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, would imply a quadratic upper bound on the reset threshold of a new class of automata.

中文翻译：

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原始矩阵集的同步概率函数

受有关同步 DFA 和原始集的最新结果的启发，我们通过研究既没有零行也没有零列的非负矩阵集的原始性现象来解决同步过程和相关的长期 Černý 猜想。我们在两人概率游戏的设置中制定了原始过程，并利用凸优化技术来描述其行为。我们开发了一种工具，用于对任何原始集的指数进行逼近和上限，并得到数值结果的支持，我们提出一个猜想，如果为真，则意味着一类新自动机的复位阈值的二次上限。