This paper presents for the first time the semantic foundations of CLP in a self-contained and complete package. The main contributions are threefold. First, we extend the original conference paper by presenting definitions and basic semantic constructs from first principles, giving new and complete proofs for the main lemmas. Importantly, we clarify which theorems depend on conditions such as solution compactness, satisfaction completeness and independence of constraints. Second, we generalize the original results to allow for incompleteness of the constraint solver. This is important since almost all CLP systems use an incomplete solver. Third, we give conditions on the (possibly incomplete) solver which ensure that the operational semantics is confluent, that is, has independence of literal scheduling.

Constraint Logic Programming (CLP) is a merger of two declarative paradigms: constraint solving and logic programming. Although a relatively new field, CLP has progressed in several quite different directions. In particular, the early fundamental concepts have been adapted to better serve in different areas of applications. In this survey of CLP, a primary goal is to give a systematic description of the major trends in terms of common fundamental concepts. The three main parts cover the theory, implementation issues, and programming for applications.

. We address the problem of determining those constraint domains A for which the traditional logic programming semantics and the constraint logic programming semantics CLP (A) coincide. This reduces to a study of non-standard models of Clark's axioms and the notion of solution compactness introduced in the CLP scheme. The results of this study include the proof of the existence of a free product in the class of algebras defined by Clark's axioms, a characterization of when Clark's axioms form a model complete theory, and a limited characterization of those models of Clark's axioms which form solution compact constraint domains. 1 Introduction Appropriate semantics for definite logic programs are now largely agreed upon [17]. They involve a completed program, SLD-resolution, a one-step consequence function, a least Herbrand model and numerous relationships between them: soundness and completeness of SLD-refutations, soundness and completeness of the negation-asfailure rule, .....