A type theory for productive coprogramming via guarded recursion

Abstract

To ensure consistency and decidability of type checking, proof assistants impose a requirement of productivity on corecursive definitions. In this paper we investigate a type-based alternative to the existing syntactic productivity checks of Coq and Agda, using a combination of guarded recursion and quantification over clocks. This approach was developed by Atkey and McBride in the simply typed setting, here we extend it to a calculus with dependent types. Building on previous work on the topos-of-trees model we construct a model of the calculus using a family of presheaf toposes, each of which can be seen as a multi-dimensional version of the topos-of-trees. As part of the model construction we must solve the coherence problem for modelling dependent types in locally cartesian closed categories simulatiously in a whole family of locally cartesian closed categories. We do this by embedding all the categories in a large one and applying a recent approach to the coherence problem due to Streicher and Voevodsky.

Guarded Dependent Type Theory with Coinductive Types

“…4 extends the system of Fig. 2 to allow the removal of clocks in such a setting, by introducing clock quantifiers ∀κ [3,18,6]. This is a binding construct with associated term constructor Λκ, which also binds κ.…”

“…The encoding of coinductive types using guarded recursive types crucially uses a family of type isomorphisms commuting ∀κ over other type formers [3,18]. By The following type isomorphisms follow by using β and η laws for the constructs involved.…”

“…More pragmatically, the bare addition of ⊲ disallows productive but acausal [15] functions such as the 'every other' function that returns every second element of a stream. Atkey and McBride proposed clock quantifiers [3] for such functions; these have been extended to dependent types [18,6], and Møgelberg [18,Thm. 2] has shown that they allow the definition of types whose denotation is precisely that of standard coinductive types interpreted in set-based semantics.…”

“…-We introduce the extensional guarded dependent type theory gDTT, and show that it gives a framework for programming and proving with guarded recursive and coinductive types. The key novel feature is the generalisation of the 'later' type-former and 'next' term-former via delayed substitutions; -We prove the soundness of gDTT via a model similar to that used in earlier work on guarded recursive types and clock quantifiers [18,6].…”

Guarded Traced Categories

“…In models of computation, various notions of guardedness serve to control cyclic behaviour by allowing only guarded cycles, with the aim to ensure properties such as solvability of recursive equations or productivity. Typical examples are guarded process algebra specifications [29,6], coalgebraic guarded (co-)recursion [34,27], finite delay in online Turing machines [9], and productive definitions in intensional type theory [1,30], but also contractive maps in (ultra-)metric spaces [24]. A highly general model for unrestricted cyclic computations, on the Fig.…”