# Computing Crisp Simulations and Crisp Directed Simulations for Fuzzy Graph-Based Structures

Like bisimulations, simulations and directed simulations are used for analyzing graph-based structures such as automata, labeled transition systems, linked data networks, Kripke models and interpretations in description logic. Simulations characterize the class of existential modal formulas, whereas directed simulations characterize the class of positive modal formulas. These notions are worth studying. For example, one may be interested in checking whether a given finite automaton simulates another or whether an object in a linked data network has all positive properties that another object has. To deal with vagueness and uncertainty, fuzzy graph-based structures are used instead of crisp ones. In this article, we design efficient algorithms with the complexity O((m+n)n) for computing the largest crisp simulation and the largest crisp directed simulation between two finite fuzzy labeled graphs, where n is the number of vertices and m is the number of nonzero edges of the input fuzzy graphs. We also adapt them to computing the largest crisp simulation and the largest crisp directed simulation between two finite fuzzy automata.

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