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 $\xRightarrow{\hat{\alpha}}\triangleq\implies\xrightarrow{\alpha}\implies$

S is a weak simulation if and only if $$\forall(p, q) \in S \space \forall p \xrightarrow{\alpha} p' \exists q' \space s.t. \space q\xRightarrow{\hat{\alpha}}q' \space and \space (p', q') \in S$$
A relation S is called weak bisimulation if both $S$ and $S^{-1}$ are weak simulations.
We say that p and q are weakly bisimilar, written $p \approx q$, if there exists a weak bisimulation $S$ such that $(p, q) \in S$.

**Prop:**
$\approx$ is a
1. equivalence
2. congruence
3. weak bisimulation
4. $\sim a$