Computable function

Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm, 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 μ-recursive functions. Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm, 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 μ-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. 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 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. 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. 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. Computability of a function is an informal notion. One way to describe it is to say that a function is computable if its value can be obtained by an effective procedure. With more rigor, a function f : N k → N {displaystyle f:mathbb {N} ^{k} ightarrow mathbb {N} } is computable if and only if there is an effective procedure that, given any k-tuple x {displaystyle mathbf {x} } of natural numbers, will produce the value f ( x ) {displaystyle f(mathbf {x} )} . In agreement with this definition, the remainder of this article presumes that computable functions take finitely many natural numbers as arguments and produce a value which is a single natural number. As counterparts to this informal description, there exist multiple formal, mathematical definitions. The class of computable functions can be defined in many equivalent models of computation, including 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, giving rise to the opinion that formal computability is both natural and not too narrow. 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. Formally speaking, a partial function f : N k → N {displaystyle f:mathbb {N} ^{k} ightarrow mathbb {N} } can be calculated if and only if there exists a computer program with the following properties: