text: 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.
  @cite_24: An important limitation of traditional logic programming as a knowledge representation tool, in comparison with classical logic, is that logic programming does not allow us to deal directly with incomplete information. In order to overcome this limitation, we extend the class of general logic programs by including classical negation, in addition to negation-as-failure. The semantics of such extended programs is based on the method of stable models. The concept of a disjunctive database can be extended in a similar way. We show that some facts of commonsense knowledge can be represented by logic programs and disjunctive databases more easily when classical negation is available. Computationally, classical negation can be eliminated from extended programs by a simple preprocessor. Extended programs are identical to a special case of default theories in the sense of Reiter.
summary: flp04 developed the theory of disjunctive logic programs with aggregates. Similarly as p04 and pdb06 , flp04 do not allow for aggregates to appear in the heads of program clauses. This is one of the differences between that approach and programs with monotone (convex) constraints we studied here. The other major difference is related to the postulate of the minimality of stable models (called answer sets in the context of the formalism considered by flp04 ). In keeping with the spirit of the original answer-set semantics @cite_24 , answer sets of disjunctive programs with aggregates, as defined by flp04 , are minimal models. Stable models of programs with abstract constraints do not have this property. However, for the class of programs with abstract monotone constraints with the heads of rules of the form @math the semantics of answer sets defined by flp04 coincides with the semantics of stable models by mt04 and mnt03,mnt06 .