In mathematics, computing, linguistics, and related subjects, an algorithm is a finite sequence of instructions, an explicit, step-by-step procedure for solving a problem, often used for calculation and data processing. It is formally a type of effective method in which a list of well-defined instructions for completing a task, will when given an initial state, proceed through a well-defined series of successive states, eventually terminating in an end-state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as probabilistic 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" (Kleene 1943:274) or "effective method" (Rosser 1939:225); 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.

Etymology

Al-Khwārizmī, Persian astronomer and mathematician, wrote a treatise in 825 AD, On Calculation with Hindu Numerals. (See algorism). It was translated into Latin in the 12th century as Algoritmi de numero Indorum (al-Daffa 1977), whose title was likely intended to mean "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's rendition of the author's name; but people misunderstanding the title treated Algoritmi as a Latin plural and this led to the word "algorithm" (Latin algorismus) coming to mean "calculation method". The intrusive "th" is most likely due to a false cognate with the Greek lang: ἀριθμός (arithmos) meaning "number".

Why algorithms are necessary: an informal definition

For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations. For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples.

While there is no generally accepted formal definition of "algorithm", an informal definition could be "a process that performs some sequence of operations." For some people, a program is only an algorithm if it stops eventually. For others, a program is only an algorithm if it stops before a given number of calculation steps.

A prototypical example of an "algorithm" is Euclid's algorithm to determine the maximum common divisor of two integers (X and Y)which are greater than one: We follow a series of steps: In step i, we divide X by Y and find the remainder, which we call R1. Then we move to step i + 1, where we divide Y by R1, and find the remainder, which we call R2. If R2=0, we stop and say that R1 is the greatest common divisor of X and Y. If not, we continue, until Rn=0.Then Rn-1 is the max common division of X and Y. This procedure is known to stop always and the number of subtractions needed is always smaller than the larger of the two numbers.