一句话：用同余刻画最小历史确定性 co-Büchi 自动机，并给出多项式时间被动学习算法。
本文研究 history-deterministic（transition-based）co-Büchi 自动机的最小化与学习。此前 Abu Radi 和 Kupferman 基于 Kuperberg 与 Skrzypczak 的结果给出了这类自动机的高效最小化。作者转而给出这些最小自动机的同余式描述，并提供该描述正确性的自包含证明。基于同余，文章构造了一种被动学习算法，可从一组带标签的示例词中学习最小 history-deterministic co-Büchi 自动机。该算法在给定示例集上以多项式时间运行；并且对每个最小 history-deterministic co-Büchi 自动机，都存在多项式规模的特征示例集。
Abu Radi and Kupferman (2019) demonstrated the efficient minimization of history-deterministic (transition-based) co-Büchi automata, building on the results of Kuperberg and Skrzypczak (2015). We give a congruence-based description of these minimal automata, and a self-contained proof of its correctness. We use this description based on congruences to create a passive learning algorithm that can learn minimal history-deterministic co-Büchi automata from a set of labeled example words. The algorithm runs in polynomial time on a given set of examples, and there is a characteristic set of examples of polynomial size for each minimal history-deterministic co-Büchi automaton.