In

LET [] = [] is the assignment instruction symbolized by ←.
5 REM Euclid's algorithm for greatest common divisor
6 PRINT "Type two integers greater than 0"
10 INPUT A,B
20 IF B=0 THEN GOTO 80
30 IF A > B THEN GOTO 60
40 LET B=B-A
50 GOTO 20
60 LET A=A-B
70 GOTO 20
80 PRINT A
90 END
''How "Elegant" works'': In place of an outer "Euclid loop", "Elegant" shifts back and forth between two "co-loops", an A > B loop that computes A ← A − B, and a B ≤ A loop that computes B ← B − A. This works because, when at last the minuend M is less than or equal to the subtrahend S (Difference = Minuend − Subtrahend), the minuend can become ''s'' (the new measuring length) and the subtrahend can become the new ''r'' (the length to be measured); in other words the "sense" of the subtraction reverses.
The following version can be used with List of C-family programming languages, programming languages from the C-family:
// Euclid's algorithm for greatest common divisor
int euclidAlgorithm (int A, int B)

ACM-SIAM Symposium On Discrete Algorithms (SODA)

, Kyoto, January 2012. See also th

sFFT Web Page

. Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.

Turing machine
A Turing machine is a mathematical model of computation
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are ...

—begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates a machine as a model of computation of numbers.
:"Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book...I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite...
:"The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite...
:"Let us imagine that the operations performed by the computer to be split up into 'simple operations' which are so elementary that it is not easy to imagine them further divided."Turing 1936–37 in Davis 1965:136
Turing's reduction yields the following:
:"The simple operations must therefore include:
::"(a) Changes of the symbol on one of the observed squares
::"(b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares.
"It may be that some of these change necessarily invoke a change of state of mind. The most general single operation must, therefore, be taken to be one of the following:
::"(A) A possible change (a) of symbol together with a possible change of state of mind.
::"(B) A possible change (b) of observed squares, together with a possible change of state of mind"
:"We may now construct a machine to do the work of this computer."
A few years later, Turing expanded his analysis (thesis, definition) with this forceful expression of it:
:"A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Though it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematical expressible definition ... [he discusses the history of the definition pretty much as presented above with respect to Gödel, Herbrand, Kleene, Church, Turing, and Post] ... We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability † with effective calculability ... .
::"† We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions".Turing 1939 in Davis 1965:160

Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was a Polish-born American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

are included; those cited in the article are listed here by author's name.
* Davis offers concise biographies of Gottfried Leibniz, Leibniz, George Boole, Boole, Gottlob Frege, Frege, Georg Cantor, Cantor, David Hilbert, Hilbert, Gödel and Turing with John von Neumann, von Neumann as the show-stealing villain. Very brief bios of Joseph-Marie Jacquard, Babbage, Ada Lovelace, Claude Shannon, Howard Aiken, etc.
*
*
*
* ,
* Yuri Gurevich

''Sequential Abstract State Machines Capture Sequential Algorithms''

ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources. * , 3rd edition 1976[?], (pbk.) * , . Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof. * Presented to the American Mathematical Society, September 1935. Reprinted in ''The Undecidable'', p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper ''An Unsolvable Problem of Elementary Number Theory'' that proved the "decision problem" to be "undecidable" (i.e., a negative result). * Reprinted in ''The Undecidable'', p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in Kleene 1952:300) and name it "Church's Thesis"(Kleene 1952:317) (i.e., the Church thesis). * * * * Kosovsky, N.K. ''Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms'', LSU Publ., Leningrad, 1981 * * A.A. Markov (1954) ''Theory of algorithms''. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS .] * Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 ''Computability, Effective Procedures and Algorithms. Infinite machines.'' * Reprinted in ''The Undecidable'', pp. 289ff. Post defines a simple algorithmic-like process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the so-calledChurch–Turing thesis
In Computability theory (computation), computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) i ...

.
*
* Reprinted in ''The Undecidable'', p. 223ff. Herein is Rosser's famous definition of "effective method": "...a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps... a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer" (p. 225–226, ''The Undecidable'')
*
*
*
*
* Cf. in particular the first chapter titled: ''Algorithms, Turing Machines, and Programs''. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an ''algorithm''" (p. 4).
*
* . Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in ''The Undecidable'', p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.
* Reprinted in ''The Undecidable'', pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton.
* United States Patent and Trademark Office (2006)

''2106.02 **>Mathematical Algorithms: 2100 Patentability''

Manual of Patent Examining Procedure (MPEP). Latest revision August 2006

Selected Papers on Analysis of Algorithms

'. Stanford, California: Center for the Study of Language and Information. * Knuth, Donald E. (2010).

'. Stanford, California: Center for the Study of Language and Information. *

Dictionary of Algorithms and Data Structures

– National Institute of Standards and Technology ; Algorithm repositories

The Stony Brook Algorithm Repository

– State University of New York at Stony Brook

Collected Algorithms of the ACM

– Association for Computing Machinery

The Stanford GraphBase

– Stanford University {{Authority control Algorithms, Articles with example pseudocode Mathematical logic Theoretical computer science

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

and computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , , and . Computer science ...

, an algorithm () is a finite sequence of well-defined
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always unambiguous
Ambiguity is a type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty
Uncertainty refers to Epistemology, epistemic sit ...

and are used as specifications for performing calculation
A calculation is a deliberate process that transforms one or more inputs into one or more results. The term is used in a variety of senses, from the very definite arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Anc ...

s, data processing
Data processing is, generally, "the collection
Collection or Collections may refer to:
* Cash collection, the function of an accounts receivable department
* Collection agency, agency to collect cash
* Collections management (museum)
** Colle ...

, automated reasoningIn computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of Algorith ...

, and other tasks.
As an effective methodIn logic, mathematics and computer science, especially metalogic and computability theory, an effective methodGeoffrey Hunter (logician), Hunter, Geoffrey, ''Metalogic: An Introduction to the Metatheory of Standard First-Order Logic'', University of ...

, an algorithm can be expressed within a finite amount of space and time, and in a well-defined formal language for calculating a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation
Computation is any type of calculation that includes both arithmetical and non-arithmetical steps and which follows a well-defined model (e.g. an algorithm).
Mechanical or electronic devices (or, History of computing hardware, historically, peop ...

that, when executed
Capital punishment, also known as the death penalty, is the State (polity), state-sanctioned killing of a person as punishment for a crime. The sentence (law), sentence ordering that someone is punished with the death penalty is called a dea ...

, 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
Determinism is the philosophical
Philosophy (from , ) is the study of general and fundamental questions, such as those about existence
Existence is the ability of an entity to interact with physical or mental reality
Reality is the ...

; some algorithms, known as randomized algorithms
A ''randomized algorithm'' is an algorithm that employs a degree of randomness as part of its logic. The algorithm typically uses Uniform distribution (discrete), uniformly random bits as an auxiliary input to guide its behavior, in the hope of ach ...

