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{{other uses}}{{short description|unambiguous specification of how to solve a class of problems}}{{Use mdy dates|date=September 2017}}
missing image!
- Euclid flowchart.svg|thumb|right| Flowchart of an algorithm ((Euclid's algorithm]]) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B â‰¥ A yields "yes" (or true) (more accurately the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B â† B âˆ’ A (meaning the number b âˆ’ a replaces the old b). Similarly, IF A > B, THEN A â† A âˆ’ B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).)File:Diagram for the computation of Bernoulli numbers.jpg -
In mathematics and computer science, an algorithm ({{IPAc-en|audio=en-us-algorithm.ogg|Ëˆ|Ã¦|l|É¡|É™|r|Éª|Ã°|É™m}}) is a sequence of instructions, typically to solve a class of problems or perform a computation. Algorithms are unambiguous specifications for performing calculation, data processing, automated reasoning, and other tasks.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),"An algorithm has zero or more inputs, i.e., quantities 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 the 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. Greek mathematicians used algorithms in the sieve of Eratosthenes for finding prime numbers, and the Euclidean algorithm for finding the greatest common divisor of two numbers.BOOK, Cooke, Roger L., The History of Mathematics: A Brief Course, 2005, John Wiley & Sons, 978-1-118-46029-0, The word algorithm itself is derived from the 9th-century mathematician Muá¸¥ammad ibn MÅ«sÄ al-KhwÄrizmÄ«, Latinized Algoritmi. A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem (decision problem) posed by David Hilbert in 1928. Later 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 GÃ¶delâ€“Herbrandâ€“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 1936â€“37 and 1939.

Informal definition

{{about||a detailed presentation of the various points of view on the definition of "algorithm"| Algorithm characterizations}}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, and (for example) any prescribed bureaucratic procedure.BOOK
, Simanowski
, Roberto
, Roberto Simanowski
, Chase
, Jefferson
, The Death Algorithm and Other Digital Dilemmas
, Untimely Meditations
, 14
, Cambridge, Massachusetts
, MIT Press
, 2018
, 147
, 9780262536370
, 27 May 2019
, [...] the next level of abstraction of central bureaucracy: globally operating algorithms.
, 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 1973:7â€“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.{{Harvtxt|Boolos|Jeffrey|1974, 1999|ref=CITEREFBoolosJeffrey1999}} offer an informal meaning of the word in the following quotation:No human being can write fast enough, or long enough, or small enoughâ€  ( â€ "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 1974,1999:19An "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 â€“ 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 1973:7â€“8 states that there must be, "...a procedure that a robot [i.e., computer] 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 formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related to 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.

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 a Turing-complete 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".{{harvnb|Minsky|1967|page=105}}

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 , an algorithm is a computational process defined by a Turing machine".Gurevich 2000:1, 3
Turing 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. This is because of a major theorem of Computability Theory known as the Halting Problem.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 structures.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 flow of control.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.

Expressing algorithms

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, drakon-charts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts, drakon-charts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in 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-charts (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.

Design

{{See also|Algorithm#By design paradigm}}Algorithm design refers to a method or mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories of operation research, such as dynamic programming and divide-and-conquer. Techniques for designing and implementing algorithm designs are also called algorithm design patterns,{{citation|url=http://ww3.algorithmdesign.net/ch00-front.html|title=Algorithm Design: Foundations, Analysis, and Internet Examples|last1=Goodrich|first1=Michael T.|author1-link=Michael T. Goodrich|last2=Tamassia|first2=Roberto|author2-link=Roberto Tamassia|publisher=John Wiley & Sons, Inc.|year=2002|isbn=978-0-471-38365-9}} such as the template method pattern and decorator pattern.One of the most important aspects of algorithm design is creating an algorithm that has an efficient run-time, also known as its Big O.Typical steps in the development of algorithms:
1. Problem definition
2. Development of a model
3. Specification of the algorithm
4. Designing an algorithm
5. Checking the correctness of the algorithm
6. Analysis of algorithm
7. Implementation of algorithm
8. Program testing
9. Documentation preparation

Implementation

missing image!
- TTL npn nand.svg|right|thumb|Logical NAND algorithm implemented electronically in 7400 chip]]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.

