# Two-Variable First Order Logic with Counting Quantifiers: Complexity Results

@inproceedings{Lodaya2017TwoVariableFO,
title={Two-Variable First Order Logic with Counting Quantifiers: Complexity Results},
author={Kamal Lodaya and A. V. Sreejith},
booktitle={International Conference on Developments in Language Theory},
year={2017}
}
• Published in
International Conference on…
7 August 2017
• Mathematics
Etessami et al. [5] showed that satisfiability of two-variable first order logic $$\mathrm {FO}^2$$[<] on word models is Nexptime-complete. We extend this upper bound to the slightly stronger logic $$\mathrm {FO}^2$$[$$<,succ ,\equiv$$], which allows checking whether a word position is congruent to r modulo q, for some divisor q and remainder r. If we allow the more powerful modulo counting quantifiers of Straubing, Therien et al. [22] (we call this two-variable fragment FOmod $$^2$$[\(<,succ…
• Computer Science, Mathematics
FSTTCS
• 2017
A small-model property of this logic is proved, which gives a technique for deciding the satisfiability problem for the two-variable fragment of the first-order logic extended with modulo counting quantifiers and interpreted over finite words or trees.
• Computer Science
FoSSaCS
• 2021
The main goal of this work is to study the effect of adding weak forms of percentage constraints to fragments of LTL, and sharpen most of the undecidability results on logics with arithmetics interpreted on words known from the literature, but also are fairly simple.

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It follows that the only sets of natural numbers which are definable are the finite sets and their complements, and the set of even numbers is not definable.