Given an arbitrary function, f, we define a function f’ which returns 1 on input n if f halts on input n. Now, for some number x we define a function g which, on input n, returns 1 if n = x, and otherwise calls f'(n).
If functional equivalence were decidable, then deciding whether g is identical to f’ decides whether f halts on input x. That would solve the Halting problem. Related to this discussion is Rice’s theorem.
Conclusion: functional equivalence is undecidable.
There is some discussion going on below about the validity of this proof. So let me elaborate on what the proof does, and give some example code in Python.
The proof creates a function f’ which on input n starts to compute f(n). When this computation finishes, f’ returns 1. Thus, f'(n) = 1 iff f halts on input n, and f’ doesn’t halt on n iff f doesn’t. Python:
def create_f_prime(f): def f_prime(n): f(n) return 1 return f_prime
Then we create a function g which takes n as input, and compares it to some value x. If n = x, then g(n) = g(x) = 1, else g(n) = f'(n). Python:
def create_g(f_prime, x): def g(n): return 1 if n == x else f_prime(n) return g
Now the trick is, that for all n != x we have that g(n) = f'(n). Furthermore, we know that g(x) = 1. So, if g = f’, then f'(x) = 1 and hence f(x) halts. Likewise, if g != f’ then necessarily f'(x) != 1, which means that f(x) does not halt. So, deciding whether g = f’ is equivalent to deciding whether f halts on input x. Using a slightly different notation for the above two functions, we can summarise all this as follows:
def halts(f, x): def f_prime(n): f(n); return 1 def g(n): return 1 if n == x else f_prime(n) return equiv(f_prime, g) # If only equiv would actually exist...
I’ll also toss in an illustration of the proof in Haskell (GHC performs some loop detection, and I’m not really sure whether the use of
seq is fool proof in this case, but anyway):
-- Tells whether two functions f and g are equivalent. equiv :: (Integer -> Integer) -> (Integer -> Integer) -> Bool equiv f g = undefined -- If only this could be implemented :) -- Tells whether f halts on input x halts :: (Integer -> Integer) -> Integer -> Bool halts f x = equiv f' g where f' n = f n `seq` 1 g n = if n == x then 1 else f' n