On Automata Recognizing Birecurrent Sets

TL;DR

This paper investigates the computational complexity of recognizing birecurrent sets by automata, proving that the decision problem is PSPACE-complete even for simple binary automata, and explores related rank computation issues.

Contribution

It establishes the PSPACE-completeness of recognizing birecurrent sets in automata, including special cases with binary and complete automata, and examines related rank problems.

Findings

Deciding birecurrence recognition is PSPACE-complete.

The problem remains PSPACE-complete for binary automata with all states accepting.

The paper explores the complexity of computing the rank of a partial DFA.

Abstract

In this note we study automata recognizing birecurrent sets. A set of words is birecurrent if the minimal partial DFA recognizing this set and the minimal partial DFA recognizing the reversal of this set are both strongly connected. This notion was introduced by Perrin, and Dolce et al. provided a characterization of such sets. We prove that deciding whether a partial DFA recognizes a birecurrent set is a PSPACE-complete problem. We show that this problem is PSPACE-complete even in the case of binary partial DFAs with all states accepting and in the case of binary complete DFAs. We also consider a related problem of computing the rank of a partial DFA.