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