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.

中文翻译：

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确定常规语言的FO可定义性

我们证明，与识别DFA A语言L（A）的FO（<）可定义性的已知PSpace相似，确定FO（<，C）-和FO（<，MOD）可定义性都是PSpace -完全的。（在这里，FO（<，C）用标准的全等模关系扩展一阶逻辑FO（<），而FO（<，MOD）用量词检查满足给定公式的位数是否可被除以在给定n> 1的情况下，已知这些FO语言定义了分别在AC0和ACC0中可确定的常规语言。）我们通过首先显示可以捕获L（A）的FO可定义性的已知代数表征来获得这些结果。通过使用A的过渡半形体的“可定位”属性。然后，使用我们的标准，推广FO（<）-可定义性的PSpace-hardness的已知证明，