  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 $\cal A$ and $\cal B$ is a
forward bisimulation if and only if its kernel and co-kernel are forward
bisimulation fuzzy equivalences on $\cal A$ and $\cal B$ and there is a special
isomorphism between factor fuzzy automata with respect to these fuzzy
equivalences. As a consequence we get that fuzzy automata $\cal A$ and $\cal B$
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 $\cal A$ and $\cal B$ with respect to their greatest forward bisimulation
fuzzy equivalences. 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.
