We study a variant of the synchronization game on finite deterministic automata. In this game, Alice chooses one input letter of an automaton $A$ on each of her moves while Bob may respond with an arbitrary finite word over the input alphabet of $A$; Alice wins if the word obtained by interleaving her letters with Bob's responses resets $A$. We prove that if Alice has a winning strategy in this game on $A$, then $A$ admits a reset word whose length is strictly smaller than the number of states of $A$. In contrast, for any $k\ge 1$, we exhibit automata with shortest reset-word length quadratic in the number of states, on which Alice nevertheless wins a version of the game in which Bob's responses are restricted to arbitrary words of length at most $k$. We provide polynomial-time algorithms for deciding the winner in various synchronization games, and we analyze the relationships between variants of synchronization games on fixed-size automata.

翻译：我们研究有限确定性自动机上同步博弈的一个变体。在该博弈中，Alice 在每一步选择自动机 $A$ 的一个输入字母，而 Bob 可以用 $A$ 输入字母表上的任意有限字进行回应；若将 Alice 的字母与 Bob 的回应交错拼接得到的字能够重置 $A$，则 Alice 获胜。我们证明，若 Alice 在该博弈中对 $A$ 存在必胜策略，则 $A$ 存在一个长度严格小于其状态数的重置字。相比之下，对于任意 $k\ge 1$，我们构造了一类最短重置字长度相对于状态数呈二次增长的自动机，在这些自动机上，Alice 仍能在 Bob 的回应被限制为长度至多 $k$ 的任意字的博弈版本中获胜。我们提供了判定多种同步博弈中获胜方的多项式时间算法，并分析了固定规模自动机上不同同步博弈变体之间的关系。