Prove linearizability of concurrent objects

How can I prove the linearizability of the shared counter?

The shared counter is an example of a concurrent system that can be used to demonstrate linearizability. To prove the linearizability of the shared counter, we must show that each invocation of an operation (increment and read) appears to take effect atomically. This means that if two operations A and B are invoked, then A must appear to have been executed before B or B must appear to have been executed before A.

To demonstrate linearizability, we must show that each invocation of the increment and read operations appears to take effect atomically. To do this, we use the CompareAndSet object, which has an int v field and two methods: read() and cas(expected, update).

The read() method returns the value of v and the cas(expected, update) method takes two arguments: an expected value and an update value. If the current v value is equal to the expected value, then it is replaced by the update value; otherwise, the value is left unchanged. The method call returns a Boolean indicating whether the value v changed.

The increment() method uses the CompareAndSet object to increment the shared counter. It does this by repeatedly executing the cas(expected, update) method until the returned Boolean is true. This indicates that the v value changed, and the counter was successfully incremented.

By using the CompareAndSet object, each invocation of the increment and read operations appears to take effect atomically, showing the linearizability of the shared counter.

To prove the linearizability of the shared counter, we must show that each invocation of the increment and read operations appears to take effect atomically. We use the CompareAndSet object, which has the read() and cas(expected, update) methods. The increment() method uses the cas(expected, update) method to increment the counter, and each operation appears to take effect atomically.