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Intuitively, a mathematical function f is computable if, given an argument value x, its image value f(x) can be computed by a purely mechanical process ("algorithm"), which could be carried out by a machine. For giving a precise meaning to this concept, several different mathematical models of computation were introduced, which all turned out to define the same set of computable functions. The Church–Turing thesis is the claim that every philosophically acceptable formalization of the intuitive concept will again define that very function set. Many authors use "computable function" refer to that formally defined function set, while some others use it to refer to the informal concept.

Computable functions are the basic objects of study in computability theory.

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.

Intuitive concept

One way to describe the intuitive concept is to say that a function is computable if its value can be obtained by an effective procedure. With more rigor, a (total) function $f:\mathbb {N} ^{k}\rightarrow \mathbb {N}$ is computable if and only if there is an effective procedure that, given any k-tuple $\mathbf {x}$ of natural numbers, will produce the value $f(\mathbf {x} )$.

Enderton [1977] gives the following characteristics of a procedure for computing a computable (partial) 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 (i.e., one of its instructions cannot be executed), 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 defined as correct if and only if it produces an outcome.

Enderton goes on to list several clarifications of these 3 requirements of the procedure for a computable function:

1. 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.
2. 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.
3. 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.

To summarise, based on this view a function is computable if: (a) given an input from its domain, possibly relying on unbounded storage space, it can give the corresponding output by following a procedure (program, algorithm) that is formed by a finite number of exact unambiguous instructions; (b) it returns such output (halts) in a finite number of steps; and (c) if given an input which is not in its domain it either never halts or it gets stuck.

Mathematical models

See also: Total Turing machine

Turing machine to compute truncated subtraction

As counterparts to this informal description, there exist multiple formal, mathematical definitions. The class of computable (partial or total) functions can be defined in many equivalent models of computation, including

• Turing machines:

a mathematically formalized equivalent of a finite-state control unit acting on a two-sided infinite memory tape (see picture). A computation step of a Turing machine consists in reading a symbol from the current tape cell, then, according to that symbol and the current machine state, writing a "new" symbol to the cell, moving the tape left or right, and entering a "new" state.

As an example, the picture shows a Turing machine to compute truncated subtraction of natural numbers in the unary numeral system. The start configuration is shown with the tape representing the pair ⟨5,3⟩ in unary notation. The computation will eventually stop in state "g", with the tape representing the number 2.

• μ-recursive functions:

mathematical functions from ℕ×...×ℕ to ℕ built from the constant $0$, the successor function $S$, and projection functions $P_{i}^{k}$ by means of function composition $\circ$, primitive recursion $\rho$, and μ-recursion $\mu$.

For example, $\rho (P_{1}^{1},S\circ P_{2}^{3})$ denotes to a function f:ℕ×ℕ→ℕ with $f(0,y)=P_{1}^{1}(y)=y$ and $f(S(x),y)=(S\circ P_{2}^{3})(x,f(x,y),y)=S(f(x,y))$; i.e., f computes the sum of its arguments.

• Lambda calculus:

a minimalist calculus to construct and to use unary anonymous functions. An expression is either a variable (e.g. "x"), or built from subexpressions M and N by lambda abstraction (written e.g. "λx.M") or application (written "(M N)"). A binary function can be simulated by a unaary function that returns another unary function, etc. ("Currying"). For example, function composition can be expressed as "λf.λg.λx.(f (g x))".

Church encoding can be used to represent natural numbers by lambda expressions, e.g. $0$, $1$, and $2$ by "λs.λz.z", "λs.λz.(sz)", and "λs.λz.(s(sz))". In this encoding, e.g. addition of two numbers can be expressed as "λa.λb.λs.λz.((a s) ((b s) z)))". The fixed-point operator "λf.(λx.(f (x x)) λx.(f (x x)))" can be used to obtain a kind of μ-recursion.

• Post machines (Post–Turing machines and tag machines).
• Register machines

Although these models use different representations for the functions, their inputs and their outputs, translations exist between any two models, and so every model describes essentially the same class of functions.

For example, one can formalize computable functions as μ-recursive functions, which are partial functions that take finite tuples of natural numbers and return a single natural number (just as above). They are the smallest class of partial functions that includes the constant, successor, and projection functions, and is closed under composition, primitive recursion, and the μ operator.

Equivalently, computable functions can be formalized as functions which can be calculated by an idealized computing agent such as a Turing machine or a register machine.

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

Computable functions are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the μ-recursive functions.

Data types

Most commonly, computability is defined for functions from n-tuples of natural numbers to natural numbers. Functions from strings to strings are also in use for Turing machines, and Post machines, since both models work with strings. Turing's original article defined computability for real numbers, rather than for functions at all. In Lambda calculus, there is no distinction between program and data; both are represented by lambda terms.

In order to establish equivalence between two models, a correspondence between their data representation has to be provided. Most authors use natural numbers as a common representation. For example, the 5-tuple $\langle 1,6,0,4,2\rangle$ may be represented by the string #1#110#0#100#10#, where # is used as a delimiting symbol, and 0 and 1 are the binary digits. It may also be represented in unary form by #I#IIIIII##IIII#II#; this is convenient in theoretical issues when computation time does not matter. Register machines and μ-recursive functions work directly on natural numbers. To represent a string or a λ-term by a number, Gödel numbering can be used.