, incorporate random input.
The concept of algorithm has existed since antiquity. Arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...

algorithms, such as a division algorithm
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software.
Divisi ...

, was used by ancient Babylonian mathematicians c. 2500 BC and Egyptian mathematicians c. 1550 BC. Greek mathematicians later used algorithms in 240 BC in the sieve of Eratosthenes
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

for finding prime numbers, and the Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...

for finding the greatest common divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

of two numbers. Arabic mathematicians such as al-Kindi
Abu Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī (; ar, أبو يوسف يعقوب بن إسحاق الصبّاح الكندي; la, Alkindus; c. 801–873 AD) was an Arab
The Arabs (singular Arab ; singular ar, عَرَب ...

in the 9th century used cryptographic
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or '' -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of third ...

algorithms for code-breaking
cipher machine
Cryptanalysis (from the Greek language, Greek ''kryptós'', "hidden", and ''analýein'', "to analyze") is the study of analyzing information systems in order to study the hidden aspects of the systems. Cryptanalysis is used to bre ...

, based on frequency analysis
In cryptanalysis, frequency analysis (also known as counting letters) is the study of the letter frequencies, frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers.
Frequency analy ...

.
The word ''algorithm'' itself is derived from the name of the 9th-century mathematician Muḥammad ibn Mūsā al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( fa, محمد بن موسی خوارزمی, Moḥammad ben Musā Khwārazmi; ), Arabized as al-Khwarizmi and formerly Latinisation of names, Latinized as ''Algorithmi'', was a Persians, Persian polymath who ...

, whose nisba
The Arabic
Arabic (, ' or , ' or ) is a Semitic language
The Semitic languages are a branch of the Afroasiatic language family originating in the Middle East
The Middle East is a list of transcontinental countries, transcontin ...

(identifying him as from Khwarazm
Khwarazm , or Chorasmia (Old Persian
Old Persian is one of the two directly attested Old Iranian languages
The Iranian or Iranic languages are a branch of the Indo-Iranian languagesIndo-Iranian may refer to:
* Indo-Iranian languages ...

) was Latinized ''as Algoritmi''. A partial formalization of what would become the modern concept of algorithm began with attempts to solve the ''Entscheidungsproblem
In mathematics and computer science, the ' (, German language, German for "decision problem") is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers ...

'' (decision problem) posed by David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, G ...

in 1928. Later formalizations were framed as attempts to define " effective calculability" or "effective method". Those formalizations included the Gödel– Herbrand–Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an United States, American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of t ...

recursive functions of 1930, 1934 and 1935, Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
The United States of America (US ...

's lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system
A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, ar ...

of 1936, Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was a Polish-born American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

's Formulation 1 of 1936, and Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...

's Turing machines
A Turing machine is a mathematical model of computation that defines an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, given any computer algorithm, a Turing machine ca ...

of 1936–37 and 1939.
Etymology

The word 'algorithm' has its roots in Latinizing the nisba, indicating his geographic origin, of the name ofPersian
Persian may refer to:
* People and things from Iran, historically called ''Persia'' in the English language
** Persians, Persian people, the majority ethnic group in Iran, not to be conflated with the Iranian peoples
** Persian language, an Iranian ...

mathematician Muhammad ibn Musa al-Khwarizmi
Muḥammad ibn Mūsā al-Khwārizmī ( fa, محمد بن موسی خوارزمی, Moḥammad ben Musā Khwārazmi; ), or al-Khwarizmi and formerly as ''Algorithmi'', was a who produced vastly influential works in , , and . Around 820 CE he w ...

to ''algorismus''. Al-Khwārizmī (Arabized
Arabization or Arabisation ( ar, تعريب ') describes both the process of growing Arab
The Arabs (singular Arab ; singular ar, عَرَبِيٌّ, ISO 233: , Arabic pronunciation: , plural ar, عَرَبٌ, ISO 233: , Arabic pronunciati ...

Persian
Persian may refer to:
* People and things from Iran, historically called ''Persia'' in the English language
** Persians, Persian people, the majority ethnic group in Iran, not to be conflated with the Iranian peoples
** Persian language, an Iranian ...

الخوارزمی c. 780–850) was a mathematician, astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...

, geographer
A geographer is a physical scientist, social scientist or humanist whose area of study is geography
Geography (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), offic ...

, and scholar in the House of Wisdom
The House of Wisdom ( ar, بيت الحكمة, Bayt al-Ḥikmah), also known as the Grand Library of Baghdad, refers to either a major Abbasid Caliphate, Abbasid public academy and intellectual center in Baghdad or to a large private library be ...

in Baghdad
Baghdad (; ar, بَغْدَاد ) is the capital of Iraq
Iraq ( ar, الْعِرَاق, translit=al-ʿIrāq; ku, عێراق, translit=Êraq), officially the Republic of Iraq ( ar, جُمْهُورِيَّة ٱلْعِرَاق '; ku, ...

, whose name means 'the native of Khwarazm
Khwarazm , or Chorasmia (Old Persian
Old Persian is one of the two directly attested Old Iranian languages
The Iranian or Iranic languages are a branch of the Indo-Iranian languagesIndo-Iranian may refer to:
* Indo-Iranian languages ...

', a region that was part of Greater Iran
( BC) at its greatest extent ()
File:Achaemenid_(greatest_extent).svg, Achaemenid Empire (550 BC–330 BC) at its greatest extent ()
Greater Iran ( fa, ایران بزرگ, translit=Irān-e Bozorg) refers to the regions of Western Asia, ...

and is now in Uzbekistan
Uzbekistan (, ; uz, Ozbekiston, italic=yes, ), officially the Republic of Uzbekistan ( uz, Ozbekiston Respublikasi, italic=yes), is a doubly landlocked country
A landlocked country is a country
A country is a distinct territory, ter ...

.
About 825, al-Khwarizmi wrote an Arabic language
Arabic (, ' or , ' or ) is a that first emerged in the 1st to 4th centuries CE.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walter de G ...

treatise on the Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Arabic numeral system or Hindu numeral system) is a positional notation, positional decimal numeral system, and is t ...

, which was translated into Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

during the 12th century. The manuscript starts with the phrase ''Dixit Algorizmi'' ('Thus spake Al-Khwarizmi'), where "Algorizmi" was the translator's Latinization
Latinisation or Latinization can refer to:
* Latinisation of names
Latinisation (or Latinization) of names, also known as onomastic Latinisation (or Latinization), is the practice of rendering a ''non''-Latin name in a Latin style. It is commonl ...

of Al-Khwarizmi's name. Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through another of his books, the Algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

. In late medieval Latin, ''algorismus'', English 'algorism
Algorism is the technique of performing basic arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Anci ...

