The Defining Power of Stratified and Hierarchical Logic Programs

We investigate the defining power of stratified and hierarchical logic programs. As an example for the treatment of negative information in the context of these structured programs we also introduce a stratified and hierarchical closed-world assumption. Our analysis tries to relate the defining power of stratified and hierarchical programs (with and without an appropriate closed-world assumption) very precisely to notions and hierarchies in classical definability theory. Stratified and hierarchical logic programs are two well-known and typical candidates of what one may more generally denote as structured programs. In both cases we have to deal with normal logic programs which satisfy certain syntactic conditions with respect to the occurrence of negative literals. Recently they have gained a lot of importance in connection with the search for nice declarative semantics for logic programs and the treatment of negative information in logic programming (e.g., Lloyd [10]). Stratified programs were introduced into logic programming by Apt, Blair, and Walker [2] and van Gelder [17] not long ago. In mathematical logic, however, theories of this kind have been studied for more than 20 years under the general theme of iterated inductive definability. Indeed, stratified programs can be understood as systems for (finitely) iterated inductive definitions where the definition clauses are of very low logical complexity. The notion of hierarchical program (e.g., Clark [6], Shepherdson [15]), on the other hand, is motivated by database theory and tries to reflect the idea of iterated explicit definability by simple principles. From a conceptual point of view we are interested in the relationship between logic programming, inductive definability and equational definability. By making use of these connections we obtain a uniform and perspicuous approach to a series of interesting questions in this area. Address correspondence to Gerhard Jäger or Robert F. Stärk, Institut für Informatik und angewandte Mathematik, Länggassstrasse 51, CH-3012 Bern, Switzerland. Email: 〈[email protected]〉 or 〈[email protected]〉 Appeared in: J.of Logic Programming, 15 (1&2): 55–77, 1993. 2 GERHARD JÄGER AND ROBERT F. STÄRK The plan of this paper is as follows: Section 1 introduces some basic notions. Sections 2 and 3 present the relevant concepts from classical definability theory and are concerned with various forms of definability over Herbrand universes of first-order languages. In Section 4 we characterize the defining power of stratified programs. Among other things we prove that the arithmetically definable subsets of the nonnegative integers comprise the defining power of suitable stratified programs with the stratified closed-world assumption. Section 5 is then devoted to the study of hierarchical programs. It is shown that definite hierarchical programs pin down exactly the so-called term-definable relations. This is in sharp contrast to the defining power of arbitrary hierarchical logic programs which is shown to be equivalent to that of definite programs. Finally the hierarchical programs with the hierarchical closed-world assumption represent a class of intermediate strength. We will see that they exactly define the equationally definable relations.