Well-founded and stable semantics of logic programs with aggregates

Abstract

In this paper, we present a framework for the semantics and the computation of aggregates in the context of logic programming. In our study, an aggregate can be an arbitrary interpreted second order predicate or function. We define extensions of the Kripke-Kleene, the well-founded and the stable semantics for aggregate programs. The semantics is based on the concept of a three-valued immediate consequence operator of an aggregate program. Such an operator approximates the standard two-valued immediate consequence operator of the program, and induces a unique Kripke-Kleene model, a unique well-founded model and a collection of stable models. We study different ways of defining such operators and thus obtain a framework of semantics, offering different trade-offs between precision and tractability. In particular, we investigate conditions on the operator that guarantee that the computation of the three types of semantics remains on the same level as for logic programs without aggregates. Other results show that, in practice, even efficient threevalued immediate consequence operators which are very low in the precision hierarchy, still provide optimal precision.

Dyna: Extending Datalog for Modern AI

“…Our goal for the Dyna engine is merely to mimic the original AI method; hence we are willing to return any supported model, accepting that the particular one we find (if any) will be sensitive to initial conditions and procedural choices, as before. This is quite different from usual practice in the logic programming community (see [54] for a review and synthesis), which when it permits non-stratified programs at all, typically identifies their semantics with one [29] or more [44] "stable models" or the intersection thereof [63,37], although in general the stable models are computationally intractable to find.…”

Strong and uniform equivalence of nonmonotonic theories – an algebraic approach

“…In a similar way, we can characterize strong and uniform equivalence of logic programs with aggregates as studied in [Pel04,PDBn04], and of modal theories with the semantics of extensions [DMT00], which yields a version of autoepistemic logic forming a precise modal match to the default logic. The reason is that in each case the semantics (stable models, extensions) is given in terms of an operator on a complete lattice and its approximating mapping.…”

“…It provides an algebraic account of several nonmonotonic logics including (normal) logic programming, default logic and autoepistemic logic, and allows one to state and prove properties of these logics in a uniform, general and abstract way [DMT03]. Recent applications of the approximation theory include the development of semantics of logic programs with aggregates [Pel04,PDBn04] and an abstract account of splitting theorems [VGD04b,VGD04a].…”

The KB Paradigm and Its Application to Interactive Configuration

“…FO(LFP) stands for the extension of FO with a least fixpoint construction (Immerman and Vardi 1997). Currently, the language of the IDP system in the project is FO(T, ID, Agg, arit, PF) (Denecker and Ternovska 2008;Pelov et al 2007): FO extended with types, definitions, aggregates, arithmetic and partial functions. Abusing notation, we will use FO(·) as an abbreviation for this language.…”