', the corruption of his name, simply meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός (''arithmos''), 'number' (''cf.'' 'arithmetic'), the Latin word was altered to ''algorithmus'', and the corresponding English term 'algorithm' is first attested in the 17th century; the modern sense was introduced in the 19th century.
In English, it was first used in about 1230 and then by Chaucer
Geoffrey Chaucer (; – 25 October 1400) was an English poet and author. Widely considered the greatest English poet of the Middle Ages
In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th ...

in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English.
Another early use of the word is from 1240, in a manual titled ''Carmen de Algorismo'' composed by Alexandre de Villedieu
Alexander of Villedieu was a French people, French author, teacher and poet, who wrote text books on Latin grammar and arithmetic, everything in verse. He was born around 1175 in Villedieu-les-Poêles in Normandy, studied in Paris, and later taught ...

. It begins with:
which translates to:
The poem is a few hundred lines long and summarizes the art of calculating with the new styled Indian dice (''Tali Indorum''), or Hindu numerals.
Informal definition

An informal definition could be "a set of rules that precisely defines a sequence of operations", which would include all computer programs (including programs that do not perform numeric calculations), and (for example) any prescribedbureaucratic
The term bureaucracy () may refer both to a body of non-elected governing officials and to an administrative policy-making group. Historically, a bureaucracy was a government administration managed by departments staffed with non-elected offi ...

procedure
or cook-book recipe
A recipe is a set of instructions that describes how to prepare or make something, especially a dish
Dish, dishes or DISH may refer to:
Culinary
* Dish (food), something prepared to be eaten
* Dishware, plates and bowls for eating, cutting bo ...

.
In general, a program is only an algorithm if it stops eventually - even though infinite loops may sometimes prove desirable.
A prototypical example of an algorithm is the Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...

, which is used to determine the maximum common divisor of two integers; an example (there are others) is described by the flowchart
A flowchart is a type of diagram that represents a workflow or process. A flowchart can also be defined as a diagrammatic representation of an algorithm, a step-by-step approach to solving a task.
The flowchart shows the steps as boxes of var ...

above and as an example in a later section.
offer an informal meaning of the word "algorithm" in the following quotation:
No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit … you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give ''explicit instructions for determining the nth member of the set'', for arbitrary finite ''n''. Such instructions are to be given quite explicitly, in a form in which ''they could be followed by a computing machine'', or by a ''human who is capable of carrying out only very elementary operations on symbols.''An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an ''arbitrary'' "input" integer or integers that, in theory, can be arbitrarily large. For example, an algorithm can be an algebraic equation such as ''y = m + n'' (i.e., two arbitrary "input variables" ''m'' and ''n'' that produce an output ''y''), but various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example): :Precise instructions (in a language understood by "the computer") for a fast, efficient, "good" process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities) to find, decode, and then process arbitrary input integers/symbols ''m'' and ''n'', symbols ''+'' and ''='' … and "effectively" produce, in a "reasonable" time, output-integer ''y'' at a specified place and in a specified format. The concept of ''algorithm'' is also used to define the notion of decidability—a notion that is central for explaining how

formal system
A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A formal system is essen ...

s come into being starting from a small set of axiom
An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s and rules. In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

, 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.
Formalization

Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform—in a specific order—to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by aTuring-complete
In computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) include ...

system. Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000):
Minsky: "But we will also maintain, with Turing … that any procedure which could "naturally" be called effective, can, in fact, be realized by a (simple) machine. Although this may seem extreme, the arguments … in its favor are hard to refute".

Gurevich: “… Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine … according to SavageTuring machines can define computational processes that do not terminate. The informal definitions of algorithms generally require that the algorithm always terminates. This requirement renders the task of deciding whether a formal procedure is an algorithm impossible in the general case—due to a major theorem of987 Year 987 ( CMLXXXVII) was a common year starting on Saturday A common year starting on Saturday is any non-leap year A leap year (also known as an intercalary year or wikt:bissextile, bissextile year) is a calendar year that contains an ad ...an algorithm is a computational process defined by a Turing machine".

computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. ...

known as the halting problem
In computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set the ...

.
Typically, when an algorithm is associated with processing information, data can be read from an input source, written to an output device and stored for further processing. Stored data are regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structure
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of ...

s.
For some of these computational processes, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. This means that any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).
Because an algorithm is a precise list of precise steps, the order of computation is always crucial to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom"—an idea that is described more formally by '' flow of control''.
So far, the discussion on the formalization of an algorithm has assumed the premises of imperative programming
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , , ...

. This is the most common conception—one which attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, which sets the value of a variable. It derives from the intuition of "memory
Memory is the faculty of the brain
A brain is an organ
Organ may refer to:
Biology
* Organ (anatomy)
An organ is a group of Tissue (biology), tissues with similar functions. Plant life and animal life rely on many organs that co-exis ...

" as a scratchpad. An example of such an assignment can be found below.
For some alternate conceptions of what constitutes an algorithm, see functional programming
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , ...

and logic programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic prog ...

.
Expressing algorithms

Algorithms can be expressed in many kinds of notation, includingnatural language
In neuropsychology
Neuropsychology is a branch of psychology. It is concerned with how a person's cognition and behavior are related to the brain and the rest of the nervous system. Professionals in this branch of psychology often focus on ...

s, pseudocode
In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine re ...

, flowchart
A flowchart is a type of diagram that represents a workflow or process. A flowchart can also be defined as a diagrammatic representation of an algorithm, a step-by-step approach to solving a task.
The flowchart shows the steps as boxes of var ...

s, drakon-charts, programming language
A programming language is a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s or control table
Control tables are tables that control the control flow
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for the ...

s (processed by interpreters
Interpreting is a Translation studies, translational activity in which one produces a first and final translation on the basis of a one-time exposure to an expression in a Source language (translation), source language.
The most common two modes ...

). 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 the statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are also often used as a way to define or document algorithms.
There is a wide variety of representations possible and one can express a given Turing machine
A Turing machine is a mathematical model of computation
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are ...

program as a sequence of machine tables (see finite-state machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation
In computer science
Computer science deals with the theoretical fou ...

