Around Don's conjecture for binary completely reachable automata

TL;DR

It is proved that every $k-element subset of states in an $n-state standardized DFA has a reaching word of length n(n-k) + n - 1, thus generalizing the conjecture for standardized DFAs with additional properties.

Abstract

A word $w$ is called a reaching word of a subset $S$ of states in a deterministic finite automaton (DFA) if $S$ is the image of $Q$ under the action of $w$. A DFA is called completely reachable if every non-empty subset of the state set has a reaching word. A conjecture states that in every $n$-state completely reachable DFA, for every $k$-element subset of states, there exists a reaching word of length at most $n(n-k)$. We present infinitely many completely reachable DFAs with two letters that violate this conjecture. A subfamily of completely reachable DFAs with two letters, is called standardized DFAs, introduced by Casas and Volkov (2023). We prove that every $k$-element subset of states in an $n$-state standardized DFA has a reaching word of length $\le n(n-k) + n - 1$. Finally, we confirm the conjecture for standardized DFAs with additional properties, thus generalizing a result of Casas and Volkov (2023).

Figures (2)

• Figure 1: The action of $a$ of the $8$-state DFA $\mathcal{B}_8$. The shortest word that reaches $\{1,2,3,5,6,7\}$ is the word $a^2b^5ab^5ab^3$.
• Figure 2: The action of $a$ in the DFA $\mathop{\mathrm{\mathcal{A}}}\nolimits_n$. The states that are fixed by $a$ are omitted.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Proposition 6
• proof
• proof : Proof of \ref{['thm:upperBound']}