We consider the two-variable fragment of first-order logic with counting, subject to the stipulation that a sin-gle distinguished binary predicate be interpreted as an equivalence. We show that the satisfiability and finite satisfiability problems for this logic are both NEXPTIME-complete. We further show that the corresponding problems for two-variable first-order logic with counting and two equivalences are both undecidable. Copyright line will be provided by the publisher