Computer algorithms

File:Euclid's algorithm structured blocks 1.png
-
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

Examples

{{further|List of algorithms}}

Algorithm example

File:Sorting quicksort anim.gif|thumb|right|An animation of the quicksort algorithm sorting an array of randomized values. The red bars mark the pivot element; at the start of the animation, the element farthest to the right-hand side is chosen as the pivot.]]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:
1. If there are no numbers in the set then there is no highest number.
2. Assume the first number in the set is the largest number in the set.
3. 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.
4. 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:{{algorithm-begin|name=LargestNumber}}
Input: A list of numbers L.
Output: The largest number in the list L.

if L.size = 0 return null
largest â† L
for each item in L, do
if item > largest, then
largest â† item
return largest
{{algorithm-end}}

Euclid's algorithm

{{further|Euclid's algorithm}}(File:Euclid's algorithm Book VII Proposition 2 2.png|250px|thumb|left|The example-diagram of Euclid's algorithm from T.L. Heath (1908), with more detail added. Euclid does not go beyond a third measuring and gives no numerical examples. Nicomachus gives the example of 49 and 21: "I subtract the less from the greater; 28 is left; then again I subtract from this the same 21 (for this is possible); 7 is left; I subtract this from 21, 14 is left; from which I again subtract 7 (for this is possible); 7 is left, but 7 cannot be subtracted from 7." Heath comments that "The last phrase is curious, but the meaning of it is obvious enough, as also the meaning of the phrase about ending 'at one and the same number'."(Heath 1908:300).)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 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." '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 âˆ’ qÃ—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 1969:293â€“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 â€œproperâ€; 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.

Computer language for Euclid's algorithm

Only a few instruction types are required to execute Euclid's algorithmâ€”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

(File:Euclid's algorithm Inelegant program 1.png|thumb|163px|right|"Inelegant" is a translation of Knuth's version of the algorithm with a subtraction-based remainder-loop replacing his use of division (or a "modulus" instruction). Derived from Knuth 1973:2â€“4. Depending on the two numbers "Inelegant" may compute the g.c.d. in fewer steps than "Elegant".)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 1973:2â€“4:INPUT:
{{vanchor|1|el1}} [Into two locations L and S put the numbers l and s that represent the two lengths]:
INPUT L, S
{{vanchor|2|el2}} [Initialize R: make the remaining length r equal to the starting/initial/input length l]:
R â† L
E0: [Ensure r â‰¥ s.]
{{vanchor|3|el3}} [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.
{{vanchor|4|el4}} L â† R (this first step is redundant, but is useful for later discussion).
{{vanchor|5|el5}} R â† S
{{vanchor|6|el6}} S â† 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.
{{vanchor|7|el7}} IF S > R THEN
done measuring so
GOTO 10
ELSE
measure again,
{{vanchor|8|el8}} R â† R âˆ’ S
{{vanchor|9|el9}} [Remainder-loop]:
GOTO 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.
{{vanchor|10|el10}} IF R = 0 THEN
done so
GOTO step 15
ELSE
CONTINUE TO 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.
{{vanchor|11|el11}} L â† R
{{vanchor|12|el12}} R â† S
{{vanchor|13|el13}} S â† L
{{vanchor|14|el14}} [Repeat the measuring process]:
GOTO 7
OUTPUT:
{{vanchor|15|el15}} [Done. S contains the greatest common divisor]:
PRINT S
DONE:
{{vanchor|16|el16}} HALT, END, STOP.

An elegant program for Euclid's algorithm

{{clarify span|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.|reason=Contrary this (unsourced) text, it is general consensus in software engineering that good programming style does not aim at minimizing the number of 'core' (whatever that means) instructions or instruction types. Software engineers during the entire recent 50 years would have ostracized the shown program, if only due to its extensive use of 'goto'. While the program should not be removed here, it could serve as an example of poor programming style, leading to hard-to-understand source code.|date=March 2019}} 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 LET [] = [] is the assignment instruction symbolized by â†.
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 â† A âˆ’ B, and a B â‰¤ A loop that computes B â† B âˆ’ A. This works because, when at last the minuend M is less than or equal to the subtrahend S ( Difference = Minuend âˆ’ 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 Object Oriented languages:// Euclid's algorithm for greatest common divisorint euclidAlgorithm (int A, int B){
A=Math.abs(A);
B=Math.abs(B);
while (B!=0){
if (A>B) A=A-B;
else B=B-A;
}
return A;
}