, state transition table
In automata theory
Automata theory is the study of abstract machines and automata, as well as the computational problem
In theoretical computer science
An artistic representation of a Turing machine. Turing machines are used to model general ...

and control table
Control tables are tables that control the control flow
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for the ...

for more), as flowcharts and drakon-charts (see state diagram
A state diagram is a type of diagram
A diagram is a symbolic representation
Representation may refer to:
Law and politics
*Representation (politics)
Political representation is the activity of making citizens "present" in public policy mak ...

for more), or as a form of rudimentary machine code
In computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, ge ...

or assembly code
In computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a specific task. Programming involves tasks such as: analysis, genera ...

called "sets of quadruples" (see Turing machine
A Turing machine is a mathematical model of computation
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are ...

for more).
Representations of algorithms can be classed into three accepted levels of Turing machine description, as follows:
; 1 High-level description
: “…prose to describe an algorithm, ignoring the implementation details. At this level, we do not need to mention how the machine manages its tape or head."
; 2 Implementation description
: “…prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level, we do not give details of states or transition function."
; 3 Formal description
: Most detailed, "lowest level", gives the Turing machine's "state table".
For an example of the simple algorithm "Add m+n" described in all three levels, see Algorithm#Examples.
Design

Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories of operation research, such asdynamic programming
Dynamic programming is both a mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lo ...

and divide-and-conquer. Techniques for designing and implementing algorithm designs are also called algorithm design patterns, with examples including the template method pattern and the decorator pattern.
One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the big O notation
Big O notation is a mathematical notation that describes the limiting behavior of a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be p ...

is used to describe e.g. an algorithm's run-time growth as the size of its input increases.
Typical steps in the development of algorithms:
# Problem definition
# Development of a model
# Specification of the algorithm
# Designing an algorithm
# Checking the correctness of the algorithm
# Analysis of algorithm
# Implementation of algorithm
# Program testing
# Documentation preparation
Implementation

Most algorithms are intended to be implemented ascomputer programs
A computer program is a collection of instructions that can be executed by a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform ...

. However, algorithms are also implemented by other means, such as in a biological neural network
A neural circuit is a population of neurons interconnected by synapses to carry out a specific function when activated. Neural circuits interconnect to one another to form large scale brain networks. Biological neural networks have inspired the ...

(for example, the human brain
The human brain is the central organ
Organ may refer to:
Biology
* Organ (anatomy)
An organ is a group of Tissue (biology), tissues with similar functions. Plant life and animal life rely on many organs that co-exist in organ systems.
...

implementing arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...

or an insect looking for food), in an electrical circuit
An electrical network is an interconnection of electrical component
An electronic component is any basic discrete device or physical entity in an electronic system
Electronic may refer to:
*Electronics, the science of how to control elect ...

, or in a mechanical device.
Computer algorithms

Incomputer systems
A computer is a machine
A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people
A people is a plurality of person
A person (plural ...

, an algorithm is basically an instance of logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

written in software by software developers, to be effective for the intended "target" computer(s) to produce ''output'' from given (perhaps null) ''input''. An optimal algorithm, even running in old hardware, would produce faster results than a non-optimal (higher time complexity
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of com ...

) algorithm for the same purpose, running in more efficient hardware; that is why algorithms, like computer hardware, are considered technology.
''"Elegant" (compact) programs, "good" (fast) programs '': The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin
Gregory John Chaitin ( ; born 25 June 1947) is an Argentina, Argentine-United States, American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, i ...

:
:Knuth: " … we want ''good'' algorithms in some loosely defined aesthetic sense. One criterion … is the length of time taken to perform the algorithm …. Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc"
:Chaitin: " … a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant—such a proof would solve the Halting problem
In computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set the ...

(ibid).
''Algorithm versus function computable by an algorithm'': For a given function multiple algorithms may exist. This is true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is ... important to distinguish between the notion of ''algorithm'', i.e. procedure and the notion of ''function computable by algorithm'', i.e. mapping yielded by procedure. The same function may have several different algorithms".
Unfortunately, there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example that uses Euclid's algorithm appears below.
''Computers (and computors), models of computation'': A computer (or human "computor") is a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions. Melzak's and Lambek's primitive models reduced this notion to four elements: (i) discrete, distinguishable ''locations'', (ii) discrete, indistinguishable ''counters'' (iii) an agent, and (iv) a list of instructions that are ''effective'' relative to the capability of the agent.
Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability
Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic
Mathematical logic, also calle ...

". Minsky's machine proceeds sequentially through its five (or six, depending on how one counts) instructions unless either a conditional IF-THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three ''assignment'' (replacement, substitution) operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1). Rarely must a programmer write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...

with only four general ''types'' of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT. However, a few different assignment instructions (e.g. DECREMENT, INCREMENT, and ZERO/CLEAR/EMPTY for a Minsky machine) are also required for Turing-completeness; their exact specification is somewhat up to the designer. The unconditional GOTO is a convenience; it can be constructed by initializing a dedicated location to zero e.g. the instruction " Z ← 0 "; thereafter the instruction IF Z=0 THEN GOTO xxx is unconditional.
''Simulation of an algorithm: computer (computor) language'': Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example". But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can ''effectively'' execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't, then the algorithm, to be effective, must provide a set of rules for extracting a square root.
This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).
But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of ''simulation'' enters". When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a " modulus" instruction available rather than just subtraction (or worse: just Minsky's "decrement").
''Structured programming, canonical structures'': Per the Church–Turing thesis
In Computability theory (computation), computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) i ...

, any algorithm can be computed by a model known to be Turing complete
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...

, and per Minsky's demonstrations, Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code
Spaghetti code is a pejorative
A pejorative or slur is a word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practica ...

", a programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language". Tausworthe augments the three Böhm-Jacopini canonical structures: SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction.
''Canonical flowchart symbols'': The graphical aide called a flowchart
A flowchart is a type of diagram that represents a workflow or process. A flowchart can also be defined as a diagrammatic representation of an algorithm, a step-by-step approach to solving a task.
The flowchart shows the steps as boxes of var ...

, offers a way to describe and document an algorithm (and a computer program of one). Like the program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only four: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm–Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. The symbols, and their use to build the canonical structures are shown in the diagram.
Examples

Algorithm example

One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as: ''High-level description:'' # If there are no numbers in the set then there is no highest number. # Assume the first number in the set is the largest number in the set. # For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set. # When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set. ''(Quasi-)formal description:'' Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm inpseudocode
In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine re ...

or pidgin code:
Input: A list of numbers ''L''.
Output: The largest number in the list ''L''.
if ''L.size'' = 0 return null
''largest'' ← ''L''[0]
for each ''item'' in ''L'', do
if ''item'' > ''largest'', then
''largest'' ← ''item''
return ''largest''
Euclid's algorithm

Euclid's algorithm to compute thegreatest common divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

(GCD) to two numbers appears as Proposition II in Book VII ("Elementary Number Theory") of his ''Euclid's Elements, Elements''. Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number [to be] a multitude composed of units": a counting number, a positive integer not including zero. To "measure" is to place a shorter measuring length ''s'' successively (''q'' times) along longer length ''l'' until the remaining portion ''r'' is less than the shorter length ''s''. In modern words, remainder ''r'' = ''l'' − ''q''×''s'', ''q'' being the quotient, or remainder ''r'' is the "modulus", the integer-fractional part left over after the division.
For Euclid's method to succeed, the starting lengths must satisfy two requirements: (i) the lengths must not be zero, AND (ii) the subtraction must be “proper”; i.e., a test must guarantee that the smaller of the two numbers is subtracted from the larger (or the two can be equal so their subtraction yields zero).
Euclid's original proof adds a third requirement: the two lengths must not be prime to one another. Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers' common measure is in fact the ''greatest''. While Nicomachus' algorithm is the same as Euclid's, when the numbers are prime to one another, it yields the number "1" for their common measure. So, to be precise, the following is really Nicomachus' algorithm.
Computer language for Euclid's algorithm

