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 of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion, thereby demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.

Sikkel: Multimode Simple Type Theory as an Agda Library

“…the base category. As such, it can in principle represent many different type theory extensions like guarded recursion [10,20], parametricity [1,2,4,33,30], univalence [6,12,15], nominal [34] and directed [36] type theory, etc., as well as combinations of these [11,41].…”

“…2 Multimode Simple Type Theory (MSTT) We first present Sikkel's syntactic layer: Multimode Simple Type Theory (MSTT), which is essentially Multimode Type Theory (MTT) by Gratzer et al [20] restricted to simple (non-dependent) types. Just like MTT, MSTT is parametrized by a mode theory specifying the available modalities, discussed in Section 2.1.…”

“…Over the past years many authors have proposed new type systems that extend a theory like MLTT or CIC with new primitives which allow users to prove more theorems or write more programs. Examples of such extensions include guarded recursion [10,20], parametricity [1,2,4,33,30,11], univalence [6,12,15], directed type theory [36,41] and nominal reasoning [34].…”

“…However, the question remains how these extended formal systems can be used in existing languages like Agda or Coq. In this paper we present Sikkel 1 , an Agda library that allows users to work in a class of extended type theories called multimode or multimodal type theories [26,20]. Such type theories are parametrized by a mode theory which specifies new primitive type constructors called modalities.…”

Normalization for Cubical Type Theory

“…Sterling et al [56] adapted Coquand's gluing argument to prove the first non-operational strict canonicity result for a cubical type theory. Gratzer et al [29] used gluing to prove canonicity for a general multi-modal dependent type theory. Sterling and Harper [55] employ a different gluing argument to establish a proof-relevant generalization of the Reynolds Abstraction Theorem for a calculus of ML modules.…”