vault backup: 2025-04-28 08:37:04

Concurrent Systems/notes/13 - Weak Bisimilarity.md

@@ -13,4 +13,15 @@ $P \implies P'$ if and only if there exist $P_{0}, P_{1},\dots,P_{k}$ (for $k \g
 
 relation $\xRightarrow{\hat{\alpha}}$:
 - if $\alpha=\tau$ then $\xRightarrow{\hat{\alpha}}\triangleq\implies$
-- otherwise $\xRightarrow{\hat{\alpha}}\triangleq\implies\xrightarr$
+- 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. $\sim a$