en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3427504

2013-03-11T20:29:46Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3427504

← Older revision		Revision as of 22:29, 11 March 2013
Line 1:		Line 1:
	~~Luke is usually a celebrity within~~ the ~~earning along with~~ the ~~vocation progress 1st 2nd~~ to ~~his 3rd stadium record,~~ ~~And~~ , ~~may be~~ the ~~resistant~~. ~~He burst~~ to the ~~picture~~ in ~~2015 together with his crazy mixture [http~~://~~lukebryantickets~~.~~neodga~~.~~com tickets~~ to see ~~luke bryan]~~ of ~~down-home accessibility~~, ~~movie legend good seems~~ and ~~words~~, is ~~scheduled t inside~~ a ~~main way~~. ~~The newest~~ a on the ~~nation chart and #2~~ in the ~~put charts~~, ~~generating it~~ the ~~next maximum very first at that time~~ of ~~2007 for the nation artist~~. <br><br>The ~~kid~~ of the , ~~knows perseverance and dedication are important elements in relation~~ to a ~~prosperous~~ ~~career~~- . ~~His first album,~~ ~~Stay Me~~, ~~generated~~ the ~~Top strikes “All My Pals Say”~~ and ~~“Country Guy~~,~~” when his energy~~, ~~Doin’~~ ~~Factor, found~~ the ~~vocalist-about three straight No~~. ~~3 single people~~: ~~Else Getting in touch~~ with ~~Is a Very good Factor~~.”<br><br>~~During~~ the ~~tumble of 2013, Concert tours~~: ~~Bryan & that have~~ a ~~remarkable listing of~~ , ~~including Urban~~. ~~“It’s much like you are receiving a approval~~ to ~~visit to a higher [http:~~//~~lukebryantickets~~.~~lazintechnologies~~.~~com tour dates~~ for ~~luke bryan] level~~, ~~claims those artists that were~~ a ~~part~~ of ~~the Concert tourmore than into a greater measure~~ of ~~performers~~.~~” It covered among the most successful~~ ~~organized tours~~ in ~~their 10~~-~~year history~~.~~<br><br>Here is my web~~-~~site~~ - [~~http~~:~~//www.banburycrossonline.com~~ the ~~today show luke bryan~~]		In [[recursion theory\|computability theory]] '''Post's theorem''', named after [[Emil Post]], describes the connection between the [[arithmetical hierarchy]] and the [[Turing degree]]s.

			== Background ==
			The statement of Post's theorem uses several concepts relating to [[definability]]{{dn\|date=August 2013}} and [[recursion theory]]. This section gives a brief overview of these concepts, which are covered in depth in their respective articles.

			The [[arithmetical hierarchy]] classifies certain sets of natural numbers that are definable in the language of Peano arithmetic. A [[formula (mathematical logic)\|formula]] is said to be <math>\Sigma^{0}_m</math> if it is an existential statement in [[prenex normal form]] (all quantifiers at the front) with <math>m</math> alternations between existential and universal quantifiers applied to a quantifier free formula. Formally a formula <math>\phi(s)</math> in the language of Peano arithmetic is a <math>\Sigma^{0}_m</math> formula if it is of the form
			:<math>\exists n_1 \forall n_2 \exists n_3 \forall n_4 \cdots Q n_m \rho(n_1,\ldots,n_m,x_1,\ldots,x_k),</math>
			where ρ is a quantifier free formula and ''Q'' is <math>\forall</math> if ''m'' is even and <math>\exists</math> if ''m'' is odd. Note that any formula of the form
			:<math>\left(\exists n^1_1\exists n^1_2\cdots\exists n^1_{j_1}\right)\left(\forall n^2_1 \cdots \forall n^2_{j_2}\right)\left(\exists n^3_1\cdots\right)\cdots\left(Q_1 n^m_1 \cdots \right)\rho(n^1_1,\ldots n^m_{j_m},x_1,\ldots,x_k)</math>
			where <math>\rho</math> contains only [[bounded quantifier]]s is provably equivalent to a formula of the above form from the axioms of [[Peano arithmetic]]. Thus it isn't uncommon to see <math>\Sigma^{0}_m</math> formulas defined in this alternative and technically nonequivalent manner since in practice the distinction is rarely important.

			A set of natural numbers ''A'' is said to be <math>\Sigma^0_m</math> if it is definable by a <math>\Sigma^0_m</math> formula, that is, if there is a <math>\Sigma^0_m</math> formula <math>\phi(s)</math> such that each number ''n'' is in ''A'' if and only if <math>\phi(n)</math> holds. It is known that if a set is <math>\Sigma^0_m</math> then it is <math>\Sigma^0_n</math> for any <math>n > m</math>, but for each ''m'' there is a <math>\Sigma^0_{m+1}</math> set that is not <math>\Sigma^0_m</math>. Thus the number of quantifier alternations required to define a set gives a measure of the complexity of the set.

			Post's theorem uses the relativized arithmetical hierarchy as well as the unrelativized hierarchy just defined. A set ''A'' of natural numbers is said to be <math>\Sigma^0_m</math> relative to a set ''B'', written <math>\Sigma^{0,B}_m</math>, if
			''A'' is definable by a <math>\Sigma^0_m</math> formula in an extended language that includes a predicate for membership in ''B''.