Testing 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 sourceWEB,weblink Euclid's Elements, Book VII, Proposition 2, Aleph0.clarku.edu, May 20, 2012, uses 3009 and 884. Knuth suggested 40902, 24140. Another interesting case is the two relatively prime numbers 14157 and 5950.But "exceptional cases"While this notion is in widespread use, it cannot be defined precisely. 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 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 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".Knuth 1973:13â€“18. He credits "the formulation of algorithm-proving in terms of assertions and induction" to R W. Floyd, Peter Naur, C.A.R. Hoare, H.H. Goldstine and J. von Neumann. Tausworth 1977 borrows Knuth's Euclid example and extends Knuth's method in section 9.1 Formal Proofs (pp. 288â€“298). Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof.Tausworthe 1997:294

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 analysiscf Knuth 1973:7 (Vol. I), and his more-detailed analyses on pp. 1969:294â€“313 (Vol II). 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"â€”that is, it computes the function intended by its authorâ€”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;Breakdown occurs when an algorithm tries to compact itself. Success would solve the Halting problem. 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 the big O notation 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 'effort' than others. For example, a binary search algorithm (with cost O(log n) ) outperforms a sequential search (cost O(n) ) when used for table lookups on sorted lists or arrays.

Formal versus empirical

The analysis, and study of algorithms is a discipline of computer science, and is often practiced abstractly without the use of a specific programming language 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 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. 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.JOURNAL, Kriegel, Hans-Peter, Hans-Peter Kriegel, Schubert, Erich, Zimek, Arthur, Arthur Zimek, The (black) art of run-time evaluation: Are we comparing algorithms or implementations?, Knowledge and Information Systems, 52, 2, 2016, 341â€“378, 0219-1377, 10.1007/s10115-016-1004-2,

Execution efficiency

To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to FFT algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging.WEB, Better Math Makes Faster Data Networks, Gillian Conahan, January 2013,weblink discovermagazine.com, 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) {{webarchive|url=https://web.archive.org/web/20130704180806weblink |date=July 4, 2013 }}, Kyoto, January 2012. See also the 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.{| style="float:right; width:200pt;"| int gcd(int A, int B) {
if (B == 0)
return A;
else if (A > B)
return gcd(A-B,B);
else
return gcd(A,B-A);
}
C (programming language)>C implementation of Euclid's algorithm from the above flowchart
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 to functional programming. Iterative algorithms use repetitive constructs like loops and sometimes additional data structures like 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 logical deduction. This notion may be expressed as: Algorithm = logic + control.Kowalski 1979 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 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 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.BOOK,weblink Knapsack Problems {{!, Hans Kellerer {{!}} Springer|language=en|isbn=978-3-540-40286-2|publisher=Springer|year=2004}}
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.

Another way of classifying algorithms is by their design methodology or 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 or exhaustive search
This is the naive method of trying every possible solution to see which is best.BOOK, Carroll, Sue, Daughtrey, Taz, Fundamental Concepts for the Software Quality Engineer,weblink July 4, 2007, American Society for Quality, 978-0-87389-720-4, 282 et seq,
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 recursively) until the instances are small enough to solve easily. One such example of divide and conquer is 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, that 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 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.For instance, the volume of a convex polytope (described using a membership oracle) can be approximated to high accuracy by a randomized polynomial time algorithm, but not by a deterministic one: see {{citation
| last1 = Dyer | first1 = Martin
| last2 = Frieze | first2 = Alan
| last3 = Kannan | first3 = Ravi
| date = January 1991
| doi = 10.1145/102782.102783
| issue = 1
| journal = J. ACM
| pages = 1â€“17
| title = A Random Polynomial-time Algorithm for Approximating the Volume of Convex Bodies
| volume = 38| citeseerx = 10.1.1.145.4600}}. Whether randomized algorithms with 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:
1. Monte Carlo algorithms return a correct answer with high-probability. E.g. RP is the subclass of these that run in polynomial time.
2. Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. ZPP.

