Logic programs with monotone abstract constraint atoms

Mathematical Programming, 1999

A mathematical programming model may contain qualitative as well as quantitative elements. One may, for example, wish to combine a rule base with numerical constraints. This raises the issue of how to represent logical constraints in inequality form so that they have a useful linear relaxation. We provide a simple recursive procedure that generates a convex hull description of any logical condition that can be written as a "cardinality rule", which seems to be a form that occurs often in practice. A cardinality rule asserts that if at least k of the propositions A 1 , . . . , A m are true, then at least of the propositions B 1 , . . . , B n are true. The main result of the paper is that the procedure in fact provides a convex hull description.

Constraints, 2000

Set constraints (SC) are logical formulae in which atoms are inclusions between set expressions. Those set expressions are built over a signature , variables and various set operators. On a semantical point of view, the set constraints are interpreted over sets of trees built from and the inclusion symbol is interpreted as the subset relation over those sets. By restricting the syntax of those formulae and/or the set of operators that can occur in set expressions, different classes of set constraints are obtained. Several classes have been proposed and studied for some problems such as satisfiability and entailment. Among those classes, we focus in this article on the class of definite SC's introduced by Heintze and Jaffar, and the class of co-definite SC's studied by Charatonik and Podelski. In spite of their name, those two classes are not dual from each other, neither through inclusion inversion nor through complementation. In this article, we propose an extension for each of those two classes by means of an intentional set construction, so called membership expression. A membership expression is an expression {x | (x)}. The formula (x) is a positive first-order formula built from membership atoms t ∈ S in which S is a set expression. We name those two classes respectively generalized definite and generalized co-definite set constraints. One of the main point concerning those so-extended classes is that the two generalized classes turn out to be dual through complementation. First, we prove in this article that generalized definite set constraints is a proper extension of the definite class, as it is more expressive in terms of sets of solutions. But we show also that those extensions preserve some main properties of the definite and co-definite class. Hence for instance, as definite set constraints, generalized definite SC's have a least solution whereas the generalized co-definite SC's have a greatest solution, just as co-definite ones. Furthermore, we devise an algorithm based on tree automata that solves the satisfiability problem for generalized definite set constraints. Due to the dualization, the algorithm solves the satisfiability problem for generalized co-definite set constraints as well. This algorithm proves first that for those generalized classes, the satisfiability problem remains DEXPTIME-complete. It provides also a proof for regularity of the least solution of generalized definite constraints and so, by dualization for the greatest solution for the generalized co-definite SC's.

Proceedings of the 35th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, 2016

Journal of Artificial Intelligence Research, 2007

In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negation-as-failure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly well-supported models, are generalizations of the ...

2006

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. Our def...

1998

Abstract We compare two (apparently) rather different set-based constraint languages, and we show that, in spite of their different origins and aims, there are large classes of constraint formulae for which both proposals provide suitable procedures for testing constraint satisfiability with respect to a given privileged interpretation.

2004

All major semantics of normal logic programs and normal logic programs with aggregates can be described as fixpoints of the one-step provability operator or of operators that can be derived from it. No such systematic operator-based approach to semantics of disjunctive logic programs has been developed so far. This paper is the first step in this direction. We formalize the concept of one-step-provability for disjunctive logic programs by means of non-deterministic operators on the lattice of interpretations. We establish characterizations of models, minimal models, supported models and stable models of disjunctive logic programs in terms of pre-fixpoints and fixpoints of non-deterministic immediateconsequence operators and their extensions to the four-valued setting. We develop our results for programs in propositional language extended with monotone aggregate atoms. For the most part, our concepts, results and proof techniques are algebraic, which opens a possibility for further generalizations to the abstract algebraic setting of non-deterministic operators on complete lattices.

In recent years, Answer Set Programming (ASP), logic programming under the stable model or answer set semantics, has seen several extensions by generalizing the notion of an atom in these programs: be it aggregate atoms, HEX atoms, generalized quantifiers, or abstract constraints, the idea is to have more complicated satisfaction patterns in the lattice of Herbrand interpretations than traditional, simple atoms. In this paper we refer to any of these constructs as generalized atoms. Several semantics with differing characteristics have been proposed for these extensions, rendering the big picture somewhat blurry. In this paper, we analyze the class of programs that have convex generalized atoms (originally proposed by Liu and Truszczyński in [10]) in rule bodies and show that for this class many of the proposed semantics coincide. This is an interesting result, since recently it has been shown that this class is the precise complexity boundary for the FLP semantics. We investigate whether similar results also hold for other semantics, and discuss the implications of our findings.

In Proceedings of the International Workshop on …, 2001

We investigate cardinality constraints of the form M,! K, where M is a set and is one of the comparison operators\=,", or\": a model of such a constraint is required to contain\ exactly, at most", or\ at least", respectively, K elements of M. Applications dealing with ...

SIAM Journal on Computing, 2008

The constraint satisfaction probem (CSP) is a well-acknowledged framework in which many combinatorial search problems can be naturally formulated. The CSP may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints, in front of which all variables are existentially quantified. The quantified constraint satisfaction problem (QCSP) is the generalization of the CSP where universal quantification is permitted in addition to existential quantification. The general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language, which is a set of relations that can be used to express constraints. This paper introduces collapsibility, a technique for deriving positive complexity results on the QCSP. In particular, this technique allows one to show that, for a particular constraint language, the QCSP reduces to the CSP. We show that collapsibility applies to three known tractable cases of the QCSP that were originally studied using disparate proof techniques in different decades: QUANTIFIED 2-SAT (Aspvall, Plass, and Tarjan 1979), QUANTIFIED HORN-SAT (Karpinski, Kleine Büning, and Schmitt 1987), and QUANTIFIED AFFINE-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals common structure among these cases, which are describable by constraint languages over a two-element domain. In addition to unifying these known tractable cases, we study constraint languages over domains of larger size.