Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T18:35:05.106Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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

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