be extended and further used for an approx- imation of extensional fuzzy relations (interpretations of fuzzy predicate formulas). By approximation we mean “partial” equivalence, i.e. the equivalence is bounded from below. We will introduce two types of normal forms, namely infinite which has been originally introduced in my [17] (under a different name) and discrete (or finite) normal forms appearing in [44] in the chapter elaborated by I. Perfilieva and further studied in [56]. We will see that the first type of normal forms can be viewed as a precise representation of the initial fuzzy relation while the second one serves us as a “universal” approximation formula. Let us be aware that on the algebraic level we do not have the notion of a limit at our disposal. This is the main argument for having normal forms of the infinite type which serve as a limit element of a sequence of finite normal forms where the number of nodes specifying them increases. Consequently, normal forms of the finite type can be viewed as a finite sub-formula of the respective normal form of the infinite type.

1.4 The outline of this work

This thesis can be divided into three main parts. The first part is devoted to the study of logical approximation by means of generalized normal forms and their contribution to drawing conclusions on the basis of feasible approximate inference method. This is studied form the syntactical point of view in Chapter 3. Further, normal forms of both types and their functional realization over lin- early ordered BL-algebras will be shown in Chapter 4. This part of the research is extended by a functional characterization of the class of formulas which are repre- sentable by normal forms. The approximation theorem, which validates the use of

20 the term “universal approximation” formula for normal form of the finite type, is placed at the end of the chapter. Finally, applications of normal forms in the automated learning methodologies are elaborated in Chapter 5. There, the parameters of normal forms are determined using genetic algorithms. The choice of genetic algorithms has been motivated by their global capability of finding approximate solutions to difficult-to-solve problems.

21 Chapter 2

Preliminaries

In order to provide the mathematical apparatus for the following studies of the consequences of the logical approximation in the frame of Basic logic or equivalently approximations by fuzzy relations represented by formulas, the basics of fuzzy sets, lattice theory and many-valued logics will be introduced. In the sequel, we will always refer to the literature in case of reproduced results.

2.1 Fuzzy sets and fuzzy relations

2.1.1 Basic notions

The concept of fuzzy sets and fuzzy logic has been introduced by Lotfi A. Zadeh in his seminal work [62] published in 1965. His idea was to generalize the concept of an ordinary set, which is uniquely determined by its characteristic function. Strictly speaking, every ordinary set A can be associated with a function

1, if x ∈ A, χA(x)= ( 0, otherwise.

The following definition is a natural generalization of the above presented associa- tion.

22 Definition 2.1.1 Let L be a scale of values having the smallest 0 and greatest

1 elements, respectively. Moreover, let U be a non-empty set. Then the L-fuzzy set

A is identified with a function

A : U −→ L assigning a value A(x) ∈ L to each element x ∈ U, and we write A ⊂ U. The value ∼ A(x) is interpreted as the degree to which x belongs to A.

Similarly, a fuzzy relation is a fuzzy subset of a Cartesian product of non-empty sets. The values of such a fuzzy relation are interpreted as the degrees in which particular individuals relate to each other.

Definition 2.1.2 Let M1,...,Mn be non-empty sets. An n-ary L-fuzzy relation R is a L-fuzzy set

R ⊂ M × . . . × M . ∼ 1 n

We will mostly assume a Cartesian product of the only set. In this connection, we prefer to introduce the following notion of L-fuzzy relation specified with respect to some set.

Definition 2.1.3 Let M be a non-empty set of objects. A function R : M n → L is called n-ary L-fuzzy relation on M.

It is very natural is to consider L = [0, 1]. In this case we will use notions fuzzy set and fuzzy relation instead [0, 1]-fuzzy set and [0, 1]-fuzzy relation, respectively.

2.1.2 T-norms and their residua as logical operations

Original motivation of K. Menger in [39] for introduction the class of generalized multiplications known as triangular norms (t-norms) was not logical. The main idea

23 was to generalize the concept of the triangular inequality into probabilistic metric spaces. Since t-norms preserve the fundamental properties of the crisp conjunction, consequently they become to be interesting for fuzzy logic as a natural generalization of conjunction in many valued reasoning systems.

The eminent role between the basic structures for truth values of many-valued logics is played by the BL-algebra, which is residuated lattice enriched by two special properties. Continuous t-norms are very important there because they determine

BL-algebras on the unit interval [0, 1] with their standard linear ordering.

Definition 2.1.4 An operation ∗ : [0, 1]2 → [0, 1] is called triangular norm (t- norm) if it is commutative, associative, non-decreasing mapping fulfilling boundary condition, i.e. if for all x, y, z ∈ [0, 1]:

x ∗ y = y ∗ x (commutativity), x ∗ (y ∗ z)=(x ∗ y) ∗ z (associativity), x ≤ y =⇒ x ∗ z ≤ y ∗ z (monotonicity), x ∗ 1= x (boundary condition).

Example 2.1.1 Below, we show the best known examples of continuous t-norms which serve as natural interpretations of a generalized conjunction:

(1) Minimum t-norm x ∗ y = x ∧ y,

(2) Product t-norm x ∗ y = x ⊙ y = x · y,

(3) Lukasiewicz t-norm x ∗ y = x ⊗ y = max(0, x + y − 1).

It follows from the definition of the t-norm that it is a monoidal operation on

[0, 1]. Furthermore, h[0, 1], ∧, ∨i is a complete lattice. Therefore, we can introduce the residual operation in the following form.

24 2 Definition 2.1.5 Let ∗ be a t-norm. The residual operation →∗: [0, 1] → [0, 1] is defined by

x →∗ y = {z ∈ [0, 1] | x ∗ z ≤ y}. (2.1.1) _ The left-continuity of ∗ guarantees that residual operation defined by (2.1.1) fulfils the adjunction property, i.e. x ∗ z ≤ y iff z ≤ x →∗ y.

Furthermore, we will use the following derived operations

n x∗ = x ∗ . . . ∗ x, n−times n | {z } xi = x1 ∗ . . . ∗ xn, i=1 O x ↔∗ y =(x →∗ y) ∗ (y →∗ x).

n n We use only x whenever it is clear which ∗ is used in the expression x∗ .

Interesting t-norms are those which have additive generators. It means that these t-norms are build from a function of one variable and its inverse. Consequently, expressions including a high amount of operations representable on the basis of additive generator are easy to manipulate.

Definition 2.1.6 Let g : [0, 1] → [0, ∞] be a continuous strictly decreasing function such that g(1) = 0 and ∗ is a t-norm. Then g is an additive generator of ∗ if

x ∗ y = g(−1)(g(x)+ g(y)), (2.1.2) holds for all x, y ∈ [0, 1]. The function g(−1) : [0, ∞] → [0, 1] such that

g−1(y) if y ∈ [0,g(0)], g(−1)(y)= ( 0 if y ∈ (g(0), ∞], is called the pseudoinverse of g.

25 Additive continuous generators of t-norms are determined uniquely up to a pos- itive multiplicative constant.

Example 2.1.2 The following are examples of additive generators for continuous t-norms:

(1) gL(x) = 1 − x generatesLukasiewicz t-norm,

(2) gP (x)= − ln x generates product t-norm.

For a t-norm generated by a continuous additive generator g, the corresponding residuum is given by

−1 x →∗ y = g (max(0,g(y) − g(x))), (2.1.3)

and the corresponding biresidual operation by

−1 x ↔∗ y = g (|g(x) − g(y)|). (2.1.4)

Definition 2.1.7 A t-norm ∗ is called Archimedean if for every x, y ∈ (0, 1) there exists n ∈ N such that xn < y.

The following theorem (see [35]) characterizes the class of generated t-norms.

Theorem 2.1.1

A t-norm ∗ : [0, 1]2 → [0, 1] is a continuous Archimedean t-norm if and only if it has a continuous additive generator.

An interesting result has appeared in [35], where the authors proved that an arbi- trary continuous t-norm can be approximated by t-norms having additive generators with arbitrary precision.

26 Definition 2.1.8 Let ∗ be a t-norm generated by the additive generator g. Then

• ∗ is called nilpotent if g(0) < +∞,

• ∗ is called strict if g(0) = +∞.

Theorem 2.1.2 demonstrates that continuous Archimedean t-norms can be di- vided in two disjoint classes, namely nilpotent and strict (see [35] or Theorem 2.10 in [44]).

Theorem 2.1.2

Let ∗ be a continuous Archimedean t-norm. Then ∗ is nilpotent if and only if ∗ is not strict.

In fact, a continuous Archimedean t-norm is nilpotent (strict) if and only if it is isomorphic withLukasiewicz (product) t-norm.

Remark 2.1.1 Let ∗ be a continuous Archimedean t-norm, then

n 1 if x ≤ y, lim (x →∗ y) = n→∞ ( 0 otherwise.

Let us denote

∞ n (x →∗ y) = lim (x →∗ y) . (2.1.5) n→∞

This operation is known as Wu implication [48]. Moreover, let

0 (x →∗ y) = 1. (2.1.6)

In the study of continuous t-norms, the basic t-norms (Lukasiewicz, product and minimum) have a distinguished role reflected in the concept of constructing new t-norms known as ordinal sum of t-norms.

27 2.1.3 Extensional fuzzy relations

Extensionality is well known notion from the classical set theory. A generalized version of this notion has been introduced by F. Klawonn and R. Kruse in [34].

There, the extensional fuzzy relations are defined w.r.t. a similarity relation on their domain, where the notion of similarity stands for generalized relation of equality between objects.

Below, we present a more general case of extensionality. The comes from the fact that extensional fuzzy relations defined w.r.t. similarity have properties related to Lipschitz continuity. Let us recall the paper [40] where it has been proved that in a t-norm based algebra, the extensionality of a fuzzy relation w.r.t. a simi- larity is equivalent to Lipschitz continuity w.r.t. the pseudo-metric induced by the similarity.

In the following generalized definition, we assume that the description of a neigh- bourhood of some node is different for different variables and in this way, we should obtain better characterization of a given fuzzy relation.

Let M be some nonempty set of objects and L be some scale of truth values such that it includes 0 as minimal element and 1 as maximal one, and ⋆ : L × L → L.

Definition 2.1.9 An L-fuzzy relation P : M n → L is extensional w.r.t. binary

L-fuzzy relations R1,...,Rn on M if for each x1,...,xn, y1,...,yn ∈ M,

R1(x1, y1) ⋆ ··· ⋆ Rn(xn, yn) ⋆ P (x1,...,xn) ≤ P (y1,...,yn). (2.1.7)

If R1 = R2 = . . . = Rn = R then we call P extensional w.r.t. one L-fuzzy relation R.

It is worth to consider also left and right extensionality. Analogously to the

28 notion of continuity at some point, right (left) extensionality expresses the exten- sionality property in this point. Informally speaking, extensionality characterizes closeness between P (x1,...,xn) and P (c1,...,cn) depending on a distance from the point [c1,...,cn].

Definition 2.1.10 An n-ary L-fuzzy relation P on M is right extensional at [c1,...,cn] ∈

n M on M w.r.t. binary L-fuzzy relations R1,...,Rn if for each x1,...,xn ∈ M

R1(x1,c1) ⋆ ··· ⋆ Rn(xn,cn) ⋆ P (x1,...,xn) ≤ P (c1,...,cn), (2.1.8)

n holds. And P is left extensional at [c1,...,cn] ∈ M on M w.r.t. binary L-fuzzy relations R1,...,Rn if for each x1,...,xn ∈ M

R1(c1, x1) ⋆ ··· ⋆ Rn(cn, xn) ⋆ P (c1,...,cn) ≤ P (x1,...,xn), (2.1.9) holds.

Analogously, if R1 = R2 = . . . = Rn = R then we call P right respectively left extensional w.r.t. R at [c1,...,cn].

Example 2.1.3 Let M = L = [0, 1] and P (x, y) = sin(x) sin(y), for all x, y ∈ M, defines a fuzzy relation on M 2.

(1) AssumingLukasiewicz t-norm ⊗, we obtain that P is extensional w.r.t. binary

fuzzy relation R1 given by

R1(x, y)= x →⊗ y

and also when we consider product t-norm ⊙, we obtain that P is extensional

w.r.t. R2 given by

R2(x, y) = sin(x) →⊙ sin(y).

29 (a) Picture of a fuzzy relation P (x,y) from (b) Violated extensionality of P (x,y) w.r.t. Example 2.1.3 R3(x,y) = sin(x) →⊗ y at point [0.8, 0.8]

Figure 2.1: Extensionality of a fuzzy relation w.r.t. different binary fuzzy relations

(2) P is not extensional w.r.t. R3 given by

R3(x, y) = sin(x) →⊗ y.

We can visualize this fact on Figure 2.1(b), where the pyramidal part above

the original fuzzy relation represents the couples of [x, y] for which the exten-

sionality does not hold at the point [0.8, 0.8], i.e.

R3(0.8, x) ⊗ R3(0.8, y) ⊗ P (0.8, 0.8) > P (x, y).

(3) If we change to the product t-norm ⊙, it is clear that P is extensional w.r.t.

′ ′ R1(x, y) = x →⊙ y, P is extensional w.r.t. R2(x, y) = sin(x) →⊙ sin(y) and

′ also P is extensional w.r.t. R3(x, y) = sin(x) →⊙ y. Lemma 2.1.3

Let P be an n-ary L-fuzzy relation on M

1. If P is extensional w.r.t. R1,...,Rn then it is left (right) extensional w.r.t.

n R1,...,Rn at [c1,...,cn] for arbitrary [c1,...,cn] ∈ M .

30 n 2. If P is left (right) extensional w.r.t. R1,...,Rn at each point [c1,...,cn] ∈ M

then it is extensional w.r.t. R1,...,Rn. : Obvious. 2

In the all previous definitions and results, the L-fuzzy relations R1,...,Rn were arbitrary ones. Later on, we will work also with L-fuzzy relations extensional w.r.t. binary L-fuzzy relations which are supposed to fulfill some of the following proper- ties.

Definition 2.1.11 Let ⋆ : L × L → L and consider binary L-fuzzy relation R on a domain M. Then

1. R is called reflexive if

R(x, x)= 1, for all x ∈ M,

2. R is called symmetric if

R(x, y)= R(y, x), for all x, y ∈ M,

3. R is called ⋆-transitive if

R(x, y) ⋆ R(y, z) ≤ R(x, z), for all x, y, z ∈ M.

The following are examples of binary fuzzy relations fulfilling different properties.

Example 2.1.4 Let ∗ be a continuous t-norm and →∗ its associated residuum. Moreover, let A(x) be an arbitrary fuzzy set on a non-empty domain M, c ∈ [0, 1] and k, l ∈ N.

k (1) R1(x, y) = c ∗ (A(x) →∗ A(y)) defines the transitive binary fuzzy relation on M,

31 k (2) R2(x, y) = c ∗ (A(x) ↔∗ A(y)) defines the symmetric and transitive binary fuzzy relation on M,

k l (3) R3(x, y)=(A(x) →∗ A(y)) ∗ (A(y) →∗ A(x)) defines the reflexive and transi- tive binary fuzzy relation on M,

(4) In this example, we will show a binary fuzzy relations which are reflexive but

not transitive.

Let ∗ be an Archimedean t-norm and T (x) : M → N be a mapping such that

there exist r, s ∈ M:

A(r) ≤ A(s) and T (r) − T (s)= n − 1,

n where n fulfills (A(r) →∗ A(t)) < (A(s) →∗ A(t)) for some t ∈ M. Then

T (x) R4(x, y)=(A(x) →∗ A(y)) ,

defines the reflexive binary fuzzy relation on M.

The reflexivity follows from the properties of →∗ and transitivity is violated at r,s,t ∈ M, since

T (r)−T (s) (A(r) →∗ A(t)) ∗ (A(r) →∗ A(t)) < (A(s) →∗ A(t)),

T (s) T (r)−T (s) T (s) (A(r) →∗ A(t)) ∗ (A(r) →∗ A(t)) < (A(s) →∗ A(t)) , i.e.

R4(r, t) < R4(s,t),

and using the fact that A(r) →∗ A(s) = 1, we obtain

T (r) R4(r, t) < (A(r) →∗ A(s)) ∗ R4(s,t),

R4(r, t) < R4(r, s) ∗ R4(s,t).

The relations R1,...,R3 in the case of product t-norm are depicted on the Figure 2.2.

32 (a) Transitive relation R1 (b) Symmetric and transitive relation R2

(c) Reflexive and transitive relation R3

Figure 2.2: Illustration of the relations from Example 2.1.4.

33 For the set of real numbers, positive real numbers, natural numbers, natural numbers with 0, and integers, we will use the standard notation R, R+, and N, N0, and Z, respectively. Moreover, the standard metric on R will be denoted by ds,

n where ds(x, y)= |x − y|. In this thesis, we will consider for R mostly the standard

n metric given by ds(x, y)= i=1 |xi − yi| and x = [x1,...,xn], y = [y1,...,yn].

We will often use the notionP of Lipschitz continuity in the following text. Let us remind this notion in the definition below.

Definition 2.1.12 Let M ⊆ R be a non-empty set. We call a function F (x1,...,xn) on M n Lipschitz continuous w.r.t. metric d if there exists K ∈ (0, ∞) such that

|F (x) − F (y)| ≤ Kd(x, y), (2.1.10) for each x, y ∈ M n.

We will speak about standard Lipschitz continuity whenever d is the standard metric.

It is clear that every Lipschitz continuous function is uniformly continuous and hence continuous. Conversely, the continuity of a function is the necessary condition for its Lipschitz continuity. But the sufficient condition is to be a differentiable function with bounded partial derivatives.

2.2 Basic fuzzy predicate logic

In [28], H´ajek defined an axiomatic system for basic propositional logic (BL) and basic predicate logic (BL∀). And later, in [12] the authors proved that the axiomatic system of BL is sound and complete with respect to the algebraic structure induced by continuous t-norms.

34 Below, we will give a brief overview of BL-predicate logic, which will be the basis of our constructions.

2.2.1 Algebraic structures for BL and BL∀

The interpretation of the logical connectives of BL and BL∀ is determined by the respective operations of the linearly ordered BL-algebra L on a set L which is an algebra

L = hL, ∧, ∨, ⋆, →⋆, 0, 1i (2.2.1) with four binary operations and two constants such that

(i) (L, ∨, ∧, 0, 1) is a lattice with 0 and 1 as the least and largest elements w.r.t.

the lattice ordering,

(ii) (L, ⋆, 1) is an Abelian monoid, i.e. the multiplication ⋆ is associative, commu-

tative and 1 ⋆ x = x for all x ∈ L,

(iii) ⋆ and →⋆ form an adjoint pair, i.e.

z ≤ (x →⋆ y) iff x ⋆ z ≤ y

for all x, y, z ∈ L,

(iv) moreover, for all x, y ∈ L

x ⋆ (x →⋆ y)= x ∧ y,

(v) and finally L is linearly ordered, i.e. for each pair x, y

x ∧ y = x or x ∧ y = y,

or equivalently

x ∨ y = x or x ∨ y = y.

35 In particular, when L = [0, 1], the operation ⋆ stands for a continuous t-norm. And apparently each continuous t-norm determines a BL-algebra on the unit interval

[0, 1] with its standard linear ordering.

Example 2.2.1 The following are examples of linearly ordered BL-algebras

• G¨odel algebra

LG = h[0, 1], ∧, ∨, →G, 0, 1i, (2.2.2)

where the multiplication ⋆ = ∧ and

1, x ≤ y; x →∧ y = ( y, otherwise.

• Standard product algebra

also known as Goguen algebra

LP = h[0, 1], ∧, ∨, ⊙, →⊙, 0, 1i, (2.2.3)

where the multiplication ⋆ = ⊙ and

1, x ≤ y; x →⊙ y = y ( x , otherwise.

• Lukasiewicz algebra

LL = h[0, 1], ∧, ∨, ⊗, →⊗, 0, 1i, (2.2.4)

where the multiplication x ⋆ y = x ⊗ y = max(0, x + y − 1) and

x →⊗ y = min(1, 1 − x + y).

• Boolean algebra for classical logic

LB = h{0, 1}, ∧, ∨, →, 0, 1i, (2.2.5)

where the multiplication ⋆ = ∧ and → is the classical implication.

36 2.2.2 Syntax and semantics

The language J of BL∀ includes a non-empty set of predicates, a set of object con- stants, object variables, a set of connectives {&, →}, truth constants 0¯, 1,¯ quantifiers

∀, ∃ and does not include functional symbols.

Terms are object variables and object constants.

By a formula, we mean a formula of BL∀ in the language J, build in the usual way, i.e. each P (t1,...,tn), where P is an n-ary predicate and t1,...,tn are terms, and truth constant is an atomic formula. Each formula results from atomic formulas by iterated use of the following rule: if ϕ, ψ are formulas and x is an object variable then (∀x)ϕ, (∃x)ϕ, ϕ&ψ and ϕ → ψ are formulas.

In BL∀ it is possible to define the following derived connectives:

¬ϕ as ϕ → 0¯ ϕ ∧ ψ as ϕ&&(ϕ → ψ) ϕ ∨ ψ as [(ϕ → ψ) → ψ] ∧ [(ψ → ϕ) → ϕ] ϕ ≡ ψ as (ϕ → ψ)&&&(ψ → ϕ)

Before we summarize logical calculus of BL∀, let us roughly recall that an object constant is always substitutable into a formula and a variable y is substitutable into

ϕ for x if the does not change any free occurrence of x in ϕ into a bound occurrence of y.

37 BL∀ consists of the following axioms of BL for connectives:

(ϕ → ψ) → [(ψ → χ) → (ϕ → χ)], (2.2.6)

(ϕ&ψ) → ϕ, (2.2.7)

(ϕ&ψ) → (ψ&ϕ), (2.2.8)

[ϕ&&(ϕ → ψ)] → [ψ&&(ψ → ϕ)], (2.2.9)

[ϕ → (ψ → χ)] → [(ϕ&ψ) → χ], (2.2.10)

[(ϕ&ψ) → χ] → [ϕ → (ψ → χ)], (2.2.11)

[(ϕ → ψ) → χ] → [((ψ → ϕ) → χ) → ϕ], (2.2.12)

0¯ → ϕ, (2.2.13) together with the axioms on quantifiers:

(∀x)ϕ(x) → ϕ(t) (t is substitutable for x in ϕ), (2.2.14)

ϕ(t) → (∃x)ϕ(x) (t is substitutable for x in ϕ), (2.2.15)

(∀x)(ϕ → ψ) → (ϕ → (∀x)ψ) (x is not free in ϕ), (2.2.16)

(∀x)(ϕ → ψ) → ((∃x)ϕ → ψ) (x is not free in ψ), (2.2.17)

(∀x)(ϕ ∨ ψ) → ((∀x)ϕ ∨ ψ) (x is not free in ψ), (2.2.18) and two deduction rules ϕ, ϕ → ψ (modus ponens - MP), ψ ϕ (generalization - GR). (∀x)ϕ

The notions of theory T , proof, provability, proof and provability in a theory over

BL∀ are defined in the same way as in classical logic.

Definition 2.2.1 • A theory T over BL∀ is a set of formulas. Elements of T

are axioms T and J(T ) denotes its language.

38 • A proof in a theory T is a sequence of formulas ϕ1 . . . , ϕk, where each member is either an axiom of BL∀ or a member of T or follows from some preceding

members of the sequence using the deduction rules.

• We say that formula ϕ is provable in T if there is a proof in T such that ϕ is

its last element. We denote this fact by T ⊢ ϕ.

An L-structure M = hM, (rP )P ∈J , (mc)c∈J i for the language J consists of a non-

n empty domain M, fuzzy relations rP : M −→ L assigned to each n-ary predicate symbol P , and designated elements mc ∈ M assigned to each object constant c.

The interpretation of the logical connectives ∧,∨,&,→ is determined by the respective operations of the BL-algebra L on a set L which is

L = hL, ∧, ∨, ⋆, →∗, 0, 1i

The other two derived connectives ¬,≡ are interpreted by the respective opera- tions ¬⋆, ↔⋆:

¬⋆a = a →⋆ 0,

a ↔⋆ b =(a →⋆ b) ∧ (b →⋆ a).

We have used the fact that the formula p ≡ q being understood as the short of

(p → q)&&&(q → p) is equivalent to (p → q) ∧ (q → p).

An interpretation of a formula ϕ is given in usual Tarskian style.

Definition 2.2.2 (1) An M-evaluation of object variables is a mapping e assigning

to each object variable x an element e(x) ∈ M. Let e1,e2 be two evaluations.

Let e1 =x e2 means that e1(y)= e2(y) for each variable y distinct from x.

(2) Let L be a BL-algebra of the form (2.2.1). The truth value of a term determined

39 by M,e is defined as follows

||x||M,e = e(x), ||c||M,e = mc.

The truth value of a formula determined by M,e we define by

||0¯||M,e = 0

||1¯||M,e = 1

L L L ||ϕ ∨ ψ||M,e = ||ϕ||M,e ∨ ||ψ||M,e

L L L ||ϕ ∧ ψ||M,e = ||ϕ||M,e ∧ ||ψ||M,e

L L L ||ϕ → ψ||M,e = ||ϕ||M,e →⋆ ||ψ||M,e

L L L ||ϕ&ψ||M,e = ||ϕ||M,e ⋆ ||ψ||M,e

L L ′ ||(∀x)ϕ(x)||M,e = inf{||ϕ(c)||M,e′ | e =x e },

L L ′ ||(∃x)ϕ(x)||M,e = sup{||ϕ(c)||M,e′ | e =x e }.

