Adversarial Synchronization

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.