Approximate bisimulations for fuzzy-transition systems

Abstract

Bisimulation techniques have been recently introduced to fuzzy transition systems to compare their behavior. When studying bisimulation, one of the most popular methods is relational lifting. In this paper, we first realize the relational lifting through a function S˜. The new relational lifting method relaxes the condition of lifting relation from using weight functions to function S˜ defined by residua (implications) in complete residuated lattices. We first discuss some properties of the lifting operation, and then define an α-bisimulation under S˜, where α∈[0,1], for nondeterministic fuzzy transition systems (NFTSs). The new bisimulation can handle the case that two states are equivalent intuitively but are simply treated as distinguishable by some behavioral measures, and so it is much more natural and robust. Then we provide some properties of α-bisimulation under S˜. The parallel composition operator of NFTSs is addressed and we show that α-bisimilarity under S˜ is nonexpansive with respect to the operator, which makes compositional verification possible. In addition, we give a fixed-point characterization of α-bisimulation under S˜ and compute α-bisimilarity under S˜. Finally, we introduce a real-valued logic to characterize α-bisimilarity under S˜.