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September 2011 A combinatory account of internal structure
Barry Jay, Thomas Given-Wilson
J. Symbolic Logic 76(3): 807-826 (September 2011). DOI: 10.2178/jsl/1309952521

Abstract

Traditional combinatory logic uses combinators S and K to represent all Turing-computable functions on natural numbers, but there are Turing-computable functions on the combinators themselves that cannot be so represented, because they access internal structure in ways that S and K cannot. Much of this expressive power is captured by adding a factorisation combinator F. The resulting SF-calculus is structure complete, in that it supports all pattern-matching functions whose patterns are in normal form, including a function that decides structural equality of arbitrary normal forms. A general characterisation of the structure complete, confluent combinatory calculi is given along with some examples. These are able to represent all their Turing-computable functions whose domain is limited to normal forms. The combinator F can be typed using an existential type to represent internal type information.

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Barry Jay. Thomas Given-Wilson. "A combinatory account of internal structure." J. Symbolic Logic 76 (3) 807 - 826, September 2011. https://doi.org/10.2178/jsl/1309952521

Information

Published: September 2011
First available in Project Euclid: 6 July 2011

zbMATH: 1248.03025
MathSciNet: MR2849246
Digital Object Identifier: 10.2178/jsl/1309952521

Rights: Copyright © 2011 Association for Symbolic Logic

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Vol.76 • No. 3 • September 2011
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