We study properties of programs with monotone and convex constraints. We extend to these formalisms concepts and results from normal logic programming. They include the notions of strong and uniform equivalence with their characterizations, tight programs and Fages Lemma, program completion and loop formulas. Our results provide an abstract account of properties of some recent extensions of logic programming with aggregates, especially the formalism of lparse programs. They imply a method to compute stable models of lparse programs by means of off-the-shelf solvers of pseudo-boolean constraints, which is often much faster than the smodels system.
p04 and pdb06 studied a more general class of aggregates and developed a systematic theory of aggregates in logic programming based on the approximation theory @cite_0 . The resulting theory covers not only the stable models semantics but also the supported-model semantics and extensions of 3-valued Kripke-Kleene and well-founded semantics. The formalism introduced and studied by p04 and pdb06 allows for arbitrary aggregates (not only monotone ones) to appear in the bodies of rules. However, it does not allow for aggregates to appear in the heads of program clauses. Due to differences in the syntax and the scope of semantics studied there is no simple way to relate p04 and pdb06 formalism to programs with monotone (convex) constraints. We note though that programs with abstract monotone constraints with the heads of rules of the form @math can be viewed almost literally as programs in the formalism by p04 and pdb06 and that they have the same stable models according to the definitions we used in this paper and those by p04 and pdb06 .
@cite_0
1598820892 2064864283 2063061546 50549667
In this paper, we propose an extension of the well-founded and stable model semantics for logic programs with aggregates. Our approach uses Approximation Theory, a fixpoint theory of stable and well-founded fixpoints of non-monotone operators in a complete lattice. We define the syntax of logic programs with aggregates and define the immediate consequence operator of such programs. We investigate the well-founded and stable semantics generated by Approximation Theory. We show that our approach extends logic programs with stratified aggregation and that it correctly deals with well-known benchmark problems such as the shortest path program and the company control problem. The addition of aggregates has been one of the most relevant enhancements to the language of answer set programming (ASP). They strengthen the modelling power of ASP in terms of natural and concise problem representations. Previous semantic definitions typically agree in the case of non-recursive aggregates, but the picture is less clear for aggregates involved in recursion. Some proposals explicitly avoid recursive aggregates, most others differ, and many of them do not satisfy desirable criteria, such as minimality or coincidence with answer sets in the aggregate-free case. In this paper we define a semantics for programs with arbitrary aggregates (including monotone, antimonotone, and nonmonotone aggregates) in the full ASP language allowing also for disjunction in the head (disjunctive logic programming - DLP). This semantics is a genuine generalization of the answer set semantics for DLP, it is defined by a natural variant of the Gelfond-Lifschitz transformation, and treats aggregate and non-aggregate literals in a uniform way. This novel transformation is interesting per se also in the aggregate-free case, since it is simpler than the original transformation and does not need to differentiate between positive and negative literals. We prove that our semantics guarantees the minimality (and therefore the incomparability) of answer sets, and we demonstrate that it coincides with the standard answer set semantics on aggregate-free programs. Moreover, we carry out an in-depth study of the computational complexity of the language. The analysis pays particular attention to the impact of syntactical restrictions on programs in the form of limited use of aggregates, disjunction, and negation. While the addition of aggregates does not affect the complexity of the full DLP language, it turns out that their presence does increase the complexity of normal (i.e., non-disjunctive) ASP programs up to the second level of the polynomial hierarchy. However, we show that there are large classes of aggregates the addition of which does not cause any complexity gap even for normal programs, including the fragment allowing for arbitrary monotone, arbitrary antimonotone, and stratified (i.e., non-recursive) nonmonotone aggregates. The analysis provides some useful indications on the possibility to implement aggregates in existing reasoning engines. We propose a semantics for aggregates in deductive databases based on a notion of minimality. Unlike some previous approaches, we form a minimal model of a program component including aggregate operators, rather than insisting that the aggregate apply to atoms that have been fully determined or that aggregate functions are rewritten in terms of negation. In order to guarantee the existence of such a minimal model we need to insist that the domains over which we are aggregating are complete lattices and that the program is in a sense mono- tonic. Our approach generalizes previous approaches based on the well-founded semantics and various forms of stratification. We are also able to handle a large variety of monotonic (or pseudo-monotonic) aggregate functions. 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. It is known that programs with generalized atoms that have monotonic or antimonotonic satisfaction patterns do not increase complexity with respect to programs with simple atoms if satisfaction of the generalized atoms can be decided in polynomial time under most semantics. It is also known that generalized atoms that are nonmonotonic being neither monotonic nor antimonotonic can, but need not, increase the complexity by one level in the polynomial hierarchy if non-disjunctive programs under the FLP semantics are considered. In this paper we provide the precise boundary of this complexity gap: programs with convex generalized atom never increase complexity, while allowing a single non-convex generalized atom under reasonable conditions always does. We also discuss several implications of this result in practice.