First steps in synthetic guarded domain theory: step-indexing in the topos of trees

TL;DR

This paper introduces the topos of trees as a model for guarded recursion, demonstrating its ability to handle recursive types and higher-order store within a synthetic, dependently-typed logic framework.

Contribution

It develops the internal logic of the topos of trees for modeling guarded recursion and extends the framework to sheaves over Heyting algebras, generalizing previous models.

Findings

The topos of trees models guarded recursion with modal operators.

Recursive type equations with dependent types are solvable in this setting.

Constructs a model of a language with higher-order store and recursive types inside the internal logic.

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. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S.