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

Foundations of Order-Sorted Fuzzy Set Logic Programming in Predicate Logic and Conceptual Graphs l ,.,' , ' by \ • '- ." Tru H Cao B Eng HCMUT (Gold Medal) M Eng AlT (Tim Kendall Memorial Prize) ! A thesis submiued in Iulfilment of the requiremcuts for the degree of Doctor of Philosophy 700 & ~ Department of Computer Science and Electrical Engineering The Univergity of Queensland Australia March 1999 Dedicated Lv the memory ofmy mother Statement of Sources 1declare that the work presented in this thesis is, to the best of my knowledge and belief, -1; original and my own work, except as acknowledged in the text, and that the material has nol been submiued, either in whole or in part, for a degree at this or any ether university Parts of the material have been reported in Cao (l997a, 1997b), Cao and Creasy (l996a, 1996b, 1997b, 1998a), published in Cao (l998a), Cao and Creasy (1997a, 1998b), Cao, Creasy and Wuwongse (l997a, 1997b), or accepted for publication in Cao (1998b) Tru H Cao Brisbane, Mareh 1999 Acknowledgements Whell drinking water, think ofits source Vietnamese proverb J wish to express my sincere gratitude to my supervisor, Dr Peter Creasy, for his kindness, advice and support throughout my PhD course rernember times he patiently listened to me presenting research issues on the whiteboard in his room These and other Limes having discussions with him have helped me to clarify, consolidate and dev elop ideas His tireless reading of my writings and commenting on every bit of them have helped me to improve my skill of scientific writing in English would like to thank the Commonwealth Government of Australia for awarding me an Overseas Postgraduate Research Scholarship, and the University of Queensland for a University of Queensland Postgraduate Research Scholarship My PhD program here would not have been feasible without thde financial supports would also like to thank the Department of Computer Science and Electrical Engineering for funding my attendance at several conferences 10 present my working papers and to have discussions with leading researchers in the field The questions and comments received l'rom these presentations and discussions have helped me to consolidate the idcus and sueugthen the arguments for many issues presented in this thesis AI this special milestone on my scholastic road, J think of teachers who have taught me since J started to learn the alphabet Although this thesis is the direct outcome of my PhI) study, behind ir are background knowledge and methodology that have been taught for years also think of people who have helped me in many different ways am grateful 10 aIl of them Nevertheless, neither this thesis nor anything else could have done without my beloved family There have been many limes when their faith in me has made the difference between giving up and trying again, many times when their love has helped me to find a strength did not know had can never thank them enough Abstract For dealing with the pervasive vagueness and imprecision of the real world as reflected in natural language, conce ptual graph programs have been extended to fuzzy conceptual graph programs However, as many fundamental issues have rern ained unaddressed and unresolved , no rigorous foundation has been established Therefore, this thesis aims at a sound and complete foundation for fuzzy conceptual graph progrumming, in particular, and for order-sorted fuzzy set logie prograrnming, in general There are three main problems 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 the fundamentals of a theorern proyer and, rnoreover, the previous definition of rnodeltheoretic 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 degree properties of object types under uucertainty and/or partial truth Third, conceptual graph theory has Iacked the formai integration of funetional relation types and the notion of conjunctive types and, furthermore, the traditional CG unification is not adequate to obtuin a complete resolution style proof procedure with dose coupling of a type hierarchy and a program We solve these three problems correspondingly as follows Firstly, viewing