Self Types for Dependently Typed Lambda Encodings

Fu, Peng; Stump, Aaron

doi:10.1007/978-3-319-08918-8_16

Cited by 6 publications

(7 citation statements)

References 11 publications

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“…One might suspect that a coinductive formulation of DOT, and of rule (T-{}-I) in particular, might allow making (D-Typ-Abs) sound without using later. However, DOT's -types (and Amin et al's refinements [2012]) differ from standard recursive types, and resemble more closely recursively defined signatures [Crary et al 1999], Cedille's -types [Fu and Stump 2014], and dependent intersections [Kopylov 2003].…”

Section: Related Workmentioning

confidence: 99%

Scala step-by-step: soundness for DOT with step-indexed logical relations in Iris

Giarrusso

Stefanesco

Timany

et al. 2020

Proc. ACM Program. Lang.

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The metatheory of Scala's core type system Ð the Dependent Object Types (DOT) calculus Ð is hard to extend, like the metatheory of other type systems combining subtyping and dependent types. Soundness of important Scala features therefore remains an open problem in theory and in practice. To address some of these problems, we use a semantics-first approach to develop a logical relations model for a new version of DOT, called guarded DOT (gDOT). Our logical relations model makes use of an abstract form of step-indexing, as supported by the Iris framework, to model various forms of recursion in gDOT. To demonstrate the expressiveness of gDOT, we show that it handles Scala examples that could not be handled by previous versions of DOT, and prove using our logical relations model that gDOT provides the desired data abstraction. The gDOT type system, its semantic model, its soundness proofs, and all examples in the paper have been mechanized in Coq.

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Section: Related Workmentioning

confidence: 99%

Scala step-by-step: soundness for DOT with step-indexed logical relations in Iris

Giarrusso

Stefanesco

Timany

et al. 2020

Proc. ACM Program. Lang.

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“…The present paper proposes two new type constructs for this, called constructor-constrained recursive types and lifting. The former deepens earlier work by Fu and Stump on System S, which solves the problem of induction using a typing construct called self types, to allow the type to refer to the subject of the typing via bound variable x in ιx.T (Fu & Stump, 2014). To prove consistency, they rely on a dependencyeliminating translation to System F ω plus positive-recursive types.…”

Section: Stumpmentioning

confidence: 99%

“…To do this, we would unfold the definition of Nat, and then before we could add local variables s and z to the context, we would be forced again to kind (P Z), since this would be the type for z. There is a circularity here, which System S avoided by using an ad-hoc form of mutually recursive types (Fu & Stump, 2014).…”

Section: Constructor-constrained Recursive Typesmentioning

confidence: 99%

The calculus of dependent lambda eliminations

Stump¹

2017

J. Funct. Prog.

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Modern constructive type theory is based on pure dependently typed lambda calculus, augmented with user-defined datatypes. This paper presents an alternative called the Calculus of Dependent Lambda Eliminations, based on pure lambda encodings with no auxiliary datatype system. New typing constructs are defined that enable induction, as well as large eliminations with lambda encodings. These constructs are constructor-constrained recursive types, and a lifting operation to lift simply typed terms to the type level. Using a lattice-theoretic denotational semantics for types, the language is proved logically consistent. The power of CDLE is demonstrated through several examples, which have been checked with a prototype implementation called Cedille.

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“…The following definitions constitute a new formulation of a system in (Fu & Stump, 2014). The syntax for kinds, types, terms, and contexts is:…”

Section: Syntaxmentioning

confidence: 99%

“…In Appendix C of (Fu & Stump, 2014), the following theorem is proved for a more declarative formulation of F rec ω : folding and unfolding of recursive types take place as part of a nonalgorithmic definitional equality, and the system is a type-assignment system (so the only term constructs are those of pure untyped lambda calculus). Similarly to the proof in (Abel & Matthes, 2004), a complete lattice ( κ , ⊆ κ , ∩ κ ) is defined, for each kind κ.…”

Section: Strong Normalizationmentioning

confidence: 99%

Efficiency of lambda-encodings in total type theory

Stump

2016

J. Funct. Prog.

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This paper proposes a new typed lambda-encoding for inductive types which, for Peano numerals, has the expected time complexities for basic operations like addition and multiplication, has a constant-time predecessor function, and requires only quadratic space to encode a numeral. This improves on the exponential space required by the Parigot encoding. Like the Parigot encoding, the new encoding is typable in System F-omega plus positive-recursive type definitions, a total type theory. The new encoding is compared with previous ones through a significant case study: mergesort using Braun trees. The practical runtime efficiency of the new encoding, and the Church and Parigot encodings, are compared by two translations, one to Racket and one to Haskell, on a small suite of benchmarks.

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Self Types for Dependently Typed Lambda Encodings

Cited by 6 publications

References 11 publications

Scala step-by-step: soundness for DOT with step-indexed logical relations in Iris

Scala step-by-step: soundness for DOT with step-indexed logical relations in Iris

The calculus of dependent lambda eliminations

Efficiency of lambda-encodings in total type theory

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