Efficiency of lambda-encodings in total type theory

Stump, Aaron; Fu, Peng

doi:10.1017/s0956796816000034

Cited by 8 publications

(9 citation statements)

References 11 publications

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“…Unfortunately, Church-style and Mendler-style encodings suffer from the linear time predecessor function. The possible alternatives are Parigot and Stump-Fu encodings [15,20]. Parigot encoding represents datatypes as their own recursors which allows to have a constant time predecessor.…”

Section: Discussionmentioning

confidence: 99%

Generic derivation of induction for impredicative encodings in Cedille

Firsov

Stump

2018

Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018

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This paper presents generic derivations of induction for impredicatively typed lambda-encoded datatypes, in the Cedille type theory. Cedille is a pure type theory extending the Curry-style Calculus of Constructions with implicit products, primitive heterogeneous equality, and dependent intersections. All data erase to pure lambda terms, and there is no built-in notion of datatype. The derivations are generic in the sense that we derive induction for any datatype which arises as the least fixed point of a signature functor. We consider Church-style and Mendler-style lambda-encodings. Moreover, the isomorphism of these encodings is proved. Also, we formalize Lambek's lemma as a consequence of expected laws of cancellation, reflection, and fusion.

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Section: Discussionmentioning

confidence: 99%

Generic derivation of induction for impredicative encodings in Cedille

Firsov

Stump

2018

Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018

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“…s 1 (s 0 z). While in theory the space required for normal forms is exponential, in practice closure-based implementations of lambda calculus compute efficiently with Parigot encodings, as has been found in several studies (Koopman et al, 2014;Stump & Fu, 2016). And Parigot encodings can be typed in a normalizing extension of System F with positive-recursive types (cf.…”

Section: The Problems In More Detailmentioning

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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“…This is not the final definition of algebra, though, because as formulated so far, there is no support for iteration. So the encoding would be more like a Scott encoding than a Church encoding (see [30] for a comparison). To support iteration, the algebra must be given a way to evaluate the value of type Trmga · Alg ·Y returned by its input.…”

Section: A Solution Using Kripke Function Spacesmentioning

confidence: 99%

A Weakly Initial Algebra for Higher-Order Abstract Syntax in Cedille

Stump

2019

Electron. Proc. Theor. Comput. Sci.

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Cedille is a relatively recent tool based on a Curry-style pure type theory, without a primitive datatype system. Using novel techniques based on dependent intersection types, inductive datatypes with their induction principles are derived. One benefit of this approach is that it allows exploration of new or advanced forms of inductive datatypes. This paper reports work in progress on one such form, namely higher-order abstract syntax (HOAS). We consider the nature of HOAS in the setting of pure type theory, comparing with the traditional concept of environment models for lambda calculus. We see an alternative, based on what we term Kripke function-spaces, for which we can derive a weakly initial algebra in Cedille. Several examples are given using the encoding.

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Efficiency of lambda-encodings in total type theory

Cited by 8 publications

References 11 publications

Generic derivation of induction for impredicative encodings in Cedille

Generic derivation of induction for impredicative encodings in Cedille

The calculus of dependent lambda eliminations

A Weakly Initial Algebra for Higher-Order Abstract Syntax in Cedille

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