Kripke-style models for typed lambda calculus

Abstract

Mitchell, J.C. and E. Moggi, Kripke-style models for typed lambda calculus, Annals of Pure and Applied Logic 51 (1991) 99–124.The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripke-style models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. To those familiar with Kripke models of modal or intuitionistic logics, Kripke lambda models are likely to seem adequately ‘semantic’. However, when viewed as cartesian closed categories, they do not have the property variously referred to as concreteness, well- pointedness or having enough points. While the traditional lambda calculus proof system is not complete for Henkin models that may have empty types, we prove strong completeness for Kripke models. In fact, every set of equations that is closed under implication is the theory of a single Kripke model. We also develop some properties of logical relations over Kripke structures, showing that every theory is the theory of a model determined by a Kripke equivalence relation over a Henkin model. We discuss cartesian closed categories but present the main definitions and results without the use of category theory.