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Computable functions are the basic objects of study in computability theory. They are also called Turing-computable functions and (total) recursive functions. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Their definition, however, must make reference to some specific model of computation.

Before the precise definition of computable function, mathematicians often used the informal term effectively computable. This term has since come to be identified with the computable functions. Note that the effective computability of these functions does not imply that they can be efficiently computed (i.e. computed within a reasonable amount of time). In fact, for some effectively computable functions it can be shown that any algorithm that computes them will be very inefficient in the sense that the running time of the algorithm increases exponentially (or even superexponentially) with the length of the input. The fields of feasible computability and computational complexity study functions that can be computed efficiently.

According to the Church-Turing thesis, computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space. Equivalently, this thesis states that any function which has an algorithm is computable.

The Blum axioms can be used to define an abstract computational complexity theory on the set of computable functions. In computational complexity theory, the problem of determining the complexity of a computable function is known as a function problem.

1 Definition
2 Characteristics of computable functions
3 Computable sets and relations
4 Formal languages
5 Examples
6 Church-Turing thesis
7 Incomputable functions and unsolvable problems
8 Extensions of computability
9 See also
10 References

Definition

There are many equivalent ways to define the class of computable functions. For concreteness, the remainder of this article will assume that computable functions have been defined as those finitary partial functions on the natural numbers that can be calculated by a Turing machine. There are many equivalent models of computation that define the same class of computable functions. These models of computation include

Turing machines
μ-recursive functions
Lambda calculus
Post machines (Post-Turing machines and tag machines).
Register machines

and others.

Each computable function $f$ takes a fixed number of natural numbers as arguments. Because the functions are partial, they may not be defined for every possible choice of input. If a computable function is defined then it returns a single natural number as output (this output can be interpreted as a list of numbers using a pairing function). These functions are also called partial recursive functions. In computability theory, the domain of a function is taken to be the set of all inputs for which the function is defined.

A function which is defined for all arguments is called total. If a computable function is total, it is called a total computable function or total recursive function.

The notation $f(x_1,\ldots,x_k) \downarrow$ indicates that the partial function $f$ is defined on arguments $x_1,\ldots,x_k$ , and the notation $f(x_1,\ldots,x_k) \downarrow = y$ indicates that $f$ is defined on the arguments $x_1,\ldots,x_k$ and the value returned is $y$ .

Characteristics of computable functions

Main article: Algorithm

The basic characteristic of a computable function is that there must be a finite procedure (an algorithm) telling how to compute the function. The models of computation listed above give different interpretations of what a procedure is and how it is used, but these interpretations share many properties. The fact that these models give equivalent classes of computable functions stems from the fact that each model is capable of reading and mimicking a procedure for any of the other models, much as a compiler is able to read instructions in one computer language and emit instructions in another language.

Enderton [1977] gives the following characteristics of a procedure for computing a computable function; similar characterizations have been given by Turing [1936], Rogers [1967], and others.

“There must be exact instructions (i.e. a program), finite in length, for the procedure.”

Thus every computable function must have a finite program that completely describes how the function is to be computed. It is possible to compute the function by just following the instructions; no guessing or special insight is required.

“If the procedure is given a k-tuple x in the domain of f, then after a finite number of discrete steps the procedure must terminate and produce f(x).”

Intuitively, the procedure proceeds step by step, with a specific rule to cover what to do at each step of the calculation. Only finitely many steps can be carried out before the value of the function is returned.

“If the procedure is given a k-tuple x which is not in the domain of f, then the procedure might go on forever, never halting. Or it might get stuck at some point, but it must not pretend to produce a value for f at x.”

Thus if a value for f(x) is ever found, it must be the correct value. It is not necessary for the computing agent to distinguish correct outcomes from incorrect ones because the procedure is always correct when it produces an outcome.

Enderton goes on to list several clarifications of these requirements of the procedure for a computable function:

The procedure must theoretically work for arbitrarily large arguments. It is not assumed that the arguments are smaller than the number of atoms in the Earth, for example.
The procedure is required to halt after finitely many steps in order to produce an output, but it may take arbitrarily many steps before halting. No time limitation is assumed.
Although the procedure may use only a finite amount of storage space during a successful computation, there is no bound on the amount of space that is used. It is assumed that additional storage space can be given to the procedure whenever the procedure asks for it.

The field of computational complexity studies functions with prescribed bounds on the time and/or space allowed in a successful computation.

Computable sets and relations

A set A of natural numbers is called computable (synonyms: recursive, decidable) if there is a computable function f such that for each number n, $f(n) \downarrow = 1$ if n is in A and $f(n) \downarrow = 0$ if n is not in A.

A set of natural numbers is called computably enumerable (synonyms: recursively enumerable, semidecidable) if there is a computable function f such that for each number n, f(n) is defined if and only if n is in the set. Thus a set is computably enumerable if and only if it is the domain of some computable function. The word enumerable is used because the following are equivalent for a nonempty subset B of the natural numbers:

B is the domain of a computable function.
B is the range of a total computable function. If B is infinite then the function can be assumed to be injective.

If a set B is the range of a function f then the function can be viewed as an enumeration of B, because the list f(0), f(1), ... will include every element of B.

