Continuation-passing style models complete for intuitionistic logic

A class of models is presented, in the form of continuation monads polymorphic for first-order individuals, that is sound and complete for minimal intuitionistic predicate logic (including disjunction and the existential quantifier). The proofs of soundness and completeness are constructive and the computational content of their composition is, in particular, a β-normalisation-by-evaluation program for simply typed lambda calculus with sum types. Although the inspiration comes from Danvyʼs type-directed partial evaluator for the same lambda calculus, the use of delimited control operators (i.e. computational effects) is avoided. The role of polymorphism is crucial – dropping it allows one to obtain a notion of model complete for classical predicate logic.