stylistic:style_transfer, quality:corruption
Definition

Given two sets A, B ⊆ N {\displaystyle A,B\subseteq \mathbb {N} } of natural numbers, we say A {\displaystyle A} is Turing reducible to B {\displaystyle B} and write

if and only 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 A {\displaystyle A} is Turing equivalent to B {\displaystyle B} and write A ≡ T B {\displaystyle A\equiv _{T}B\,} if both A ≤ T B {\displaystyle A\leq _{T}B} and B ≤ T A. {\displaystyle B\leq _{T}A.} The equivalence classes of Turing equivalent sets are called Turing degrees. The Turing degree of a set X {\displaystyle X} is written deg (X) {\displaystyle {\textbf {deg}}(X)}.

Given a set X ⊆ P (N) {\displaystyle {\mathcal {X}}\subseteq {\mathcal {P}}(\mathbb {N})}, a set A ⊆ N {\displaystyle A\subseteq \mathbb {N} } is called Turing hard for X {\displaystyle {\mathcal {X}}} if X ≤ T A {\displaystyle X\leq _{T}A} for all X ∈ X {\displaystyle X\in {\mathcal {X}}}. If additionally A ∈ X {\displaystyle A\in {\mathcal {X}}} then A {\displaystyle A} is called Turing complete for X {\displaystyle {\mathcal {X}}}.