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In mathematics, computer science, and related subjects, an algorithm (derived from the name of mathematician alKhwārizmī and transformed to match the Greek "arithmos"number) is an effective method for solving a problem expressed as a finite sequence of steps. Algorithms are used for calculation, data processing, and many other fields. (In more advanced or abstract settings, the instructions do not necessarily constitute a finite sequence, and even not necessarily a sequence; see, e.g., "nondeterministic algorithm".)
Each algorithm is a list of welldefined instructions for completing a task. Starting from an initial state, the instructions describe a computation that proceeds through a welldefined series of successive states, eventually terminating in a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate randomness.
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"^{[1]} or "effective method";^{[2]} 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.
The adjective "continuous" when applied to the word "algorithm" can mean:
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