Foundations of order sorted fuzzy set logic programming in predicate logic and conceptual graphs

First, there has been no complete fuzzy set logic programming system with thefundamentals of a theorern proyer and, rnoreover, the previous definition of rnodel-theoretic sernanti cs of

l Foundations of

Predicate Logic and Conceptual Graphs

,.,'

by

'-

Tru H Cao

B.Eng HCMUT (Gold Medal)

.M Eng AlT (Tim Kendall Memorial Prize)

1

.•

700 & ~ 3

Department of Computer Science and Electrical Engineering

The Univergity of Queensland

Australia

Mar ch 1999

Dedicated Lvthe memory ofmy mother

Tru H.Cao

Brisbane, Mareh 1999

1 would like to thank the Commonwealth Government of Australia for awarding me

an Overseas Postgraduate Research Scholarship, and the University of Queensland for aUniversity of Queensland Postgraduate Research Scholarship My PhD program herewould not have been feasible without thde financial supports

1 would also like to thank the Department of Computer Science and ElectricalEngineering for funding my attendance at several conferences 10 present my workingpapers and to have discussions with leading researchers in the field The questions and

comments received l'rom these presentation s and discussions have helped me toconsolidate the idcus and sueugthen the arguments for man y issues presented in thisthesis

AI this special milestone on my scholastic road,Jthink of teachers who have taught

me sinceJstarted to learn the alphabet Although this thesis is the direct outcome of myPhI) study, behind ir are background knowledge and methodology that 1 have been taughtfor years.1also think of people who have helped me in man y different ways.1am grateful

10 aIl of them

Nevertheless, neither this thesis nor anything else could 1 have done without mybeloved family There have been many limes when their faith in me has made thedifference between giving up and trying again, many times when their love has helped me

to find a strength 1 did not know 1 had 1 can never thank them enough

For dealing with the pervasive vagueness and imprecision of the real world as reflected innatural language, conce p tual graph programs have been extended to fuzzy conceptualgraph programs However, as many fundamental issues have rernained unaddressed andunresolved , no rigorous foundation has been established.Therefore, this thesis aims at asound and complete foundation for fuzzy conceptual graph progrumming, in particular,and for order-sorted fuzzy set logie prograrnming, in general There are three mainproblems in both the fuzzy logic area and the conceptual graph ureu to be solved in order

ta attain this objective

First, there has been no complete fuzzy set logic programming system with thefundamentals of a theorern proyer and, rnoreover, the previous definition of rnodel-theoretic sernanti cs of fuzzy set logic programs cannet deal with local inconsistency.Second, there has been no fuzzy type framework to study lattice and mismatching degreeproperties of object types under uucertainty and/or partial truth.Third, conceptual graphtheory has Iacked the formai integration of funetional relation types and the notion ofconjunctive types and, furthermore, the traditional CG unification is not adequate toobtuin a complete resolution style proof procedure with dose coupling of a type hierarchyand a program We solve these three problems correspond ingly as follows

Firstly, viewing fuzzy sets as lauice-based values, we extend classical annotatedlogic progruius to annotated fuzzy logic programs as a general framework for fuzzy setIogic progranuning that can deal with local inconsistency.Secondly,we propose a generalframework of fuzzy types, also viewed as Iattice-based values, and extend annotatedfuzzy logic programs ta order-sorted ones Thirdly, we formally integrute funetionalrelation types into couceptual gruph theory radieally as a part of its signature, introducethe nution of conjuuctive types into conceptual graph theory, and develop fuzzyconceptual graph prograrns based on order-sorted annotated fuzzy Iogic programs as anabstract framework

We believe that this thesis is the first sound and complete theoretical foundation fororder-sorted fuzzy set logic programming in the two complementary logic notations,numely, predicute logicand conceptual graphs The results obtained for fuzzy couceptualgraph programs are applicable to conceptual graph programs as special fuzzy ones andusetul for extending conceptual graphs with lattice-based annotations to enhance theirscruautics It adds to efforts of the fusfn of couccptual graphs and uncertuinty logicstowards a kuowledge representation and reasoning language thut upprouchcs humanexpression and reasoning,

2.1 Introduction 132.2 Fuzzy Sets and Possibility Distributions 15

2.4 Partial Truth- Valued Logic, Possibilistic Logic andFuzzy Set Logic 25

2.5 Fuzzy Set Logic Programming 312.6 Conclusion ~ : 36

3.1 Introduction " 39

3.2 AFLP Syntax 42

3.3 AFLP Model-Theoretic Semantics 4g3.4 AFLP Fixpoint Semantics 55

3.5 AFLP Reductants and Constraints : 59

3.6 AFLP Procedural Semantics 66

4.1 Introduction 714.2 l'ruth-Value Set Structures 764.3 Single Fuzzy Types H24.4 Conjunctive Fuzzy Types H7

9095102

105

108

Fun ctional Relation Types Conjunctive Concept Types and

5.15.2

5.3

5.6 Fuz zy Con ceptual Graph Pr oj ection and Normalization , 133

6.3 F C Gp Model-Theoret ic S eman tics " 150

6.5 G en er al I ssu es o f C G U n ific atio n and C G p R esoluti on 1606.6 F eG Unificati on and Fe G p Redu ctants 1676.7

o R

C o ncl usio n .

171176

Generalizr tion and Specialization

/

Chapter 7

7.1

7.27.37.4

Int rodu cti on .Gcneral izati on .... . .

C onclusion .

179179180184191

245

249

xiii

List of Figures

11718284346

3.2.2 The notion of ideals applied toa fuz zy set lattice

3.2.3 Fuzzy set v alues for cx cmpli f y i ng the noti on of rcstri ctcd AFLPs 47

3.3.1 Fu zzy set misrnatching and relative ncccs sit yd egrees 514.2.1 A gen eral s truc ture o f truth-value lauiccs 79

4.2.2 TRUE-ch aract erist ic and FAL5E-characl erisli c truth-valu es 794.2.3 The arnbiguityard er 81

5.2.1 An exa rnple CG 10 8

5.2.3 A CG p roj e cti on 1135.2.'1 N on -anti s ymm e tr y o f C G projecti on 113

5.2.5 CG n ormal Iorm f o r the co rn p le tc ness o f C G p roj e cti on 114

5.3.1 A CG w ith a fun ct ion al rel ati on Ils

5.3.2 A C G with a fun cti on al relati on type 1165.3.3 A fun eti onal rel ati on t yp e and o ne of its fun eti on al s u btypes 117

5.3.4 A f un ct ional relati on type and on e of its no n-fu n ctiona l supcrtypcs 1175.3.5 Latti ce-theoreti c and o rde r- theo re tic int erpretati ons of a type latti ce 1 185.3.6 A CG with conjun ct ive concept types a nd co nj unc tive relati on types 120

5.4.1 A CG projecti on w ith fun cti onal rel ati on typ e s conj unc tive

conce p t typ es a nd c o nj unc tive relati on t y p es 1235.4.2 Fun cti onal rel ati on t y pe a nd C G j o in 1245.4.3 Fu n ct ion al re la tio n s u bt ype a nd CG join

12 9

5.5.4 An FCG with fuzzy types 130

5.5.5' FCGs with fuzzy referents 133

5.6.1 An FCG projection 134

5.6.2 A functional relation under uncertainty : 135

5.6.3 An FCG join 135

5.6.4 A redundant infinite CG and one of its strict subgraphs 138 5.6.5 Two equivalent irredundant infinite CGs 139

6.2.1 An example CGP , , 146 6.2.2 CG representation of terms 148 6.2.3 An example FCGP 148 6.2.4 A definite FCGP 150

6.5.1 A CGP to exemplify the effect of a type lattice interpretation in resolution ' 162

6.5.2 Acap that realizes close coupling 164

6.6.1 An FCGP reductant 170

1.3.1 A possibilistic CG projection 190

'

xv

Chapter 1

Prologue

Naturallanguage is a principal and important means of human communication.ft is used

to express information as inputs to be processed by human brains then, very often, outputsare a'lsoexpressed in natural language How humans process information represented innaturallanguage is still a challenge to science, in general, and to Artificial Intelligence, inparticular However, it is clear that, for a computer with the conventional processingparadigm to process natural language, a formalism is required For reasoning, it isdesirable that such a formalism be a logical one

A logic for hundling naturul language should have not only a structure of formulasclose to that of natural language sentences, but also a capability to deal with the sernantics

of vague linguistic terms pervasive in naturaI language expressions.Currently, conceptualgraphs (Cas) (Sowa 1984) and fuzzy logic (Zad eh 1975) are two logical formalisms thatemphasize the target of natural language, each of which is focused on one of the twomentioned desired features of a logic for handling natural language While a smoothmapping between logic and naturallanguage has been regarded as the main motivation ofconceptual graphs (Sowa 199( 1997),a methodology for computing with words has beenregarded as the main contribution of fuzzy logic (Zadeh 1978b, 1996)

2 Chapter 1. Prologue

Coneeptual graphs, based 011 sernantic networks and Peirce's existential graphS,combine the visual advantage of graphieal languages and the expressive power of logie

On the one hand, the graphical notation of co neeptua grap is IS a va, '

representation of nested information w ose representation 111 mear 1 •

ta Iollow On the other hand, the formai order-sorted logie foundation of conceptualgraphs provides a rigarous basis not only for reasoning proccsscs performed directly onthem, but also for justifyingthe soundncss and the completcncss of a reasoningprocedure.Conceptual graphshave beenused for snlvingproblerns in severa!are as such as, butnot limited ta, natural language proccssing, knowlcdge acquisition and management,database design and interface,and information systems This language has been proposed

as a normative conccptual schema language by the ANS 1 standards committee onInformation Resource Dictionary Systems (ANSI Report No X3H4193-l96), and as aknowledge representation language in conceptual rnodels by the ANSl standardseommittee on Information Proeessing Systems (ANSI Report No.X3T2/95-019r2).Meanwhile, fuzzy lagie, based on fuzzy set theory and possibility theory, has beendeveloped for approxirn ate representation of, and reasoning with, imprecise information

on en encountered in the real worlel as refl cctcd in natural language In particular, fuzzyIogic deals with the partial truth as weil as the possibility and necessity measures ofuncertainty, contrasting with,and complementary ta,the probability measures

Vague linguistic terms, considered as information granules, can be denoted byfuzzy sets and then cornputcd through fuzzy set numcrical operations.Whilethere are stillmany unresolvcd theorctical issues rcgarding the uncertainty management problcm, ingeneral, and fuzzy logic in particular,fuzzy lagic has been succcssfully applied ta severalareas, sueh as expert systems , knowledge acquisition and fusion, decision making, andinformation retrieval, among ethers

However, although eonceptual graphs and fuzzy logie have the cornmon target ofnatural language, they have been studied and developed quite separarely sa far Only aIew researchers have reeognized the great advantage of their combination t~wards a

knowledge representati on language that ean approach the expressive power of natural

language At this juncture, conceptual graphs provide a syntactic structure for a srnooth

mapping la and from naturnl language whi\c fuzzy \, . ogie provid se. a scmantlC processor

for approx ima tc rcas onu, ith words having vague meanings

Section 1.1 scope, Motivation and'Objective 3

Sowa (1991) showed that naturallanguage expressions with quantifying words like

many, few or most could be smoothly represented in conceptual graphs but not inpredicate logic However, the classical logic semantics of conceptual graphs cannotinterpret and reason with such generalized quantifiers, whieh are intrinsically vague Itwas fuzay logic that provided a methodology to deal with the vagueness and imprecision

of sueh words (Zadeh 1983)

A medium for human expression is one of the five roles that Davis, Shrobe andSzolovits (1993) argued for a knowledge representation, as "a language in whieh we saythings about the world" While natural language is a language thut humans essentiallyuse in daily life not only for expressing things but also for thinking and reasoning, it istao informal to befully processed by computers Thus a logic with a smooth mapping toand from natural language like coneeptual g~aphs is really an advanee in knowledgerepresentation Il compromises the medium for human expression role and the medium for machine computation role, which is another of the five roles that the authors arguedfor a knowledge representation

Evaluating conceptual graphs for knowledge representation 111 the abovementioned five-role framework, Mann (1996) concluded that eonceptual graphs perhapsbest of ail fulfilled the medium for human expression role However, we argue thatconceptual graphs need to be extended with the incorporation of logics of uncertain andupproximate reusoning, such as fuzzy logic and probabilistic Iogic (Nilsson 1986), tofulfilthis role.This is not only because human expressions often contain imprecision anduncertainty, which 'j ust reftect reality, but also because human reasoning is oftenappraximate rather than precise, especially under uncertainty

-Morton (1987) early recognized the advantages of both eonceptual graphs and fuzzylogic and eombined them into fuzzy eonceptual graphs (FCGs), eoncluding thatconceptual graphs were for representing knowledge at the symbolic level of human brainsand fuzzy logic for enhancing CG semantics to deal with imprecise and uncertaininformation Fuzzy conceptual graphs ~erethen si~died further and applied in Manzano'(1991), Wuwongse and Manzano (1993), Maher (1991, 1993) and Ho (1994)

ln another aspect, while elassical order-sorted logie has been mueh studied forautomated reasoning ~ystems with taxonomie information, researeh on fuzzy logic in asimilur direction appears to be sporudic This shows another significance of the work on

4 Chapter 1 Prologue

fuzzy conceptual graphs as an order-sorted fuzzy ogre sys em

and/or partial truth about types of objects

As logic programming has made logic applicable to computer science as aprogramming language, Cao (1995) first introduced and studied FCG programs (FCGPs).Although this work addresscd, and proposed solutions to, sorne fundamental issues ofFCG programming, the system developed therein was not complete ft is worth notingthat, during that tirne, CG prograrnming was still under study as weil (e.g Ghosh andWuwongse 1995b).Coniinuing that work, this thesis is to establish a sound and completefoundation for FCG programming

There were three main problems in altaining this objective The first problem wasthat, although several fuzzy set logic progrumming systems had been developed, e.g

Baldwin, Martin and Pilsworth (1995),Goda and Vila (1995), and Virtanen (1996), whichwill be surveyed in Chapter 2, there was not acom plete system with the fundamentals of

a theorem proyer This needed, at first, a formai logical framework for studying fuzzy setlogic programming Moreovcr, the previous definitions of model-theoretic sernantics offuzzy set logic prograrns cou Id not deal with local inconsistency, so a proposedfrarnework should not have this drawback

Here, since the tennjilzzy[agie has becn used for different logic systems that haveoriginated from the theory of fuzzy sets but with very different characteristics, we use theterm [uzzy set [agi e for one that involves fuzzy set values in formulas This is to bedistinguished from partial truth-valued logic and possibilistic logic, where formulas areassociated with real numbcrs in the intcrval [0, II interprcted as truth degrees oruncertainty degrees, respcctively

The second problem was the lack of a fuzzy type framework in the previous works

on fuzzy conceptual graphs, to deal with uncertainty and/or partial truth about types ofobjects or relations betwecn objects That Ied to the following shortcornings Firstly,lattice and mismatching degree properties of concept types and relation types underuncertainty and/or partial truth were not studied completely Secondly, the notion ofconjunctive fuzzy types for joining FCGs was not supported Thirdly, and moreirnportantly, the view of FCG programming as annotated logie programming was notrecognized As a result, the FCG programming system first developed in Cao (1995) andWuwongse and Cao (199 0) was notcompleted ,

