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

You could check in your compiler to see if they are "exactly" identical, sure, but determining if they return identical values would be difficult and time consuming. You would have to basically call that method and perform its routine over an infinite number of possible calls and compare the value with that from the other routine.

Even if you could do the above, you would have to account for what global values change within the function, what objects are destroyed / changed in the function that do not affect the outcome.

You can really only compare the compiled code. So compile the compiled code to refactor?

Imagine the run time on trying to compile the code with "that" compiler. You could spend a LOT of time on here answering questions saying: "busy compiling..." :)

Answer 2

It depends on what you mean by "function."

If the functions you are talking about are guaranteed to terminate -- for example, because they are written in a language in which all functions terminate -- and operate over finite domains, it's "easy" (although it might still take a very, very long time): two functions are equivalent if and only if they have the same value at every point in their shared domain.

This is called "extensional" equivalence to distinguish it from syntactic or "intensional" equivalence. Two functions are extensionally equivalent if they are intensionally equivalent, but the converse does not hold.

(All the other people above noting that it is undecidable in the general case are quite correct, of course, this is a fairly uncommon -- and usually uninteresting in practice -- special case.)

Answer 3

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!