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

There are multiple avenues to explaining this, one of which is the Halting Problem:

In computability theory, the halting problem can be stated as follows: "Given a description of an arbitrary computer program, decide whether the program finishes running or continues to run forever". This is equivalent to the problem of deciding, given a program and an input, whether the program will eventually halt when run with that input, or will run forever.

Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist.

If I write a program that looks like this:

do tons of complex stuff
if (condition on result of complex stuff)
{
    change value of x
}
else
{
    do not change value of x
}

Does the value of x change? To determine this, you would first have to determine whether the do tons of complex stuff part causes the condition to fire - or even more basic, whether it halts. That's something the compiler can't do.