Abstract To measure the similarity of nodes in the neighboring subgraphs, Milner introduced the notion of k-limited bisimilarity. Recently, as a weaker vision of k-limited bisimilarity, the notion of k-limited similarity was proposed and applied to graph pattern matching. (Bi)simulations have been widely used in comparing the behavior of fuzzy transition systems. In order to study the (bi)simulation semantics of labeled fuzzy transition systems in the residuated lattice-valued logic setting, we introduce an extension of labeled approximation spaces, called the quantitative fuzzy approximation spaces (QFASs), whose labels are equipped with a residuated lattice-valued equality relation. In a QFAS, we define a new notion of limited approximate similarity, to quantify to what extent one state is simulated by another in the neighboring subgraphs, and provide its properties. Based on the new notion, we give a definition of limited approximate simulation and discuss its properties. Then we introduce an ordered pair of relations, one on the state (vertex) set (limited approximate simulation) and one on the edge (transition) set induced by the relation on state set, called VE limited approximate simulation in this paper. We also present a new notion of limited approximate bisimilarity in a QFAS, to quantify to what extent two states are similar in the neighboring subgraphs, and give its properties. One main contribution of the paper is to give a condition for two states to be limited approximate bisimilar and investigate the degree of similarity between two states in a QFAS. Finally, we discuss the relationships between the rough approximations based on the underlying crisp relation induced by underlying labeled fuzzy relation and the rough approximations based on limited approximate bisimilarity.

中文翻译：

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有限的近似互模拟和相应的粗略近似

摘要 为了衡量相邻子图中节点的相似度，Milner 引入了 k-limited bisimilarity 的概念。最近，作为k-limited双相似性的一个较弱的愿景，k-limited相似性的概念被提出并应用于图模式匹配。(Bi) 模拟已广泛用于比较模糊过渡系统的行为。为了研究剩余格值逻辑设置中标记模糊转换系统的（双）模拟语义，我们引入了标记近似空间的扩展，称为定量模糊近似空间（QFAS），其标签配备了剩余的格值等式关系。在 QFAS 中，我们定义了一个新的有限近似相似性概念，以量化相邻子图中一种状态被另一种状态模拟的程度，并提供其属性。基于新的概念，我们给出了有限近似模拟的定义并讨论了它的性质。然后我们引入了一对有序的关系，一个在状态（顶点）集（有限近似模拟）上，一个在边缘（转移）集上，由状态集上的关系引起，本文称为 VE 有限近似模拟。我们还在 QFAS 中提出了有限近似双相似性的新概念，以量化相邻子图中两个状态的相似程度，并给出其属性。该论文的一个主要贡献是给出了两个状态有限近似双相似的条件，并研究了 QFAS 中两个状态之间的相似程度。最后，