Dependent Types and Fibred Computational Effects

Ahman, Danel; Ghani, Neil; Plotkin, Gordon

doi:10.1007/978-3-662-49630-5_3

Cited by 26 publications

(70 citation statements)

References 35 publications

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“…A promising venue to reconcile dependent types and effects is to study dependent variants of call-by-push-value (CBPV) [Levy 2001], as recently done by Ahman et al [2016] and Vákár [2015]. While the CBPV setting can accommodate any effect described in monadic style, these approaches also need to impose a purity restriction for dependency.…”

Section: Related Workmentioning

confidence: 99%

A reasonably exceptional type theory

Pédrot

Tabareau

Fehrmann

et al. 2019

Proc. ACM Program. Lang.

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Traditional approaches to compensate for the lack of exceptions in type theories for proof assistants have severe drawbacks from both a programming and a reasoning perspective. Pédrot and Tabareau recently extended the Calculus of Inductive Constructions (CIC) with exceptions. The new exceptional type theory is interpreted by a translation into CIC, covering full dependent elimination, decidable type-checking and canonicity. However, the exceptional theory is inconsistent as a logical system. To recover consistency, Pédrot and Tabareau propose an additional translation that uses parametricity to enforce that all exceptions are caught locally. While this enforcement brings logical expressivity gains over CIC, it completely prevents reasoning about exceptional programs such as partial functions. This work addresses the dilemma between exceptions and consistency in a more flexible manner, with the Reasonably Exceptional Type Theory (RETT). RETT is structured in three layers: (a) the exceptional layer, in which all terms can raise exceptions; (b) the mediation layer, in which exceptional terms must be provably parametric; (c) the pure layer, in which terms are non-exceptional, but can refer to exceptional terms. We present the general theory of RETT, where each layer is realized by a predicative hierarchy of universes, and develop an instance of RETT in Coq: the impure layer corresponds to the predicative universe hierarchy, the pure layer is realized by the impredicative universe of propositions, and the mediation layer is reified via a parametricity type class. RETT is the first full dependent type theory to support consistent reasoning about exceptional terms, and the CoqRETT plugin readily brings this ability to Coq programmers.

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Section: Related Workmentioning

confidence: 99%

A reasonably exceptional type theory

Pédrot

Tabareau

Fehrmann

et al. 2019

Proc. ACM Program. Lang.

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show abstract

“…(2) and a function translating function symbols i 2 : Σ 2 → Θ 2 that respects typing in that t(i 2 (f )) = i 0 (t(f )) and s(i 2 (f )) = i 0 (s(f )) where i 0 (s(f )) is the pointwise application of i 0 . Such that axioms are translated to axioms in that (1)…”

Section: Vol 16:1mentioning

confidence: 99%

“…Next, we define a functor that takes a CPM and produces the "complete signature": one that includes all types as base types, all multi-arrows as function symbols, and all true dynamism facts as axioms. (1) (U (C)) 0 = C 0 (2) (U (C)) 1 3 = {(f (x 0 , . .…”

Section: Vol 16:1mentioning

confidence: 99%

“…Additionally, the combination of dependent typing with gradual typing is worth exploring, especially because while dependent contract checking been used for some time and was explicitly inspired in part by dependent typing, no semantic connection has been established between dependent contracts and category-theoretic models of dependent typing. The main difficulty will be the combination of dependent typing with effects, but there has been much recent work in this area [1,41,27].…”

Section: Contracts As Coreflectionsmentioning

confidence: 99%

“…We will sometimes refer to (x(·; A; ·)) ∈ C 1 (A; A) as id A , the identity multiarrow. (5) A monotone "composition" function • : 1 denotes a pullback, which is explicitly given by…”

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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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A General Semantic Construction of Dependent Refinement Type Systems, Categorically

Kura

2021

Lecture Notes in Computer Science

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Dependent refinement types are types equipped with predicates that specify preconditions and postconditions of underlying functional languages. We propose a general semantic construction of dependent refinement type systems from underlying type systems and predicate logic, that is, a construction of liftings of closed comprehension categories from given (underlying) closed comprehension categories and posetal fibrations for predicate logic. We give sufficient conditions to lift structures such as dependent products, dependent sums, computational effects, and recursion from the underlying type systems to dependent refinement type systems. We demonstrate the usage of our construction by giving semantics to a dependent refinement type system and proving soundness.

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Dependent Types and Fibred Computational Effects

Cited by 26 publications

References 35 publications

A reasonably exceptional type theory

A reasonably exceptional type theory

Gradual type theory

A General Semantic Construction of Dependent Refinement Type Systems, Categorically

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