Is it possible to know if two python functions are functionally equivalent?

Question

Let's say I have two python functions f and g:

def f(x):
    y = x**2 + 1
    return y

def g(x):
    a = x**2
    b = a + 1
    return b

These two functions are clearly functionally equivalent (both return x**2 + 1).

My definition of functionally equivalent is as follows:

If two functions f and g always produce the same output given the same input, then f and g are functionally equivalent.

Further, let's say no global variables are involved in f and g.

Is it possible to automatically determine (without human inspection) if python functions f and g are functionally equivalent?

Answer 1

By Rice's Theorem, no. If you could do this, you could solve the halting problem. (This is true even if f and g are always guaranteed to halt.)