Intensional Type Theory with Guarded Recursive Types qua Fixed Points on Universes

  • Authors:

  • Lars Birkedal;Rasmus Ejlers Mogelberg

  • Affiliations:

  • -;-

  • Venue:
  • LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
  • Year:
  • 2013

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Abstract

Guarded recursive functions and types are useful for giving semantics to advanced programming languages and for higher-order programming with infinite data types, such as streams, e.g., for modeling reactive systems. We propose an extension of in tensional type theory with rules for forming fixed points of guarded recursive functions. Guarded recursive types can be formed simply by taking fixed points of guarded recursive functions on the universe of types. Moreover, we present a general model construction for constructing models of the in tensional type theory with guarded recursive functions and types. When applied to the groupoid model of in tensional type theory with the universe of small discrete groupoids, the construction gives a model of guarded recursion for which there is a one-to-one correspondence between fixed points of functions on the universe of types and fixed points of (suitable) operators on types. In particular, we find that the functor category ωop from the preordered set of natural numbers to the category of groupoids is a model of intensional type theory with guarded recursive types.