A nondeterministic well-founded semantics

Abstract

In order to define the meaning of a collection of rules forming a database or logic program, we have to consider a number of alternative interpretations. In the case of stratified and locally stratified programs, researchers agree that a single such interpretation captures the intended semantics [16]. Nevertheless, nonstratified datalog databases and logic programs can be seen as descriptions of a set of “alternative scenarios”. Sacca and Zaniolo [20] have shown that the nondeterministic nature of these programs can be used to model several useful queries such as those involving thechoice operator. They introducedpartial stable models, a semantics that exhibits the desired nondeterministic behavior. This work opened the problem of finding efficient ways to compute these nondeterministic scenarios. Papadimitriou and Yannakakis [18] introduced a tie-breaking procedure that nondeterministically computes fixpoints for some programs and has polynomial data complexity. However, this algorithm does not handle many programs includingchoice programs. In this paper, we introduce the notion ofwell-founded hypothesis, an intuitive account based on hypothetical reasoning that captures the same semantics as Sacca and Zaniolo's partial stable models. We introduce a notion of linearity that can be used in a skeptical or nondeterministic fashion. We show that the skeptical case corresponds to thewell-founded semantics and that the nondeterministic case computes a sound subclass of well-founded hypotheses. We show that this latter subclass has polynomial data complexity, correctly handleschoice programs, and is universally defined. We develop a simple nondeterministic procedure that computes these linear hypotheses, and we extend it to compute a strict superclass of the well-founded tie-breaking fixpoints in polynomial time.