On the relationship between CWA, minimal model, and minimal herbrand model semantics

Abstract

The purpose of this article is to compare three types of nonmonotonic 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 nonmonotonicity of commonsense reasoning, that is, 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 article 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 always coincide for existential queries. This result is particularly significant in view of the fact that in many applications existential queries are of main interest. It also plays an important role in the problem of finding a suitable declarative semantics for logic programs. In the second part we investigate arbitrary universal theories and we show that subtle differences exist between the three approaches and therefore no straightforward generalization of the results from the first part can be obtained.