In the sequel we will work with L-safe L-structures M only. It means that the value

L ||ϕ||M,e is defined for all ϕ and evaluation e.

Definition 2.2.3 Let T be a theory over BL∀, let L be a BL-algebra of the form

(2.2.1) and M be an L-structure for the language J(T ). Moreover, let ϕ be a formula of T .

(1) The truth value of ϕ in M is

L L ||ϕ||M = inf{||ϕ||M,e| e is M-evaluation}.

L (2) A formula ϕ is an L- if ||ϕ||M = 1L for each L-structure M.

L (3) M is an L-model of T if all axioms of T are 1L-true in M, i.e. ||ϕ||M = 1L for each ϕ ∈ T .

40 The completeness of the above logical system with respect to evaluations in lin- early ordered BL-algebras has been proved in [28] and is formulated in the following theorem. Theorem 2.2.1

Let T be a theory over BL∀ and let ϕ be a formula of the language J(T ).

T proves ϕ if and only if for each linearly ordered BL-algebra L and each safe

L L-model M of T , ||ϕ||M = 1L.

Note that that standard BL-tautologies, i.e. formulas true in all standard interpre- tations w.r.t. all continuous t-norms are not recursively axiomatizable and thus for

BL∀ we do not have standard completeness, see [29, 41]. For the detailed presenta- tion of predicate calculus BL∀ see [28].

2.2.3 Provable propositional and predicate tautologies

Here, we are going to summarize tautologies of BL and BL∀ expressing proper- ties of connectives to which we will refer later when proving properties of logical approximation.

The following are properties of implication. Lemma 2.2.2

BL proves the following formulas:

(1¯ → ϕ) → ϕ, (2.2.19)

ϕ → ϕ, (2.2.20)

ϕ → (ψ → ϕ), (2.2.21)

[ϕ → (ψ → χ)] → [ψ → (ϕ → χ)]. (2.2.22)

Moreover, we will use in the sequel the following properties of implication with respect to other connectives, min-conjunction and max-disjunction.

41 Lemma 2.2.3

BL proves the following formulas:

[ϕ&&(ϕ → ψ)] → ψ, (2.2.23)

(ϕ → ψ) → [(ϕ&χ) → (ψ&χ)], (2.2.24)

[(ϕ1 → ψ1)&&&(ϕ2 → ψ2)] → [(ϕ1&ϕ2) → (ψ1&ψ2)], (2.2.25)

ϕ → (ϕ&1)¯ , (2.2.26)

ϕ → (ϕ ∨ ψ), (2.2.27)

(ϕ ∧ ψ) → ϕ, (2.2.28)

(ϕ&ψ) → (ϕ ∧ ψ). (2.2.29)

Properties of connections applied on the finite sets of formulas are listed below. Lemma 2.2.4

Let I and J be finite index sets. The following are theorems of BL:

(ϕi → ψ) → ( ϕi → ψ), (2.2.30) i∈I i∈I ^ _ (ϕ → ψi) → (ϕ → ψi), (2.2.31) i∈I i∈I ^ ^ ϕi& ψi ≡ (ϕi&ψj), (2.2.32) i∈I j∈J i∈I,j∈J ^ ^ ^ ϕi& ψi ≡ (ϕi&ψj), (2.2.33) i∈I j∈J i∈I,j∈J _ _ _ ϕi ∨ ψi ≡ (ϕi ∨ ψj), (2.2.34) i∈I j∈J i∈I,j∈J ^ ^ ^ ϕi ∧ ψi ≡ (ϕi ∧ ψj), (2.2.35) i∈I j∈J i∈I,j∈J _ _ _ ϕi → ψi ≡ (ϕi → ψj), (2.2.36) i∈I j∈J i∈I,j∈J _ ^ ^ ϕi → ψi ≡ (ϕi → ψj). (2.2.37) i∈I j∈J i∈I,j∈J ^ _ _ Finally, we summarize properties of quantifiers with respect to the connectives.

42 Lemma 2.2.5

The following are theorems of BL∀:

(∀x)(ϕ → ψ) → ((∀x)ϕ → (∀x)ψ), (2.2.38)

(∀x)(ϕ → ψ) ≡ (ϕ → (∀x)ψ) (x is not free in ϕ), (2.2.39)

(∀x)(ϕ → ψ) → ((∃ x)ϕ → (∃ x)ψ), (2.2.40)

(∀x)(ϕ → ψ) ≡ ((∃ x)ϕ → ψ) (x is not free in ψ), (2.2.41)

[(∀x)ϕ&&(∃ x)ψ] → (∃ x)(ϕ&ψ), (2.2.42)

(∃ x)(ϕ&ψ) ≡ ((∃ x)ϕ&ψ) (x is not free in ψ), (2.2.43)

(∀x)(ϕ ∧ ψ) ≡ [(∀x)ϕ ∧ (∀x)ψ], (2.2.44)

(∃ x)(ϕ ∨ ψ) ≡ [(∃ x)ϕ ∨ (∃ x)ψ]. (2.2.45)

2.2.4 Normal Forms in BL∀

In this section, we recall the definitions of disjunctive and conjunctive normal forms as special formulas of BL∀ (for details see [56]). Suppose that the language J is extended by a finite number of truth constants. Moreover, let us use the following priority of the connectives: &,∨,∧ have higher priority then →. In this sense, we will not use parenthesis when unnecessary.

Since two conjunctions are at disposal then there are more possibilities how to define generalized notion of normal forms. Let us confine ourselves to the following concept.

Definition 2.2.4 Let P1,...,Pk be unary predicate symbols and Ei1...in , 1 ≤ ij ≤ k, 1 ≤ j ≤ n, n ≥ 1, be closed instances of some formula. We say that a formula is in disjunctive normal form (DNF) if it has the following form

k

(Pi1 (x1)&& ···&Pin (xn)&&Ei1...in ), (2.2.46) i ,...,i =1 1 _n 43 and analogously, a formula is in conjunctive normal form (CNF) if it has the fol- lowing form k

(Pi1 (x1)&& ···&Pin (xn) →→ Ei1...in ). (2.2.47) i ,...,i =1 1 ^n Naturally, we would expect the internal parts of CNF joined by strong disjunc- tion, i.e. the dual notion to &&. This is possible inLukasiewicz logic since the existence of strong disjunction ▽ is ensured by axiom ϕ ≡≡¬¬¬¬¬ϕ and then ϕ ▽ ψ is introduced as ¬ϕ → ψ.

In analogy with classical logic we can ask, which formulas of predicate BL-logic can be equivalently transformed to their disjunctive and conjunctive normal forms.

In [56], I. Perfilieva has shown that extensional formulas w.r.t. some similarity have this property only ”partially”. It means that obtained equivalence between initial formula and its normal form is implied by some condition. Later, we will extend this class of formulas to the class of extensional formulas w.r.t. reflexive binary predicates.

Let R be a binary predicate symbol. Furthermore, we will refer to any of the following properties of R:

(∀x)R(x, x), (reflexivity) (2.2.48)

(∀x, y)(R(x, y) → R(y, x)), (symmetry) (2.2.49)

(∀x, y, z)(R(x, y)&&R(y, z) → R(x, z)). (transitivity) (2.2.50)

If, in the sequel, we will say that the language J(T ) of some theory T is extended by a reflexive, symmetric or transitive predicate R then this will mean extending T by the corresponding formula (2.2.48), (2.2.49) or (2.2.50). Lemma 2.2.6

If hM, ri is a model of the axioms of reflexivity, symmetry or transitivity over a BL- algebra L, then the fuzzy relation r is reflexive, symmetric or transitive, respectively.

44 In axiomatic set theory, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory. In this theory, the axiom of extensionality is in the following form

(∀A, B)(A = B) ≡ (∀x)(x ∈ A ≡ x ∈ B). (2.2.51)

For better understanding, note that the symbolic above simply expresses that A and B have precisely the same members. Thus, the axiom is really saying that two sets are equal if and only if they have precisely the same members, or in another words, a set is determined by its elements.

The identity = in (2.2.51) is a name of a predicate in predicate logic. Substituting it by a vague predicate we will obtain the generalized axiom of extensionality (also called congruence by some authors).

Definition 2.2.5 Let P be an n-ary predicate. The extensionality axiom for P with respect to the binary predicates R1,...,Rn is the formula

R1(x1, y1)&& ···&Rn(xn, yn) → (P (x1,...,xn) → P (y1,...,yn)). (2.2.52)

If R1 = . . . = Rn = R then we say that the extensionality axiom for P with respect to the binary predicate R is the formula

R(x1, y1)&& ···&R(xn, yn) → (P (x1,...,xn) → P (y1,...,yn)). (2.2.53) Lemma 2.2.7

Let M = hM, r1,...,rn,pi be an interpretation of the predicates R1,...,Rn and P from Definition 2.2.5. M is a model of the extensionality axiom for P w.r.t.

R1,...,Rn if and only if p is extensional w.r.t. r1,...,rn.

Note that the order of variables on the left side of (2.2.52) and (2.2.53) should be respected. The following lemma shows when this may not be taken into account.

45 Lemma 2.2.8

Let T be a theory over BL∀ containing the axiom of extensionality (2.2.52) for n-ary predicate P w.r.t. R1 ...,Rn. Then

T ⊢ R1(x1, y1)&&R1(y1, x1)&& ···&Rn(xn, yn)&&Rn(yn, xn) →

(P (x1,...,xn) ≡ P (y1,...,yn)).

If T contains also the axioms of symmetry (2.2.49) for R1 ...,Rn then

T ⊢ R1(x1, y1)&& ···&Rn(xn, yn) → (P (x1,...,xn) ≡ P (y1,...,yn)). (2.2.54)

We may also consider a weaker property of local extensionality of n-ary predicate

P if one of the n-tuples of variables (x1,...,xn)or(y1,...,yn) in (2.2.52) is replaced by an n-tuple of object constants. For this purpose, we will extend the language J by object constants c1,..., cn, n ≥ 1.

Definition 2.2.6 We say that the following axioms introduce right extensionality

R1(x1, c1)&& ···&Rn(xn, cn) → (P (x1,...,xn) → P (c1,..., cn)) (2.2.55) and left extensionality

R1(c1, x1)&& ···&Rn(cn, xn) → (P (c1,..., cn) → P (x1,...,xn)) (2.2.56)

of an n-ary predicate P with respect to a binary predicates R1,...,Rn at [c1,..., cn].

Analogously, if R1,...,Rn are equivalent to R then we speak about right (left) extensionality axiom for P w.r.t. R at [c1,..., cn].

Analogy of Lemma 2.2.7 is valid also for the right and left extensionality. Lemma 2.2.9

Let M = hM, r1,...,rn,p,c1,...,cni be an interpretation of the predicates R1,...,Rn,

P and n-tuple of object constants [c1,..., cn] from Definition 2.2.6. M is a model

46 of the left (right) extensionality axiom for P w.r.t. R1,...,Rn at [c1,..., cn] if and only if p is left (right) extensional w.r.t. r1,...,rn at [c1,...,cn].

An interesting result about the extensionality w.r.t. similarity has been shown in [28] (Lemma 5.6.8).

Lemma 2.2.10

Let T be a theory over BL∀ containing the similarity axioms for S (i.e. reflexivity, symmetry and transitivity axioms for S) and extensionality axioms for predicates

P1,...,Pn w.r.t. S. Let ϕ be a formula built from P1,...,Pn and possibly S and let x1,...,xn be all free variables of ϕ. Then there exists k ∈ N such that

k k T ⊢ (S (x1, y1)&& . . .&S (xn, yn)) → (ϕ(x1,...,xn) ≡ ϕ(y1,...,yn)).

The value k is called syntactic degree of the formula ϕ and its calculation follows by induction on the length of ϕ.

47 Chapter 3

Logical approximation

In this chapter we will introduce special logical formulas which we call normal forms and used them for the approximate representation of extensional formulas in (fuzzy)

BL-logic. By the approximate representation in logical framework we mean that any extensional formula is equivalent of its normal form (disjunctive or conjunctive) under the special condition. The schematic representation to which we refer as conditional equivalence or logical approximation, looks as follows:

Condition → (Extensional Formula ≡ Normal Form). (3.0.1)

Informally, using the argumentation of the theory of approximation, we can say that the “Normal Form” is approximately equal to the “Extensional Formula” and the

“Condition” characterizes a quality of such approximation. Moreover, it is worth noticing that the conditional equivalence in the form (3.0.1) is more meaningful in fuzzy logic than in classical one because the precision of the approximation can be evaluated from above by the truth value of the “Condition”. This fact has been proved formally in predicate BL-logic by I. Perfilieva in [56] with similarity with respect to which extensionality property is determined. It opened a further area for investigation, whether a similar conditional equivalence can be obtained for other formulas. The partial progress has been done by M. Daˇnkov´ain [17]. This work

48 is a contribution to this investigation and brings the exhaustive analysis of this problematic.

First, let us discuss the construction of normal forms and its connection with similarity on the algebraical level. In essence, normal forms representing a certain formula can be viewed as an aggregated description of an available local informa- tion about a fuzzy relation corresponding to this formula in a given interpretation

(see also [37, 44, 51, 52]). The local information consists of two parts combined by conjunction: the first part characterizes a local domain by its membership function, and the second part describes a value of the fuzzy relation provided that its argu- ments lie inside the respective local domain. In [56], I. Perfilieva characterized the local domain as a neighborhood of a some node and used a similarity relation for the description of the neighborhood. In this work we will use a binary relation more general than the similarity for the description of a neighborhood of a node.

Formally, if c denotes a node then a neighborhood of c can be represented by

R(x, c) where R is arbitrary binary predicate (cf. [56] where R is a similarity predicate, i.e. R is reflexive, symmetric and transitive). In this way, a wider class of formulas can be represented by normal forms. Moreover, improving the quality of the approximation at the same expenses in the form of complexity of representation is highly expected.

The last part of this chapter includes investigation of logical aspects of formalized fuzzy control based on generalized normal forms. For this purpose, two distinct inferences will be introduced with the aim at modeling different ways approaching the precise description of some physical system. Evidently, given a physical system, an approximate model can be represented by the normal forms. This leads to an analogous question about logical approximation based on the fixed inference method.

Roughly speaking, we want to know how close is our approximate description to the

49 precise solution.

It will appear that conditional equivalence between a conclusion of the inference and the given functional dependency described by some predicate F is influenced by the input observation and the “Condition” for logical approximation between F and a normal form. This fact is significant since it shows that such inference method do not brings any additional distortion to the result consequence.

We chose fuzzy BL-logic introduced by P. H´ajek in [28] as the background logic.

The reason is its generality. In fact, once being proved in BL-logic, the result will be valid in any of its special cases. Recall that the most known ones are G¨odel logic, product logic andLukasiewicz logic. The other reason of our choice is the richness of the structure of truth values which allows us to express, among others, the error of an approximation.

3.1 Normal Forms for Extensional Formulas

Let us specify the normal forms (2.2.46) and (2.2.47) by choosing their constituents:

unary predicates P1,...,Pk and atomic subformulas Ei1...in , where 1 ≤ ij ≤ k, 1 ≤ j ≤ n and n ≥ 1. We will introduce two types of normal forms related to a concrete formula, namely finite (discrete) and infinite.

The following definition will introduce a special theories of approximations {NFi}i∈N over BL∀ by adding special axioms giving the concrete form to the expressions DNF and CNF (cf. (2.2.46) and (2.2.47)).

Definition 3.1.1 Let k ∈ N. The language Jk extends J(BL∀) by a finite set of object constants {ci | i = 1 ...k}, binary predicate symbols R1,...,Rn and n-ary

∃ ∀ predicate symbols DNFϕ,k, CNFϕ,k and DNFϕ(x1,...,xn), CNFϕ(x1,...,xn).

Let ϕ be a formula over Jk with n-free variables.

50 Then, the theory NFϕ,k (over BL∀) over the language Jk is specified by the fol- lowing special axioms defining n-ary predicate DNFϕ,k

DNFϕ,k(x1,...,xn) ≡ k

(R1(ci1 , x1)&& ···&Rn(cin , xn)&&ϕ(ci1 ,..., cin )), (3.1.1) i ,...,i =1 1 _n and CNFϕ,k

CNFϕ,k(x1,...,xn) ≡ k

(R1(x1, ci1 )&& ···&Rn(xn, cin ) → ϕ(ci1 ,..., cin )). (3.1.2) i ,...,i =1 1 ^n Finally, the theory NFϕ = k∈N NFϕ,k together with special axioms defining ∃ DNFϕ(x1,...,xn) by S

∃ DNFϕ(x1,...,xn) ≡

(∃y1,...,yn)(R1(y1, x1)&& ···&Rn(yn, xn)&&ϕ(y1,...,yn)), (3.1.3)

∀ and CNFϕ(x1,...,xn) by

∀ CNFϕ(x1,...,xn) ≡

(∀y1,...,yn)(R1(x1, y1)&& ···&Rn(xn, yn) → ϕ(y1,...,yn)). (3.1.4)

The introduced expressions DNFϕ,k and CNFϕ,k related to the formula ϕ can be understood as having a finite or discrete type. The other two expressions of normal forms related to ϕ can be characterized by saying that their type is infinite.

The last remark is to the notion of normal forms of the infinite type. The reader may disagree with using of this name since it does not reflect the standard concept.

Originally, the formulas (3.1.3) and (3.1.4) have been introduced under a different name in my [17], and later as normal forms in [20]. Nevertheless, we keep the latter notation since it has been already accepted and published in papers of I. Perfilieva

(see e.g. [53, 54]).

51 3.1.1 Characterization of extensional formulas

In this subsection we will investigate first, the relationship between normal forms of different types and second, how extensional formulas can be represented by infinite normal forms.

The following formulas express the fact that adding new disjuncts (conjuncts) to the disjunctive (conjunctive) normal form will increase (decrease) its truth values with respect to the particular models. Lemma 3.1.1

The theory NFϕ proves

DNFϕ,k(x1,...,xn) → DNFϕ,k+1(x1,...,xn),

CNFϕ,k+1(x1,...,xn) → CNFϕ,k(x1,...,xn). for each k ∈ N. proof: Obvious. 2

The next result shows how normal forms of different types relate to each other. Lemma 3.1.2

For arbitrary k ∈ N, the theory NFϕ proves

∃ DNFϕ,k(x1,...,xn) → DNFϕ(x1,...,xn),

∀ CNFϕ(x1,...,xn) → CNFϕ,k(x1,...,xn). proof: For simplicity, we will consider the case n = 1. It follows from the substitution axioms (2.2.14) and (2.2.15) that for all i

NFϕ ⊢(R1(ci, x)&&ϕ(ci)) → (∃y)(R1(y, x)&&ϕ(y)),

NFϕ ⊢(∀y)(R1(x, y) → ϕ(y)) → (R1(x, ci) → ϕ(ci))

52 which by the properties (2.2.30) and (2.2.31) of BL-logic gives

k

NFϕ ⊢ (R1(ci, x)&&ϕ(ci)) → (∃y)(R1(y, x)&&ϕ(y)), i=1 _ k

NFϕ ⊢(∀y)(R1(x, y) → ϕ(y)) → (R1(x, ci) → ϕ(ci)), i=1 ^ and hence,

∃ NFϕ ⊢ DNFϕ,k(x) → DNFϕ(x),

∀ NFϕ ⊢ CNFϕ(x) → CNFϕ,k(x).

2

Our next goal is to prove the representation theorem for extensional formulas.

First, we prove one-way implications between an extensional formula and its normal forms of infinite type. Theorem 3.1.3

If R1,...,Rn are reflexive then

∃ NFϕ ⊢ϕ(x1,...,xn) → DNFϕ(x1,...,xn),

∀ NFϕ ⊢ CNFϕ(x1,...,xn) → ϕ(x1,...,xn). proof: Consider the case n = 1. Let us start with a proof of the second formula.

NFϕ ⊢ (∀y)(R1(x, y) → ϕ(y)) → (R1(x, x) → ϕ(x)) (instance of (2.2.14)),

NFϕ ⊢ R1(x, x) → ((∀y)(R1(x, y) → ϕ(y)) → ϕ(x)) (by (2.2.22)),

NFϕ ⊢ (∀y)(R1(x, y) → ϕ(y)) → ϕ(x) (from reflexivity of R1), ∀ NFϕ ⊢ CNFϕ(x) → ϕ(x).

The proof of the first formula is analogous:

NFϕ ⊢ ϕ(x) → ϕ(x)&&R1(x, x) (because of reflexivity and (2.2.26)),

NFϕ ⊢ (R1(x, x)&&ϕ(x)) → (∃y)(R1(y, x)&&ϕ(y)) (from (2.2.15)),

53 and finally,

NFϕ ⊢ϕ(x) → (∃y)(R1(y, x)&&ϕ(y)), i.e.

∃ NFϕ ⊢ϕ(x) → DNFϕ(x).

2

The following lemma can be called a representation theorem for extensional formulas. Lemma 3.1.4

If ϕ is extensional w.r.t. reflexive R1,...,Rn then

∃ NFϕ ⊢ DNFϕ(x1,...,xn) ≡ ϕ(x1,...,xn) and

∀ NFϕ ⊢ CNFϕ(x1,...,xn) ≡ ϕ(x1,...,xn). proof: Let n = 1. Taking into account the previous lemma, it is sufficient to show that

∃ NFϕ ⊢ DNFϕ → ϕ,

∀ NFϕ ⊢ϕ → CNFϕ .

Consider the following instances of the extensionality axiom for ϕ:

NFϕ ⊢R1(x, y) → (ϕ(x) → ϕ(y)),

NFϕ ⊢(R1(y, x)&&ϕ(y)) → ϕ(x).

On the basis of this we obtain first,

NFϕ ⊢ϕ(x) → (R1(x, y) → ϕ(y)),

NFϕ ⊢ϕ(x) → (∀y)(R1(x, y) → ϕ(y)) (by GR and (2.2.16))

∀ NFϕ ⊢ϕ(x) → CNFϕ(x).

54 and second,

NFϕ ⊢(∀y)((R1(y, x)&&ϕ(y)) → ϕ(x)),

NFϕ ⊢(∃y)(R1(y, x)&&ϕ(y)) → ϕ(x) (by GR and (2.2.17))

∃ NFϕ ⊢ DNFϕ(x) → ϕ(x).

2

The theorem below joins the extensionality of a formula with its representability by normal forms. Theorem 3.1.5

A formula ϕ(x1,...,xn) is extensional w.r.t. reflexive R1,...,Rn if and only if

∃ NFϕ ⊢ DNFϕ(x1,...,xn) ≡ ϕ(x1,...,xn) or (3.1.5)

∀ NFϕ ⊢ CNFϕ(x1,...,xn) ≡ ϕ(x1,...,xn). (3.1.6) proof: Let n = 1. Considering Lemma 3.1.4, it is necessary to prove only the

∃ ∃ extensionality from DNFϕ(x) ≡ ϕ(x) (CNFϕ(x) ≡ ϕ(x)).

∃ (a) At first, suppose that NFϕ ⊢ DNFϕ(x) ≡ ϕ(x). Then

∃ NFϕ ⊢ DNFϕ(x) → ϕ(x), i.e.

NFϕ ⊢(∃y)(R1(y, x)&&ϕ(y)) → ϕ(x),

NFϕ ⊢(∀y)(R1(y, x)&&ϕ(y) → ϕ(x)) (by (2.2.41)),

NFϕ ⊢R1(y, x)&&ϕ(y) → ϕ(x) (using (2.2.14)),

which implies the extensionality of ϕ(x) w.r.t. R1.

55 ∀ (b) Secondly, suppose that ϕ(x) ≡ CNFϕ(x). Then

∀ NFϕ ⊢ϕ(x) → CNFϕ(x), i.e.

NFϕ ⊢ϕ(x) → (∀y)(R1(x, y) → ϕ(y)),

NFϕ ⊢(∀y)(ϕ(x) → (R1(x, y) → ϕ(y))) (by (2.2.39)),

NFϕ ⊢(ϕ(x) → (R1(x, y) → ϕ(y))) (using (2.2.14)),

which again implies the extensionality of ϕ(x) w.r.t. reflexive R1. This com- pletes the proof.

2

3.1.2 Logical Approximation by Finite Normal Forms

Taking into account Theorem 3.1.5 we cannot expect that finite normal forms will be equivalent with the original formula for all cases. However, we can prove that a formula and its finite normal forms are conditionally equivalent which means that equivalence between them is implied by a condition (characterizing how “precise” this equivalence may be). The truth value of this condition in each interpretation depends on the number and the choice of objects described by constants {ci | i = 1,...,k}. The proof of the conditional equivalence will be obtained in two steps.

At the first step we will prove that DNFϕ,k and CNFϕ,k represent lower and upper approximations of a given extensional formula ϕ. Theorem 3.1.6

Let k ∈ N and ϕ(x1,...,xn) be a formula of NFϕ,k left and right extensional w.r.t.

