A shared variable $x$, initialized to zero, is operated on by four concurrent processes $W, X, Y, Z$ as follows. Each of the processes $W$ and $X$ reads $x$ from memory, increments by one, stores it to memory, and then terminates. Each of the processes $Y$ and $Z$ reads $x$ from memory, decrements by two, stores it to memory, and then terminates. Each process before reading $x$ invokes the $P$ operation (i.e., wait) on a counting semaphore $S$ and invokes the $V$ operation (i.e., signal) on the semaphore $S$ after storing $x$ to memory. Semaphore $S$ is initialized to two. What is the maximum possible value of $x$ after all processes complete execution?

We are given:

• Initial value: $x = 0$

• Initial semaphore: $S = 2$ (allowing at most 2 processes in their critical regions simultaneously)

• Processes $W, X$: read $x$, compute $x + 1$, write $x$

• Processes $Y, Z$: read $x$, compute $x - 2$, write $x$

To achieve the maximum possible value of $x$, we want the decrement operations of $Y$ and $Z$ to be overwritten by the increment operation of $W$, and then let $X$ perform a clean increment:

1. Process $Y$ executes $P(S)$ $\implies S = 1$. It reads local value $x = 0$.

2. Process $W$ executes $P(S)$ $\implies S = 0$. It reads local value $x = 0$.

3. Process $Y$ resumes, calculates $0 - 2 = -2$, writes $x = -2$, and executes $V(S) \implies S = 1$, then terminates.

4. Process $Z$ executes $P(S)$ $\implies S = 0$. It reads local value $x = -2$.

5. Process $Z$ calculates $-2 - 2 = -4$, writes $x = -4$, executes $V(S) \implies S = 1$, and terminates.

6. Process $W$ resumes (having already read $x = 0$), computes $0 + 1 = 1$, writes $x = 1$, executes $V(S) \implies S = 2$, and terminates. (This completely overwrites the decrements done by $Y$ and $Z$.)

7. Process $X$ executes $P(S) \implies S = 1$, reads $x = 1$, computes $1 + 1 = 2$, writes $x = 2$, executes $V(S) \implies S = 2$, and terminates.

Thus, the final and maximum possible value of $x$ is 2, which corresponds to option (D).