Set Constraints in Logic Programming

Abstract

In this note we will investigate a form of logic programming with constraints. The constraints that we consider will not be restricted to statements on real numbers as in CLP(R), see [15]. Instead our constraints will be arbitrary global constraints. The basic idea is that the applicability of a given rule is not predicated on the fact that individual variables satisfy certain constraints, but rather on the fact that the least model of the set rules that are ultimately applicable satisfy the constraint of the rule. Thus the role of clauses will be slightly different than in the usual Logic Programming with constraints. In fact, the paradigm we present is closely related to stable model semantics of general logic programming [13]. We will define the notion of a constraint model of our constraint logic program and show that stable models of logic programs as well as the supported models of logic programs are just special cases of constraint models of constraint logic programs.

Disjunctive Programs with Set Constraints

“…Set constraints were introduced by the authors in [13] and were further studied in [9,12]. The semantics for programs with set constraints introduced in [13] was a natural generalization of the proposal of [18]. An alternative semantics for those constraints (with the name abstract constraints) was studied in [19].…”

“…That is, propositional disjunctive programs capture the class of Σ P 2 problems [3]. In this paper, we investigate a class of programs, called set constraint disjunctive (SCD) logic programs, which generalize disjunctive logic programs [11] and set constraint logic programs [13]. Clauses in SCD logic programs are allowed to have heads which are a disjunction of monotone set constraints and bodies which are conjunctions of monotone and antimonotone set constraints.…”

On the Continuity of Gelfond-Lifschitz Operator and Other Applications of Proof-Theory in ASP

“…This extension has been further studied in [MR04,MNT08]. To keep things simple, we will limit our discussion to cardinality constraints only, although it is possible to extend our arguments to any class of convex constraints [LT05].…”

“…The stable semantics for CC -programs is defined via fixpoints of an analogue of the Gelfond-Lifschitz operator GL P ; see the details in [SNS02] and [MR04]. The operator in question is neither monotone nor antimonotone.…”

Logic Programs with Abstract Constraint Atoms: The Role of Computations

“…if we weaken requirements on computations and consider the class of weak computations, the general schema of defining f -models of programs yields characterizations of supported models and mr-answer sets [15] and so also deserves further attention.…”