A Demand-driven Algorithm for Generating Minimal Models

The task of generating minimal models of a knowledge base is a significant computational problem in artificial intelligence. This task is at the computational heart of diagnosis systems like truth maintenance systems, and of nonmonotonic systems like autoepistemic logic, default logic, and disjunctive logic programs. Unfortunately, it is NP-hard. In this paper we present a hierarchy of classes of knowledge bases, Ψ₁, Ψ₂, ..., with the following properties: first, Ψ₁ is the class of all Horn knowledge bases; second, if a knowledge base T is in Ψ_k, then T has at most k minimal 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 T; third, for an arbitrary knowledge base T, we can find the minimum k such that T belongs to Ψ_k in time polynomial in the size of T; and, last, where K is the class of all knowledge bases, it is the case that U^∞_i=1Ψ_i = K, that is, every knowledge base belongs to some class in the hierarchy. The algorithm is demand-driven, that is, it is capable of generating one model at a time.