Recognizing Completely Reachable Automata in Quadratic Time

A complete deterministic finite (semi)automaton (DFA) with a set of states

\( Q \)

is
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 Černý 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 whether 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.