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According to Wikipedia:

Semaphores are a useful tool in the prevention of race conditions; however, their use is by no means a guarantee that a program is free from these problems. Semaphores which allow an arbitrary resource count are called counting semaphores, while semaphores which are restricted to the values 0 and 1 (or locked/unlocked, unavailable/available) are called binary semaphores and are used to implement locks.

Is a semaphore always initialized to the maximum value it can take?

Is the difference between a counting semaphore and a binary semaphore only their initial value: a semaphore is binary if and only if it is initialized to 1, and counting but not binary if and only if it is initialized to an integer larger than 1?

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The difference is more about the purpose: Binary semaphores are usually used to implement mutual exclusion, whereas counting semaphores to limit access. But in the end, the binary semaphore is just a special case of the counting semaphore.

The details remain implementation dependent. In the POSIX standard for example, the semaphores are counting semaphores. So indeed, the initial value (defined with sem_init() in the case of an unnamed semaphore) would de facto define if it's binary or not. However, a counting sempahore remains a counting semaphore, and using more sem_post() (release) than sem_wait()(acquire) will increase the value of the semaphore beyond the initial value (demo). By the way, this allows also to create a semaphore in an already locked state (initial value of 0).

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