Section 1.1 Scope, Motivation and Objective 5

The third problem was of CG theory and CG programming themselves Firstly,CGtheory lacked a formaI and complete integration of functional relation types forrepresenting functional dependency and predicate logic terms Most CG systems did notmention or consider functional relations, whereas some others used them irnplicitlywithout an adequate formaI basis.There were attempts to formally incorporate the notion

of functions into CG theory, e.g Mineau (1994), Sowa (1994), and Cao and Creasy(1996a, 1997a), but it was not radical and complete In particular, Ghosh (1996) andSalvat (1997) did not consider functional relations in CG programs (CGPs), which couldcorrespond to terms in predicate logic programs

Secondly.iwhen a concept type lattice is order-theoretically interpreted or only apartially ordered set of concept types is assumed, it is not sound to join two concepts ofdifferent types into one concept with the maximal common subtype of the types of thetwo concepts as.the type of the resulting concept This needs the notion of conjunctiveconcept types With the same intuition of conjunctive concept types, conjunctive relationtypes are to join relations of the same arity and coinciding respective neighbour conceptsinto one relation

Thirdly, CG unification and CGP resolution are closely related to the way a typehierarchy is interpreted, which can be lattice-theoretic or order-theoretic, and to the waythe type hierarchy and the axiomatic part of a knowledge base are coupled, which can beloose or close However, these issues were not discussed in previous works on CGprograms, e.g Ghosh (1996) and Salvat (1997), or those on FCG programs, e.g Cao(1995) and Wuwongse and Cao (1996) We will show that, with close coupling, thetraditional CG unification is not adequate to obtain a complete CGP resolution proofprocedure

Therefore, to achieve the stated objective, firstly, we propose a framework for fuzzyset logic programrning.The pivotal idea is that we view fuzzy sets as lattice-based values

to study fuzzy set logic programming in the annotated logic programming framework.We

ex tend the classical annotated logic program formalism for fuzzy set annotations andfuzzy reasoning, resulting in annotated fuzzy logic programs (A~LPs) Secondly, wepropose a framework of fuzzy types with their latticeand mismatching degree propertiesstudied thoroughly,then ex tend annotated fuzzy logic programs to order-sorted ones withfuzzy types also viewed as lattice-based values

6 Chapler1. Prologue

In fact, anno tated logic p rogramm ing pr ovid es an abstract frarnew or k ra ther than a

co nc re te la nguage for stud ying la tt ice- based reasoning That is, even tho ugh in a

p ar ticular language lattice-based data ma y n ot be sy ntac tically so cIearly separa te d, they can still be ab stractly cons ide red as annota tio ns In p art icular, fu zzy typ es in FCGs can be reg arded as lattice-based annotation s, whenc e F CG pr ogramming c an be studied from the annotated logic programming point of view.

The study of order-sorted AFLPs th us has a two-Iold purpos e On the one hand, it provides a theoretical foundation for order-s orted fuz zy set logic programming in the predicate logic notati on On the other hand , it providcs an abstract framcwork to dcvelop FCG programs , for whi ch th cre are additional issu es of CG th e ory and CG programming

se ts in the case o f fuzzy l ogic T his m appin g is not one - ta-o ne b ut approxim ate o n ly, fir stly because a set o f lin gui sti c terms is t ypi c al ly cou nrab lc, wh er e as a set o f informati on granules w hose definiti on s are rootcd 1 num cri cal valu es is typ ic ally un countabl e The ether component in the buck - end lay er is to compute with information granules as denotations of liuguistic r crm s, with the rcsuli s m appcd back to linguisti c tenns for human reading.

This thesis is co nc e med wi th o nly the bac k-e nd co m po ne nt o f the described s yste m with only fuzzy logi c bei ng ap plie d That is , our atten tio n is Iocu sed on rea soning with fuz zy set values, w hic h l'a n ap prox ima te a c lass of im preci se a nd vagu e ling uis tio terrns Also, th is thesis s tudies o nly fuzzy se t logi c pr og rarns that co ns is t o f H orn -lil: e c lauses whi c h correspond to de fini te pred ica te log i c programs a nd definite CGPs D ue to the

ab o ve-rne nti o ned problerns- o f fuzz y se t l ogi c progra mmi ng , in gene ra I, a nd FCG progr amming in pa rti cul ar , w e ar e m ain l y c on ce rn ed wit h a the or e tic al fou nd at ion , ra th er

th an a n im pl e me ntati on In o ur o pin ion a th e or cti ca l f oundat io n shou ld be estab lis he d Iirst so that th c s ound n ess and co m pletc ne ss o f i m plcmc nt auo n » c a b justifie d

Section 1.2 Summary of Major Contributions

1.2 Summary of Major Contributions

7

The following are the major contributions of this thesis to the fuzzy logic area and theconceptual graph area We list them in bottom-up order, i.e., one supports its suceessors,which are almost in parallel to the contents of Chapters 3 to 6:

1 A general framework for studying and developing fuzzy set logic programmingsystems The AFLP framework is more advantageous than previous fuzzy setlogic programming ones in both the semantic and the syntactic aspects TheAFLP rnodel-theoretic semantics approach can deal with local inconsistencyand the AFLP syntax facilitates fuzzy set deduction performed and studiedindependently From symbolic manipulation

2 A general fuzzy type framework as a logical basis for the development of sorted fuzzy set logic programming systems, in order to deal with uncertaintyand/or partial truth about types of objects in knowledge-based systems withtaxonomie information The new formulation of fuzzy conceptual graphs withfuzzy types provides the unified structure and treatment for both conceptual

order-graphs and fuzzy conceptual graphs

3 An AFLP framework-based order-sorted fuzzy set logic programming language

in the predicated logic notation, with the syntax and the declarative semantics

,

formally defined An SLD-resolution style proof procedure is developed fororder-sorted AFLPs and proved to be sound and complete with respect to theirdeclarative semantics This is, to our knowleclge, not only the first sound andcomplete fuzzy set logic programming system, but also the first order-sortedone

4 Functional relation types are formally and completely integrated into CG theory

to represent functional dependen cy and predicate logic terms Conjunctiveconcept types and conjunctive relation types are introduced into CG theory forjoining CGs, when the interpretation of a type lattice is order-theoretic or when

concept types and relation types are just partially ordered

8 Chapter 1 Prologue

5 General Issues of FCG prog ramrrung as we as

and solved , regarding the u se of fun ctlOna re atton

a type lattice, the couphng of the taxonOlTIlC an

knowledge base , and the graph structure of COs

6 An extension of CG prograrns to fuzz y ones to deal with imprec ise and uncertain

kn owledge, with the syn tax a nd the declarative scmantics fonnally defined A graph-based SLD -reso lutio n sty le proof procedure for FCG progra rns is develo ped an d proved to be so un d a nd comp lete wit h respect to th e ir d ecl arati ve semantics T his is , to our know ledge, the first sound a nd complete graph-based order-sorted fuzzy set logic progrnrnming syst em.

After Chapter l , the rern aining chapters of th is th es is ar e to present in detail our ideas and soluti ons to allain the above -stated objectiv e 111 e or ganization of the se chapters a re as follo ws

Chapter 2 presents th e b ac kgro und of this th esi s on fuzzy set theory and fuzzy logic Sections 2 2 and 2 3 summari ze basic notions and prin ciples applied to this thesis of fuzzy set theory and possibi lity theory Section 2 4 c harac te r izes and dis tinguishes th ree ty p ic a l fuzzy lo g ic s, w hic h are partia l tr uth-va lued lo g ic , possibi lis tic lo g ic and Iu zz y set lo gi c Section 2.5 surveys previous fuzzy se t Io gic pr ogramming systems.

