The equivalence studied up to now is quite discriminating, in the sense that it distinguishes, for example, τ.P and τ.τ.P.

- If an external observer can count the number of non-observable actions (i.e., the τ’s), this distinction makes sense.
- If we assume that an observer cannot access any internal information of the system, then this distinction is not acceptable.

The idea of the new equivalence is to ignore (some) τ’s:
- a visible action must be replied to with the same action, possibly together with some internal actions
- an internal action must be replied to by a (possibly empty) sequence of internal actions.

We define the relation $\implies$ as:

$P \implies P'$ if and only if there exist $P_{0}, P_{1},\dots,P_{k}$ (for $k \geq 0$) such that $P=P_{0} \xrightarrow{\tau} P_{1} \xrightarrow{\tau}\dots\xrightarrow{\tau}Pk=P'$

relation $\xRightarrow{\hat{\alpha}}$:
- if $\alpha=\tau$ then $\xRightarrow{\hat{\alpha}}\triangleq\implies$
- otherwise $\xRightarrow{\hat{\alpha}}\triangleq\implies\xrightarr$