The query topology in logic programming

Abstract

We define a topology called the query topology on each of two sets — the set of interpretations of a first order language and the set of models of any sentence in the language. We show that in each of these cases, the resulting topology is a perfectly normal, T ₄-space. In addition, the query topology on the set of interpretations is compact. We derive a necessary and sufficient condition for the query topology on the space of models of sentences to be compact and show, in addition, that the completions of canonical logic programs (cf. Jaffar and Stuckey [12]) have a compact space of models. The familiar T _P operator (cf. Lloyd [16]) may now be viewed as a function from a compact Hausdorff space to a compact Hausdorff space. We show that if P is either covered or function-free, then T _P is continuous in the query topology. The fact that the space of interpretations of a language is compact Hausdorff allows us to use the well-known theorems in topological fixed point theory to obtain heretofore unknown results on the semantics of logic programming. We present one such result — viz. a necessary and sufficient topological condition that guarantees the J-consistency (a notion defined in the paper) of completions of general logic programs.