Chapter 3 presents the AFLP framework for fuzzy set logic program m ing Sections

3 2 and 3 3 formally define the s y nta x and the mod el -theoretic semantics of AFLPs

Section 3.4 studies the f ixpoint sernantics of Af-LPs a s a bridg e bctween their declarative and procedural sernanti cs. In Se cti on 3 5 , A f-LP r cductants , whi ch are used in stead of

clauses in AFLP resoluti on step s , and fuzzy set valu e constraints are defined and their

pr operties studied Then , in- S ection 3.6, an SLD -resolutian style proof procedure far AFLPs is de veloped an d pr ov ed t o be s o un d a nd ca m p iete with respect ta AFLP

d eclarative s ernanti cs This c hapter i s a r efin c mcnt of Cao (1997b, 1998a 1998b).

Section 1 3 Structure Overview 9

Chapter 4 presents a fuzzy type framework for order-sorted fuzzy set logicprogramming and FCG formulation Section 4.2 surveys differentdefinitions of a truth-value set in existing fuzzy logic systems, then proposes a general structure of fuzzy truth-values with different truth-characteristics and various degrees Sections 4.3 and 4.4formula te fuzzy types and conjunctive fuzzy types, then study their lattice properties.Section 4.5 defines fuzzy type mismatching degrees and studies their properties forreasoning with fuzzy types Then, in Section 4.6, annotated fuzzy logic programs areextended with fuzzy types to be order-sorted ones This chapter is an extension, with theintroduction of conjunctive fuzzy types and the definition of fuzzy type rnisrnatchingdegrees, of Cao, Creasy and Wuwongse (1997a), which was partially reported in Cao andCreasy (l997b)

Chapter 5 presents the extension of simple CGs with functional relation types,

conj unctive concept types and conjunctive relation types, and the new formulation offuzzy conceptual graphs with fuzzy types Section 5.2 presents the basic notions of CGtheory Sections 5.3and 5.4 discuss and extend simple CGs with functional relation types,conjunctive concept types and conjunctive relation types Section 5.5 formulates simpleFCGs with fuzzy types,then Section 5.6defines FCG projection and FCG normalization

accordingly.For the definitions of FCGP and CGP rnodel-theoretic semantics, a note oninfinite FCGs as weil as CGs is given.This chapter is based on Cao and Creasy (l997a,1997b)

Chapter 6 presents FCG programs as an extension of CG programs and as a based version of order-sorted fuzzy set logicprograms FCGP syntax and model-theoreticsemantics are formally defined in Sections 6.2 and 6.3, respectively.Section 6.4 studiesthe fixpoint semantics of FCG programs as a bridge between their declarative andprocedural sernantics Section 6.5 discusses general issues that are common to both FCGprogramming and CG programming Section 6 ~ defines FCG unification and FCGPreductants Then, in Section 6.7, an SLD-resolution style proof procedure using graphoperations for FCG programs is developed and proved to be sound and complete withrespect to their declarative semantics This chapter is a refinement of Cao and Creasy(l997b, 1998a, 1998b)

graph-Chapter? presents a generalization and specialization of order-sorted AFLPs andFCGPs Section 7.2generalizes order-sorted AFLPs and FCGPs with clauses weighted by

10 .Chapler 1 Prologue

certainty degrees in [0, 1J Section 7.3 specializes order- sorted AFLPs and FCGPs withonly special fuzzy truth-valu es representing truth, possibility or necessity degrees asfuzzy set values involved in programs.111ey are less expressive than general order-sortedAFLPs and FCGPs, but have simpler computation and require less data storage

Each chapter from Chapter 2 to Chapter 7 has an introdu cti on and a conclusion.Chapter 8 surnrnarizes the thesis and sugges ts future research The proofs of thepropositions and theorems in Chapters 2 to 7 are present ed in Appendixes A to F,respectively,except for those that are obvions from, or are direct conseq ue nce s of, otherpropositions and the or ern s. Appendi xG for r-norms and r-conorms ,and Appendix H , forrnultiple-valued logic implicati on s are basc d on Klir and Yuan (1995)

1.4 SymboI and Abbreviation Conventions

The following gener al sym bols are used thraugh out this thesis:

ç;; :the classi cal/Iuzzy subs e trelation

( l : the class ica l/ fuzz y set intersection operator

U : the class ical/ fuzzy set union opcrator

:s; :the rcal number les s-than-or-cqual-to relation

max :the real number maximum Iunction

min :the r eal number minimum fu ncti on

sup :the real numb er s up re m um Iun cti on

inf :the real number infirnurn function

[lib : the lani ce Jeast upper bound function

g lb :the lattieegrcat est lower boun d functi on

N :theset of ail non-negative intcger s

Z :thesetof ail integers

We especially use :s;\as the corn rnonsymbol far ail orde rs usee! in this thesis , under

the same umbrell a of i nform a tion o rde.rin vl't. whereby• 1\ A <- \ B (or e,qmva en 1 IIy,B >- \ Il)

means Bis morc inform ative Ill' morc s <'ciflc than A III partie J • A '

is a subtype of11 Il will he cle r ina Sl""'iflc contex t whicl 1 II'

1orecr nx cornmon syrnbo l

Section 1.4 Symbol and Abbreviation Convent ions 11

denotes Also, we will write A <1 B (or, equivalently, B >1A) to indicate that A ~IBand

A :F- B

The following abbreviations are used throughout this thesis:

if! :"if and only if'

wrt :"with respect to"

Also, for simplicity: we may use'sto denote a sequence of indexed expressions, e.g E;'s

denoting El> E 2•

is no clear-cut boundary between them and Ilot young orIlot old, Ilot shortor Ilot ta//, Ilot eheapor Ilot expensive,respectively (Honderich 1995) In other words, the membership

of an object in the extension of such a concept is not a matter of "to be or not to be", butrather a matter of degree Classical set theory, in which the membership grade of anelement in a set can only be either 0 or l, is thus inadequate to deal with vague concepts.This was the main motivation of Zadeh (1965) founding fuzzy set theory that generalizesclassical set theory by defining membership grades to be real numbers in the interval [0,1] Goguen (1967) then extended fuzzy sets to L-fuzzy sets where membership gradescould be values in a lattice other than [0, 1]

Fuzzy set theory then gave birth to fuzzy logic.In the literature, the termjUzzy logic

has been used for different logic systems that have originated from the theory of fuzzysets However, they may have so different characteristics that they need to bedistinguished to avoid confusion Actually, fuzzy logic has grown up with two maintrends One trend is a continuation of multiple-valued logic (Rescher 1969) to deal with

14 Chapter 2 Fuzzy Logics

d h f la has the meaning of a trUth partial truth , where a value ln [0, 1] associate wrt a ormu

degree We cali fuzzy logic in this trend part ial trlltlz -valued l ogic,e.g those of Pavelka (1979), Novâk (1987) and Hâjek (1995) The other trend is a development of fuzzy logic

as a logic to deal with possibility in contrast to probability , which is based on possibility theory also founded by Zadeh (1978a).

Key notions of possibility theory are ones of possibility degrees and possibility distributions Possibilistic logic, devcloped by Dubois and Prade (1988), also has formulas associated with values in [0, 1], but their meaning is possibility or neccssity degrees instead of trUth degrees Zadeh himself coined the name fuzry logic and developed fuzzy logic as a logic to deal with fuzzy propositions, i e., on es that involve vague linguistic terms represented by fuzzy sets (Zad eh 1975) His main concern was approximate reasoning with possibility distributions induced by fuzzy propositions (Zadeh 1979), rather than forrnal syntax or model-theoretic sernantics of the logic In Esteva, Garcia-Calvés and Godo (1994), a model-theorctic semantics for fuzzy propositions was proposed,

We cali a fuzzy logic whose formulas involve fuzzy sets.fuzzyset logic. Fuzzy logic programming systems, with respect to their underlying fuzzy logics, can also be roughly c1assified into two groups depending on wheth er they involve fuzzy sets in programs or not Systems that do not involve fuzzy sets usually have formulas weighted by real numbers in the interval [0, 1), e g Mukaidono, Shen and Ding (19R9), Dubois, Lang and Prade (1991 b), and Klawonn (1995) Systems that involve fuzzy sets include those of Umano (1987), Baldwin, Martin and Pilsworth (1995), Godo and Vila (1995), and Virtanen (1996) This thesis is concerned with fuzzy set logic programming, for which there has been no complete system with the fundamentals of a theorem proyer.

