Models and equality for logical programming

We argue that some standard tools from model theory provide a better semantic foundation than the more syntactic and operational approaches usually used in logic programming. In particular, we show how initial models capture the intended semantics of both functional and logic programming, as well as their combination, with existential queries having logical variables (for both functions and relations) in the presence of arbitrary user-defined abstract data types, and with the full power of constraint languages, having any desired built-in (computable) relations and functions, including disequality (the negation of the equality relation) as well as the usual ordering relations on the usual built-in types, such as numbers and strings. These results are based on a new completeness theorem for order-sorted Horn clause logic with equality, plus the use of standard interpretations for fixed sorts, functions and relations. Finally, we define “logical programming,” based on the concept of institution, and show how it yields a general framework for discussions of this kind. For example, this viewpoint suggests that the natural way to combine functional and logic programming is simply to combine their logics, getting Horn clause logic with equality.