Abductive Disjunctive Logic Programming. Ullrich Hustadt

Size: px

Start display at page:

Download "Abductive Disjunctive Logic Programming. Ullrich Hustadt"

Cecil Spencer
5 years ago
Views:

1 Abductive Disjunctive Logic Programming Ullrich Hustadt Max-Planck-Institut fur Informatik, Im Stadtwald, W-6600 Saarbrucken, Germany Phone: (+9 681) , Fax: (+9 681) , 1 Introduction [EK89] introduced the notion of an abductive framework and proposed stable models as a semantics for abduction. They showed that abductive frameworks can be used to provide an alternative basis for negation-as-failure in logic programming. [KM90] introduced the notion of generalized stable models by suitably extending the denition of stable models. The semantics of generalized stable models claries the meaning of integrity constraints within an abductive framework. In ([SI92]) a goal-directed method for computing the generalized stable models of an abductive framework has been proposed. Their method is correct for any consistent abductive framework. Whereas abductive frameworks correspond to normal logic programs with integrity constraints, I propose an extension to disjunctive normal logic programs. Disjunctive normal logic programs extend normal logic programs to disjunctive logic programs and therefore, provide full rst-order expressibility. 2 Abductive Frameworks Denition 2.1 (Abductive framework) An atom is an expression P (t 1 ; : : :; t n ), where P is a predicate symbol and t 1,: : :, t n are terms. A positive literal is an atom, a negative literal is an expression not(a 1 ), where A 1 is an atom. A literal is either a positive or a negative literal. Let L be a literal. Then L c denotes the complement of L. A clause is either of the form A 1 _ : : : _ A m L 1 ^ : : : ^ L n ; where A 1,: : :, A m, m 1 are atoms, and L 1,: : :, L n are literals, or? L 1 ^ : : : ^ L n ; where L 1,: : :, L n are literals. The left hand side of a clause is the head, denoted by head(c), the right hand side is the body of the clause, denoted by body(c). A program is a set of clauses. An abductive framework is a pair ht; Ai where A is a set of predicate symbols, called abducible predicates, and T is a set of clauses such that no predicate symbols of head atoms are in A. A set of ground atoms for predicates in A is called abducibles. The set of all abducibles is denoted by A. Given an abductive framework ht; Ai, pos(c) is the set of positive literals in the body of a clause C which are not abducibles, neg(c) is the set of negative literals in the body of C, abd(c) is the set of abducibles in the body of C. The Herbrand base of a program T is denoted by HB(T ), its Herbrand universe by HU(T ). We impose the restriction that the clauses of a program must be range-restricted, i.e. any variable in a clause C must occur in pos(c). Any clause can be transformed to a range-restricted clause by 1

2 inserting for every variable violating the range-restrictedness condition a predicate dom describing the Herbrand universe. Denition 2.2 (Minimal Model) An interpretation I for a program T is a subset of HB(T ). An interpretation I satises a ground atom A 1 i A 1 2 I. It satises a ground literal not(a 1 ) i A 1 62 I. No interpretation satises?. An interpretation I satises a clause A 1 _ : : : _ A m L 1 ^ : : : ^ L n ; i for every ground substitution either one of A 1,: : :, A m is satised by I or one of L 1,: : :, L n is not satised by I. An interpretation I is a model of T if I satises every clause in T. An model I of T is minimal if there is no interpretation I 0 I such that I 0 is a model of T. Denition 2.3 (Gelfond-Lifschitz Transformation) Let T be a program and I be an interpretation. The Gelfond-Lifschitz Transformation GL(T; I) of T is dened by GL(T; I) = f(a 1 _ : : : _ A m B 1 ^ : : : ^ B n ) j A 1 _ : : : _ A m B 1 ^ : : : ^ B n ^ not(c 1 ) ^ : : : ^ not(c k ) 2 T; is a ground substitution, and C 1 ; : : :; C k 62 Ig Denition 2. (Generalized Stable Model) An interpretation I is a stable model of T i I is a minimal model of GL(T; I). Let ht; Ai be an abductive framework and be a set of abducibles. A generalized stable model M() of ht; Ai is a stable model of T [ fh j H 2 g. An abductive framework ht; Ai is consistent if there exists a generalized stable model M() of ht; Ai for some set. In the following, we restrict our intention to consistent abductive frameworks. 3 Proof Procedure for Abductive Frameworks Denition 3.1 (Goal) A goal is a disjunction of conjunctions of literals, written (L 1 1 ^ : : : ^ L 1 n 1 ) _ : : : _ (L m 1 ^ : : : ^ L m nm ): An interpretation I satises a goal if there exists a ground substitution such that for some i, 1 i m, I satises L i j for every 1 j n i. Let D be a disjunction of atoms A 1 _ : : : _ A m. Then D c denotes the conjunction of negative literals A c 1 ^ : : : ^ A c m. Denition 3.2 (Abductive Explanation) Let ht; Ai be an abductive framework and G a goal. We call a set of abducibles an abductive explanation for G if there exists a generalized stable model M() that satises G. 2

