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A combinatory account of internal structure

Published online by Cambridge University Press:  12 March 2014

Barry Jay  and
Thomas Given-Wilson
Barry Jay
Affiliation:
University of Technology Sydney, School of Software, P.O. Box 123, Broadway, 2007, Australia, E-mail: Barry.Jay@uts.edu.au, E-mail: Thomas.Given-Wilson@uts.edu.au
Thomas Given-Wilson
Affiliation:
University of Technology Sydney, School of Software, P.O. Box 123, Broadway, 2007, Australia, E-mail: Barry.Jay@uts.edu.au, E-mail: Thomas.Given-Wilson@uts.edu.au
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 807 - 826
Copyright
Copyright © Association for Symbolic Logic 2011

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