Modalities are everywhere in programming and mathematics! Despite this, however, there are still significant technical challenges in formulating a core dependent type theory with modalities. We present a dependent type theory MLTT supporting the connectives of standard Martin-Löf Type Theory as well as an S4-style necessity operator. MLTT supports a smooth interaction between modal and dependent types and provides a common basis for the use of modalities in programming and in synthetic mathematics. We design and prove the soundness and completeness of a type checking algorithm for MLTT , using a novel extension of normalization by evaluation. We have also implemented our algorithm in a prototype proof assistant for MLTT , demonstrating the ease of applying our techniques. CCS Concepts: • Theory of computation → Modal and temporal logics; Type theory; Proof theory.

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The theory of program modules is of interest to language designers not only for its practical importance to programming, but also because it lies at the nexus of three fundamental concerns in language design: the phase distinction , computational effects , and type abstraction . We contribute a fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi (kinds, constructors, dynamic programs) are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction. We simplify the account of type abstraction (embodied in the generativity of module functors) through a lax modality that encapsulates computational effects, placing projectibility of module expressions on a type-theoretic basis. Our main result is a (significant) proof-relevant and phase-sensitive generalization of the Reynolds abstraction theorem for a calculus of program modules, based on a new kind of logical relation called a parametricity structure . Parametricity structures generalize the proof-irrelevant relations of classical parametricity to proof- relevant families, where there may be non-trivial evidence witnessing the relatedness of two programs—simplifying the metatheory of strong sums over the collection of types, for although there can be no “relation classifying relations,” one easily accommodates a “family classifying small families.” Using the insight that logical relations/parametricity is itself a form of phase distinction between the syntactic and the semantic, we contribute a new synthetic approach to phase separated parametricity based on the slogan logical relations as types , by iterating our modal account of the phase distinction. We axiomatize a dependent type theory of parametricity structures using two pairs of complementary modalities (syntactic, semantic) and (static, dynamic), substantiated using the topos theoretic Artin gluing construction. Then, to construct a simulation between two implementations of an abstract type, one simply programs a third implementation whose type component carries the representation invariant.

Nakano's later modality can be used to specify and define recursive functions which are causal or synchronous; in concert with a notion of clock variable, it is possible to also capture the broader class of productive (co)programs. Until now, it has been difficult to combine these constructs with dependent types in a way that preserves the operational meaning of type theory and admits a hierarchy of universes U i .We present an operational account of guarded dependent type theory with clocks called CTT , featuring a novel clock intersection connective {k ÷ clk} → A that enjoys the clock irrelevance principle, as well as a predicative hierarchy of universes U i which does not require any indexing in clock contexts. CTT is simultaneously a programming language with a rich specification logic, as well as a computational metalanguage that can be used to develop semantics of other languages and logics. *