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Concurrent Systems/notes/12b - CCS cose varie.md
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@ -2,10 +2,10 @@ An n-ary semaphore S(n)(p,v) is a process used to ensure that there are no more
The specification of a unary semaphore is the following:
$$S^{(1)} \triangleq p \cdot S_{1}^{(1)}$$
$$S_{1}^{(1)} \triangleq p \cdot S_{1}^{(1)}$$
$$S_{1}^{(1)} \triangleq p \cdot S_{}^{(1)}$$
The specification of a binary semaphore is the following:
$$S_{}^{(2)} \triangleq p \cdot S_{1}^{(2)}$$
$$S_{1}^{(2)} \triangleq p \cdot S_{1}^{(2)}+v\cdot S^{(2)}$$
$$S_{1}^{(2)} \triangleq p \cdot S_{2}^{(2)}+v\cdot S^{(2)}$$
$$S_{2}^{(2)} \triangleq v \cdot S_{1}^{(2)}$$
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 $$S^{(1)}|S^{(1)} \space \textasciitilde \space S^{2}$$