Optimal compression of propositional Horn knowledge bases: complexity and approximation

Abstract

Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge bases. The standard approach to this problem, consisting in the removal of redundant rules from a knowledge base, leads to an “irredundant” but not necessarily optimal knowledge base. We prove here that the number of rules in any irredundant Horn knowledge base involving n propositional variables is at most n − 1 times the minimum possible number of rules. Therefore, the quadratic time transformation of an arbitrary Horn production rule base to an equivalent irredundant and prime one (presented in [9]) provides a reasonable approximation algorithm. In order to formalize the optimal compression problem, we define a Boolean function of a knowledge base as being the function whose set of true points is the set of models of the knowledge base. In this way the optimal compression of production rule knowledge bases becomes a problem of Boolean function minimization. In this paper we prove that the minimization of Horn functions (i.e. Boolean functions associated to Horn knowledge bases) is NP-complete.