  We prove that, similarly to known PSpace-completeness of recognising
FO(<)-definability of the language L(A) of a DFA A, deciding both FO(<,C)- and
FO(<,MOD)-definability are PSpace-complete. (Here, FO(<,C) extends the
first-order logic FO(<) with the standard congruence modulo n relation, and
FO(<,MOD) with the quantifiers checking whether the number of positions
satisfying a given formula is divisible by a given n>1. These FO-languages are
known to define regular languages that are decidable in AC0 and ACC0,
respectively.) We obtain these results by first showing that known algebraic
characterisations of FO-definability 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 two-way
NFAs.
