Capture-avoiding substitution as a nominal algebra

Authors:
Murdoch J. Gabbay;Aad Mathijssen
Affiliations:
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, Scotland, Great Britain;Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, MB, The Netherlands
Venue:
ICTAC'06 Proceedings of the Third international conference on Theoretical Aspects of Computing
Year:
2006

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Abstract

Substitution is fundamental to computer science, underlying for example quantifiers in predicate logic and beta-reduction in the lambda-calculus. So is substitution something we define on syntax on a case-by-case basis, or can we turn the idea of ‘substitution’ into a mathematical object? We exploit the new framework of Nominal Algebra to axiomatise substitution. We prove our axioms sound and complete with respect to a canonical model; this turns out to be quite hard, involving subtle use of results of rewriting and algebra.