A Behavioral Distance for Fuzzy-Transition Systems

In contrast with the existing approaches to exact bisimulation for fuzzy systems, we introduce a robust notion of behavioral distance to measure the behavioral similarity of nondeterministic fuzzy-transition systems which are a generalization of fuzzy automata. This behavioral distance provides a quantitative analogue of bisimilarity and is defined as the greatest fixed point of a suitable monotonic function. The behavioral distance has the important property that two systems are at zero distance if and only if they are bisimilar. Moreover, for any given threshold, we find that systems with behavioral distances bounded by the threshold are equivalent. In addition, we show that two system combinators-parallel composition and product-are nonexpansive with respect to our behavioral distance, which makes compositional verification possible. The theory developed here is applicable to the quantitative verification, approximate reduction, and reliability analysis of fuzzy-transition systems.