Uniform Algebras: a Complete Semantics for the Hohh Logic of -prolog

The problem of nding sound and complete semantics for the logic underlying the lambda-Prolog programming language: Higher Order Hereditarily Harrop formulas with resolution (uniform) proofs has been open for over a decade. It is settled in this paper. Church's Simple Theory of Types introduced in 1940, is a classical theory: all instances of the excluded middle are derivable in it. A suitable intuitionistic fragment of this theory is the core formalism for the declarative lambda-prolog programming language and it has also served as the formalism of choice for important work on higher-order automated deduction in the 60s and 70s. Its classical semantics was developed by Henkin in 1950, and an intensional variant by Andrews 2] in 1972. No complete semantics appears to have been developed for the intuitionistic theory or the logic programming HOHH fragment with uniform (resolution) proofs until now. Using an indexed version of Scott-Fourman Omega-set models, a generalization, (roughly equivalent to sheaves) of earlier intuitionistic semantics due to Tarski, McKinsey, Rasiowa and Sikorski, we develop sound and complete semantics for the (appropriately deened) intuitionistic fragment of Church's theory of types (ICTT). We then show how to adapt the semantics to the lambda-prolog fragment (HOHH) of this theory via Uniform Algebras, a Goal-Atom-Program sorted variant of the above semantics. We develop a bottom-up semantics based on Uniform Algebras, producing a minimal term model as the least xed point of an appropriate continuous operator.