🤖 AI Summary
This paper investigates the problem of determining whether a given completely reachable deterministic finite automaton (DFA) has the property that every nonempty subset of its state set is the image of the full state set under some input word. First, we present the first exact algorithm for this problem with time complexity $O(|Sigma| cdot n^2)$, which is quadratically optimal. Second, we prove the weak Don conjecture, thereby establishing a tight upper bound of $O(n^2)$ on the reset threshold. Moreover, for any nonempty subset $S$, we derive a sharp upper bound $2n(n-|S|) - n cdot H_{n-|S|}$ on the length of the shortest word mapping the full state set onto $S$, where $H_k$ denotes the $k$-th harmonic number—unifying and generalizing prior results for several classes of automata. Our approach integrates combinatorial automata theory, directed graph modeling, inductive analysis, and harmonic inequality estimation.
📝 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.
