Apt, K., Blair, H., and Walker, A., Towards a theory of declarative knowledge. Foundations of Deductive Databases and Logic Programming, J. Minker (ed.), Morgan Kau#man, Los Altos, USA, 1988, 89-142.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... y] does there exist a stable p model M for P such that M(A) x; y] Brave p Reasoning) We have analyzed the complexity of the above decisional problems for the classes of QDLP which are generalizations of the corresponding DLP classes, the other fragments The notion of Stratified Negation [1] and of Head Cycle Free Disjunction [4, 5] are extended from traditional DLP to QDLP in a straightforward manner. Their formal definitions are given in Appendix A. 10 being of low interest from the practical point of view. The results for non disjunctive and disjunctive quantitative programs are ....

K.R. Apt, H.A. Blair, and A. Walker. Towards a Theory of Declarative Knowledge. Foundations of Deductive Databases and Logic Programming, Minker, J. (ed.), Morgan Kaufmann, Los Altos, 1987.

....lesser path(X, Y, C) a lesser path(X, Y, C) path(X, Y, C) path(X, Y, C # ) C # C. where a lesser path is a new predicate symbol not appearing elsewhere in the program. This has formal semantics because, by rewriting the min predicates by means of negation, we get a stratified program [3,36]. However, a straightforward evaluation of such a stratified program would materialize the predicate path and then choose the smallest cost tuple for every pair of nodes x and y. There are two problems with this approach: first, it is very ine#cient, and second, the computation could be ....

....a program P whose head predicate symbols belong to the same recursive component of the graph G P is called sub program of P . Sub programs can be (partially) ordered on the base of the dependencies among predicate symbols. A program P is stratified if G P does not contain cycles with marked arcs [3]. Stratified programs have a total well founded model which coincides with the unique stable model; this model is also called perfect model or stratified model [25] The perfect model of a stratified program can be computed by partitioning the program into an ordered number of suitable subprograms ....

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Apt, K., Blair, H., and Walker, A., Towards a theory of declarative programming. Foundations of Deductive Databases and Logic Programming (J. Minker ed.), 89--148 (1988).

....semantics for Horn clause logic programming has been developed, based on classical logic ( 16] 2] But it can not deal adequately with negations when they are allowed in clause bodies. Two kinds of generalizations have been proposed to deal with this problem. The best known is stratification [1], 17] Here the kind of logic programs one is allowed to write is restricted; recursions through negations are forbidden. For such programs there is a fixed point semantics generalizing pure Horn clause semantics, yielding a classical, two valued semantics. The other kind of generalization ....

....gives us the following. Proposition 5.1 Suppose the lower upper pair W # W # and the three valued work space W are associated, and P is a program that is admissible in W. Then the P extension of W and the P extension of W # W # are also associated. 6 Weak stratified semantics In [1] and [17] and earlier in [3] stratification was introduced into logic programming, based on the idea that relations must be completely defined before they can be used negatively. Not all programs are stratifiable, and for those that are the stratification need not be unique. Associated with the ....

K. R. Apt, H. A. Blair, A. Walker, Towards a theory of declarative knowledge, Foundations of Deductive Databases and Logic Programming, J. Minker ed, Morgan Kaufmann, Los Altos (1987).

....to these) The instance of the proof theory for partial stable models can also be used to compute stable models for those cases when the set of all partial stable models coincides with the set of all stable models. In LP, this is the case, in particular, when the given logic program is strati ed [2] or, more generally, order consistent [42] The abstract proof theory thus provides a common basic computational framework for the various argumentation semantics. We use the abstract proof theory as a basis for developing top down, goal oriented, abstract proof procedures for argumentation. ....

....However, any proof procedure for computing the admissibility semantics can serve as a proof procedure for computing stable models in all cases where the set of all partial stable models is the same as the set of all stable models. This is the case, e.g. when the given logic program is strati ed [2] or order consistent [42] Note that the programs in examples 7 and 12 are neither strati ed nor order consistent. 2 A attacks not p i A attacks fnot pg. 11 The program in example 4 is order consistent but not strati ed. Therefore, our proof procedure for computing the admissibility semantics ....

K.R. Apt, H. Blair, A. Walker. Towards a theory of declarative knowledge. Foundations of deductive databases and logic programming, J. Minker, editor, Morgan Kaufmann, Los Altos, CA (1988)

.... y] does there exist a stable p model M for P such that M(A) x; y] Brave p Reasoning) We have analyzed the complexity of the above decisional problems for the classes of QDLP which are generalizations of the corresponding DLP classes, the other fragments 2 The notion of Stratified Negation [1] and of Head Cycle Free Disjunction [4, 5] are extended from traditional DLP to QDLP in a straightforward manner. Their formal definitions are given in Appendix A. being of low interest from the practical point of view. The results for non disjunctive and disjunctive quantitative programs are ....

K.R. Apt, H.A. Blair, and A. Walker. Towards a Theory of Declarative Knowledge. Foundations of Deductive Databases and Logic Programming, Minker, J. (ed.), Morgan Kaufmann, Los Altos, 1987.

....rules have at most one subgoal. A unit uni rule program (UU) is both unit and uni rule. Likewise, a unit uni appearance program (UA) is unit and uni appearance. The algorithms we present to compute the stable and well founded semantics for these classes operate on their dependency digraphs as in [10] and [1] The algorithms are closely related to one another in that each deals with the acyclic and the cyclic sections of the digraph, separately. The syntactic restrictions on these classes of programs induce obvious properties on their corresponding digraphs. The two classes we present for ....

