Turing degree


In computer science and mathematical logic the Turing degree or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set.

Overview

The concept of Turing degree is fundamental in computability theory, where sets of natural numbers are often regarded as decision problems. The Turing degree of a set is a measure of how difficult it is to solve the decision problem associated with the set, that is, to determine whether an arbitrary number is in the given set.
Two sets are Turing equivalent if they have the same level of unsolvability; each Turing degree is a collection of Turing equivalent sets, so that two sets are in different Turing degrees exactly when they are not Turing equivalent. Furthermore, the Turing degrees are partially ordered so that if the Turing degree of a set X is less than the Turing degree of a set Y then any procedure that correctly decides whether numbers are in Y can be effectively converted to a procedure that correctly decides whether numbers are in X. It is in this sense that the Turing degree of a set corresponds to its level of algorithmic unsolvability.
The Turing degrees were introduced by Emil Leon Post, and many fundamental results were established by Stephen Cole Kleene and Post. The Turing degrees have been an area of intense research since then. Many proofs in the area make use of a proof technique known as the priority method.

Turing equivalence

For the rest of this article, the word set will refer to a set of natural numbers. A set X is said to be Turing reducible to a set Y if there is an oracle Turing machine that decides membership in X when given an oracle for membership in Y. The notation X ≤T Y indicates that X is Turing reducible to Y.
Two sets X and Y are defined to be Turing equivalent if X is Turing reducible to Y and Y is Turing reducible to X. The notation X ≡T Y indicates that X and Y are Turing equivalent. The relation ≡T can be seen to be an equivalence relation, which means that for all sets X, Y, and Z:
A Turing degree is an equivalence class of the relation ≡T. The notation denotes the equivalence class containing a set X. The entire collection of Turing degrees is denoted.
The Turing degrees have a partial order ≤ defined so that ≤ if and only if X ≤T Y. There is a unique Turing degree containing all the computable sets, and this degree is less than every other degree. It is denoted 0 because it is the least element of the poset.
For any sets X and Y, X join Y, written X ⊕ Y, is defined to be the union of the sets and . The Turing degree of X ⊕ Y is the least upper bound of the degrees of X and Y. Thus is a join-semilattice. The least upper bound of degrees a and b is denoted a ∪ b. It is known that is not a lattice, as there are pairs of degrees with no greatest lower bound.
For any set X the notation X′ denotes the set of indices of oracle machines that halt when using X as an oracle. The set X′ is called the Turing jump of X. The Turing jump of a degree is defined to be the degree ; this is a valid definition because X′ ≡T Y′ whenever X ≡T Y. A key example is 0′, the degree of the halting problem.

Basic properties of the Turing degrees

A great deal of research has been conducted into the structure of the Turing degrees. The following survey lists only some of the many known results. One general conclusion that can be drawn from the research is that the structure of the Turing degrees is extremely complicated.

Order properties

A degree is called recursively enumerable if it contains a recursively enumerable set. Every r.e. degree is less than or equal to 0′ but not every degree less than 0′ is an r.e. degree.
studied the r.e. Turing degrees and asked whether there is any r.e. degree strictly between 0 and 0′. The problem of constructing such a degree became known as Post's problem. This problem was solved independently by Friedberg and Muchnik in the 1950s, who showed that these intermediate r.e. degrees do exist. Their proofs each developed the same new method for constructing r.e. degrees which came to be known as the priority method. The priority method is now the main technique for establishing results about r.e. sets.
The idea of the priority method for constructing a r.e. set X is to list a countable sequence of requirements that X must satisfy. For example, to construct a r.e. set X between 0 and 0′ it is enough to satisfy the requirements Ae and Be for each natural number e, where Ae requires that the oracle machine with index e does not compute 0′ from X and Be requires that the Turing machine with index e does not compute X. These requirements are put into a priority ordering, which is an explicit bijection of the requirements and the natural numbers. The proof proceeds inductively with one stage for each natural number; these stages can be thought of as steps of time during which the set X is enumerated. At each stage, numbers may be put into X or forever prevented from entering X in an attempt to satisfy requirements. Sometimes, a number can be enumerated into X to satisfy one requirement but doing this would cause a previously satisfied requirement to become unsatisfied. The priority order on requirements is used to determine which requirement to satisfy in this case. The informal idea is that if a requirement is injured then it will eventually stop being injured after all higher priority requirements have stopped being injured, although not every priority argument has this property. An argument must be made that the overall set X is r.e. and satisfies all the requirements. Priority arguments can be used to prove many facts about r.e. sets; the requirements used and the manner in which they are satisfied must be carefully chosen to produce the required result.