# Programming and Reasoning with Guarded Recursion for Coinductive Types

@inproceedings{Clouston2015ProgrammingAR, title={Programming and Reasoning with Guarded Recursion for Coinductive Types}, author={Ranald Clouston and Ale{\vs} Bizjak and Hans Bugge Grathwohl and Lars Birkedal}, booktitle={FoSSaCS}, year={2015} }

We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. [... ] Key Method 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 demonstrate the expressiveness of the calculus by showing the definability of solutions to Rutten's behavioural differential equations. We introduce a… Expand

## 46 Citations

A model of Clocked Cubical Type Theory

- Computer ScienceArXiv
- 2021

This paper presents the first denotational model of a type theory combining multi-clocked guarded recursion with the features of Cubical Type Theory, which allows for simple programming and reasoning about coinductive types that are traditionally hard to represent in type theory.

Denotational semantics for guarded dependent type theory

- Computer ScienceMathematical Structures in Computer Science
- 2020

A new model of guarded dependent type theory (GDTT), a type theory with guarded recursion and multiple clocks in which one can program with and reason about coinductive types, and shows how to model these universes in such a way that inclusions of clock contexts give rise toInclusions of universes commuting with type operations on the nose.

Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks

- Computer Science
- 2021

A new principle of induction under clocks is provided, providing computational content to one of the main axioms required for encoding coinductive types in type theory.

The clocks are ticking: no more delays!: reduction semantics for type theory with guarded recursion

- Computer ScienceLICS 2017
- 2017

Clocked Type Theory is presented, a new type theory for guarded recursion that is more suitable for reduction semantics than the existing ones, and it is proved confluence, strong normalisation and canonicity for its reduction semantics, constructing the theoretical basis for a future implementation.

Guarded Recursive Types in Type Theory

- Computer Science
- 2015

The main result is a model based on relational parametricity for the dependently typed calculus the authors designed, a result about strong normalization for a small language extended with guarded recursive types.

Guarded Cubical Type Theory: Path Equality for Guarded Recursion

- Mathematics, Computer ScienceCSL
- 2016

This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubicaltype theory (CTT), and provides a computational interpretation of extensionality for guarded recursive types.

Sequent Calculus in the Topos of Trees

- Philosophy, Computer ScienceFoSSaCS
- 2015

A sound and cut-free complete sequent calculus for KM lin is given via a strategy that decomposes implication into its static and irreflexive components and yields decidability of the logic and the coNP-completeness of its validity problem.

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

- Computer ScienceESOP
- 2018

We extend the simply-typed guarded \(\lambda \)-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations.…

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