Higher-Order Logic Programming Languages with Constraints: A Semantics

A Kripke Semantics is defined for a higher-order logic programming language with constraints, based on Church’s Theory of Types and a generic constraint formalism. Our syntactic formal system, hoHH(C) (higher-order hereditary Harrop formulas with constraints), which extends λProlog’s logic, is shown sound and complete. A Kripke semantics for equational reasoning in the simply typed lambda-calculus (Kripke Lambda Models) was introduced by Mitchell and Moggi in 1990. Our model theory extends this semantics to include full impredicative higher-order intuitionistic logic, as well as the executable hoHH fragment with typed lambdaabstraction, implication and universal quantification in goals and constraints. This provides a Kripke semantics for the full higher-order hereditarily Harrop logic of λProlog as a special case (with the constraint system chosen to be β,η-conversion).