Completely reachable automata: a quadratic decision algorithm and a quadratic upper bound on the reaching threshold

TL;DR

This paper introduces a quadratic-time algorithm to determine if a DFA is completely reachable and proves a quadratic upper bound on the length of the shortest words reaching any subset, advancing understanding of automata synchronization.

Contribution

It provides the first efficient algorithm for deciding complete reachability and establishes a quadratic bound on the reaching threshold for this class of automata.

Findings

Algorithm runs in O(|Σ|·n^2) time for deciding complete reachability.

Proves a quadratic upper bound on the length of shortest words reaching subsets.

Generalizes bounds for subclasses of completely reachable automata.

Abstract

A complete deterministic finite (semi)automaton (DFA) with a set of states $Q$ is \emph{completely reachable} if every nonempty subset of $Q$ is the image of the action of some word applied to $Q$. The concept of completely reachable automata appeared, in particular, in connection with synchronizing automata; the class contains the \v{C}ern{\'y} automata and covers several distinguished subclasses. The notion was introduced by Bondar and Volkov (2016), who also raised the question about the complexity of deciding if an automaton is completely reachable. We develop an algorithm solving this problem, which works in ${\mathcal{O}(|\Sigma|\cdot n^2)}$ time and $\mathcal{O}(|\Sigma|\cdot n)$ space, where $n=|Q|$ is the number of states and $|\Sigma|$ is the size of the input alphabet. In the second part, we prove a weak Don's conjecture for this class of automata: a nonempty subset of states $S \subseteq Q$ is reachable with a word of length at most $2n(n-|S|) - n \cdot H_{n-|S|}$, where $H_i$ is the $i$-th harmonic number. This implies a quadratic upper bound in $n$ on the length of the shortest synchronizing words (reset threshold) for the class of completely reachable automata and generalizes earlier upper bounds derived for its subclasses.