In mathematics and computer science , an ALGORITHM (/ˈælɡərɪðəm/ ( listen ) AL-gə-ridh-əm ) is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation , data processing and automated reasoning tasks.
An algorithm is an effective method that can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function . Starting from an initial state and initial input (perhaps empty ), the instructions describe a computation that, when executed , proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic ; some algorithms, known as randomized algorithms , incorporate random input.
The concept of algorithm has existed for centuries; however, a
partial formalization of what would become the modern algorithm began
with attempts to solve the
Entscheidungsproblem (the "decision
problem") posed by
* 1 Etymology * 2 Informal definition
* 3 Formalization
* 3.1 Expressing algorithms
* 4 Implementation * 5 Computer algorithms
* 6 Examples
* 6.2 Euclid\'s algorithm
* 6.2.1 Computer language for Euclid\'s algorithm * 6.2.2 An inelegant program for Euclid\'s algorithm * 6.2.3 An elegant program for Euclid\'s algorithm
* 7 Algorithmic analysis
* 7.1 Formal versus empirical * 7.2 Execution efficiency
* 8 Classification
* 8.1 By implementation * 8.2 By design paradigm * 8.3 Optimization problems * 8.4 By field of study * 8.5 By complexity
* 9 Continuous algorithms * 10 Legal issues
* 11 History: Development of the notion of "algorithm"
* 11.1 Ancient Near East
* 11.2 Discrete and distinguishable symbols
* 11.3 Manipulation of symbols as "place holders" for numbers:
* 11.4 Mechanical contrivances with discrete states
* 12 See also * 13 Notes * 14 References * 15 Bibliography * 16 Secondary references * 17 Further reading * 18 External links
The word 'algorithm' is a combination of the
In English, it was first used in about 1230 and then by 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:
Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris.
which translates as:
Algorism is the art by which at present we use those Indian figures, which number two times five.
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.
For a detailed presentation of the various points of view on the definition of "algorithm", see Algorithm characterizations .
An informal definition could be "a set of rules that precisely defines a sequence of operations." which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually.
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 flow chart above and as an example in a later section.
Boolos, Jeffrey 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 .
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
Representations of algorithms can be classed into three accepted
For an example of the simple algorithm "Add m+n" described in all three levels, see Algorithm#Examples .
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.
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" Chaitin: " . . . a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant'"—such a proof would solve the Halting problem (ibid).
Unfortunately there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example that uses Euclid's algorithm appears below.
Computers (and computors), models of computation: A computer (or human "computor" ) is a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions. Melzak's and Lambek's primitive models reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent.
Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability ". Minsky\'s machine proceeds sequentially through its five (or six, depending on how one counts) instructions, unless either a conditional IF–THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution) operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1). Rarely must a programmer write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT.
Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example". But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't, then the algorithm, to be effective, must provide a set of rules for extracting a square root.
This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).
But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters". When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus " instruction available rather than just subtraction (or worse: just Minsky's "decrement").
Structured programming, canonical structures: Per the Church–Turing thesis , 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". Tausworthe augments the three Böhm-Jacopini canonical structures : SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction .
Canonical flowchart symbols : The graphical aide called a flowchart 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.
Further information: List of algorithms
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 English prose, as:
* 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 :
ALGORITHM LargestNumber Input: A list of numbers L. Output: The largest number in the list L.
IF L.size = 0 RETURN null largest ← L FOR EACH item IN L, DO IF item > largest, THEN largest ← item RETURN largest
* "←" is a shorthand for "changes to". For instance, "largest ← item" means that the value of largest changes to the value of item. * "RETURN" terminates the algorithm and outputs the value that follows.
Further information: Euclid\'s algorithm The example-diagram of
Euclid's algorithm from T.L. Heath (1908), with more detail added.
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.
1599 = 650×2 + 299 650 = 299×2 + 52 299 = 52×5 + 39 52 = 39×1 + 13 39 = 13×3 + 0
Computer Language For Euclid\'s Algorithm
Only a few instruction types are required to execute Euclid's algorithm—some logical tests (conditional GOTO), unconditional GOTO, assignment (replacement), and subtraction.
* A location is symbolized by upper case letter(s), e.g. S, A, etc. * The varying quantity (number) in a location is written in lower case letter(s) and (usually) associated with the location's name. For example, location L at the start might contain the number l = 3009.
An Inelegant Program For Euclid\'s Algorithm
"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 1973:2–4. 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 1973:2–4:
1 : INPUT L, S 2 : R ← L
3 : 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. 4 L ← R (this first step is redundant, but is useful for later discussion). 5 R ← S 6 S ← L
E1: : 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.