Only a few instruction ''types'' are required to execute Euclid's algorithm—some logical tests (conditional GOTO), unconditional GOTO, assignment (replacement), and subtraction. * A ''location'' is symbolized by upper case letter(s), e.g. S, A, etc. * The varying quantity (number) in a location is written in lower case letter(s) and (usually) associated with the location's name. For example, location L at the start might contain the number ''l'' = 3009.An inelegant program for Euclid's algorithm

The following algorithm is framed as Knuth's four-step version of Euclid's and Nicomachus', but, rather than using division to find the remainder, it uses successive subtractions of the shorter length ''s'' from the remaining length ''r'' until ''r'' is less than ''s''. The high-level description, shown in boldface, is adapted from Knuth 1973:2–4: INPUT: [Into two locations L and S put the numbers ''l'' and ''s'' that represent the two lengths]: INPUT L, S [Initialize R: make the remaining length ''r'' equal to the starting/initial/input length ''l'']: R ← L E0: [Ensure ''r'' ≥ ''s''.] [Ensure the smaller of the two numbers is in S and the larger in R]: IF R > S THEN the contents of L is the larger number so skip over the exchange-steps #4, 4, #5, 5 and #6, 6: GOTO step #7, 7 ELSE swap the contents of R and S. L ← R (this first step is redundant, but is useful for later discussion). R ← S S ← L E1: [Find remainder]: Until the remaining length ''r'' in R is less than the shorter length ''s'' in S, repeatedly subtract the measuring number ''s'' in S from the remaining length ''r'' in R. IF S > R THEN done measuring so GOTO #10, 10 ELSE measure again, R ← R − S [Remainder-loop]: GOTO #7, 7. E2: [Is the remainder zero?]: EITHER (i) the last measure was exact, the remainder in R is zero, and the program can halt, OR (ii) the algorithm must continue: the last measure left a remainder in R less than measuring number in S. IF R = 0 THEN done so GOTO #15, step 15 ELSE CONTINUE TO #11, step 11, E3: [Interchange ''s'' and ''r'']: The nut of Euclid's algorithm. Use remainder ''r'' to measure what was previously smaller number ''s''; L serves as a temporary location. L ← R R ← S S ← L [Repeat the measuring process]: GOTO #7, 7 OUTPUT: [Done. S contains thegreatest common divisor
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

]:
PRINT S
DONE:
HALT, END, STOP.
An elegant program for Euclid's algorithm

The flowchart of "Elegant" can be found at the top of this article. In the (unstructured) Basic language, the steps are numbered, and the instructionTesting the Euclid algorithms

Does an algorithm do what its author wants it to do? A few test cases usually give some confidence in the core functionality. But tests are not enough. For test cases, one source uses 3009 and 884. Knuth suggested 40902, 24140. Another interesting case is the two relatively prime numbers 14157 and 5950. But "exceptional cases" must be identified and tested. Will "Inelegant" perform properly when R > S, S > R, R = S? Ditto for "Elegant": B > A, A > B, A = B? (Yes to all). What happens when one number is zero, both numbers are zero? ("Inelegant" computes forever in all cases; "Elegant" computes forever when A = 0.) What happens if ''negative'' numbers are entered? Fractional numbers? If the input numbers, i.e. the domain of a function, domain of the function computed by the algorithm/program, is to include only positive integers including zero, then the failures at zero indicate that the algorithm (and the program that instance (computer science), instantiates it) is a partial function rather than a total function. A notable failure due to exceptions is the Ariane 5 Flight 501 rocket failure (June 4, 1996). ''Proof of program correctness by use of mathematical induction'': Knuth demonstrates the application of mathematical induction to an "extended" version of Euclid's algorithm, and he proposes "a general method applicable to proving the validity of any algorithm". Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof.Measuring and improving the Euclid algorithms

''Elegance (compactness) versus goodness (speed)'': With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. However, "Inelegant" is ''faster'' (it arrives at HALT in fewer steps). Algorithm analysis indicates why this is the case: "Elegant" does ''two'' conditional tests in every subtraction loop, whereas "Inelegant" only does one. As the algorithm (usually) requires many loop-throughs, ''on average'' much time is wasted doing a "B = 0?" test that is needed only after the remainder is computed. ''Can the algorithms be improved?'': Once the programmer judges a program "fit" and "effective"—that is, it computes the function intended by its author—then the question becomes, can it be improved? The compactness of "Inelegant" can be improved by the elimination of five steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; rather, it can only be done heuristically; i.e., by exhaustive search (examples to be found at Busy beaver), trial and error, cleverness, insight, application of inductive reasoning, etc. Observe that steps 4, 5 and 6 are repeated in steps 11, 12 and 13. Comparison with "Elegant" provides a hint that these steps, together with steps 2 and 3, can be eliminated. This reduces the number of core instructions from thirteen to eight, which makes it "more elegant" than "Elegant", at nine steps. The speed of "Elegant" can be improved by moving the "B=0?" test outside of the two subtraction loops. This change calls for the addition of three instructions (B = 0?, A = 0?, GOTO). Now "Elegant" computes the example-numbers faster; whether this is always the case for any given A, B, and R, S would require a detailed analysis.Algorithmic analysis

It is frequently important to know how much of a particular resource (such as time or storage) is theoretically required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers (estimates); for example, the sorting algorithm above has a time requirement of O(''n''), using thebig O notation
Big O notation is a mathematical notation that describes the limiting behavior of a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be p ...

with ''n'' as the length of the list. At all times the algorithm only needs to remember two values: the largest number found so far, and its current position in the input list. Therefore, it is said to have a space requirement of ''O(1)'', if the space required to store the input numbers is not counted, or O(''n'') if it is counted.
Different algorithms may complete the same task with a different set of instructions in less or more time, space, or 'algorithmic efficiency, effort' than others. For example, a binary search algorithm (with cost O(log n) ) outperforms a sequential search (cost O(n) ) when used for lookup table, table lookups on sorted lists or arrays.
Formal versus empirical

The analysis of algorithms, analysis, and study of algorithms is a discipline ofcomputer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , , and . Computer science ...

, and is often practiced abstractly without the use of a specific programming language
A programming language is a formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

or implementation. In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation. Usually pseudocode
In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine re ...

is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware/software platforms and their algorithmic efficiency is eventually put to the test using real code. For the solution of a "one off" problem, the efficiency of a particular algorithm may not have significant consequences (unless n is extremely large) but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.
Empirical testing is useful because it may uncover unexpected interactions that affect performance. Benchmark (computing), Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization.
Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner.
Execution efficiency

To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to Fast Fourier transform, FFT algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging. In general, speed improvements depend on special properties of the problem, which are very common in practical applications.Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price,ACM-SIAM Symposium On Discrete Algorithms (SODA)

, Kyoto, January 2012. See also th

sFFT Web Page

. Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.

