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The range property fails for H

Published online by Cambridge University Press:  12 March 2014

Andrew Polonsky
Andrew Polonsky*
Affiliation:
University of Nijmegen, Faculty of Science, Neijendaalseweg 135, 6525AJ Nijmegen, The Netherlands, E-mail: andrew.polonsky@gmail.com
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Abstract

We work in , the untyped λ-calculus in which all unsolvables are identified. We resolve a conjecture of Barendregt asserting that the range of a definable map is either infinite or a singleton. This is refuted by constructing a λ-term Ξ such that ΞM = ΞI ⇔ ΞM ≠ ΞΩ. The construction generalizes to ranges of any finite size, and to some other sensible lambda theories.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 4 , December 2012 , pp. 1195 - 1210
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCE

Barendregt, Henk [1984], The lambda calculus: Its syntax and semantics, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 103, North-Holland, Amsterdam.Google Scholar
Barendregt, Henk [1993], Constructive proofs of the range property in lambda calculus, Theoretical Computer Science, vol. 121, no. 1-2, pp. 59–69, Corrado Böhm Festschrift.CrossRefGoogle Scholar
Barendregt, Henk [2008], Towards the range property for the lambda theory H, Theoretical Computer Science, vol. 398, no. 1-3, pp. 12–15, Calculi, Types and Applications: Essays in honour of M. Coppo, M. Dezani-Ciancaglini and S. Ronchi Della Rocca.CrossRefGoogle Scholar
Barendregt, Henk, Bergstra, Jan, Klop, Jan Willem, and Volken, Henri [1976], Degrees, reductions and representability in the Lambda Calculus, Preprint no. 22, University of Utrecht, Department of Mathematics.Google Scholar
Böhm, Corrado [1968], Alcune proprieta delle forme βη-normali nel λΚ-calculus, Pubblicazioni 696, Instituto per le Applicazioni del Calcolo, Roma.Google Scholar
Intrigila, Benedetto and Statman, Richard [2007], On Henk Barendregt's Favorite Open Problem, Reflections on type theory, lambda calculus, and the mind (Barendsen, Eriket al., editors), Radboud University, Henk Barendregt Festschrift.Google Scholar
Intrigila, Benedetto and Statman, Richard [2011], Solution to the range problem for combinatory logic, Fundamenta Informaticae, vol. 111, no. 2, pp. 203–222.CrossRefGoogle Scholar
Statman, Richard [1993], Does the range property hold for the λ-theory H?, TLCA list of open problems, http://tlca.di.unito.it/opltlca/.Google Scholar
Visser, Albert [1980], Numerations, λ-calculus, and Arithmetic, To H. B. Curry: Essays on combinatory logic, lambda calculus, and formalism (Seldin, J.P. and Hindley, J.R., editors). Academic Press, pp. 259–284.Google Scholar