An incremental algorithm for generating all minimal models

The task of generating minimal models of a knowledge base 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 Ψk, then T has at most k minimal models, and all of them may be found in time O(lk2), where l is the length of the knowledge base; third, for an arbitrary knowledge base T, we can find the minimum k such that T belongs to Ψk in time polynomial in the size of T; and, last, where K is the class of all knowledge bases, it is the case that ⋃i=1∞Ψi=K, that is, every knowledge base belongs to some class in the hierarchy. The algorithm is incremental, that is, it is capable of generating one model at a time.