In computability theory relative computability or relativized computability is the study of computability concepts relative to a given set. Most concepts in regular computability can be generalized to relative computability. Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ...

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 1948 Emil Post extended his canonical systems to define a similar concept. Alan Turing is often considered the father of modern computer science. ... In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. ... Stephen Cole Kleene (January 5, 1909 - January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ... It has been suggested that this article or section be merged into computable function. ... Emil Leon Post (February 11, 1897 - April 21, 1954) was a Polish-American mathematician and logician. ...

Definition

Given a set of natural numbers A, a function A natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...

is called A-computable if there exists an oracle machine with an oracle for A which can compute f. In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. ...

A set B is called A-recursive if its indicator function 1_B is A-computable. In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ...

Examples

• For any recursive set A, a function f as above is A-computable iff it is computable in the usual sense.

In computability theory a countable set is called recursive, computable or decidable if we can construct an algorithm which terminates after a finite amount of time and decides whether or not a given element belongs to the set. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... It has been suggested that recursive function be merged into this article or section. ...

See also

• Turing equivalence (recursion theory)
• Turing reduction
• Turing degrees

In computational complexity theory, a Turing reduction from a problem A to a problem B is, intuitively, a reduction which easily solves B, assuming A is easy to solve. ...

References

• R. I. Soare, "Computability and Recursion", Bulletin of Symbolic Logic 2 (1996), 284-321.