7 IF S > R THEN done measuring so GOTO 10 ELSE measure again, 8 R ← R − S 9 : GOTO 7 .
E2: : 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.
10 IF R = 0 THEN done so GOTO step 15 ELSE CONTINUE TO step 11 ,
E3: : The nut of Euclid's algorithm. Use remainder r to measure what was previously smaller number s; L serves as a temporary location.
11 L ← R 12 R ← S 13 S ← L 14 : GOTO 7
15 : PRINT S
16 HALT, END, STOP.
An Elegant Program For Euclid\'s Algorithm
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 LET  =  is the assignment instruction symbolized by ←.
5 REM Euclid's algorithm for greatest common divisor 6 PRINT "Type two integers greater than 0" 10 INPUT A,B 20 IF B=0 THEN GOTO 80 30 IF A > B THEN GOTO 60 40 LET B=B-A 50 GOTO 20 60 LET A=A-B 70 GOTO 20 80 PRINT A 90 END
The following version can be used with Object Oriented languages:
How "Elegant" works: In place of an outer "
TESTING THE EUCLID ALGORITHMS
Does an algorithm do what its author wants it to do? A few test cases usually suffice to confirm core functionality. One source uses 3009 and 884. Knuth suggested 40902, 24140. Another interesting case is the two relatively prime numbers 14157 and 5950.
But exceptional cases must be identified and tested. Will "Inelegant" perform properly when R > S, S > R, R = S? Ditto for "Elegant": B > A, A > B, A = B? (Yes to all). What happens when one number is zero, both numbers are zero? ("Inelegant" computes forever in all cases; "Elegant" computes forever when A = 0.) What happens if negative numbers are entered? Fractional numbers? If the input numbers, i.e. the domain of 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". Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof.
MEASURING AND IMPROVING THE EUCLID ALGORITHMS
Elegance (compactness) versus goodness (speed): With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. However, "Inelegant" is faster (it arrives at HALT in fewer steps). Algorithm analysis indicates why this is the case: "Elegant" does two conditional tests in every subtraction loop, whereas "Inelegant" only does one. As the algorithm (usually) requires many loop-throughs, on average much time is wasted doing a "B = 0?" test that is needed only after the remainder is computed.
Can the algorithms be improved?: Once the programmer judges a program "fit" and "effective"—that is, it computes the function intended by its author—then the question becomes, can it be improved?
The compactness of "Inelegant" can be improved by the elimination of five steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; rather, it can only be done heuristically ; i.e., by exhaustive search (examples to be found at Busy beaver ), trial and error, cleverness, insight, application of inductive reasoning , etc. Observe that steps 4, 5 and 6 are repeated in steps 11, 12 and 13. Comparison with "Elegant" provides a hint that these steps, together with steps 2 and 3, can be eliminated. This reduces the number of core instructions from thirteen to eight, which makes it "more elegant" than "Elegant", at nine steps.
The speed of "Elegant" can be improved by moving the "B=0?" test outside of the two subtraction loops. This change calls for the addition of three instructions (B = 0?, A = 0?, GOTO). Now "Elegant" computes the example-numbers faster; whether this is always the case for any given A, B and R, S would require a detailed analysis.
Analysis of algorithms
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.
Main article: Algorithmic 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. In general, speed improvements depend on special properties of the problem, which are very common in practical applications. Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.
There are various ways to classify algorithms, each with its own merits.
One way to classify algorithms is by implementation means. Recursion
A recursive algorithm is one that invokes (makes reference to) itself
repeatedly until a certain condition (also known as termination
condition) matches, which is a method common 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:
BY DESIGN PARADIGM
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 include 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. 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 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 . 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:
* Monte Carlo algorithms return a correct answer with high-probability. E.g. RP is the subclass of these that run in polynomial time . * Las Vegas algorithms always return the correct answer, but their running time is only probabilistically bound, e.g. 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 .
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
Linear programming When searching for optimal solutions
to a linear function bound to linear equality and inequality
constraints, the constraints of the problem can be used directly in
producing the optimal solutions. There are algorithms that can solve
any problem in this category, such as the popular simplex algorithm .
Problems that can be solved with linear programming include the
maximum flow problem for directed graphs. If a problem additionally
requires that one or more of the unknowns must be an integer then it
is classified in integer programming . A linear programming algorithm
can solve such a problem if it can be proved that all restrictions for
integer values are superficial, i.e., the solutions satisfy these
restrictions anyway. In the general case, a specialized algorithm or
an algorithm that finds approximate solutions is used, depending on
the difficulty of the problem.
