First Steps in Synthetic Guarded Domain Theory: Step-Indexing in the Topos of Trees

Authors:
Lars Birkedal;Rasmus Ejlers Mogelberg;Jan Schwinghammer;Kristian Stovring
Affiliations:
-;-;-;-
Venue:
LICS '11 Proceedings of the 2011 IEEE 26th Annual Symposium on Logic in Computer Science
Year:
2011

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Abstract

We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S.