  Description logics (DLs) are a suitable formalism for representing knowledge
about domains in which objects are described not only by attributes but also by
binary relations between objects. Fuzzy extensions of DLs can be used for such
domains when data and knowledge about them are vague and imprecise. One of the
possible ways to specify classes of objects in such domains is to use concepts
in fuzzy DLs. As DLs are variants of modal logics, indiscernibility in DLs is
characterized by bisimilarity. The bisimilarity relation of an interpretation
is the largest auto-bisimulation of that interpretation. In DLs and their fuzzy
extensions, such equivalence relations can be used for concept learning. In
this paper, we define and study fuzzy bisimulation and bisimilarity for fuzzy
DLs under the G\"odel semantics, as well as crisp bisimulation and strong
bisimilarity for such logics extended with involutive negation. The considered
logics are fuzzy extensions of the DL $\mathcal{ALC}_{reg}$ (a variant of PDL)
with additional features among inverse roles, nominals, (qualified or
unqualified) number restrictions, the universal role, local reflexivity of a
role and involutive negation. We formulate and prove results on invariance of
concepts under fuzzy (resp. crisp) bisimulation, conditional invariance of
fuzzy TBoxex/ABoxes under bisimilarity (resp. strong bisimilarity), and the
Hennessy-Milner property of fuzzy (resp. crisp) bisimulation for fuzzy DLs
without (resp. with) involutive negation under the G\"odel semantics. Apart
from these fundamental results, we also provide results on using fuzzy
bisimulation to separate the expressive powers of fuzzy DLs, as well as results
on using strong bisimilarity to minimize fuzzy interpretations.
