On the Relationship Between CWA, Minimal Model and Minimal Herbrand Model Semantics

The purpose of this paper is to compare three types of non-monotonic semantics: (a) proof-theoretic semantics based on the closed world assumption, (b) model-theoretic semantics based on the notion of a minimal model and (c) model-theoretic semantics based on the notion of a minimal Herbrand model. All of these semantics capture the non-monotonicity of common sense reasoning, i.e. the ability to withdraw conclusions after some new information is added to the original theories, and proved to be powerful enough to handle most examples of such reasoning presented in the literature. However, since these formalizations are based on different intuitions and often produce different results, the problem of understanding the relationship between them is especially important. In the first part of the paper we concentrate on the class of positive logic programs, also called definite theories. Although the three semantics usually differ for universal sentences, our main result shows that they alwa..