A Model of Guarded Recursion With Clock Synchronisation

Bizjak, Aleš; Møgelberg, Rasmus Ejlers

doi:10.1016/j.entcs.2015.12.007

Cited by 12 publications

(26 citation statements)

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“…Clock quantification allows for the controlled elimination of the later type-former, and hence the encoding of first-class coinductive types via guarded recursive types. The generality of our approach to semantics in this paper should allow us to build a model by combining cubical sets with the presheaf model of GDTT with multiple clocks [9]. The main challenges lie in ensuring decidable type checking (GDTT relies on certain rules involving clock quantifiers which seem difficult to implement), and solving the coherence problem for clock substitution.…”

Section: Resultsmentioning

confidence: 99%

Guarded Cubical Type Theory

et al. 2018

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This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-Löf type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, enjoys canonicity for the natural numbers type, and is conjectured to support decidable type-checking. Our new type theory, guarded cubical type theory (GCTT), provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation. We show that CTT can be given semantics in presheaves on C × D, where C is the cube category, and D is any small category with an initial object. We then show that the category of presheaves on C × ω provides semantics for GCTT. arXiv:1611.09263v2 [cs.LO] 6 Oct 2017Clouston et al. [11] developed guarded recursive types in a simply-typed setting, following earlier work [27,3,1], with semantics in the presheaf category ω known as the topos of trees, and also presented a logic for reasoning about programs with guarded recursion. For large examples, such as models of program logics, we would like to be able to formalise such reasoning. A major approach to formalisation is via dependent types, used for example in the proof assistants Coq [24] and Agda [28]. Bizjak et al. [10], following earlier work [6,26], introduced guarded dependent type theory (GDTT), integrating the type-former into a dependently typed calculus, and supporting the definition of guarded recursive types as fixed-points of functions on universes, and guarded recursive operations on these types.We wish to formalise non-trivial theorems about equality between guarded recursive constructions, but such arguments often cannot be accommodated within intensional Martin-Löf type theory. For example, we may need to be able to reason about the extensions of streams in order to prove the equality of different stream functions. Hence GDTT includes an equality reflection rule, which is well known to make type checking undecidable. This problem is close to well-known problems with functional extensionality [16, Section 3.1.3], and indeed this analogy can be developed. Just as functional extensionality involves mapping terms of type (x : A) → Id B (f x) (gx) to proofs of Id (A → B) f g, extensionality for guarded recursion requires an extensionality principle for later types, namely the ability to m...

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Section: Resultsmentioning

confidence: 99%

Guarded Cubical Type Theory

et al. 2018

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“…is "topos of clocks" defined above inherits a rich internal logic which corresponds to a combination of cartesian/structural nominal logic 4 and guarded recursion. e topos S is related to the models considered by Bizjak and Møgelberg [2015], except that rather than constructing a family of presheaf toposes fibered over clock contexts, we combine clock contexts with time assignments into a single base category, and take the topos of presheaves over that; our topos is nearly identical to the presheaf category considered independently in Bizjak and Møgelberg [2017].…”

Section: Remarkmentioning

confidence: 99%

“…is topos can be regarded as a denotational model for a variant of Martin-Löf's extensional type theory equipped with the ◮ modality. By indexing this topos over a category of clock contexts ∆, it is possible to develop a model of extensional type theory with clock quantification called GDTT Møgelberg, 2015]. In order to justify a crucial clock irrelevance principle, it is necessary to index universes in clock contexts, i.e.…”

Section: Examples: Revisiting Streamsmentioning

confidence: 99%

Guarded Computational Type Theory

Sterling

Harper

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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Nakano's later modality can be used to specify and define recursive functions which are causal or synchronous; in concert with a notion of clock variable, it is possible to also capture the broader class of productive (co)programs. Until now, it has been difficult to combine these constructs with dependent types in a way that preserves the operational meaning of type theory and admits a hierarchy of universes U i .We present an operational account of guarded dependent type theory with clocks called CTT , featuring a novel clock intersection connective {k ÷ clk} → A that enjoys the clock irrelevance principle, as well as a predicative hierarchy of universes U i which does not require any indexing in clock contexts. CTT is simultaneously a programming language with a rich specification logic, as well as a computational metalanguage that can be used to develop semantics of other languages and logics. *

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…-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].…”

Section: Introductionmentioning

confidence: 99%

Guarded Dependent Type Theory with Coinductive Types

Bizjak

Grathwohl

Clouston

et al. 2016

Lecture Notes in Computer Science

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We present guarded dependent type theory, gDTT, an extensional dependent type theory with a 'later' modality and clock quantifiers for programming and proving with guarded recursive and coinductive types. The later modality is used to ensure the productivity of recursive definitions in a modular, type based, way. Clock quantifiers are used for controlled elimination of the later modality and for encoding coinductive types using guarded recursive types. Key to the development of gDTT are novel type and term formers involving what we call 'delayed substitutions'. These generalise the applicative functor rules for the later modality considered in earlier work, and are crucial for programming and proving with dependent types. We show soundness of the type theory with respect to a denotational model.

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A Model of Guarded Recursion With Clock Synchronisation

Cited by 12 publications

References 3 publications

Guarded Cubical Type Theory

Guarded Cubical Type Theory

Guarded Computational Type Theory

Guarded Dependent Type Theory with Coinductive Types

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