The Turing degree $\mathbf{0}'$ is defined as all languages Turing-equivalent to the halting problem. In fact any recursively enumerable language is polynomial-time reducible to the halting problem.

My question is, are all languages with Turing degree $\mathbf{0}'$ Turing-reducible to each other in polynomial time?
If not, can the fastest reduction between two $\mathbf{0}'$ languages get arbitrarily slow, and are there known subsets of $\mathbf{0}'$ that are polynomial-time reducible to each other?


1 Answer 1


Gödel's incompleteness theorem can be thought of as a reduction from the Halting problem to the language $\langle \varphi \mid \varphi \text{ is a true sentence in number theory}\rangle$, and a careful analysis of the running time would show that it is indeed a polynomial time reduction.

Not every such reduction is polynomial time, however. You can observe that a reduction from the Halting problem $\langle i \mid TM_i \text{ halts on input } i\rangle$ to the unary language $\langle 1^i \mid TM_i \text{ halts on input } i\rangle$ has exponential blowup in instance size.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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