On the problem of computing the well-founded semantics

TL;DR

This paper investigates the computation of well-founded semantics for logic programs, proposing a new top-down algorithm for a specific class of programs that improves efficiency and achieves linear time complexity in many cases.

Contribution

It introduces a novel top-down implementation of the alternating-fixpoint algorithm for programs with at most one positive atom per rule, enhancing speed and optimality.

Findings

The new algorithm is faster than existing methods.

For a broad class of programs, the algorithm runs in linear time.

The approach is particularly effective for programs with restricted rule structures.

Abstract

The well-founded semantics is one of the most widely studied and used semantics of logic programs with negation. In the case of finite propositional programs, it can be computed in polynomial time, more specifically, in O(|At(P)|size(P)) steps, where size(P) denotes the total number of occurrences of atoms in a logic program P. This bound is achieved by an algorithm introduced by Van Gelder and known as the alternating-fixpoint algorithm. Improving on the alternating-fixpoint algorithm turned out to be difficult. In this paper we study extensions and modifications of the alternating-fixpoint approach. We then restrict our attention to the class of programs whose rules have no more than one positive occurrence of an atom in their bodies. For programs in that class we propose a new implementation of the alternating-fixpoint method in which false atoms are computed in a top-down fashion. We show that our algorithm is faster than other known algorithms and that for a wide class of programs it is linear and so, asymptotically optimal.