Computable functions are the basic objects of study in computability theory. The set of computable functions is equivalent to the set of Turing-computable functions and partial 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. 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 with the length of the input. The fields of feasible computability and computational complexity study functions that can be computed efficiently.
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