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Concurrent Systems/notes/13 - Weak Bisimilarity.md

@@ -27,7 +27,7 @@ $\approx$ is a
 4. $\sim \subset \approx$
 
 #### Examples of weakly bisimilar processes
-![](../../Pasted%20image%2020250428083727.png)
+![](images/Pasted%20image%2020250428083727.png)
 
 **Theorem:** given any process P and any sum M, N, then:
 1. $P \approx \tau.{P}$
@@ -41,7 +41,7 @@ take the symmetric closure of the following relations, that can be easily shown
 3. $S=\{ ((M+\alpha.P+\alpha.(N+\tau.P), M+\alpha.(N+\tau.P)) \} \cup Id$
 
 #### Weak bisimilarity abstracts from any $\tau$
-![](../../Pasted%20image%2020250428084340.png)
+![](images/Pasted%20image%2020250428084340.png)
 
 **There exists no weak bisimulation S that contains (P, Q).**
 *Proof:*
@@ -71,7 +71,7 @@ A possible implementation of this specification is obtained by having two worker
 - For difficult works, they have to use the special machine.
 There is only one special and only one general machine that the workers have to share.
 
-![](../../Pasted%20image%2020250428085410.png)
+![](images/Pasted%20image%2020250428085410.png)
 
 where rg and rs are used to require the general/special machine, lg and ls are used to leave the general/special machine, and S and G implement a semaphore on the two different machines.
 
@@ -83,7 +83,7 @@ i.e., that the specification and the implementation of the factory behave the sa
 
 Let N denote {rg,rs,lg,ls} and x,y ∊ {E,M,D}
 We can prove that the following relation is a weak bisimulation:
-![](../../Pasted%20image%2020250428085837.png)
+![](images/Pasted%20image%2020250428085837.png)
 
 This is a family of relations:
 - 3 pairs of the second form (one for every x)
@@ -101,7 +101,7 @@ We want to model a lottery L where we can select any ball from a bag that contai
 
 The specification is: $$L \triangleq \tau.\bar{p_{1}}L+\tau.\bar{p_{2}}.L+\dots+\tau.\bar{p_{n}}.L$$
 where $\tau$'s represent ball extractions and $\tilde{p_{i}}$ is the action that communicates with the value of the extracted ball. The LTS resulting from this specification is:
-![](../../Pasted%20image%2020250428175449.png)
+![](images/Pasted%20image%2020250428175449.png)
 
 We now build a system with n components, one for every ball.