An incremental algorithm for generating all minimal models

AbstractThe 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

Reasoning with minimal models: efficient algorithms and applications

AbstractReasoning with minimal models is at the heart of many knowledge-representation systems. Yet it turns out that this task is formidable, even when very simple theories are considered. In this paper, we introduce the elimination algorithm, which performs, in linear time, minimal model finding and minimal model checking for a significant subclass of positive CNF theories which we call positive head-cycle-free (HCF) theories. We also prove that the task of minimal entailment is easier for positive HCF theories than it is for the class of all positive CNF theories. Finally, we show how variations of the elimination algorithm can be applied to allow queries posed on disjunctive deductive databases and disjunctive default theories to be answered in an efficient way