This chapter presente; a background of fuzzy set theory and fuzzy logic for the rernaining chapters of the thesis, Section 2.2presents the definition and basic operations

of fuzzy sets , then the notion of possibility distributions Section 2.1 presents th e basic principles for reasoning with possibility distributions, then discusses two different views

of fuzzy rules, narnely, the object-Ievel and the rneta-levcl ones , with their corresponding deduction rnechanisrns Secti'on 2.4 characterizes three main groups of Iuzzy logic, namely, partial truth-valu ed logic, possibilistic logic and fuzzy set logic Section 2.5 surveys previous works o n [UlZY set logi c programming and discusses their shortcornings

Finally, Section 2 6 pr esents concluding icmarks of the chapter.

Section 2.2 Fuzzy Sets and Possibility Distributions

2.2 Fuzzy Sets and Possibility Distributions

15

For a classical set, an element is to be or not to be in the set or, in other words, themembership grade of an element in the set is binary For a fuzzy set, the membershipgrade of an element in the set is expressed by a real number in the interval [0, 1] Thefollowing definition is adapted from Zadeh (1965), with the tenns normal fuzzy set and

subnormal fuzzy set from Klir and Yuan (1995)

Definition 2.2.1 Afuzzy set A on a domain U is defined by a membership function IlA

from U to [0,1] It is said to benormal ifsUP ueU{IlA(u)} = 1 or subnormalotherwise

As such, a c1assical subsetA of Ucan be considered as a special fuzzy set whosemembership function is defined by VuE U : IlA (u)=1 ifuE Aor IlA(u) = °otherwise.Anempty classical set is a fuzzy set whose membership function has only value 0, denoted

by 0, and a universal c1assical set is a fuzzy set whose membership function has onlyvalue l, denoted by Uitself In practice, for efficient computation, membership functions

of trapezoidal diagrams are commonly used, Figure 2.2.1 illustrates two fuzzy setsA and

B on a domain U being the interval [0, 100), where A is a normal fuzzy set and B is asubnormal one

o

\ \ - 8

Figure 2.2.1 Normal and subnormal fuzzy sels

As their motivation, fuzzy sets represent classes of objects where the boundary for

16 Chapter2 Fuzzy L oç ics

an object to be or not to be in a class is not clear-cut due to the vagueness of the conceptassociated with the class Such vague concepts are frequently encountered in the realworld, likeyoung oro/d, shortor tall, cheap or e xpensive. For example,if the interval [0,100] represents a range of human ages,then the fuzzy setAin Figure 2.2.1 can represent

a c1ass ofy oung persons Here, a person of°to 25 years of age is definitely in the class,

a person of 40 to 100 years of age is definitely not, and the membership grade in the c1ass

of a person of 25 to 40 years of age is linearly decreasing from 1 to O Just as theinterpretation of a vague concept like Y O III IR is context-dependent, the mernbershipfunction of a fuzzy set rcprcsenting it is nlso coutcxt-rlcpcndcnt rOT instance, in onecontext, the range of ages for a persan to be definitely youn g is 10, 25) but, in anothercontext, the range may be 10.30]

When classical sets are considered as special fuzzy sets whose mernbershipfunctions have values in {O, I} only, the subset relation and the basic operationsintersection, union and complement can be defined in terrns of operations on {O, 1).Thefuzzy subset relation and fuzzy set operations generalize those of classical sets withoperations on 10, 1) instead.The following definition is adapted from Zadeh (1965)

Definition2.2.2 LetA and B be two fuzzy sel s on a domain U.Then:

l A is said to be afu zzy subset ofB ,den otcd byAc B ,iffVil E U : ~A(u) ~ ~lJ(u).

2 Thefu zzy inters ection ofAand B is a fuzzy set denotcdby AriBand defined by:

Vil E U : llA nn(lI) =mill{IlA(Il) , J In (II )}

3 The fuzzy union of A and B is a fuzzy set denoted by A u Band defined by:

Vu E U: ~AvLJ(u) =max{llA(u) , J ILJ (Il)}.

4 Thefuz zy compl ement ofA is a fuzzy set denoted by Aand defined by:

Vu E U: ll (Il) = 1-IlA(u).

The definitions above of fuzzy intersection ,union and complement are the standardones ln general, other functions with similar characteristics can be used instead of the

min, max and minus ones (Klir and Yuan 1995) In this thesis,we apply only the standarddefinitions, for which it is straight forward from Definition 2.2.2 that the folIowingproposition holds

Section 2.2 Fuzzy Sets and Possibility Distributions 17

Proposition 2.2.1The set of aIl fuzzy sets on a domain U forms a complete lattice withthe fuzzy subset relation as the partial order For a set S of fuzzy sets, glb(S) is a fuzzy setdefined by VuE U : Ilgtb(S )(U)=infAd{ IlA(u)} and lub(S) is a fuzzy set defined by Vu E

U: Iltub(S)(U) =suPAeS{IlA(u)} ,The least and greatest elements are 0 and U,respectively

' " - - _ - _~" -'"

10060

for instance, then  represents the concept Ilot young Whereas the opposite concept to

young is old, which can be represented by a fuzzy set denoted by -,A and defined by

Vu E U : Il-.A(u) = IlA(l - u). Figure 2.2.2 illustrates the difference between fuzzycomplement and fuzzy opposition

Figure 2.2.2 Fuzzy c omplement and fuzzy oppos ition

Fuzzy sets then served as a basis for possibility theory (Zadeh 1978a), which is atheory to deal with possibility and necessity, in contrast to probability.A key notion ofpossibility theory is possibility distribution Given a variable x and a fuzzy set A on a

domain U,ifx takes a valueUE U then IlA(u) measures the membership grade ofx in theextension of the concept represented byA. In contrast, if the value ofxis iIl-defined byA

as a fuzzy restriction on the values thatx can take, then IlA(u) measures the possibility for

u E U to be the value ofx.

Thereby, ts fuzzy proposition "xis A", i.e.,one that involves fuzzy set values (Zadeh

18 Chapler 2. Fuzzy Logics

1975), induces a possibility distribution 1txassociated with x on V and ùefined by:

VuE V: 1ti u )=IlA(u)

where1t

x ( u )=0means thatx=u is impossible and1txCu)=1means thatx=uis completelypossible Then, we write 1t x := A

For ex ample,given theconceptyousg represented by the fuzzy setAin Figure2.2.2,

if John's age is 30, then the membership grade of John in the set of young persons is

IlA(30) = 1 - (30 - 25)/(40 - 25) = 2/3 In contrast, with the fuzzy proposition "John is

young" ,the possibility degree for John's age 10 be 30 is1t agc (John)(30)= ~lA(30) =2/3. Asnoted in Dubois and Prade (1980).although a possibility distribution funetion is the sa me

as the membership function of the fuzzy set inducing it, the two functions have differentunderlying notions as explained above

As for a fuzzy set, a possibility distribution 1t xof a variablex on a domain Vis said

to be normal ifsUP U E v( 1tiu)} = 1 or subnormal otherwise.That 1t x is subnormal irnplies

a partial inconsistency aboutx.because it méans that it is not completely possible forx totake any U E Vas its value which is assumed to be in V Also, in general x can be a tuple

ofIl variablesxI, x2 , •XII and V the Cartesian productVIx V 2X X VII where, for every

