Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T20:18:37.227Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Barendregt, H. P. [1984], The lambda calculus: its syntax and semantics, 2nd ed., North-Holland, Amsterdam.Google Scholar
Curry, H. B. [1930], Grundlagen der kombinatorischen Logik, American Journal of Mathematics, vol. 52, pp. 509–536, 789–834.CrossRefGoogle Scholar
Curry, H. B. [1933], Apparent variables from the standpoint of combinatory logic, Annals of Mathematics, ser. 2, vol. 34, pp. 381–404.CrossRefGoogle Scholar
Curry, H. B. [1941], Consistency and completeness of the theory of combinators, this Journal, vol. 6, pp. 54–61.Google Scholar
Curry, H. B. [1949], A simplification of the theory of combinators, Synthese, vol. 7, pp. 7–391.Google Scholar
Curry, H. B. and Feys, R. [1958], Combinatory logic. Vol I, North-Holland, Amsterdam.Google Scholar
Curry, H. B., Hindley, J. R. and Seldin, J. P. [1972], Combinatory logic. Vol. II, North-Holland, Amsterdam.Google Scholar
Hindley, J. R. and Seldin, J. P. [1986], Introduction to combinators and λ-calculus, Cambridge University Press, Cambridge.Google Scholar
Lambek, J. [1980], From λ-calculus to cartesian closed categories, To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism (Seldin, J. P. and Hindley, J. R., editors), Academic Press, London, pp. 375–402.Google Scholar
Rosser, J. B. [1935], A mathematical logic without variables. I, II, Annals of Mathematics, ser. 2, vol. 36, pp. 127–150; Duke Mathematical Journal, vol. 1, pp. 1–328.CrossRefGoogle Scholar
Seldin, J. P. [1983], Remarks on β-strong equality, this Journal, vol. 48, pp. 902–903 (abstract).Google Scholar