Fully abstract compositional semantics for an algebra of logic programs

Abstract

A simple extension of logic programming consists of introducing a set of basic program composition operations, which form an algebra of logic programs with interesting properties for reasoning about programs and program compositions. From a programming perspective, the operations enhance the expressive power of the logic programing paradigm by supporting a wealth of programming techniques, ranging from software engineering to artificial intelligence applications. This paper focuses on the semantics of program composition operations. It is shown that the immediate consequence operator T( P) properly characterises the intended meaning of a program P when considering compositions of programs. More precisely, it is shown that the T( P) semantics is both compositional and fully abstract w.r.t. the set of composition operations of the algebra. This implies that the T( P) semantics induces the coarsest equivalence relation on programs (subsumption-equivalence) and that any other semantics of programs must induce the same equivalence relation to be both compositional and fully abstract w.r.t. the whole set of operations of the algebra. The T( P) semantics is then related to other well known semantics for logic programs which induce coarser equivalence relations. In particular, three equivalence relations, originally studied by Maher (1988), are considered: Weak subsumption-equivalence, logical equivalence and least Herbrand model equivalence. It is shown that the chain of equivalence relations composed by weak subsumption-equivalence, logical equivalence and least Herbrand model equivalence coincides with the chain of fully abstract compositional equivalence relations for proper subsets of the operations of the algebra, obtained by dropping one operation at a time from the set of compositions.