vault backup: 2025-04-15 08:30:18

Concurrent Systems/notes/13 -.md

@@ -7,3 +7,9 @@ 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)
+
+## Congruence