In mutual exclusion, we should satisfy the safety and progress properties. However, if we have a spurious wake-ups, is the safety property still satisfied?
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If your system has lock semantics that allow for spurious wakeups, then the programmer has to handle them accordingly. At each wakeup, check whether it was real or spurious. This information could be returned by the wait function, or the program could just check whether the desired event has occurred (e.g. do we hold the mutex we were waiting on). If yes, it's safe to enter the critical section. If no, then go back to sleep, or maybe do some other work.
So pseudo-code to protect a critical section could look like: