Are two functions equal?

Question

[Edit]

The general question seems incredibly hard to solve. Here is a significantly restricted version of this question.

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How do I determine equality of functions?

lets say we have

function f() {
    // black box code.
}

function g() {
    // black box code.
}

We take a mathematical definition of a function. So

if for all x in domain, f(x) === g(x) then f === g

• How do we handle domains?
• How can we otherwise determine if f === g

Checks by source code is silly because

function f(i) {
     return i % 2;
}

function g(i) {
     var returnVal = i % 2;
     return returnVal;
}

Are obvouisly equal. These are trivial examples but you can imagine more complex functions being equal but not source-equal.

You may assume that f and g have no side effects that we care about.

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[Edit]

As @Pointy mentioned it's probably best to constrain the domain. Rather then having the equality function try and guess the domain, the user of the equality function should supply a domain.

It doesn't make sense to ask whether two functions are equal without defining their domain somewhere.

To simply the problem we can assume to domain is the set of all integers or a subset of that so we need a function:

function equal (f, g, domain) {

}

The structure of the domain is irrelevant and can be made to make the problem as easy as possible. You can also assume that f and g act nicely on the domain of integers and don't crash&burn.

You may assume that f and g halt!

Again @Pointy points out a good example of non-deterministic functions

What if we limit f & g to be deterministic.

Answer 1

This is impossible according to Rice's theorem:

In computability theory, Rice's theorem states that, for any non-trivial property of partial functions, there is no general and effective method to decide whether an algorithm computes a partial function with that property.

If you restrict the domain of the functions to be finite, then you can trivially verify if ∀x: f(x) = g(x) using brute force, but this is not possible with an infinite domain.