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

Etessami, Vardi and Wilke [5] showed that satisfiability of two-variable first order logic FO[<] on word models is Nexptime-complete. We extend this upper bound to the slightly stronger logic FO[<, 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 and Thomas [22] (we call this two-variable fragment FOmod[<, succ]), satisfiability becomes Expspace-complete. A more general counting quantifier, FOunC[<, succ], makes the logic undecidable.