Deciding FO-definability of regular languages
Abstract
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.
Keywords
Cite
@article{arxiv.2105.06202,
title = {Deciding FO-definability of regular languages},
author = {Agi Kurucz and Vladislav Ryzhikov and Yury Savateev and Michael Zakharyaschev},
journal= {arXiv preprint arXiv:2105.06202},
year = {2021}
}