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Concurrent Systems/notes/13 - Weak Bisimilarity.md

@@ -7,4 +7,10 @@ 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.
 
-$P \implies P'$ if and only if there exist $P_{0}, P_{1},\dots,P_{k}$ (for $k \geq 0$) such that $P=P_{0} \xrightarrow{\tau} P_{1} \xrightarrow{\tau}$
+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 $\xRightarrow{\hat{\alpha}}\triangleq\implies\xrightarr$