Assuming I have functions $f, g : \mathbb{N} \to \mathbb{N}$ and I know that $g \circ f$ is a primitive recursive function. What can I tell about $f$ and $g$? Are they primitive recursive as well? Or at least one of them?

  • $\begingroup$ What have you tried and where did you get stuck? Hint: you don't know anything. Find examples. $\endgroup$
    – Raphael
    Jan 28 '15 at 11:50
  • 2
    $\begingroup$ Hint: $g(x) = 1$ $\endgroup$
    – Pseudonym
    Jan 28 '15 at 13:21

Let $f$ be a bijective function, which is not primitive recursive. We know, that such a function exists. Let further be $g=f^{-1}$ the inverse function of $f$. Therefore $f\circ g$ is the identity function, which is clearly primitive recursive.

  • 3
    $\begingroup$ Another fun example is if one function is constant, the other can be arbitrary (i.e. non-primitive recursive, or even noncomputable) $\endgroup$
    – Mike B.
    Jan 28 '15 at 15:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.