Turing reducibility

A relation between sets of natural numbers. We say that $A$ is Turing reducible to $B$, $A \le_{\mathrm{T}} B$, if there is a Turing machine for the decision problem $x \in A$ given an auxiliary tape which encodes the answers to all questions $y \in B$. Such a tape is often described as an "oracle" or "black box" for the problem of membership of $B$. This defines a pre-order on sets of natural numbers:Turing reduction may also be regarded as a relation between functions on natural numbers: in this case, the characteristic functions of sets of natural numbers. The equivalence relation $A \equiv_{\mathrm{T}} B$ if $A \le_{\mathrm{T}} B$ and $B \le_{\mathrm{T}} A$ is Turing equivalence and the equivalence classes are the Turing degrees. Turing reducibility defines an partial order on the Turing degrees.

References

• Nies, André Computability and randomness Oxford Logic Guides 51 Oxford University Press (2009) ISBN 0-19-923076-1 Zbl 1169.03034
• Pippenger, Nicholas Theories of Computability Cambridge University Press (1997) ISBN 0-521-55380-6 Zbl 1200.03025