# Turing jump

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.

## Definition

Given a set X and a Gödel numbering φiX of the X-computable functions, the Turing jump X of X is defined as

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

The nth Turing jump X(n) is defined inductively by

$X^{(0)} = X, \,$
$X^{(n+1)}=(X^{(n)})'. \,$

The ω jump X(ω) of X is the effective join of the sequence of sets X(n) for nN:

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

where pi denotes the ith 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 nth 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 < ω1CK, where ω1CK is the Church-Kleene ordinal, are closely related to the hyperarithmetic hierarchy. Beyond ω1CK, 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 0′ of the empty set is Turing equivalent to the halting problem.
• For each n, the set 0(n) 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(ω).

## Properties

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 Unpublished.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
• 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". Mathematical Research Letters 6 (5–6): 711–722. 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.