TITLE: Non-existence of algorithm converting NP algorithm to P algorithm?
QUESTION [9 upvotes]: [Edit: in the light of Nate Eldredge's answer below I rephrase the question]
P=NP is equivalent to the existence of a map of the following form:

Input: a polynomial-time non-deterministic Turing machine which accepts some language (call the language L) [Edit: we are not to assume these NDTMs come with any certificate proving they run in polynomial time -- Ryan requested this clarification, below]
Output: a polynomial-time deterministic Turing machine which accepts the language L

Is it known that if such a map exists then it cannot be computable?

REPLY [3 votes]: The question has basically been answered via the comments but it may help to summarize the conclusion.  If you insist that the input be unclocked NP machines then nothing useful can possibly be computed from the input, as explained in the answer to this related MO question by Joel David Hamkins.  But this kind of uncomputability result is, I would argue, completely uninteresting and irrelevant to your intended question, because it has absolutely nothing at all to do with P or NP.  It just amounts to the fact that arbitrary Turing machines are intractable objects.  On the other hand, if the input is a clocked NP machine, then Cook's reduction shows how to construct a P machine that solves your problem (assuming P = NP).  This is really what we care about in practice.  If I have a problem that I know is in NP, then I want a mechanical way of producing a polytime algorithm for it (assuming P = NP).  It's really irrelevant that there are all kinds of other, bizarre NP machines that accept the same language, and that it's an uncomputable task to sift through them.