变换幺半群的同步阈值

On the interplay between Babai and Cerny's conjectures

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 $\tfrac{n^2}{4} + o(n^2)$.

状态数较少的极小二元部分有限自动机

Extremal Binary PFAs with Small Number of States

本文研究状态数固定时二元部分有限自动机的最大重置阈值，证明其严格大于DFA的$(n-1)^2$当且仅当$n\geq6$，并通过分析Černý族揭示与斐波那契及Padovan序列相关的显式公式，同时改进Martyugin构造证明该族在$n\geq41$时非极值。

The Power of Programs over Monoids in J and Threshold Dot-depth One Languages

The model of programs over (finite) monoids, introduced by Barrington and Thérien, gives an interesting way to characterise the circuit complexity class $\mathsf{NC^1}$ and its subclasses and showcases deep connections with algebraic automata theory. In this article, we investigate the computational power of programs over monoids in $\mathbf{J}$, a small variety of finite aperiodic monoids. First, we give a fine hierarchy within the class of languages recognised by programs over monoids from $\mathbf{J}$, based on the length of programs but also some parametrisation of $\mathbf{J}$. Second, an…

Černý猜想的继承性问题

The hereditariness problem for the Černý conjecture

本文探讨了Černý猜想的提升问题，证明只需验证三种特定重置自动机子类（radical, simple, quasi-simple）即可。通过建立同余和转换单oid理想 lattice之间的Galois连接，提供了计算radical理想和结构洞察的系统方法。

同步自动机：开放问题

Synchronizing Automata: Open Problems

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。

On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs

We tackle the problem of the randomized generation of slowly synchronizing deterministic automata (DFAs) by generating random primitive sets of matrices. We show that when the randomized procedure is too simple the exponent of the generated sets is O(n log n) with high probability, thus the procedure fails to return DFAs with large reset threshold. We extend this result to random nondeterministic automata (NDFAs) by showing, in particular, that a uniformly sampled NDFA has both a 2-directing word and a 3-directing word of length O(n log n) with high probability. We then present a more involved…

The Černy Conjecture for aperiodic automata

A word w is called a synchronizing (recurrent, reset) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some state; a DFA that has a synchronizing word is said to be synchronizing. Cerny conjectured in 1964 that every n-state synchronizing DFA possesses a synchronizing word of length at most (n -1)2. We consider automaton with aperiodic transition monoid (such automaton is called aperiodic). We show that every synchronizing n-state aperiodic automaton has a synchronizing word of length at most n(n-2)+1. Thus, for aperiodic automaton as well as for automa…

Subset Synchronization and Careful Synchronization of Binary Finite Automata

We present a strongly exponential lower bound that applies both to the subset synchronization threshold for binary deterministic automata and to the careful synchronization threshold for binary partial automata. In the later form, the result finishes the research initiated by Martyugin (2013). Moreover, we show that both the thresholds remain strongly exponential even if restricted to strongly connected binary automata. In addition, we apply our methods to computational complexity. Existence of a subset reset word is known to be PSPACE-complete; we show that this holds even under the restricti…

非齐次随机自动机的快速同步

Fast synchronization of inhomogenous random automata

本文研究随机确定性自动机的重置阈值，证明在部分独立条件下，一个随机映射加两个随机置换字母的自动机以高概率具有 $\mathcal{O}(\sqrt{n \log^3 n})$ 的重置阈值，改进了相关结果。

完全可达自动机：二次决策算法与可达阈值的二次上界

Completely reachable automata: a quadratic decision algorithm and a quadratic upper bound on the reaching threshold

针对完全可达自动机的判定复杂度问题，提出时间与空间复杂度分别为${\rm{\textit{O}(|Σ|\cdot n^2)}}$和$\textit{O}(|Σ|\cdot n)$的二次决策算法，并证明弱Don猜想，给出该类自动机最短同步单词长度的二次上界。

通过环形谱下降法证明单簇自动机的切尔尼猜想

The Černý Conjecture for One-Cluster Automata via Annular Spectral Descent

研究同步单簇自动机的切尔尼猜想，通过特定方法证明对于相关条件下存在特定单词，给出复位单词长度上界，构造示例说明界的紧性，证明借助与OpenAI Codex交互并经作者验证。

The Synchronizing Probability Function for Primitive Sets of Matrices

Motivated by recent results relating synchronizing DFAs and primitive sets, we tackle the synchronization process and the related longstanding Černý conjecture by studying the primitivity phenomenon for sets of nonnegative matrices having neither zero-rows nor zero-columns. We formulate the primitivity process in the setting of a two-player probabilistic game and we make use of convex optimization techniques to describe its behavior. We develop a tool for approximating and upper bounding the exponent of any primitive set and supported by numerical results we state a conjecture that, if true, w…