fuzzy sets as lauice-based values, we extend classical annotated logic progruius to annotated fuzzy logic programs as a general framework for fuzzy set Iogic progranuning that can deal with local inconsistency Secondly, we propose a general framework of fuzzy types, also viewed as Iattice-based values, and extend annotated fuzzy logic programs ta ord er-sorted ones Thirdly, we formally integrute funetional relation types into couceptual gruph theory radieally as a part of its signature, introduce the nution of conjuuctive types into conceptual graph theory , and develop fuzzy conceptual graph prograrns based on order-sorted annotated fuzzy Iogic programs as an abstract framework We believe that this thesis is the first sound and complete the oretical foundation for ord er-sorted fuzzy set logic programming in the two complementary logic notations , numely, predic ute logic and conceptual graphs The results obtained for fuzzy couceptual graph programs are applicable to conceptual graph programs as special fuzzy ones and usetul for extending conceptual graphs with lattice-based annotations to enhance their scruautics It adds to efforts of the fusfn of couccptual gr aphs and uncertuinty logics towards a kuowledge representation and reasoning language thut upprouchcs human expression and reasoning, Table of Contents xiv List of Figures Chapter Prologue 1.1 Scope, Motivation and Objective 1.2 Summary of Major Contributions 1.3 Structure Overview 1.4 Symbol and Abbreviation Conventions ' ' 10 Chapter Fuzzy Logics 13 2.1 Introduction 13 2.2 Fuzzy Sets and Possibility Distributions 15 2.3 Reasoning with Possibility Distributions 19 2.4 Partial Truth- Valued Logic, Possibilistic Logic and Fuzzy Set Logic 25 2.5 Fuzzy Set Logic Programming 2.6 Conclusion ~ : Chapter Annotaled Fuzzy Logic Programs Cha pter 31 36 39 3.1 Introduction " 39 3.2 AFLP Syntax 42 3.3 AFLP Model-Theoretic Semantics 4g 3.4 AFLP Fixpoint Semantics 55 3.5 AFLP Redu ctants and Constraints 3.6 AFLP Procedural Semantics 66 3.7 Conclusion 69 : 59 Fuzzy Types 71 4.1 Introduction 71 4.2 l'ruth-Value Set Structures 76 4.3 Single Fuzzy Types H2 4.4 Conjunctive Fuzzy Types H7 Chapter 4.5 Fuzzy Type Mismatching Degrees 4.6 Order-Sorted AFLPs 4.7 C onclu sion Chapter 102 105 5.1 Introduction 5.2 B asic Notions of Conceptual Graph s 5.3 Functional Relation Types Conjunctive Concept Types and , 108 114 5.4 E xtcnded Simple Conceptual Graphs 121 5.5 Simpl e Fuzzy Conceptual Graphs 127 5.6 Fuzzy Conceptual Graph Pr oj ection and Normalization 5.7 Conc lusio n , 133 · · · ·· · · · · · 139 Fuzzy C onceptua l Graph Progrnms 141 6.1 In trodu c tion 141 6.2 Fe Gp S ynta x 145 6.3 FCGp Model-The oret ic S eman tics 6.4 Fccrr Fixpoint Sern antics 156 6.5 G ener al Issu es o f CG Un ific atio n and CG p Resoluti on 160 6.6 FeG Unificati on and FeG p Redu ctants 167 6.7 FeGP Pr oc edural Sernantics o.R C o ncl usio n 176 Generalizr tion and Specialization 7.1 7.2 Chapter 95 lOS Fuzzy Conceptual Graphs C on jun ct ive Relation Types Chapter 90 150 " 171 179 Int roducti on 179 / Gcn eral izati on 180 7.3 Spec ia lization 7.4 C onclusion 184 191 Epilogue 193 R I Summary R.2 Su ggest io ns for Future Rc sc arch 193 195 197 Appendixes A B Proofs for Chapter 197 Proofs for Chapter 198 C Proofs for Chapter ~ 207 D Proofs for Chapter 214 E Proofs for Chapter 216 F Proofs for Chapter 222 G T-norms and T-conorms 223 H Multiple-Valued Logic Implications 225 References 227 Author Index 245 Subject Index 249 xiii List of Figures 2.2 Normal and subnorrnal fu zz y sets 15 17 2.2.2 Fuzzy c omplemen t a nd fuzzy oppo siti on 2.2 Specificity of possi bility distributions 2.4.1 Typ ical fuzzy truth -values 3.2.1 Information ordering o n fu zzy se ts 43 3.2.2 The notion of ideals applied to a fuz zy set lattice 46 3.2.3 Fuzzy set values for cxcmpli fying the noti on of rcstri ctcd AFLPs 47 3.3.1 51 18 28 Fu zzy set misrnatching and relative ncccssit y degrees 4.2.1 A gen eral s truc ture o f truth -valu e laui ccs 79 4.2 TRUE-ch aract erist ic and FAL5E-characl erisli c truth -valu es 79 4.2.3 The arnbiguit y ard er 81 An ab norma l fuzzy tru th-val uc 92 5.2 An exarnple CG 10 5.2.2 A CG w ith a corefere