# Datalog Program Equivalence with Fixed Universe Size

It is well known that to compute equivalence of two Datalog programs (or equivalently of two first order formulas with the least fixed point operator) wrt all possible inputs and universes, is undecidable. If we fix the universe size to be N, it is clearly decidable in time exponential wrt N. My question is whether one can do better, maybe even in time logarithmic in N (and maybe much worse wrt the length of the programs). Note that equivalence of two nonrecursive Datalog rules can be done in constant time for all inputs, let alone for fixed N, but is NP-Complete wrt the rules' size.