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 a successively harder decision problem with the property that is not decidable by an oracle machine with an oracle for.
The operator is called a jump operator because it increases the Turing degree of the problem. That is, the problem is not Turing-reducible to. 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

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 and a Gödel numbering of the -computable functions, the Turing jump of is defined as
The th Turing jump is defined inductively by
The jump of is the effective join of the sequence of sets for :
where denotes the th prime.
The notation or is often used for the Turing jump of the empty set. It is read zero-jump or sometimes zero-prime.
Similarly, is the th jump of the empty set. For finite, these sets are closely related to the arithmetic hierarchy.
The jump can be iterated into transfinite ordinals: the sets for, where is the Church–Kleene ordinal, are closely related to the hyperarithmetic hierarchy. Beyond, the process can be continued through the countable ordinals of the constructible universe, using set-theoretic methods. The concept has also been generalized to extend to uncountable regular cardinals.

Examples

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