Is finding the equivalence of two functions undecidable?

Question

Is it impossible to know if two functions are equivalent? For example, a compiler writer wants to determine if two functions that the developer has written perform the same

Answer 1

Is it impossible to know if two functions are equivalent?

No. It is possible to know that two functions are equivalent. If you have f(x), you know f(x) is equivalent to f(x).

If the question is "it is possible to determine if f(x) and g(x) are equivalent with f and g being any function and for all functions g and f", then the answer is no.

However, if the question is "can a compiler determine that if f(x) and g(x) are equivalent that they are equivalent?", then the answer is yes if they are equivalent in both output and side effects and order of side effects. In other words, if one is a transformation of the other that preserves behavior, then a compiler of sufficient complexity should be able to detect it. It also means that the compiler can transform a function f into a more optimal and equivalent function g given a particular definition of equivalent. It gets even more fun if f includes undefined behavior, because then g can also include undefined (but different) behavior!