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A connection between Similarity Logic Programming and Gödel Modal Logic

Luciano Blandi Dept. Matematica & Informatica University of Salerno 84084 Fisciano (Salerno) ITALY lblandi@unisa.it. A connection between Similarity Logic Programming and Gödel Modal Logic. Lluis Godo Institut d’Investigaci ó en intel ·ligència Artificial

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A connection between Similarity Logic Programming and Gödel Modal Logic

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  1. Luciano BlandiDept. Matematica & InformaticaUniversity of Salerno 84084 Fisciano (Salerno) ITALY lblandi@unisa.it A connection between Similarity Logic Programming and Gödel Modal Logic Lluis GodoInstitut d’Investigació en intel·ligència Artificial Consejo Superior de Investigaciones Ccientificas Campus UAB, 08193 Bellaterra, Spain godo@iiia.csic.es Ricardo Oscar Rodriguez Dpto. de Computación Fac. Ciencias Exactas y Naturales Universidad de Buenos Aires Ciudad de Buenos Aires, Argentina ricardo@dc.uba.ar

  2. Overview • Two approaches to Similarity-based Approximate Reasoning: • Extended Logic Programming • Extended Modal Logic • Correspondencesbetween the twoapproaches

  3. Similarity relation (*) Definition A Similarity on a domain U is a fuzzy subset R: U×U[0,1] of U×U such that the following properties hold: i) R(x, x) = 1 for any x є U (reflexivity) ii) R(x, y) = R(y, x) for any x, y є U (symmetry) iii) R(x, z)  R(x, y) *R(y, z) for any x, y, z є U (transitivity) where * : [0,1] × [0,1]  [0,1] is a t-norm We say that R is strict if the following implication is also verified iv) R(x, z) = 1  x = z (*) L. Valverde, “On the structure of F-indistinguishability operators”. Fuzzy Sets Syst 1985;17:313–328.

  4. I. Approximate reasoning by extension of the Logic Programming paradigm Similarity-based SLD Resolution • SLD Resolution inferential engine as model of logic deduction • Similarity relation embedded in the SLD Resolution model as approach to the approximate reasoning (*) M.I. Sessa, “Approximate Reasoning by Similarity-based SLD Resolution.”, Theoretical Computer Science, 275 (2002) 389-426.

  5. MATCH I. Approximate reasoning by extension of the Logic Programming paradigm SLD Resolution: an example Goal:meeting(‘Data Mining’, date(D, M,Y)) X = ‘Data Mining’ • Program: meeting(X, date(D, M,Y))  conference(X, date(D, M, Y)). • conference(Information Retrieval’, date(22, 6, 2000)). • conference('Neural Network', date(17, 9, 2000)). • conference('Artificial Intelligence', date(10, 1, 2002)). • conference('Data Mining', date(12, 5, 2001)). • conference('Data Discovery', date(22, 9, 2001)). Solutions: meeting(‘Data Mining’, date(12, 5, 2001))

  6. R(‘Data Mining’,‘Information Retrieval’) = 0.6 > 0 R(‘Data Mining’,‘Data Mining’) = 1 > 0 R(‘Data Mining’, ‘Data Discovery’) = 0.4 > 0 I. Approximate reasoning by extension of the Logic Programming paradigm Similarity-based SLD Resolution: an example Goal:  conference(‘Data Mining’, date(D, M,Y)) Program: conference(‘Information Retrieval’, date(22, 6, 2000)). conference('Neural Network', date(17, 9, 2000)). conference('Artificial Intelligence', date(10, 1, 2002)). conference('Data Mining', date(12, 5, 2001)). conference('Data Discovery', date(22, 9, 2001)). Solutions: Approximation degree: conference(‘Data Mining’,date(22, 6, 2000))  = 0.6 conference(‘Data Mining’, date(12, 5, 2001))  = 1 exact solution conference(‘Data Mining’,date(22, 9, 2001))  = 0.4

