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Concept
Reduction (computability theory)
Summary
In computability theory, many reducibility relations (also called reductions, reducibilities, and notions of reducibility) are studied. They are motivated by the question: given sets and of natural numbers, is it possible to effectively convert a method for deciding membership in into a method for deciding membership in ? If the answer to this question is affirmative then is said to be reducible to .
The study of reducibility notions is motivated by the study of decision problems. For many notions of reducibility, if any noncomputable set is reducible to a set then must also be noncomputable. This gives a powerful technique for proving that many sets are noncomputable.
A reducibility relation is a binary relation on sets of natural numbers that is
Reflexive: Every set is reducible to itself.
Transitive: If a set is reducible to a set and is reducible to a set then is reducible to .
These two properties imply that reducibility is a preorder on the powerset of the natural numbers. Not all preorders are studied as reducibility notions, however. The notions studied in computability theory have the informal property that is reducible to if and only if any (possibly noneffective) decision procedure for can be effectively converted to a decision procedure for . The different reducibility relations vary in the methods they permit such a conversion process to use.
Every reducibility relation (in fact, every preorder) induces an equivalence relation on the powerset of the natural numbers in which two sets are equivalent if and only if each one is reducible to the other. In computability theory, these equivalence classes are called the degrees of the reducibility relation. For example, the Turing degrees are the equivalence classes of sets of naturals induced by Turing reducibility.
The degrees of any reducibility relation are partially ordered by the relation in the following manner. Let be a reducibility relation and let and be two of its degrees. Then if and only if there is a set in and a set in such that .
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