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arXiv · 2605.02957

Nondeterministic state complexity of square root

Sergey Onishchenko
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Abstract

We investigate the nondeterministic state complexity of the square-root operation $\sqrt{L}=\{\,w \mid ww\in L\,\}$ on regular languages represented by nondeterministic finite automata. For an $n$-state NFA accepting $L$, it was previously known that $\sqrt{L}$ can be accepted by an NFA with at most $n^{3}$ states, while the best lower bound was only (n-1)(n-2)(n-3). In this paper, we close this gap completely and prove that $n^{3}$ states are sufficient and necessary in the worst case.

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Sergey Onishchenko. 2026-05-02. Nondeterministic state complexity of square root. https://arxiv.org/abs/2605.02957

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