Analysis of cryptographic protocols in a symbolic model is relative to a
deduction system that models the possible actions of an attacker regarding an
execution of this protocol. We present in this paper a transformation algorithm
for such deduction systems provided the equational theory has the finite
variant property. the termination of this transformation entails the
decidability of the ground reachability problems. We prove that it is necessary
to add one other condition to obtain the decidability of non-ground problems,
and provide one new such criterion
