Abstract
We prove that, similarly to known \(\textsc {PSpace}\)-completeness of recognising \(\mathsf {FO}(<)\)-definability of the language \({\boldsymbol{L}}(\mathfrak A)\) of a DFA \(\mathfrak A\), deciding both \(\mathsf {FO}(<,\equiv )\)- and \(\mathsf {FO}(<,\mathsf {MOD})\)-definability (corresponding to circuit complexity in \({\textsc {AC}^0}\) and \({\textsc {ACC}^0}\)) are \(\textsc {PSpace}\)-complete. We obtain these results by first showing that known algebraic characterisations of FO-definability of \({\boldsymbol{L}}(\mathfrak A)\) can be captured by ‘localisable’ properties of the transition monoid of \(\mathfrak A\). Using our criterion, we then generalise the known proof of \(\textsc {PSpace}\)-hardness of \(\mathsf {FO}(<)\)-definability, and establish the upper bounds not only for arbitrary DFAs but also for 2NFAs.
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This work was supported by UK EPSRC EP/S032282.
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Kurucz, A., Ryzhikov, V., Savateev, Y., Zakharyaschev, M. (2021). Deciding FO-definability of Regular Languages. In: Fahrenberg, U., Gehrke, M., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science(), vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_15
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