The Clocks They Are Adjunctions: Denotational Semantics for Clocked Type Theory

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

Denotational semantics for guarded dependent type theory

We present 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. Productivity of recursively defined coinductive programs and proofs is encoded in types using guarded recursion, and can therefore be checked modularly, unlike the syntactic checks implemented in modern proof assistants. The model is based on a category of covariant presheaves over a category of time objects, and quantification over clocks is modelled using a presheaf of clocks. To model the clock irrelevance axiom, crucial for programming with coinductive types, types must be interpreted as presheaves orthogonal to the object of clocks. In the case of dependent types, this translates to a lifting condition similar to the one found in homotopy theoretic models of type theory, but here with an additional requirement of uniqueness of lifts. Since the universes defined by the standard Hofmann-Streicher construction in this model do not satisfy this property, the universes in GDTT must be indexed by contexts of clock variables. We show how to model these universes in such a way that inclusions of clock contexts give rise to inclusions of universes commuting with type operations on the nose.Comment: This is the third version of the paper representing a minor revision over the second version. The paper has 40 page

Diamonds are not forever: Liveness in reactive programming with guarded recursion

When designing languages for functional reactive programming (FRP) the main challenge is to provide the user with a simple, flexible interface for writing programs on a high level of abstraction while ensuring that all programs can be implemented efficiently in a low-level language. To meet this challenge, a new family of modal FRP languages has been proposed, in which variants of Nakano's guarded fixed point operator are used for writing recursive programs guaranteeing properties such as causality and productivity. As an apparent extension to this it has also been suggested to use Linear Temporal Logic (LTL) as a language for reactive programming through the Curry-Howard isomorphism, allowing properties such as termination, liveness and fairness to be encoded in types. However, these two ideas are in conflict with each other, since the fixed point operator introduces non-termination into the inductive types that are supposed to provide termination guarantees. In this paper we show that by regarding the modal time step operator of LTL a submodality of the one used for guarded recursion (rather than equating them), one can obtain a modal type system capable of expressing liveness properties while retaining the power of the guarded fixed point operator. We introduce the language Lively RaTT, a modal FRP language with a guarded fixed point operator and an `until' type constructor as in LTL, and show how to program with events and fair streams. Using a step-indexed Kripke logical relation we prove operational properties of Lively RaTT including productivity and causality as well as the termination and liveness properties expected of types from LTL. Finally, we prove that the type system of Lively RaTT guarantees the absence of implicit space leaks

Ticking clocks as dependent right adjoints: Denotational semantics for clocked type theory

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 the equational theory. 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 on a clock used in CloTT for coinductive reasoning about coinductive types. We build on a category previously used to model guarded recursion with multiple clocks. In this category there is an object of clocks but no object of ticks, and so tick-assumptions in a context can not be modelled using standard tools. Instead we model ticks using dependent right adjoint functors, a generalisation of the category theoretic notion of adjunction to the setting of categories with families. Dependent right adjoints are known to model Fitch-style modal types, but in the case of CloTT, the modal operators constitute a family indexed internally in the type theory by clocks. We model this family using a dependent right adjoint on the slice category over the object of clocks. Finally we show how to model the tick constant of CloTT using a semantic substitution. This work improves on a previous model by the first two named authors which not only had a flaw but was also considerably more complicated.Comment: 31 pages. Second version is a minor revision. arXiv admin note: text overlap with arXiv:1804.0668

Greatest HITs: Higher Inductive Types in Coinductive Definitions via Induction under Clocks

Guarded recursion is a powerful modal approach to recursion that can be seen as an abstract form of step-indexing. It is currently used extensively in separation logic to model programming languages with advanced features by solving domain equations also with negative occurrences. In its multi-clocked version, guarded recursion can also be used to program with and reason about coinductive types, encoding the productivity condition required for recursive definitions in types. This paper presents the first type theory combining multi-clocked guarded recursion with the features of Cubical Type Theory, as well as a denotational semantics. Using the combination of Higher Inductive Types (HITs) and guarded recursion allows for simple programming and reasoning about coinductive types that are traditionally hard to represent in type theory, such as the type of finitely branching labelled transition systems. For example, our results imply that bisimilarity for these imply path equality, and so proofs can be transported along bisimilarity proofs. Among our technical contributions is a new principle of induction under clocks. This allows universal quantification over clocks to commute with HITs up to equivalence of types, and is crucial for the encoding of coinductive types. Such commutativity requirements have been formulated for inductive types as axioms in previous type theories with multi-clocked guarded recursion, but our present formulation as an induction principle allows for the formulation of general computation rules.Comment: 29 page

The enriched effect calculus: syntax and semantics

This paper introduces the enriched effect calculus, which extends established type theories for computational effects with primitives from linear logic. The new calculus provides a formalism for expressing linear aspects of computational effects; for example, the linear usage of imperative features such as state and/or continuations. The enriched effect calculus is implemented as an extension of a basic effect calculus without linear primitives, which is closely related to Moggi’s computational metalanguage, Filinski’s effect PCF and Levy’s call-by-push-value. We present syntactic results showing: the fidelity of the behaviour of the linear connectives of the enriched effect calculus; the conservativity of the enriched effect calculus over its non-linear core (the effect calculus); and the non-conservativity of intuitionistic linear logic when considered as an extension of the enriched effect calculus. The second half of the paper investigates models for the enriched effect calculus, based on enriched category theory. We give several examples of such models, relating them to models of standard effect calculi (such as those based on monads), and to models of intuitionistic linear logic. We also prove soundness and completeness.