Definition
Given two sets of natural numbers, we say is Turing reducible to and write
if there is an oracle machine that computes the characteristic function of A when run with oracle B. In this case, we also say A is B-recursive and B-computable.
If there is an oracle machine that, when run with oracle B, computes a partial function with domain A, then A is said to be B-recursively enumerable and B-computably enumerable.
We say is Turing equivalent to and write if both and The equivalence classes of Turing equivalent sets are called Turing degrees. The Turing degree of a set is written .
Given a set, a set is called Turing hard for if for all . If additionally then is called Turing complete for .
Read more about this topic: Turing Reduction
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