Translator Disclaimer
December 2007 On the complexity of alpha conversion
Rick Statman
J. Symbolic Logic 72(4): 1197-1203 (December 2007). DOI: 10.2178/jsl/1203350781


We consider three problems concerning alpha conversion of closed terms (combinators).

1. Given a combinator M find the an alpha convert of M with a smallest number of distinct variables.

2. Given two alpha convertible combinators M and N find a shortest alpha conversion of M to N.

3. Given two alpha convertible combinators M and N find an alpha conversion of M to N which uses the smallest number of variables possible along the way.

We obtain the following results.

1. There is a polynomial time algorithm for solving problem (1). It is reducible to vertex coloring of chordal graphs.

2. Problem (2) is co-NP complete (in recognition form). The general feedback vertex set problem for digraphs is reducible to problem (2).

3. At most one variable besides those occurring in both M and N is necessary. This appears to be the folklore but the proof is not familiar. A polynomial time algorithm for the alpha conversion of M to N using at most one extra variable is given.

There is a tradeoff between solutions to problem (2) and problem (3) which we do not fully understand.


Download Citation

Rick Statman. "On the complexity of alpha conversion." J. Symbolic Logic 72 (4) 1197 - 1203, December 2007.


Published: December 2007
First available in Project Euclid: 18 February 2008

zbMATH: 1130.03011
MathSciNet: MR2371200
Digital Object Identifier: 10.2178/jsl/1203350781

Rights: Copyright © 2007 Association for Symbolic Logic


This article is only available to subscribers.
It is not available for individual sale.

Vol.72 • No. 4 • December 2007
Back to Top