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Topic: Turing reduction

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	Turing reduction - Encyclopedia, History, Geography and Biography
		More formally, a Turing reduction is a function computable by an oracle machine with an oracle for B. If such a reduction exists, then every algorithm for M immediately yields an algorithm for L, formed by inserting a "call" to that algorithm at each place where the oracle machine uses it.
		Demonstrating a Turing reduction from a problem A to a problem in such a class C shows that A ∈ C. Many important complexity classes such as NP are not believed to be closed under Turing reductions.
		Turing reductions are often subjected to additional resource restrictions, for example that the oracle machine runs in polynomial time or logarithmic space; see polynomial-time reduction and log-space reduction for details.
	www.arikah.com /encyclopedia/Turing_reduction (464 words)

	Turing reduction (Site not responding. Last check: 2007-11-02)
		In computational complexity theory, a Turing reduction from a problem A to a problem B is, intuitively, a reduction which easily solves B, assuming A is easy to solve.
		More formally, a Turing reduction is a function computable by an oracle machine with an oracle for A. If such a reduction exists, then every algorithm for M immediately yields an algorithm for L, formed by inserting a "call" to that algorithm at each place where the oracle machine uses it.
		Demonstrating a Turing reduction from a problem A to a problem in such a class C shows that A ∈ C. Many important complexity classes such as NP are not closed under Turing reductions.
	www.kiwipedia.com /turing-reduction.html (336 words)

	Alan Turing (Stanford Encyclopedia of Philosophy)
		Turing's motivations were scientific rather than industrial or commercial, and he soon returned to the theoretical limitations of computation, this time focussing on the comparison of the power of computation and the power of the human brain.
		Turing's underlying argument was that the human brain must somehow be organised for intelligence, and that the organisation of the brain must be realisable as a finite discrete-state machine.
		Turing was in fact sensitive to the difficulty of separating ‘intelligence’ from other aspects of human senses and actions; he described ideas for robots with sensory attachments and raised questions as to whether they might enjoy strawberries and cream or feel racial kinship.
	plato.stanford.edu /entries/turing (9730 words)

	NP-complete
		This holds because by their definition the classes of NP-complete and co-NP-complete problems under Turing reductions are the same and because these classes are both supersets of the same classes defined with many-one reductions.
		Another type of reduction that is also often used to define NP-completness is the logarithmic-space many-one reduction[?] which is a many-one reduction that can be computed with only a logarithmic amount of space.
		This type of reduction is more refined then the more usual polynomial-time many-one reductions and it allows us to distinguish more classes such as P-complete.
	www.ebroadcast.com.au /lookup/encyclopedia/np/NP-complete.html (949 words)

	Reduction (complexity)
		A reduction is a preordering, that is a reflexive and transitive relation, on P(N)×P(N), where P(N) is the power set of the natural numbers.
		Many-one reductions map instances of one problem to instances of another; Turing reductions compute the solution to one problem, assuming the other problem is easy to solve.
		Reduction is also used in computability theory to show whether problems are or are not solvable by machines at all; in this case, reductions are restricted only to computable functions.
	www.xasa.com /wiki/en/wikipedia/r/re/reduction__complexity_.html (895 words)

	PlanetMath: Cook reduction
		Cook reductions, which are often just called Cook reductions.
		Note that it is a stronger condition than a Cook reduction.
		This is version 4 of Cook reduction, born on 2002-09-06, modified 2005-03-03.
	planetmath.org /encyclopedia/CookReduction.html (150 words)

	Polynomial-time Turing reduction: Definition and Links by Encyclopedian.com
		...Polynomial-time Turing reduction Polynomial-time Turing reduction In...theory a polynomial-time Turing reduction or Cook reduction of a decision problem L to a...a decision problem M is an oracle machine that has an oracle for M and can decide L in polynomial...
		If such a reduction exists, then every algorithm for M immediately yields an algorithm for L, with only a modest (i.e.
		The intuitive notion of reducibility can be formalized in different ways: see polynomial-time many-one reduction and logarithmic-space many-one reduction[?].
	www.encyclopedian.com /po/Polynomial-time-Turing-reduction.html (258 words)

	Tools for Thought by Howard Rheingold: Chapter One
		Turing proved that his hypothetical machine is an automated version of a formal system specified by the starting position (the pattern of Os and Xs on the tape at the beginning of the computation) and the rules (the instructions given by the instruction tables).
		Turing then asks us to substitute a machine for one of the unknown players and make a new object for the game: This time, the interrogator is to guess, on the basis of the teletyped conversation, which inhabitant of the other room is a human being and which one is a machine.
		Turing was arrested and convicted of "gross indecency," and sentenced to probation on the condition that he submit to humiliating and physically debilitating female hormone injections.
	www.well.com /user/hlr/texts/tft3.html (6936 words)

