Nondeterministic state complexity of square root

TL;DR

This paper determines the exact nondeterministic state complexity of the square root operation on regular languages, proving that n^3 states are both sufficient and necessary for an n-state NFA.

Contribution

It establishes the tight bound of n^3 states for the nondeterministic state complexity of the square root operation, closing previous gaps in the understanding.

Findings

n^3 states are sufficient for the square root operation.

n^3 states are necessary in the worst case.

The result precisely characterizes the nondeterministic state complexity.

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.