In computability theory, a Turing reduction from a decision problem A to a decision problem B is an oracle machine that decides problem A given an oracle for B (Rogers 1967, Soare 1987). It can be understood as an algorithm that could be used to solve A if it had available to it a subroutine for solving B. The concept can be analogously applied to function problems. If a Turing reduction from A to B exists, then every algorithm for B can be used to produce an algorithm for A, by inserting the algorithm for B at each place where the oracle machine computing A queries the oracle for B. However, because the oracle machine may query the oracle a large number of times, the resulting algorithm may require more time asymptotically than either the algorithm for B or the oracle machine computing A. A Turing reduction in which the oracle machine runs in polynomial time is known as a Cook reduction. The first formal definition of relative computability, then called relative reducibility, was given by Alan Turing in 1939 in terms of oracle machines. Later in 1943 and 1952 Stephen Kleene defined an equivalent concept in terms of recursive functions. In 1944 Emil Post used the term "Turing reducibility" to refer to the concept.

Definition

Given two sets of natural numbers, we say A is Turing reducible to B and write if and only if there is an oracle machine that computes the characteristic function of A when run with oracle B. In this case, we also say A is B-recursive and B-computable. If there is an oracle machine that, when run with oracle B, computes a partial function with domain A, then A is said to be B-recursively enumerable and B-computably enumerable. We say A is Turing equivalent to B and write if both A \leq_T B and B \leq_T A. The equivalence classes of Turing equivalent sets are called Turing degrees. The Turing degree of a set X is written. Given a set, a set is called Turing hard for \mathcal{X} if X \leq_T A for all. If additionally then A is called Turing complete for \mathcal{X}.

Relation of Turing completeness to computational universality

Turing completeness, as just defined above, corresponds only partially to Turing completeness in the sense of computational universality. Specifically, a Turing machine is a universal Turing machine if its halting problem (i.e., the set of inputs for which it eventually halts) is many-one complete for the set \mathcal{X} of recursively enumerable sets. Thus, a necessary but insufficient condition for a machine to be computationally universal, is that the machine's halting problem be Turing-complete for \mathcal{X}. Insufficient because it may still be the case that, the language accepted by the machine is not itself recursively enumerable.

Example

Let W_e denote the set of input values for which the Turing machine with index e halts. Then the sets and are Turing equivalent (here (-,-) denotes an effective pairing function). A reduction showing A \leq_T B can be constructed using the fact that. Given a pair (e,n), a new index i(e,n) can be constructed using the smn theorem such that the program coded by i(e,n) ignores its input and merely simulates the computation of the machine with index e on input n. In particular, the machine with index i(e,n) either halts on every input or halts on no input. Thus holds for all e and n. Because the function i is computable, this shows B \leq_T A. The reductions presented here are not only Turing reductions but many-one reductions, discussed below.

Properties

The use of a reduction

Since every reduction from a set A to a set B has to determine whether a single element is in A in only finitely many steps, it can only make finitely many queries of membership in the set B. When the amount of information about the set B used to compute a single bit of A is discussed, this is made precise by the use function. Formally, the use of a reduction is the function that sends each natural number n to the largest natural number m whose membership in the set B was queried by the reduction while determining the membership of n in A.

Stronger reductions

There are two common ways of producing reductions stronger than Turing reducibility. The first way is to limit the number and manner of oracle queries. The second way to produce a stronger reducibility notion is to limit the computational resources that the program implementing the Turing reduction may use. These limits on the computational complexity of the reduction are important when studying subrecursive classes such as P. A set A is polynomial-time reducible to a set B if there is a Turing reduction of A to B that runs in polynomial time. The concept of log-space reduction is similar. These reductions are stronger in the sense that they provide a finer distinction into equivalence classes, and satisfy more restrictive requirements than Turing reductions. Consequently, such reductions are harder to find. There may be no way to build a many-one reduction from one set to another even when a Turing reduction for the same sets exists.

Weaker reductions

According to the Church–Turing thesis, a Turing reduction is the most general form of an effectively calculable reduction. Nevertheless, weaker reductions are also considered. Set A is said to be arithmetical in B if A is definable by a formula of Peano arithmetic with B as a parameter. The set A is hyperarithmetical in B if there is a recursive ordinal \alpha such that A is computable from, the α-iterated Turing jump of B. The notion of relative constructibility is an important reducibility notion in set theory.