Synchronizing Automata: Open Problems

2026-08-25

AI Summary

该文探讨了同步自动机理论中的未解决问题，主要围绕著名的Černý猜想，讨论了避免词、状态压缩、子集同步等问题及线性代数方法。

Abstract

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.

Problem

• synchronizing automata
• Černý conjecture
• reset word

Innovation

• synchronizing automata
• Černý conjecture
• reset word
• state compression
• linear-algebraic methods