AI 中文总结
本文围绕Černý猜想，综述同步自动机理论的开放问题，探讨相关方法、判定复杂度等内容并给出新辅助结果，丰富该领域研究进展。
AI 中文摘要
本文综述了同步自动机理论中选定的开放问题，核心围绕著名的Černý猜想展开。确定有限自动机若存在一个重置字，其作用能将所有状态映射至单一状态，则称该自动机是同步的。Černý猜想指出，每个含n个状态的同步自动机都拥有一个长度不超过(n-1)²的重置字。文中还探讨了规避字、用另一状态压缩某一状态、（给定或任意）子集的同步性、同步性判定的复杂度、平均重置阈值以及线性代数方法，并给出了一些新的辅助结果。
英文摘要
We survey selected open problems in the theory of synchronizing automata, centered around the famous Černý conjecture. A deterministic finite automaton is called synchronizing if it admits a reset word whose action maps all states to a single state. The Černý conjecture states that every synchronizing automaton with n states possesses a reset word of length at most (n-1)^2. We discuss avoiding words, compressing a state with another, synchronization of a (given or any) subset, complexity of deciding the synchronizability, average reset threshold, and linear-algebraic methods. Some new auxiliary results are also presented.
CommentsIn Proceedings AFL 2026, arXiv:2608.23071
Journal refEPTCS 451, 2026, pp. 33-47
