As an effective method, an algorithm can be expressed within a finite amount of space and time"Any classical mathematical algorithm, for example, can be described in a finite number of English words" (Rogers 1987:2). and in a well-defined formal languageWell defined with respect to the agent that executes the algorithm: "There is a computing agent, usually human, which can react to the instructions and carry out the computations" (Rogers 1987:2). for calculating a function."an algorithm is a procedure for computing a function (with respect to some chosen notation for integers) ... this limitation (to numerical functions) results in no loss of generality", (Rogers 1987:1). Starting from an initial state and initial input (perhaps Empty string),"An algorithm has zero or more inputs, i.e., quantity which are given to it initially before the algorithm begins" (Knuth 1973:5). the instructions describe a computation that, when executed, proceeds through a finite"A procedure which has all the characteristics of an algorithm except that it possibly lacks finiteness may be called a 'computational method'" (Knuth 1973:5). number of well-defined successive states, eventually producing "output""An algorithm has one or more outputs, i.e. quantities which have a specified relation to the inputs" (Knuth 1973:5). and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.Whether or not a process with random interior processes (not including the input) is an algorithm is debatable. Rogers opines that: "a computation is carried out in a discrete stepwise fashion, without use of continuous methods or analogue devices . . . carried forward deterministically, without resort to random methods or devices, e.g., dice" Rogers 1987:2.
The concept of algorithm has existed for centuries and the use of the concept can be ascribed to Greek mathematicians, e.g. the sieve of Eratosthenes and Euclid's algorithm;
Roger L. Cooke (2018). 9781118460290, John Wiley & Sons. ISBN 9781118460290
the term algorithm itself derives from the 9th Century mathematician Muḥammad ibn Mūsā al'Khwārizmī, latinized 'Algoritmi'. A partial formalization of what would become the modern notion of algorithm began with attempts to solve the Entscheidungsproblem (the "decision problem") posed by David Hilbert in 1928. Subsequent formalizations were framed as attempts to define "effective calculability"Kleene 1943 in Davis 1965:274 or "effective method";Rosser 1939 in Davis 1965:225 those formalizations included the Gödel–Jacques Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, and Alan Turing's Turing machines of 1936–7 and 1939.
The word 'algorithm' has its roots in latinizing the name of Muhammad ibn Musa al-Khwarizmi in a first step to algorismus. Al-Khwārizmī (خوارزمی, c. 780–850) was a Persian people mathematician, astronomer, geographer, and scholar in the House of Wisdom in Baghdad, whose name means 'the native of Khwarezm', a region that was part of Greater Iran and is now in Uzbekistan.
About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, which was translated into Latin during the 12th century under the title Algoritmi de numero Indorum. This title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
Corona Brezina (2018). 9781404205130, The Rosen Publishing Group. . ISBN 9781404205130
Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through another of his books, the Al-Jabr. Foremost mathematical texts in history, according to Carl B. Boyer. In late medieval Latin, algorismus, English 'algorism', the corruption of his name, simply meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός '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.Oxford English Dictionary, Third Edition, 2012 s.v.
In English, it was first used in about 1230 and then by Geoffrey Chaucer 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. It begins thus:
which translates as:
The poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
An informal definition could be "a set of rules that precisely defines a sequence of operations."Stone 1973:4 which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually.Stone simply requires that "it must terminate in a finite number of steps" (Stone 1973:7–8).
A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers; an example (there are others) is described by the flowchart above and as an example in a later section.
offer an informal meaning of the word 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.Boolos and Jeffrey 1974,1999:19
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. Thus an algorithm can be an algebraic equation such as y = m + n – 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 language understood by "the computer")cf Stone 1972:5 for a fast, efficient, "good"Knuth 1973:7 states: "In practice we not only want algorithms, we want good algorithms ... one criterion of goodness is the length of time taken to perform the algorithm ... other criteria are the adaptability of the algorithm to computers, its simplicity and elegance, etc." process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities)cf Stone 1973:6 to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and = ... and "effectively"Stone 1973:7–8 states that there must be, "...a procedure that a robot i.e., can follow in order to determine precisely how to obey the instruction." Stone adds finiteness of the process, and definiteness (having no ambiguity in the instructions) to this definition. produce, in a "reasonable" time,Knuth, loc. cit 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. That notion is central for explaining how come into being starting from a small set of and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our 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.
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 a Turing reduction 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 Savage [1987], an algorithm is a computational process defined by a Turing machine".Gurevich 2000:1, 3
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 .
For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, 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 control flow.
So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, setting the value of a variable. It derives from the intuition of "memory" as a scratchpad. There is an example below of such an assignment.
For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming.
Algorithms can be expressed in many kinds of notation, including , pseudocode, , DRAKON, programming languages or (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, DRAKON and control tables are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are 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 program as a sequence of machine tables (see more at finite-state machine, state transition table and control table), as flowcharts and DRAKON (see more at state diagram), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see more at Turing machine).
