Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Deciding FO-definability of regular languages   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.

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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org