A language that is RE complete with respect to polynomial-time reductions?

Question

Is there any language in RE that is complete with regard to polynomial-time reductions?

I think that A_TM will be a good example,but I'm not sure...

Answer 1

Yes, A_TM is RE-complete with respect to polynomial-time reductions. Given any RE language L, let M be a recognizer for it. Then the function f(w) = can be computed in polynomial time (for some reasonable representation of tuples) because M is a fixed machine and the length of w in the encoded version should certainly be at most polynomially larger than the original input w. We also have that w ∈ L if and only if M accepts w if and only if ∈ A_TM, so f is a polynomial-time reduction from an arbitrary RE language L to A_TM, making A_TM RE-complete with respect to polynomial-time reductions.

I'm not sure why you'd be interested in this particular notion of RE-completeness, since RE is mostly useful for notions of computability (can you solve this problem at all?) while polynomial-time reductions are usually for complexity (can you solve this problem efficiently?) If you do have an interesting use case for them, though, I'd love to hear about it!