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| In [[computability theory]], a '''truth-table reduction''' is a [[reduction (complexity)|reduction]] from one set of [[natural number]]s to another.
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| As a "tool", it is weaker than [[Turing reduction]], since not every Turing reduction between sets can be performed by a truth-table reduction, but every truth-table reduction can be performed by a Turing reduction. For the same reason it is said to be a stronger reducibility than Turing reducibility, because it implies Turing reducibility. A '''weak truth-table reduction''' is a related type of reduction which is so named because it weakens the constraints placed on a truth-table reduction, and provides a weaker equivalence classification; as such, a "weak truth-table reduction" can actually be more powerful than a truth-table reduction as a "tool", and perform a reduction which is not performable by truth table.
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| A Turing reduction from a set ''B'' to a set ''A'' computes the membership of a single element in ''A'' by asking questions about the membership of various elements in ''B'' during the computation; it may adaptively determine which questions it asks based upon answers to previous questions. In contrast, a truth-table reduction or a weak truth-table reduction must present all of its (finitely many) [[oracle (computer science)|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 which may depend on the given answers but may not ask further questions of the oracle.
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| Equivalently, a weak truth-table reduction is a Turing reduction for which the [[Turing reduction#The use of a reduction|use]] of the reduction is bounded by a [[computable function]]. For this reason, they are sometimes referred to as '''bounded Turing''' (bT) reductions rather than as weak truth-table (wtt) reductions.
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| == Properties ==
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| As every truth-table reduction is a Turing reduction, if ''A'' is truth-table reducible to ''B'' (''A'' ≤<sub>tt</sub> ''B''), then ''A'' is also Turing reducible to ''B'' (''A'' ≤<sub>T</sub> ''B''). Considering also one-one reducibility, many-one reducibility and weak truth-table reducibility, one gets the following chain of implications:
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| * <math>A \leq_1 B \Rightarrow A \leq_m B \Rightarrow A \leq_{tt} B \Rightarrow A \leq_{wtt} B \Rightarrow A \leq_T B</math>; one-one reducibility implies many-one reducibility, which implies truth-table reducibility, which in turn implies weak truth-table reducibility, which in turn implies Turing reducibility.
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| == References ==
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| * [[Hartley Rogers, Jr.|H. Rogers, Jr.]], 1967. ''The Theory of Recursive Functions and Effective Computability'', second edition 1987, MIT Press. ISBN 0-262-68052-1 (paperback), ISBN 0-07-053522-1
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| [[Category:Computational complexity theory]]
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| [[Category:Computability theory]]
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| {{mathlogic-stub}}
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