[Paper Review] The clocks they are adjunctions:Denotational semantics for Clocked Type Theory

Bassel Mannaa, Rasmus Ejlers Møgelberg|arXiv (Cornell University)|Apr 18, 2018

Logic, programming, and type systems4 citations

TL;DR

This paper presents a denotational semantics for Clocked Type Theory (CloTT) by modeling ticks—essential for coinductive reasoning—using adjoint functors and semantic substitutions. It extends prior models of guarded recursion to handle tick assumptions in contexts, enabling sound semantics for future extensions like path types in Guarded Cubical Type Theory.

ABSTRACT

Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for programming with coinductive types, allowing productivity to be encoded in types, and for reasoning about advanced programming language features using an abstract form of step-indexing. CloTT has previously been shown to enjoy a number of syntactic properties including strong normalisation, canonicity and decidability of type checking. In this paper we present a denotational semantics for CloTT useful, e.g., for studying future extensions of CloTT with constructions such as path types. The main challenge for constructing this model is to model the notion of ticks used in CloTT for coinductive reasoning about coinductive types. We build on a category previously used to model guarded recursion, but in this category there is no object of ticks, so tick-assumptions in a context can not be modelled using standard tools. Instead we show how ticks can be modelled using adjoint functors, and how to model the tick constant using a semantic substitution.

Motivation & Objective

• To provide a denotational semantics for Clocked Type Theory (CloTT) that supports coinductive reasoning via ticks.
• To address the challenge of modeling tick assumptions (e.g., α:κ) in contexts, which lack a corresponding object in standard models.
• To extend existing guarded recursion models—based on presheaves over time categories—to incorporate ticks using adjoint functors.
• To enable semantic reasoning for future extensions of CloTT, such as path types in Guarded Cubical Type Theory.
• To formalize tick handling via a left adjoint to the delay functor, ensuring soundness and compositional substitution.

Proposed method

• Model contexts using a category with families, where tick extension Γ, α:κ is interpreted as L^κ⟦Γ⟧ via a left adjoint L^κ to the delay functor R^κ.
• Use an adjunction L^κ ⊣ R^κ between endofunctors on a presheaf category to model the tick and delay modalities.
• Interpret the tick constant as a semantic substitution via natural transformations from L^κ to the identity functor.
• Construct a model of the tick calculus as a category with families equipped with an adjunction and a natural transformation satisfying coherence conditions.
• Prove soundness of substitution and typing by verifying that semantic substitutions commute with term interpretations under context extension.
• Leverage existing models of guarded recursion (e.g., Bizjak and Møgelberg’s model over time categories) as a foundation for the new semantics.

Experimental results

Research questions

• RQ1 How can tick assumptions in contexts be modeled denotationally when no object of ticks exists in the base category?
• RQ2 What categorical structure is needed to model the tick constant and its interaction with type families?
• RQ3 Can the tick calculus be soundly interpreted in a category with families using adjoint functors?
• RQ4 How does the adjoint functor structure ensure coherence of substitution and typing in the presence of ticks?
• RQ5 Can this semantics support future extensions of CloTT, such as path types, by providing a foundation for higher-dimensional type theory?

Key findings

• Ticks in contexts are modeled using a left adjoint functor L^κ to the delay functor R^κ, enabling context extension via L^κ⟦Γ⟧.
• The tick constant is interpreted as a natural transformation from L^κ to the identity functor, capturing the idea that a tick allows access to a delayed value.
• The model ensures that substitution of tick variables commutes with semantic interpretation, proving soundness of the typing rules.
• The construction generalizes prior models of guarded recursion by incorporating tick handling via adjunctions.
• The semantics supports future extensions such as path types by providing a compositional and sound categorical foundation.
• The model is sound for the tick calculus fragment, with substitution and typing preserved under semantic interpretation.