Modal logics and behavioral equivalences play an important role in the specification and verification of concurrent systems. In this paper, we first present a new notion of bisimulation for nondeterministic fuzzy transition systems, which is distribution based and coarser than state-based bisimulation appeared in the literature. Then, we define a distribution-based bisimilarity metric as the least fixed point of a suitable monotonic function on a complete lattice, which is a behavioral distance and is a more robust way of formalizing behavioral similarity between states than bisimulations. We also propose an on-the-fly algorithm for computing the bisimilarity metric. Moreover, we present a fuzzy modal logic and provide a sound and complete characterization of the bisimilarity metric. Interestingly, this characterization holds for a class of fuzzy modal logics. In addition, we show the nonexpansiveness of a typical parallel composition operator with respect to the bisimilarity metric, which makes compositional verification possible.

中文翻译：

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非确定性模糊转换系统的基于分布的行为距离

模态逻辑和行为等价在并发系统的规范和验证中发挥着重要作用。在本文中，我们首先提出了一种新的非确定性模糊转换系统的互模拟概念，它是基于分布的，比文献中出现的基于状态的互模拟更粗糙。然后，我们将基于分布的双相似性度量定义为完整格上合适的单调函数的最小不动点，这是一种行为距离，是一种比双模拟更稳健的形式化状态之间行为相似性的方法。我们还提出了一种用于计算双相似性度量的即时算法。此外，我们提出了一个模糊模态逻辑，并提供了双相似性度量的合理和完整的表征。有趣的是，这种表征适用于一类模糊模态逻辑。此外，我们展示了典型并行组合算子相对于双相似性度量的非扩展性，这使得组合验证成为可能。