论巴巴伊与塞尔尼著作的相互作用
On the interplay between Babai and Cerny's conjectures
摘要：
受Babai猜想和Cerny猜想的启发，本文研究了状态集的变换幺半群等于变换幺半群的自动机的复位阈值。对于具有$n$状态的自动机，我们证明了重置阈值的上界为$2n^2-6n+5$，并且可以达到值$\tfrac{n(n-1)}{2}$。此外，我们还研究了置换自动机的偶有向图的直径，构造了具有直径$\tfrac{n^2}{4} + o(n^2)$的$n$ -状态置换自动机。
英文摘要：
Motivated by the Babai conjecture and the Cerny conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with $n$ states in this class, we prove that the reset thresholds are upper-bounded by $2n^2-6n+5$ and can attain the value $\tfrac{n(n-1)}{2}$. In addition, we study diameters of the pair digraphs of permutation automata and construct $n$-state permutation automata with diameter $\tfrac{n^2}{4} + o(n^2)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