Figure 2.2 3 Spccificity of possibiliry distributions

Given two possibility distributions 11: x and 11: x '"of a variablexon a domain V, ifVu

E V: 11: xC II ) ~11: x"'(u) ,then 11: x is said 10be morespe cifiethan1t x"' .The intuition is that, every

Section 2.3 Reasoning with Possibility Distributions 19

UE Uis then more possible to be a value ofx with 1tx*than with 1tx'and thus 1tx*is lessspecifie or less informative than1txwith respect to the question "Which is the actual value

ofx?". For example, let1t1 :==Al'1t2 :== A2and 1t3:==A 3whereAl, A2and A 3are the fuzzysets in Figure 2.2.3, representing the concepts exactly 40, about 40 and between 30 and

50 years of age, respectively Then,due to Al ç A 2 C A 3,1t1 is more specifie than 1t2 and1t2 is more specifie than1t3' As such, given two fuzzy sets A and B whereA c B ,we alsosay A is more specifie than B, in terms of the possibility distributions that the y induce

2.3 Reasoning with Possibility Distributions

On the basis of possibility distributions, a pro cess of approximate reasoning (Zadeh1979) has three basic steps First, the premise fuzzy propositions are translated topossibility distributions Second, computation on those possibility distributions isperformed to obtain new possibility distributions of concemed variables Third, theobtained possibility distributions are translated back to fuzzy propositions as theconclusion

The translation of primitive fuzzy propositions of the form "x is A" is already

presented in Section 2.2 The translation of composite fuzzy propositions constructed bythe basic logical connectives conjunction, disjunction and implication were defined inZadeh (1979) as follows A conjunction " x is A 1\ Y is B", where A and B are fuzzy sets

respectively on domains U and V,is translated to the joint possibility distribution 1t(x ,y)defincd by:

v 'u E UVvE V : 1t(x,y)(u, v) ==mill{IlA(u), Iln(v)}.

The disjunction"x isA v yisB" is translated ro1t(x,y)defined by:

VUE UVVE V :1t(x,y)(u,v)==max{IlA(U),lln(v)}.

The implication"x isA -7Y is B" is translated to 1t(x,y )defined by:

Vu E UVvE V: 1t(x , y)(u, v) ==1(IlA(u) , IlB(v» ==minIl , 1-Il A(u) +IlB(v)}

where 1is the Lukasiewicz implication (see Appendix H)

An example of a fuzzy rule is the rule "If a tomate is red , then it is ripe " Assurveyed in Magrez and Smets (19H9) and Dubois and Prade (1991), differentimplications were proposed instead of the Lukasiewicz implication for the joint

20 " Chapter2 Fuzzy Logics

possibility distribution induced by a fuzzy rule prese~tedabove, so that the rule had some

h a fact"x

expected behaviours when being fired For instance, one behaviourISt at,glven

is A*"where A* c A, the conclusion "y is B" is inferred, to be consistent with c1assicalmodus ponens That is, for the rule above, if the fact "T orna to #78 is very red" is given,then the conclusion "Tomate #78 isripe" is inferred, provided that very redc red Here,

we use vague linguistic terms, like very r ed and red, also as labels of the fuzzy sets

defining them

Another behaviour is that, no conclusion "y is B *" where B *ç B can be inferrcd,

that is, a fuzzy rule does not produce any conclusion more informative than itsconsequent For example, if this behaviour is assurned,the mie above does not imply "Themore a tomato is red, the more it isripe" In Dubois and Prade (1991, 1996), the authorsclassified fuzzy rule models not by implications applied to them, but by characteristics ofqualification, namely, truth or possibility or certainty, on the consequent of a rule Eachqualification characteristic led to a different joint possibility distribution induced by afuzzy rule

For reasoning with possibility distributions, the basic principles are the entai/ment prin ciple,theproj ection prin cipleand theco nj unc tion principl e. The entailmentprinciplestates that, a possibility distribution 1txof a variablex entails any possibility distribution 1tx* that is less specifie than 1tx' The general statements of the projection and theconjunction principles were given in Zadeh (1979), with joint possibility distributions oftwo or more variables Here, we present only their main ideas with joint possibilitydistributions of two variables.The projection principle states that, given a joint possibilitydistribution 1t(x Y) ' where x and y are variables respectively on demains U and V , the

ma rginal possibility distributions 1t xofx and 1t y ofycan be derived as follows:

VuE U : 1tiu)=SIlP VE v{ 1t(x, y )( u , v)}

VvE V : 1tyCv) =SllPIlE u { 1t(x, y)(u , v)}

The conjunction principle states that,given joint possibility distributions TI: andrt

• (x,y ) (y,z),

where x,y and zare variables respectively on demains U, V,and W ,the joint possibilitydistribution 1t(x, y ,z)can be derived as follows:

Vu E U VI'E V VwE W : 1t( x,y,z )( u, v, w) = mill (1t(x, y)(u, v) , 1t(y , 1)(V, w»).

A special case of the conjunction principle is when thex and zarguments are empty,thar is, two possibility distributions 1t1 and 1t2 of y entails the possibility distribution 1t

Y

Section 2.3 Reasoning with Possibility Distributions 21

defined by:

\IvE V: 1tyCv):=min(1t1(v), 1t2(v)}

This is consistent with the principle of minimum specificity (Dubois and Prade 1987,1991) that1t ydefined above is selected as the least specifie solution subsuming ail otherpossible solutions, because the "actual" possibility distribution ofy must be more specifiethan both 1t1and1t2'

Consequently, by the entailment principle,if Ac Bthen"x isA"entails"x isB"and,

by the principle of minimum specificity, if"x is A" and"x is B" then"x is A riB", where

x is a variable and A and B are fuzzy sets on the same domain For example, let theconceptsyoung, not young, not old and middle-aged be represented by the fuzzy sets such

thatoldc not young, as in Figure 2.2.2,andmiddle-aged:=not youngrinot old Then, by

the entailment principle, "John is old" entails "John is not young" and, by the principle ofminimum specificity,if "John isnot young" and "John is not old" then "John is middle- aged".

Meanwhile, fuzzy modus ponens is performed by application of the conjunctionprinciple and then the projection principle as follows Suppose a cule"x is A ~ y is B"

and a fact "xis A"".wherexis a variable and A and A*are fuzzy sets on a domain U and

y is a variable and B is a fuzzy set on a domain V.The cule induees a joint possibilitydistribution 1t(x y)and the faet induces a possibility distribution1tx ' An application'oftheconjunction principle to1t(x, y)and1txderives1t(x.y)* defined by\luE U \IvE V: 1t(x.y)*(u, v) :=min (1t(x y)(u v), 1txCu)} Then, an application of the projection principle toy derives7ty* defined by \Iv E V: 7ty*(v) := SUPUE U{7t(x, y)*(u, v)}. Thus, the inferred fuzzyproposition is "y is B*" where 7t/ ::= B* This deduetion mechanism is called Zadeh's

cornpositional rule of inference or generalized modus ponens

The view of the meaning of a fuzzy cule"x isA ~ y is B"presented so far is tharthe cule implies a relation between the possibility ofx taking a value uE U and that ofy

taking a value v E Vand, thus, defines a joint possibility distribution of (x, y) on U x V.