			While the arithmetical hierarchy measures definability of sets of natural numbers, [[Turing degrees]] measure the level of uncomputability of sets of natural numbers. A set ''A'' is said to be [[Turing reduction\|Turing reducible]] to a set ''B'', written <math>A \leq_T B</math>, if there is an [[oracle Turing machine]] that, given an oracle for ''B'', computes the [[Indicator function\|characteristic function]] of ''A''.
			The [[Turing jump]] of a set ''A'' is a form of the [[Halting problem]] relative to ''A''. Given a set ''A'',
			the Turing jump <math>A'</math> is the set of indices of oracle Turing machines that halt on input ''0'' when run with oracle ''A''. It is known that every set ''A'' is Turing reducible to its Turing jump, but the Turing jump of a set is never Turing reducible to the original set.

			Post's theorem uses finitely iterated Turing jumps. For any set ''A'' of natural numbers, the notation
			<math>A^{(n)}</math> indicates the ''n''-fold iterated Turing jump of ''A''. Thus <math>A^{(0)}</math> is just ''A'', and <math>A^{(n+1)}</math> is the Turing jump of <math>A^{(n)}</math>.

			== Post's theorem and corollaries ==

			Post's theorem establishes a close connection between the arithmetical hierarchy and the Turing degrees of the form <math>\emptyset^{(n)}</math>, that is, finitely iterated Turing jumps of the empty set. (The empty set could be replaced with any other computable set without changing the truth of the theorem.)

			Post's theorem states:
			#A set ''B'' is <math>\Sigma^0_{n+1}</math> if and only if <math>B</math> is [[recursively enumerable]] by an oracle Turing machine with an oracle for <math>\emptyset^{(n)}</math>, that is, if and only if ''B'' is <math>\Sigma^{0,\emptyset^{(n)}}_1</math>.
			#The set <math>\emptyset^{(n)}</math> is <math>\Sigma^0_n</math> complete for every <math>n > 0</math>. This means that every <math>\Sigma^0_n</math> set is [[many-one reduction\|many-one reducible]] to <math>\emptyset^{(n)}</math>.

			Post's theorem has many corollaries that expose additional relationships between the arithmetical
			hierarchy and the Turing degrees. These include:
			#Fix a set ''C''. A set ''B'' is <math>\Sigma^{0,C}_{n+1}</math> if and only if ''B'' is <math>\Sigma^{0,C^{(n)}}_1</math>. This is the relativization of the first part of Post's theorem to the oracle ''C''.
			#A set ''B'' is <math>\Delta_{n+1}</math> if and only if <math>B \leq_T \emptyset^{(n)}</math>. More generally, ''B'' is <math>\Delta^C_{n+1}</math> if and only if <math>B \leq_T C^{(n)}</math>.
			#A set is defined to be arithmetical if it is <math>\Sigma^0_n</math> for some ''n''. Post's theorem shows that, equivalently, a set is arithmetical if and only if it is Turing reducible to <math>\emptyset^{(m)}</math> for some ''m''.
			<!--

			== Proof of Post's theorem ==

			{{Expand section\|date=May 2012}}

			-->

			== References ==

			Rogers, H. ''The Theory of Recursive Functions and Effective Computability'', MIT Press. ISBN 0-262-68052-1; ISBN 0-07-053522-1

			Soare, R. ''Recursively enumerable sets and degrees.'' Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1987. ISBN 3-540-15299-7

			[[Category:Theorems in the foundations of mathematics]]
			[[Category:Computability theory]]
			[[Category:Mathematical logic hierarchies]]

en>Rich Farmbrough at 00:47, 12 December 2011

2011-12-12T00:47:10Z

New page

Luke is usually a celebrity within the earning along with the vocation progress 1st 2nd to his 3rd stadium record, And , may be the resistant. He burst to the picture in 2015 together with his crazy mixture [http://lukebryantickets.neodga.com tickets to see luke bryan] of down-home accessibility, movie legend good seems and words, is scheduled t inside a main way. The newest a on the nation chart and #2 in the put charts, generating it the next maximum very first at that time of 2007 for the nation artist. The kid of the , knows perseverance and dedication are important elements in relation to a prosperous career- . His first album, Stay Me, generated the Top strikes “All My Pals Say” and “Country Guy,” when his energy, Doin’ Factor, found the vocalist-about three straight No. 3 single people: Else Getting in touch with Is a Very good Factor.” During the tumble of 2013, Concert tours: Bryan & that have a remarkable listing of , including Urban. “It’s much like you are receiving a approval to visit to a higher [http://lukebryantickets.lazintechnologies.com tour dates for luke bryan] level, claims those artists that were a part of the Concert tourmore than into a greater measure of performers.” It covered among the most successful organized tours in their 10-year history. Here is my web-site - [http://www.banburycrossonline.com the today show luke bryan]

Projective frame - Revision history

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3427504

en>Rich Farmbrough at 00:47, 12 December 2011