Rule Ordering in Bottom-up Fixpoint Evaluation of Logic Programs

Abstract

Logic programs can be evaluated bottom-up by repeatedly applying all rules, in "iterations", until the fixpoint is reached. However, it is often desirable--- and in some cases, e.g. programs with stratified negation, even necessary to guarantee the semantics--- to apply the rules in some order. An important property of a fixpoint evaluation algorithm is that it does not repeat inferences; we say that such algorithms have the semi-naive property. The semi-naive algorithms in the literature do not address the issue of how to apply rules in a specified order while retaining the semi-naive property. We present two algorithms; one (GSN) is capable of dealing with a wide range of rule orderings but with a little more overhead than the usual semi-naive algorithm (which we call BSN). The other (OSN, and a variant, PSN) handles a smaller class of rule orderings, but with no overheads beyond those in BSN. This smaller class is sufficiently powerful to enforce the ordering required to implement stratified programs. We demonstrate that rule orderings can offer another important benefit: by choosing a good ordering, we can reduce the number of rule applications (and thus joins). We present a theoretical analysis of rule orderings. In particular, we identify a class of orderings, called cycle-preserving orderings, that minimize the number of rule applications (for all possible instances of the base relations) with respect to a class of orderings called fair orderings. We also show that