Computing intersections of Horn theories for reasoning with models

Abstract

Model-based reasoning has been proposed as an alternative form of representing and accessing logical knowledge bases. In this approach, a knowledge base is represented by a set of characteristic models. In this paper, we consider computational issues when combining logical knowledge bases, which are represented by their characteristic models; in particular, we study taking their logical intersection. We present low-order polynomial time algorithms or prove intractability for the major computation problems in the context of knowledge bases which are Horn theories. In particular, we show that a model of the intersection Σ of Horn theories Σ 1,…, Σ l , represented by their characteristic models, can be found in linear time, and that some characteristic model of Σ can be found in polynomial time. Moreover, we present an algorithm which enumerates all the models of Σ with polynomial delay. The analogous problem for the characteristic models is proved to be intractable, even if the possible exponential size of the output is taken into account. Furthermore, we show that approximate computation of the set of characteristic models is difficult as well. Nonetheless, we show that deduction from Σ is possible for a large class of queries in polynomial time, while abduction turns out to be intractable. We also consider a generalization of Horn theories, and prove negative results for the basic questions, indicating that an extension of the positive results beyond Horn theories is not immediate.