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Journal of Logic and Computation Advance Access published online on September 13, 2009
Journal of Logic and Computation, doi:10.1093/logcom/exp049
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© The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

A Nominal Axiomatization of the Lambda Calculus

Murdoch J. Gabbay 1

School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, Great Britain.

Aad Mathijssen

Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
E-mail: a.h.j.mathijssen{at}tue.nl

Received 12 June 2008.


  Abstract

The lambda calculus is fundamental in computer science. It resists an algebraic treatment because of capture-avoidance sideconditions. Nominal algebra is a logic of equality designed for specifications involving binding. We axiomatize the lambda calculus using nominal algebra, demonstrate how proofs with these axioms reflect the informal arguments on syntax and we prove the axioms to be sound and complete. We consider both non-extensional and extensional versions (alpha-beta and alpha-beta-eta equivalence). This connects the nominal approach to names and binding with the view of variables as a syntactic convenience for describing functions. The axiomatization is finite, close to informal practice and it fits into a context of other research such as nominal rewriting and nominal sets.

Keywords: Lambda calculus; equational logic; nominal techniques

1Homepage: www.gabbay.org.uk


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