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

TL;DR

The paper shows that determining whether a DFA is completely reachable can be done in $O(|\Sigma|\cdot n^2)$ time using a reduction-graph framework and complementary set representations. It introduces the witness concept as a certificate of non-complete reachability and develops efficient procedures to verify witnesses and to extract reductions, enabling a polynomial-time decision algorithm. Furthermore, it proves a quadratic upper bound on the length to reach any nonempty subset, namely $2n(n-|S|) - n\cdot H_{n-|S|}$, which implies a quadratic bound on the reset threshold for completely reachable automata and strengthens results across several subclasses. The results yield practical insights into the structure of completely reachable automata, including an alphabet-size bound of at most $2n-2$ for minimal complete alphabets and connections to the Černý conjecture under certain orbit-size conditions.

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 Čern{ý} 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 ${\mathcal{O}(|Σ|\cdot n^2)}$ time and $\mathcal{O}(|Σ|\cdot n)$ space, where $n=|Q|$ is the number of states and $|Σ|$ 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.

Figures (5)

• Figure 1: Two completely reachable automata; the left one is from the Černý series Cerny1964. The transitions of permutational letters are dashed and those of singular letters are solid.
• Figure 2: Two automata that are not completely reachable. The transitions of permutational letters are dashed and those of singular letters are solid.
• Figure 3: A possible sequence of computed reduction graphs for the not completely reachable automaton from \ref{['fig:examples-notcompletely']} (right) (also shown above). In each step, the dashed line denotes the added edge or the dashed circle denotes the unified vertex.
• Figure 4: The scheme of the reduction construction in the proof of \ref{['thm:PEW-PSPACE']}.
• Figure 5: An example construction from the reduction in the proof of \ref{['thm:PEW-PSPACE']}. Two input automata are shown above and the construction is shown below. Some transitions are omitted: for all $x \in Q_i$, $y \in Q_j$ with $i \neq j$, the transitions of the transitional letters $c_{\alpha,x}$ fix the states $y$ and $t_{\alpha,y}$; the other omitted transitions go to $\mathit{trash}$. We have $I = \{q_1,p_1\}$ and $F = \{q_2,p_1\}$. The shortest accepted word is $bab$, and its corresponding properly extending word from the proof is: $\mathit{done}\;b\;c_{b,q_1}\;c_{b,p_1}\;a\;c_{a,q_2}\;c_{a,p_2}\;b\;c_{b,q_1}\;c_{b,p_3}$.

Theorems & Definitions (75)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Example 2.4
• Example 2.5
• proof : Proof of the example
• Theorem 3.1
• proof
• Lemma 3.2
• proof