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.

Definition

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

The nth 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 nN:

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

Properties

Many properties of the Turing jump operator are discussed in the article on Turing degrees.

References

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