Multimodal Dependent Type Theory

Abstract

We introduce MTT, a dependent type theory which supports multiple modalities.MTT is parametrized by a mode theory which specifies a collection of modes,modalities, and transformations between them. We show that different choices ofmode theory allow us to use the same type theory to compute and reason in manymodal situations, including guarded recursion, axiomatic cohesion, andparametric quantification. We reproduce examples from prior work in guardedrecursion and axiomatic cohesion, thereby demonstrating that MTT constitutes asimple and usable syntax whose instantiations intuitively correspond toprevious handcrafted modal type theories. In some cases, instantiating MTT to aparticular situation unearths a previously unknown type theory that improvesupon prior systems. Finally, we investigate the metatheory of MTT. We prove theconsistency of MTT and establish canonicity through an extension of recenttype-theoretic gluing techniques. These results hold irrespective of the choiceof mode theory, and thus apply to a wide variety of modal situations