3 We can dene an abductive proof procedure generating abductive explanations by combining the proof procedure given in ([SI92]) with a proof procedure for programs. Such a proof procedure is described in ([RLS91]). We need the following denition for the description of the abductive proof procedure. Denition 3.3 Let ht; Ai be an abductive framework and L be a ground literal. Then the set of resolvents with respect to L and T, resolve(l; T ), is dened by resolve(l; T ) = f (H 1 _ : : : _ H i 1 _ H i+1 _ : : : _ H k f (H 1 _ : : : _ H k L 1 ^ : : : ^ L m ) j L is negative and (H 1 _ : : : _ H k L 1 ^ : : : ^ L m ) 2 T and L c = H i by a ground substitution g [ L 1 ^ : : : ^ L i 1 ^ L i+1 ^ : : : ^ L m ) j (H 1 _ : : : _ H k L 1 ^ : : : ^ L m ) 2 T and L = L i by a ground substitution g The set of deleted clauses with respect to L and T, delete(l; T ), is dened by delete(l; T ) = f(h 1 _ : : : _ H k L 1 ^ : : : ^ L m ) j (H 1 _ : : : _ H k L 1 ^ : : : ^ L m ) 2 T and L c = L i by a ground substitution g Denition 3. (Deduction rules) Instead of using a kind of pseudo-code to describe the abductive proof procedure, we will provide inference rules for deriving judgements of the form hht; Ai; 1 i `a hg; ; 2 i; where ht; Ai is an abductive framework, 1, 2 are sets of abducibles, G is a goal, and is a substitution. Intuitively, the judgement above means that 2 is an abductive explanation for G. To dene the inference rules for `a, we need additional judgements of the form hht; Ai; 1 i `p hg; ; 2 i; hht; Ai; 1 i `r hc; 2 i; hht; Ai; 1 i `l hl 1 ; 2 i; hht; Ai; 1 i `d hc; 2 i; where L 1 is a literal and C is a set of clause. We will provide inference rules for these judgements too. Abductive Inference hht; Ai; i `a hk 1 _ : : : _ K n ; ; i if hht; Ai; [ f?gi `p h?; ; [ f?gi hht; Ai; i `a hk 1 _ : : : _ K n ; ; 0 i if hht [ fq(x) K 1 ; : : :; q(x) K n g; Ai; i `p hq(x); ; 0 i where K 1,: : :, K n are conjunctions of literals, q is a fresh predicate symbol, and x are the free variables of K 1,: : :, K n. Hypothesis Rule 3

