Why is it impossible to build a compiler that can determine if a C++ function will change the value of a particular variable?

Question

I read this line in a book:

It is provably impossible to build a compiler that can actually determine whether or not a C++ function will change the val

Answer 1

Really surprised that there isn't an answer that using the halting problem directly! There's a very straightforward reduction from this problem to the halting problem.

Imagine that the compiler could tell whether or not a function changed the value of a variable. Then it would certainly be able to tell whether the following function changes the value of y or not, assuming that the value of x can be tracked in all the calls throughout the rest of the program:

foo(int x){
   if(x)
       y=1;
}

Now, for any program we like, let's rewrite it as:

int y;
main(){
    int x;
    ...
    run the program normally
    ...
    foo(x);
}

Notice that, if, and only if, our program changes the value of y, does it then terminate - foo() is the last thing it does before exiting. This means we've solved the halting problem!

What the above reduction shows us is that the problem of determining whether a variable's value changes is at least as hard as the halting problem. The halting problem is known to be incomputable, so this one must be also.