Bisimulations for fuzzy automata

Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata AA and BB is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalence relations on AA and BB and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations. As a consequence we get that fuzzy automata AA and BB are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of AA and BB with respect to their greatest forward bisimulation fuzzy equivalence relations. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward–forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward–backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.

► We study the equivalence between fuzzy automata.
► Conjunction of concepts of uniform fuzzy relation and bisimulations is established.
► The characterization results for bisimulations are provided.
► We proved the main results for uniform bisimulation equivalent fuzzy automata.
► An overview of various concepts related to bisimulations is given.