R1,...,Rn at each n-tuple [ci1 ,..., cin ], 1 ≤ ij ≤ k. Then

NFϕ,k ⊢ DNFϕ,k(x1,...,xn) → ϕ(x1,...,xn), (3.1.7)

56 and

NFϕ,k ⊢ ϕ(x1,...,xn) → CNFϕ,k(x1,...,xn), (3.1.8) respectively. proof: Consider the case of one free variable (n = 1). From the left and right extensionality axioms (2.2.55) and (2.2.56) for ϕ it follows that

NFϕ,k ⊢ R1(ci, x) → (ϕ(ci) → ϕ(x)),

NFϕ,k ⊢ (R1(ci, x)&&ϕ(ci)) → ϕ(x), and

NFϕ,k ⊢ R1(x, ci) → (ϕ(x) → ϕ(ci)),

NFϕ,k ⊢ ϕ(x) → (R1(x, ci) → ϕ(ci)), respectively. Hence, using the properties of & and → (see (2.2.30) and (2.2.31)) we obtain

NFϕ,k ⊢ (R1(ci, x)&&ϕ(ci)) → ϕ(x), i _ NFϕ,k ⊢ ϕ(x) → (R1(x, ci) → ϕ(ci)), i ^ which gives us

NFϕ,k ⊢ DNFϕ,k(x) → ϕ(x), (3.1.9)

NFϕ,k ⊢ ϕ(x) → CNFϕ,k(x). (3.1.10)

2

Corollary 3.1.7

Under the assumptions of Theorem 3.1.6 it can be proved that

NFϕ,k ⊢ (∀x1,...,xn)(DNFϕ,k(x1,...,xn) → CNFϕ,k(x1,...,xn)). (3.1.11)

57 Before proving a conditional equivalence between normal forms and initial for- mula, let us inspect the properties of normal forms themselves. Lemma 3.1.8

NFϕ,k proves

Ck(x1,...,xn) → (CNFϕ,k(x1,...,xn) → DNFϕ,k(x1,...,xn)), (3.1.12)

where Ck(x1,...,xn) stands for the formula

k

[R1(x1, ci1 )&&R1(ci1 , x1)&& ···&Rn(xn, cin )&&Rn(cin , xn)], (3.1.13) i ,...,i =1 1 _n for each k ∈ N. proof: It is sufficient to consider n = 1. From (2.2.23), it follows

NFϕ,k ⊢ R1(x, ci)&&&(R1(x, ci) → ϕ(ci)) → ϕ(ci), then by (2.2.24)

NFϕ,k ⊢ R1(x, ci)&&R1(ci, x)&&&(R1(x, ci) → ϕ(ci)) → R1(ci, x)&&ϕ(ci), and hence

k

NFϕ,k ⊢ R1(x, ci)&&R1(ci, x)&& (R1(x, ci) → ϕ(ci)) → i=1 ^ k

(R1(ci, x)&&ϕ(ci)), (3.1.14) i=1 _ thus k

NFϕ,k ⊢ R1(x, ci)&&R1(ci, x) → (CNFϕ,k(x) → DNFϕ,k(x)). i=1 _ 2

Provided that Ck is 1-true with respect to some model M then from the formula

(3.1.12) it follows that || CNFϕ,k ||M ≤ || DNFϕ,k ||M.

58 At the second step we will prove the proclaimed conditional equivalence. It is worth to be mentioned that unlike the case when normal forms are of infinite type, in finite case we do not need reflexivity of the relations R1,...,Rn with respect to which the extensionality is considered. Theorem 3.1.9

Let k ∈ N and ϕ(x1,...,xn) be a formula of NFϕ,k left and right extensional w.r.t.

R1,...,Rn at each n-tuple [ci1 ,..., cin ], 1 ≤ ij ≤ k. Then

NFϕ,k ⊢Ck(x1,...,xn) → (DNFϕ,k(x1,...,xn) ≡≡ ϕ(x1,...,xn)), (3.1.15)

NFϕ,k ⊢Ck(x1,...,xn) → (CNFϕ,k(x1,...,xn) ≡≡ ϕ(x1,...,xn)), (3.1.16)

where Ck(x1,...,xn) stands for the formula (3.1.13). proof: Obviously, it follows from (3.1.12) and Theorem 3.1.6. 2

Corollary 3.1.10

Under the assumptions of Theorem 3.1.9

NFϕ,k ∪ {(∀x1,...,xn)Ck(x1,...,xn)}⊢

(∀x1,...,xn)(DNFϕ,k(x1,...,xn) ≡≡ ϕ(x1,...,xn)), and

NFϕ,k ∪ {(∀x1,...,xn)Ck(x1,...,xn)}⊢

(∀x1,...,xn)(CNFϕ,k(x1,...,xn) ≡≡ ϕ(x1,...,xn)),

where Ck(x1,...,xn) stands for the formula (3.1.13).

The results from above show that we can logically approximate extensional for- mulas by normal forms of the finite type with arbitrary precision which follows from the interpretation of the antecedent of the formula (3.1.15) and (3.1.16), i.e. the

59 condition (3.1.13). Observe that (3.1.13) is independent of the initial formula, thus, the quality of logical approximation can be adjusted by requiring additional proper- ties from c1,..., ck with respect to R1,...,Rn. Moreover, the normal forms of the infinite type represent the fact that with growing number of the nodes (interpreting the constants ci) we approach the considered extensional formula. Meaning of the condition of conditional equivalence will be discussed in the next chapter in detail.

The rest of this section will be devoted to an investigation of the extensionality property of normal forms of the finite type.

For the sake of brevity, we will use the following notation: let π be a variations of order n out of the set Ik = {1,...,k}, then

R1(cπ1 , x1)&& . . .&Rn(cπn , xn)&&ϕ(cπ1 ,..., cπn ), (3.1.17)

we denote by Aπ(x1,...,xn) and

R1(x1, cπ1 )&& . . .&Rn(xn, cπn ) → ϕ(cπ1 ,..., cπn ), (3.1.18)

we denote by Bπ(x1,...,xn). Moreover, let Pn,k be a set of all variations of order n out of the set Ik.

First, let us analyze the internal parts of the normal forms. Lemma 3.1.11

Let π ∈ Pn,k, where k ∈ N. If a formula ϕ(x1,...,xn) over Jk is left and right

extensional w.r.t. reflexive R1,...,Rn at [cπ1 ,..., cπn ] then

• BL∀ proves the right extensionality of Aπ(x1,...,xn),

• BL∀ proves the left extensionality of Bπ(x1,...,xn),

with respect to R1,...,Rn at [cπ1 ,..., cπn ].

60 proof: It is sufficient to give the proof for the case n = 1. Let us consider some ci. The following formulas expressing the left and right extensionality of ϕ(x) w.r.t.

R1 at ci

⊢(R1(ci, x)&&ϕ(ci)) → ϕ(x),

⊢ϕ(x) → (R1(x, ci) → ϕ(ci)), from which it follows

⊢ (R1(ci, x)&&ϕ(ci)) → (R1(x, ci) → ϕ(ci)). (3.1.19)

By the property of → (2.2.19) together with the reflexivity of R1 we obtain

⊢ [R1(ci, x)&&&(R1(ci, ci) → ϕ(ci))] → (R1(x, ci) → ϕ(ci)), and using the axiom (2.2.22)

⊢ R1(ci, x) → [(R1(ci, ci) → ϕ(ci)) → (R1(x, ci) → ϕ(ci))], or equivalently

⊢ R1(ci, x) → (Bi(ci) → Bi(x)), which proves the left extensionality of Bi w.r.t. R1 at ci.

Analogously, from (3.1.19) using (2.2.26) and reflexivity of R1

⊢ R1(x, ci) → [(R1(ci, x)&&ϕ(ci)) → (R1(ci, ci)&&ϕ(ci))],

⊢ R1(x, ci) → (Ai(x) → Ai(ci)),

which completes the proof of the right extensionality of Ai w.r.t. R1 at ci. 2

In fact, we are interested to have “approximating” formulas (normal forms) from the same class of extensional formulas. As we will see, normal forms are extensional formulas.

61 Theorem 3.1.12

Let ϕ be a formula over Jk with n free variables and k ∈ N.

• If ϕ(x1,...,xn) is left extensional w.r.t. R1,...,Rn at [cπ1 ,..., cπn ] for each

π ∈ Pk, then DNFϕ,k(x1,...,xn) over NFϕ,k is extensional w.r.t. D1,...,Dn given by k

Di(x, y) ≡ Ri(ci, x) → Ri(ci, y), (3.1.20) i=1 ^ for all i = 1,...,n.

• If ϕ(x1,...,xn) is right extensional w.r.t. R1,...,Rn at [cπ1 ,..., cπn ] for each

π ∈ Pk, then CNFϕ,k(x1,...,xn) over NFϕ,k is extensional w.r.t. E1,...,En given by k

Ei(x, y) ≡ Ri(y, ci) → Ri(x, ci), (3.1.21) i=1 ^ for all i = 1,...,n. proof: Let us consider the case n = 1. We start to prove the extensionality of

DNFϕ,k. From (2.2.24), it follows

NFϕ,k ⊢ (R1(ci, x) → R1(ci, y)) → [(R1(ci, x)&&ϕ(ci)) → (R1(ci, y)&&ϕ(ci))], using changing of assumptions

NFϕ,k ⊢ (R1(ci, x)&&ϕ(ci)) → [(R1(ci, x) → R1(ci, y)) → (R1(ci, y)&&ϕ(ci))], then by (2.2.27) and (2.2.37), we obtain

k

NFϕ,k ⊢(R1(ci, x)&&ϕ(ci)) → [(R1(ci, x) → R1(ci, y)) → (R1(ci, y)&&ϕ(ci))], i=1 _k k

NFϕ,k ⊢(R1(ci, x)&&ϕ(ci)) → (R1(ci, x) → R1(ci, y)) → (R1(ci, y)&&ϕ(ci)) , "i=1 i=1 # ^ _

62 and finally by (2.2.30) together with (2.2.29)

k k k

NFϕ,k ⊢ (R1(ci, x)&&ϕ(ci)) → (R1(ci, x) → R1(ci, y)) → (R1(ci, y)&&ϕ(ci)) , i=1 "i=1 i=1 # _k ^ k _k

NFϕ,k ⊢ (R1(ci, x) → R1(ci, y)) → (R1(ci, x)&&ϕ(ci)) → (R1(ci, y)&&ϕ(ci)) , i=1 "i=1 i=1 # ^k _ _

NFϕ,k ⊢ (R1(ci, x) → R1(ci, y)) → (DNFϕ,k(x) → DNFϕ,k(y)), i=1 ^ which proves the extenstionality of DNF(x) w.r.t. D1.

Let us prove the extensionality of CNFϕ,k. From (2.2.6), it follows

NFϕ,k ⊢ (R1(x, ci) → R1(y, ci)) → [(R1(y, ci) → ϕ(ci)) → (R1(x, ci) → ϕ(ci))], (3.1.22) then by the property of ∧ (2.2.28)

k

NFϕ,k ⊢ [(R1(x, ci) → R1(y, ci))&&&(R1(y, ci) → ϕ(ci))] → (R1(x, ci) → ϕ(ci)). i,j=1 ^ and finally by (2.2.32),

k k

NFϕ,k ⊢ (R1(x, ci) → R1(y, ci))&& (R1(y, ci) → ϕ(ci)) → (R1(x, ci) → ϕ(ci)), "i=1 j=1 # ^ ^ k k k

NFϕ,k ⊢ (R1(x, ci) → R1(y, ci))&& (R1(y, ci) → ϕ(ci)) → (R1(x, ci) → ϕ(ci)), "i=1 j=1 # i=1 ^ ^ ^ which implies

k

NFϕ,k ⊢ (R1(x, ci) → R1(y, ci)) → (CNFϕ,k(y) → CNFϕ,k(x)), i=1 ^ or equivalently

NFϕ,k ⊢ E1(x, y) → (CNFϕ,k(x) → CNFϕ,k(y)).

63 The proof for the case n> 1 differs from the above one only at the final stage, where we use the fact that

k

NFϕ,k ⊢ [(R1(x1, ci1 ) → R1(y1, ci1 ))&& . . .&&(Rn(xn, cin ) → Rn(yn, cin ))] ≡ i ...,i =1 1 ^n k k

(R1(x1, ci) → R1(y1, ci))&& . . .& (Rn(xn, ci) → Rn(yn, ci)) , "i=1 i=1 # ^ ^ which immediately implies the result extensionality for DNFϕ,k w.r.t. D1,...,Dn.

Analogously, we come to the extensionality of CNFϕ,k w.r.t. E1,...,En. 2

In full generality, normal forms from Theorem 3.1.12 based on arbitrary transitive binary predicate R1 are extensional. Corollary 3.1.13

Let all assumptions of Theorem 3.1.12 be valid. If R1,...,Rn are transitive then

• DNFϕ,k(x1,...,xn) is extensional w.r.t. R1,...,Rn,

• CNFϕ,k(x1,...,xn) is extensional w.r.t. R˜1,..., R˜n, whereas NFϕ,k ⊢ R˜i(x, y) ≡

Ri(y, x), for all i = 1,...,n.

proof: The claim easily follows from the transitivity of Ri

NFϕ,k ⊢ (Ri(x, y)&&Ri(ci, x)) → Ri(ci, y),

NFϕ,k ⊢ Ri(x, y) → (Ri(ci, x) → Ri(ci, y)), k

NFϕ,k ⊢ Ri(x, y) → (Ri(ci, x) → Ri(ci, y)), i.e. i=1 ^ NFϕ,k ⊢ Ri(x, y) → Di(x, y),

for each i = 1,...,n, and the extensionality of DNFϕ,k w.r.t. R1,...,Rn simply follows from Theorem 3.1.12. Analogously for R˜i(x, y) we have NFϕ,k ⊢ R˜i(x, y) →

Ei(x, y). 2

64 3.2 Logical approximation in approximate infer- ence

Let us pose a question: What do we understand under the notion of approximate inference? To infer means to derive a conclusion. An inference mechanism draw the conclusion from a set of facts or circumstances. Much of the study of logic explores the or invalidity of inferences, i.e. whether an application to true premisses leads to true or false conclusions.

There may exists inference from a set of formulas to another, without absolute certainty. This follows from the fact that inference is a weaker notion then deduction.

Unlike as in the case of deduction, for which it is impossible that are true and conclusion is false at the same time.

In logic, especially in mathematical logic, rules of inference are schemes for con- structing valid inferences, and in fact, all of them are deductions as well. These schemes establish syntactic relations between a set of formulas called premises and an assertion called a conclusion. Based on such schemes, new true assertions are de- rived from other already known ones. As eminent examples of rules of inference let us recall the rules of modus ponens and modus tollens of the classical propositional logic. Both examples are deductions as well. From the examples of inferences which are not deductions, we can name induction (used in specific propositions such as:

“All swans we have discovered are white, ergo, all swans, that will ever be there, are white.”) or abduction, i.e. the inference to the best explanation (actually introduced by C. S. Peirce to explain a fact, e.g. the observation “The grass is wet.” and the rule “If it rains, the street will get wet.” induce “It has been raining.”).

A long list of can be assembled when people apply ”common sense” to certain vague concepts, e.g. of the heap (or the ) which

65 arises from using modus ponens to vague properties. What the most of paradoxes of this type show is that the idea of a crisp border between ”having” and ”not having” some property is insufficient.

A solution to such paradoxes has been brought by generalized rule of modus ponens as a particular case of compositional rule of inference in the global concept of many-valued logics introduced by L. Zadeh in [63]. The analysis of logical aspects of Zadeh’s compositional rules of inference was done by P. H´ajek in [28] or V. Nov´ak in [45, 44] (for evaluated syntax). But most of the known works investigate and introduce new generalized inferences as a special operations over some algebraic structure, e.g. see [46, 23].

The generalized rule of inference in the original form can be viewed as a particu- lar instruction how to calculate a conclusion from the set of assertions. But general schematic notation for generalized as well as classical inference can be written as follows Assumption ,..., Assumption 1 n . Conclusion A significant class of classical inferences can be inscribed in the following form:

A, A £ B , B where £ stands for some of the introduced connective of the given logical calculus.

Hence, our knowledge about a certain relationship between input A and output B of the inference is well known, i.e. is determined by formula A £ B.

On the other side, approximate inference stems from the idea of making decision even in the case when the input of the inference is not included in the antecedent

A £ B. This approximate knowledge can be formally written as

A∗,A £ B . B∗

66 The most eminent task is how to handle the information contained in premisses and draw a conclusion from them. Formally, we may specify what is derived either by replacing B∗ by a concrete formula or by adding a formula introducing the conclusion into the set of premisses. As an example, we can put

B∗ ≡ [A∗&&(A £ B)] or B∗ ≡ [A∗ → (A £ B)].

If we denote a formula introducing the conclusion by RI then it will reflect on the inference schema as follows A∗,A £ B,RI , B∗ which can be translated: Our knowledge lies in A £ B, we have an observation A∗, and finally, RI says what is to be obtained, i.e. B∗, from A∗ on the basis of A £ B.

Further generalization is straightforward

A∗,A £ B ,...,A £ B ,RI 1 1 k k . B∗

This structure of inference rule is considered as basic for dealing with formalized knowledge contained in fuzzy IF-THEN rules.

3.2.1 Approximate inferences based on Normal Forms

As it was explicated above, approximate inference is able to handle non-precise information and provide the conclusion from a observation based on approximate knowledge. The question arises: is it also suitable for formalization of approximate reasoning based on a collection of fuzzy IF-THEN rules? These rules are understood as non-precise descriptions of some real physical system using the natural language.

A partial knowledge formalized by normal forms of the form (3.1.1) or (3.1.2) can be interpreted in fuzzy control as a rule-base consisting of kn rules in the following

67 form

IF (x1 is Ri1 ) AND . . . AND (xn is Rin ) THEN (xn+1 is Fi1...in ), (3.2.1) or generally

IF (x1 is Ri1 ) AND . . . AND (xp is Rip )

THEN [(xp+1 is Ri(p+1) ) AND . . . AND (xn is Rin )] ◦ Fi1...in , (3.2.2) for each 1 ≤ ij ≤ k and 1 ≤ p < n, n ≥ 2.

The first rule-base (3.2.1) is of the Takagi-Sugeno type (see [1]) and can be understand as an aggregated description of an available local information about some function, while the second one (3.2.2) describes some relational dependency between two sets of objects. In some special cases a rule-base (3.2.2) can be said as of the Mamdani type. Consequently, normal forms are suitable to formalize significant sub-class of fuzzy IF-THEN rules.

The above introduced rules differs from the traditional ones in values Fi1...in . If we

are able to find representatives c1,...,ck such that values Fi1...in = F (ci1 ,...,cin ) = 1 then a disjunctive rule-base, i.e. the connection between particular rules is inter- preted by disjunction, can be regarded as traditional one. Analogous claim for con- junctive rule-bases works only in the case of the interpretation of connectives given

byLukasiewicz algebra (or any other MV-algebra) and values Fi = F (ci1 ,...,cin )=

0. It follows from the formula (R1&R2) ↔ ¬(R1 → ¬R2) being valid just in Lukasiewicz logic.

Assuming some L-structure, the above formulated fuzzy IF-THEN rules consist of the following parts:

• xi being just of the attributes determined by its domain e.g. the fever with a domain [37 ◦C, 42 ◦C] or the intoxication with a domain [0 0%, 5 0%].

68 • The symbols Rij , representing L-valued fuzzy sets lij (xj) or pij (xj), are asso-

ciated with binary predicates Ri(cij , xj) or Ri(xj, cij ), 1 ≤ ij ≤ k, where cij are constant symbols. These fuzzy sets are related to properties of the given

attributes e.g. small, strong, more or less tall, approximately 7.

• Fuzzy control deals with partial knowledge about some f(x1,...,xn) associ-

ated with n-ary predicate F (x1,...,xn). In the case of functional interpreta- tion, i.e. when the rule-base has the form (3.2.1), f is an ordinary function on

M n, otherwise, it is a fuzzy relation.

• And Fi1...in standing for the names of values from L corresponding to the

fragments of knowledge formalized as F (ci1 ,..., cin ) which modify fuzzy sets

li(p+1) ,...,lin or pi(p+1) ,...,pin in the way designated by an operation ◦, for

all i = 1,...,k. On the basis of the rule (3.2.1), we would say that Fi1...in

corresponds with the value of f in the node [ci1 ,...,cin ]. In the case of (3.2.2),

Fi1...in would express the degree in which objects [ci1 ,...,cip ] and [cip+1 ,...,cin ] relates to each other.

Example 3.2.1 We will demonstrate the way of reading a particular rule of the rule-bases (3.2.1) and (3.2.2) for special problems of fuzzy control.

• Let us consider a robot lacquering car-bodies using different color saturations.

Then a rule of the rule-base (3.2.1) can look as follows

IF X-Coordinate is greater then 1.5 AND Y-Coordinate is arbitrary AND Z-Coordinate is less then 0.5 THEN Saturation is 0.7,

assuming that the contexts of linguistic variables X,Y,Z-Coordinate are the

69 Figure 3.1: Fuzzy sets associated with the linguistic expressions from Example 3.2.1.

same. The fuzzy sets are associated with the above linguistic expression ac-

cording to Figure 3.1, where

R1(x) “greater then 1.5”,

R2(x) is assigned to “arbitrary”,

R3(x) “less then 0.5”.

• A rule from the rule-base of the type (3.2.2) for a heating boiler can be read as

follows

IF Air temperature is medium AND Air temperature variation is negatively big THEN Boiler pressure is ratherbig AND Water temperature is medium

In this particular case the value Fi = 1.

In order to give an answer to the question formulated at the beginning of this subsection, we will introduce new rules of approximate inference (as formulas) based on normal forms. Later on, we will discuss logical aspects of fuzzy control systems based on the principles of these new inferences.

A fuzzy control system contains of a rule-base (including available knowledge) consisting of fuzzy IF-THEN rules and an inference method that applies the fuzzy rules to the fuzzy input variables, generating values of fuzzy output variables accord- ing to Figure 3.2. Inputs and outputs of fuzzy control are crisp, therefore, methods

70 Figure 3.2: A fuzzy control system based on DNF or CNF. of fuzzification and defuzzification are necessarily included in the case of rule-base

(3.2.2). The consequent parts of a rule-base (3.2.1) are described using ordinary, crisp, numbers. Hence, the defuzzification is already built into the inference.

Below, we will introduce special theories for fuzzy control systems, which we will call fuzzy control theories. In the following sub-section we will investigate properties of approximate inferences and then we will show consequences of these inferences to the sub-part of fuzzy control, i.e. to fuzzy control systems, without taking into account fuzzification and defuzzification methods.

3.2.2 Rules of inference and the theory of fuzzy control

In the sequel, we will analyze fuzzy IF-THEN rules of the form (3.2.2) since the rule-base of the form (3.2.1) is just the special case of the former one.

Let us suppose that F , DNFF,k and CNFF,k are n-ary predicates and p ∈ N such that 1 ≤ p ≤ n. Then, we can consider the following formula

∗ (∀ xp+1 ...xn)[BDNF(xp+1,...,xn) ≡

∗ (∃ x1,...,xp)(A (x1,...,xp)&& DNFF,k(x1,...,xn))], (3.2.3)

∗ ∗ which defines BDNF from A using the compositional rule of inference in Zadeh’s sense. An alternative approach to an inference based on CNFF,k can be introduced

71 as follows

∗ (∀ xp+1 ...xn)[BCNF(xp+1,...,xn) ≡

∗ (∀ x1,...,xp)(A (x1,...,xp) → CNFF,k(x1,...,xn))], (3.2.4) which is not a compositional rule of inference in Zadeh’s sense, but generally, it behaves analogously to the inference introduced for DNFF,k. For the sake of brevity, we will denote formula (3.2.3) by RIDNF and formula (3.2.4) by RICNF.

Remark 3.2.1 The generalized inferences can be visualized as inference rules of the following forms

∗ ∗ A , DNFF,k,RIDNF A , CNFF,k,RICNF ∗ and ∗ . BDNF BCNF

For the simplicity, we will denote the n-tuple of symbols for variables [x1,...,xn]

by ¯xn, n-tuple of constant symbols [ci1 ,..., cin ] by c¯in and the formula

R1(ci1 , x1)&& . . .&Rn(cin , xn),

by the symbol R(c¯in , x¯n). Analogous shortening we use for the other formulas or predicates as well.

Definition 3.2.1 Let k ∈ N. A theory of fuzzy control TDNF,k (over BL∀) based on DNF is specified by

• the language JFC = J(TDNF,k) extended by binary predicates R1,...,Rn, n-ary

predicates F , DNFF,k and CNFF,k, constant symbols c1,..., ck and additionally

has n constants X1,...,Xn,

• the axioms are formulas (3.1.1) (defining DNFF,k), (3.2.3) for each p = 1,...,n−

∗ ∗ 1 (defining BDNF from A and DNFF,k), and additionally, it contains axioms

for left and right extensionality of F w.r.t. R1,...,Rn at each [ci1 ,..., cin ],

where 1 ≤ ij ≤ k, j = 1,...,n.

