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Flowcharts are often used to graphically represent algorithms.
Flowcharts are often used to graphically represent algorithms. A flowchart is a Schematic representation of an Algorithm or a stepwise process, showing the steps as boxes of various kinds and their order by connecting

In mathematics, computing, linguistics and related disciplines, an algorithm is a sequence of instructions, often used for calculation, data processing. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Computing is usually defined like the activity of using and developing Computer technology Computer hardware and software. Linguistics is the scientific study of Language, encompassing a number of sub-fields A calculation is a deliberate process for transforming one or more inputs into one or more results with variable change It is formally a type of effective method in which a list of well-defined instructions for completing a task will, when given an initial state, proceed through a well-defined series of successive states, eventually terminating in an end-state. An effective method (also called an effective procedure) for a class of problems is a method for which each step in the method may be described as a mechanical operation and The transition from one state to the next is not necessarily deterministic; some algorithms, known as probabilistic algorithms, incorporate randomness. Determinism is the philosophical Proposition that every event including human cognition and behaviour decision and action is causally determined A randomized algorithm or probabilistic algorithm is an Algorithm which employs a degree of randomness as part of its logic

A partial formalization of the concept began with attempts to solve the Entscheidungsproblem (the "decision problem") posed by David Hilbert in 1928. In Mathematics, the Entscheidungsproblem ( German for ' Decision problem ' is a challenge posed by David Hilbert in 1928 David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Subsequent formalizations were framed as attempts to define "effective calculability" (Kleene 1943:274) or "effective method" (Rosser 1939:225); those formalizations included the Gödel-Herbrand-Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's "Formulation I" of 1936, and Alan Turing's Turing machines of 1936-7 and 1939. Recursion in computer science is a way of thinking about and solving problems Alonzo Church ( June 14, 1903 – August 11, 1995) was an American Mathematician and logician In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function Emil Leon Post, PhD, ( February 11 1897, Augustów – April 21 1954, New York City) was a Mathematician Alan Mathison Turing, OBE, FRS (ˈt(jʊ(ərɪŋ (23 June 1912 &ndash 7 June 1954 was an English Mathematician Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm

Contents

Etymology

Al-Khwārizmī, Persian astronomer and mathematician, wrote a treatise in Arabic in 825 AD, On Calculation with Hindu Numerals. layout and formatting it should ensure no clashes with the top of the infobox Historically Astronomy was more concerned with the classification and description of phenomena in the sky while Astrophysics attempted to explain these phenomena A mathematician is a person whose primary area of study and research is the field of Mathematics. A treatise is a formal lengthy systematic Discourse on some subject Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language (See algorism). Algorism is the technique of performing basic Arithmetic by writing numbers in Place value form and applying a set of memorized rules and facts to the digits It was translated into Latin in the 12th century as Algoritmi de numero Indorum (al-Daffa 1977), which title was likely intended to mean "[Book by] Algoritmus on the numbers of the Indians", where "Algoritmi" was the translator's rendition of the author's name in the genitive case; but people misunderstanding the title treated Algoritmi as a Latin plural and this led to the word "algorithm" (Latin algorismus) coming to mean "calculation method". Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. The intrusive "th" is most likely due to a false cognate with the Greek αριθμος (arithmos) meaning "number". False cognates are pairs of words in the same or different Languages that are similar in form and meaning but have different roots. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly

Why algorithms are necessary: an informal definition

No generally accepted formal definition of "algorithm" exists yet. We can, however, derive clues to the issues involved and an informal meaning of the word from the following quotation from Boolos & Jeffrey (1974, 1999) (boldface added):

No human being can write fast enough, or long enough, or small enough 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 & Jeffrey 1974, 1999, p.  19)

The words "enumerably infinite" mean "countable using integers perhaps extending to infinity". 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 chosen from 0 to infinity. Thus we might expect an algorithm to be an algebraic equation such as y = m + n — two arbitrary "input variables" m and n that produce an output y. As we see in Algorithm characterizations — the word algorithm implies much more than this, something on the order of (for our addition example):

Precise instructions (in language understood by "the computer") for a "fast, efficient, good" process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally-contained information and capabilities) to find, decode, and then munch arbitrary input integers/symbols m and n, symbols + and = . The word Algorithm does not have a generally accepted definition . . and (reliably, correctly, "effectively") produce, in a "reasonable" time, output-integer y at a specified place and in a specified format. For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of

The concept of algorithm is also used to define the notion of decidability. In Logic, the term decidable refers to the existence of an Effective method for determining membership in a set of formulas That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules. In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension. Logic is the study of the principles of valid demonstration and Inference. 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.

For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations. The word Algorithm does not have a generally accepted definition For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples. The word Algorithm does not have a generally accepted definition This article Algorithm examples supplements Algorithm and Algorithm characterizations.

