In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is an operation that assigns to each decision problem  a successively harder decision problem  with the property that  is not decidable by an oracle machine with an oracle for .
The operator is called a jump operator because it increases the Turing degree of the problem .
That is, the problem  is not Turing-reducible to .
Post's theorem establishes a relationship between the Turing jump operator and the arithmetical hierarchy of sets of natural numbers..
Informally, given a problem, the Turing jump returns the set of Turing machines that halt when given access to an oracle that solves that problem.
Definition
The Turing jump of X can be thought of as an oracle to the halting problem for oracle machines with an oracle to X.
Formally, given a set  and a Gödel numbering  of the -computable functions, the Turing jump  of  is defined as
X'= \{x \mid \varphi_x^X(x) \ \mbox{is defined} \}.
The th Turing jump  is defined inductively by
X^{(0)} = X,
X^{(n+1)} = (X^{(n)})'.
The  jump  of  is the effective join of the sequence of sets  for :
X^{(\omega)} = \{p_i^k \mid i \in \mathbb{N} \text{ and } k \in X^{(i)}\},
where  denotes the th prime.
The notation  or  is often used for the Turing jump of the empty set.
It is read zero-jump or sometimes zero-prime.
Similarly,  is the th jump of the empty set.
For finite , these sets are closely related to the arithmetic hierarchy.
The jump can be iterated into transfinite ordinals: the sets  for , where  is the Church–Kleene ordinal, are closely related to the hyperarithmetic hierarchy.
Beyond , the process can be continued through the countable ordinals of the constructible universe, using set-theoretic methods (Hodes 1980).
The concept has also been generalized to extend to uncountable regular cardinals (Lubarsky 1987).
Examples
The Turing jump  of the empty set is Turing equivalent to the halting problem.
For each , the set  is m-complete at level \Sigma^0_n in the arithmetical hierarchy (by Post's theorem).
The set of Gödel numbers of true formulas in the language of Peano arithmetic with a predicate for  is computable from .
Properties
is -computably enumerable but not -computable.
If  is Turing-equivalent to , then  is Turing-equivalent to .
The converse of this implication is not true.
(Shore and Slaman, 1999) The function mapping  to  is definable in the partial order of the Turing degrees.
Many properties of the Turing jump operator are discussed in the article on Turing degrees.
References
Ambos-Spies, K. and Fejer, P.  Degrees of Unsolvability.
Unpublished.
http://www.cs.umb.edu/~fejer/articles/History_of_Degrees.pdf