Because each finitary relation on the natural numbers can be identified with a corresponding set of finite sequences of natural numbers, the notions of computable relation and computably enumerable relation can be defined from their analogues for sets.

Formal languages

Main article: Formal language

In computability theory in computer science, it is common to consider formal languages. An alphabet is an arbitrary set. A word on an alphabet is a finite sequence of symbols from the alphabet; the same symbol may be used more than once. For example, binary strings are exactly the words on the alphabet ${0,1}$ . A language is a subset of the collection of all words on a fixed alphabet. For example, the collection of all binary strings that contain exactly 3 ones is a language over the binary alphabet.

A key property of a formal language is the level of difficulty required to decide whether a given word is in the language. Some coding system must be developed to allow a computable function to take an arbitrary word in the language as input; this is usually considered routine. A language is called computable (synonyms: recursive, decidable) if there is a computable function $f$ such that for each word w over the alphabet, $f(w) \downarrow = 1$ if the word is in the language and $f(w)\downarrow = 0$ if the word is not in the language. Thus a language is computable just in case there is a procedure that is able to correctly tell whether arbitrary words are in the language.

A language is computably enumerable (synonyms: recursively enumerable, semidecidable) if there is a computable function f such that $f (w)$ is defined if and only if the word w is in the language. The term enumerable has the same etymology as in computably enumerable sets of natural numbers.

Examples

The following functions are computable:

Each function with a finite domain; e.g., any finite sequence of natural numbers.

Each constant function f : N^k → N, f(n₁,...n_k) := n.

Addition f : N²→ N, f(n₁,n₂) := n₁ + n₂

The function which gives the list of prime factors of a number.

The greatest common divisor of two numbers is a computable function.

Bézout's identity, a linear Diophantine equation

If f and g are computable, then so are: f + g, f * g, $f \circ g$ if f is unary, max(f,g), min(f,g), argmin{y≤f(x)} and many more combinations.

The following examples illustrate that a function may be computable though it is not known which algorithm computes it.

The function f such that f(n) = 1 if there is a sequence of n consecutive fives in the decimal expansion of π, and f(n) = 0 otherwise, is computable. (The function f is either the constant 1 function, which is computable, or else there is a k such that f(n) = 1 if n < k and f(n) = 0 if n ≥ k. Every such function is computable. It is not known whether there are arbitrarily long runs of fives in the decimal expansion of π, so we don't know which of those functions is f. Nevertheless, we know that the function f must be computable.)

Each finite segment of an incomputable sequence of natural numbers (such as the Busy Beaver function Σ) is computable. E.g., for each natural number n, there exists an algorithm that computes the finite sequence Σ(0), Σ(1), Σ(2), ..., Σ(n) — in contrast to the fact that there is no algorithm that computes the entire Σ-sequence, i.e. Σ(n) for all n.

Church-Turing thesis

Main article: Church-Turing thesis

The Church-Turing thesis states that any function computable from a procedure possessing the three properties listed above is a computable function. Because these three properties are not formally stated, the Church-Turing thesis cannot be proved. The following facts are often taken as evidence for the thesis:

Many equivalent models of computation are known, and they all give the same definition of computable function (or a weaker version, in some instances).
No stronger model of computation which is generally considered to be effectively calculable has been proposed.

The Church-Turing thesis is sometimes used in proofs to justify that a particular function is computable by giving a concrete description of a procedure for the computation. This is permitted because it is believed that all such uses of the thesis can be removed by the tedious process of writing a formal procedure for the function in some model of computation.

Incomputable functions and unsolvable problems

Main article: List of undecidable problems

Because every computable function has a finite procedure telling how to compute it, there are only countably many computable functions. There are uncountably many finitary functions on the natural numbers, so most such functions are not computable. The Busy beaver function is a concrete example of such a function.

Similarly, most subsets of the natural numbers are not computable. The Halting problem was the first such set to be constructed. The Entscheidungsproblem, proposed by David Hilbert, asked whether there is an effective procedure to determine which mathematical statements (coded as natural numbers) are true. Turing and Church independently showed in the 1930s that this set of natural numbers is not computable. According to the Church-Turing thesis, there is no effective procedure (with an algorithm) which can perform these computations.

Extensions of computability

The notion of computability of a function can be relativized to an arbitrary set of natural numbers A, or equivalently to an arbitrary function f from the naturals to the naturals, by using Turing machines (or any other model of computation) extended by an oracle for A or f. Such functions may be called A-computable or f-computable respectively.

Although the Church-Turing thesis states that the computable functions include all functions with algorithms, it is possible to consider broader classes of functions that relax the requirements that algorithms must possess. The field of Hypercomputation studies methods and procedures that can solve non-algorithmically decidable problems. Hyperarithmetical theory studies a different extension of standard computability. Even more general recursion theories have been studied, such as E-recursion theory in which any set can be used as an argument to an E-recursive function.

References

Enderton, H.B. Elements of recursion theory. Handbook of Mathematical Logic (North-Holland 1977) pp. 527–566.

Rogers, H. Theory of recursive functions and effective computation (McGraw-Hill 1967).

Turing, A. (1936), On Computable Numbers, With an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, Series 2, Volume 42 (1936). Reprinted in M. Davis (ed.), The Undecidable, Raven Press, Hewlett, NY, 1965.