Abstract

We show how the topos of trees, i.e., the category of presheaves over the ordered natural numbers, models guarded recursive terms, predicates and types. We have two motivations for this work: one is as a model of computation with streams, using guards to ensure well-definedness of recursive definitions. The other is the construction of models of programming languages with higher-order store, which involves solving recursive domain equations. But since guarded recursion is commonplace in computer science we expect many more applications.

In the talk I will give a computational intuition for the topos of trees and show how the internal logic of this model can be used as an expressive language combining dependent types and subset types with guarded recursion. I will also sketch how to construct a model of higher-order store entirely inside this language, using a combination of set-like constructions and guarded recursion. Relations to step-indexing and metric space models of guarded recursion will be discussed.

This talk is based on joint work with Lars Birkedal, Kristian Stovring and Jan Schwinghammer