Classification

There are various ways to classify algorithms, each with its own merits.By implementation

One way to classify algorithms is by implementation means. ; Recursion : A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition (also known as termination condition) matches, which is a method common tofunctional programming
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , ...

. Iteration, Iterative algorithms use repetitive constructs like Program loops, loops and sometimes additional data structures like Stack (data structure), stacks to solve the given problems. Some problems are naturally suited for one implementation or the other. For example, towers of Hanoi is well understood using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.
; Logical
: An algorithm may be viewed as controlled Deductive reasoning, logical deduction. This notion may be expressed as: ''Algorithm = logic + control''. The logic component expresses the axioms that may be used in the computation and the control component determines the way in which deduction is applied to the axioms. This is the basis for the logic programming
Logic programming is a programming paradigm which is largely based on formal logic. Any program written in a logic programming language is a set of sentences in logical form, expressing facts and rules about some problem domain. Major logic prog ...

paradigm. In pure logic programming languages, the control component is fixed and algorithms are specified by supplying only the logic component. The appeal of this approach is the elegant Formal semantics of programming languages, semantics: a change in the axioms produces a well-defined change in the algorithm.
; Serial, parallel or distributed
: Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time. Those computers are sometimes called serial computers. An algorithm designed for such an environment is called a serial algorithm, as opposed to parallel algorithms or distributed algorithms. Parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time, whereas distributed algorithms utilize multiple machines connected with a computer network. Parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. The resource consumption in such algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Some sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable. Some problems have no parallel algorithms and are called inherently serial problems.
; Deterministic or non-deterministic
: Deterministic algorithms solve the problem with exact decision at every step of the algorithm whereas non-deterministic algorithms solve problems via guessing although typical guesses are made more accurate through the use of heuristics.
; Exact or approximate
: While many algorithms reach an exact solution, approximation algorithms seek an approximation that is closer to the true solution. The approximation can be reached by either using a deterministic or a random strategy. Such algorithms have practical value for many hard problems. One of the examples of an approximate algorithm is the Knapsack problem, where there is a set of given items. Its goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. Total weight that can be carried is no more than some fixed number X. So, the solution must consider weights of items as well as their value.
; Quantum algorithm
: They run on a realistic model of quantum computation. The term is usually used for those algorithms which seem inherently quantum, or use some essential feature of Quantum computing such as quantum superposition or quantum entanglement.
By design paradigm

Another way of classifying algorithms is by their design methodology or algorithmic paradigm, paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories includes many different types of algorithms. Some common paradigms are: ; Brute force search, Brute-force or exhaustive search : This is the Naïve algorithm, naive method of trying every possible solution to see which is best. ; Divide and conquer : A divide and conquer algorithm repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually recursion, recursively) until the instances are small enough to solve easily. One such example of divide and conquer is mergesort, merge sorting. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in the conquer phase by merging the segments. A simpler variant of divide and conquer is called a ''decrease and conquer algorithm'', which solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms. An example of a decrease and conquer algorithm is the binary search algorithm. ; Search and enumeration : Many problems (such as playing chess) can be modeled as problems on graph theory, graphs. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. This category also includes search algorithms, branch and bound enumeration and backtracking. ;Randomized algorithm : Such algorithms make some choices randomly (or pseudo-randomly). They can be very useful in finding approximate solutions for problems where finding exact solutions can be impractical (see heuristic method below). For some of these problems, it is known that the fastest approximations must involve some randomness. Whether randomized algorithms with P (complexity), polynomial time complexity can be the fastest algorithms for some problems is an open question known as the P versus NP problem. There are two large classes of such algorithms: # Monte Carlo algorithms return a correct answer with high-probability. E.g. RP (complexity), RP is the subclass of these that run in polynomial time. # Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. Zero-error Probabilistic Polynomial time, ZPP. ; Reduction (complexity), Reduction of complexity : This technique involves solving a difficult problem by transforming it into a better-known problem for which we have (hopefully) asymptotically optimal algorithms. The goal is to find a reducing algorithm whose Computational complexity theory, complexity is not dominated by the resulting reduced algorithm's. For example, one selection algorithm for finding the median in an unsorted list involves first sorting the list (the expensive portion) and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as ''Transform and conquer algorithm, transform and conquer''. ; Back tracking : In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution.Optimization problems

For optimization problems there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following: ; Linear programming : When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular simplex algorithm. Problems that can be solved with linear programming include the maximum flow problem for directed graphs. If a problem additionally requires that one or more of the unknowns must be an integer then it is classified in integer programming. A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway. In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem. ; Dynamic programming : When a problem shows optimal substructures—meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems—and overlapping subproblems, meaning the same subproblems are used to solve many different problem instances, a quicker approach called ''dynamic programming'' avoids recomputing solutions that have already been computed. For example, Floyd–Warshall algorithm, the shortest path to a goal from a vertex in a weighted graph (discrete mathematics), graph can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and memoization go together. The main difference between dynamic programming and divide and conquer is that subproblems are more or less independent in divide and conquer, whereas subproblems overlap in dynamic programming. The difference between dynamic programming and straightforward recursion is in caching or memoization of recursive calls. When subproblems are independent and there is no repetition, memoization does not help; hence dynamic programming is not a solution for all complex problems. By using memoization or maintaining a Mathematical table, table of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity. ; The greedy method : A greedy algorithm is similar to a dynamic programming algorithm in that it works by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution, which may be given or have been constructed in some way, and improve it by making small modifications. For some problems they can find the optimal solution while for others they stop at local optimum, local optima, that is, at solutions that cannot be improved by the algorithm but are not optimum. The most popular use of greedy algorithms is for finding the minimal spanning tree where finding the optimal solution is possible with this method. Huffman coding, Huffman Tree, kruskal's algorithm, Kruskal, Prim's algorithm, Prim, Sollin's algorithm, Sollin are greedy algorithms that can solve this optimization problem. ;The heuristic method :In optimization problems, heuristic algorithms can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical. These algorithms work by getting closer and closer to the optimal solution as they progress. In principle, if run for an infinite amount of time, they will find the optimal solution. Their merit is that they can find a solution very close to the optimal solution in a relatively short time. Such algorithms include local search (optimization), local search, tabu search, simulated annealing, and genetic algorithms. Some of them, like simulated annealing, are non-deterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the non-optimal solution is known, the algorithm is further categorized as an approximation algorithm.By field of study

Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical analysis, numerical algorithms, graph theory, graph algorithms, string algorithms, computational geometry, computational geometric algorithms, combinatorial, combinatorial algorithms, medical algorithms, machine learning, cryptography, data compression algorithms and parsing, parsing techniques. Fields tend to overlap with each other, and algorithm advances in one field may improve those of other, sometimes completely unrelated, fields. For example, dynamic programming was invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields.By complexity

Algorithms can be classified by the amount of time they need to complete compared to their input size: * Constant time: if the time needed by the algorithm is the same, regardless of the input size. E.g. an access to an Array data structure, array element. * Logarithmic time: if the time is a logarithmic function of the input size. E.g. binary search algorithm. * Linear time: if the time is proportional to the input size. E.g. the traverse of a list. * Polynomial time: if the time is a power of the input size. E.g. the bubble sort algorithm has quadratic time complexity. * Exponential time: if the time is an exponential function of the input size. E.g. Brute-force search. Some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them.Continuous algorithms

The adjective "continuous" when applied to the word "algorithm" can mean: * An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations—such algorithms are studied in numerical analysis; or * An algorithm in the form of a differential equation that operates continuously on the data, running on an analog computer.Legal issues

Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), and hence algorithms are not patentable (as in Gottschalk v. Benson). However practical applications of algorithms are sometimes patentable. For example, in Diamond v. Diehr, the application of a simple feedback algorithm to aid in the curing of synthetic rubber was deemed patentable. The Software patent debate, patenting of software is highly controversial, and there are highly criticized patents involving algorithms, especially data compression algorithms, such as Unisys' Graphics Interchange Format#Unisys and LZW patent enforcement, LZW patent. Additionally, some cryptographic algorithms have export restrictions (see export of cryptography).History: Development of the notion of "algorithm"

Ancient Near East

The earliest evidence of algorithms is found in the Babylonian mathematics of ancient Mesopotamia (modern Iraq). A Sumerian clay tablet found in Shuruppak nearBaghdad
Baghdad (; ar, بَغْدَاد ) is the capital of Iraq
Iraq ( ar, الْعِرَاق, translit=al-ʿIrāq; ku, عێراق, translit=Êraq), officially the Republic of Iraq ( ar, جُمْهُورِيَّة ٱلْعِرَاق '; ku, ...

and dated to circa 2500 BC described the earliest division algorithm
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software.
Divisi ...

. During the First Babylonian dynasty, Hammurabi dynasty circa 1800-1600 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 circa 1550 BC. Algorithms were later used in ancient Hellenistic mathematics. Two examples are the Sieve of Eratosthenes, which was described in the ''Introduction to Arithmetic'' by Nicomachus, and the Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...

, which was first described in ''Euclid's Elements'' (c. 300 BC).
Discrete and distinguishable symbols

Tally-marks: To keep track of their flocks, their sacks of grain and their money the ancients used tallying: accumulating stones or marks scratched on sticks or making discrete symbols in clay. Through the Babylonian and Egyptian use of marks and symbols, eventually Roman numerals and the abacus evolved (Dilson, p. 16–41). Tally marks appear prominently in unary numeral system arithmetic used inTuring machine
A Turing machine is a mathematical model of computation
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are ...

and Post–Turing machine computations.
Manipulation of symbols as "place holders" for numbers: algebra

Muhammad ibn Mūsā al-Khwārizmī, a Mathematics in medieval Islam, Persian mathematician, wrote the ''Al-jabr'' in the 9th century. The terms "algorism
Algorism is the technique of performing basic arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Anci ...

" and "algorithm" are derived from the name al-Khwārizmī, while the term "algebra" is derived from the book ''Al-jabr''. In Europe, the word "algorithm" was originally used to refer to the sets of rules and techniques used by Al-Khwarizmi to solve algebraic equations, before later being generalized to refer to any set of rules or techniques. This eventually culminated in Gottfried Leibniz, Leibniz's notion of the calculus ratiocinator (ca 1680):
Cryptographic algorithms

The firstcryptographic
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or '' -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of third ...

algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Mathematics in medieval Islam, Arab mathematician, in ''A Manuscript On Deciphering Cryptographic Messages''. He gave the first description of cryptanalysis by frequency analysis
In cryptanalysis, frequency analysis (also known as counting letters) is the study of the letter frequencies, frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers.
Frequency analy ...

, the earliest codebreaking algorithm.
Mechanical contrivances with discrete states

''The clock'': Bolter credits the invention of the weight-driven clock as "The key invention [of Europe in the Middle Ages]", in particular, the verge escapement that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata theory, 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. ''Logical machines 1870 – Stanley Jevons' "logical abacus" and "logical machine"'': The technical problem was to reduce Boolean equations when presented in a form similar to what is now known as Karnaugh maps. Jevons (1880) describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the [logical] combinations can be picked out mechanically ... More recently, however, I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a ''Logical Machine''" His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano [etc] ...". With this machine he could analyze a "syllogism or any other simple logical argument". This machine he displayed in 1870 before the Fellows of the Royal Society. Another logician John Venn, however, in his 1881 ''Symbolic Logic'', turned a jaundiced eye to this effort: "I have no high estimate myself of the interest or importance of what are sometimes called logical machines ... it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines"; see more at Algorithm characterizations. But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's ''abacus'' ... [And] [a]gain, corresponding to Prof. Jevons's logical machine, the following contrivance may be described. I prefer to call it merely a logical-diagram machine ... but I suppose that it could do very completely all that can be rationally expected of any logical machine". ''Jacquard loom, Hollerith punch cards, telegraphy and telephony – the electromechanical relay'': 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 (ca 1870s) was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the teleprinter (ca. 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". Davis (2000) observes the particular importance of the electromechanical relay (with its two "binary states" ''open'' and ''closed''): : It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had envisioned."Mathematics during the 19th century up to the mid-20th century

''Symbols and rules'': In rapid succession, the mathematics of George Boole (1847, 1854), Gottlob Frege (1879), and Giuseppe Peano (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's ''The principles of arithmetic, presented by a new method'' (1888) was "the first attempt at an axiomatization of mathematics in a Symbolic language (programming), symbolic language". But Heijenoort gives Frege (1879) this kudos: Frege's is "perhaps the most important single work ever written in logic. ... in which we see a " 'formula language', that is a ''lingua characterica'', a language written with special symbols, "for pure thought", that is, free from rhetorical embellishments ... constructed from specific symbols that are manipulated according to definite rules". The work of Frege was further simplified and amplified by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (1910–1913). ''The paradoxes'': At the same time a number of disturbing paradoxes appeared in the literature, in particular, the Burali-Forti paradox (1897), the Russell paradox (1902–03), and the Richard Paradox. The resultant considerations led to Kurt Gödel's paper (1931)—he specifically cites the paradox of the liar—that completely reduces rules of recursion to numbers. ''Effective calculability'': In an effort to solve theEntscheidungsproblem
In mathematics and computer science, the ' (, German language, German for "decision problem") is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers ...

defined precisely by Hilbert in 1928, mathematicians first set about to define what was meant by an "effective method" or "effective calculation" or "effective calculability" (i.e., a calculation that would succeed). In rapid succession the following appeared: Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
The United States of America (US ...