  7. I. Approximate reasoning by extension of the Logic Programming paradigm Similarity-based SLD Resolution To determine the computed answer substitutions by means similarity-based SLD derivation G0C1,1,U1 G1  … Cm,m,Um Gm … Approximation degree associated to the computed answer  = min{U1, U2, …, Uk}

  8. I. Approximate reasoning by extension of the Logic Programming paradigm Weak Unification Algorithm (*) Given two atoms A = p(s1,... , sn) and B = q(t1,... , tn) of the same arity with no common variables to be unified, construct the associated set of equation W = {p = q, s1 = t1, ... , sn = tn}. If R(p,q) = 0, halts with failure, otherwise, set U = R(p, q) and W = W-{p=q}. Until the current set of equation W does not change, non deterministically choose from W an equation of a form below and perform the associated action. (1) f(s1, ..., sn) = g(t1, ...,tn) where R(f, g) > 0: replace by the equations s1=t1, ..., sn=tn, and set U = min{U, R(f, g)}; (2) f(s1,... , sn) = g(t1,... , tm) where R(f, g) = 0: halts with failure; (3) x = x: delete the equation; (4) t = x where t is not a variable; replace by the equation x = t; (5) x = t where x  t and x has another occurrence in the set of equations: if x appears in t then halt with failure, otherwise perform the substitution {x/t} in every other equations. R.K. Apt, “Logic Programming”, in: J. van Leeuwen (Ed.), Haridbook of Theoretical Computer Science, vol. B, (Elsevier, Amsterdam, 1990) 492-574. (*) M.I. Sessa, “Approximate Reasoning by Similarity-based SLD Resolution.”, Theoretical Computer Science, 275 (2002) 389-426.

  9. I. Approximate reasoning by extension of the Logic Programming paradigm An extended Prolog system (*) meeting(‘Data Mining’, date(D, M,Y)) meeting(X, date(D, M,Y)) :- conference(X, date(D, M, Y)), check(D,M,Y,Agenda). … conference(‘Information Retrieval', date(12, 5, 01)). conference('Neural Network', date(17, 9, 00)). conference('Artificial Intelligence', date(10, 1, 02)). conference('Data Mining', date(12, 5, 01)). conference('Data Discovery', date(22, 9, 01)). check(D,M,Y,Agenda) :- ….. … Weak Solutions: meeting('Data Mining',date(12, 5,01)) (=0.4) meeting('Data Mining', date(22, 9, 01)) (=0.8) Exact Solutions: meeting('Data Mining',date(12,5,01)) Weak Solutions: meeting('Data Mining',date(12, 5, 01)) ( =0.4) meeting('Data Mining', date(22, 9, 01)) (=0.8) Exact Solution: meeting('Data Mining',date(12,5,2001)) (*) V. Loia, S. Senatore, M.I. Sessa, Similarity-based SLD Resolution and its implementation in an Extended Prolog System, Proc. 10th IEEE International Conference on Fuzzy Systems, 2001, Melbourne, Australia.

  10. I. Approximate reasoning by extension of the Logic Programming paradigm Similarity interface of Silog

  11. II. Approximate reasoning by extension of the Modal Logic RGS5◊: The rational Gödel similarity-based S5 modal logic Language: Set of the symbols of propositional constant: Const = {p1, p2, …} Logic connectives: (conjunction),  (implication) Modal operator: (approximation) Truth-constants: for each r  [0, 1] Axioms: Axioms(BL) + {G} + {Bookkeeping axioms} + {Approximation axioms} Inference rules: From  and   infer  From   infer   

  12. II. Approximate reasoning by extension of the Modal Logic BL: The basic many-valued propositional logic Language: Set of the symbols of propositional constant: Const = {p1, p2, …} Logic connectives: (strong conjunction),  (implication) Truth-constant: Axioms: (A1) ()  (()()) (A2) ()   (A3) () () (A4) (()) (()) (A5a) (()) (()) (A5b) (()) (()) (A6) (())  ((()))(A7)  Inference rule Modus ponens:From  and   infer 

