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Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do the job of the function, i. e. given an input of the function domain it can return the corresponding output. 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 general recursive functions. According to the Church–Turing thesis, computable functions are exactly the functions that can be calculated using a mechanical (that is, automatic) calculation device given unlimited amounts of time and storage space. More precisely, every model of computation that has ever been imagined can compute only computable functions, and all computable functions can be computed by any of several models of computation that are apparently very different, such as Turing machines, register machines, lambda calculus and general recursive functions. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable. This term has since come to be identified with the computable functions. 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 calculable 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. 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.

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, if the word is in the language and 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.

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.

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.

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 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) = fn(m), where fn is the n th primitive recursive function (for k ary functions, this will be set to fn(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 = fj, we would have got g(j) = en(j,j)+1 = fj (j)+1= g(j)+1, a contradiction. (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 the 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 ).

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 fn(n) (where fn 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.

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 }). 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.

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.

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.

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.