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

Abstract

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.

Citation

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Rick Statman. "On the complexity of alpha conversion." J. Symbolic Logic 72 (4) 1197 - 1203, December 2007. https://doi.org/10.2178/jsl/1203350781

Information

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

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Vol.72 • No. 4 • December 2007
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