  13. II. Approximate reasoning by extension of the Modal Logic Extended BL with rationals Language: Language(BL) + { for each r  [0, 1] } Axioms:Axioms(BL) + {Book-keeping axioms} Book-keeping axioms: Inference rule: Modus ponens:From  and   infer  Notation: (, r) is ( ) Derived inference rule: From (, r) and ( , s) infer (, r * s)

  14. II. Approximate reasoning by extension of the Modal Logic RGL: Rational Gödel Logic Language: Language(BL) + { for each r  [0, 1] } Axioms:Axioms(BL) + {Book-keeping axioms}+ {G} Axiom G:   ( & ) Inference rule: Modus ponens:From  and   infer  Remark: Axioms(G) = Axioms (BL) + {G}

  15. II. Approximate reasoning by extension of the Modal Logic RGS5◊: The rational Gödel similarity-based S5 modal logic Language: Language(RGL) + {  (approximation modal operator) } Axioms:Axioms(RGL) + {Approximation axioms} Approximation axioms: D◊: ◊(  )  (◊ ◊) F◊: ¬◊0 T◊:  ◊ Z◊+ : ◊¬¬ ¬¬◊ 4◊: ◊◊ ◊ B◊:  ¬◊¬◊ R1:  ◊ R2:  ◊  ◊(  ) Inference rules: From  and   infer  From   infer   

  16. II. Approximate reasoning by extension of the Modal Logic Many-valued similarity-based Kripke modelsM=W, S, e W   is a set of possible worlds S: W × W  [0, 1 ] is a similarity relation on W e: Const × W  [0, 1 ] represents an evaluation assigning to each atomic formula pi Const and each interpretation w  W a truth value e(pi, w)  [0, 1] of pi in w. e is extended to formulas by defining e(  , w) = min{e(, w), e(, w)} e(  , w) = e(, w) G e(, w) e( , w) = r for all r   [0, 1] e(◊, w) = supw’W min{S(w, w’), e(, w’)}

  17. Correspondencesbetween the twoapproaches (1) Let R : Const × Const  [0, 1] be a similarity relation on Const. Let P = Facts  Rules  L be a logic program Define a mapping * : L  LGby (q  p1, . . . , pn)* = p1pnq (p1, . . . , pn)* = p1 pn Then, we can define Rules* = {* |  Rules} Crisp = {p  ¬p | p  Const} Sim = {  ((p q) (q p)) | p, q  Const} and the following theory in the language LG P= Facts  Rules* Crisp  Sim

  18. Correspondencesbetween the twoapproaches (2) Sim = {  ((p q)  (q p)) | p, q  Const} (*) In RGS5◊, We note that IS(p | p) = 1 (reflexivity) IS(r | q)  min{IS(r | p), IS(p | q)} (min-transitivity) RS(p, q) = min{IS(p |q), IS(q | p)} is a min-similarity (*) E.H. Ruspini, “On the semantics of fuzzy logic”, in: Int. Journal of Approximate Reasoning, 5, 1991, pp. 45-88.

  19. Correspondencesbetween the twoapproaches (3) Proposition Let R : Const × Const  [0, 1] be a similarity relation, P = Rules  Facts a definite program on a propositional language L and q’ a goal.. If there exists a similarity-based SLD refutation with approximation degree  for P  { q’} D = G0C1,1 G1 · · · Ck−1,k−1 Gk−1Ck,k⊥ where G0 =  q’ and  = min{1, . . . , k}, then P◊⊢S◊q’

  20. Conclusions and related works We have shown and put into relation two approaches to Similarity-based Approximate Reasoning: • Framework: Extended Logic Programming Tool: Similarity-based SLD Resolution (Similarity relation defined on symbols of the language) • Framework: Many-valued Rational Gödel Modal Logic Tool: Similarity relation as accessibility relation (Similarity relation defined on possible worlds) • Future tasks: • To prove the full relationship among the two approaches • To prove the relative counterpart for predicate languages. • To study possible links with respect to other approaches to similarity-based reasoning

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