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

Abstract

Etessami et al. [5] showed that satisfiability of two-variable first order logic \(\mathrm {FO}^2\)[<] on word models is Nexptime-complete. We extend this upper bound to the slightly stronger logic \(\mathrm {FO}^2\)[\(<,succ ,\equiv \)], 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, Therien 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.