reduction from VC to IS

Question

$\ L= \{ \langle M \rangle | $ the function f that M computes is a polynomial reduction from VC to IS $\ \} $

Is the above language in RE? in coRE?

Answer 1

Hint:

Since VC and IS are both NP-complete problems, you know that at least one polynomially-computable reduction $f$ exists. Now for some machine $T$ consider the two machines

M1:
  read X
  simulate T with empty input tape for |X| steps
  if T has halted yet:
      loop forever
  else:
      compute and return f(X)

and

M2:
  read X
  simulate T with empty input tape until it halts
  compute and return f(X)

Answer 2

Such an $L$ would neither be RE nor co-RE, because $L$ can many-one compute the $\Pi_2$-complete language $\mathrm{Tot} = \{M : M\ \mathrm{halts\ on\ every\ possible\ input}\}$.