The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

Abstract

Abstract. We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We introduce a program logic with Löb induction for reasoning about the contextual equivalence of programs. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Rutten's behavioural differential equations.

A Generalized Modality for Recursion

“…Such a type is in effect not subject to the context restriction in rule W , since for any p we have * ω τ <: * p * ω τ . Thus, * ω corresponds to the constant ( ) modality used in some guarded type theories [13].…”

“…On the other hand Core λ * lacks many features present in other guarded type theories (including the gλ-calculus). It would be useful, for instance, to replace the fixed stream type with general guarded recursive types [7,13]; this requires designing a guardedness criterion in the presence of the warping modality. Clock variables [3] would allow types to express that unrelated program pieces may operate within disjoint time scales.…”

“…Nakano's original insight has led to a flurry of proposals [3, 4, 6-8, 13, 20, 21, 26, 31]. Recent developments have integrated several advances-such as clock variables [3] or the constant ( ) modality [13]-into expressive languages capturing many recursive definitions out of reach of more syntactic productivity checks. The topos of trees [7] provides an elegant categorical setting where such languages find their natural home.…”

Monadic refinements for relational cost analysis

“…However, this limitation does not seem to be fundamental. In recent work [Aguirre et al 2017a], a subset of the authors of this paper (and others) have re-worked a version of RHOL/UHOL based on the guarded λ-calculus [Clouston et al 2016] and a model in the topos of trees, which generalizes sets. This version supports infinite computations, including computations over infinite streams, and allows proving properties of all finite prefixes of the computations.…”

Modalities, cohesion, and information flow

“…Nevertheless, the implicit properties of these systems have not been subjected to a detailed treatment. Given the recent resurgence of interest in modal type theory-as exemplified by cohesive homotopy type theory [Shulman 2018], and guarded type theory [Clouston et al 2016] on the theoretical side, but also calculi for functional reactive programming [Krishnaswami 2013], effects [Curien et al 2016] or coeffects [Petricek et al 2014] on the more application-driven end-we believe that there is a need for a more universal approach to modal type theory. There have been major recent advances on the syntactical side, particularly through the fibrational framework of Licata et al [2017].…”