  In this paper, we propose several opacity-preserving (bi)simulation relations
for general nondeterministic transition systems (NTS) in terms of initial-state
opacity, current-state opacity, K-step opacity, and infinite-step opacity. We
also show how one can leverage quotient construction to compute such relations.
In addition, we use a two-way observer method to verify opacity of
nondeterministic finite transition systems (NFTSs). As a result, although the
verification of opacity for infinite NTSs is generally undecidable, if one can
find such an opacity-preserving relation from an infinite NTS to an NFTS, the
(lack of) opacity of the NTS can be easily verified over the NFTS which is
decidable.
