Morse–Kelley set theory: Difference between revisions
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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. | |||
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. | |||
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. | |||
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. | |||
== Properties == | |||
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: | |||
* <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. | |||
== References == | |||
* [[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 | |||
[[Category:Computational complexity theory]] | |||
[[Category:Computability theory]] | |||
{{mathlogic-stub}} | |||
Revision as of 17:42, 17 November 2013
In computability theory, a truth-table reduction is a reduction from one set of natural numbers to another. 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.
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 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.
Equivalently, a weak truth-table reduction is a Turing reduction for which the 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.
Properties
As every truth-table reduction is a Turing reduction, if A is truth-table reducible to B (A ≤tt B), then A is also Turing reducible to B (A ≤T B). Considering also one-one reducibility, many-one reducibility and weak truth-table reducibility, one gets the following chain of implications:
- ; 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.
References
- 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