Limited approximate bisimulations and the corresponding rough approximations

Abstract

To measure the similarity of nodes in the neighboring subgraphs, Milner introduced the notion of k-limited bisimilarity. Recently, as a weaker version 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.