72 A theory of fuzzy control TCNF,k (over BL∀) based on CNF is specified as follows

• the language J(TCNF,k) is identical with J(TDNF,k),

• the axioms are formulas (3.1.2) (defining CNFF,k), (3.2.4) for each p = 1,...,n−

∗ ∗ 1 (defining BCNF from A and CNFF,k), and additionally, it contains axioms

for left and right extensionality of F w.r.t. R1,...,Rn at each [ci1 ,..., cin ],

where 1 ≤ ij ≤ k, j = 1,...,n.

Moreover, a theory TFC over BL∀ with the language JF K contains axioms defining

∗ ∗ DNFF,k, CNFF,k and BDNF, BCNF, i.e. axioms (3.1.1), (3.1.2) and (3.2.3), (3.2.4) for each p = 1,...,n − 1, respectively.

For the sake of brevity, we will denote TDNF,k, TCNF,k, DNFF,k and CNFF,k by

TDNF, TCNF, DNF and CNF, respectively, whenever it is evident that we work with the same fixed k.

3.2.3 Properties of approximate inferences

As the first step in the analysis of approximate inferences based on normal forms, we will take a glance at relationship between output of RIs and its inputs. In other words, we want to know if the conclusion is implied and the way how the implication works. Lemma 3.2.1

Under the present notation

∗ ∗ TFC ⊢ (A (X1,...,Xp)&& DNF(X1,...,Xn)) → BDNF(Xp+1,...,Xn),

∗ ∗ TFC ⊢ BCNF(Xp+1,...,Xn) → (A (X1,...,Xp) → CNF(X1,...,Xn)).

73 Consequently, for each model DFC of TFC and

∗ ∗ ||A (X1,...,Xp)&& DNF(X1,...,Xn)||DFC ≤ ||B (Xp+1,...,Xn)||DFC ,

∗ ∗ ||B (Xp+1,...,Xn)||DFC ≤ ||A (X1,...,Xp) → CNF(X1,...,Xn)||DFC , for arbitrary p ∈ N, 1 ≤ p < n. proof: It is sufficient to make the proof for p = n − 1. The first formula is a simple consequence of Lemma 7.1.4. from [28]. We start the proof of the second formula applying axiom of substitution to the formula RICNF, i.e.

∗ ∗ TFC ⊢ BCNF(Xn) ≡ (∀ x1,...,xn−1)(A (x1,...,xn−1) → CNF(x1,...,Xn)), and substituting once more

∗ ∗ TFC ⊢ BCNF(Xn) → (A (X1,...,Xn−1) → CNF(X1,...,Xn)).

The rest is evident. 2

Considering extensionality property, we obtain the following relationship between

∗ ∗ consequences BDNF and BCNF. Theorem 3.2.2

TDNF ∪ TCNF proves

[(∀x1,...,xp, y1,...,yp)(D1(x1, y1)&& . . .&Dp(xp, yp))&&

(Dp+1(xp+1, yp+1)&& . . .&Dn(xn, yn))] →

∗ ∗ (BDNF(xp+1,...,xn) → BCNF(yp+1,...,yn)), (3.2.5)

for each p ∈ N, 1 ≤ p < n, where Di(x, y) are given by (3.1.20), for all i = 1,...,n. proof: For the simplicity, let us consider p = n − 1. We know that

TDNF ∪ TCNF ⊢ D(¯xn, y¯n) → (DNF(¯xn) → DNF(¯yn)),

74 and

TDNF ∪ TCNF ⊢ DNF(¯xn) → CNF(¯xn)), which gives

TDNF ∪ TCNF ⊢ D(¯xn, y¯n) → (DNF(¯xn) → CNF(¯yn)), this implies

∗ ∗ TDNF ∪ TCNF ⊢ D(¯xn, y¯n) → [(A (¯xn−1)&&A (¯yn−1)&& DNF(¯xn)) → CNF(¯yn)],

∗ ∗ TDNF ∪ TCNF ⊢ D(¯xn, y¯n) → [(A (¯xn−1)&& DNF(¯xn)) → (A (¯yn−1) → CNF(¯yn))], thus by (2.2.38)

TDNF ∪ TCNF ⊢ (∀x¯n−1, y¯n−1)D(¯xn, y¯n) →

∗ ∗ (∀x¯n−1, y¯n−1)[(A (¯xn−1)&& DNF(¯xn)) → (A (¯yn−1) → CNF(¯yn))], form (2.2.39) and (2.2.41) it follows

TDNF ∪ TCNF ⊢ (∀x¯n−1, y¯n−1)D(¯xn, y¯n) →

∗ ∗ (∃x¯n−1)(A (¯xn−1)&& DNF(¯xn)) → (∀y¯n−1)(A (¯yn−1) → CNF(¯yn)), and hence

∗ ∗ TDNF ∪ TCNF ⊢ (∀x¯n−1, y¯n−1)D(¯xn, y¯n) → (BDNF(xn) → BCNF(yn)).

2

Roughly speaking, if p = n − 1 and for some model D of theory TDNF ∪ TCNF are

∗ ||xi||D close enough to ||yi||D, for all i = 1,...,n, then ||BDNF(xn)||D is sufficiently

∗ included in ||BCNF(yn)||D. More precisely, if ||(∀xi, yi)Di(xi, yi)||D ≤ di, for all i =

∗ ∗ 1,...,n − 1, and ||Dn(xn, yn)||D ≤ dn then ||BDNF(xn) → BCNF(yn)||D ≥ d1 ∗ . . . ∗ dn

75 ∗ ∗ (where ∗ is the truth function of &&), or equivalently, formula BDNF(xn) → BCNF(yn) is at least (d1 ∗ . . . ∗ dn)-true in the model D.

∗ The reflexivity of Di(x, y) implies that the degree of “BDNF(x) being smaller

∗ then BCNF(x)” is independent of x. Corollary 3.2.3

TDNF ∪ TCNF proves

(∀x1,...,xp, y1,...,yp)(D1(x1, y1)&& . . .&Dp(xp, yp)) →

∗ ∗ (∀xp+1,...,xn)(BDNF(xp+1,...,xn) → BCNF(xp+1,...,xn)). (3.2.6) for each p ∈ N, 1 ≤ p < n.

proof: Obviously (3.2.6) follows from (3.2.5) and by reflexivity of Di(x, y) for all i = 1,...,n. 2

3.2.4 Consequences to fuzzy control

In this subsection, we will present logical formulas which refer to the properties of approximate inference aimed at handling and operating with partial knowledge about some function. In other words, we want to express formally a particular part of a process of fuzzy control inclusive of an exhaustive study of relationships between inference output and the rule-base with respect to given input.

At first, we are interested in a relationship between a precise value of F and conclusion B∗ of the inferences based on the normal forms. Lemma 3.2.4

TDNF proves

∗ ∗ (A (X1,...,Xp)&&C(X1,...,Xn)) → (F (X1,...,Xn) → BDNF(Xp+1,...,Xn)), (3.2.7)

76 and TCNF proves

∗ ∗ (A (X1,...,Xp)&&C(X1,...,Xn)) → (BCNF(Xp+1,...,Xn) → F (X1,...,Xn)) (3.2.8) for each p ∈ N, 1 ≤ p < n, where n-ary predicate C is given by (3.1.13). proof: For simplicity, we will consider the case p = n − 1. From (3.1.12) and using (2.2.24)

∗ ∗ TDNF ⊢ [C(X¯n)&& CNF(X¯n)&&A (X¯n−1)] → (A (X¯n−1)&& DNF(X¯n)), applying (2.2.15), (2.2.6) and MP

∗ ∗ TDNF ⊢ [C(X¯n)&& CNF(X¯n)&&A (X¯n−1)] → ((∃ x¯n−1)A (¯xn−1)&& DNF(X¯n)),

¯ ¯ ∗ ¯ ∗ TDNF ⊢ (C(Xn)&& CNF(Xn)) → [A (Xn−1) → BDNF(Xn)], since F is extensional w.r.t. R consequently ⊢ F (X¯n) → CNFF (X¯n) that gives

¯ ¯ ∗ ¯ ∗ TDNF ⊢ (C(Xn)&&F (Xn)) → [A (Xn−1) → BDNF(Xn)],

¯ ∗ ¯ ¯ ∗ TDNF ⊢ (C(Xn)&&A (Xn−1)) → [F (Xn) → BDNF(Xn)], which proves the first claim.

The proof of (3.2.8) is analogous. Modification of (3.1.12) is

TCNF ⊢ CNF(X¯n) → (C(X¯n) → DNF(X¯n)), and by the transitivity axiom for → (2.2.6)

∗ ∗ TCNF ⊢ (A (X¯n−1) → CNF(X¯n)) → [A (X¯n−1) → (C(X¯n) → DNF(X¯n))],

substituting x1,...,xn−1 to the formula above we obtain

∗ TCNF ⊢ (∀x¯n−1)(A (¯xn−1) → CNF(¯xn−1, Xn)) →

∗ → [A (X¯n−1) → (C(X¯n) → DNF(X¯n))],

77 which gives

∗ ∗ TCNF ⊢ (B (Xn)&&A (X¯n−1)) → (C(X¯n) → DNF(X¯n)), additionally, extensionality implies that ⊢ DNFF (X¯n) → F (X¯n) and consequently

∗ ∗ ¯ ¯ ¯ TCNF ⊢ (BCNF(Xn)&&A (Xn−1)) → (C(Xn) → F (Xn)),

¯ ∗ ¯ ∗ ¯ TCNF ⊢ (C(Xn)&&A (Xn−1)) → (BCNF(Xn) → F (Xn)), that gives formula (3.2.8). 2

Let us discuss the meaning of the formulas (3.2.7) and (3.2.8). Both formulas

∗ ∗ reflect the way how the conclusions BDNF and BCNF are obtained. Roughly speaking, we could understand the antecedents of (3.2.7) and (3.2.8) as lower bound for the

∗ truth values of ”F being greater then BDNF” respective of ”F being smaller then

∗ BCNF”.

Taking into account Corollary 3.2.3, we obtain the following result relating to a

∗ ∗ boundary of an equivalence between solutions BDNF, BCNF and F . Theorem 3.2.5

For each p ∈ N, 1 ≤ p < n, TDNF ∪ TCNF proves

∗ ∗ (A (X1,...,Xp)&&C(X1,...,Xn)) → (F (X1,...,Xn) ≡ BDNF(Xp+1,...,Xn)), (3.2.9) ∗ ∗ (A (X1,...,Xp)&&C(X1,...,Xn)) → (F (X1,...,Xn) ≡ BCNF(Xp+1,...,Xn)) (3.2.10) where n-ary predicate C is abbreviation for (3.1.13). proof: For the simplicity, let us consider p = n − 1. By (3.2.6) and (3.2.8), we obtain that

∗ TDNF ∪ TCNF ⊢ (A (X1,...,Xn−1)&&C(X1,...,Xn)) →

∗ (BDNF(Xn) → F (X1,...,Xn)),

78 together with (3.2.7) we derive (3.2.9). Analogously for (3.2.10). 2

These formulas say even more: they provide an idea of how can be estimated a

∗ “closeness” between F and B . Having on mind some model D of TDNF ∪ TCNF, let us remind that ||C||D is the lower boundary of equivalences between F and normal

∗ ∗ forms. If A is 1-true in D then ||F ≡ BDNF||D is influenced only by ||C||D, i.e.

∗ knowing that ||C||D ≥ p follows ||F ≡ BDNF||D ≥ p.

Furthermore, we shall investigate “closeness” of the input A∗ with respect to

descriptions of the neighborhoods of nodes [ci1 ,..., cip ] and its influence on the consequence B∗ of compositional rule of inference based on the respective normal form. Proposition 3.2.6

The theory TFC proves

L(c¯ip , x¯p) → (∀xp+1,...,xn)[Rp+1(cip+1 , xp+1)&& . . .&Rn(cin , xn)&&F (c¯in ) →

∗ BDNF(xp+1,...,xn)], (3.2.11) and

∗ L(¯xp, c¯ip ) → (∀xp+1,...,xn)[BCNF(xp+1,...,xn) →

(Rp+1(xp+1, cip+1 )&& . . .&Rn(xn, cin ) → F (ci1 ,..., cin ))], (3.2.12)

where L(¯xp, y¯p) stands for the formula

∗ (∀y1,...,yp)(R1(x1, y1)&& . . .&Rp(xp, yp) → A (y1,...,yp))&&

2 &&(∃ y1,...,yp)(R1(x1, y1)&& . . .&Rp(xp, yp)) , (3.2.13) for each k ∈ N and p ∈ N, 1 ≤ p < n. proof: It is sufficient to make the proof for p = n − 1. Let us start with the first

79 formula. An instance of (2.2.7)

∗ ∗ TFC ⊢ [A (¯xn−1)&&R(c¯in , x¯n)&&F (c¯in )&&&(A (¯xn−1) → R(c¯in−1 , x¯n−1))] →

∗ → (A (¯xn−1)&&R(c¯in , x¯n)&&F (c¯in )), then using commutativity of ∧ (2.2.9) and property of ∨ (2.2.27)

∗ TFC ⊢ [R(c¯in−1 , x¯n−1)&&R(c¯in , x¯n)&&F (c¯in )&&&(R(c¯in−1 , x¯n−1) → A (¯xn−1))] → k ∗ → [A (¯xn−1)&& (R(c¯in , x¯n)&&F (c¯in ))], i ,...,i =1 1 _n together with (2.2.40) and MP

2 TFC ⊢ (∃ x¯n−1)[R (c¯in−1 , x¯n−1)&&R(cin , xn)&&F (c¯in )&&

∗ &&(R(c¯in−1 , x¯n−1) → A (¯xn−1))] →

∗ → (∃ x¯n−1)(A (¯xn−1)&& DNF(¯xn)), and then by (2.2.42), (2.2.43) and transitivity of → (2.2.6)

2 TFC ⊢ [(∃ x¯n−1)R (c¯in−1 , x¯n−1)&&

∗ &&(∀x¯n−1)(R(c¯in−1 , x¯n−1) → A (¯xn−1))&&R(cin , xn)&&F (c¯in )] →

∗ → (∃ x¯n−1)(A (¯xn−1)&& DNF(¯xn)), which gives finally

2 TFC ⊢ [(∃ x¯n−1)R (c¯in−1 , x¯n−1)&&

∗ &&(∀x¯n−1)(R(c¯in−1 , x¯n−1) → A (¯xn−1))] →

∗ → [(R(cin , xn)&&F (c¯in )) → BDNF(xn)].

Now we prove (3.2.12) starting from the instance of (2.2.21)

∗ TFC ⊢ [A (¯xn−1) → (R(c¯in , x¯n) → F (c¯in ))] →

∗ ∗ → [(A (¯xn−1) → R(c¯in−1 , x¯n−1)) → (A (¯xn−1) → (R(c¯in , x¯n) → F (c¯in )))],

80 then by (2.2.28)

∗ TFC ⊢ [A (¯xn−1) → CNF(¯xn)] →

∗ ∗ → [(A (¯xn−1) → R(c¯in−1 , x¯n−1)) → (A (¯xn−1) → (R(c¯in , x¯n) → F (c¯in )))], from (2.2.10) and (2.2.9) we have

∗ TFC ⊢ (A (¯xn−1) → CNF(¯xn)) →

∗ → [((R(c¯in−1 , x¯n−1) → A (¯xn−1))&&R(c¯in−1 , x¯n−1)) → (R(c¯in , x¯n) → F (c¯in ))], which gives

∗ TFC ⊢ (A (¯xn−1) → CNF(¯xn)) →

∗ 2 → [((R(c¯in−1 , x¯n−1) → A (¯xn−1))&&R (c¯in−1 , x¯n−1)) → (R(cin , xn) → F (c¯in ))],

GR and (2.2.17) implies

∗ TFC ⊢ (∃ x¯n−1)[(A (¯xn−1) → CNF(¯xn))&&

∗ 2 &&(R(c¯in−1 , x¯n−1) → A (¯xn−1))&&R (c¯in−1 , x¯n−1)] →

→ (R(cin , xn) → F (c¯in )), applying two-time (2.2.42) together with the transitivity axiom for → expressed by

(2.2.6)

∗ TFC ⊢ [(∀x¯n−1)(A (¯xn−1) → CNF(¯xn))&&

∗ 2 &&(∀x¯n−1)(R(c¯in−1 , x¯n−1) → A (¯xn−1))&&&(∃ x¯n−1)R (c¯in−1 , x¯n−1)] →

→ (R(cin , xn) → F (c¯in )),

finally by (2.2.11) we obtain required formula (3.2.12). 2

Informatively saying, the above formulas formally express the natural observa- tion about the relationship of input and output of an inference with respect to the particular rule of a rule-base.

81 Let us read the antecedents of formulas (3.2.11) and (3.2.12) for p = n − 1 as

1-true in some model of TDNF and TCNF respectively. Then, we can say that

∗ 2 (a) if R(c¯in−1 , x¯n−1) is “sufficiently included” in A (¯xn−1) and R (c¯in−1 , x¯n−1) is

∗ non-empty then Rn(cin , xn)&&F (c¯in ) is “sufficiently included” in BDNF(xn),

∗ (b) and under the analogous assumptions as above, we can conclude that BCNF(xn)

is “sufficiently included” in Rn(xn, cin ) → F (c¯in ).

Strictly speaking,

∗ 2 (a) if R(c¯in−1 , x¯n−1) → A (¯xn−1) is p -true and (∀x¯n−1)R (c¯in−1 , x¯n−1) is r-true in

∗ D then Rn(cin , xn)&&F (c¯in ) → BDNF(xn) is p ∗ r-true in D,

∗ 2 (b) and analogously, if R(¯xn−1, c¯in−1 ) → A (¯xn−1) is p -true and (∀x¯n−1)R (¯xn−1, c¯in−1 )

∗ is r-true in D then BCNF(xn) → Rn(xn, cin ) → F (c¯in ) is p ∗ r-true in D.

All results of this subsection provide a quality estimation of fuzzy control sys- tems consisting of rule-bases interpreted as normal forms. Moreover, these results enlighten the problematic of approximate reasoning based on an imprecise descrip- tion of some physical system, which change the blind belief that considered system

(whenever it is known) can work properly into the well-defined and estimated fuzzy control system.

82 Chapter 4

Approximations over linearly ordered BL-algebras

Below, we are going to study approximation abilities of normal forms from the algebraical point of view.

First of all, formulas aimed at approximating a given fuzzy relation will be in- troduced over a BL-algebra. By virtue of the completeness of BL∀ with respect to linearly ordered BL-algebras, the results relating to the properties of such approxi- mation contained in Sections 4.1.1 and 4.1.2 can be obtained as direct consequences of the analysis of the normal forms made in the previous chapter.

Then, we shall investigate what kind of fuzzy relations satisfy the extensionality property. The extensionality expresses a relationship between functional values of a fuzzy relation w.r.t. the closeness of the input values.

Main part of this investigation lies in the construction of fuzzy relations r1,...,rn describing neighborhoods of a fixed n-dimensional point. The concrete shape of these relations has been motivated by special relations used in normal forms having an approximating character for functions from the functional systems of the three particular extensions of Basic Logic, i.e.Lukasiewicz, Product, and G¨odel logic. It

83 is necessary to present these functional systems together with special normal forms for understanding of the following construction.

Finally, we will formulate an approximation theorem based on the results of this chapter. It shows that Lipschitz continuity is a fundamental property which enables to approximate by normal forms with an arbitrary precision.

4.1 Normal Forms for extensional fuzzy relations

Throughout this section, let L be a linearly ordered BL-algebra in the form (2.2.1) with a support L and M be some nonempty set of objects. Moreover, let f(x1,...,xn) be an n-ary L-fuzzy relation on M, r1,...,rn are binary L-fuzzy relations on M,

n and Nk = {c1,...,ck}, where ci ∈ M for each i = 1,...,k. Let us denote by V (A) the set of all variations of order n from the elements of a set A.

The following definition introduces the normal forms of a finite (discrete) type for the L-fuzzy relation f based on the set of nodes {c1,...,ck}.

Definition 4.1.1 We say that fDNF,k is the disjunctive normal form for f w.r.t.

Nk if

fDNF,k(x)= (r1(c1, x1) ∗···∗ rn(cn, xn) ∗ f(c1,...,cn)), (4.1.1) n c∈V_(Nk) and analogously fCNF,k is the conjunctive normal form for f w.r.t. Nk if

fCNF,k(x)= (r1(x1,c1) ∗···∗ rn(xn,cn) →∗ f(c1,...,cn)). (4.1.2) n c∈V^(Nk) Below, we introduce the normal forms for the fuzzy relation f of an infinite type.

Definition 4.1.2 We say that fDNF is the disjunctive normal form for f if

fDNF(x)= (r1(c1, x1) ∗···∗ rn(cn, xn) ∗ f(c1,...,cn)), (4.1.3) c∈M n _ 84 and analogously fCNF is the conjunctive normal form for f if

fCNF(x)= (r1(x1,c1) ∗···∗ rn(xn,cn) →∗ f(c1,...,cn)). (4.1.4) c∈M n ^ The results of this section can be viewed as direct consequences of the logical formulas from the previous chapter which relate to representation and approximation properties of normal forms for extensional formulas. Why this is so, we shall show in the following lemma. Lemma 4.1.1

Let L,M,f,r1,...,rn,c1,...,ck are as above and let the theory NFϕ over BL∀ be given by Definition 3.1.1. Moreover, let

M = hM,f,r1,...,rn,c1,...,ck, fDNF,k, fCNF,k, fDNF, fCNFi

be a structure for NFϕ over L interpreting formula ϕ and predicates R1,...,Rn,

∃ ∀ individual constants c1,..., ck, and predicates DNFϕ,k, CNFϕ,k,DNFϕ,CNFϕ, re- spectively.

M is a model of the axioms of NF if and only if fDNF,k, fCNF,k, fDNF, fCNF are given by (4.1.1)–(4.1.4). proof: Obvious. 2

In the sequel, only L-safe structures M will be considered.

4.1.1 Approximating abilities of Normal Forms

Left and right extensionality gives us a special relationship between normal forms and initial fuzzy relation. We can say that fDNF,k (fCNF,k) for each k ∈ N represents lower (upper) approximation of a given left (right) extensional L-fuzzy relation.

85 Proposition 4.1.2

n If an L-fuzzy relation f(x) on M is left respective right extensional w.r.t. r1,...,rn

n at each c ∈ V (Nk), then

fDNF,k(x) ≤ f(x), (4.1.5) and

f(x) ≤ fCNF,k(x), (4.1.6) respectively.

In the sequel, we are interested in approximation abilities of normal forms over linearly ordered BL-algebras. Since the operation ↔∗ is dual to a pseudo-metric, it can serve as a measure of the approximation quality as well. Proposition 4.1.3

n If an L-fuzzy relation f(x) on M is left and right extensional w.r.t. r1,...,rn at

n each c ∈ V (Nk), then

P (x) ≤ fDNF,k(x) ↔∗ f(x) (4.1.7) and

P (x) ≤ fCNF,k(x) ↔∗ f(x), (4.1.8) where

P (x)= r1(c1, x1) ∗ r1(x1,c1) ∗ . . . ∗ rn(cn, xn) ∗ rn(xn,cn). n c∈V_(Nk) The structure of bounding function P (x) implies that the closeness of normal forms and the initial fuzzy relation can be adjusted by a suitable choice of c1,...,cn, whenever r1,...,rn are fixed. In other words, the quality of approximation depends

n on fuzzy partition of M specified by the nodes c1,...,ck. Having in mind that r1,...,rn may stand for different relations, we can obtain various fuzzy partitions.

86 Normal forms as approximating formulas constructed for extensional L-fuzzy relations should satisfy extensionality as well. The following proposition states that this natural requirement is fulfilled. Proposition 4.1.4

If an L-fuzzy relation f(x) is left respective right extensional w.r.t. r1,...,rn at each

n c ∈ V (Nk), then fDNF,k(x) is extensional w.r.t. binary L-fuzzy relations d1,...,dn on M given by

di(x, y)= ri(c, x) → ri(c, y), (4.1.9) c∈N ^k respective fCNF,k(x) is extensional w.r.t. e1,...,en given by

ei(x, y)= ri(y,c) → ri(x, c), (4.1.10) c∈N ^k for all i = 1,...,n.

4.1.2 Representation of extensional fuzzy relations

Now we can present the main result showing the representability of L-fuzzy relations by normal forms. Theorem 4.1.5

n An L-fuzzy relation f(x1,...,xn) on M is extensional w.r.t. reflexive binary L- fuzzy relations r1,...,rn on M if and only if

fDNF(x1,...,xn)= f(x1,...,xn) or (4.1.11)

fCNF(x1,...,xn)= f(x1,...,xn). (4.1.12)

This result shows that only extensional L-fuzzy relations can be approximated by normal forms of the discrete type with arbitrary precision provided that the limit case of fDNF,k (fCNF,k) for k → ∞ is fDNF (fCNF). Moreover, fDNF (fCNF) represents f if and only f is extensional.