We call this view an object-level one Another view, which we call ameta-levet one, isthat the cule implies a relation between a degree of sorne measure of"x is A" as a wholeand that of"y isB" as a whole

These two different views of fuzzy ru les leud to different deduction mechanisms.For the object-level view, deduction is perforrned through conjunction and projection of

22 C hapter 2 Fuzzy Logics

possibility distributions as shown above whereas , for the meta-level view , it is through propagation, qualification and modification of degrees of sorne measure The former involves more complex computation due to operations on joint possibility distributions on Cartesian products of fuzzy set demains, whereas the latter requires only operation s on possibility distributions on smgle fuzzy set demains (Magrez and Smets 1989)

For the mensure of unccrtainty of a fuzzy proposition with respect 10 anothcr Olle,

there are Iwo cornplerncnrary notions that ar c relati ve possibilit y degree and relative necessity d egree. In Magrez and Smcts (1989) ( cf Du bois and Prade 1991), the relative possibility degree of"x is A" given" is A"''', denoted by n(A 1A* ), was defined by:

(2 3 1 ) where U was the domain of A and A'" and ® was a l -n01711 (see Appendix G) This definition generalized thatof Zadeh (1978a) where ® was themin function.

The relative necessity degree of " is A" given "x is A"" tis denoted by N(A 1A"')

with the intuition that, ifN(A 1A "') = 1 then "x isA "'''fully entails"x isA" There are two definitions of the relativ e necessity degree In Magrez and Smets (1989), it was definedby:

In Dubois and Prade (1992) the definition W<lS :

N(A 1A*) = ill/liE V'(J1.11 (U),~tll (Il»

(2 3.2)

(2 3 3)

where ?was the recipro cal of the Gëdcl implication (s ec Appendix Il) d efined by\:fa, hE

[U , 1] : '(a, b) = 1 if a ~ b, or 1 - a otherwi se Both the definitions sati sfy the two following properties :

l N(A 1A *) = ill/liEvI J - J1.1I'(u) 1u 2: A} when A is non-fuzzy, i.e., a classical set, and

2 N(A 1A"')= 1 when A'" cA

where property 1 is to he consistent with possibilistic logic and property 2 with the entailrnent principle

From (2 3 3), Godo and Vila (1995) analysed further that, among multiple-valued logic implications , only R -implications or th eir recipr ocals (see Appendix 1-1) satisfied prop ert y 2 Als o , ex ccp t I or th e Lll bsiew îcz impli c ation and its identi cal reciprocal,

Section 2.3 Reasoning with Possibility Distributions 23

other R-implications did not satisfy property 1 Meanwhile, the reciprocals of

R-implications different from the Lukasiewicz one had a counter-intuitive behaviour that

N(A 1A*) =0 due to just one UE Usuch thatIlA*(U) =land IlA(u) '* l, no matter howclose IlA*(u) and IlA(U) were Therefore, the authors chose the Lukasiewicz implication,whence (2.3.3)'became:

(2.3.4)

This definition coincides with (2.3.2) when ~ is the bounded difference r-norm defined

byVa, bE [0, 1]:®(a, b)=max{O, a+b - l} (see Appendix G), as proposed in Magrezand Smets (1989)

Based on the notion of relative necessity degrees, Magrez and Smets (1989)established a mathematical framework to devise a fuzzy modus panens model Themodel, as discussed by the authors and Dubois and Prade (1991), acts at the meta-Ievel,rather than the object-level, of combination of a rule and a fact in deriving a conclusion

In accordance with this, the basis of the model is that the certainty degree of the head(i.e., consequent) of a rule is determined by the certainty degree of the rule and thecertainty degree of its body (i.e., antecedent):

where Band H are two fuzzy sets on domains U and V. respectively, CCp) stands for the

certaintydegree of a fuzzy proposition p and ~ is ar -norrn, GivenC(yis Hf- X isB)= 1,(2.3.5) becomes:

because Va E [0, 1]:~(I,a) =~(a, 1)=a, for every r-norrn ® (see Appendix G)

Supposing that a fact "x is B*" is given and "y is H*" is the conclusion to beinferred, C(y is H) is defined to be the relative necessity degree of"y is H' given "y is

H*":

C(yis H) =N(H 1H*)= 1 - sUP V E v{ max{0,IlfJ*(u) -IlH(u)} }

From (2.3.6) and (2.3.7), one has:

N(H 1f/*) =C(xis B)

, (2.3.7)

(2.3.8)whence the least specifie solution (in accordance with the principle of minimumspecificity) for H*is:

24 Chapter2. Fuzzy Logics

(2.3.9)

(2.3.10)

H* =H +(1- C(x is B»=H +(1-N (B1B*»

'.I " BH given "x is where C(x is B) is also defined to be the relative necessity degree u t IS

B*" and, for E E [0, 1], H+E is defined by :

Vv E V: ~H+E(V)= minI1, ~1I(v)+El

representing H beingpcrvaded overall with an indetermination degree E E [0, 1]

We have a remark on the authon;' derivation of (2.3.9) that, for a E (0, 1], there may not cxistH" such that N(1I 1Il''') is exacrly a. lndccd, by (2.3.4), one has:

N(ll III"') =1 - ' ''l ' I ' Ey(nl(u(O, ~lI'(I')- PlI(I')}}

= il!f vr: y( mill (l, 1 - ~II'( I')+~lf(v)1}

~ il!fvEy(mill( 1,1 - 1+PlI (v)}}

= ;l!fvr:vl~Llf(l')} · Thus, if CL < il!fvEy(~lI(v)J. there is no 1/* such that N(H 1 1/*) = CL Meanwhile, the following proposition holds (see Appendix 1\for the proof)

Proposition 2.3.1 For every fuzzy set A and Cl E [0, 1J, A +(1 - CL) is the least specifie solution for A*such that N(A 1A*) ~ CL.

Thcrefore, we propose a refinement of(2.3.5) by using thegreater-than-or-equal-torelation instead of the stri ct equal relation, that is :

This model is consistent with the entailrnent principle and classical modus ponens, that is, when the body of a rule fully matches a fact, the head of the rule can be inferred When the body rnismat ches the fact by sorne mismatching degree, one has a degree of indetermination in rea soniiig, and the conclusion becornes less informative than it is when there is no rnismatching As explained by the authors, in this case, there is no information which could allow one to assign different degrees of indetermination to different elements ofVand, thus , a constant additi on is applied to li as in (2.3.9).

Section 2.4 Partial Truth-Valued Logic Poss ibilislic Logic and Fuzzy Set Logic

2.4 Partial Truth-Valued Logic, Possibilistic

Logic and Fuzzy Set Logic

25

In the literature, the termfuzzy log ie has been used for different logic systems that haveoriginated from the theory of fuzzy sets However, they may have very differentcharacteristics that need to be distinguished to avoid confusion.In this thesis, we classifythem into three main groups, namely, partial truth-valued logic, possibilistic logic andfuzzy set logic as presented below

We calI a fuzzy logic that deals with partial truth partial truth-valued logic, which

is a special multiple-valuedlogic, whose formulas are associated with real numbers in theinterval [0, 1]. The works on partial truth-valued logic includes, for instance, those ofPavelka (1979), Novàk (1987) and Hajek (1995) The distinct characteristic of partialtruth-valued logic is that a value in [0, 1] has the meaning of a truth degree of a formula

An interpretation of partial truth-valued logic generalizes that of classicallogic bymapping each ground atomic formula to a value in [0, 1] and, rhus, can be regarded as afuzzy set on the domain consisting of ail ground atomic formulas (cf Novak 1987) Adenotation function T then de fines the truth degree of a formula with respect to aninterpretation by the following basic rules:

T(p /\ q) =mill(T(p), T(q)}

T(p v q) =max( T(P), T(q)}

T(-,p) = 1 - T(P)

where /\ and v are the conjunction and the disjunction connectives, respectively As such,

in partial truth-valued logic, neither the non-contradiction law nor the excluded-middlelaw holds, because neitherT(p /\ -,p) =0 norT(p v -,p) =1 holds for everyp.

