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On the complexity of alpha conversion

Published online by Cambridge University Press:  12 March 2014

Rick Statman
Rick Statman*
Affiliation:
Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA 15213, USA. E-mail: [email protected]
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1197 - 1203
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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