Adversarial Synchronization

TL;DR

This work studies adversarial synchronization games on finite deterministic automata, introducing a robust hierarchy of variants $A_k$ (for $k\in\mathbb{N}\cup\{\omega\}$) that constrain Bob's responses and analyze how quickly synchronization can be forced. The authors prove that if Alice can win the $\omega$-game, then the automaton has a reset word shorter than the number of states, i.e., $rt(\mathrsfs{A})<|Q|$, and they show a quadratic lower bound for $A_k$-automata, demonstrating that fixed-$k$ adversaries can force much longer resets. They develop polynomial-time algorithms to decide the winner in these games, including the $\omega$-game and $A_k$-games, and explore how the variants relate on fixed-size automata, showing, for example, collapse phenomena and precise thresholds for when $A_k$-automata become $A_\omega$-automata. Collectively, the results deepen the understanding of synchronization under adversarial interference and yield efficient tools for classifying automata by their game-theoretic synchronization behavior.

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.

Figures (13)

• Figure 1: The automaton $\mathrsfs{C}_n$
• Figure 2: Moves in a synchronization game on $\mathrsfs{C}_5$
• Figure 3: Automata $\mathrsfs{E}_3\in\mathbf{A}_2\setminus\mathbf{A}_3$ and $\mathrsfs{E}_4\in\mathbf{A}_3\setminus\mathbf{A}_4$
• Figure 4: A graph from $\mathcal{T}_C$ with components from $\mathcal{V}(C)$ shown by dashed contours. Blue edges belong to $E_{\ell-1}$ (loops from $E_0$ are not shown); red edges are from $E_\ell\setminus E_{\ell-1}$.
• Figure 5: An $n$-state DFA with reset threshold $n-1$ on which Bob wins by keeping tokens (shown in gray) on states 1, 2, …, $n-2$.

Theorems & Definitions (57)

• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• proof
• Proposition 3.1
• proof