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Co-inductive logic programming and its applications

Co-inductive logic programming and its applications. Overview. Induction vs. co-induction Co-inductive logic programming A goal-directed approach for Answer Set Computing. Induction. Inductive definitions have 3 components: Initiality (e.g., [] is a list)

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Co-inductive logic programming and its applications

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  1. Co-inductive logic programming and its applications

  2. Overview • Induction vs. co-induction • Co-inductive logic programming • A goal-directed approach for Answer Set Computing

  3. Induction Inductive definitions have 3 components: • Initiality (e.g., [] is a list) • Iteration (e.g., [H|T] is a list if T is a list, and H is a number) • Minimality (e.g., nothing else is a list) Inductive definitions correspond to least fixed point interpretations of recursive definitions.

  4. Co-induction • Eliminate the initiality condition • Replaces the minimality condition with maximality • Iteration: [H|T] is a list if T is a list and H is a number • Maximality: the set of lists is the maximal set of such lists Co-induction corresponds to the greatest point interpretation of recursive definition

  5. Example - list list([]). list([1|T]) :- list(T). • What is its inductive semantics? • What is its co-inductive semantics?

  6. Co-inductive logic programming • Operational semantics relies on a co-inductive hypothesis set (CHS); • During execution, if the current resolvent R contains a call C’ that unifies with a call C encountered earlier, then the call C’ succeeds; the new resolvent is R’where  = mgu(C, C’) and R’ is obtained by deleting C’ from R.

  7. Example - list :- coinductive list/1. list([]). list([1|T]) :- list(T). ?- list(X)X = [1|X] list(X) X = [1|T] list(T)

  8. Example - list :- coinductive list/1. list([]). list([1|T]) :- list(T). ?- list(X) X = [] X = [1|X] X = [1] X = [1, 1] … list(X) X = [] X = [1|T] list(T) T= [] T= [1] … …

  9. Example - Stream :- coinductive stream/1. stream([H|T]) :- number(H), stream(T). number(0). number(s(N)) :- number(N). ?- stream([0, s(0), s(s(0)) | T]).

  10. Example – list membership (1) member(H, [H|_]). (2) member(H, [_|T]) :- member(H, T). the desired element is the last element of some prefix of the list membera(X, L) :- drop(X, L, _). drop(H, [H|T], T). drop(H, [_|T], T1) :- drop(H, T, T1).

  11. Example - comember :- coinductivecomember/2. comember(X, L) :- drop(X, L, L1), comember(X, L1). ?- X = [1, 2, 3 | X], comember(2, X). ?- X = [1, 2, 3, 1, 2, 3], comember(2, X). ?- X = [1, 2, 3 | X], comember(Y, X). comember/2 is true if and only if the desired element does occur in an infinite number of times in the list.

  12. Application on computing answer sets (stable models)

  13. Why does a stable model matter? p. r :-p, q. s :- p, not q. With negation as failure p. r :-p, q. s :- p, not q. Another model!! What makes the left model so special?

  14. Stable model • [Gelfond and Lifschitz, 1988] • For a program P not containing any negation, the stable model is unique, defined as its least fixed point. • E.g., a model I= {p, s} • the reduct of P relative to I is the set of rules without negation obtained from P by GL-transformation: • dropping each rule s.t.C in Iand ‘not C’ in the body of the rule • dropping all the rest negative atom ‘not C’ from the bodies of the remaining rules • I is a stable model of P if I is the stable model of the reduct of P relative to I.

  15. Is a Stable Model p. r :-p, q. s :- p, not q. I = {p, s} The reduct p. r :-p, q. s :- p. I is a stable model

  16. NOT a Stable Model p. r :-p, q. s :- p, not q. I = {p, q, r} The reduct p. r :-p, q. I is NOT a stable model

  17. Non-monotonic reasoning (nmr) // {q} is a stable model q. p :- q, not p. Is {q} a stable model? Is {p, q} a stable model? r :- not s.s :- not r. p :- s, not p. // {r} or {s} is a stable model

  18. Goal-directed ASP • Ordinary Rules • all non-cyclical rules • Cyclical rules which when used to expand a call to a subgoalG lead to a recursive call to G through an even (but non-zero) number of negations. E.g.,(1) p :- not q.(2) q :- not p.(3) r. (4) s :- r. :- p CHS = {} :- not q CHS = {p} :- not not p CHS = {p, not q} :- p CHS = {p, not q}

  19. Goal-directed ASP • Odd Loops Over Negation (OLON) • Cyclical rules which when used to expand a call to subgoal G lead to a recursive call to G that is in the scope of an odd number of negations.E.g., (1) p :- q, not r.(2) r :- not p.(3) q :- t, not p.

  20. OLON rules • p :- q, r, not p. • If p is true through other parts of the program, then it is useless. • If p is not true through the rest of the program, then q or r has to be false. chk_p :- p. chk_p :- not q. chk_p :- not r.

  21. Goal-directed execution p :- q, not r. (od & olon)r :- not p. (od)q :- t, not p. (olon) q. (od) p :- q, not r.r :- not p.q. chk_p :- p.chk_p :- not q.chk_p :- r.chk_q :- q.chk_q :- not t. nmr_chk :- chk_p, chk_q. :- p, nmr_chk. {} :- q, not r, nmr_chk. {p, q} :- not r, nmr_chk. {p, q} :- not not p, nmr_chk {p, q, not r} :- p, nmr_chk {p, q, not r} :- nmr_chk {p, q, not r}

  22. Issues • Identifying OLON and ordinary rules • Through a graph travel algorithm in O(|P| * n) • Partial answer set • If cyclical rules not through any number of negations, then the recursive call fails.E.g., p :- p.

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