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.

翻译：我们证明,类似于已知的承认FO( < )-DFA语言L(A)定义的FO( < )-决定FO( <,C)-和FO( <,MOD)-确定性决定FO( <,C)-决定FO()-决定FO( <,MOD)-确定性决定的FO()-决定FO( <,C)-决定FO()-决定FO()-确定性决定的FO( <,OD)-确定性是PSMFO( < )-确定性能的完整性能,与已知的承认FOFO( <,MOD)-确认FO()-确定性能(FO)-确定性能(FO)-确定性能(FO)-确定性能(FOFO)-确定性能(FFO)-确定性能性能(L)的确定性能(FOFO( < )- 和FO( < )- )- 和FOC0/ACC0)- 确定性能(FO(FO)- 确定性能定义性能定义性能(FOFOFO)的确定性能特性)的确定性能性能性能特性的确定性能(PFO)的确定性特征特性特性特性)的确定性能性(PFO(PFO)的确定性能特性的确定性能特征)的确定性能性能性能的确定性能特性的确定性能(PFOFO( < )的确定性能性能性能性能性能特征)的确定性( < )的确定性能( < )的确定性能特征特性)的确定性( < )的确定性( < )的确定性( < )的确定性( < )的确定性( < )的确定性(PFOFO( < )的确定性( < )的确定性能特征)的确定性( < )-C)-C)的确定性能性能( < 和FOFOFOFO( < )性能性能性能