# Turing jump

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Turing jump

In computability theory, the Turing jump or Turing jump operator, named for Alan Turing, is intuitively described as an operation that assigns to each decision problem "X" a successively harder decision problem "X"&prime; with the property that "X"&prime; is not decidable by an oracle machine with an oracle for "X".

The operator is called a "jump operator" because it increases the Turing degree of the problem "X". That is, the problem "X"&prime; is not Turing reducible to "X".
Post's theorem establishes a relationship between the Turing jump operator and the arithmetical hierarchy of sets of natural numbers.

Definition

Given a set "X" and a Gödel numbering $varphi_i^X$ of the "X"-computable functions, the Turing jump "X"&prime; of "X" is defined as

:$X\text{'}= \left\{x mid varphi_x^X\left(x\right) mbox\left\{is defined\right\} \right\}.$

The "n"th Turing jump "X"("n") is defined inductively by:$X^\left\{\left(0\right)\right\} = X, ,$:$X^\left\{\left(n+1\right)\right\}=\left(X^\left\{\left(n\right)\right\}\right)\text{'}. ,$

The &omega; jump "X"(&omega;) of "X" is the effective join of the sequence of sets $langle X^\left\{\left(n\right)\right\}mid n in mathbb\left\{N\right\} angle$:

:$X^\left\{\left(omega\right)\right\} = \left\{p_i^k mid k in X^\left\{\left(i\right)\right\}\right\},,$

where $p_i$ denotes the "i"th prime.

The notation 0&prime; is often used for the Turing jump of the empty set, which can also be written as

:$emptyset\text{'}.$

Similarly, $0^\left\{\left(n\right)\right\}$ is the "n"th jump of the empty set.

Examples

* The Turing jump $varnothing\text{'}$ of the empty set is Turing equivalent to the halting problem.
* For each "n", the set $varnothing^\left\{\left(n\right)\right\}$ is m-complete at level $Sigma^0_n$ in the arithmetical hierarchy.
* The set of Gödel numbers of true formulas in the language of Peano arithmetic with a predicate for "X" is computable from $X^\left\{\left(omega\right)\right\}$.

Properties

* "X"&prime; is "X"-computably enumerable but not "X"-computable.
* If "A" is Turing equivalent to "B" then "A"&prime; is Turing equivalent to "B"&prime;. The converse of this implication is not true.
* (Shore and Slaman, 1999) The function mapping "X" to "X"&prime; 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

cite book
author = Lerman, M.
year = 1983
title = Degrees of unsolvability: local and global theory
publisher = Berlin; New York: Springer-Verlag
isbn = 3-540-12155-2

cite book
author = Rogers Jr, H.
year = 1987
title = Theory of recursive functions and effective computability
publisher = MIT Press Cambridge, MA, USA
isbn = 0-07-053522-1

cite journal
author = Shore, R.A.
coauthors = Slaman, T.A.
year = 1999
title = Defining the Turing jump
journal = Math. Res. Lett.
volume = 6
issue = 5-6
pages = 711-722
url = http://www.mrlonline.org/mrl/1999-006-006/1999-006-006-010.pdf
accessdate = 2008-07-13

cite book
author = Soare, R.I.
year = 1987
title = Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets
publisher = Springer
isbn = 3-540-15299-7

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