87 4.1.3 Illustrations

For illustration, let us consider n = 1. Figure 4.1 illustrates how the functions represented by normal forms based on a reflexive fuzzy relation approximate the given one (sin(x)). Figure 4.1(c) illustrates that for the chosen function (sin(x)) and the fixed number of nodes normal forms based on the reflexive fuzzy relation

(x →L y) enable more precise approximation in comparison with normal forms based on the similarity (x ↔L y).

On Figure 4.1(d), we see the function P (x) which bounds the values of f(x) ↔L fDNF(CNF),5(x) from below. We can say dually that |f(x)−fDNF(CNF),5(x)| ≤ 1−P (x).

4.2 Functions associated with formulas of BL-logic

In this part we will present unified normal forms for a special sub-class of functions associated with formulas of the propositionalLukasiewicz, Product, and G¨odel logic.

We will see that formulas of BL are associated with functions having a unified form on each region specified by inequalities between functions of the same type.

Consequently, the representation of such functions leads to recursive algorithms, as has been shown in [44, 25] forLukasiewicz formulas, in [13] for Product formulas, and in [27] for G¨odel formulas. Remark that a characterization of functional systems of all the three particular extensions of Basic logic can be found in [27]. Below, we will show another characterization of these functional systems. Moreover, we will propose unified formula which has representing character for functions associated with G¨odel formulas and approximating character for the functions associated with the formulas ofLukasiewicz as well as Product formulas.

In fact, we are not interested in having a complex formula in most cases. We wish to find a simplified expression for the given formula (fuzzy relation), but the

88 0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

(a) r1(x,y) = x →L y (b) r1(x,y) = x ↔L y

1 0.2

0.95 0.15

0.9 0.1

0.85

0.05

0.8

0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

(c) Comparison of approximation errors (d) Bounding function P (x) for given as |f(x) − fDNF(CNF),5(x)| fDNF,5(x) ↔L f(x) as well as fCNF,5(x) ↔L f(x), where r1(x,y) = x →L y

Figure 4.1: Approximation of f(x) = sin(x) (black line) on [0, 1] by functions rep- resented by fDNF,5 (violet line) and fCNF,5 (blue line).

89 price we have to pay for it is that such a representation is not precise. Nevertheless, we shall show that an approximation of the functions associated with formulas of the propositional BL-calculus is possible with an arbitrary precision.

For the sake of brevity, we will use the notion of FL-function instead of fuzzy re- lation on [0, 1]. Further, we assume the language of BL expanded by truth constants a interpreted by a ∈ [0, 1].

The following definition establishes the connection between formulas of BL and the associated functional system in an iterative way.

Definition 4.2.1 We say that an FL-function fA is a function represented by a formula A or equivalently, a formula A represents the FL-function fA if their con- nection is based on the following inductive procedure:

1. If A is an atomic formula A := xi or A := a, a ∈ [0, 1] then fA(xi) ≡ xi or

fA ≡ a.

2. If A, B are formulas and fA, fB are FL-functions represented by them then A ◦ B, where ◦ ∈ {&,→,∧,∨∨}, represents

fA◦B ≡ fA • fB, (4.2.1)

where • ∈ {∗, →∗, ∧, ∨}, respectively. Moreover, ¬A represents ¬fA.

Moreover, consider BL+△ obtained from BL by adding the connective △ to- gether with the list of axioms introduced by M. Baaz in [2] (it can be also found in

[28], Definition 2.4.5). Interpretation is given as follows

1, if fA = 1; f△A ≡△fA = ( 0, otherwise.

90 4.2.1 Normal Forms for functions represented byLukasiewicz formulas

Below, we are going to investigate functions associated with formulas ofLukasiewicz logic. Until McNaughton Theorem, the formulations and results are taken from the monograph [44]. There, I. Perfilieva gave a constructive proof of the already known result by McNaughton, which concerns functions associated toL ukasiewicz formulas and their relationship to piecewise linear functions.

Definition 4.2.2 A non-empty set D ⊆ [0, 1]n is a convex polyhedron if it can be given by the system of linear inequalities

n T D = {x ∈ [0, 1] | Ax + b ≤ 0m}, (4.2.2)

where A is an integer (m×n)-matrix, b is a real (m×1)-vector, 0m is (m×1)-vector of zeros and ≤ is defined component-wise.

Definition 4.2.3 A function f(x) : [0, 1]n → [0, 1] is said to be piecewise lin- ear (with integer coefficients) if there exists a finite number of convex polyhedra

k n (1) (k) D1,...,Dk such that i=1 Dk = [0, 1] and f = f|D1,...,f = f|Dk are linear functions with integerS coefficients, i.e.

T f(x)= aix + bi if x ∈ Di, (4.2.3)

where ai is an n-dimensional vector of integers and bi is a real number for i =

1,...,k. The convex polyhedra D1,...,Dk will be called the linearity domains of f. Theorem 4.2.1

(McNaughton Theorem) An FL-function f(x) is represented by a formula of propositionalLukasiewicz logic if and only if f(x) is piecewise linear function with integer coefficients.

91 In the sequel, we will investigate the way of approximating of arbitrary piecewise linear function with integer coefficients by a function associated with the formula of propositionalLukasiewicz logic.

As the first step, we show another equivalent representation of a hyperplane. Lemma 4.2.2

T n T Let f(x)= ax +b, for all x ∈ R , then f(x)= a(x−c) +cn+1, where cn+1 = f(c). proof: Obvious. 2

Let us extend the powers of implications to the case when k ∈ Z

k k (x →⊗ y) if k ≥ 0, (x →⊗ y) = (4.2.4) (y → x)|k| otherwise.  ⊗

Next, we look at the transformation of truncated linear functions into some other equivalent form. We can pose the question: how to represent a truncated linear function on its sub-domain? Lemma 4.2.3

Let f¯ be a truncated linear function f with integer coefficients, i.e.

f¯(x) = 0 ∨ (a1x1 + . . . + anxn + b) ∧ 1, (4.2.5)

where ai ∈ Z and b ∈ R.

n Then, for each c = [c1,...,cn] ∈ [0, 1] there exists cn+1 ∈ [0, 1] such that

a1 an f¯(x)= f¯T,c(x)= cn+1 ⊗ (c1 →⊗ x1) ⊗ . . . ⊗ (cn →⊗ xn) , (4.2.6) for each

c ≤ y if a ≤ 0, x ∈ B = y ∈ [0, 1]n i i i for each i = 1,...,n . (4.2.7) ( yi ≤ ci otherwise, )

92 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2 1 0.1 0.8 0.6 0 0 0.4 0.2 0.4 0.2 0.6 0.8 1 0

Figure 4.2:Lukasiewicz implication expressed using (4.2.6) in c = [0.5, 0.5].

proof: For the simplicity, let us consider ai ≥ 0 for i = 1,...,n. Let us take

n+1 some [c1,...,cn,cn+1 = f(c1,...,cn)] ∈ [0, 1] and by Lemma 4.2.2 we have that

f¯(x) = 0 ∨ (a1(x1 − c1)+ . . . + an(xn − cn)+ cn+1) ∧ 1 n

= 0 ∨ (cn+1 + [ai(xi − ci + 1) − ai]) ∧ 1 i=1 Xn n ai = 0 ∨ (cn+1 + (ci →⊗ xi) + [(ai − 1) − ai]) ∧ 1 i=1 i=1 Xn X ai = 0 ∨ (cn+1 + (ci →⊗ xi) + [(n − 1) − n]) ∧ 1 i=1 n O ai = cn+1 ⊗ (ci →⊗ xi) = f¯T,c(x). i=1 O 2

From the above lemma, we conclude that it is impossible to represent arbitrary truncated linear function with integer coefficients in the form (4.2.6), see Figure 4.2.

Having this in mind, we can further ask, how to approximate a given FL-function?

An answer will be given at the end of this chapter. Here, we will only illustrate a possible solution on Figure 4.3.

93 1

0.8

0.6

0.4

0.2

0 1

0.8 1 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0 0

3 ¯ Figure 4.3:Lukasiewicz implication approximated by i=1 fT,ci (x), where ci = [ci,ci], i = 1, 2, 3, and ci’s are from {0.25, 0.5, 0.8}. W 4.2.2 Functions associated with Product (Goguen) formulas

In this subsection, we will search for analogous results for the case of propositional

Product logic. The ordering of the text in the sequel will follow the previous sub- section.

Definition 4.2.4 A function f(x) : [0, 1]n → [0, 1] is said to be monomial with

(positive) integer coefficients if

a1 an f(x)= b x1 ... xn , (4.2.8)

where ai ∈ Z (ai ∈ N0), i = 1,...,n, and b ∈ R+.

Definition 4.2.5 A non-empty set D ⊆ [0, 1]n is a monomial region if there ex- ist monomial functions f1(x),...,fk(x) and h1(x),...,hk(x) with positive integer coefficients such that D is specified by the following system of inequalities

n i D = {x ∈ [0, 1] | fi(x) 6 hi(x), i ∈ I}, (4.2.9) where 6i∈ {<, ≤} for each i ∈ I = {1,...,k}.

94 Figure 4.4: Monomial domains specified by →⊙.

Definition 4.2.6 A function f(x) : [0, 1]n → [0, 1] is said to be piecewise mono- mial with integer coefficients if there exists a finite number of monomial regions

k n (1) (k) D1,...,Dk such that i=1 Dk = [0, 1] and f = f|D1,...,f = f|Dk are mono- mial functions with integerS coefficients, i.e.

ai1 ain f(x)= bi x1 ... xn if x ∈ Di, (4.2.10)

n where ai = (ai1,...,ain) ∈ Z and bi ∈ R+ for all i = 1,...,k. The monomial regions D1,...,Dk will be called the monomial domains of f.

In the sequel, we will write only piecewise monomial function instead of piecewise monomial function with integer coefficients. Lemma 4.2.4

Let ⊙ be the product t-norm. Then, each formula of the form x&y, x → y, x ∧ y, x ∨ y, ¬x represents a picewise monomial function equal to x⊙y, x →⊙ y, x∧y, x∨y, ¬⊙x, respectively.

95 proof: Let us prove the claim for →⊙, ∧ and ∨.

1 if x ≤ y, x →⊙ y = x−1y otherwise,  and thus 

2 D1 = {[x, y] ∈ [0, 1] | x ≤ y, },

2 D2 = {[x, y] ∈ [0, 1] | y < x},

create the monomial domains of →⊙ and also of ∧ and ∨, see Figure 4.4.

From the above, we have that

1 if x = 0, ¬⊙x = x →⊙ 0= 0 otherwise,  hence 

D1 = {x ∈ [0, 1]| 0 < x} and D2 = {x ∈ [0, 1]| x ≤ 0}.

Finally, the operation ⊙ is specified by D = {x ∈ [0, 1]2| xy ≤ 1}. 2

A set of piecewise monomial functions has the same unique monomial domains. Lemma 4.2.5

Let f1,...,fk are n-ary piecewise monomial functions. Then there exist monomial

p n regions D1,...,Dp such that i=1 Di = [0, 1] and each Di, i = 1,...,p, is a monomial domain for all f1,...,fS k. proof: Obvious. 2

Below, the characterization of the functions represented by formulas of Product logic is given. Theorem 4.2.6

Let fA be an FL-function represented by a formula A of Product logic. Then, fA is piecewise monomial.

96 proof: The proof proceeds by induction on the number of connectives in the formula C.

1. If C is an atomic formula then fC (x) ≡ x or fC (x) ≡ a, a ∈ [0, 1], and thus

fC is piecewise monomial.

2. Suppose that C := A ◦ B, where ◦ ∈ {&, →, ∧, ∨}, and fA, fB are piecewise

monomial. Without loss of generality, we will assume that fA, fB are of the same arity equal to n.

Then fC = fA • fB, where • ∈ {⊙, →⊙, ∧, ∨}, respectively. By Lemma 4.2.4, • is piecewise monomial function. In accordance with Lemma 4.2.5, choose a

monomial region D1 being a monomial domain for both fA, fB and monomial

region D2 being a monomial domain for •. Then, there exist integer matrix (1) (2) (2) (1) (1) R2 A1 = (aij )(2×n), integer vector A2 = (a1 ,a2 ), b1 = (b1 , b2 ) ∈ +, and

(2) b ∈ R+ such that n (1) (1) a1i fA(x)= b1 xi , if x ∈ D1, i=1 Yn (1) (1) a2i fB(x)= b2 xi , if x ∈ D1, i=1 Y (2) a(2) a(2) x • y = b x 1 y 2 , if (x, y) ∈ D2.

−1 −1 Let us define D = D1 ∩ (fA , fB )D2 and prove that it is a monomial domain

for both fA, fB. Suppose that D1,D2 are specified as follows

n (1) n (1) n (1) cji j (1) gji D1 = {x ∈ [0, 1] | dj xi 6 hj xi , j = 1,...,m1}, i=1 i=1 Y(2) (2) Y(2) (2) n (2) ci1 ci2 i (2) ci1 gi2 D2 = {x ∈ [0, 1] | di x1 x2 6 hi x1 x2 , i = 1,...,m2},

then D is specified by the system of inequalities and equalities specifying D1 together with

(2) (2) (2) (2) (2) (2) cj1 cj2 j gj1 gj2 dj (fA(x)) (fB(x)) 6 hj (fA(x)) (fB(x)) ,

97 and equivalently n n rji j sji pj xi 6 qj xi , i=1 i=1 Y Y where (2) (1) (2) (1) (2) (1) (2) (1) (2) cj1 cj2 pj = dj (b1 ) (b2 ) , rjk = a1k cj1 + a2k cj2 , (2) (1) (2) (1) (2) (1) (2) (1) (2) cj1 cj2 qj = hj (b1 ) (b2 ) , sjk = a1k gj1 + a2k gj2 ,

for j = 1,...,m2, k = 1,...,n. It leads to the following specification

(1) (1) (1) n cji j (1) n gji n d x 6 h x , j = 1,...,m D = x ∈ [0, 1] j i=1 i j i=1 i 1 , n rji j n sji ( pj x 6 qj x , j = 1,...,m ) Q i=1 i Qi=1 i 2

using a simple transformation, Q where all negativeQ rjk,sjk we will obtain the

monomial domain of fA as well as fB.

Assume that D =6 ∅ and x ∈ D. Then

(2) (2) n a1 n a2 (1) (1) (2) (1) a1i (1) a2i fA(x) • fB(x)= b b1 xi b2 xi i=1 ! i=1 ! Y n Y (1) (2) (1) (2) (1) (2) (1) (2) (a a +a a ) (2) a1 a2 1i 1 2i 2 = b (b1 ) (b2 ) xi i=1 Y t1 tn = s x1 ...xn ,

(1) (2) (1) (2) (1) (2) (1) (2) (2) a1 a2 where s = b (b1 ) (b2 ) and ti = a1i a1 + a2i a2 for i = 1,...,n.

n The union over all couples (D1,D2) described above is equal [0, 1] , i.e.

−1 −1 n −1 −1 2 n (D1 ∩ (fA , fB )D2) = [0, 1] ∩ (fA , fB )([0, 1] ) = [0, 1] , D ,D [1 2 which proves that (fA • fB)(x) is a piecewise monomial function.

2

Now, we are going to show how a special class of piecewise monomial functions can be represented by formulas of propositional Product logic.

Analogously to the case of linear function, any monomial function can be trans- formed into another more suitable form.

98 Lemma 4.2.7

a a1 an Let f(x)= b x = b x1 ... xn then for each c = [c1,...,cn]

a a1 an x x1 xn f(x)= cn+1 = cn+1 . . . , c c1 cn       where cn+1 = f(c). proof: Obvious. 2

Let us extend the powers of implication to the case when k ∈ Z

k k (x →⊙ y) if k ≥ 0, (x →⊙ y) = (4.2.11) (y → x)|k| otherwise.  ⊙ The following lemma relates to the problem of representation of a monomial function on its special sub-domain. Lemma 4.2.8

Let f¯ be a truncated monomial function f with integer coefficients, i.e.

¯ a1 an f(x) = 1 ∧ (b x1 ... xn ), (4.2.12) where ai ∈ Z and b ∈ R+.

n Then, for each c ∈ [0, 1] there exists cn+1 ∈ [0, 1] such that

a1 an f¯(x)= f¯T,c(x)= cn+1 ⊙ (c1 →⊙ x1) ⊙ . . . ⊙ (cn →⊙ xn) , (4.2.13) for each y ≤ c if a ≤ 0, x ∈ B = y ∈ [0, 1]n i i i for all i = 1,...,n . (4.2.14) ( ci ≤ yi otherwise, )

proof: For the simplicity, let us consider ai ≥ 0 for i = 1,...,n. By Lemma n+1 4.2.7 there exists [c1,...,cn,cn+1] ∈ [0, 1] such that

a1 an x1 xn f¯(x) = 0 ∨ cn+1 . . . ∧ 1 c1 cn n    ai = cn+1 ⊙ (ci →⊙ xi) = f¯T,c(x), i=1 K 99 1

0.8

0.6

0.4

0.2

0 1

0.8 1 0.6 0.8 0.4 0.6 0.4 0.2 0.2 0 0

3 ¯ Figure 4.5: Product implication approximated by i=1 fT,ci (x) in the form (4.2.13), where ci = [ci,ci], i = 1, 2, 3, and ci’s are from {0.25, 0.5, 0.8}. W for arbitrary x ∈ B by properties of →⊙. 2

Thus, we have come to the analogous conclusion as in the previous subsection.

It means that, for the case of piecewise monomial functions, the formula (4.2.6) has only an approximating character, see Figure 4.5.

4.2.3 Functions associated with G¨odel (Minimum) formulas

Analogously as in the previous subsections, we will characterize functions associated with formulas of propositional G¨odel logic.

Definition 4.2.7 A function f(x) : [0, 1]n → [0, 1] is said to be minimial if

a1 an f(x)= b ∧ x1 ∧ . . . ∧ xn , (4.2.15)

where ai ∈ {0, 1} for i = 1,...,n and b ∈ [0, 1].

Definition 4.2.8 A non-empty set D ⊆ [0, 1]n is a minimial region if there exist minimial functions f1(x),... ,fk(x) and h1(x),... ,hk(x) such that D is specified

100 by the following system of inequalities

n i D = {x ∈ [0, 1] | fi(x) 6 hi(x), i ∈ I = {1,...,k}}, (4.2.16) where 6i∈ {<, ≤} for each i ∈ I.

Definition 4.2.9 A function f(x) : [0, 1]n → [0, 1] is said to be piecewise minimial

k if there exists a finite number of minimial regions D1,...,Dk such that i=1 Dk =

n (1) (k) [0, 1] and f = f|D1,...,f = f|Dk are minimial functions, i.e. S

(i) ai1 ain f(x)= f (x)= bi ∧ x1 ∧ . . . ∧ xn if x ∈ Di, (4.2.17) where aij ∈ {0, 1} and bi ∈ [0, 1] for i = 1,...,k, j = 1,...,n. The minimial regions

D1,...,Dk will be called the minimial domains of f. Lemma 4.2.9

Each formula of the form x → y, x∧y, x∨y, ¬x is represented by a piecewise minimial function equal to x →∧ y, x ∧ y, x ∨ y, ¬∧x, respectively. proof: Obvious. 2

Similarly to the case of piecewise linear (monomial) functions, a set of piecewise minimial functions has the same unique minimial domain. Lemma 4.2.10

Let f1,...,fk are n-ary piecewise minimial functions. Then there exist minimial

p n regions D1,...,Dp such that i=1 Di = [0, 1] and each Di, i = 1,...,p, is a minimial domain for all f1,...,fSk.

Below, the characterization of the functions represented by formulas of G¨odel logic is given. Theorem 4.2.11

Let fA be an FL-function represented by a formula A of G¨odel logic. Then, fA is piecewise minimial.

101 proof: The proof proceeds by induction on the number of connectives in the formula C.

1. If C is an atomic formula then fC (x) ≡ x or fC (x) ≡ a, a ∈ [0, 1], and thus

fC is piecewise minimial.

2. Suppose that C := ¬A or C := A ◦ B, where ◦ ∈ {∧, ∨, →}, and fA, fB are

piecewise minimial. Without loss of generality, we will assume that fA, fB are of the same arity equal to n.

The assertion is evidently true for fC = ¬∧fA. Suppose that fC = fA • fB,

where •∈{∧, ∨, →∧}. By Lemma 4.2.9, • is piecewise minimial function. In

accordance with Lemma 4.2.10, choose a minimial region D1 being a minimial

domain for both fA, fB and minimial region D2 being a minimial domain for

(1) •. Then there exist {0, 1}-matrix A1 = (aij )(2×n) and {0, 1}-vector A2 =

(2) (2) (1) (1) 2 (2) (a1 ,a2 ) and vector b1 =(b1 , b2 ) ∈ [0, 1] and b ∈ [0, 1] such that

(1) (1) a (1) 11 a1n fA(x)= b1 ∧ x1 ∧ . . . ∧ xn , if x ∈ D1,

(1) (1) a (1) 21 a2n fB(x)= b2 ∧ x1 ∧ . . . ∧ xn , if x ∈ D1,

(2) a(2) a(2) x • y = b ∧ x 1 ∧ y 2 , if (x, y) ∈ D2.

−1 −1 Let us define D = D1 ∩ (fA , fB )(D2) and prove that it is minimial domain

for both fA, fB. Suppose that D1,D2 are specified as follows

n (1) n (1) n (1) cji j (1) gji D1 = {x ∈ [0, 1] | dj ∧ xi 6 hj ∧ xi , j = 1,... ,m1}, i=1 i=1 ^(2) (2) ^ (2) (2) n (2) ci1 ci2 i (2) gi1 gi2 D2 = {x ∈ [0, 1] | di ∧ x1 ∧ x2 6 hi ∧ x1 ∧ x2 ,i = 1,... ,m2},

102 then D is specified by inequalities of D1 and by

(2) (2) n cj1 n cj2 (1) (1) (2) (1) a1i (1) a2i dj ∧ b1 ∧ xi ∧ b2 ∧ xi i=1 ! i=1 ! ^ ^ (2) (2) n gj1 n gj2 (1) (1) j (2) (1) a1i (1) a2i 6 hj ∧ b1 ∧ xi ∧ b2 ∧ xi , i=1 ! i=1 ! ^ ^ which gives n (1) n (2) (1) rji j (2) rji pj ∧ xi 6 pj ∧ xi , i=1 i=1 ^ ^ where

(1) (2) (1) (2) (1) (2) (1) (1) (2) (1) (2) cj1 cj2 pj = dj ∧ (b1 ) ∧ (b2 ) , rjk = a1k cj1 ∨ a2k cj2 , (2) (2) (1) (2) (1) (2) (2) (1) (2) (1) (2) gj1 gj2 pj = hj ∧ (b1 ) ∧ (b2 ) , rjk = a1k gj1 ∨ a2k gj2 , for j = 1,... ,m2, k = 1,... n. Thus

(1) (1) (1) n cji j (1) n gji n dj ∧ i=1 xi 6 hj ∧ i=1 xi , j = 1,...,m1 D = x ∈ (0, 1] (1) (2) . r r  (1) n ji 6j (2) n ji   pj ∧ Vi=1 xi pj ∧ Vi=1 xi , j = 1,...,m2  V V Assume that D =6 ∅ and x ∈ D. Then 

(2) (2) n a1 n a2 (1) (1) (2) (1) a1i (1) a2i fA(x) • fB(x)= b ∧ b1 ∧ xi ∧ b2 ∧ xi i=1 ! i=1 ! ^ n ^ (1) (2) (1) (2) (1) (2) (1) (2) (a a ∨a a ) (2) a1 a2 1i 1 2i 2 = b ∧ (b1 ) ∧ (b2 ) ∧ xi i=1 ^ t1 tn = s ∧ x1 ∧ . . . ∧ xn ,

(1) (2) (1) (2) (1) (2) (1) (2) (2) a1 a2 where s = b ∧ (b1 ) ∧ (b2 ) and ti = a1i a1 ∨ a2i a2 for i = 1,...,n.

n The union over all couples (D1,D2) described above is equal [0, 1] , i.e.

−1 −1 n −1 −1 2 n (D1 ∩ (fA , fB )(D2)) = [0, 1] ∩ (fA , fB )([0, 1] ) = [0, 1] , (D[1,D2) which proves that (fA • fB)(x) is a piecewise minimial function.

103 2

Now, we are going to show that each piecewise minimial function can be repre- sented by a formula of propositional G¨odel logic.

As the first step, let us look over the transformation of minimial functions into some other equivalent form. Lemma 4.2.12

a a1 an Let f(x)= b ∧ x = b ∧ x1 ∧ . . . ∧ xn be a minimial function then

a1 an f(x)= fT (x)= b ∧ (b →∧ x1) ∧ . . . ∧ (b →∧ xn) . (4.2.18)

proof: Since b = b ∧ b and b ∧ x = b ∧ (b →∧ x), we can write

a1 an f(x)= b ∧ x1 ∧ . . . ∧ xn

a1 an =(b ∧ x1 ) ∧ . . . ∧ (b ∧ xn )

a1 an = b ∧ (b →∧ x1 ) ∧ . . . ∧ (b →∧ xn )

a1 an = b ∧ (b →∧ x1) ∧ . . . ∧ (b →∧ xn) .

