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In mathematics and computer science, an algorithm is an effective method expressed as a finite list[1] of well-defined instructions[2] for calculating a function[3]. Algorithms are used for calculation, data processing, and automated reasoning. It represents simplified Mathematics ideas solving complicated problems. For example, sorting the order of customers processing, calculating the income of a company need algorithm.
Starting from an initial state and initial input (perhaps null)[4], the instructions describe a computation that, when executed, will proceed through a finite [5] number of well-defined successive states, eventually producing "output"[6] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input[7].
A partial formalization of the concept began with attempts to solve the Entscheidungsproblem (the "decision problem") posed by David Hilbert in 1928. Subsequent formalizations were framed as attempts to define "effective calculability"[8] or "effective method";[9] those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's "Formulation 1" of 1936, and Alan Turing's Turing machines of 1936–7 and 1939.