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 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.
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.
George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions. Springer-Verlag. 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 substructuresâ€”meaning the optimal solution to a problem can be constructed from optimal solutions to subproblemsâ€”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, Floydâ€“Warshall algorithm, the shortest path to a goal from a vertex in a weighted 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 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 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 Tree, Kruskal, Prim, 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, 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

{{See also|List of algorithms}}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 algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, medical algorithms, machine learning, cryptography, data compression algorithms and 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

{{See also|Complexity class| Parameterized 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 element.
• Linear time: if the time is proportional to the input size. E.g. the traverse of a list.
• Logarithmic time: if the time is a logarithmic function of the input size. E.g. binary search algorithm.
• 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 approximationsâ€”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.BOOK, Tsypkin, Adaptation and learning in automatic systems,weblink 1971, Academic Press, 978-0-08-095582-7, 54,

Legal issues

{{see also|Software patent}}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 patenting of software is highly controversial, and there are highly criticized patents involving algorithms, especially data compression algorithms, such as Unisys' 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 near Baghdad and dated to circa 2500 BC described the earliest division algorithm.BOOK, Chabert, Jean-Luc, A History of Algorithms: From the Pebble to the Microchip, 2012, Springer Science & Business Media, 9783642181924, 7â€“8, During the Hammurabi dynasty circa 1800-1600 BC, Babylonian clay tablets described algorithms for computing formulas.JOURNAL, Knuth, Donald E., Ancient Babylonian Algorithms, Commun. ACM, 1972, 15, 7, 671â€“677, 10.1145/361454.361514,weblink 0001-0782, 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.{{Citation | last = Aaboe | first = Asger | author-link = Asger Aaboe | date = 2001 | title = Episodes from the Early History of Astronomy | publisher = Springer | place = New York | pages = 40â€“62 | isbn = 978-0-387-95136-2 }}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,WEB,weblink Eratosthenes, Wichita State University: Department of Mathematics and Statistics, Courtney, Ast, {{rp|Ch 9.2}} and the Euclidean algorithm, which was first described in Euclid's Elements (c. 300 BC).{{rp|Ch 9.1}}

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. 16â€“41). Tally marks appear prominently in unary numeral system arithmetic used in Turing machine and Postâ€“Turing machine computations.

Manipulation of symbols as "place holders" for numbers: algebra

Muhammad ibn MÅ«sÄ al-KhwÄrizmÄ«, a Persian mathematician, wrote the Al-jabr in the 9th century. The terms "algorism" and "algorithm" are derived from the name al-KhwÄrizmÄ«, 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.BOOK, Chabert, Jean-Luc, A History of Algorithms: From the Pebble to the Microchip, 2012, Springer Science & Business Media, 9783642181924, 2, This eventually culminated in Leibniz's notion of the calculus ratiocinator (ca 1680):

Cryptographic algorithms

The first cryptographic algorithm for deciphering encrypted code was developed by Al-Kindi, a 9th-century Arab mathematician, in A Manuscript On Deciphering Cryptographic Messages. He gave the first description of cryptanalysis by frequency analysis, the earliest codebreaking algorithm.BOOK, Dooley, John F., A Brief History of Cryptology and Cryptographic Algorithms, 2013, Springer Science & Business Media, 9783319016283, 12â€“3,

Mechanical contrivances with discrete states

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."Davis 2000:14

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 (1888â€“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".van Heijenoort 1967:81ffBut 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".van Heijenoort's commentary on Frege's Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought in van Heijenoort 1967:1 The work of Frege was further simplified and amplified by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (1910â€“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 (1902â€“03), and the Richard Paradox.Dixon 1906, cf. Kleene 1952:36â€“40 The resultant considerations led to Kurt GÃ¶del's paper (1931)â€”he specifically cites the paradox of the liarâ€”that completely reduces rules of recursion to numbers.Effective calculability: In an effort to solve the Entscheidungsproblem 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, Stephen Kleene and J.B. Rosser's Î»-calculuscf. footnote in Alonzo Church 1936a in Davis 1965:90 and 1936b in Davis 1965:110 a finely honed definition of "general recursion" from the work of GÃ¶del acting on suggestions of Jacques Herbrand (cf. GÃ¶del's Princeton lectures of 1934) and subsequent simplifications by Kleene.Kleene 1935â€“6 in Davis 1965:237ff, Kleene 1943 in Davis 1965:255ff Church's proofChurch 1936 in Davis 1965:88ff that the Entscheidungsproblem was unsolvable, Emil Post'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.cf. "Finite Combinatory Processes â€“ formulation 1", Post 1936 in Davis 1965:289â€“290 Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"Turing 1936â€“37 in Davis 1965:116ffâ€”in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine".Rosser 1939 in Davis 1965:226 S.C. Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I",Kleene 1943 in Davis 1965:273â€“274 and a few years later Kleene's renaming his Thesis "Church's Thesis"Kleene 1952:300, 317 and proposing "Turing's Thesis".Kleene 1952:376

