Guarded Computational Type Theory

Abstract

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 Ui. 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 Ui 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.