E is NDC if and only if , .
This definition of NDC is difficult to use in practice, because of the universal quantification on high level processes. We need an alternative formulation of BNDC which avoids universal quantification, exploiting local information only. It is possible to prove [FG95] that NDC is equivalent to Strong Non-deterministic Non Interference (SNNI, for short) which is defined as follows:
This property is a natural generalization of NI to non-deterministic systems. Intuitively, the high level does not interfere with the low level if the effects of high level actions are not visible by a low level user. In order to check this, we consider every trace of the system containing some high level actions. Then, we look if there exists another trace with the same subsequence of low level actions and without any high actions. A low level user, which can only observe low level actions, is not able to distinguish and . As both and are legal traces, we can conclude that the possible execution of the high level actions in has no effect on the low level view of the system. As a matter of fact, in all the high level actions are hidden, hence giving the low level view of the system; instead removes all the traces containing high level actions. So, if the two terms are equivalent, then for every trace with high level actions we can find another trace without such actions but with the same subsequence of low level actions.
We have the following result [FG95]:
NDC = SNNI
It is interesting to observe that this characterization of NDC through the SNNI property is possible since we are using an equivalence notion which is not able to detect deadlocks. In [FG95,FG] it is shown that under finer equivalence notions (testing, bisimulation) this result does not hold and a good (i.e., easy to check) characterization of NDC is still to be found.