|
|
| Line 1: |
Line 1: |
| In [[computability theory]], the '''Turing jump''' or '''Turing jump operator''', named for [[Alan Turing]], is an operation that assigns to each [[decision problem]] {{math|''X''}} a successively harder decision problem {{math|''X'' ′}} with the property that {{math|''X'' ′}} is not decidable by an [[oracle machine]] with an [[oracle (computer science)|oracle]] for {{math|''X''}}.
| | Hi, everybody! <br>I'm German male :). <br>I really like Seaglass collecting!<br><br>Feel free to surf to my page - [http://Livebabble.com/activity/p/62111/ Fifa 15 coin generator] |
| | |
| The operator is called a ''jump operator'' because it increases the [[Turing degree]] of the problem {{math|''X''}}. That is, the problem {{math|''X'' ′}} is not [[Turing reducible]] to {{math|''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 {{math|''X''}} and a [[Gödel numbering]] {{math|φ<sub>''i''</sub><sup>''X''</sup>}} of the [[relative computability|{{math|''X''}}-computable]] functions, the '''Turing jump''' {{math|''X'' ′}} of {{math|''X''}} is defined as
| |
| | |
| :<math>X'= \{x \mid \varphi_x^X(x) \ \mbox{is defined} \}.</math> | |
| | |
| The '''{{math|''n''}}th Turing jump''' {{math|''X''<sup>(''n'')</sup>}} is defined inductively by
| |
| :<math>X^{(0)} = X, \,</math>
| |
| :<math>X^{(n+1)}=(X^{(n)})'. \,</math>
| |
| | |
| The '''{{math|ω}} jump''' {{math|''X''<sup>(ω)</sup>}} of {{math|''X''}} is the [[effective join]] of the sequence of sets {{math|''X''<sup>(''n'')</sup>}} for {{math|''n'' ∈ '''[[Natural numbers|N]]'''}}: <!-- <math>\langle X^{(n)}\mid n \in \mathbb{N}\rangle</math> -->
| |
| | |
| :<math>X^{(\omega)} = \{p_i^k \mid k \in X^{(i)}\},\,</math>
| |
| | |
| where {{math|''p''<sub>''i''</sub>}} denotes the {{math|''i''}}th prime.
| |
| | |
| The notation {{math|0′}} or {{math|∅′}} is often used for the Turing jump of the empty set. It is read ''zero-jump'' or sometimes ''zero-prime''.
| |
| | |
| Similarly, {{math|0<sup>(''n'')</sup>}} is the {{math|''n''}}th jump of the empty set. For finite {{math|''n''}}, these sets are closely related to the [[arithmetic hierarchy]].
| |
| | |
| The jump can be iterated into transfinite ordinals: the sets {{math|0<sup>(α)</sup>}} for {{math|α < ω<sub>1</sub><sup>CK</sup>}}, where {{math|ω<sub>1</sub><sup>CK</sup>}} is the [[Church-Kleene ordinal]], are closely related to the [[hyperarithmetic hierarchy]]. Beyond {{math|ω<sub>1</sub><sup>CK</sup>}}, 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 cardinal]]s (Lubarsky 1987).
| |
| | |
| == Examples ==
| |
| | |
| * The Turing jump {{math|0′}} of the empty set is Turing equivalent to the [[halting problem]].
| |
| * For each {{math|''n''}}, the set {{math|0<sup>(''n'')</sup>}} is [[m-complete]] at level <math>\Sigma^0_n</math> in the [[arithmetical hierarchy]].
| |
| * The set of Gödel numbers of true formulas in the language of [[Peano arithmetic]] with a predicate for {{math|''X''}} is computable from {{math|''X''<sup>(ω)</sup>}}.
| |
| | |
| == Properties ==
| |
| | |
| * {{math|''X'' ′}} is {{math|''X''}}-[[computably enumerable]] but not {{math|''X''}}-[[computable function|computable]].
| |
| * If {{math|''A''}} is [[Turing degree|Turing equivalent]] to {{math|''B''}} then {{math|''A''′}} is Turing equivalent to {{math|''B''′}}. The converse of this implication is not true.
| |
| * ([[Richard Shore|Shore]] and [[Theodore Slaman|Slaman]], 1999) The function mapping {{math|''X''}} to {{math|''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 degree]]s.
| |
| | |
| == References ==
| |
| | |
| *Ambos-Spies, K. and Fejer, P. Degrees of Unsolvability. Unpublished. http://www.cs.umb.edu/~fejer/articles/History_of_Degrees.pdf
| |
| *{{cite journal
| |
| | title=Jumping Through the Transfinite: The Master Code Hierarchy of Turing Degrees
| |
| | last=Hodes
| |
| | first=Harold T.
| |
| | journal=Journal of Symbolic Logic
| |
| | volume=45
| |
| |date=June 1980
| |
| | pages=204–220
| |
| | publisher=Association for Symbolic Logic
| |
| | jstor=2273183
| |
| | issue=2
| |
| | doi=10.2307/2273183
| |
| }}
| |
| *{{cite book
| |
| | author = Lerman, M.
| |
| | year = 1983
| |
| | title = Degrees of unsolvability: local and global theory
| |
| | publisher = Berlin; New York: Springer-Verlag
| |
| | isbn = 3-540-12155-2
| |
| }}
| |
| *{{cite article
| |
| | author = Lubarsky, Robert S.
| |
| | title = Uncountable Master Codes and the Jump Hierarchy
| |
| | journal = Journal of Symbolic Logic
| |
| | volume = 52
| |
| | issue = 4
| |
| | year = 1987
| |
| | month = Dec.
| |
| | pages = 952–958
| |
| | jstor = 2273829
| |
| }}
| |
| *{{cite book
| |
| | author = Rogers Jr, H.
| |
| | year = 1987
| |
| | title = Theory of recursive functions and effective computability
| |
| | publisher = MIT Press Cambridge, MA, USA
| |
| | isbn = 0-07-053522-1
| |
| }}
| |
| *{{cite journal
| |
| | author = Shore, R.A.
| |
| | coauthors = Slaman, T.A.
| |
| | year = 1999
| |
| | title = Defining the Turing jump
| |
| | journal = Mathematical Research Letters
| |
| | volume = 6
| |
| | issue = 5–6
| |
| | pages = 711–722
| |
| | url = http://math.berkeley.edu/~slaman/papers/jump.pdf
| |
| | accessdate = 2008-07-13
| |
| }}
| |
| *{{cite book
| |
| | author = Soare, R.I.
| |
| | year = 1987
| |
| | title = Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets
| |
| | publisher = Springer
| |
| | isbn = 3-540-15299-7
| |
| | |
| }}
| |
| | |
| [[Category:Computability theory]]
| |
Hi, everybody!
I'm German male :).
I really like Seaglass collecting!
Feel free to surf to my page - Fifa 15 coin generator