Representations of algorithms can be classed into three accepted levels of Turing machine description:Sipser 2006:157
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.
Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.
In computer systems, an algorithm is basically an instance of logic 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) 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:
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"Knuth 1973:7
Chaitin: " . . . a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"Chaitin 2005:32
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 (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".Rogers 1987:1–2
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"In his essay "Calculations by Man and Machine: Conceptual Analysis" Seig 2002:390 credits this distinction to Robin Gandy, cf Wilfred Seig, et al., 2002 Reflections on the foundations of mathematics: Essays in honor of Solomon Feferman, Association for Symbolic Logic, A. K Peters Ltd, Natick, MA.) is a restricted type of machine, a "discrete deterministic mechanical device"cf Gandy 1980:126, Robin Gandy Church's Thesis and Principles for Mechanisms appearing on pp. 123–148 in Jon Barwise et al. 1980 The Kleene Symposium, North-Holland Publishing Company. that blindly follows its instructions.A "robot": "A computer is a robot that performs any task that can be described as a sequence of instructions." cf Stone 1972:3 Melzak's and Lambek's primitive modelsLambek's "abacus" is a "countably infinite number of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads, etc). The locations are distinguishable, the counters are not". The holes have unlimited capacity, and standing by is an agent who understands and is able to carry out the list of instructions" (Lambek 1961:295). Lambek references Melzak who defines his Q-machine as "an indefinitely large number of locations . . . an indefinitely large supply of counters distributed among these locations, a program, and an operator whose sole purpose is to carry out the program" (Melzak 1961:283). B-B-J (loc. cit.) add the stipulation that the holes are "capable of holding any number of stones" (p. 46). Both Melzak and Lambek appear in The Canadian Mathematical Bulletin, vol. 4, no. 3, September 1961. reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable countersIf no confusion results, the word "counters" can be dropped, and a location can be said to contain a single "number". (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent."We say that an instruction is effective if there is a procedure that the robot can follow in order to determine precisely how to obey the instruction." (Stone 1972:6)
Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability".cf Minsky 1967: Chapter 11 "Computer models" and Chapter 14 "Very Simple Bases for Computability" pp. 255–281 in particularMinsky 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)cf Knuth 1973:3. 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).But always preceded by IF–THEN to avoid improper subtraction. 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 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".Knuth 1973:4 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.Stone 1972:5. Methods for extracting roots are not trivial: see Methods of computing square roots.
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, any algorithm can be computed by a model known to be Turing complete, 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", 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".John G. Kemeny and Thomas E. Kurtz 1985 Back to Basic: The History, Corruption, and Future of the Language, Addison-Wesley Publishing Company, Inc. Reading, MA, . Tausworthe augments the three Böhm-Jacopini canonical structures:Tausworthe 1977:101 SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.Tausworthe 1977:142 An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction.Knuth 1973 section 1.2.1, expanded by Tausworthe 1977 at pages 100ff and Chapter 9.1
Canonical flowchart symbolscf Tausworthe 1977: The graphical aide called a flowchart offers a way to describe and document an algorithm (and a computer program of one). Like 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.
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 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 in pseudocode or pidgin code:
Input: A list of numbers ''L''.
Output: The largest number in the list ''L''.
Euclid's algorithm to compute the greatest common divisor (GCD) to two numbers appears as Proposition II in Book VII ("Elementary Number Theory") of his Elements.Heath 1908:300; Hawking's Dover 2005 edition derives from Heath. Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number to 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." 'Let CD, measuring BF, leave FA less than itself.' This is a neat abbreviation for saying, measure along BA successive lengths equal to CD until a point F is reached such that the length FA remaining is less than CD; in other words, let BF be the largest exact multiple of CD contained in BA" (Heath 1908:297) 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 modern treatments using division in the algorithm, see Hardy and Wright 1979:180, Knuth 1973:2 (Volume 1), plus more discussion of Euclid's algorithm in Knuth 1969:293–297 (Volume 2).
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 (alternately, 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.Euclid covers this question in his Proposition 1. 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.
[[File:Euclids-algorithm-example-1599-650.gif|350px|thumb|right|A graphical expression of Euclid's algorithm to find the greatest common divisor for 1599 and 650.
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.
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
[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, 5 and 6:
GOTO step 6
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: 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
ELSE
measure again,
R ← R − S
[Remainder-loop]:
GOTO 7.
E2: Is: 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 step 15
ELSE
CONTINUE TO step 11,
E3: Interchange: 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
OUTPUT:
[Done. S contains the greatest common divisor]:
PRINT S
The following version of Euclid's algorithm requires only six core instructions to do what thirteen are required to do by "Inelegant"; worse, "Inelegant" requires more types of instructions. The flowchart of "Elegant" can be found at the top of this article. In the (unstructured) Basic language, the steps are numbered, and the instruction