Emil Post (1936) and Alan Turing (1936â€“37, 1939)

Emil Post (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]".Turing 1936â€“37 in Davis 1965:289â€“290 See more at Postâ€“Turing machine
File:Alan Turing.jpg|thumb|200px|Alan Turing's statue at Bletchley ParkBletchley ParkAlan Turing's workTuring 1936 in Davis 1965, Turing 1939 in Davis 1965:160 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'".Hodges, p. 96 Given the prevalence of Morse code and telegraphy, ticker tape machines, and teletypewriters we{{Who|date=March 2017}} might conjecture that all were influences.Turingâ€”his model of computation is now called a Turing machineâ€”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.Turing 1936â€“37:116
"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 1936â€“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 GÃ¶del, 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 â€  with effective calculability ... .
"â€  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

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 1939:225â€“226)
Rosser's footnote No. 5 references the work of (1) Church and Kleene and their definition of Î»-definability, in particular Church's use of it in his An Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand and GÃ¶del and their use of recursion in particular GÃ¶del's use in his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931); and (3) Post (1936) and Turing (1936â€“37) in their mechanism-models of computation.Stephen C. Kleene defined as his now-famous "Thesis I" known as the Churchâ€“Turing thesis. 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 the Churchâ€“Turing thesis) and philosophy of mind (especially arguments about artificial intelligence). For more, see Algorithm characterizations.

