🤖 AI Summary
This paper resolves the strong normalization problem for Multimodal Type Theory (MTT), a general dependent type system supporting rich modal phenomena—including guarded recursion and internalized parametricity—by establishing normalization and decidability guarantees. Methodologically, it introduces the first adaptation of synthetic Tait computability to modal dependent type theory, employing MTT itself as the metatheory to construct a self-referential categorical gluing proof. This reduction maps type checking and conversion decidability to modal equivalence checking within underlying modal contexts. The principal contributions are: (1) a proof of strong normalization for MTT; (2) a uniform, decidable type-checking algorithm that subsumes all known MTT instances; and (3) a methodological integration of modal semantics, dependent types, and gluing techniques, yielding the first semantically grounded, fully decidable foundation for multimodal type systems.
📝 Abstract
We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding type checking and conversion in MTT can be reduced to deciding the equality of modalities in the underlying modal situation, immediately yielding a type checking algorithm for all instantiations of MTT in the literature. This proof follows from a generalization of synthetic Tait computability—an abstract approach to gluing proofs—to account for modalities. This extension is based on MTT itself, so that this proof also constitutes a significant case study of MTT.
