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Denotational semantics for guarded dependent type theory

Published online by Cambridge University Press:  08 May 2020

Aleš Bizjak Open the ORCID record for Aleš Bizjak [Opens in a new window]  and
Rasmus Ejlers Møgelberg Open the ORCID record for Rasmus Ejlers Møgelberg [Opens in a new window]
Aleš Bizjak
Affiliation:
Department of Computer Science, Aarhus University, Aarhus N, Denmark
Rasmus Ejlers Møgelberg*
Affiliation:
Department of Computer Science, IT University of Copenhagen, Copenhagen, Denmark
*
*Corresponding author. Email: mogel@itu.dk
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Abstract

We present a new model of guarded dependent type theory (GDTT), a type theory with guarded recursion and multiple clocks in which one can program with and reason about coinductive types. Productivity of recursively defined coinductive programs and proofs is encoded in types using guarded recursion and can therefore be checked modularly, unlike the syntactic checks implemented in modern proof assistants. The model is based on a category of covariant presheaves over a category of time objects, and quantification over clocks is modelled using a presheaf of clocks. To model the clock irrelevance axiom, crucial for programming with coinductive types, types must be interpreted as presheaves internally right orthogonal to the object of clocks. In the case of dependent types, this translates to a lifting condition similar to the one found in homotopy theoretic models of type theory, but here with an additional requirement of uniqueness of lifts. Since the universes defined by the standard Hofmann–Streicher construction in this model do not satisfy this property, the universes in GDTT must be indexed by contexts of clock variables. We show how to model these universes in such a way that inclusions of clock contexts give rise to inclusions of universes commuting with type operations on the nose.

Keywords

Type theorydenotational semanticsguarded recursioncoinductive types
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 4 , April 2020 , pp. 342 - 378
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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