Kripke semantics for dependent type theory and realizability interpretations

Abstract

Constructive reasoning has played an increasingly important role in the development of provably correct software. Both typed and type-free frameworks stemming from ideas of Heyting, Kleene, and Curry have been developed for extracting computations from constructive specifications. These include Realizability, and Theories based on the Curry-Howard isomorphism. Realizability — in its various typed and type-free formulations — brings out the algorithmic content of theories and proofs and supplies models of the “recursive universe”. Formal systems based on the propositions-as-types paradigm, such as Martin-Löf's dependent type theories, incorporate term extraction into the logic itself.

Another, major tradition in constructive semantics originated in the model theory developed by Gödel, Herbrand and Tarski, resulting in the interpretations developed by Kripke and Beth, and in subsequent categorical generalizations. They provide a complete semantics for constructive logic. These models are a powerful tool for building counterexamples and establishing independence and conservativity results, but they are often less constructive and less computationally oriented.

It is highly desirable to combine the power of these approaches to constructive semantics, and to elucidate some connections between them. We define modified Kripke and Beth models for syntactic Realizability and Dependent Type theory, in particular for the one-universe Intensional Martin-Löf Theory ML ⁱ₀ and prove a completeness theorem for the latter theory. These models provide a new framework for reasoning about computational evidence and the process of term-extraction. They are defined over a constructive type-free metatheory based on the Feferman-Beeson theories of abstract applicative structure.

Our models have a feature which is shared by all published constructive completeness theorems for intuitionistic logic, known in the literature as “fallibility”: there may be worlds in which some sentences are both false and true, a phenomenon which corresponds to the presence of empty types in various type disciplines. We also identify a natural lattice of truth values associated with type theory and realizability: the degrees of inhabitation.