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On adding (ξ) to weak equality in combinatory logic

Published online by Cambridge University Press:  12 March 2014

Martin W. Bunder ,
J. Roger Hindley  and
Jonathan P. Seldin
Martin W. Bunder
Affiliation:
Department of Mathematics, University of Wollongong, Wollongong, New South Wales, Australia Department of Mathematics, Concordia University, Montréal, Québec, Canada
J. Roger Hindley
Affiliation:
Mathematics Division, University CollegeSwansea, Wales Department of Mathematics, Concordia University, Montréal, Québec, Canada
Jonathan P. Seldin
Affiliation:
Odyssée Recherches Appliquées Montréal, Québec H2V 1K6, Canada Department of Mathematics, Concordia University, Montréal, Québec, Canada
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Abstract

Because the main difference between combinatory weak equality and λβ-equality is that the rule

is valid for the latter but not the former, it is easy to assume that another way of defining combinatory β-equality is to add rule (ξ) to the postulates for weak equality. However, to make this true, one must choose the definition of combinatory abstraction in (ξ) very carefully. If one tries to use one of the more common abstraction algorithms, the result will be an equality, =ξ, that is either equivalent to βη-equality (and so strictly stronger than β-equality) or else strictly weaker than β-equality. This paper will study the relations =ξ for several commonly used abstraction-algorithms, distinguish between them, and axiomatize them.

Keywords

Combinatory logiclambda-calculusequalityrule (ξ)monotonic ruleabstraction algorithm.
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 590 - 607
Copyright
Copyright © Association for Symbolic Logic 1989

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References

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