Reduction (computability theory)

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.

Reducibility relations

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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.

Degrees of a reducibility relation

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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  . This is equivalent to the property that for every set   in   and every set   in  ,  , because any two sets in C are equivalent and any two sets in   are equivalent. It is common, as shown here, to use boldface notation to denote degrees.

Turing reducibility

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The most fundamental reducibility notion is Turing reducibility. A set   of natural numbers is Turing reducible to a set   if and only if there is an oracle Turing machine that, when run with   as its oracle set, will compute the indicator function (characteristic function) of  . Equivalently,   is Turing reducible to   if and only if there is an algorithm for computing the indicator function for   provided that the algorithm is provided with a means to correctly answer questions of the form "Is   in  ?".

Turing reducibility serves as a dividing line for other reducibility notions because, according to the Church-Turing thesis, it is the most general reducibility relation that is effective. Reducibility relations that imply Turing reducibility have come to be known as strong reducibilities, while those that are implied by Turing reducibility are weak reducibilities. Equivalently, a strong reducibility relation is one whose degrees form a finer equivalence relation than the Turing degrees, while a weak reducibility relation is one whose degrees form a coarser equivalence relation than Turing equivalence.

Reductions stronger than Turing reducibility

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The strong reducibilities include

  • One-one reducibility:   is one-one reducible to   if there is a computable one-to-one function   with   for all  .
  • Many-one reducibility:   is many-one reducible to   if there is a computable function   with   for all  .
  • Truth-table reducible:   is truth-table reducible to   if   is Turing reducible to   via a single (oracle) Turing machine which produces a total function relative to every oracle.
  • Weak truth-table reducible:   is weak truth-table reducible to   if there is a Turing reduction from   to   and a computable function   which bounds the use. Whenever   is truth-table reducible to  ,   is also weak truth-table reducible to  , since one can construct a computable bound on the use by considering the maximum use over the tree of all oracles, which will exist if the reduction is total on all oracles.
  • Positive reducible:   is positive reducible to   if and only if   is truth-table reducible to   in a way that one can compute for every   a formula consisting of atoms of the form   such that these atoms are combined by and's and or's, where the and of   and   is 1 if   and   and so on.
  • Enumeration reducibility: Similar to positive reducibility, relating to the effective procedure of enumerability from   to  .
  • Disjunctive reducible: Similar to positive reducible with the additional constraint that only or's are permitted.
  • Conjunctive reducibility: Similar to positive reducibility with the additional constraint that only and's are permitted.
  • Linear reducibility: Similar to positive reducibility but with the constraint that all atoms of the form   are combined by exclusive or's. In other words,   is linear reducible to   if and only if a computable function computes for each   a finite set   given as an explicit list of numbers such that   if and only if   contains an odd number of elements of  .

Many of these were introduced by Post (1944). Post was searching for a non-computable, computably enumerable set which the halting problem could not be Turing reduced to. As he could not construct such a set in 1944, he instead worked on the analogous problems for the various reducibilities that he introduced. These reducibilities have since been the subject of much research, and many relationships between them are known.

Bounded reducibilities

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A bounded form of each of the above strong reducibilities can be defined. The most famous of these is bounded truth-table reduction, but there are also bounded Turing, bounded weak truth-table, and others. These first three are the most common ones and they are based on the number of queries. For example, a set   is bounded truth-table reducible to   if and only if the Turing machine   computing   relative to   computes a list of up to   numbers, queries   on these numbers and then terminates for all possible oracle answers; the value   is a constant independent of  . The difference between bounded weak truth-table and bounded Turing reduction is that in the first case, the up to   queries have to be made at the same time while in the second case, the queries can be made one after the other. For that reason, there are cases where   is bounded Turing reducible to   but not weak truth-table reducible to  .

Strong reductions in computational complexity

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The strong reductions listed above restrict the manner in which oracle information can be accessed by a decision procedure but do not otherwise limit the computational resources available. Thus if a set   is decidable then   is reducible to any set   under any of the strong reducibility relations listed above, even if   is not polynomial-time or exponential-time decidable. This is acceptable in the study of computability theory, which is interested in theoretical computability, but it is not reasonable for computational complexity theory, which studies which sets can be decided under certain asymptotical resource bounds.

The most common reducibility in computational complexity theory is polynomial-time reducibility; a set A is polynomial-time reducible to a set   if there is a polynomial-time function f such that for every  ,   is in   if and only if   is in  . This reducibility is, essentially, a resource-bounded version of many-one reducibility. Other resource-bounded reducibilities are used in other contexts of computational complexity theory where other resource bounds are of interest.

Reductions weaker than Turing reducibility

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Although Turing reducibility is the most general reducibility that is effective, weaker reducibility relations are commonly studied. These reducibilities are related to the relative definability of sets over arithmetic or set theory. They include:

  • Arithmetical reducibility: A set   is arithmetical in a set   if   is definable over the standard model of Peano arithmetic with an extra predicate for  . Equivalently, according to Post's theorem, A is arithmetical in   if and only if   is Turing reducible to  , the  th Turing jump of  , for some natural number  . The arithmetical hierarchy gives a finer classification of arithmetical reducibility.
  • Hyperarithmetical reducibility: A set   is hyperarithmetical in a set   if   is   definable (see analytical hierarchy) over the standard model of Peano arithmetic with a predicate for  . Equivalently,   is hyperarithmetical in   if and only if   is Turing reducible to  , the  th Turing jump of  , for some  -recursive ordinal  .
  • Relative constructibility: A set   is relatively constructible from a set   if   is in  , the smallest transitive model of ZFC set theory containing   and all the ordinals.

References

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  • K. Ambos-Spies and P. Fejer, 2006. "Degrees of Unsolvability." Unpublished preprint.
  • P. Odifreddi, 1989. Classical Recursion Theory, North-Holland. ISBN 0-444-87295-7
  • P. Odifreddi, 1999. Classical Recursion Theory, Volume II, Elsevier. ISBN 0-444-50205-X
  • E. Post, 1944, "Recursively enumerable sets of positive integers and their decision problems", Bulletin of the American Mathematical Society, volume 50, pages 284–316.
  • 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
  • G. Sacks, 1990. Higher Recursion Theory, Springer-Verlag. ISBN 3-540-19305-7
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