4 hht; Ai; [ fa 1 gi `p hb 1 ^ B 2 ^ : : : ^ B k ; ; 2 i if hht; Ai; [ fa 1 gi `p hb 2 ^ : : : ^ B k ; ; 2 i where is the most general unier of A 1 and B 1. Resolution Rule hht [ fa 1 L 1 ^ : : : ^ L n g; Ai; 1 i `p hb 1 ^ B 2 ^ : : : ^ B k ; ; 3 i if hht [ fa 1 L 1 ^ : : : ^ L n g; Ai; 1 i `p hl 1 ^ : : : ^ L n ^ B 2 ^ : : : ^ B k ; ; 2 i and hht; Ai; 2 i `l ha 1 ; 3 i; where is the most general unier of A 1 and B 1. Abduction Rule hht; Ai; 1 i `p ha 1 ^ L 2 ^ : : : ^ L m ; ; 3 i if hht; Ai; 1 i `l ha 1 ; 2 i, hht; Ai; 2 i `p hl 2 ^ : : : ^ L m ; ; 3 i, and A 1 is in A. Negation Rule hht; Ai; 1 i `p hnot(a 1 ) ^ L 2 : : : ^ L m ; ; 3 i if hht; Ai; 1 i `l hnot(a 1 ); 2 i and hht; Ai; 2 i `p hl 2 ^ : : : ^ L m ; ; 3 i. Splitting Rule hht [ fa 1 _ : : : _ A m L 1 ^ : : : ^ L n g; Ai; 1 i `p hg; m+1 i if hht [ fa i L 1 ^ : : : ^ L n g; Ai; 1 i `p hg; 2 i for some 1 i m, hht [ fa j g; Ai; j+1 i `a hg; j+2 i for each j = 1; : : :; i 1, and hht [ fa j g; Ai; j i `a hg; j+1 i for each j = i + 1; : : :; m. Consistency of literals `l hht; Ai; 1 [ f?gi `l hl; 1 [ f?gi hht; Ai; 1 [ flgi `l hl; 1 [ flgi hht; Ai; 1 i `l hl; 3 i if hht; Ai; 1 [ flgi `r hresolve(l; T ); 2 i, hht; Ai; 2 i `d hdelete(l; T ); 3 i, and L is not?. Consistency of rule deletions `d hht; Ai; 1 i `d hfcg [ C; 3 i if hht; Ai; 1 i `p hhead(c); 1 ; 2 i and hht; Ai; 1 i `d hfcg [ C; 3 i if hht; Ai; 1 i `l hhead(c) c ; 2 i and hht; Ai; 1 i `d h;; 1 i 1 The identity substitution is denoted by

5 Consistency of rules `r hht; Ai; 1 i `r hfcg [ C; 3 i if hht; Ai; 1 i `p hl c ; ; 2 i for some literal L in the body of C and hht; Ai; 1 i `r hfcg [ C; i if hht; Ai; 1 i `p hbody(c); ; 2 i, hht; Ai; 2 i `l hh 1 ; 3 i for some atom H 1 in the head of C, and hht; Ai; 3 i `d hc; i. hht; Ai; 1 i `r h;; 1 i Theorem 3.5 Let ht; Ai be an consistent abductive framework and G a goal. Then G has an abductive explanation i hht; Ai; ;i `p hg; ; 0 i can be derived for some substitution and a set of literals 0, such that 0 \ A. Future Work [SI92] introduced the notion of the relevant ground program T for a normal logic program T which is a subset of the set of ground rules obtainable from T. Using the relevant ground program it is possible to reduce the size of the sets resolve(l; T ) and delete(l; T ). Only if these two sets are nite, the proposed abductive proof procedure is applicable. Although there is a notion of the relevent ground program for a disjunctive normal logic program, it is not obvious that it can be used to reduce the size of resolve(l; T ) and delete(l; T ) without loosing correctness of the abductive proof procedure. References [EK89] K. Eshghi and R.A. Kowalski. Abduction compared with negation by failure. In Georgio Levi and Maurizio Martelli, editors, Proceedings of the Sixth International Conference on Logic Programming, pages 23{25, Lisabon, Portugal, June 19{ IEEE, MIT Press. [KM90] A.C. Kakas and P. Mancarella. Generalized stable models: A semantics for abduction. In Luigia Carlucci Aiello, editor, Proceeding of the 9th European Conference on Artical Intelligence, pages 385{391, Stockholm, Sweden, August, 6{ Pitman Publishing. [RLS91] David W. Reed, Donald W. Loveland, and Bruce T. Smith. The near-horn approach to disjunctive logic programming. In L.-H. Eriksson, Hallnas, and P. Schroeder-Heister, editors, Proceedings of the Second International Workshop on Extension of Logic Programming (ELP '91), volume 596 of LNCS, pages 35{369, Stockholm, Sweden, January, 27{ Springer-Verlag. [SI92] Ken Satoh and Noboru Iwayama. A query evaluation method for abductive logic programming. In Krzysztof R. Apt, editor, Proceedings of the Joint International Conference and Symposium on Logic Programming, pages 671{685, Washington, D.C., USA, November, 9{ The MIT Press. 5

Reduction of abductive logic programs. In this paper we study a form of abductive logic programming which combines

Reduction of abductive logic programs to normal logic programs Francesca Toni, Robert A. Kowalski Department of Computing, Imperial College 18 Queen's Gate, London SW7 2BZ, UK fft, rakg@doc.ic.ac.uk Abstract