A hierarchy of tractable subsets for computing stable models

Abstract

Finding the stable models of a knowledge base is a significant computational problem in artificial intelligence. This task is at the computational heart of truth maintenance systems, autoepistemic logic, and default logic. Unfortunately, it is NP-hard. In this paper we present a hierarchy of classes of knowledge bases, Ω1,Ω2,..., with the following properties: first, Ω1 is the class of all stratified knowledge bases; second, if a knowledge base II is in Ωk, then II has at most k stable models, and all of them may be found in time O(lnk), where l is the length of the knowledge base and n the number of atoms in II; third, for an arbitrary knowledge base II, we can find the minimum k such that II belongs to Ωk in time polynomial in the size of II; and, last, where K is the class of all knowledge bases, it is the case that Ui=1∞Ωi = K, that is, every knowledge base belongs to some class in the hierarchy.