Concept logics with function symbols

Abstract

Constrained resolution allows the incorporation of domain specific problem solving methods into the classical resolution principle. Firstly, the domain specific knowledge is represented by a restriction theory. One then starts with formulas containing so-called restricted quantifiers, written as forall_{X:R} F and exists_{X:R} F, where X is a set of variables and the restriction R is used to encode domain specific knowledge by filtering out some assignments to the variables in X. Formulas with restricted quantifiers can be translated into clauses which consist of a (classical) clause together with a restriction. In order to attain a refutation procedure which is based on such clauses one needs algorithms to decide satisfiability and validity of restrictions w.r.t. the given restriction theory. Recently, concept logics have been proposed where the restriction theory is defined by terminological logics. However, in this approach problems have been assumed to be given as sets of clauses with restrictions and not in terms of formulas with restricted quantifiers. For this special case algorithms to decide satisfiability and validity of restrictions have been given. In this paper we will show that things become much more complex if problems are given as sets of formulas with restricted quantifiers. The reason for this is due to the fact that Skolem function symbols are introduced when translating such formulas into clauses with restrictions. While we will give a procedure to decide satisfiability of restrictions containing function symbols, validity of such restrictions turns out to be undecidable. Nevertheless, we present an application of concept logics with function symbols, namely their usef or generating (partial) answers to queries.