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Concurrent Systems/notes/13 - CCS cose varie.md

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+An n-ary semaphore S(n)(p,v) is a process used to ensure that there are no more than n istances of the same activity concurrently in execution. An activity is started by action p and is terminated by action v.
+
+The specification of a unary semaphore is the following:
+$$S^{(1)} \triangleq p \cdot S_{1}^{(1)}$$
+$$S_{1}^{(1)} \triangleq p \cdot S_{1}^{(1)}$$
+The specification of a binary semaphore is the following:
+$$S_{}^{(2)} \triangleq p \cdot S_{1}^{(2)}$$
+$$S_{1}^{(2)} \triangleq p \cdot S_{1}^{(2)}+v\cdot S^{(2)}$$
+$$S_{2}^{(2)} \triangleq v \cdot S_{1}^{(2)}$$
+
+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)
+
+## Restrictions
+**Proposition:** $a.P \textbackslash a ∼ 0$
+*Proof:*
+- S = {(a.P\a , 0)} is a bisimulation
+
+Which challenges can (a.P)\a have?
+- a.P can only perform a (and become P)
+- however, because of restriction, a.P\a is stuck
+
+ No challenge from a.P\a, nor from 0 -> bisimilar!
+
+**Proposition:** $\bar{a}.P \textbackslash a ∼ 0$
+*Proof is similar.*
+
+## Idempotency of Sum
+**Proposition:** $α.P+α.P+M ∼ α.P+M$
+*Proof:*