Variants in the Infinitary Unification Wonderland

Abstract

So far, results about variants, the finite variant property (FVP), and variant unification have been developed for equational theories \(E \cup B\) where B is a set of axioms having a finitary unification algorithm, and the equations E, oriented as rewrite rules \(\vec {E}\), are convergent modulo B. The extension to the case when B has an infinitary unification algorithm, for example because of non-commutative symbols having associative axioms, seems undeveloped. This paper takes a first step in developing such an extension. In particular, the relationships between the FVP and the boundedness properties, the identification of conditions on \(E \cup B\) ensuring FVP, and the effective computation of variants and variant unifiers are explored in detail. The extension from the finitary to the infinitary case includes both surprises and opportunities.