nce link 5.2 109 A CG proj ecti on 113 5.2.'1 Non -anti symme tr y o f C G pr ojecti on 113 5.2.5 CG normal Iorm fo r the co rn ple tc ness o f CG p roj ecti on 11 5.3 A CG w ith a fun ct ion al rel ati on Ils 5.3.2 A CG with a fun cti on al relati on type 116 5.3.3 11 A fun eti onal rel ati on typ e and o ne of its fun eti on al s u btypes 5.3.4 A fun ctional relati on type and on e of its no n-fu nctiona l supcrtypcs 117 5.3.5 Latti ce-theoreti c and o rde r-theore tic int erpretati ons of a type latti ce 18 5.3.6 A CG with conjunctive concept types and conj unc tive relati on types 120 5.4.1 A C G projecti on w ith fun cti onal rel ati on typ es co nj unc tive conce p t typ es and c o nj unc tive relati on ty pes 123 5.4.2 Functi onal rel ati on ty pe a nd CG j o in 124 5.4 Fu nction al re latio n s ubt ype and CG join 125 5.4.4 CG normal fon n 126 5.5.1 A n ex amp le FCG 5.5.2 An FCG w ith a fu zz y quantifi er 5.5 127 ln An FCG w ith fu zzy truth -va luc s 12 .~ 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 A cap that realizes close coupling 164 6.6.1 An FCGP reductant 170 1.3.1 A possibilistic CG projection 190 ' xv References 241 119 Reitsma, E 1998 From requirements to implementation; Applying conceptual graphs to intelligent networks PhD Thesis, University of Twente, 120 Rescher, N 1969 Many-valued Logic Mc Graw-Hill: New York .i 121 Rose, A 1951 Systems of logic whose truth-values form lattices Mathmatische Annalen, 123, 152-165 .i 122 Salvat, E 1997 Raisonner avec des opération de graphes: graphes conceptuels 'et règles d'inférence PhD Thesis, Université Montpellier Il ' 123 Salvat, E and Mugnier, M.L 1996 Sound and complete forward and backward chainings of graph mies In Eklund, P.W et al (eds), Conceptual Structures - Knowledge Representation as Interlingua, Lecture Notes in Artificial Intelligence, Vol 1115, Springer-Verlag, Berlin, pp 248-262 124 Schmidt-Schauû, M 1989 Computational aspects of ail order-sorted logic with term declarations Lecture Notes in Artificial Intelligence, Vol 395, SpringerVerlag, Berlin 125 Schwartz, D.G 1991 A system for reasoning with imprecise linguistic information International Journal ofApproximate Reasoning, 5, 463-488 126 Sowa, 1.F 1984 Conceptual Stru ctures - Information Processing in Mind and Machine Addison-Wesley Publishing Company, Massachusetts 127 Sowa, J.F 1988 In discussion following the invited talk "Knowledge representation in databases, expert systems and natural language", at an IFIP conference on the role of artificial intelligence in databases and information systems, Guangzhou, China, July, 19H8 128 Sowa, 1.F 1991 Towards the expressive power of natural language In Sowa, References 242 J.F (ed ), Princip les of Semantic Networks - Explorations ill the Representation of Knowledge, Morgan Kaufmann Publishers, San M ateo, CA, pp 157-189 129 Sowa, J.E 1994 Knowledge Representation: Logical, Philosophical and Computational Foundations A draft book in preparation 130 Sowa, J.F 1997 M at ching logical structure to linguistic structure In Hauser, N and Roberts , 0 and Van Evra, J (eds), Studies in the Logic of Charles Sanders Peirce , Indiana University Press, B1oomington , pp 418-444 131 Stumme, G 1996 Loca l scaling in conceptual data systems In Eklund, P.W et al (ed s) , Conc eptuel Structures - Knowledge Repres entation as Interlingua, Lecture Notes in Artificial Intelligence , Vol 1115 , Springer-Verlag, Berlin, pp 308-320 132 Subrahmanian, V.S 1987 On the sernantics of quantitative logic programs In Proceedings of the 4th IEEE Symposium on Logic Programming , IEEE Computer Society Press, Wa shington, D.C , pp 173 -1 R2 133 Subrahmanian, V.S 19 90 M ech anical pr oof procedures for rnany-valued latticebased Iogic programrning Journal of Non -Classical Logic, 7, 73-107 134 Turunen , E 1992 Algebraic structures in fuzzy logic International Journal for Fu zzy Sets and Systems , 52 , 181-188 135 Ullman , J.O 1988 Prin cip/es of Database and Knowleâge-Base Systems, Vol Computer Science Press , Ro ckville, Maryland 136 U mano , M 19 87 Fuzzy set Pr olog In Preprjnt~ of the 2nd Internationai Fuzzy S ystems Ass ociati on Cà ngress , pp 750-753 137 Van E m de n, M.II 1