Well-founded and Stable Semantics of Logic Programs with Aggregates

TL;DR

This paper develops a comprehensive framework for defining and computing well-founded and stable semantics in logic programs with aggregates, accommodating various trade-offs between precision and computational efficiency.

Contribution

It introduces a new semantics framework for aggregate logic programs based on three-valued consequence operators, extending classical semantics to aggregates.

Findings

Unique models for Kripke-Kleene, well-founded, and stable semantics are established.

Conditions are identified under which aggregate semantics computation remains as efficient as standard logic programs.

Low-precision operators can still yield optimal results in practice.

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 three-valued immediate consequence operators which are very low in the precision hierarchy, still provide optimal precision.