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.
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.
The adjective "continuous" when applied to the word "algorithm" can mean:
* An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations—such algorithms are studied in numerical analysis ; or * An algorithm in the form of a differential equation that operates continuously on the data, running on an analog computer .
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
Additionally, some cryptographic algorithms have export restrictions (see export of cryptography ).
Researcher, Andrew Tutt , argues that algorithms should be overseen by a specialist regulatory agency, similar to FDA. His academic work emphasizes that the rise of increasingly complex algorithms calls for the need to think about the effects of algorithms today. Due to the nature and complexity of algorithms, it will prove to be difficult to hold algorithms accountable under criminal law. Tutt recognizes that while some algorithms will be beneficial to help meet technological demand, others should not be used or sold if they fail to meet safety requirements. Thus, for Tutt, algorithms will require "closer forms of federal uniformity, expert judgment, political independence, and pre-market review to prevent the introduction of unacceptably dangerous algorithms into the market". The issue of algorithmic accountability (the responsibility of algorithm designers to provide evidence of potential or realised harms) is of particular relevance in the field of dynamic and non-linearly programmed systems, e.g. artificial neural networks, deep learning, and genetic algorithms (see Explainable AI ).
HISTORY: DEVELOPMENT OF THE NOTION OF "ALGORITHM"
ANCIENT NEAR EAST
Algorithms were used in ancient Greece. Two examples are the Sieve of
Eratosthenes , which was described in Introduction to
DISCRETE AND DISTINGUISHABLE SYMBOLS
TALLY-MARKS: To keep track of their flocks, their sacks of grain and
their money the ancients used tallying: accumulating stones or marks
scratched on sticks, or making discrete symbols in clay. Through the
Babylonian and Egyptian use of marks and symbols, eventually Roman
numerals and the abacus evolved (Dilson, p. 16–41). Tally marks
appear prominently in unary numeral system arithmetic used in Turing
machine and Post–
MANIPULATION OF SYMBOLS AS "PLACE HOLDERS" FOR NUMBERS: ALGEBRA
The work of the ancient Greek geometers (
Euclidean algorithm ), the
A good century and a half ahead of his time, Leibniz proposed an algebra of logic, an algebra that would specify the rules for manipulating logical concepts in the manner that ordinary algebra specifies the rules for manipulating numbers.
MECHANICAL CONTRIVANCES WITH DISCRETE STATES
The clock: Bolter credits the invention of the weight-driven clock as
"The key invention ", in particular the verge escapement that
provides us with the tick and tock of a mechanical clock. "The
accurate automatic machine" led immediately to "mechanical automata "
beginning in the 13th century and finally to "computational
machines"—the difference engine and analytical engines of Charles
Logical machines 1870— Stanley Jevons ' "logical abacus" and "logical machine": The technical problem was to reduce Boolean equations when presented in a form similar to what are now known as Karnaugh maps . Jevons (1880) describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the combinations can be picked out mechanically . . . More recently however I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a Logical Machine" His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano . . .". With this machine he could analyze a "syllogism or any other simple logical argument".
This machine he displayed in 1870 before the Fellows of the Royal Society. Another logician John Venn , however, in his 1881 Symbolic Logic, turned a jaundiced eye to this effort: "I have no high estimate myself of the interest or importance of what are sometimes called logical machines ... it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines"; see more at Algorithm characterizations . But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's abacus ... gain, corresponding to Prof. Jevons's logical machine, the following contrivance may be described. I prefer to call it merely a logical-diagram machine ... but I suppose that it could do very completely all that can be rationally expected of any logical machine".
Jacquard loom, Hollerith punch cards, telegraphy and telephony—the
electromechanical relay: Bell and Newell (1971) indicate that the
Telephone-switching networks of electromechanical relays (invented 1835) was behind the work of George Stibitz (1937), the inventor of the digital adding device. As he worked in Bell Laboratories, he observed the "burdensome' use of mechanical calculators with gears. "He went home one evening in 1937 intending to test his idea... When the tinkering was over, Stibitz had constructed a binary adding device".
Davis (2000) observes the particular importance of the
electromechanical relay (with its two "binary states" open and
closed): It was only with the development, beginning in the 1930s, of
electromechanical calculators using electrical relays, that machines
were built having the scope
MATHEMATICS DURING THE 19TH CENTURY UP TO THE MID-20TH CENTURY
Symbols and rules: In rapid succession the mathematics of George Boole (1847, 1854), Gottlob Frege (1879), and Giuseppe Peano (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's The principles of arithmetic, presented by a new method (1888) was "the first attempt at an axiomatization of mathematics in a symbolic language".