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{{Reflist}}

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• BOOK, Hodges, Andrew, Andrew Hodges, Alan Turing: The Enigma, Physics Today, 37, 11, 107, 1983, Simon and Schuster, New York, 978-0-671-49207-6, Alan Turing: The Enigma, 1984PhT....37k.107H, 10.1063/1.2915935, , {{ISBN|0-671-49207-1}}. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
• JOURNAL, Kleene, Stephen C., Stephen Kleene, General Recursive Functions of Natural Numbers, Mathematische Annalen, 112, 727â€“742,weblink 1936, 10.1007/BF01565439, 5, September 30, 2013,weblink" title="web.archive.org/web/20140903092121weblink">weblink September 3, 2014, dead, 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).
• JOURNAL, Kleene, Stephen C., Stephen Kleene, Recursive Predicates and Quantifiers, American Mathematical Society Transactions, 54, 41â€“73, 1943, 10.2307/1990131, 1, 1990131, free, 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).
• BOOK, Kleene, Stephen C., Kleene, Introduction to Metamathematics, Tenth, 1991, 1952, North-Holland Publishing Company, 978-0-7204-2103-3,
• BOOK, Knuth, Donald, Donald Knuth, Fundamental Algorithms, Third Edition, 1997, Addisonâ€“Wesley, Reading, Massachusetts, 978-0-201-89683-1,
• BOOK, Knuth, Donald, Donald Knuth, Volume 2/Seminumerical Algorithms, The Art of Computer Programming First Edition, Addisonâ€“Wesley, Reading, Massachusetts, 1969,
• Kosovsky, N.K. Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms, LSU Publ., Leningrad, 1981
• JOURNAL, Kowalski, Robert, Robert Kowalski, Algorithm=Logic+Control, Communications of the ACM, 22, 7, 424â€“436, 1979, 10.1145/359131.359136,
• 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 60-51085.]
• BOOK, Minsky, Marvin, Marvin Minsky, Computation: Finite and Infinite Machines, First, 1967, Prentice-Hall, Englewood Cliffs, NJ, 978-0-13-165449-5, Minsky expands his "...idea of an algorithm â€“ an effective procedure..." in chapter 5.1 Computability, Effective Procedures and Algorithms. Infinite machines.
• JOURNAL, Post, Emil, Emil Post, Finite Combinatory Processes, Formulation I, The Journal of Symbolic Logic, 1, 1936, 103â€“105, 10.2307/2269031, 3, 2269031, 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 Churchâ€“Turing thesis.
• BOOK, Rogers, Jr, Hartley, Theory of Recursive Functions and Effective Computability, The MIT Press, 1987, 978-0-262-68052-3,
• JOURNAL, Rosser, J.B., J. B. Rosser, An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem, Journal of Symbolic Logic, 4, 2, 1939, 10.2307/2269059, 53â€“60, 2269059, 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. 225â€“226, The Undecidable)
• BOOK, Santos-Lang, Christopher, 2014, Simon, van Rysewyk, Matthijs, Pontier, Machine Medical Ethics, 74, Springer, Switzerland, 111â€“127, Moral Ecology Approaches to Machine Ethics,weblink PDF, 10.1007/978-3-319-08108-3_8, Intelligent Systems, Control and Automation: Science and Engineering, 978-3-319-08107-6,
• BOOK, Scott, Michael L., Programming Language Pragmatics, 3rd, Morgan Kaufmann Publishers/Elsevier, 2009, 978-0-12-374514-9,
• BOOK, Sipser, Michael, Introduction to the Theory of Computation, 2006, PWS Publishing Company, 978-0-534-94728-6,weblink
• BOOK, Sober, Elliott, Wilson, David Sloan, 1998, Unto Others: The Evolution and Psychology of Unselfish Behavior, Cambridge, Harvard University Press,
• BOOK, Stone, Harold S., Introduction to Computer Organization and Data Structures, 1972, McGraw-Hill, New York, 978-0-07-061726-1, 1972, 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).
• BOOK, Tausworthe, Robert C, Standardized Development of Computer Software Part 1 Methods, 1977, Prenticeâ€“Hall, Inc., Englewood Cliffs NJ, 978-0-13-842195-3,
• JOURNAL, Turing, Alan M., A. M. Turing, On Computable Numbers, With An Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, Series 2, 42, 230â€“265, 1936â€“37, 10.1112/plms/s2-42.1.230, . Corrections, ibid, vol. 43(1937) pp. 544â€“546. Reprinted in The Undecidable, p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.
• JOURNAL, Turing, Alan M., A. M. Turing, Systems of Logic Based on Ordinals, Proceedings of the London Mathematical Society, 45, 161â€“228, 1939, 10.1112/plms/s2-45.1.161, 21.11116/0000-0001-91CE-3, 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

• BOOK, Bellah, Robert Neelly, 1985, Robert N. Bellah, Habits of the Heart: Individualism and Commitment in American Life, Berkeley, 978-0-520-25419-0, University of California Press,weblink harv,
• BOOK, Berlinski, David, The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer, 2001, Harvest Books, 978-0-15-601391-8,weblink
• BOOK, Chabert, Jean-Luc, A History of Algorithms: From the Pebble to the Microchip, 1999, Springer Verlag, 978-3-540-63369-3,
• BOOK, Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein, Introduction To Algorithms, 3rd, 2009, MIT Press, 978-0-262-03384-8,
• BOOK, Harel, David, Feldman, Yishai, Algorithmics: The Spirit of Computing, 2004, Addison-Wesley, 978-0-321-11784-7,
• BOOK, Hertzke, Allen D., McRorie, Chris, 1998, Lawler, Peter Augustine, McConkey, Dale, The Concept of Moral Ecology, Community and Political Thought Today, Westport, CT, Praeger Publishers, Praeger, harv,
• Knuth, Donald E. (2000). Selected Papers on Analysis of Algorithms. Stanford, California: Center for the Study of Language and Information.
• Knuth, Donald E. (2010). Selected Papers on Design of Algorithms. Stanford, California: Center for the Study of Language and Information.
• BOOK, Wendell, Wallach, Colin, Allen, November 2008, Moral Machines: Teaching Robots Right from Wrong, 978-0-19-537404-9, Oxford University Press, US, harv,

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