A closer look at declarative interpretations

Abstract

Three semantics have been proposed as the most promising candidates for a declarative interpretation for logic programs and pure Prolog programs: the least Herbrand model, the least term model, i.e., the C -semantics, and the I -semantics. Previous results show that a strictly increasing information ordering between these semantics exists for the class of all programs. In particular, the I -semantics allows us to model the computed answer substitutions, which is not the case for the other two. We study here the relationship between these three semantics for specific classes of programs. We show that for a large class of programs (which is Turing complete), these three semantics are isomorphic. As a consequence, given a query, we can extract from the least Herbrand model of a program in this class all computed answer substitutions. However, for specific programs the least Herbrand model is tedious to construct and reason about because it contains “ill-typed” facts. Therefore, we propose a fourth semantics that associates with a “correctly typed” program the “well-typed” subset of its least Herbrand model. This semantics is used to reason about partial correctness and absence of failures of correctly typed programs. The results are extended to programs with arithmetic.