Алгоритмдер: қадамдық нұсқаулар мен есептеулер тізбегі

Sequence of operations for a task

In mathematics and computer science, an algorithm (/audio=en us algorithm. ogg/'/æ//l//ɡ//ə//r//ɪ//ð//əm/) is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision making) and deduce valid inferences (referred to as automated reasoning), achieving automation eventually. Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well defined correct or optimal result. For example, social media recommender systems rely on heuristics in such a way that, although widely characterized as "algorithms" in 21st century popular media, cannot deliver correct results due to the nature of the problem. As an effective method, an algorithm can be expressed within a finite amount of space and time and in a well defined formal language for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well defined successive states, eventually producing "output" 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.

Around 825 AD, Persian scientist and polymath Muḥammad ibn Mūsā al Khwārizmī wrote kitāb al ḥisāb al hindī ("Book of Indian computation") and kitab al jam' wa'l tafriq al ḥisāb al hindī ("Addition and subtraction in Indian arithmetic"). Both of these texts are lost in the original Arabic at this time. However, his other book on algebra remains. which would include all computer programs (including programs that do not perform numeric calculations), and (for example) any prescribed bureaucratic procedure
or cook book recipe. In general, a program is an algorithm only if it stops eventually—even though infinite loops may sometimes prove desirable. define an algorithm to be a set of instructions for determining an output, given explicitly, in a form that can be followed by either a computing machine, or a human who could only carry out specific elementary operations on symbols. The concept of algorithm is also used to define the notion of decidability—a notion that is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related to the customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term. Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.

Since antiquity, step by step procedures for solving mathematical problems have been attested. This includes Babylonian mathematics (around 2500 BC), The Ifa Oracle (around 500 BC), Greek mathematics (around 240 BC, e. g. sieve of Eratosthenes and Euclidean algorithm), and Arabic mathematics (9th century, e. g. cryptographic algorithms for code breaking based on frequency analysis). The first cryptographic algorithm for deciphering encrypted code was developed by Al Kindi, a 9th century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm. During the Hammurabi dynasty c. 1800 – c. 1600 BC, Babylonian clay tablets described algorithms for computing formulas. Algorithms were also used in Babylonian astronomy. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events. Algorithms for arithmetic are also found in ancient Egyptian mathematics, dating back to the Rhind Mathematical Papyrus c. 1550 BC. that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata" beginning in the 13th century and finally to "computational machines"—the difference engine and analytical engines of Charles Babbage and Countess Ada Lovelace, mid 19th century. Lovelace is credited with the first creation of an algorithm intended for processing on a computer—Babbage's analytical engine, the first device considered a real Turing complete computer instead of just a calculator—and is sometimes called "history's first programmer" as a result, though a full implementation of Babbage's second device would not be realized until decades after her lifetime.

Bell and Newell (1971) indicate that the Jacquard loom (1801), precursor to Hollerith cards (punch cards, 1887), and "telephone switching technologies" were the roots of a tree leading to the development of the first computers. By the mid 19th century the telegraph, the precursor of the telephone, was in use throughout the world, its discrete and distinguishable encoding of letters as "dots and dashes" a common sound. By the late 19th century, the ticker tape (c. 1870s) was in use, as was the use of Hollerith cards in the 1890 U. S. census. Then came the teleprinter (c. 1910) with its punched paper use of Baudot code on tape. Telephone switching networks of electromechanical relays (invented 1835) was behind the work of George Stibitz (1937), the inventor of the digital adding device. As he worked in Bell Laboratories, he observed the "burdensome' use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea When the tinkering was over, Stibitz had constructed a binary adding device". The mathematician Martin Davis supported the particular importance of the electromechanical relay.

In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the Entscheidungsproblem (decision problem) posed by David Hilbert. Later formalizations were framed as attempts to define "effective calculability" or "effective method". 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 1936–37 and 1939.

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon charts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, drakon charts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but they are also often used as a way to define or document algorithms.