2

The following lemma relates to representation of a minimial domain. Lemma 4.2.13

Let D ⊆ [0, 1]n be a minimial region given by the system of minimial inequalities

(4.2.16), i.e.

D = {x ∈ [0, 1]n| f (i)(x) 6i h(i)(x), i = 1,...,m}.

Moreover, let △(x → y), if ¡ is ≤; d(x, y, ¡)= ∧ ( ¬(△(y →∧ x)), otherwise. Then, the function defined by m (i) (i) i χD(x)= d(fT (x),hT (x), 6 ), (4.2.19) i=1 ^ 104 is the characteristic function of D, i.e.

n D = {x ∈ [0, 1] | χD(x) = 1}.

The following theorem is fundamental for this subsection as it gives the required representation of piecewise minimial functions. Theorem 4.2.14

Let f(x) be a piecewise minimial function given by (4.2.17) with the minimial do- main D1,...,Dk.

Then, f(x) can be represented by formulas in the following two forms

k (i) fDNF(x)= [χDi ∧ fT (x)], (4.2.20) i=1 _ and k (i) fCNF(x)= [χDi →∧ fT (x)], (4.2.21) i=1 ^ called disjunctive and conjunctive normal form, respectively, such that

f(x)= fDNF(x)= fCNF(x). proof: It follows from Lemmas 4.2.12 and 4.2.13. 2

Theorems 4.2.11 and 4.2.14 imply the following characterization theorem. Corollary 4.2.15 f(x) is represented by a formula of propositional G¨odel logic if and only if f(x) is piecewise minimial function.

Disjunctive and conjunctive normal forms for a piecewise minimial function f given by (4.2.20) and (4.2.21), respectively, are the canonical forms of f. The fol- lowing corollary comprises the results about transformation of the class of piecewise minimial functions into the canonical forms.

105 Corollary 4.2.16

Let f be an n-ary piecewise minimial function with minimial domains D1,...,Dk,

(i) n aij where f|Di = f (x) = bi ∧ j=1 xi , aij ∈ {0, 1}, bi ∈ [0, 1] for i = 1,...,k, j = 1,...,n. Moreover, each DVi is specified by inequalities

n (l) n (l) eij ij dij gij ∧ xl 6 hij ∧ xl , l l ^=1 ^=1 j = 1,... ,mi, i = 1,...,k.

Then f(x) can be represented in canonical forms (4.2.20) and (4.2.21), where

(i) ai1 ain fT (x)= bi ∧ (bi →∧ x1) ∧ . . . ∧ (bi →∧ xn) , mi n n (l) (l) eij dij ij χDi (x)= d(gij ∧ (gij →∧ x1) ,hij ∧ (hij →∧ xn) , 6 ), j=1 l l ^ ^=1 ^=1 for all i = 1,... ,k.

The representation of an arbitrary minimial function by formulas having form of implications joined by conjunction proved above was possible since the minimum operation is idempotent.

4.2.4 Normal forms for functions associated with formulas of BL-logic

The main result of this section shows that each uniformly continuous FL-function can be approximated by normal forms associated with formulas over BL-logic with an arbitrary precision. This result is a direct generalization of Theorem 5.3 from [44], where the universal approximation formula has been introduced overLukasiewicz algebra and only in disjunctive form. Lemma 4.2.17

Let ∗ be a continuous t-norm and ci1 ≤ ci2, for all i = 1,...,n, where cij ∈ [0, 1].

106 Moreover, let D ⊆ [0, 1]n be a region given by the system of inequalities

n D = {x ∈ [0, 1] | ci1 ≤ xi and xi ≤ ci2, for i ∈ I ⊆ {1,...,n}}. (4.2.22)

Then, the function defined by

χD(x)= △ (xi →∗ ci2) ∗ (ci1 →∗ xi), (4.2.23) i∈I ^ is the characteristic function of D, i.e.

n D = {x ∈ [0, 1] | χD(x) = 1}. proof: Obvious. 2

The following theorem shows that each uniformly continuous FL-function can be approximated by normal forms with arbitrary precision. Theorem 4.2.18

Let f be a uniformly continuous n-ary FL-function and ∗ be a continuous t-norm. N Then, for arbitrary ε> 0 there exist k ∈ , Di1,...,in of the form (4.2.22) and dij , for

1 ≤ ij ≤ k, such that

k f (x)= (χ ∗ f(d ,...,d )), (4.2.24) DNF,k Di1,...,in i1 in i ,...,i =1 1 _n k f (x)= (χ → f(d ,...,d )), (4.2.25) CNF,k Di1,...,in ∗ i1 in i ,...,i =1 1 ^n where each χ is of the form (4.2.23), ε-approximate f w.r.t. d(x, y)= |x − y|, Di1,...,in i.e.

|fDNF,k(x) − f(x)| ≤ ε and |fCNF,k(x) − f(x)| ≤ ε. proof: The uniform continuity of f means that for arbitrary ε > 0 there ex- ists δ > 0 such that for all xi, yi ∈ [0, 1]: |xi − yi| < δ, i = 1,...,n, implies

|f(x1,...,xn) − f(y1,...,yn)| <ε.

107 Let us fix some ε then we find δ satisfying the above definition of continuity of

1 f. Next, we set k = [ δ ] + 1, where [d] is the truncation of d, and take c1 = 0 and ci+1 = ci + δ, for i = 1,...,k. Then, we specify Di1,...,in as follows

n−1 xi ≥ cij , xi ≤ cij+1 if ij ∈ {1,...,k − 1}, Di1,...,in = x ∈ [0, 1] i = 1,...,n . ( xi ≥ cij otherwise, )

We associate with these regions the characteristic functions χD in accordance i1,...,in with formula (??). Finally, we choose dij so that [di1 ,...,din ] ∈ Di1,...,in , for all

1 ≤ ij ≤ k. The continuity of f implies that |f(x1,...,xn) − f(di1 ,...,din )| <ε, for

each [x1,...,xn] ∈ Di1,...,in , and consequently

|f(x1,...,xn) − (χDi1,...,in ∗ f(di1 ,...,din ))| < ε, and

|f(x1,...,xn) − (χDi1,...,in →∗ f(di1 ,...,din ))| < ε,

for each [x1,...,xn] ∈ Di1,...,in , which gives

k

|f(x1,...,xn) − (χDi1,...,in ∗ f(di1 ,...,din ))| < ε, and i ,...,i =1 1 _n k

|f(x1,...,xn) − (χDi1,...,in →∗ f(di1 ,...,din ))| < ε, i ,...,i =1 1 ^n n for arbitrary [x1,...,xn] ∈ [0, 1] . 2

Note that approximating formulas (4.2.24) and (4.2.25) are partial constant func- tions (discontinuous).

4.3 Characterization of extensional fuzzy relations

As we have seen in the previous section the representation of the functions associated with formulas of BL is not possible by formulas in the form (4.1.1) or (4.1.2) (except the 1-dimensional case). Thus it is necessary to involve a property natural to all formulas of BL and waive one’s hope to precise representation. Extensionality is the

108 best candidate for such property since all formulas of propositional BL are exten- sional and obviously each formula build from extensional predicates is extensional as well, see Lemma 2.2.10.

The question is: what kind of functions satisfies extensionality property? Note that extensionality of fuzzy relation refers to the relationship between objects and their properties expressed by considered relation. Roughly speaking, if two objects are e.g. similar and the first one has some property then the second object has the same property.

In [22], it was proved that a pseudo-metric induces similarity relation and vice versa. This relation is based on an additive generator of a given t-norm.

The above cited construction of pseudo-metrics from similarities based on ad- ditive generator of a t-norm was used by I. Perfilieva in [54] for the study of rela- tionship between extensionality and continuity of fuzzy relations. It was shown that extensionality w.r.t. similarity leads to Lipschitz continuity.

In this work, we will focus on much general case of extensionality. We will investigate extensionality of a fuzzy relation w.r.t. reflexive binary fuzzy relations.

We expected to show that this generalized extensionality is equivalent to continuity.

But the results of Section 4.3.2 lead to conclusion that it is much stronger property.

A construction of the reflexive binary fuzzy relations which make considered fuzzy function extensional will be shown there. And it follows that continuity is necessary but not sufficient condition on a fuzzy relation to be extensional w.r.t. reflexive binary fuzzy relations with continuous membership functions.

109 4.3.1 Extensionality w.r.t. Reflexivity means Lipschitz Con- tinuity

In this section, the class of extensional fuzzy relations will be characterized.

Let us remind the results from the paper [16] where it has been proved that for

Lukasiewicz t-norm, extensionality of a fuzzy relation w.r.t. similarity is equivalent to Lipschitz continuity w.r.t. the pseudo-metric induced by the similarity.

Relationship between Lipschitz continuity and extensionality of fuzzy relation was also investigated in [54]. It was shown that extensionality of a fuzzy relation w.r.t. similarity leads to generalized Lipschitz continuity w.r.t. the pseudo-metric induced by this similarity and vice versa.

Let us remind that a similarity is a reflexive, symmetric and t-transitive binary fuzzy relation. In the following sections, we are interested in relationship between extensionality and continuity w.r.t. reflexive binary fuzzy relations.

Theorem 4.3.1 is the generalization of the above cited results. We will consider n-ary extensional fuzzy relations which are defined on R and prove their continuity.

We will use the following abbreviation in the sequel

R(x, y)= r1(x1, y1) ∗···∗ rn(xn, yn), (4.3.1)

n for arbitrary x = [x1,...,xn], y = [y1,...,yn] ∈ M .

Remark 4.3.1 In the sequel, we will analyze mappings from X ⊆ Rn onto [0,g(0)], where g is the additive generator of a t-norm. Thus, it may happen that ∞ is included. For the full generality of Theorem 4.3.1, we put by the definition:

|g(0) − x| = g(0), for each x ∈ [0,g(0)),

g(0) − g(0) = 0.

110 Hence, de(x, y)= |x − y| is the extended standard metric on [0,g(0)]. Theorem 4.3.1

Let M ⊆ R be a closed interval and a t-norm ∗ has the continuous additive gen- erator g(x). Moreover, let r1,...,rn be reflexive binary fuzzy relations on M and

2 g(r1(x, y)),...,g(rn(x, y)) : M → [0,g(0)] are Lipschitz continuous w.r.t. the ex-

n tended standard metric. If an n-ary fuzzy relation F (x1,...,xn) : M → [0, 1] is extensional w.r.t. r1,...,rn on M then g(F (x1,...,xn)) is Lipschitz continuous w.r.t. the extended standard metric.

proof: We will prove the theorem by . Suppose that F (x1,...,xn) is extensional w.r.t. r1,...,rn with Lipschitz continuous functions g(r1),...,g(rn). Suppose that F is not Lipschitz continuous, i.e. for each K > 0 there exist c =

[c1,...,cn] and d = [d1,...,dn] such that n

|g(F (c)) − g(F (d))| >K |ci − di|. (4.3.2) i=1 X Lipschitz continuity of g(r1),...,g(rn) means that there exist α1,...,αn > 0 such that 2

|g(rj(x1, x2)) − g(rj(y1, y2))| ≤ αj |xi − yi|, (4.3.3) i=1 X for all xi, yi ∈ M, i = 1, 2 and j = 1,...,n.

Now, let us take K = max {αi} and find c and d satisfying inequality (4.3.2). i=1,...,n Substituting the concrete values into (4.3.3), we obtain for i = 1,...,n

|g(ri(ci,di)) − g(ri(di,di))| ≤ αi|ci − di|, by reflexivity

g(ri(ci,di)) ≤ αi|ci − di|.

Let us define x + g(0) = g(0), for each x ∈ [0,g(0)], and hence n n

g(ri(ci,di)) ≤ max {αi} |ci − di|. (4.3.4) i=1,...,n i=1 i=1 X X 111 From (4.3.4) and (4.3.2) follow that

n n

g(ri(ci,di)) ≤ K |ci − di| < |g(F (c)) − g(F (d))|. i=1 i=1 X X We will distinguish the following cases:

1. If g(F (c)) − g(F (d)) > 0 then

n

g(ri(ci,di))

ri(ci,di) > F (d) →∗ F (c)), i=1 O which contradicts the assumption of the extensionality of F .

2. If g(F (c)) − g(F (d)) < 0 then

n

g(ri(ci,di))

ri(ci,di) > F (c) →∗ F (d)), i=1 O which again contradicts the assumption of extensionality of F .

3. If g(F (c)) − g(F (d)) = 0 then

n

g(ri(ci,di)) < 0, i=1 X it would means that at least some g(ri(ci,di)) < 0. Whereas g : [0, 1] → [0,g(0)].

2

The above result links extensionality with Lipschitz continuity assuming that the considered t-norm is generated. Analogous result can be obtain even for minimum t-norm.

112 Theorem 4.3.2

Let M ⊆ R be a closed interval and ∗ be the t-norm of the minimum. Moreover, let

2 r1,...,rn be reflexive binary fuzzy relations on M which are Lipschitz continuous w.r.t. the metric d(x, y) = maxi=1,...,n |xi − yi|.

n If an n-ary fuzzy relation F (x1,...,xn) : M → [0, 1] is extensional w.r.t. r1,...,rn on M then F (x1,...,xn) is Lipschitz continuous w.r.t. the metric d.

proof: From the Lipschitz continuity and reflexivity of r1,...,rn

|1 − ri(x, y)| ≤ αi|x − y|, for all i = 1,...,n at each x, y ∈ M. Moreover, the extensionality of F (x) implies

|F (x) − F (y)| ≤ 1 − F (y) ≤ |1 − R(x, y)| = max |1 − ri(xi, yi)|, i=1,...,n whenever F (x) > F (y). And hence

|F (x) − F (y)| ≤ K max |xi − yi|, i=1,...,n where K = maxi=1,...,n αi. The case F (x) < F (y) is analogous, we just exchange variables, and the case F (x)= F (y) is obvious. 2

Notice that in the case of similarity relation R(x, y) on M the operation d(x, y)=

1 − R(x, y), ∀x, y ∈ M defines ultrapseudometric on M, which is a pseudometric such that, for any x, y, z ∈ M, d(x, y) ≤ d(x, z) ∨ d(z, y). For the details see [26].

However, the analogous relationship between left (right) extensionality and left

(right) continuity does not hold. There exist examples of left (right) extensional relations which are not left (right) continuous.

113 4.3.2 From Lipschitz Continuity to Extensionality w.r.t. Re- flexivity

From the previous section, we see that if a fuzzy relation is extensional w.r.t. Lip- schitz continuous reflexive binary fuzzy relations then it is Lipschitz continuous.

In the following, we will try to solve the reverse problem. We will search for suitable reflexive binary fuzzy relations which make a given continuous fuzzy relation extensional.

It is clear that if each binary fuzzy relation ri, i = 1,...,n, is the ordinary equal- ity on M then arbitrary n-ary fuzzy relation is extensional on M w.r.t. r1,...,rn. Ordinary equality is a binary fuzzy relation characterized by the following member- ship function: 1 if x = y, R(x, y)= (4.3.5) 0 otherwise.  We will further refer to this fuzzy relation as trivial one. In Lemma 4.3.3, we will be interested in non-trivial binary fuzzy relations.

Let us consider an arbitrary n-ary fuzzy relation F (x1,...,xn) and define binary fuzzy relations r1,...,rn by the formulas

k˜i l˜i ri(x, y)=(T (x) →∗ T (y)) ∗ (T (y) →∗ T (x)) , (4.3.6)

for all x, y ∈ [a, b] and i = 1,...,n. The parameters ki, li are dependent on the variables x, y and will be specified later. Here, we denoted

max{0, [p] + 1} if p> [p], p˜ = (4.3.7) max{0,p} otherwise.  and fuzzy sets T (x) are specified as follows

x − a T (x)= . (4.3.8) b − a

114 Figure 4.6: Illustration of a combination r1(x1,c1) ⊗ r2(x2,c2) of the type (4.3.6).

Note that the transformation T may have different forms. In fuzzy control it is called scaling function.

It may happen that some of the parameters ki, li, i = 1,...,n, are equal to zero. Then, the membership functions of such a relations will be given by (2.1.6).

An example of ∗-combination of relations of the type (4.3.6) in 2-dimensional case is shown on Figure 4.6. On this figure, it can be seen that the relations r1 and r2 are reflexive. It easily follows also from the properties of implication. Lemma 4.3.3

Let g(x) be a continuous additive generator of a t-norm ∗ and F (x) be an n-ary FL-

−1 −1 function. Moreover, let g(F (g (x1),...,g (xn))) has bounded partial derivatives w.r.t. each variable on (0,g(0))n. Then for any c ∈ [0, 1]n

115 1

0.8

0.6

0.4

0.2

0 1 1 0.8 0.5 0.6 0.4 0.2 0 0

Figure 4.7: Relation from Figure 4.6 for ⊙.

1. F (x) is left extensional w.r.t. r1,...,rn given by (4.3.6) and c, i.e.

n k˜i ˜li (ci →∗ xi) ∗ (xi →∗ ci) ≤ F (c) →∗ F (x), (4.3.9) i=1 O

for all xi ∈ [0, 1], i = 1,...,n. The parameters ki, li, i = 1,...,n in the

relations r1,...,rn are as follows

∂(g ◦ F ◦ g−1) ki = max (g(x)), x,xi≤ci ∂xi ∂ − (g ◦ F ◦ g−1) li = max (g(x)), x,ci≤xi ∂xi

2. F (x) is right extensional w.r.t. r1,...,rn given by (4.3.6) and c, i.e.

n k˜i ˜li (xi →∗ ci) ∗ (ci →∗ xi) ≤ F (x) →∗ F (c), (4.3.10) i=1 O

for all xi ∈ [0, 1], i = 1,...,n. The parameters ki, li, i = 1,...,n in the

116 relations r1,...,rn are as follows

∂(g ◦ F ◦ g−1) ki = max (g(x)) x,ci≤xi ∂xi ∂ − (g ◦ F ◦ g−1) li = max (g(x)). x,xi≤ci ∂xi

Moreover, r1(x, y),...,rn(x, y) are left-continuous in the first variable and right- continuous in the second one for the case of strict ∗ and continuous for nilpotent

∗.

proof: For shortness, let us denote an n-tuple [x1,...,xn] by x and [g(x1),...,g(xn)] by g(x). We will use the following sub-domains of [0, 1]n w.r.t. c:

−1 n Di = {x ∈ [0, 1] | xi ≤ ci},

1 n Di = {x ∈ [0, 1] | xi ≥ ci},

Letp ¯ = (p1,...,pn) be an n-dimensional vector such that pi ∈ {−1, 1} for each i = 1,...,n. Then we define the following sub-domain w.r.tp ¯

n pi Dp¯ = Di . i=1 \

Let P = {p¯ | pi ∈ {−1, 1},i = 1,...,n} then

n [0, 1] = Dp¯. p¯∈P [ 1. Let us consider arbitrary c ∈ [0, 1]n and prove the left extensionality of F (x)

w.r.t. r1,...,rn and c. From the generalized Taylor’s theorem we obtain

′ ′ ′ n ′ ′ ′ that for any c = (c1,...,cn) ∈ (0,g(0)) and arbitrary x = (x1,...,xn) ∈

n n (0,g(0)) there exist points Pi ∈ (0,g(0)) , i = 1,...,n such that

n ∂(g ◦ F ◦ g−1) g(F (g−1(x′))) − g(F (g−1(c′))) = (x′ − c′ ) (P′), (4.3.11) i i ∂x′ i i=1 i X 117 ′ ′ ′ ′ and moreover d(Pi, c )

The restriction for the partial derivatives implies that for all x′ ∈ (0,g(0))n

−1 ∂(g ◦ F ◦ g ) ′ | ′ (x )| ≤ Mi, ∂xi and therefore, ∂(g ◦ F ◦ g−1) | (g(x))| ≤ Mi, ∂xi for all x ∈ [0, 1]n and i = 1,...,n.

At first, let us consider the case x ∈ Dp¯, wherep ¯ = (−1, −1,..., −1). It follows that there exist 0 ≤ Qi ≤ Mi, i = 1,...,n, such that ∂(g ◦ F ◦ g−1) (g(x)) ≤ Qi for all x ∈ Dp1 , ∂xi ∂(g ◦ F ◦ g−1) ki = max (g(x)) ≤ Qi. x,xi≤ci ∂xi ′ ′ ′ Substituting x = g(x), c = g(c) and Pi = g(Pi) into (4.3.11) we get n ∂(g ◦ F ◦ g−1) g(F (x)) − g(F (c)) = (g(xi) − g(ci)) (g(Pi)), (4.3.12) ∂xi i=1 Xn

g(F (x)) − g(F (c)) ≤ (g(xi) − g(ci))k˜i. (4.3.13) i=1 X Using the properties of the generator g, we can rewrite the last inequality as follows n (−1) (−1) (−1) g [ g(g ((g(xi) − g(ci))))k˜i] ≤ g [g(F (x)) − g(F (c))], (4.3.14) i=1 X n k˜i (ci →∗ xi) ≤ F (c) →∗ F (x). (4.3.15) i=1 O Ki It is clear that xi ≤ ci implies (xi →∗ ci) = 1 for arbitrary integer Ki ≥ 0.

Because ˜li are bounded for all i = 1,...,n, analogously as in the case of ki then we can set Ki = ˜li and rewrite (4.3.15) equivalently n ˜li k˜i (xi →∗ ci) ∗ (ci →∗ xi) ≤ F (c) →∗ F (x), (4.3.16) i=1 O which proves the left extensionality of F w.r.t. r1 ...,rn and c for all X ∈ Dp¯.

118 2. In order to prove the right extensionality of F (x) w.r.t. r1 ...,rn and c on Dp¯ we will use (4.3.12)

n −∂(g ◦ F ◦ g−1) g(F (c)) − g(F (x)) = (g(xi) − g(ci)) (g(Pi)). ∂xi i=1 X Then n

g(F (c)) − g(F (x)) ≤ (g(xi) − g(ci))˜ni, i=1 X where ∂ − (g ◦ F ◦ g−1) li = max (g(x)) ≤ Si, x,xi≤ci ∂xi and analogously n ˜li (ci →∗ xi) ≤ F (x) →∗ F (c). i=1 O Let us set Ki = k˜i for all i = 1,...,n, then we can rewrite the last inequality as follows n k˜i ˜li (xi →∗ ci) ∗ (ci →∗ xi) ≤ F (x) →∗ F (c), i=1 O which is the right extensionality or F w.r.t. r1,...,rn and c on Dp¯.

The left and right extensionality on the others 2n − 1 sub-areas determined by the point c are proved analogously.

The proof for the cases where ci ∈ {0, 1}, i = 1,...,n are analogous to the given one.

Since for arbitrarily chosen c we obtain finite parameters ki, li,i = 1,...,n then the relations (4.3.6) is continuous for nilpotent ∗ and left(right)-continuous in the

fist(second) variable for strict ∗. 2

In the following theorem, a generalized claim for fuzzy relations is formulated and proved.

119 Theorem 4.3.4

Let g(x) be a continuous additive generator of a t-norm ∗ and F (x) be an n-ary

−1 −1 −1 −1 fuzzy relation. Moreover, let g(F (T ◦ g (x1),...,T ◦ g (xn))) has bounded partial derivatives w.r.t. each variable on (0,g(0))n. Then for any c ∈ [a, b]n

1. F (x) is left extensional w.r.t. r1,...,rn given by (4.3.6) at c and the parame-

ters ki, li, i = 1,...,n are as follows

∂(g ◦ F ◦ T −1 ◦ g−1) ki = max (g ◦ T (x)), x,xi≤ci ∂xi ∂ − (g ◦ F ◦ T −1 ◦ g−1) li = max (g ◦ T (x)), x,ci≤xi ∂xi

2. F (x) is right extensional w.r.t. r1,...,rn given by (4.3.6) at c and the para-

meters ki, li, i = 1,...,n are as follows

∂(g ◦ F ◦ T −1 ◦ g−1) ki = max (g ◦ T (x)) x,ci≤xi ∂xi ∂ − (g ◦ F ◦ T −1 ◦ g−1) li = max (g ◦ T (x)). x,xi≤ci ∂xi

−1 −1 −1 −1 Moreover, r˜i(x, y)= g(ri(T ◦g (x), T ◦g (y))) for all i are Lipschitz continuous on (0,g(0))2 w.r.t. the standard metric.

proof: The claim about extensionality of F w.r.t. r1 ...,rn easily follows from the previous lemma.

