We present a generic partition refinement algorithm that quotients coalgebraic systems by behavioural equivalence, an important task in system analysis and verification. Coalgebraic generality allows us to cover not only classical relational systems but also, e.g. various forms of weighted systems and furthermore to flexibly combine existing system types. Under assumptions on the type functor that allow representing its finite coalgebras in terms of nodes and edges, our algorithm runs in time $\mathcal{O}(m\cdot \log n)$ where $n$ and $m$ are the numbers of nodes and edges, respectively. The generic complexity result and the possibility of combining system types yields a toolbox for efficient partition refinement algorithms. Instances of our generic algorithm match the run-time of the best known algorithms for unlabelled transition systems, Markov chains, deterministic automata (with fixed alphabets), Segala systems, and for color refinement.

中文翻译：

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高效的模块化代数划分细化

我们提出了一种通用的分区细化算法，该算法通过行为等价对多代数系统进行商数，这是系统分析和验证中的一项重要任务。代数一般性使我们不仅可以涵盖经典的关系系统，还可以涵盖各种形式的加权系统，此外还可以灵活地组合现有的系统类型。在允许用节点和边表示其有限余代数的类型函子的假设下，我们的算法在时间 $\mathcal{O}(m\cdot \log n)$ 中运行，其中 $n$ 和 $m$ 是数字分别为节点和边。通用复杂性结果和组合系统类型的可能性为有效的分区细化算法提供了一个工具箱。我们的通用算法的实例与用于未标记转换系统的最著名算法的运行时间相匹配，