Don's conjecture for binary completely reachable automata: an approach and its limitations

TL;DR

Don's conjecture asks whether, in a completely reachable DFA with $n$ states, every nonempty subset of size $k>0$ is the image of the full state set under a word of length at most $n(n-k)$. The paper develops an expandability framework and analyzes the restricted orbit digraph for binary circular DFAs, proving the conjecture for standardized DFAs whose orbit subgroup equals the full cyclic group $(\mathbb{Z}_n,\oplus)$ and thus generalizing Don's coprime result. It further shows that almost all proper subsets are $n$-expandable, yielding the length bound via an induction on $n-k$, but also presents limitations through examples and a discussion of the Rystsov digraph; the 2024 revision by Zhu provides counterexamples to Don's conjecture in the general case, while confirming the standardized orbit-$2\mathbb{Z}_n$ situation. Overall, the work connects expandability with orbit structure and Rystsov digraphs, contributing to the broader understanding of completely reachable and synchronizing automata, and highlighting both the promise and the limits of expansion-based methods.

Abstract

A deterministic finite automaton in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word is called completely reachable. It was conjectured that in each completely reachable automaton with $n$ states, every set of $k>0$ states is the image of a word of length at most $n(n-k)$. We confirm the conjecture for completely reachable automata with two input letters satisfying certain restrictions on the action of the letters.

Figures (7)

• Figure 1: The DFA $\mathrsfs{E}_{6}$. Solid and dashed edges show the action of $a$ and, resp., $b$; if $a$ fixes a state, the corresponding loop is omitted to improve readability.
• Figure 2: The DFA $\mathrsfs{E}_{48}=\langle\mathbb{Z}_{48},\{a,b\}\rangle$. Solid and dotted edges show the action of $a$ and, resp., $b$; if $a$ fixes a state, the corresponding loop is omitted to improve readability.
• Figure 3: One of the strongly connected components of the digraph $\Gamma^{(1)}$ constructed for the DFA $\mathrsfs{E}_{48}$ from Figure \ref{['fig:e48']}. The solid edges are inherited from $\Gamma^{(0)}$; the dashed edges are newly added.
• Figure 4: The DFA $\mathrsfs{E}_{12}=\langle\mathbb{Z}_{12},\{a,b\}\rangle$. Solid and dashed edges show the action of $a$ and, resp., $b$.
• Figure 5: The restricted Rystsov digraph of the DFA $\mathrsfs{E}_{12}$. Each edge is labeled by the shortest word of defect 1 forcing it.

Theorems & Definitions (17)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• Lemma 5
• proof
• Proposition 6
• proof