Two-Variable First Order Logic with Counting Quantifiers: Complexity Results.

Etessami et al. [5] showed that satisfiability of two-variable first order logic FO^2 [<] on word models is Nexptime-complete. We extend this upper bound to the slightly stronger logic FO^2 [ <,succ ,≡ ], which allows checking whether a word position is congruent to r modulo q, for some divisor q and remainder r. If we allow the more powerful modulo counting quantifiers of Straubing, Thérien et al. [22] (we call this two-variable fragment FOmod ^2 [ <,succ ]), satisfiability becomes Expspace-complete. A more general counting quantifier, FOunC ^2 [ <,succ ], makes the logic undecidable.