Canonization for Lk-equivalence is Hard

Let Lk be the k-variable fragment of first-order logic, for some k 3. We prove that equivalence of finite structures in Lk has no P-computable canonization function unless NP P=poly. The latter assumption is considered as highly unlikely; in particular it implies a collapse of the polynomial hierarchy. The question for such a canonization function came up in the context of the problem of whether there is a logic for P. Slight modifications of our result yield answers to questions of Dawar, Lindell, and Weinstein [4] and Otto [16] concerning the inversion of the so-called Lk-invariants.