Deciding FO-definability of regular languages

Abstract

We prove that, similarly to known PSpace-completeness of recognising FO(<)-definability of the language L(A) of a DFA A, deciding both FO(<, ≡)-and FO(<, MOD)-definability (corresponding to circuit complexity in AC 0 and ACC 0 ) are PSpace-complete. We obtain these results by first showing that known algebraic characterisations of FOdefinability of L(A) can be captured by 'localisable' properties of the transition monoid of A. Using our criterion, we then generalise the known proof of PSpace-hardness of FO(<)-definability, and establish the upper bounds not only for arbitrary DFAs but also for 2NFAs.