	Science Fair Projects - Polynomial-time Turing reduction
		In computational complexity theory a polynomial-time reduction is a reduction which is computable by a deterministic Turing machine in polynomial time.
		Polynomial-time reductions are important and widely-used because they are powerful enough to perform many transformations between important problems, but still weak enough that polynomial-time reductions from problems in NP or co-NP to problems in P are considered unlikely to exist.
		Within the class P, however, polynomial-time reductions are inappropriate, because any problem in P can be polynomial-time reduced (both many-one and Turing) to any other problem in P. Thus, for classes within P such as L, NL, NC, and P itself, log-space reductions are used instead.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Polynomial-time_Turing_reduction (418 words)

	Alan Turing and the Turing Test - Andrew Hodges
		Turing's 1950 paper did not arise in isolation, and the purpose of this biographical sketch is to set Turing's test in the context of his life and work.
		Turing's post-war argument (the point of view he probably arrived at in about 1941) is, however, that human beings do not reliably see the truth of such statements.
		Turing also tellingly described 1940 as the date after which machines were no longer restricted to 'extremely straightforward, possibly even to repetitious, jobs.' He must have had his own Enigma-breaking Bombe, and other highly sophisticated codebreaking operations, in mind.
	www.turing.org.uk /publications/testbook.html (4287 words)

	Alan Turing: a short biography - 8
		This short on-line biography of Alan Turing is based on the entry I wrote for the British Dictionary of National Biography in 1995.
		Alan Turing was arrested and came to trial on 31 March 1952, after the police learned of his sexual relationship with a young Manchester man. He made no serious denial or defence, instead telling everyone that he saw no wrong with his actions.
		Besides this he refreshed his youthful interest in quantum physics, studying the problem of wave-function reduction in quantum mechanics, with a hint that he was considering a non-linear mechanism for it.
	www.turing.org.uk /bio/part8.html (592 words)

	[No title]
		In the case of Turing transducers, however, none of the constraints made on memory is significant, because they can all be removed and still the transducers acquire no more definition power.
		The intuitive explanation for this phenomenon is that each Turing transducer is a description of a language (i.e., a set of strings), which itself has a description by a string.
		The variant of the technique used here is called a proof by diagonalization, owing to the employment of the diagonal of a table for choosing the language that provides the contradiction.
	www.cse.ohio-state.edu /~gurari/theory-bk/theory-bk-fourse5.html (1665 words)

	Foundations of Mathematics
		In the period very roughly from the beginnings of modern physics (1905) up to Alan Turing's description of the Turing machine in 1938, one of the focal points of dispute in the theory of knowledge was the foundations of mathematics.
		Turing championed the theory that computers could be constructed that would be capable of human thought and his writing on this subject show considerable affinity with behaviourist psychology.
		Turing's work introduced new concepts of complexity in language which have provided the basis for Noam Chomsky's Kantian structural psychology and the foundations of complexity theory.
	www.marxists.org /reference/subject/philosophy/help/maths.htm (3348 words)

	NP-easy
		This means that given a fl box that solves instances of Y in unit time, there exists an algorithm that solves X in polynomial time by repeatedly using that fl box.
		The definition of NP-easy used a Turing reduction rather than a many-one reduction[?] because the answers to problem Y are only TRUE or FALSE, but the answers to problem X can be more general.
		NP-hard also uses a Turing reduction, for the same reason.
	www.ebroadcast.com.au /lookup/encyclopedia/np/NP-easy.html (210 words)

	Turing degree - RSCI, The Science Classification Index (Site not responding. Last check: 2007-11-02)
		In computability theory, the Turing degree of a subset $X of the natural numbers, \omega, is the equivalence class of all subsets of \omega equivalent to X under$ Turing reducibility.
		A recursively enumerable Turing degree (computably enumerable Turing degree) is one containing a recursively enumerable set (computably enumerable set).
		The recursively enumerable Turing degrees under the partial order induced by Turing reducibility form an upper semi-lattice and are an object that has been much studied by the logic community.
	www.scienceindex.org /Turing_degree.html (174 words)

	turing reducibility
		This is usually done with Turing machines that are allowed to ask arbitrary questions about a given set A and to use the results in their computations as they please.
		The last clause of the proposition shows that Turing reduction is also somewhat strange, for we can no longer rely on Turing computability to preserve one-sided notions of success.
		Turing degrees (or simply "degrees") are conventionally denoted by bold-faced letters: a, b, c,...
	www.andrew.cmu.edu /user/kk3n/recursionclass/11tdegrees.html (1178 words)

	COSC5313FINAL-1999SPRING
		If P is polynomially (Turing) reduced to Q, the reduction process involved may take longer than a polynomial time in n where n is the size of the instance of either P or Q, whichever is longer.
		If P is polynomially (Turing) reduced to Q, then P cannot be more difficult to solve than Q. This is because any solution to P, no matter how tractable or intractable it may be, suggests a solution to Q of no more complexity.
		PARTITION problem (on page 470) is polynomially (Turing) reduced to the 0/1 Knapsack problem in which the value of every item is equal to its weight and the total weight limit is exactly the half of the sum of all item weights.
	hal.lamar.edu /~koh/5313/5313FINAL.HTML (1017 words)