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.

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 a function is computable if and only if it has an algorithm. Note that an algorithm in this sense is understood to be a sequence of steps a person with unlimited time and an unlimited supply of pen and paper could follow.

Every of the above formal models describes essentially the same class of functions, giving rise to the opinion that formal computability is both natural and not too narrow.

Computable sets and relations

A set A of natural numbers is called computable (synonyms: recursive, decidable) if there is a computable, total function f such that for any natural number n, f(n) = 1 if n is in A and f(n) = 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) = 1 if the word is in the language and f(w) = 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), arg max{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 at least 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 uncomputable 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. Thus, "Print 0, 1, 4, 6, 13" is a trivial algorithm to compute Σ(0), Σ(1), Σ(2), Σ(3), Σ(4); similarly, for any given value of n, such a trivial algorithm exists (even though it may never be known or produced by anyone) to compute Σ(0), Σ(1), Σ(2), ..., Σ(n).

Provability

Given a function (or, similarly, a set), one may be interested not only if it is computable, but also whether this can be proven in a particular proof system (usually first order Peano arithmetic). A function that can be proven to be computable is called provably total.

The set of provably total functions is recursively enumerable: one can enumerate all the provably total functions by enumerating all their corresponding proofs, that prove their computability. This can be done by enumerating all the proofs of the proof system and ignoring irrelevant ones.

Relation to recursively defined functions

In a function defined by a recursive definition, each value is defined by a fixed first-order formula of other, previously defined values of the same function or other functions, which might be simply constants. A subset of these is the primitive recursive functions. Every such function is provably total: For such a k-ary function f, each value $f(n_{1},n_{2}...n_{k})$ can be computed by following the definition backwards, iteratively, and after finite number of iteration (as can be easily proven), a constant is reached.

The converse is not true, as not every provably total function is primitive recursive. Indeed, one can enumerate all the primitive recursive functions and define a function en such that for all n, m: en(n,m) = f_n(m), where f_n is the n-th primitive recursive function (for k-ary functions, this will be set to f_n(m,m...m)). Now, g(n) = en(n,n)+1 is provably total but not primitive recursive, by a diagonalization argument: had there been a j such that g = f_j, we would have got g(j) = en(j,j)+1 = f_j (j)+1= g(j)+1, a contradiction. (It should be noted that the Gödel numbers of all primitive recursive functions can be enumerated by a primitive recursive function, though the primitive recursive functions' values cannot).

One such function, which is provable total but not primitive recursive, is Ackermann function: since it is recursively defined, it is indeed easy to prove its computability (However, a similar diagonalization argument can also be built for all functions defined by recursive definition; thus, there are provable total functions that cannot be defined recursively ).

Total functions that are not provably total

In a sound proof system, every provably total function is indeed total, but the converse is not true: in every first-order proof system that is strong enough and sound (including Peano arithmetic), one can prove (in another proof system) the existence of total functions that cannot be proven total in the proof system.

If the total computable functions are enumerated via the Turing machines that produces them, then the above statement can be shown, if the proof system is sound, by a similar diagonalization argument to that used above, using the enumeration of provably total functions given earlier. One uses a Turing machine that enumerates the relevant proofs, and for every input n calls f_n(n) (where f_n is n-th function by this enumeration) by invoking the Turing machine that computes it according to the n-th proof. Such a Turing machine is guaranteed to halt if the proof system is sound.

Uncomputable functions and unsolvable problems

Main article: List of undecidable problems

Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it. Furthermore, this procedure has to be encoded in the finite alphabet used by the computational model, so there are only countably many computable functions. For example, functions may be encoded using a string of bits (the alphabet Σ = {0, 1}).

The real numbers are uncountable so most real numbers are not computable. See computable number. The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin's constant.

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

Relative computability

The notion of computability of a function can be relativized to an arbitrary set of natural numbers A. A function f is defined to be computable in A (equivalently A-computable or computable relative to A) when it satisfies the definition of a computable function with modifications allowing access to A as an oracle. As with the concept of a computable function relative computability can be given equivalent definitions in many different models of computation. This is commonly accomplished by supplementing the model of computation with an additional primitive operation which asks whether a given integer is a member of A. We can also talk about f being computable in g by identifying g with its graph.

Higher recursion theory

Hyperarithmetical theory studies those sets that can be computed from a computable ordinal number of iterates of the Turing jump of the empty set. This is equivalent to sets defined by both a universal and existential formula in the language of second order arithmetic and to some models of Hypercomputation. 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.

Hyper-computation

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 models of computation that go beyond normal Turing computation.

See also

• Computable number
• Theory of computation
• Recursion theory
• Turing degree
• Arithmetical hierarchy
• Super-recursive algorithm
• Semicomputable function

References

• Cutland, Nigel. Computability. Cambridge University Press, 1980.
• Rogers, H. Theory of recursive functions and effective computation (McGraw–Hill 1967).