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Concept
Many-one reduction
Résumé
In computability theory and computational complexity theory, a many-one reduction (also called mapping reduction) is a reduction which converts instances of one decision problem (whether an instance is in ) to another decision problem (whether an instance is in ) using an effective function. The reduced instance is in the language if and only if the initial instance is in its language . Thus if we can decide whether instances are in the language , we can decide whether instances are in its language by applying the reduction and solving . Thus, reductions can be used to measure the relative computational difficulty of two problems. It is said that reduces to if, in layman's terms is harder to solve than . That is to say, any algorithm that solves can also be used as part of a (otherwise relatively simple) program that solves .
Many-one reductions are a special case and stronger form of Turing reductions. With many-one reductions, the oracle (that is, our solution for B) can be invoked only once at the end, and the answer cannot be modified. This means that if we want to show that problem A can be reduced to problem B, we can use our solution for B only once in our solution for A, unlike in Turing reduction, where we can use our solution for B as many times as needed while solving A.
This means that many-one reductions map instances of one problem to instances of another, while Turing reductions compute the solution to one problem, assuming the other problem is easy to solve. The many-one reduction is more effective at separating problems into distinct complexity classes. However, the increased restrictions on many-one reductions make them more difficult to find.
Many-one reductions were first used by Emil Post in a paper published in 1944. Later Norman Shapiro used the same concept in 1956 under the name strong reducibility.
Suppose and are formal languages over the alphabets and , respectively. A many-one reduction from to is a total computable function that has the property that each word is in if and only if is in .
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