Realizability Interpretation and Normalization of Typed Call-by-Need $$\lambda $$-calculus with Control

Miquey, Étienne; Herbelin, Hugo

doi:10.1007/978-3-319-89366-2_15

Cited by 7 publications

(36 citation statements)

References 29 publications

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“…Proof. The first two conditions are already verified for the λ [l τ ⋆] -calculus [21]. The third one is straightforward, since if a closure µt p.c ||e τ is not normalizing, it is easy to verify that c[e/t p] is not normalizing either.…”

Section: Realizability Interpretation (Proofs Of Section 4)mentioning

confidence: 82%

“…On the one hand, we set out in [20] the difficulties related to the definition of a sequent calculus with dependent types. On the other hand, building on [21], we developed a variant of Krivine realizability adapted to a lazy calculus where delayed substitutions are stored in an explicit environment. The sound combination of both frameworks led us to the definition of dLPA ω together with its realizability interpretation.…”

Section: Discussionmentioning

confidence: 99%

“…First, we define dLPA ω (Section 2), a sequent calculus with classical control, dependent types, inductive and coinductive fixpoints and lazy evaluation made available thanks to the presence of stores. This calculus can be seen as a sound combination of dLt p [20] and the λ [l τ ⋆] -calculus [1,21] extended with the expressive power of dPA ω [12]. Second, we prove the properties of normalization and soundness for dLPA ω thanks to a realizability interpretation à la Krivine, which we obtain by applying Danvy's methodology of semantic artifacts (Sections 3 and 4).…”

Section: Contributions Of the Papermentioning

confidence: 91%

“…Thanks to Danvy's methodology of semantics artifacts [8], which consists in successively refining the reduction system until getting context-free reduction rules 1 , they obtained an untyped CPS translation for the λ [l τ ⋆] -calculus. By pushing one step further this methodology, we showed with Herbelin how to obtain a realizability interpretation à la Krivine for this framework [21]. The main idea, in contrast to usual models of Krivine realizability [15], is that realizers are defined as pairs of a term and a substitution.…”

Section: Realizability Interpretation Of Classical Call-by-needmentioning

confidence: 99%

“…We recall two key properties of the interpretation, whose proofs are similar to the proofs for the corresponding statements in the λ [l τ ⋆]calculus [21]:…”

Section: Realizability Interpretation (Proofs Of Section 4)mentioning

confidence: 99%

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A sequent calculus with dependent types for classical arithmetic

Miquey¹

2018

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

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In a recent paper [12], Herbelin developed dPA ω , a calculus in which constructive proofs for the axioms of countable and dependent choices could be derived via the encoding of a proof of countable universal quantification as a stream of it components. However, the property of normalization (and therefore the one of soundness) was only conjectured. The difficulty for the proof of normalization is due to the simultaneous presence of dependent types (for the constructive part of the choice), of control operators (for classical logic), of coinductive objects (to encode functions of type N → A into streams (a 0 , a 1 , . . .)) and of lazy evaluation with sharing (for these coinductive objects).Elaborating on previous works, we introduce in this paper a variant of dPA ω presented as a sequent calculus. On the one hand, we take advantage of a variant of Krivine classical realizability that we developed to prove the normalization of classical call-byneed [21]. On the other hand, we benefit from dLt p , a classical sequent calculus with dependent types in which type safety is ensured by using delimited continuations together with a syntactic restriction [20]. By combining the techniques developed in these papers, we manage to define a realizability interpretation à la Krivine of our calculus that allows us to prove normalization and soundness.

