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The semaphore doesn't issue those. A semaphore is an object that can be used by processes to coordinate between themselves. Thus, a process will call semSignal or semWait on a shared semaphore. We can't describe the conditions under which it will do that in general, because it's up to each process how it wants to use semaphores. Semaphores are a ...

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Looks like this modification is correct. Proof is similar to proof of Lamport algorithm and follows. Consider two processes $P_i$ and $P_k$ who entered critical sections at moments $e$ and $f$, correspondingly, requested them at moments $e'$ and $f'$, and exited at moments $e''$ and $f''$. At moment $e$ process $P_i$ have received a confirmation from $P_k$....

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I believe the purpose is to fulfill the first condition of entering the critical section (site $S_i$ must receive a message from all other sites with timestamp greater than its own request) in case some other site never requests to enter the critical section. If $S_i$ sends a critical section request to $S_j$, but $S_j$ never requests the critical section ...

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