Now we have to prove Lipschitz continuity ofr ˜i(x, y) for all i = 1,...,n. Let us fix some i then for arbitrary x, y ∈ (0,g(0)) we obtain

−1 −1 −1 −1 −1 −1 k˜i −1 −1 l˜i ri(T ◦ g (x), T ◦ g (y))=(g (x) →∗ g (y)) ∗ (g (y) →∗ g (x))

˜ ˜ =(g(−1)(y − x))ki ∗ (g(−1)(x − y))li

(−1) (−1) = g (k˜i(y − x)) ∗ g (l˜i(x − y))

(−1) = g (k˜i(y − x)+ l˜i(x − y)),

120 and after applying g we have

−1 −1 −1 −1 g(ri(T ◦ g (x), T ◦ g (y))) = k˜i(y − x)+ l˜i(x − y)

≤ k˜i|y − x| + l˜i|x − y|

=(k˜i + l˜i)|x − y|,

∂(g◦F ◦T −1◦g−1) where k˜i, l˜i are bounded. Now, let us denote boundary of | | by Mi then ∂xi

˜1 ˜1 ˜2 ˜2 |r˜i(x1, y1) − r˜i(x2, y2)| ≤ (ki + li )|x1 − y1| − (ki + li )|x2 − y2|

˜1 ˜1 ˜2 ˜2 ≤ (ki + li )|x1 − y1| +(ki + li )|x2 − y2|

≤ 2Mi(|x1 − y1| + |x2 − y2|), which verifies Lipschitz continuity ofr ˜i(x, y). 2

Generally, for arbitrary fuzzy relation we can find coefficients of the reflexive binary fuzzy relations given by (4.3.6) making this fuzzy relation extensional. Then ri need not be necessary continuous. Observe that in this case ki, li ∈ [0, ∞] where ∞ is included.

The following corollary is the direct consequence of Theorem 4.3.4 for the case of relations given by (4.3.6) with ki = li for all i = 1,...,n, i.e. relations of similarities. Corollary 4.3.5

Let all the assumptions of Theorem 4.3.4 be valid. Then, F (x) is extensional w.r.t. similarities s1,...,sn given by

k˜i si(x, y)=(T (x) ↔ T (y)) . (4.3.17)

The parameters in the relations s1,...,sn are as follows ∂(g ◦ F ◦ T −1 ◦ g−1) ki = max | (g ◦ T (x))| x∈[0,1]n ∂xi for all i = 1,...,n. Moreover, g(s1(x, y)),...,g(sn(x, y)) are Lipschitz continuous w.r.t. the standard metric.

121 A relationship between the reflexive relations given by (4.3.6) and the similarities given by (4.3.17) is presented in the following corollary. Corollary 4.3.6

Let all the assumptions of Theorem 4.3.4 and Corollary 4.3.5 be valid. Then

si(x, y) ≤ ri(x, y) (4.3.18) for all i = 1,...,n and x, y ∈ [0, 1].

In the theory of approximations of fuzzy relations, we are able to propose a general approximating formula for the class of extensional fuzzy relations. This connection is closely associated with a choice of binary fuzzy relations in (2.1.7).

From inequality (4.3.18), it follows that the choice of reflexive binary fuzzy relations is preferable to the similarities given above.

Example 4.3.1 Let us consider the case n = 1 and arbitrary FL-function F (x) where g(F (g−1(x))) is Lipschitz continuous. Then, we obtain the following reflexive binary fuzzy relation

k˜1 ˜l1 r1(x, y)=(x →∗ y) ∗ (y →∗ x) . (4.3.19)

Theorem 4.3.4 states that for arbitrary c ∈ [0, 1], F (x) is left extensional w.r.t.

(4.3.19) and c with the respective coefficients

−1 ′ k1 := max (g ◦ F ◦ g ) (g(x)), x≤c −1 ′ l1 := max − (g ◦ F ◦ g ) (g(x)), c≤x and F (x) is right extensional w.r.t. (4.3.19) and c with the following coefficients

−1 ′ k1 := max (g ◦ F ◦ g ) (g(x)), c≤x −1 ′ l1 := max − (g ◦ F ◦ g ) (g(x)). x≤c

122 (a) Combinations R1 = r1(0.6,x) ⊗ F (0.6), (b) r1(0.6,x) and r1(x, 0.4) R2 = F (0.6) →⊗ r1(x, 0.4) and F (x)

Figure 4.8: Illustration of the relation of the type (4.3.6) making FL-function F given by (4.3.20) extensional

Let us illustrate this by considering the following FL-function

F (x) = sin(2x) (4.3.20) and g(x) = 1 − x, i.e. ∗ isLukasiewicz t-norm. The result is shown on Figure

4.9. The following table relates to this figure and shows the obtained coefficients for r1(c, x) (left extensionality) where c is a fixed constant and x ∈ [0, 1]

c k1 l1 0.6 2 1 and for r1(x, c) (right extensionality)

c k1 l1 0.4 2 0

Example 4.3.2 Let F (x, y) = xy and g(x) = 1 − x then F (x, y) is extensional w.r.t. r1, r2 given by (4.3.6) with the following parameters

k1, k2 l1, l2 1 0

123 1 1 0.8 0.8 0.6 1 0.6 1 0.4 0.8 0.4 0.8 0.2 0.2 0 0.6 0 0.6 0 0 0.2 0.4 0.2 0.4 0.4 0.4 0.6 0.2 0.6 0.2 0.8 0.8 10 10

(a) Combination r1(0.6,x) ⊗ r2(0.5,y) ⊗ (b) Combination r1(x, 0.6) ⊗ r2(y, 0.5) →⊗ F (0.6, 0.5) F (0.6, 0.5)

Figure 4.9: Illustration of the relation making FL-function F (x, y)= xy extensional

The main goal of this result lies in clarification of the meaning of the generalized extensionality property. We considered t-norms having additive generators. It was proved that the generalized extensionality w.r.t. reflexive binary fuzzy relations with Lipschitz continuous membership functions implies Lipschitz continuity of the transformed fuzzy relation (transformation by an additive generator).

Moreover, it was shown that for a Lipschitz continuous fuzzy relations we can

find non-trivial reflexive binary fuzzy relations which make the considered relation extensional.

4.3.3 Notes on ∧-extensionality

Since the minimum t-norm ∧ is idempotent, it would be useless to search for anal- ogous fuzzy relation of the form (4.3.6), for which the problem is reduced to the choice between 0 and 1-powers of the particular implications. In this connection, it is more suitable to work with similarities.

124 Proposition 4.3.7

A fuzzy relation F (x1,...,xn) on M is extensional w.r.t. similarity s if and only if F is 1-Lipschitz continuous w.r.t. pseudo-ultrametric d(x, y) = 1 − s(x, y).

proof: Let us consider the case f(x) =6 f(y) then f(x) ↔∧ f(y)= f(x) ∧ f(y) and so

|f(x) − f(y)| =(f(x) − f(y)) ∨ (f(y) − f(x)) ≤ (1 − f(x)) ∨ (1 − f(y)) =

1 − (f(x) ∧ f(y))=1 − (f(x) ↔ f(y)) ≤ n n

1 − s(xi, yi)= (1 − s(xi, yi)) = d(x, y). i=1 i=1 ^ _ The proof is obvious for f(x)= f(y). 2

This proposition generalizes the result of L. Bˇehounek from [5], where the exten- sionality has been studied from the point of view of fuzzy set theory.

4.3.4 Approximation Theorem

The following theorem shows that class of fuzzy relations characterized by modified

Lipschitz property can be approximated by the normal forms with an arbitrary precision. Theorem 4.3.8

Let g(x) be a continuous additive generator of a t-norm ∗, f(x) be an n-ary fuzzy

n −1 −1 −1 −1 relation on [a, b] . Moreover, let g(f(T ◦ g (x1),...,T ◦ g (xn))) has bounded partial derivations w.r.t. each variable on (0,g(0))n. Then, for arbitrary ε there exist reflexive binary fuzzy relations ri(x, y), i = 1,...,n, and Nk = {c1,...,ck}, ci ∈ [0, 1], k ∈ N such that

|g(fDNF,k(x)) − g(f(x))| ≤ ε, (4.3.21)

|g(fCNF,k(x)) − g(f(x))| ≤ ε, (4.3.22)

125 n for arbitrary x ∈ [a, b] , where fDNF,k and fDNF,k are given by (4.1.1) and (4.1.2), respectively.

proof: Let us fix some ε > 0 and consider reflexive binary relations r1,...,rn

−1 in the form (4.3.6) then by Theorem 4.3.4 we know thatr ˜i(x, y) = g(ri(T ◦ g−1(x), T −1 ◦ g−1(y))) are Lipschitz continuous with the constant equal to boundary

∂(g◦f◦T −1◦g−1) −1 of | |, which we will denote by Mi for all i = 1,...,n. Since g(f(T ◦ ∂xi −1 −1 −1 g (x1),...,T ◦ g (xn))) has bounded partial derivations then it is Lipschitz continuous, i.e. n −1 −1 −1 −1 |g ◦ f ◦ T ◦ g (x) − g ◦ f ◦ T ◦ g (y)| ≤ M |xi − yi|, i=1 X for all xi, yi ∈ (0,g(0)), or equivalently n

|g(f(x1,...,xn)) − g(f(y1,...,yn))| ≤ M |g(T (xi)) − g(T (yi))|, i=1 X n for all xi, yi ∈ (a, b) and hence i=1 Mi ≤ M.

ε W Let δ = 2·n·M and Nk = {c1,...,ck} satisfies the following properties

ci ≤ ci+1, ci+1 − ci = δ and ∀x ∈ (a, b) ∃ i : |g(T (x)) − g(T (ci))| ≤ δ.

n Since f is left and right extensional w.r.t. r1,...,rn at each c ∈ V (Nk), thus by conditional equivalence (4.1.7) and (4.1.8), we have

|g(fDNF,k(x)) − g(f(x))| ≤ g(P (x)),

|g(fCNF,k(x)) − g(f(x))| ≤ g(P (x)), where

k n

g(P (x)) = g(rj(xj,cij )) + g(rj(cij , xj)) i ...,i =1 j=1 ! 1 _n X k n

≤ 2M |g(T (xj)) − g(T (cij ))| ≤ 2nMδ = ε. i ...,i =1 j=1 ! 1 _n X

126 2

The following corollary says that each function associated to a formula ofLukasiewicz as well as Product logic can be approximated with an arbitrary precision. Corollary 4.3.9

Let ∗ be a continuous Archimedean t-norm and fA be a function representing a formula A(x) due to Definition 4.2.1. Then for arbitrary ε> 0 there exists Nk such that fDNF,k and fCNF,k ε-approximate fA.

proof: Based on Lemma 2.2.10, an arbitrary function fA representing a formula

m A is extensional w.r.t. (x ↔∗ y) for some m ∈ N, m< ∞. Thus, by Theorem 4.3.1, g(fA) (g is an additive generator of ∗) is Lipschitz continuous w.r.t. the standard metric ds with the constant m. Finally, the ε-approximation of fDNF,k and fCNF,k follows from Theorem 4.3.8. 2

Note that this result may be proved directly using Proposition 4.1.3 and on the basis of Lemma 2.2.10, see the proof of Theorem 4.3.8.

127 Chapter 5

Genetic Algorithm + Logical Approximation = Optimized Rule-Base

This chapter can be characterized as an application part, where the previous theoret- ical results contribute to the theory of fuzzy control and make possible to introduce new algorithms for automated learning of the rule-bases.

Rule-bases are supposed to be in disjunctive or conjunctive form (corresponding to fuzzy relations in normal form). As we have seen, normal forms depend on the following parameters: the number of nodes, the distribution of nodes over the universe, and the shape of relations R1,...,Rn describing neighborhoods of these nodes. Assuming relations R1,...,Rn in the form (4.3.6) reduces the number of parameters to specify. The first two parameters should be chosen with the aim to fulfill some criterion. We wish to minimize an error of approximation which can be estimated on the basis of condition P (x) of the conditional equivalence (4.1.7) and

(4.1.8).

This naturally leads to an optimization search process. In this field the Genetic

Algorithms (GA) are the best known and most widely used global search technique

128 capable of finding near optimal solutions in complex search spaces, see [30]. Integra- tion of GA and Fuzzy Systems (FS) leads to a concept of designing fuzzy systems that receives the general name of Genetic Fuzzy Systems (GFS). The most prominent type of GFSs are genetic fuzzy rule-based systems (see [14]) whose genetic processes modify a given rule-base to achieve the desired result. In this work, only rule-based

GFSs are considered, thus whenever we speak about GFS we mean rule-based GFS.

Note that GAs can be used in logical approximation where the performance measure is based on the pseudo-metric dual to biresiduation of a t-norm. But this is not the only possibility. In general, we can apply an arbitrary criterion which would not have such a nice interpretation from the logical point of view. In fact,

GAs exploit a priori knowledge about the given FS to tune the positions of nodes.

Finally, the use of a relevant inference introduced in Chapter 3 leads to the FS

(GFS) which behaves in accordance with the results of Section 3.2.

The structure of this chapter is the following: In Section 5.1, the basic principles of GAs are introduced. Then the algorithm for logical approximation is proposed.

Finally, the automated learning algorithm creating a rule-base (or equivalently GFS) from the database of samples is elaborated.

5.1 Introduction to Genetic Algorithms

A genetic algorithm (GA) is an algorithm able to find approximate solutions of complicated problems applying the principles of evolutionary biology to computer science. It uses techniques inspired by natural evolution such as natural selection, recombination, and mutation. The class of GAs is a particular subclass of evolu- tionary algorithms which are perceived as the generalization of GAs.

Generally speaking, genetic algorithms are probabilistic optimization methods

129 which are based on the principles of evolution. They are typically implemented as a computer simulation in which a population of abstract representations (chro- mosomes) of candidate solutions (individuals) to an optimization problem evolves toward better solutions. Usually, solutions are represented as a binary string of the fixed length. In this case, we speak about Standard genetic algorithm. Of course, different encodings are also possible. The process of evolution starts from a population of completely random individuals and develops in generations. Each generation consists of multiple individuals, which are stochastically selected from the current population, modified by recombination and possibly mutation to form a new population being current in the next iteration of the algorithm, i.e. the next generation.

John Holland is considered as the pioneer of much of today’s work in genetic algorithms, which has moved on from a purely theoretical ideas of J.D. Bagley in

[4] to methods used to solve difficult practical problems nowadays. However, the beginnings of genetic algorithms can be detected already in the early 1950’s, when several biologists simulated biological systems using computers. But the roots of the genetic algorithms date back to 1920’s to the classical work [15] of S. Cetverikovˇ showing the complementary character of Mendel’s genetics and Darwin’s theory of natural selection.

5.1.1 Basic principles

As stated above, genetic algorithms can be viewed as optimization methods. Let us restrict ourselves to this approach at the moment and introduce an optimization problem to be solved:

130 Let X ⊆ R be a non-empty set and f : Xn → R be an arbitrary real-

valued function.

Find anx ¯ ∈ X such that f(¯x) = minx∈X f(x), i.e. the minimum of f.

Since it is very difficult to find an exact solution of such a problem in practice, we are interested in solutions for which the values of f (objective function) are “as small as possible”.

This optimization problem is represented by a list of parameters which can be used to drive an evaluation procedure, called chromosomes or genomes associated with certain competing individuals from the search space X. Chromosomes are typically represented as simple strings of data and instructions.

Initially several such parameter lists or chromosomes are generated. This may happen totally at random, or the user may seed the genes of individuals with “hints” to form an initial collection of possible solutions called the initial generation.

The next step consists in generating a second collection of genetic pool for new in- dividuals, which is done using any or all of the genetic operators: selection, crossover

(or recombination), and mutation.

Selection: Usually, a pair of organisms are selected for breeding of new individu-

als. This operation is biased towards elements of the initial generation which

have better fitness (objective function value). There are several well known

and popular selection methods e.g. roulette wheel selection and tournament

selection.

Crossover: The crossover (or recombination) operation is performed upon the se-

lected chromosomes. It is expected that crossover of two good individuals

should produce also a good individual in the sense of objective function. Most

131 genetic algorithms exert a crossover operation on a basis of a single

(pC ), typically between 0.6 and 1.0, which encodes the probability that two selected individuals will actually breed. Crossover results in two new child

chromosomes, which are added to the next generation. The chromosomes of

the parents are mixed in some way during crossover, typically by simply swap-

ping a portion of the underlying data structure. This process is applied on the

different parent individuals until an appropriate number of candidate solutions

in the next generation is created.

Mutation: The final step is to mutate the newly created offspring. Analogously

to the case of crossover operation also mutation happen only with very small

probability of mutation (pM ). A random number between 0 and 1 is generated

and if it falls within the pM range then the new child individual’s chromosome is randomly mutated in some way. Usually, mutation randomly altering bits

in the chromosome data structure.

This process results in a new generation represented by a collection of chromo- somes that is different from the initial generation. Generally, the average of fitness values will increase by this procedure for the new generation, since only the best organisms from the previous generation are selected for breeding. This fact has been proved formally in [31] by J. Holland and it is known under the name Schema

Theorem.

The entire process continues according to the following algorithm.

Pseudo-code algorithm: begin Choose initial population; Evaluate each individual’s fitness;

132 Repeat Select best-ranking individuals to reproduce; Apply crossover operator; Apply mutation operator; Evaluate each individual’s fitness; Until terminating condition (see below); end

Terminating conditions are fulfilled in most cases if the fixed number of gener- ations is reached, allocated computation time is used up, or an individual is found that satisfies the minimum criteria. It may include also a condition on the highest ranking individual’s fitness, which is satisfied when reaching or having reached a plateau such that successive iterations are not producing better results anymore.

5.1.2 Observations

The subject of this thesis does not include an investigation of GAs and their compar- ison with classical optimizing methods. But there are several significant observations about the generation of solutions created via a GA summarizing their most obvious pros and cons, and moreover, they help understand the choice of genetic algorithms for application to the problem of logical approximation. Most of the following ob- servations are taken from [10, 42] and [3]:

• Even if the fitness function is handled properly, GA may have a tendency to

converge towards local optima rather than the global optimum of the problem.

Analysis and partial solution to the problem of the premature convergence has

been given in [38].

• Nevertheless, compared with traditional continuous optimization methods,

such as Newton or gradient descent methods, we can state that GAs feature

133 the robustness. It is due to the fact that GAs always operate on a whole pop-

ulation. Consequently, it improves the chance of reaching the global optimum

and reduces the risk of trapping in local optima.

• GAs can locate good solutions, even for difficult search spaces.

• Normal GAs do not use any auxiliary information about the objective func-

tion value such as derivatives. Therefore, they can be applied to any kind of

continuous or discrete optimization problem. The only thing to be done is to

specify a meaningful decoding function.

• GAs cannot effectively solve problems in which there is no information about

the fitness other than given by two values possibly interpreted as good or bad

individual. Such problems are like finding a needle in a haystack and the

effectiveness of GA is the same as in the case of the random-search algorithm.

• It is not a trivial task to tune the parameters such as pM and pC to find a reasonable setting for the given problem class. The theoretical upper and lower

bounds for these parameters, that can help guide selection, can be found in

the literature e.g. [42, 3].

Generally, genetic algorithms are known to produce good results for a wide class of problems and their greatest advantage lies in extreme robustness. The major disadvantage is that they are relatively slow, being very computationally exhaustive compared with other methods.

5.1.3 Simple genetic algorithm

Below, we are going to introduce a very common GA belonging to the class of the

Standard genetic algorithms that we will call Simple genetic algorithm operating

134 over the language S = {0, 1}. It means that a particular generation consists of a list of m strings

Ii =(si1,...,sin), sij ∈ S, for each i = 1,...,m, j = 1,...,n, where m is the population size and n is obviously the length of the strings.

Simple genetic algorithm (SGA): Inputs: pC , pM begin t = 0;

Generate initial population P (t)=(I1,t,..., Im,t) randomly; Evaluate P (t); While (not finished) do begin t = t + 1; Select P (t) from P (t − 1) proportionally to its fitness f; for i =1 to m − 1 step 2 do

i f Random[0, 1] ≤ pC then

Reproduce Ii,t with Ii+1,t using one-point crossover; for i =1 to m do for j =1 to n do

i f Random[0, 1] ≤ pM then Invert sij; end end

The proportional selection can be viewed as a random experiment with the proba- bility that Ij is selected, i.e. f(Ij) Pj = m . i=1 f(Ii) There are a lot of different genetic operatorsP and variants of selection which give numerous possibilities how to determine genetic algorithm, see [49] or [42]. But for the problems formulated below, it will be sufficient to use the Simple genetic algorithm.

135 5.2 Genetic algorithms in logical approximation

The problem to solve in this subsection can be formulated as follows: let f(x) be a fuzzy relation on M n ⊆ Rn, then find optimized normal forms approximating f with error ε.

The optimization task consists in the proper distribution of the nodes used in the construction of normal forms introduced in Chapter 4 (the distribution is different for each of them). On the basis of results from that chapter, we can formulate objective functions

eDNF = max g(fDNF,k(x) ↔∗ f(x)), (5.2.1) x∈M n

eCNF = max g(fCNF,k(x) ↔∗ f(x)), (5.2.2) x∈M n where g(x) is an additive generator of the t-norm ∗ and fDNF,k, fCNF,k are given by (4.1.1), (4.1.2), respectively. We have chosen the genetic algorithms as an optimiza- tion tool for this particular problem because of their advantageous behavior in case of complex problems. At this stage of investigation the simple genetic algorithm

(SGA) is considered and classical binary coding is used (see [3, 42]).

Throughout the whole section we are working with extensional fuzzy relations.

Let us recall that this property allows us to express a closeness between objects x and y in the form of fuzzy relations Ri(x, y) with respect to the closeness of their functional values. The characterization theorem for fuzzy relations leads us to the concretization of their shapes. More precisely, fuzzy relations Ri(x, y) are given by (4.3.6) and the related parameters are determined on the basis of Theorem 4.3.4.

Now, we are able to describe formally an algorithm searching for a normal form approximating the given fuzzy relation f(x) with desired accuracy ε and optimized distribution of the nodes over the universe M.

136 Logical Approximation Algorithm (LAA):

Inputs: f, ε, MaxNumber begin NumberOfNodes = 0; BestNodes = [ ];

eD(C)NF = 1 ;

while (eD(C)NF >ε) and (NumberOfNodes < MaxNumber) do begin NumberOfNodes = NumberOfNodes + 1;

[BestNodes , eD(C)NF ] = SGA(NumberOfNodes); end end

Output: BestNodes

Remark 5.2.1 The standard genetic algorithm called in the algorithm above has the following attributes:

• The initial population is generated randomly.

• An array of nodes is coded as multidimensional point (see [42]) representing

one individual.

• Evaluation function (fitness) is identical with 1 − eDNF (1 − eCNF) in the case of theLukasiewicz t-norm with the additive generator g(x) = 1 − x,

• the best representative of a population is the one with the maximal evaluation

value.

• The number of disjuncts or conjuncts of the normal form is determined on the

n basis of {c1,...,ck}, ci ∈ M. It means that we construct normal forms in k n-dimensional nodes.

137 • Stopping condition in SGA can be analyzed from p previous generations (in-

cluding the actual one) on the basis of the following characteristic

j−1 i i+1 △(p)= |eD(C)NF − eD(C)NF|, (5.2.3) i=j−p X i where j is the index of actual generation and eD(C)NF is the error of approxi- mation of the best individuum in the i’th population. Then, SGA runs either

until △(p)

reaches its maximum value.

Example 5.2.1 For simplicity, let us consider n = 1, f(x)=(1+ e−x2 cos(36x))/2 on [0, 1] and theLukasiewicz t-norm ⊗ with the generator g(x) = 1−x. Applying the genetic algorithm LAA in the form described above with ε = 0.1 and MaxNumber =

6, we have obtained k = 6 with

c = {0.1795, 0.5039, 0.689, 0.3307, 0.83, 0.0264} and

c′ = {0.4374, 0.2605, 0.957, 0.7866, 0.0873, 0.6054} for which we have

12 14 11 11  13 7   12 13  10 7 ′ ′ 4 6 [kili]=   , [k l ]=   ,   i i    15 8   7 8       10 2   13 13       3 18   8 11          ˜ ˜ ′ ˜ ′ where the coefficients are computed as ki = k(ci, x), li = l(ci, x), ki = k(x, ci), li =

˜l(x, ci)

f(x) − f(y) f(x) − f(y) k(x, y)= , l(x, y)= , x − y y − x x>y ! x

138 Plot of DNF and original function 1

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Error of approximation 0.4 Best = 0.31423 0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of nodes = 6, time = 13.156 1

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) fDNF,6(x) with error 0.31

Plot of CNF and original function 1

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Error of approximation 0.2 Best = 0.14064 0.15

0.1

0.05

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of nodes = 6, time = 13.5 1

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(b) fCNF,6(x) with error 0.18

Figure 5.1: Result of the approximation of FL-function f(x)=(1+e−x2 cos(36x))/2 by fDNF,k(x) using genetic algorithm LAA.

139 (a) fDNF,2(x,y) (b) The error of approximation

Figure 5.2: Application of LAA to f(x, y)= xy with ε = 0.1 and MaxNumber = 3. and x˜ is given according to (4.3.7). The approximation formulas are

6 ki li fDNF,6(x)= (ci →⊗ x) ⊗ (x →⊗ ci) ⊗ f(ci), i=1 _6 ′ ′ ′ li ′ ki ′ fCNF,6(x)= (ci →⊗ x) ⊗ (x →⊗ ci) →⊗ f(ci), i=1 _ depicted on Figure 5.1. The final error of approximation is greater than 0.1, which is more than the required accuracy. It follows from the fact that SGA terminates when reaching the maximal number of nodes (set to 6) independently from the required accuracy.

