Contextual Modal Type Theory with Polymorphic Contexts

Abstract

Through the Curry-Howard isomorphism between logics and calculi, necessity modality in logic is interpreted as types representing program code. Particularly, λ ◯ , which was proposed in influential work by Davies, and its successors have been widely used as a logical foundation for syntactic meta-programming. However, it is less known how to extend calculi based on modal type theory to handle more practical operations including manipulation of variable binding structures.This paper constructs such a modal type theory in two steps. First, we reconstruct contextual modal type theory by Nanevski, et al. as a Fitchstyle system, which introduces hypothetical judgment with hierarchical context. The resulting type theory, Fitch-style contextual modal type theory λ [] , is generalized to accommodate not only S4 but also K, T, and K4 modalities, and proven to enjoy many desired properties. Second, we extend λ [] with polymorphic context, which is an internalization of contextual weakening, to obtain a novel modal type theory λ ∀[] . Despite the fact that it came from observation in logic, polymorphic context allows both binding manipulation and hygienic code generation. We claim this by showing a sound translation from λ ◯ to λ ∀[] .