ln contras t, in possibilisticlogie (Dubois and Prade 1988; Dubois, Lang and Prade

1994), although formulas are also associated with real numbers in the interva1 [0, 1], a

valuein [0, 1] has the meaning of an uncertainty degree An interesting examp1e to clarifythe difference between partial truth and uncertainty by Dubois and Prade (1994) is asfollows Saying that the truth degree of "The bottle is full" is 1/2 can mean that the boule

is half full,but saying that the probability of"T he bottle is full" is 1/2 does not reflect theamount of liquid in the boule Consequently, the cules of combination of uncertaintydegrees are different from those of truth degrees, as presented below for possibilisticlogic

26 Chaoter 2. Fuzzy Logics

In possibilistic logic, each classica ogre orrn

, l ' [0 1] An interpretation 1 in

li o t' re on the set n of al1 possiblepossibilistic logic is defined by a possibility cnstn u Ion 1

worlds, which are classical logic interpretations The intuition is that, for each œ E Q,

7t/(w) IS the possibility degree for œ to e t le ac ua

denotation functions n and N then respectively define the possibility degree and thenecessity degree of a classical logic formulafJ wrt 1as fol1ows:

N(P) =1 - n(-,p)

where -, is the negation connective in classical logic and œ 1=P means that (0 satisfiesP

with respect to c1assical logic semantics

Consequently, n is cornpositional with respect to v but not /\, whereas N iscompositional with respect to /\ but not v, that is:

valued logic, col1apsed to two-valued logic However, as pointed out in Dubois and Prade(1994), that proof was not correct due to wrongly assuming the excluded-middle law inpartial truth-valued logic

As presented above, partial lruth-valued logie and possibilistic logic do not dealwith fuzzy propositi ons.Wecali a fuzzy logic whose formulas involve fuzzy setsfuzzy set

Section2.4 Partial Truth -Valued Logic Possibilistic Log ic and Fuzzy Set Logic 27

/ogie, which is more expressive than partial truth-valued logic and possibilistic logic.Nevertheless, partial truth-valued logic and possibilistic logic are a basis for fuzzy setlogic, because computing with fuzzy sets eventually resorts to computing with values in[D, 1], interpreted as truth or uncertainty degrees In fact,the namefuzzy /ogie coined by

Zadeh and the fuzzy logic developed by him (Zadeh 1975) were for fuzzy set logic

However, his main concern was approxima te reasoning rules, rather than formal syntaxormodel-theoretic semantics of the logic

In Dubois, Lang and Prade (1994), the authors attempted to extend possibilisticlogic presented above to deal with fuzzy propositions, but the model-theoretic semanticsfor it was only outlined and not clearly defined Meanwhile, Esteva, Garcia-Calvés and

Godo (1994) introduced fuzzy truth -valued /ogi e as a natural extension of possibilisticlogic that used partial truth-valued logic interpretations, instead of classicallogic ones,aspossible worlds, to which a possibility distribution is assigned

That is, in fuzzy truth-valued logic,an interpretation1is also defined by a possibilitydistribution 1t1on theset.0 of ail possible worlds.However,while in possibilistic logic aproposition p is to be or not to be satisfied by a possible world œE n, in fuzzy truth-

valued logic P has a truth degree Tp(ro)E lO, 1] Then, with respect toI, the possibilitydegree forp to "actually" have the truth degreeTp(ro)is also the possibility degree 1tJ(ro)

for CI) to be the "actual world".Thereby,the denotation ofp wrt1is defined to be a'fuzzyset 1:on [0,1] such that Vu E [D, 1]:~(u) =sUPweU{1t/(ro) 1u =Tp(ro)}. Since values in[D, 1] that1:is defined on has the meaning of truth degrees,1:is actuallyefuzzy truth -va/ue

(Zadeh 1975), hence the name of the logic

For a fuzzy truth-value diagram, the horizontal axis represents truth degrees and thevertical axis represents possibility degrees Figure 2.4.1 illustrates some typical fuzzytruth-values For a TRUE-characteristic value like true, the closer to 1 a truth degree is, the

higher its possibility degree is Whereas, for a FALSE-characteristic value like fa/se, the

closer toaa truth degree is, the higher its possibility degree is.The membership functions

oftrue and fa/se as in the figure are defined as follows:

Vu E [a, 1]:Illme(u)=U

Vu E [a, 1]:~lfalse(U) = 1 - u

The four extrerne values absolutety {rue, absolutely fa/se, absolutely inconsistent

and absolutely unknown correspond lO the four-valued truth-values in Belnap (1977),

28 Chapler 2. Fuzzy LogicS

1 t to the binary truth-values true

where absolutely true and absolut ely fa Ise are eqUiva en

f these four values are defined as andfalse in c1assicallogic The rnembership functlOns 0 •

VUE [0, Il :Ilab sollllely UllkIlO WII(ll)=1

abs olut ely fals e

-oFi~lIre 2.4.1 Typi cnl Iu zzy truth -valucs

- ab solutely tru e

As such, fuzzy truth -values can e xpress b oth partial truth and uncertainty, because

a truth degree, a possibility d egree or a necessity c1egree in [0, 1]can be represented as a special fuz zy truth -value Ind eed , a truth degree œcan be represented by the fuzzy truth- val ue '(a defi ned by:

Il (œ) = 1and Il,; (Il) = °for ail u"# U

/""'ta a

Meanwhile , a necessitydegree Pof a pr opositionp can be considered as the relative neces sity degree of" p i s ab solutely true " gi ven " p is '(\3", where '(p is s orne fuzzy truth- value Thus, "p is certain at Icast to degree P "can be represented by "p is 'tp" such that

Ntab solutely true l'(p)~ p B y Proposition 2 3.1, the leastspecifie solution (in accordance with the principle ofminimum spccificity) for "Il isab solutely tru c+(1 - B),that is:

Il, (l) =1 and Il,; ( Il)=1 - Pf or ailu"# 1

It s ati sfi es the intuiti on th at , if P= 1 th en "f} = ab solutely tru e and if P= a th en

Section2.4 Partial Truth-Valued Logic Possibilistic Logic and Fuzzy Set Logic 29

't~=absolutely unknown Similarly," p is possible at most to degreey'can be representedby"p is'Cy"such thatn(absolutely true l't'Y) ~"f, whence the least specifie solution for'Cyis:

~ (1) ="fand ~ (u) =1 for all u*1

Given a fuzzy propositionpof the form" isA", where A is a fuzzy set on a domain

U, a possible world œforpcon tains a proposition "x isu"for sorne UE U, whenceTp(û))

is defined to be ~A(u) For example, if pis "John isyoung", whereyoung is a fuzzy set on[0, 100] as illustrated in Figure 2.2.2, andœcontains"John is 30 years of age", thenTp(û))

= ~young<30) = 2/3 However, in Godo and Vila (1995), the denotation of a fuzzyproposition pwith respect to an interpretation 1was not the fuzzy truth-value defined by

T pand1[/as in Bsteva,Garcia-Calvés and Godo (1994) as presented above Rather,T pwassaid to induce a possibility distribution 1[1' := Tp on the setilof aIl possible worlds Then,the denotation of p wrt1was defined to be the relative necessity degreeN(1[p 1 1[/) ,as thecertainty degree thar1satisfiedp.

As argued by Dubois, Lang and Prade (1994), ilis not fully satisfactory to have alogic that deals with uncertainty but does not allow inconsistency However, the model-theoretic semantics approach of fuzzy truth-valued logic as well as possibilistic logic,where an interpretation is defined by a single possibility distribution on the set of aUpossible worlds, has a disadvantage that it cannot actually deal with local inconsistency.That is because, when such a possibility distribution is subnormal, it imposes a globalinconsistency on ail propositions involved in possible worlds As such, it rules out