Observe that the optimal solution tends to locate nodes near local maxima for the case of fDNF,k contrary to the case of fCNF,k where nodes are near local minima.

Example 5.2.2 Let n = 2, f(x, y) = xy on [0, 1] and consider theLukasiewicz t-norm ⊗. Using LAA with ε = 0.1 and MaxNumber = 4 we have obtained the following c = {0.4, 0.93} determining the approximating formula

2

fDNF,2(x, y)= (ci →⊗ x) ⊗ (cj →⊗ y) ⊗ f(ci,cj), i,j=1 _ depicted on Figure 5.2.

140 This result has been received in 40 seconds within 22 fired generations (for the

final number of nodes) in all 64 generations. The computational time increases exponentially.

The greatest advantage of this approximation method together with the above given algorithm lies in automatization of building an approximation formula for the given fuzzy relation without the necessity to code into the genetic information either fuzzy sets covering the domain or fuzzy relations comprising of structural information. It follows from the fact that we are working with a pre-set description

n of neighborhoods for each node ci ∈ M which eliminates time-consuming process of tricky coding and decoding of fuzzy sets describing the domain of approximated fuzzy relation f.

The characterization theorem together with estimation of the error of approxima- tion (conditional equivalence in [20]) leads to the conclusion that a given extensional fuzzy relation can be approximated by normal forms with arbitrary accuracy con- trary to the case of language containing a finite number of linguistic expressions.

Moreover, the generalized notion of extensionality brings the simplification in the sense of number of nodes used in normal forms. On the other hand an approxima- tion formula has to be created for each individuum in the population to estimate its fitting value, which increases computational complexity especially for the case of higher number of dimensions.

5.3 Learning fuzzy IF-THEN rules

To find a suitable description in the form of fuzzy IF-THEN rules for the physical system known only partially (by a set of crisp data stored in database) will be the main task of this subsection. For the given data, which are not necessarily precise

141 or are fouled by errors of measurement, we will construct an approximation formula in the sense of some criterion.

Our data comprise of a collection of pairs (xi, yi), i = 1,...,m. These pairs can be viewed as an available information about some function f such that f(xi)= yi.

An approximating formula is supposed to have a unified form, i.e. disjunctive fDNF or conjunctive fCNF form, such that the equalities fD(C)NF(xj) = yj hold for some j ∈ J = {1,...,m}. Whenever fD(C)NF(xj) = yj for all j ∈ J, we speak about interpolation.

Moreover, we assume our fuzzy system endowed with a scaling function on the

n+1 input as well as the output. Thus, it is sufficient to work with data(xi, yi) ∈ [0, 1] and in this connection to speak about [0, 1]-valued function f on [0, 1] (FL-function for short). Let us consider data (xi, yi), i ∈ J about some FL-function f(x) fulfilling the conditions from Lemma 4.3.3 and recall the shape of reflexive binary fuzzy relations R1,...,Rn, which are given by formula

ki li Ri(x, y)=(x → y) ∗ (y → x) , (5.3.1)

with special coefficients ki, li. Let us fix some c = xp, p ∈ J, then (4.3.6) turns to

k˜i ˜li Ri(x, ci)=(x → ci) ∗ (ci → x) , (5.3.2) and

′ ′ k˜ i l˜i Ri(ci, x)=(ci → x) ∗ (x → ci) , (5.3.3) for all i ∈ I = {1,...,n} with

g(f(x1,...,xn)) − g(f(c1,...,cn)) ki = , g(xi) − g(ci) xj,j∈J,j=6 i x_i>ci

g(f(x1,...,xn)) − g(f(c1,...,cn)) li = , g(ci) − g(xi) xj,j∈J,j=6 i x_i

142 and

′ g(f(x1,...,xn)) − g(f(c1,...,cn)) ki = , g(xi) − g(ci) xj,j∈J,j=6 i x_i

′ g(f(x1,...,xn)) − g(f(c1,...,cn)) li = , g(ci) − g(xi) xj,j∈J,j=6 i x_i>ci wherex ˜ is according to (4.3.7). Remark that the coefficients can be computed more efficiently but the procedure is rather long and tricky to express formally, it can be found in [18].

The SGA uis used to determine a set of nodes C = {c1,...,ck}, where k, the number of fuzzy IF-THEN rules, is fixed. Analogously to the continuous case, the objective function is calculated as

eDNF = max g(fDNF,k(xi) ↔∗ yi), (5.3.4) i∈J

eCNF = max g(fCNF,k(xi) ↔∗ yi). (5.3.5) i∈J

Note that the set C is different for fDNF,k as well as for fCNF,k. We will emphasize this fact using distinct labels, i.e. CDNF,k will denote the set of nodes for the particular fDNF,k and analogously with CCNF,k.

The above described process of a rule-base constructing together with appropri- ate inference method establishes Genetic fuzzy system depicted on Figure 5.3. The fuzzy IF-THEN rules are of the Takagi-Sugeno type in the form (3.2.1). The output of an inference is computed on the basis of the following formulas

BDNF(x)= (A(x) ∗ fDNF,k(x)), (5.3.6) n x∈_[0,1] BCNF(x)= (A(x) →∗ fCNF,k(x)), (5.3.7) n x∈^[0,1] where A(x)= A1(x1)∗. . .∗An(xn). The properties of such a GFS has been analyzed in Chapter 3. It has been proved that GFS based on a disjunctive (conjunctive)

143 Figure 5.3: Normal form based Genetic fuzzy system. normal form in the combination with the compositional rule of inference (5.3.6)

(alternatively by (5.3.7)) does not bring any additional noise into the system and thus it is the most fitting combination.

Let us summarize the whole procedure below. A particular population consists of m different individuals I1,t,..., Im,t, which represents m different sets of nodes

1 m CD(C)NF,k,..., CD(C)NF,k.

Algorithm of fuzzy IF-THEN rules learning (AFL):

Inputs: D = {(xi, yi)| i ∈ J}, k, pC , pM begin Scaling D; t = 0; Generate initial population P (t) randomly; While (not finished) do begin t = t + 1; (−1) Select P (t) from P (t − 1) proportionally to its fitness g (eD(C)NF);

Reproduce couples from P (t) using one-point crossover based on pC ;

Mutate sij on the basis pM ;

144 Plot of DNF and original function 0.8

0.6

0.4

0.2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Error of approximation 0.2 Best = 0.19554

0.1

0

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fired generations = 9 in all generations = 46, Number of nodes = 6, time = 0.891 1

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.4: Application of AFL to the input data (blue dots) assumingLukasiewicz t-norm.

end BestNodes is represented by individual with the highest fitness ; end

Output: BestNodes

Figure 5.4 illustrates the application of AFL to the input data of the form

(xi, yi), i = 1,..., 51. The algorithm has created an approximation formula in the disjunctive normal form on the basis of theLukasiewicz t-norm. Observe that the fuzzy sets R1,...,R6 assemble the set of membership functions of the final rule-base

(interpreted as fDNF,6) which has the following form

IF (x is Ri) THEN (y is Fi), for i = 1,..., 6, according to (3.2.1). The values F1,...,F6 are those of yi, i ∈ J assigned by AFL.

145 Plot of DNF and original function 1

0.8

0.6

0.4

0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Error of approximation 0.4 Best = 0.20644 0.3

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fired generations = 22 in all generations = 40, Number of nodes = 4, time = 0.859 1

0.5

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.5: Application of AFL to the input data (blue dots) assuming the t-norm with the additive generator g(x)=(1 − x)2.

When using a different t-norm with an additive generator, we receive nodes dis- tinct from those in Figure 5.4, since AFL optimizes on the basis of another objective function as can be seen at Figure 5.5. There, we have assumed g(x)=(1 − x)2.

146 Chapter 6

Conclusions

The general objective of this work is to show a global approach to fuzzy systems.

We start from the investigation of approximation properties of generalized normal forms in the first-order fuzzy logic BL, which are considered in two variants: infinite and finite. The infinite variant of normal forms has been introduced with the aim to show formally that the logical approximation by normal forms of the finite variant with an arbitrary precision is possible only in the case of extensional formulas, see

Theorem 3.1.5.

The theoretical results of Chapter 3 give us an overview of normal form based

Fuzzy Systems, and in this connection, make a contribution to fuzzy control. Es- pecially, the most common observations concerning the behavior of a fuzzy system under the special conditions have been proved formally. Moreover, the best possible integration of a normal form based rule-base and an inference engine into the fuzzy system has been proposed.

Since all the formulas of propositional BL are extensional (see Lemma 2.2.10), consequently normal forms are suitable for logical approximation in the framework of BL. The complexity reduction in the sense of the number of disjuncts (conjuncts) of the given normal form led us to the generalized notion of extensionality, see

147 Definition 2.2.5. This notion has been analyzed from the functional point of view in

Section 4.3, where it has been proved that extensionality is equivalent with Lipschitz continuity whenever related binary fuzzy relations are Lipschitz continuous as well.

The form of these relations given by (4.3.6) arose from the analysis of functional systems of the three particular logics extending BL, namelyL ukasiewicz, Product, and G¨odel.

Finally, the normal form based genetic fuzzy systems were introduced. There, the genetic algorithms are used to fine-tune the parameters of the normal forms to

find approximating formula which minimize the given criterion. Such genetic fuzzy systems are known as Knowledge-base genetic fuzzy systems. The algorithms for the “best” approximation of Lipschitz continuous functions as well as discrete data were shown. This generalized attempt to the problematic of fuzzy approximations provide the basis for reducing the size of fuzzy rule-bases compering to e.g. linguistic approximation. On the other hand, the interpretability in the sense of natural language worse; but it is certainly worth taking a closer look at some other criteria to be minimized or different fuzzy relations under consideration.

148 List of Author’s Publications

Publications in international journals

• Daˇnkov´a, M.: Extensionality and Continuity of Fuzzy Relations. Jour. of

Electrical Engineering, 2000, Vol. 51, Nr. 12, pp. 33–35.

• Daˇnkov´a, M.: Representation of Logic Formulas by Normal Forms. Kyber-

netika, 2002, Vol. 38, Nr. 6, pp. 717–728.

• Daˇnkov´a, M., Perfilieva, I.: Logical Approximation II. Soft Computing, 2003,

Vol. 7, Nr. 4, pp. 228–233.

• Daˇnkov´a, M.: Normal forms for fuzzy logic functions. Journal of Electrical

Engineering, 2003, Vol. 54, Nr. 12, pp. 80–84.

• Daˇnkov´a, M.: Generalized extensionality of fuzzy relations. Fuzzy Sets and

Systems, 2004, Vol. 148, pp. 291–304.

• Daˇnkov´a, M., Dvoˇr´ak, A.: Characterization and approximate representation of

extensional fuzzy relations. Journal of Electrical Engineering, 2004, Vol. 12/s,

Nr. 55, pp. 51–55.

• Daˇnkov´a, M., Stˇepniˇcka,ˇ M.: Fuzzy Transform as an Additive Normal Form.

Fuzzy Sets and Systems, submitted.

149 Bibliography

[1] Arz´en,˚ K.-E., Johansson, M., Babuˇska, R.: Fuzzy control versus conventional

control. In: Verbruggen, H., Zimmermann, H.-J., Babuˇska, R. (Eds.) Fuzzy Al-

gorithms for Control, Boston, Kluwer, 1999, pp. 111–125.

[2] Baaz, M.: Infinite-valued G¨odel logics with 0-1-projections and relativizations.

In: H´ajek, P. (Eds.) GODEL’96-Logical¨ Foundations of Mathematics, Computer

Science and Physics, Lecture Notes in Logic 6, Springer-Verlag, 1996, pp. 23–33.

[3] B¨ack, T.: Evolutionary Algorithms in Theory and Practice. Oxford University

Press, Oxford, 1996.

[4] Bagley, J. D.: The Behaviour of Adaptive Systems which Employ Genetic and

Correlative algorithms. PhD thesis, University of Michigan, Ann Arbor, 1967.

[5] Bˇehounek, L.: Teorie mnoˇzin v G¨odelovˇelogice. Msc. thesis, Charles University,

Prague, 1999.

[6] Bˇehounek, L., Cintula, P.: Fuzzy logics as the logics of chains. Fuzzy Sets and

Systems, to appear.

[7] Bodenhofer, U.: Applications of Fuzzy Orderings: An Overview. In: Atanassov,

K. T., Hryniewicz, O., Kacprzyk, J. (Eds.) Soft Computing - Foundations and

Theoretical Aspects, EXIT, Warsaw, 2004, to appear.

150 [8] Bodenhofer, U., De Cock, M., Kerre, E.E.: Openings and Closures of Fuzzy

Preorderings: Theoretical basics and Applications to Fuzzy Rule-based Systems.

International Journal of General Systems, 2003, Vol. 32 (4), pp. 343–360.

[9] Bodenhofer, U., Haslinger, J., Burger, M.: Data-Driven Construction of Sugeno

Controllers: Analytical Aspects and New Numerical Methods. Proc. Joint 9th

IFSA World Congress and 20th NAFIPS Int. Conf., Vancouver, 2001, pp. 239–

244.

[10] Bodenhofer, U.: Genetic Algorithms: Theory and Applications. Lecture Notes,

Second Edition - WS 2001/2002.

[11] Cignoli, R., D’Ottaviano, I. M. L., and Mundici, D.: Algebraic Foundations of

Many-valued Reasoning. Boston, Kluwer, 2000.

[12] Cignoli, R., Esteva, F., Godo, L., Torrens, A.: Basic fuzzy logic is the logic of

continuous t-norms and their residua. Soft Computing, 2000, Vol. 2, pp. 106–112.

[13] Cintula, P., Gerla, B.: Semi-normal forms and functional representation of

product fuzzy logic. Fuzzy Sets and Systems, 2004, Vol. 143, pp. 89–110.

[14] Cord´on, O., Gomide, F., Herrera, F., Hoffmann, F., Magdalena, L.: Ten years

of genetic fuzzy systems: current framework and new trends. Fuzzy Sets and

Systems, 2004, Vol. 141, pp. 5–31.

[15] Cetverikov,ˇ S. S.: On some aspects of the evolutionary process from the view-

poind of modern genetics (in Russian), Journal Exper. Biol., 1926, Vol. 2, No.

1, pp. 3–54.

[16] Daˇnkov´a, M.: Extensionality and Continuity of Fuzzy Relations. Jour. of Elec-

trical Engineering, 2000, Vol. 51, Nr. 12, pp. 33–35.

151 [17] Daˇnkov´a, M.: Representation of Logic Formulas by Normal Forms. Kybernetika,

2002, Vol. 38, Nr. 6, 717–728.

[18] Daˇnkov´a, M.: Normal forms for fuzzy logic functions. Journal of Electrical

Engineering, 2003, Vol. 54, Nr. 12, pp. 80–84.

[19] Daˇnkov´a, M.: Generalized extensionality of fuzzy relations. Fuzzy Sets and

Systems, 2004, Vol. 148, pp. 291–304.

[20] Daˇnkov´a, M., Perfilieva I.: Logical Approximation II. Soft Computing, 2003,

Vol. 7, Nr. 4, pp. 228–233.

[21] Daˇnkov´a, M.; Stˇepniˇcka,ˇ M.: Fuzzy Transform as an Additive Normal Form.

Fuzzy Sets and Systems, submitted.

[22] De Baets, B., Mesiar, R.: Pseudo-Metrics and T-Equivalences. The Journal of

Fuzzy Mathematics, 1997, Vol. 5, pp. 471–481.

[23] Dubois, D., Prade, H., Ughetto, L.: Fuzzy Logic, Control Engineering and

Artificial Intelligence. In: Verbruggen, H.B., Zimmermann, H.-J., Babuˇska, R.

(Eds.) Fuzzy Algorithms for control, Kluwer Academic Publishers, Dortrecht,

1999, pp. 17–58.

[24] Dubois, D., Prade, H.: Fuzzy sets in approximate reasoning, Part 1: Inference

with possibility distributions. Fuzzy Sets and Systems, 1999, Vol. 100, pp. 73–

132.

[25] Di Nola, A. and Lettieri, A.: On normal forms inLukasiewicz logic. Archive

for Mathematical Logic, 2004, Vol. 73, pp. 795–823.

[26] Formato, F., Gerla, G., Scarpati, L.: Fuzzy subgroups and similarities. Soft

Computing, 1999, Vol. 3, Nr. 1, pp. 1–6.

152 [27] Gerla, B.: Many-valued logics based on continuous t-norms and their functional

representation. Ph.D. dissertation, Univeristy of Milano, 2002.

[28] H´ajek, P.: of fuzzy logic. Kluwer, Dordrecht, 1998.

[29] H´ajek, P.: Fuzzy logic and arithmetical hierarchy III. Studia Logica, 2001, Vol.

68, pp. 129–142.

[30] Herrera, F., Magdalena, L.: Genetic fuzzy systems. In: Mesiar, R., Rieˇcan,

B. (Eds.) Fuzzy Structures - Current Trends, Tatra Mountains Mathematical

Publications, Bratislava, 1997, Vol. 13, pp. 93–122.

[31] Holland, J. H.: Adaptation in Natural and Artificial Systems. University of

Michigan Press, Ann Arbor, 1975.

[32] Klawonn, F.: Fuzzy sets and vague environments. Fuzzy Sets and Systems,1994,

Vol. 66, Iss. 2, pp. 207-221.

[33] Klawonn, F.: Fuzzy Points, Fuzzy Relations and Fuzzy Functions. In: Nov´ak,

V., Perfilieva, I. (Eds.) Discovering the World with Fuzzy Logic, Heidelberg,

Physica-Verlag, 2000, pp. 431–453.

[34] Klawonn, F., Kruse, R.: Equality relations as a basis for fuzzy control. Fuzzy

sets and systems, 1993, Vol. 54, Nr. 2, pp. 147–156.

[35] Klement, P. E., Mesiar, R., Pap, E.: Triangular Norms. Kluwer, Dordrecht,

2000.

[36] Kosko, B.: Fuzzy systems as universal approximators. Proceedings IEEE Inter-

national Conference on Fuzzy Systems, San Diego, 1992, pp. 1153–1162.

153 [37] Kreinovich, V., Nguyen, H.T., Sprecher, D.A.: Normal forms for fuzzy logic –

an application of Kolmogorov’s theorem. International Journal of Uncertainty,

Fuzziness and Knowledge-Based Systems, 1996, Vol. 4, pp. 331–349.

[38] Leung, Y., Gao, Y., Xu, Z.: Degree of Population Diversity - A Perspective on

Premature Convergence in Genetic Algorithms and its Markov Chain Analysis.

IEEE Transactions on Pattern Analysis and Machine Intelligence, IEEE Com-

puter Society, Washington, 2001, Vol. 23, Iss. 8, pp. 786–799.

[39] Menger, K.: Statistical metrics. Proceedings, National Academy of Science,

U.S.A., 1942, Vol. 28, pp. 535–537.

[40] Mesiar, R., Nov´ak, V.: Operations fitting triangular-norm-based biresiduation.

Fuzzy Sets and Systems, 1999, Vol. 104, pp. 77–84.

[41] Montagna, F.: Three complexity problems in quantified fuzzy logic. Studia

Logica, 2001, Vol. 68, 143-152.

[42] Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolution Programs.

Springer-Verlag, Berlin Heidelberg, 1996.

[43] Moser, B., Navara, M.: Fuzzy Controllers With Conditionally Firing Rules.

IEEE TRANSACTIONS ON FUZZY SYSTEMS, 2002, Vol. 10, Nr. 3.

[44] Nov´ak, V., Perfilieva, I., Moˇckoˇr, J.: Mathematical Principles of Fuzzy Logic.

Kluwer, Boston Dordrecht, 1999.

[45] Nov´ak, V.: On the logical basis of approximate reasoning. In: Nov´ak, V.,

Ram´ık, J., Mareˇs, M., Cern´y, M., and Nekola, J. (Eds.) Fuzzy Approach to

Reasoning and Decision Making, Kluwer, Dordrecht, 1992.

154 [46] Klawonn, F., Nov´ak, V.: The relation between inference and interpolation in

the framework of fuzzy systems. Fuzzy sets and systems, 1996, Vol. 81, pp. 331–

354.

[47] Klir, G.: Fuzzy Sets: an overview of fundamentals, applications and personal

views. Beijing Normal University Press, Beijing, 2000.

[48] Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy logic - Theory and Applications.

Prentice Hall, Upper Saddle River, 1995.

[49] Koza, J.: Genetic Programming: On the Programing of Computers by Means

of Natural Selection. The MIT Press, 1992.

[50] Pavelka, J.: On fuzzy logic I, II, III. Zeitsch. f. Math. Logik und Grundl. der

Math., 1979, Vol. 25, 45–52, 119–134, 447–464.

[51] Perfilieva, I.: Fuzzy logic normal forms for control law representation. In: Ver-

bruggen H., Zimmermann H.-J., Babuˇska R. (Eds.) Fuzzy Algorithms for Control,

Boston, Kluwer, 1999, pp.111–125.

[52] Perfilieva, I.: Normal Forms for Fuzzy Logic Functions and Their Approxima-

tion Ability. Fuzzy Sets and Systems, 2001, Vol. 124, pp. 371–384.

[53] Perfilieva, I.: Normal forms for fuzzy relations and their contirbution to univer-

sal approximation. In: Bouchon-Meunier B., Foulloy L.; Yager, Ronald (Eds.)

Inteligent Systems for Information Processing: From Representation to Applica-

tions, Amsterdam, Elsevier, 2003, pp. 381–392.

[54] Perfilieva, I.: Normal Forms in BL-algebra of functions and their contribution

to universal approximation of functions. Fuzzy Sets and Systems, 2004, Vol. 143,

pp. 111–127.

155 [55] Perfilieva, I.: Functions represented by BL-algebra formulas: characterization

and approximate representation. Fuzzy Sets and Systems, submitted.

[56] Perfilieva, I.: Logical Approximation. Soft Computing, 2001, Vol. 2, 73–78.

[57] Perfilieva, I.: Fuzzy transforms. In: Dubois D., Inuiguchi M., (Eds.) Fuzzy Sets

and Rough Sets, Springer-Verlag, to appear.

[58] Sudkamp, T.: Similarity, interpolation, and fuzzy rule construction. Fuzzy Sets

and Systems, 1993, Vol. 58, Iss. 1, 73–86.

[59] T¨urk¸sen, I.B.: Fuzzy normal forms. Fuzzy Sets and Systems, 1995, Vol. 69, pp.

319–346.

[60] Valverde, L.: On the structure of F-indistinguishability operators. Fuzzy Sets

and Systems, 1985, Vol. 17, pp. 313–328.

[61] Wang, L.; Mendel, J.: Fuzzy basis functions, universal approximations and

orthogonal least square learning. IEEE Transactions on Neural Neworks, 1992,

Vol. 3, pp. 807–813.

[62] Zadeh, L.A.: Fuzzy sets. Inf. Control, 1965, Vol. 8, pp. 338–353.

[63] Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and

decision processes. IEEE Transactions on Systems, Man and Cybernetics, 1973,

Vol. 3, pp. 28–44.

[64] Zadeh, L.A.: Fuzzy logic and approximate reasoning. Synthese, 1975, Vol. 30,

pp. 407–428.

[65] Zadeh, L.A.: Preface. In: Marks, R. J. (Ed.) Fuzzy Logic Technology and Ap-

plications, 1994, IEEE Technical Activities Board.

156 [66] Yager, R. R.: Fuzzy logics and artificial intelligence. Fuzzy Sets and Systems,

1997, Vol. 90, pp. 193–198.

[67] Yam, Y., Nguyen, H. T., Kreinovich, V.: Multi-Resolution Techniques in

the Rules-Based Intelligent Control Systems: A Universal Approximation Re-

sult. Proceedings of the 14th IEEE International Symposium on Intelligent

Control/Intelligent Systems and Semiotics ISIC/ISAS’99, Cambridge, Massa-

chusetts, pp. 213–218.

157 Index

additive generator, 24 logic, 14 approximate inference, 59 relation, 22 axiom, 34 extensional, 26

BL, 34 left extensional, 27

extensionality, 41 reflexive, 28

left extensionality, 42 right extensional, 27

reflexivity, 40 symmetric, 28

right extensionality, 42 transitive, 29

symmetry, 40 set, 21

transitivity, 40 fuzzy control system, 64 biresidual operation, 24 genetic algorithm, 118

BL-algebra, 32 Simple, 123 Standard, 119 characteristic function, 21 Genetic fuzzy system, 118 conditional equivalence, 43 IF-THEN rules, 15, 61 evaluation, 36 evolutionary algorithm, 118 language, 33 extensionality, 26 Lipschitz continuity, 31 logical approximation, 43 FL-function, 82 model, 37 formula, 34 fuzzy normal form

158 conjunctive, 39, 45, 77

disjunctive, 39, 45, 77 proof, 35 provability, 35 pseudoinverse, 24 residual operation, 23 rule of inference, 60, 65 rule-base, 61 structure, 35 t-norm

Lukasiewicz, 23

Archimedean, 24

Minimum, 23

nilpotent, 25

ordinal sum, 25

Product, 23

strict, 25 tautology, 36 term, 33 theory, 35 truth value, 36

159