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.^[1] 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 for X.^[1]

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

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

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

${\displaystyle X^{(0)}=X,}$

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

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

${\displaystyle X^{(\omega )}=\{p_{i}^{k}\mid i\in \mathbb {N} {\text{ and }}k\in X^{(i)}\},}$

where p_i 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,^[2] and is in particular connected to Post's theorem.

The jump can be iterated into transfinite ordinals: there are jump operators ${\displaystyle j^{\delta }}$ for sets of natural numbers when ${\displaystyle \delta }$ is an ordinal that has a code in Kleene's ${\displaystyle {\mathcal {O}}}$ (regardless of code, the resulting jumps are the same by a theorem of Spector),^[2] in particular the sets 0^(α) for α < ω₁^CK, where ω₁^CK is the Church–Kleene ordinal, are closely related to the hyperarithmetic hierarchy.^[1] Beyond ω₁^CK, the process can be continued through the countable ordinals of the constructible universe, using Jensen's work on fine structure theory of Gödel's L.^[3][2] The concept has also been generalized to extend to uncountable regular cardinals.^[4]

Examples

• The Turing jump 0′ of the empty set is Turing equivalent to the halting problem.^[5]
• For each n, the set 0⁽ⁿ⁾ is m-complete at level ${\displaystyle \Sigma _{n}^{0}}$ 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).^[6]

Properties

• 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.^[5]

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
• 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. Vol. 52, no. 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.