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Concurrent Systems/notes/12b - CCS cose varie.md

@@ -10,7 +10,7 @@ $$S_{2}^{(2)} \triangleq v \cdot S_{1}^{(2)}$$
 
 If we consider S(2) as the specification of the expected behavior of a binary semaphore and S(1) | S(1) as its concrete implementation, we can show that $$S^{(1)}|S^{(1)} \space \textasciitilde \space S^{2}$$
 This means that the implementation and the specification do coincide. To show this equivalence, it suffices to show that following relation is a bisimulation:
-![](../../Pasted%20image%2020250415082906.png)
+![](images/Pasted%20image%2020250415082906.png)
 
 ## Restrictions
 **Proposition:** $a.P \textbackslash a ∼ 0$
@@ -50,7 +50,7 @@ One of the main aims of an equivalence notion between processes is to make equat
 **This feature on an equivalence makes it a *congruence***
 Not all equivalences are necessarily congruences (even though most of them are).
 To properly define a congruence, we first need to define an execution context, and then what it means to run a process in a context. Intuitively:
-![200](../../Pasted%20image%2020250415090109.png)
+![200](images/Pasted%20image%2020250415090109.png)
 
 where C is a context (i.e., a process with a hole ☐), P is a process, and $C[P]$ denotes filling the hole with P
 

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.
 

Concurrent Systems/notes/14.md

@@ -7,19 +7,19 @@ Inference system = axioms + inference rules
 - completeness: whatever is bisimilar, it can be inferred
 
 #### Axioms & Rules for Strong Bisimilarity
-![350](../../Pasted%20image%2020250429082812.png)
+![350](images/Pasted%20image%2020250429082812.png)
  quite obvious.
 
-![350](../../Pasted%20image%2020250429082905.png)
+![350](images/Pasted%20image%2020250429082905.png)
 basically we can let the left or the right process evolve, leaving the other unchanged, or they can synchronize.
 
-![350](../../Pasted%20image%2020250429083129.png)
+![350](images/Pasted%20image%2020250429083129.png)
 - if a process does not perform any action, a restriction won't do anything
 - ...
 
-![350](../../Pasted%20image%2020250429083455.png)
+![350](images/Pasted%20image%2020250429083455.png)
 
-![](../../Pasted%20image%2020250429083535.png)
+![](images/Pasted%20image%2020250429083535.png)
 $P$ is in standard form if and only if $P \triangleq \sum_{i}\alpha_{i}P_{i}$ and $\forall_{i}P_{i}$ is in standard form.
 
 **Lemma:** $\forall P \exists P'$* in standard form such that $\vdash P = P'$
@@ -28,15 +28,15 @@ $P$ is in standard form if and only if $P \triangleq \sum_{i}\alpha_{i}P_{i}$ an
 **Base case:** $P \triangleq 0$. It suffices to consider $P' \triangleq 0$ and conclude reflexivity.
 **Inductive step:** we have to consider three cases.
 
-![](../../Pasted%20image%2020250429084358.png)
-![](../../Pasted%20image%2020250429084921.png)
-![](../../Pasted%20image%2020250429084950.png)
+![](images/Pasted%20image%2020250429084358.png)
+![](images/Pasted%20image%2020250429084921.png)
+![](images/Pasted%20image%2020250429084950.png)
 replacing one by one every continuation with its standard form, obtaining standard form.
 
-![](../../Pasted%20image%2020250429085319.png)
+![](images/Pasted%20image%2020250429085319.png)
 
 ### Axioms & Rules for Weak Bisimilarity
-![](../../Pasted%20image%2020250429091029.png)
+![](images/Pasted%20image%2020250429091029.png)
 
 #### Example
 A server for exchanging messages, in its minimal version, receives a request for sending messages and delivers the confirmation of the reception
@@ -55,9 +55,9 @@ Let us consider the parallel of processes M and R, by using the axiom for parall
 By using the same axiom to the parallel of the three processes, we obtain
 $$\vdash S|(M|R)=send.(\overline{put}|(M|R))+put.(\overline{go}|R|S)+go.(\overline{rcv}|S|M)$$
 By restricting *put* and *go*, and by using the second axiom for restriction, we have that:
-![](../../Pasted%20image%2020250429091959.png)
+![](images/Pasted%20image%2020250429091959.png)
 
 We now apply the third axiom for restriction to the three summands:
-![](../../Pasted%20image%2020250429092055.png)
-![](../../Pasted%20image%2020250429092305.png)
-![](../../Pasted%20image%2020250429092543.png)
+![](images/Pasted%20image%2020250429092055.png)
+![](images/Pasted%20image%2020250429092305.png)
+![](images/Pasted%20image%2020250429092543.png)