Fuzzy structures such as fuzzy automata, fuzzy transition systems, weighted social networks and fuzzy interpretations in fuzzy description logics have been widely studied. For such structures, bisimulation is a natural notion for characterizing indiscernibility between states or individuals. There are two kinds of bisimulations for fuzzy structures: crisp bisimulations and fuzzy bisimulations. While the latter fits to the fuzzy paradigm, the former has also attracted attention due to the application of crisp equivalence relations, for example, in minimizing structures. Bisimulations can be formulated for fuzzy labeled graphs and then adapted to other fuzzy structures. In this article, we present an efficient algorithm for computing the partition corresponding to the largest crisp bisimulation of a given finite fuzzy labeled graph. Its complexity is of order $O((m\log{l} + n)\log{n})$, where $n$, $m$ and $l$ are the number of vertices, the number of nonzero edges and the number of different fuzzy degrees of edges of the input graph, respectively. We also study a similar problem for the setting with counting successors, which corresponds to the case with qualified number restrictions in description logics and graded modalities in modal logics. In particular, we provide an efficient algorithm with the complexity $O((m\log{m} + n)\log{n})$ for the considered problem in that setting.

翻译：模糊自动机、模糊转移系统、加权社交网络以及模糊描述逻辑中的模糊解释等模糊结构已被广泛研究。对于这类结构，互模拟是刻画状态或个体间不可区分性的自然概念。模糊结构存在两种互模拟：清晰互模拟和模糊互模拟。后者符合模糊范式，而前者因清晰等价关系在结构最小化等应用中的价值同样受到关注。互模拟可针对模糊标记图进行形式化定义，并进一步推广至其他模糊结构。本文提出一种高效算法，用于计算给定有限模糊标记图的最大清晰互模拟对应的划分。其复杂度为$O((m\log{l} + n)\log{n})$，其中$n$、$m$和$l$分别表示输入图的顶点数、非零边数及边中不同模糊度值的数量。此外，我们研究了带计数后继者设置下的类似问题，该设置对应于描述逻辑中的受限数量限定与模态逻辑中的分级模态。针对该场景，我们提供了复杂度为$O((m\log{m} + n)\log{n})$的高效算法。