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| In [[computability theory]], a '''Turing reduction''' from a problem ''A'' to a problem ''B'', is a [[reduction (complexity)|reduction]] which solves ''A'', assuming ''B'' is already known (Rogers 1967, Soare 1987). It can be understood as an [[algorithm]] that could be used to solve ''A'' if it had available to it a [[subroutine]] for solving ''B''. More formally, a Turing reduction is a function computable by an [[oracle machine]] with an oracle for ''B''. Turing reductions can be applied to both [[decision problem]]s and [[function problem]]s.
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| If a Turing reduction of ''A'' to ''B'' exists then every [[algorithm]] for ''B'' can be used to produce an algorithm for ''A'', by inserting the algorithm for ''B'' at each place where the oracle machine computing ''A'' queries the oracle for ''B''. However, because the oracle machine may query the oracle a large number of times, the resulting algorithm may require more time asymptotically than either the algorithm for ''B'' or the oracle machine computing ''A'', and may require as much space as both together.
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| The first formal definition of relative computability, then called relative reducibility, was given by [[Alan Turing]] in 1939 in terms of [[oracle machine]]s. Later in 1943 and 1952 [[Stephen Kleene]] defined an equivalent concept in terms of [[Mu-recursive function|recursive function]]s. In 1944 [[Emil Post]] used the term "Turing reducibility" to refer to the concept.
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| A [[polynomial-time reduction|polynomial-time]] Turing reduction is known as a '''Cook reduction''', after [[Stephen Cook]].
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| == Definition ==
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| Given two sets <math>A,B \subseteq \mathbb{N}</math> of natural numbers, we say <math>A</math> is '''Turing reducible''' to <math>B</math> and write
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| :<math>A \leq_T B</math>
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| if there is an [[oracle machine]] that computes the [[Indicator function|characteristic function]] of ''A'' when run with oracle ''B''. In this case, we also say ''A'' is '''''B''-recursive''' and '''''B''-computable'''.
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| If there is an oracle machine that, when run with oracle ''B'', computes a partial function with domain ''A'', then ''A'' is said to be '''''B''-[[recursively enumerable set|recursively enumerable]]''' and '''''B''-computably enumerable'''.
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| We say <math>A</math> is '''Turing equivalent''' to <math>B</math> and write <math>A \equiv_T B\,</math> if both <math>A \leq_T B</math> and <math>B \leq_T A.</math> The [[equivalence class]]es of Turing equivalent sets are called '''[[Turing degree]]s'''. The Turing degree of a set <math>X</math> is written <math>\textbf{deg}(X)</math>.
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| Given a set <math>\mathcal{X} \subseteq \mathcal{P}(\mathbb{N})</math>, a set <math>A \subseteq \mathbb{N}</math> is called '''Turing hard''' for <math>\mathcal{X}</math> if <math>X \leq_T A</math> for all <math>X \in \mathcal{X}</math>. If additionally <math>A \in \mathcal{X}</math> then <math>A</math> is called '''Turing complete''' for <math>\mathcal{X}</math>.
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| ===Relation of Turing completeness to computational universality===
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| Turing completeness, as just defined above, corresponds only partially to [[Turing completeness]] in the sense of computational universality. Specifically, a Turing machine is a [[universal Turing machine]] if its [[halting problem]] (i.e., the set of inputs for which it eventually halts) is [[Many-one reduction|many-one complete]]. Thus, a necessary ''but insufficient'' condition for a machine to be computationally universal, is that the machine's halting problem be Turing-complete for the set <math>\mathcal{X}</math> of recursively enumerable sets.
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| == Example ==
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| Let <math>W_e</math> denote the set of input values for which the Turing machine with index ''e'' halts. Then the sets <math>A = \{e \mid e \in W_e\}</math> and <math>B = \{(e,n) \mid n \in W_e \}</math> are Turing equivalent (here <math>(e,n)</math> denotes an effective pairing function). A reduction showing <math>A \leq_T B</math> can be constructed using the fact that <math>e \in A \Leftrightarrow (e,e) \in B</math>. Given a pair <math>(e,n)</math>, a new index <math>i(e,n)</math> can be constructed using the [[Smn theorem|s<sub>mn</sub> theorem]] such that the program coded by <math>i(e,n)</math> ignores its input and merely simulates the computation of the machine with index ''e'' on input ''n''.
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| In particular, the machine with index <math>i(e,n)</math> either halts on every input or halts on no input. Thus <math>i(e,n) \in A \Leftrightarrow (e,n) \in B</math> holds for all ''e'' and ''n''. Because the function ''i'' is computable, this shows <math>B \leq_T A</math>. The reductions presented here are not only Turing reductions but ''many-one reductions'', discussed below.
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| == Properties ==
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| * Every set is Turing equivalent to its complement
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| * Every [[computable set]] is Turing reducible to every other computable set. Because these sets can be computed with no oracle, they can be computed by an oracle machine that ignores the oracle it is given.
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| * The relation <math>\leq_T</math> is transitive: if <math>A \leq_T B</math> and <math>B \leq_T C</math> then <math>A \leq_T C</math>. Moreover <math>A \leq A</math> holds for every set ''A'', and thus the relation <math>\leq_T</math> is a [[preorder]] (it is not a [[partial order]] because <math>A \leq_T B</math> and <math>B \leq_T A </math> does not necessarily imply <math>A = B</math>).