H. Blair K. R. Apt and A. Walker. Towards a theory of declarative knowledge. Foundations of Deductive Databases and Logic Programming, pages 89--148, 1988.

....corresponding strongly optimal worlds. Since optimization is a non monotonic notion for example, adding a new edge to the graph could change the shortest path we need to restrict programs so that recursion through optimization and relaxation predicates is well founded (or locally stratified) [1]. While there are close connections with semantics of negation as failure, we prefer a more direct semantics, especially one that takes into account the ordering among solutions. One contribution of this paper is in showing to give such a semantics using simple semantic concepts from modal logic. ....

....the operational characterizations of various semantics for negation. A standard approach in those contexts is to investigate syntactic restrictions on programs for which the operational semantics can be proven to be sound and complete. A very popular syntactic restriction is that of stratification [1]. Definition 4.20 A program P is said to be stratified if the following holds: There is a mapping f from the set of O predicates to the set f1; ng, for some least n, such that if an instance of an O predicate P 1 appears in the body of a clause defining an O predicate P 2 , then f(P 1 ....

K.R. Apt, H.A. Blair, and A. Walker. Foundations of Deductive Databases and Logic Programming, chapter Towards a Theory of Declarative Knowledge. MIT Press, 1988.

....and we use the following for IDB relations: ffi; d j= R i (a) iff there is a rule r with head R i and ffi; d j= OE r (a) The output of a Datalog query is called the least model. 2.2.4 Stratified Datalog Queries. Each stratified Datalog [Abiteboul et al. 1995; Chandra and Harel 1982; Apt et al. 1988; Doets 1994; Ullman 1989] program is a composition through negation of Datalog programs. We explain that composition in the following way. We call semipositive those Datalog programs that allow negation of EDBs. Semantically each semipositive Datalog program Pi is a mapping from input databases ....

Apt, K., Blair, H., and Walker, A. 1988. Foundations of Deductive Databases and Logic Programming. Morgan Kaufmann.

....to these) The instance of the proof theory for partial stable models can also be used to compute stable models for those cases when the set of all partial stable models coincides with the set of all stable models. In LP, this is the case, in particular, when the given logic program is stratified [2] or, more generally, order consistent [42] The abstract proof theory thus provides a common basic computational framework for the various argumentation semantics. We use the abstract proof theory as a basis for developing top down, goal oriented, abstract proof procedures for argumentation. These ....

....However, any proof procedure for computing the admissibility semantics can serve as a proof procedure for computing stable models in all cases where the set of all partial stable models is the same as the set of all stable models. This is the case, e.g. when the given logic program is stratified [2] or order consistent [42] Note that the programs in examples 7 and 12 are neither stratified nor order consistent. 2 A attacks not p iff A attacks fnot pg. The program in example 4 is order consistent but not stratified. Therefore, our proof procedure for computing the admissibility semantics ....

K.R. Apt, H. Blair, A. Walker. Towards a theory of declarative knowledge. Foundations of deductive databases and logic programming, J. Minker, editor, Morgan Kaufmann, Los Altos, CA (1988)

....and contains Delta as a proper subset. q:e:d: It follows directly from the definitions that the abstract notions of stratification and order consistency generalise the notions of stratification and order consistency for logic programming: Theorem 7. 4 If P is a stratified logic program [4], then the corresponding assumptionbased framework hP; Ab; i is stratified. Similarly, if P is an order consistent logic program [51] then the corresponding assumption based framework hP; Ab; i is orderconsistent. 8 Related work The role of argumentation in human reasoning has been studied both ....

K.R. Apt, H. Blair, A. Walker, Towards a theory of declaratice knowledge. Foundations of deductive databases and logic programming, J. Minker, editor, Morgan Kaufmann, Los Altos, CA (1988)

....the form R) with i 0. In a Datalog : rule as above, A 0 (t 0 ) is called the head and A 1 (t 1 ) An (t n ) the body of the rule. There are several versions of Datalog : differing in the semantics for negation. The main semantics for negation are the stratified semantics [CH85, ABW88, Lif88, Gel86] and the well founded semantics [GRS88, GRS91, Gel89, BF88, Prz89, Prz90] which we do not describe here. The following is a Datalog : program which defines the complement of transitive closure under both semantics: T (x; y) G(x; y) T (x; y) G(x; z) T (z; y) T (x; y) T ....

K.R. Apt, H. Blair, and A. Walker, Towards a theory of declarative knowledge, Foundations of Deductive Databases and Logic Programming (Los Altos, CA) (J. Minker, ed.), Morgan Kaufmann, Los Altos, CA, 1988, pp. 89--148.

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Apt, K., Blair, H., and Walker, A., Towards a theory of declarative knowledge. Foundations of Deductive Databases and Logic Programming, J. Minker (ed.), Morgan Kau#man, Los Altos, USA, 1988, 89-142.

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Apt, K., Blair, H., and Walker, A., Towards a theory of declarative knowledge. Foundations of Deductive Databases and Logic Programming, J. Minker (ed.), Morgan Kau#man, Los Altos, USA, 1988, 89-142.

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