Adversarial Synchronization

TL;DR

This paper investigates a synchronization game on finite automata, establishing conditions for winning strategies, bounds on reset word lengths, and polynomial algorithms for deciding game outcomes, revealing new insights into automata synchronization.

Contribution

It introduces a novel synchronization game variant, proves bounds on reset word lengths when winning strategies exist, and provides polynomial algorithms for game outcome determination.

Findings

Winning strategies imply existence of short reset words

Automata can have quadratic shortest reset words despite winning strategies

Polynomial algorithms can decide game outcomes efficiently

Abstract

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.