But Heijenoort gives Frege (1879) this kudos: Frege's is "perhaps the
most important single work ever written in logic. ... in which we see
a " 'formula language', that is a lingua characterica, a language
written with special symbols, "for pure thought", that is, free from
rhetorical embellishments ... constructed from specific symbols that
are manipulated according to definite rules". The work of Frege was
further simplified and amplified by
Alfred North Whitehead and
The paradoxes: At the same time a number of disturbing paradoxes appeared in the literature, in particular the Burali-Forti paradox (1897), the Russell paradox (1902–03), and the Richard Paradox . The resultant considerations led to Kurt Gödel 's paper (1931)—he specifically cites the paradox of the liar—that completely reduces rules of recursion to numbers.
Effective calculability: In an effort to solve 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 ,
EMIL POST (1936) AND ALAN TURING (1936–37, 1939)
Here is a remarkable coincidence of two men not knowing each other but describing a process of men-as-computers working on computations—and they yield virtually identical definitions.
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 being marked with a stroke. Likewise the
answer 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 ) ". See more at Post–
Turing—his model of computation is now called a Turing machine —begins, as did Post, with an analysis of a human computer that he whittles down to a simple set of basic motions and "states of mind". But he continues a step further and creates a machine as a model of computation of numbers. "Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book...I assume then that the computation is carried out on one-dimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite... "The behaviour 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'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 . . . . . . We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability † with effective calculability . . . . "† We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions".
J. B. ROSSER (1939) AND S. C. KLEENE (1943)
J. Barkley Rosser defined an 'effective 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. . The simplest of these to state (due to Post and Turing) says essentially that an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." (Rosser 1939:225–6)
Rosser's footnote No. 5 references the work of (1) Church and Kleene
and their definition of λ-definability, in particular Church's use of
it in his An Unsolvable Problem of Elementary Number Theory (1936);
(2) Herbrand and
Gödel and their use of recursion in particular
Gödel's use in his famous paper On Formally Undecidable Propositions
Stephen C. Kleene defined as his now-famous "Thesis I" known as the
Church–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?"" (
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 Church–Turing thesis ) and philosophy of mind (especially arguments about artificial intelligence ). For more, see Algorithm characterizations .
* ^ "Any classical mathematical algorithm, for example, can be
described in a finite number of English words" (Rogers 1987:2).
* ^ Well 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).
* ^ "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
* ^ "An algorithm has zero or more inputs, i.e., quantities which
are given to it initially before the algorithm begins" (Knuth 1973:5).
* ^ "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).
* ^ "An algorithm has one or more outputs, i.e. quantities which
have a specified relation to the inputs" (Knuth 1973:5).
* ^ 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.
* Axt, P (1959). "On a Subrecursive Hierarchy and Primitive
Recursive Degrees". Transactions of the American Mathematical Society.
92: 85–105. doi :10.2307/1993169 .
* Bolter, David J. (1984). Turing's Man: Western Culture in the Computer Age (1984 ed.). The University of North Carolina Press, Chapel Hill NC. ISBN 0-8078-1564-0 . , ISBN 0-8078-4108-0 pbk. * Dilson, Jesse (2007). The Abacus ((1968,1994) ed.). St. Martin's Press, NY. ISBN 0-312-10409-X . , ISBN 0-312-10409-X (pbk.) * van Heijenoort, Jean (2001). From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931 ((1967) ed.). Harvard University Press, Cambridge, MA. ISBN 0-674-32449-8 . , 3rd edition 1976, ISBN 0-674-32449-8 (pbk.) * Hodges, Andrew (1983). Alan Turing: The Enigma . New York: Simon and Schuster . ISBN 0-671-49207-1 . , ISBN 0-671-49207-1 . Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
* Jean Luc Chabert (1999). A History of Algorithms: From the Pebble to the Microchip. Springer Verlag. ISBN 978-3-540-63369-3 . * Algorithmics.: The Spirit of Computing. Addison-Wesley. 2004. ISBN 978-0-321-11784-7 . * 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. * Berlinski, David (2001). The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer. Harvest Books. ISBN 978-0-15-601391-8 . * Thomas H. Cormen; Charles E. Leiserson; Ronald L. Rivest; Clifford Stein (2009). Introduction To Algorithms, Third Edition. MIT Press. ISBN 978-0262033848 .
Look up ALGORITHM in Wiktionary, the free dictionary.
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* Hazewinkel, Michiel , ed. (2001) , "Algorithm", Encyclopedia of