, Stephen Kleene and J.B. Rosser's λ-calculus a finely honed definition of "general recursion" from the work of Gödel acting on suggestions of Jacques Herbrand (cf. Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene. Church's proof that the Entscheidungsproblem was unsolvable, Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was a Polish-born American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

's definition of effective calculability as a worker mindlessly following a list of instructions to move left or right through a sequence of rooms and while there either mark or erase a paper or observe the paper and make a yes-no decision about the next instruction. Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"—in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine". Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I", and a few years later Kleene's renaming his Thesis "Church's Thesis" and proposing "Turing's Thesis".
Emil Post (1936) and Alan Turing (1936–37, 1939)

Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was a Polish-born American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

(1936) described the actions of a "computer" (human being) as follows:
:"...two concepts are involved: that of a ''symbol space'' in which the work leading from problem to answer is to be carried out, and a fixed unalterable ''set of directions''.
His symbol space would be
:"a two-way infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.
:"One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes [i.e., INPUT] being marked with a stroke. Likewise, the answer [i.e., OUTPUT] is to be given in symbolic form by such a configuration of marked boxes...
:"A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process terminates only when it comes to the direction of type (C ) [i.e., STOP]". See more at Post–Turing machine
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...

's work preceded that of Stibitz (1937); it is unknown whether Stibitz knew of the work of Turing. Turing's biographer believed that Turing's use of a typewriter-like model derived from a youthful interest: "Alan had dreamt of inventing typewriters as a boy; Mrs. Turing had a typewriter, and he could well have begun by asking himself what was meant by calling a typewriter 'mechanical'". Given the prevalence of Morse code and telegraphy, ticker tape machines, and teletypewriters we might conjecture that all were influences.
Turing—his model of computation is now called a J.B. Rosser (1939) and S.C. Kleene (1943)

J. Barkley Rosser defined an 'effective [mathematical] method' in the following manner (italicization added): :"'Effective method' is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a finite number of steps. With this special meaning, three different precise definitions have been given to date. [his footnote #5; see discussion immediately below]. The simplest of these to state (due to Post and Turing) says essentially that ''an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer''. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." (Rosser 1939:225–226) Rosser's footnote No. 5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular Church's use of it in his ''An Unsolvable Problem of Elementary Number Theory'' (1936); (2) Herbrand and Gödel and their use of recursion in particular Gödel's use in his famous paper ''On Formally Undecidable Propositions of Principia Mathematica and Related Systems I'' (1931); and (3) Post (1936) and Turing (1936–37) in their mechanism-models of computation. Stephen C. Kleene defined as his now-famous "Thesis I" known as theChurch–Turing thesis
In Computability theory (computation), computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) i ...

. But he did this in the following context (boldface in original):
:"12. ''Algorithmic theories''... In setting up a complete algorithmic theory, what we do is to describe a procedure, performable for each set of values of the independent variables, which procedure necessarily terminates and in such manner that from the outcome we can read a definite answer, "yes" or "no," to the question, "is the predicate value true?"" (Kleene 1943:273)
History after 1950

A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is on-going because of issues surrounding, in particular, foundations of mathematics (especially theChurch–Turing thesis
In Computability theory (computation), computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) i ...

) and philosophy of mind (especially arguments about artificial intelligence). For more, see Algorithm characterizations.
See also

* Abstract machine * Algorithm engineering * Algorithm characterizations * Algorithmic composition * Algorithmic entities * Algorithmic synthesis * Algorithmic technique * Algorithmic topology * Garbage in, garbage out * ''Introduction to Algorithms'' (textbook) * List of algorithms * List of algorithm general topics * List of important publications in theoretical computer science#Algorithms, List of important publications in theoretical computer science – Algorithms * Regulation of algorithms * Theory of computation ** Computability theory ** Computational complexity theoryNotes

Bibliography

* * Bell, C. Gordon and Newell, Allen (1971), ''Computer Structures: Readings and Examples'', McGraw–Hill Book Company, New York. . * Includes an excellent bibliography of 56 references. * , * : cf. Chapter 3 ''Turing machines'' where they discuss "certain enumerable sets not effectively (mechanically) enumerable". * * Campagnolo, M.L., Cris Moore, Moore, C., and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In ''Proc. of the 4th Conference on Real Numbers and Computers'', Odense University, pp. 91–109 * Reprinted in ''The Undecidable'', p. 89ff. The first expression of "Church's Thesis". See in particular page 100 (''The Undecidable'') where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc. * Reprinted in ''The Undecidable'', p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes. * * Davis gives commentary before each article. Papers of Gödel,Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American
American(s) may refer to:
* American, something of, from, or related to the United States of America, commonly known as the United States
The United States of America (US ...

, Alan Turing, Turing, J. Barkley Rosser, Rosser, Kleene, and ''Sequential Abstract State Machines Capture Sequential Algorithms''

ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources. * , 3rd edition 1976[?], (pbk.) * , . Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof. * Presented to the American Mathematical Society, September 1935. Reprinted in ''The Undecidable'', p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) was used by Church in his 1935 paper ''An Unsolvable Problem of Elementary Number Theory'' that proved the "decision problem" to be "undecidable" (i.e., a negative result). * Reprinted in ''The Undecidable'', p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in Kleene 1952:300) and name it "Church's Thesis"(Kleene 1952:317) (i.e., the Church thesis). * * * * Kosovsky, N.K. ''Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms'', LSU Publ., Leningrad, 1981 * * A.A. Markov (1954) ''Theory of algorithms''. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS .] * Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 ''Computability, Effective Procedures and Algorithms. Infinite machines.'' * Reprinted in ''The Undecidable'', pp. 289ff. Post defines a simple algorithmic-like process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the so-called

''2106.02 **>Mathematical Algorithms: 2100 Patentability''

Manual of Patent Examining Procedure (MPEP). Latest revision August 2006

Further reading

* * * * * * * Donald Knuth, Knuth, Donald E. (2000).Selected Papers on Analysis of Algorithms

'. Stanford, California: Center for the Study of Language and Information. * Knuth, Donald E. (2010).

'. Stanford, California: Center for the Study of Language and Information. *

External links

* * *Dictionary of Algorithms and Data Structures

– National Institute of Standards and Technology ; Algorithm repositories

The Stony Brook Algorithm Repository

– State University of New York at Stony Brook

Collected Algorithms of the ACM

– Association for Computing Machinery

The Stanford GraphBase

– Stanford University {{Authority control Algorithms, Articles with example pseudocode Mathematical logic Theoretical computer science