	[No title]
		Reductions and Recursive Languages Theorem: If there is a reduction from L1 to L2 and L2 is recursive, then L1 is recursive.
		Reductions and RE Languages Theorem: If there is a reduction from L1 to L2 and L2 is RE, then L1 is RE.
		Declare that the reduction proves that your “to” language is not recursive.
	www.cs.utexas.edu /~cline/ear/Slides/Turing/TuringSlides4.doc (2761 words)

	308-506 Lecture Notes for 30 October 2001 (Site not responding. Last check: 2007-11-02)
		This reduction is a pseudo-polynomial time transformation as defined on GJ page 101, but this fact does not help us immediately because PARTITION is pptime.
		Using PARTITION twice would not be possible in the Karp reductions we have been studying so far, and which we used to define NP-completeness.
		Thus TAUTOLOGY, a co-NP-complete problem, is NP-hard because the Turing reduction can ask whether the negation of the formula is in SAT and reverse the answer.
	www.cs.mcgill.ca /~barring/notes/14.htm (1620 words)

	CSCI546+646: Session 05 -- introducing the Turing Machine
		According to the "Direction of Reduction" box at the top of page 316, if we are trying to find out if P2 is undecidable, we need to reduce a known undecidable program like P1 to P2.
		In this section they said the Turing Machine was similar to the finite automata.
		Turing's Definition of a TM In Turing's paper and in early papers and books the transition function is described as being defined in a table of quintuples.
	www.csci.csusb.edu /dick/cs546/05.html (1759 words)

	Tu
		In computational complexity theory, a '''Turing reduction''' from a problem A to a problem B is, intuitively, a reduction (complexity)reduction which easily solves B, assuming A is easy to solve.
		Demonstrating a Turing reduction from a problem A to a problem in such a class C shows that A ∈ C. Many important complexity classes such as NP (complexity)NP are not closed under Turing reductions.
		However, a number of classes within P (complexity)P, such as L (complexity)L, NL (complexity)NL, SL (complexity)SL, and P (complexity)P itself, are closed under Turing reductions.
	www.gateserver.net /Topicdetails.aspx?Topicid=51806&name=&catid=543&topicname=Turing_reduction (365 words)

	Randomized Turing Reductions
		It is a long-standing open problem whether Turing reductions (or truth-table reductions) are provably more powerful than many-one reductions on NP problems.
		If in a p-time (ap-time) randomized Turing reduction all queries can be generated at the beginning independent of other queries, then the reduction is called a p-time (ap-time) randomized truth-table reduction.
		Randomized Turing reductions are closed for randomized ap-time solvable problems and are transitive in the special case where on input
	www.uncg.edu /mat/avg/avgcomp/node17.html (367 words)

	URCS Theory Technical Reports
		We prove that query-increasing and query-decreasing Turing reductions are incomparable with (that is, are neither strictly stronger than nor strictly weaker than) truth-table reductions and are strictly weaker than Turing reductions.
		Despite the fact that we prove query-increasing and query-decreasing Turing reductions to in the general case be strictly weaker than Turing reductions, we identify a broad class of sets A for which any set that Turing reduces to A will also reduce to A via both query-increasing and query-decreasing Turing reductions.
		Generalizing robust reductions, we note that robustly strong reductions are built from two restrictions, robust underproductivity and robust overproductivity, both of which have been separately studied before in other contexts.
	www.cs.rochester.edu /trs/theory-trs.html (16135 words)

	NP easy (Site not responding. Last check: 2007-11-02)
		In complexity theory, the complexity class NP-easy is the set of function problems that are solvable in polynomial time by a deterministic Turing machine with an oracle for some decision problem in NP.
		In other words, a problem X is NP-easy if and only if there existssome problem Y in NP such that X is polynomial-time Turing reducible to Y. This means that given an oracle for Y, there exists an algorithm that solves X in polynomial time(possibly by repeatedly using that oracle).
		The definition of NP-easy uses a Turing reduction rather than a many-one reduction because the answers to problem Y are only TRUE or FALSE, but the answers toproblem X can be more general.
	www.therfcc.org /np-easy-329366.html (240 words)

	(E. Hemaspaandra) On the Power of Positive Turing Reductions (Site not responding. Last check: 2007-11-02)
		In the early 1980s, Selman's seminal work on positive Turing reductions showed that positive Turing reduction to NP yields no greater computational power than NP itself.
		Thus, positive Turing and Turing reducibility to NP differ sharply unless the polynomial hierarchy collapses.
		Thus, positive Turing and Turing reducibility to DP yield the same class.
	www.jucs.org /jucs_5_12/on_the_power_of (125 words)

	Intractability
		As digital computers were developed in the 1940s and 1950s, the Turing machine served as the theoretical model of computation.
		Turing machines and proved that some problems have "an inherent complexity that cannot be circumvented by clever programming." They also proved a formal version (time hierarchy theorem) of the intuitive idea that if given more time or space, Turing machines can compute more things.
		Reduction or simulation is a powerful idea in computer science....
	www.cs.princeton.edu /introcs/78intractability (7500 words)

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