A construction of one-point bases in extended lambda calculi

Goldberg, Mayer

doi:10.1016/j.ipl.2003.12.005

Cited by 2 publications

(3 citation statements)

References 5 publications

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“…Derivation of the Arity-Generic One-Point Basis Maker. In a previous work [12], we have shown that for any n λ-terms E 1 , . .…”

Section: 10mentioning

confidence: 86%

Ellipses and Lambda Definability

Goldberg

2015

Logical Methods in Computer Science

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Ellipses are a meta-linguistic notation for denoting terms the size of which are specified by a meta-variable that ranges over the natural numbers. In this work, we present a systematic approach for encoding such meta-expressions in the \^I-calculus, without ellipses: Terms that are parameterized by meta-variables are replaced with corresponding \^I-abstractions over actual variables. We call such \^I-terms arity-generic. Concrete terms, for particular choices of the parameterizing variable are obtained by applying an arity-generic \^I-term to the corresponding numeral, obviating the need to use ellipses. For example, to find the multiple fixed points of n equations, n different \^I-terms are needed, every one of which is indexed by two meta-variables, and defined using three levels of ellipses. A single arity-generic \^I-abstraction that takes two Church numerals, one for the number of fixed-point equations, and one for their arity, replaces all these multiple fixed-point combinators. We show how to define arity-generic generalizations of two historical fixed-point combinators, the first by Curry, and the second by Turing, for defining multiple fixed points. These historical fixed-point combinators are related by a construction due to B\~Ahm: We show that likewise, their arity-generic generalizations are related by an arity-generic generalization of B\~Ahm's construction. We further demonstrate this approach to arity-generic \^I-definability with additional \^I-terms that create, project, extend, reverse, and map over ordered n-tuples, as well as an arity-generic generator for one-point bases.Comment: 31 page

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“…Derivation of the Arity-Generic One-Point Basis Maker. In a previous work [12], we have shown that for any n λ-terms E 1 , . .…”

Section: 10mentioning

confidence: 86%

Ellipses and Lambda Definability

Goldberg

2015

Logical Methods in Computer Science

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“…It is shown in [9] that a countable number of (somewhat artificially constructed) 1-point bases exist for combinator calculi, but we will focus here on Rosser's X-combinator, one of the simplest 1-point bases that is naturally connected through mutual definitions to the combinators K and S.…”

Section: X-combinator Treesmentioning

confidence: 99%

“…One-point combinator bases, together with a derivation of the X-combinator are described in [7]. In [9] the existence of a countable number of 1-point bases is proven. While esoteric programming languages exist based on similar 1-point bases [24], we have not seen any such development centered around Rosser's X-combinator, or type inference and normalization algorithms designed specifically for it, as described in this paper.…”

Section: Related Workmentioning

confidence: 99%

On a uniform representation of combinators, arithmetic, lambda terms and types

Tarau

2015

Proceedings of the 17th International Symposium on Principles and Practice of Declarative Programming

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A uniform representation, as binary trees with empty leaves, is given to expressions built with Rosser's X-combinator, natural numbers, lambda terms and simple types. Type inference, normalization of combinator expressions and lambda terms in de Bruijn notation, ranking/unranking algorithms and tree-based natural numbers are described as a literate Prolog program.With sound unification and compact expression of combinatorial generation algorithms, logic programming is shown to conveniently host a declarative playground where interesting properties and behaviors emerge from the interaction of heterogenous but deeply connected computational objects.

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A construction of one-point bases in extended lambda calculi

Cited by 2 publications

References 5 publications

Ellipses and Lambda Definability

Ellipses and Lambda Definability

On a uniform representation of combinators, arithmetic, lambda terms and types

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