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| * There are pairs of sets <math>(A,B)</math> such that ''A'' is not Turing reducible to ''B'' and ''B'' is not Turing reducible to ''A''. Thus <math>\leq_T</math> is not a linear order.
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| * There are infinite decreasing sequences of sets under <math>\leq_T</math>. Thus this relation is not [[well-founded]].
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| * Every set is Turing reducible to its own [[Turing jump]], but the Turing jump of a set is never Turing reducible to the original set.
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| == The use of a reduction ==
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| Since every reduction from a set ''B'' to a set ''A'' has to determine whether a single element is in ''A'' in only finitely many steps, it can only make finitely many queries of membership in the set ''B''. When the amount of information about the set ''B'' used to compute a single bit of ''A'' is discussed, this is made precise by the use function. Formally, the ''use'' of a reduction is the function that sends each natural number ''n'' to the largest natural number ''m'' whose membership in the set ''B'' was queried by the reduction while determining the membership of ''n'' in ''A''.
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| == Stronger reductions ==
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| There are two common ways of producing reductions stronger than Turing reducibility.
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| The first way is to limit the number and manner of oracle queries.
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| * A set ''A'' is '''[[many-one reduction|many-one reducible]]''' to ''B'' if there is a [[Computable function|total computable function]] ''f'' such that an element ''n'' is in ''A'' if and only if ''f''(''n'') is in ''B''. Such a function can be used to generate a Turing reduction (by computing ''f''(''n''), querying the oracle, and then interpreting the result).
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| * A '''[[truth table reduction]]''' or a '''[[truth table reduction|weak truth table reduction]]''' must present all of its oracle queries at the same time. In a truth table reduction, the reduction also gives a boolean function (a '''truth table''') which, when given the answers to the queries, will produce the final answer of the reduction. In a weak truth table reduction, the reduction uses the oracle answers as a basis for further computation depending on the given answers (but not using the oracle). Equivalently, a weak truth table reduction is one for which the use of the reduction is bounded by a computable function. For this reason, weak truth table reductions are sometimes called "bounded Turing" reductions.
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| The second way to produce a stronger reducibility notion is to limit the computational resources that the program implementing the Turing reduction may use. These limits on the [[computational complexity]] of the reduction are important when studying subrecursive classes such as [[P (complexity)|P]]. A set ''A'' is '''[[Polynomial-time reduction|polynomial-time reducible]]''' to a set ''B'' if there is a Turing reduction of ''A'' to ''B'' that runs in polynomial time. The concept of '''[[log-space reduction]]''' is similar.
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| These reductions are stronger in the sense that they provide a finer distinction into equivalence classes, and satisfy more restrictive requirements than Turing reductions. Consequently, such reductions are harder to find. There may be no way to build a many-one reduction from one set to another even when a Turing reduction for the same sets exists.
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| == Weaker reductions ==
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| According to the [[Church-Turing thesis]], a Turing reduction is the most general form of an effectively calculable reduction. Nevertheless, weaker reductions are also considered. A set ''A'' is said to be '''[[arithmetical set|arithmetical]] in''' ''B'' if ''A'' is definable by a formula of [[Peano arithmetic]] with ''B'' as a parameter. The set ''A'' is '''[[hyperarithmetical hierarchy|hyperarithmetical]] in''' ''B'' if there is a [[recursive ordinal]] α such that ''A'' is computable from <math>B^{(\alpha)}</math>, the α-iterated Turing jump of ''B''. The notion of '''[[Constructible universe|relative constructibility]]''' is an important reducibility notion in set theory.
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| == References ==
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| * M. Davis, ed., 1965. ''The Undecidable—Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions'', Raven, New York. Reprint, Dover, 2004. ISBN 0-486-43228-9.
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| * S. C. Kleene, 1952. Introduction to Metamathematics. Amsterdam: North-Holland.
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| * S. C. Kleene and E. L. Post, 1954. "The upper semi-lattice of degrees of recursive unsolvability". ''Annals of Mathematics'' v. 2 n. 59, 379—407.
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| * E. Post, 1944. "Recursively enumerable sets of positive integers and their decision problems." Bulletin of the American Mathematical Society, v. 50, pp. 284–316. Reprinted in "The Undecidable", M. Davis ed., 1965.
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| * A. Turing, 1939. "Systems of logic based on ordinals." ''Proceedings of the London Mathematics Society'', ser. 2 v. 45, pp. 161–228. Reprinted in "The Undecidable", M. Davis ed., 1965.
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| * H. Rogers, 1967. Theory of recursive functions and effective computability. McGraw-Hill.
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| * R. Soare, 1987. Recursively enumerable sets and degrees, Springer.
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| * {{ cite journal
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| | last = Davis
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| | first = Martin
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| | authorlink = Martin Davis
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| | title = What is...Turing Reducibility?
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| | journal = [[Notices of the American Mathematical Society]]
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| |date=November 2006
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| | volume = 53
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| | issue = 10
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| | pages = pp.1218–1219
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| | url = http://www.ams.org/notices/200610/whatis-davis.pdf
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| | format = [[PDF]]
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| | accessdate = 2008-01-16 }}
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| == External links ==
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| * [http://www.nist.gov/dads/HTML/turingredctn.html NIST Dictionary of Algorithms and Data Structures: Turing reduction]
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| [[Category:Computability theory]]
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| [[Category:Computational complexity theory]]
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| [[Category:Alan Turing]]
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| [[he:רדוקציה חישובית]]
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