Gradual type theory

New, Max S.; Licata, Daniel R.; Ahmed, Amal

doi:10.1017/s0956796821000125

Cited by 6 publications

(6 citation statements)

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“…Graduality is also based on important underlying structural properties of precision on terms, namely that term precision is stable by reduction (if t ⊑ t' and t reduces to v and t ' to v ' , then v ⊑ v'), and that the term and type constructors of the language are monotone (e.g., if t ⊑ t' and u ⊑ u' then t u ⊑ t' u ' ). These technical conditions, natural in a categorical setting [New et al 2019], coincide with the programmer-level interpretation of precision and the DGG.…”

Section: Conservativitymentioning

confidence: 95%

Gradualizing the Calculus of Inductive Constructions

Lennon-Bertrand,

Maillard,

Tabareau

et al. 2020

Preprint

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Acknowledging the ordeal of a fully formal development in a proof assistant such as Coq, we investigate gradual variations on the Calculus of Inductive Construction (CIC) for swifter prototyping with imprecise types and terms. We observe, with a no-go theorem, a crucial tradeoff between graduality and the key properties of normalization and closure of universes under dependent product that CIC enjoys. Beyond this Fire Triangle of Graduality, we explore the gradualization of CIC with three different compromises, each relaxing one edge of the Fire Triangle. We develop a parametrized presentation of Gradual CIC that encompasses all three variations, and develop their metatheory. We first present a bidirectional elaboration of Gradual CIC to a dependently-typed cast calculus, which elucidates the interrelation between typing, conversion, and the gradual guarantees. We use a syntactic model into CIC to inform the design of a safe, confluent reduction, and establish, when applicable, normalization. We also study the stronger notion of graduality as embeddingprojection pairs formulated by New and Ahmed, using appropriate semantic model constructions. This work informs and paves the way towards the development of malleable proof assistants and dependently-typed programming languages.

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Section: Conservativitymentioning

confidence: 95%

Gradualizing the Calculus of Inductive Constructions

Lennon-Bertrand,

Maillard,

Tabareau

et al. 2020

Preprint

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“…Since the original conference publication of this article [24], we have developed [25] a version of gradual type theory based on call-by-push-value, which extends call-by-value and call-byname, and generalizes the type theory we have presented here. The implementations of the dynamic value type and dynamic computation type in [25] are based on the two models in Section 6.2. We also developed operational models of CBPV gradual type theory, based on an interpretation of term dynamism as a kind of contextual approximation, following [23].…”

Section: Ur(s)'mentioning

confidence: 99%

“…We have chosen call-by-name because it is a simple setting with the necessary η principles (for negative types) to illustrate our technique. We have in other work considered application to other evaluation orders ( [25]), which we discuss in more detail in Section 7.…”

Section: Introductionmentioning

confidence: 99%

“…This article is an extended version of [24]. The primary differences are that we (1) expanded Section 3 to include proofs, illustrating the process of reasoning about gradually typed programs in the logic of GTT, (2) expanded Section 4 to include additional details of the initiality theorems, and (3) expanded Section 6 to more fully describe the construction of our denotational models.…”

Section: Introductionmentioning

confidence: 99%

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Gradual type theory

New

Licata

Ahmed

2019

Proc. ACM Program. Lang.

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We present gradual type theory, a logic and type theory for call-by-name gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism. These dynamism judgments build on prior work in blame calculi and on the "gradual guarantee" theorem of gradual typing. Combined with the ordinary extensionality (η) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that non-standard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2-category representing non-gradual types and programs. This construction is a semantic analogue of the interpretation of gradual typing using contracts, and use it to build some concrete domain-theoretic models of gradual typing. CC Creative Commons 7:2 M.S. New and D.R. Licata Vol. 16:1"strictly less than", and similarly for other terms such as "greater than", "more dynamic", etc. 7:4 M.S. New and D.R. Licata Vol. 16:1 7:6 M.S. New and D.R. Licata Vol. 16:1 7:8 M.S. New and D.R. Licata Vol. 16:1 2.2. PTT Signatures. While gradual type theory proves that most operational rules of gradual typing are equivalences, some must be added as axioms. Compare Moggi's monadic metalanguage [22]: since it is a general theory of monads, it is not provable that an effect is 7:12 M.S. New and D.R. Licata Vol. 16:1

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“…However, adding recursive types is part of our future work so that we would be able to describe more precisely cases of the like. Recursive types have been addressed previously in gradual typing implementations [9,25,12,13], and in semantics formulations [20,15]. We plan on building on this body of work.…”

Section: Related Work and (More) Future Workmentioning

confidence: 99%

Towards Gradually Typed Capabilities in the Pi-Calculus

Cimini

2019

Electron. Proc. Theor. Comput. Sci.

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Gradual typing is an approach to integrating static and dynamic typing within the same language, and puts the programmer in control of which regions of code are type checked at compile-time and which are type checked at run-time.In this paper, we focus on the π-calculus equipped with types for the modeling of input-output capabilities of channels. We present our preliminary work towards a gradually typed version of this calculus. We present a type system, a cast insertion procedure that automatically inserts run-time checks, and an operational semantics of a π-calculus that handles casts on channels. Although we do not claim any theoretical results on our formulations, we demonstrate our calculus with an example and discuss our future plans.

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Gradual type theory

Cited by 6 publications

References 46 publications

Gradualizing the Calculus of Inductive Constructions

Gradualizing the Calculus of Inductive Constructions

Gradual type theory

Towards Gradually Typed Capabilities in the Pi-Calculus

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