The Finite Variant Property: How to Get Rid of Some Algebraic Properties

We consider the following problem: Given a term t, a rewrite system $\cal R$, a finite set of equations E′ such that $\cal R$ is E′-convergent, compute finitely many instances of t: t1,...,tn such that, for every substitution σ, there is an index i and a substitution θ such that $t\sigma\mathord\downarrow =_{E'} t_i\theta$ (where $t\sigma\mathord\downarrow$ is the normal form of tσ w.r.t. $\to_{E'\mathord{\setminus}\mathcal R}$). The goal of this paper is to give equivalent (resp. sufficient) conditions for the finite variant property and to systematically investigate this property for equational theories, which are relevant to security protocols verification. For instance, we prove that the finite variant property holds for Abelian Groups, and a theory of modular exponentiation and does not hold for the theory ACUNh (Associativity, Commutativity, Unit, Nilpotence, homomorphism).