There is a Turing reduction from every problem to its complement problem.

Many-one reductions are a special case and stronger form of Turing reductions.

When defining classes in the polynomial hierarchy, polynomial-time Turing reduction is used.

A many-one reduction is weaker than a Turing reduction.

This kind of reduction corresponds to Turing reduction.

Another type of reduction is polynomial-time Turing reduction.

The second way to produce a stronger reducibility notion is to limit the computational resources that the program implementing the Turing reduction may use.

The reductions presented here are not only Turing reductions but many-one reductions, discussed below.

G. Japaridze, The logic of interactive Turing reduction.

As well as polynomial time Turing reductions, other types of reducibility have been considered as well.

As described in the example above, there are two main types of reductions used in computational complexity, the many-one reduction and the Turing reduction.

According to the Church-Turing thesis, a Turing reduction is the most general form of an effectively calculable reduction.

A polynomial-time Turing reduction is known as a Cook reduction, after Stephen Cook.

The closure of any complexity class under Turing reductions is a superset of that class which is closed under complement.

However, other subclasses of P such as NC may not be closed under Turing reductions, and so many-one reductions must be used.

Equivalently, a weak truth-table reduction is a Turing reduction for which the use of the reduction is bounded by a computable function.

Thus Toda's theorem implies that for any problem in the polynomial hierarchy there is a deterministic polynomial-time Turing reduction to a counting problem.

It therefore suffices to show that if limit computability is preserved by Turing reduction, as this will show that all sets computable from are limit computable.

Although not all languages in P/poly are sparse, there is a polynomial-time Turing reduction from any language in P/poly to a sparse language.

An alternative definition of NP-hard that is often used restricts NP-hard to decision problems and then uses polynomial-time many-one reduction instead of Turing reduction.

If it is a Turing reduction, it is called a polynomial-time Turing reduction or Cook reduction.

Many-one reductions map instances of one problem to instances of another; Turing reductions compute the solution to one problem, assuming the other problem is easy to solve.

Even for a relaxed definition of NP-completeness using Turing reductions, the existence of a sparse NP-complete language would imply an unexpected collapse of the polynomial hierarchy.

Because there are Turing reductions from every problem to its complement, any class which is closed under Turing reductions is closed under complement.

This gives a Turing reduction from SAT to Unambiguous-SAT since an assumed algorithm for Unambiguous-SAT can be invoked on the .