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

The general case is undecidable by Rice's Theorem, as others have already said (Rice's Theorem essentially says that any nontrivial property of a Turing-complete formalism is undecidable).

There are special cases where equivalence is decidable, the best-known example is probably equivalence of finite state automata. If I remember correctly equivalence of pushdown automata is already undecidable by reduction to Post's Correspondence Problem.

To prove that two given functions are equivalent you would require as input a proof of the equivalence in some formalism, which you can then check for correctness. The essential parts of this proof are the loop invariants, as these cannot be derived automatically.