# Turing jump

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In computability theory, the **Turing jump** or **Turing jump operator**, named for Alan Turing, is an operation that assigns to each decision problem *X* a successively harder decision problem *X* ′ with the property that *X* ′ 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* ′ 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. Informally, given a problem, the Turing jump returns the set of Turing machines which halt when given access to an oracle that solves that problem.

## Contents

## Definition[edit]

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 *X* and a Gödel numbering φ_{i}^{X} of the *X*-computable functions, the **Turing jump** *X* ′ of *X* is defined as

The *n*th Turing jump*X*^{(n)} is defined inductively by

The **ω jump** *X*^{(ω)} of *X* is the effective join of the sequence of sets *X*^{(n)} for *n* ∈ **N**:

where *p*_{i} denotes the *i*th prime.

The notation 0′ or ∅′ is often used for the Turing jump of the empty set. It is read *zero-jump* or sometimes *zero-prime*.

Similarly, 0^{(n)} is the *n*th jump of the empty set. For finite *n*, these sets are closely related to the arithmetic hierarchy.

The jump can be iterated into transfinite ordinals: the sets 0^{(α)} for α < ω_{1}^{CK}, where ω_{1}^{CK} is the Church–Kleene ordinal, are closely related to the hyperarithmetic hierarchy. Beyond ω_{1}^{CK}, 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[edit]

- The Turing jump 0′ of the empty set is Turing equivalent to the halting problem.
- For each
*n*, the set 0^{(n)}is m-complete at level 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
*X*is computable from*X*^{(ω)}.

## Properties[edit]

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

- Ambos-Spies, K. and Fejer, P. Degrees of Unsolvability. Unpublished. http://www.cs.umb.edu/~fejer/articles/History_of_Degrees.pdf
- Hodes, Harold T. (June 1980). "Jumping Through the Transfinite: The Master Code Hierarchy of Turing Degrees".
*Journal of Symbolic Logic*. Association for Symbolic Logic.**45**(2): 204–220. doi:10.2307/2273183. JSTOR 2273183. - Lerman, M. (1983).
*Degrees of unsolvability: local and global theory*. Berlin; New York: Springer-Verlag. ISBN 3-540-12155-2. - Lubarsky, Robert S. (Dec 1987). "Uncountable Master Codes and the Jump Hierarchy".
*Journal of Symbolic Logic*.**52**(4). pp. 952–958. JSTOR 2273829. - Rogers Jr, H. (1987).
*Theory of recursive functions and effective computability*. MIT Press, Cambridge, MA, USA. ISBN 0-07-053522-1. - Shore, R.A.; Slaman, T.A. (1999). "Defining the Turing jump" (PDF).
*Mathematical Research Letters*.**6**(5–6): 711–722. doi:10.4310/mrl.1999.v6.n6.a10. Retrieved 2008-07-13. - Soare, R.I. (1987).
*Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets*. Springer. ISBN 3-540-15299-7.