The Complexity of Finding Reset Words in Finite Automata

TL;DR

This paper investigates the computational complexity of finding reset words in deterministic finite automata, establishing new complexity classifications and answering a previously open question.

Contribution

It proves that deciding if a shortest reset word has length k is DP-complete and classifies related search problems within FP^NP and FP^NP[log], providing new insights into their computational difficulty.

Findings

Deciding shortest reset word length k is DP-complete.

Finding a shortest reset word is in FP^NP.

Computing a reset word of a given length is FNP-complete.

Abstract

We study several problems related to finding reset words in deterministic finite automata. In particular, we establish that the problem of deciding whether a shortest reset word has length k is complete for the complexity class DP. This result answers a question posed by Volkov. For the search problems of finding a shortest reset word and the length of a shortest reset word, we establish membership in the complexity classes FP^NP and FP^NP[log], respectively. Moreover, we show that both these problems are hard for FP^NP[log]. Finally, we observe that computing a reset word of a given length is FNP-complete.