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Section: Realizability Interpretation (Proofs Of Section 4)mentioning

confidence: 82%

Section: Discussionmentioning

confidence: 99%

Section: Contributions Of the Papermentioning

confidence: 91%

Section: Realizability Interpretation Of Classical Call-by-needmentioning

confidence: 99%

“…We recall two key properties of the interpretation, whose proofs are similar to the proofs for the corresponding statements in the λ [l τ ⋆]calculus [21]:…”

Section: Realizability Interpretation (Proofs Of Section 4)mentioning

confidence: 99%

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A sequent calculus with dependent types for classical arithmetic

Miquey¹

2018

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

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Evidenced Frames: A Unifying Framework Broadening Realizability Models

Cohen

Miquey

Tate

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Constructive foundations have for decades been built upon realizability models for higher-order logic and type theory. However, traditional realizability models have a rather limited notion of computation, which only supports non-termination and avoids many other commonly used effects. Work to address these limitations has typically overlaid structure on top of existing models, such as by using powersets to represent non-determinism, but kept the realizers themselves deterministic. This paper alternatively addresses these limitations by making the structure underlying realizability models more flexible. To this end, we introduce evidenced frames: a general-purpose framework for building realizability models that support diverse effectful computations. We demonstrate that this flexibility permits models wherein the realizers themselves can be effectful, such as λ-terms that can manipulate state, reduce non-deterministically, or fail entirely. Beyond the broader notions of computation, we demonstrate that evidenced frames form a unifying framework for (realizability) models of higher-order dependent predicate logic. In particular, we prove that evidenced frames are complete with respect to these models, and that the existing completeness construction for implicative algebras-another foundational framework for realizability-factors through our simpler construction. As such, we conclude that evidenced frames offer an ideal domain for unifying and broadening realizability models.

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A Classical Sequent Calculus with Dependent Types

Miquey

2019

ACM Trans. Program. Lang. Syst.

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Dependent types are a key feature of the proof assistants based on the Curry-Howard isomorphism. It is well-known that this correspondence can be extended to classical logic by enriching the language of proofs with control operators. However, they are known to misbehave in the presence of dependent types, unless dependencies are restricted to values. Moreover, while sequent calculi naturally support continuation-passing style interpretations, there is no such presentation of a language with dependent types. The main achievement of this paper is to give a sequent calculus presentation of a call-by-value language with a control operator and dependent types, and to justify its soundness through a continuation-passing style translation.We start from the call-by-value version of the λµμ-calculus. We design a minimal language with a value restriction and a type system that includes a list of explicit dependencies to maintains type safety. We then show how to relax the value restriction and introduce delimited continuations to directly prove the consistency by means of a continuation-passing-style translation. Finally, we relate our calculus to a similar system by Lepigre, and present a methodology to transfer properties from this system to our own. c b n a This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. 1:2 Étienne MiqueyNevertheless, dependent types are known to misbehave in the presence of control operators, and lead to logical inconsistencies [17]. Since the same problem arises with a wider class of effects, it seems that we are facing the following dilemma: either we choose an effectful language (allowing us to write more programs) while accepting the lack of dependent types, or we choose a dependently typed language (allowing us to write finer specifications) and give up effects.Many works have tried to fill the gap between effectful programming languages and logic, by accommodating weaker forms of dependent types with computational effects (e.g., divergence, I/O, local references, exceptions). Amongst other works, we can cite the recent works by Ahman et al.[1], by Vákár [35,36] or by Pédrot and Tabareau who proposed a systematical way to add effects to type theory [33]. Side effects-that are impure computations in functional programming-are often interpreted by means of monads. Interestingly, control operators can be interpreted similarly through the continuation monad, but the continuation monad generally lacks the properties necessary to fit these frameworks.Although dependent types and classical logic have been deeply studied separately, the problem of accommodating both features 1 in one and the same system has not found a completely satisfying answer yet. Recent works from Herbelin [18] and Lepigre [22] proposed some restrictions on dependent types to make them compatible with a classical proof system, while Blot [5] designed a hybrid realizability model where dependent types are restricted to an intuitionistic fragment. Call-by-value and value restrictionIn lang...

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Realizability Interpretation and Normalization of Typed Call-by-Need $$\lambda $$-calculus with Control

Cited by 7 publications

References 29 publications

A sequent calculus with dependent types for classical arithmetic

A sequent calculus with dependent types for classical arithmetic

Evidenced Frames: A Unifying Framework Broadening Realizability Models

A Classical Sequent Calculus with Dependent Types

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