Formalization of algorithms

Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in order to carry out a specified task, such as calculating employees’ paychecks or printing students’ report cards. A computer is a Machine that manipulates data according to a list of instructions. Computer programs (also software programs, or just programs) are instructions for a Computer. Thus, an algorithm can be considered to be any sequence of operations that can be performed by a Turing-complete system. In computability theory, several closely-related terms are used to describe the "computational power" of a computational system (such as an Abstract machine or Authors who assert this thesis include Savage (1987) and Gurevich (2000):

. . . Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine (Gurevich 2000:1). . . according to Savage [1987], an algorithm is a computational process defined by a Turing machine. (Gurevich 2000:3)

Typically, when an algorithm is associated with processing information, data are read from an input source or device, written to an output sink or device, and/or 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 a data structure, but an algorithm requires the internal data only for specific operation sets called abstract data types. A data structure in Computer science is a way of storing Data in a computer so that it can be used efficiently In Computing, an abstract data type ( ADT) is a specification of a set of data and the set of operations that can be performed on the data

For any 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 will almost always be critical 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. In Computer science control flow (or alternatively flow of control refers to the order in which the individual statements, instructions or Function

So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. In Computer science, imperative programming is a Programming paradigm that describes computation in terms of statements that change a program state 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. In Computer science the assignment statement sets or re-sets the value stored in the storage location(s denoted by a Variable Name. It derives from the intuition of "memory" as a scratchpad. In Psychology, memory is an organism's ability to store retain and subsequently retrieve information There is an example below of such an assignment.

For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming . In Computer science, functional programming is a Programming paradigm that treats Computation as the evaluation of mathematical functions and Logic programming is in its broadest sense the use of mathematical logic for computer programming

Termination

Some writers restrict the definition of algorithm to procedures that eventually finish. In such a category Kleene places the "decision procedure or decision method or algorithm for the question" (Kleene 1952:136). Others, including Kleene, include procedures that could run forever without stopping; such a procedure has been called a "computational method" (Knuth 1997:5) or "calculation procedure or algorithm" (Kleene 1952:137); however, Kleene notes that such a method must eventually exhibit "some object" (Kleene 1952:137).

Minsky makes the pertinent observation, in regards to determining whether an algorithm will eventually terminate (from a particular starting state):

But if the length of the process is not known in advance, then "trying" it may not be decisive, because if the process does go on forever — then at no time will we ever be sure of the answer (Minsky 1967:105).

As it happens, no other method can do any better, as was shown by Alan Turing with his celebrated result on the undecidability of the so-called halting problem. Alan Mathison Turing, OBE, FRS (ˈt(jʊ(ərɪŋ (23 June 1912 &ndash 7 June 1954 was an English Mathematician In computability theory, the halting problem is a Decision problem which can be stated as follows given a description of a program and a finite input There is no algorithmic procedure for determining of arbitrary algorithms whether or not they terminate from given starting states. The analysis of algorithms for their likelihood of termination is called termination analysis. In Computer science, termination analysis attempts to determine whether the evaluation of a given program will definitely terminate

See the examples of (im-)"proper" subtraction at partial function for more about what can happen when an algorithm fails for certain of its input numbers — e. Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function g. , (i) non-termination, (ii) production of "junk" (output in the wrong format to be considered a number) or no number(s) at all (halt ends the computation with no output), (iii) wrong number(s), or (iv) a combination of these. Kleene proposed that the production of "junk" or failure to produce a number is solved by having the algorithm detect these instances and produce e. g. , an error message (he suggested "0"), or preferably, force the algorithm into an endless loop (Kleene 1952:322). Davis does this to his subtraction algorithm — he fixes his algorithm in a second example so that it is proper subtraction (Davis 1958:12-15). Along with the logical outcomes "true" and "false" Kleene also proposes the use of a third logical symbol "u" — undecided (Kleene 1952:326) — thus an algorithm will always produce something when confronted with a "proposition". The problem of wrong answers must be solved with an independent "proof" of the algorithm e. g. , using induction:

We normally require auxiliary evidence for this (that the algorithm correctly defines a mu recursive function), e. g. , in the form of an inductive proof that, for each argument value, the computation terminates with a unique value (Minsky 1967:186).

Expressing algorithms

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, and programming languages. In the Philosophy of language, a natural language (or ordinary language) is a Language that is spoken or written in phonemic-alphabetic or phonemically-related Pseudocode is a compact and informal high-level description of a Computer programming Algorithm that uses the structural conventions of some Programming language A flowchart is a Schematic representation of an Algorithm or a stepwise process, showing the steps as boxes of various kinds and their order by connecting A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode and flowcharts are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements, while remaining independent of a particular implementation language. 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. A computer is a Machine that manipulates data according to a list of instructions.

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 and state transition table), as flowcharts (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). Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm In Automata theory and Sequential logic, a state transition table is a table showing what state (or states in the case of a nondeterministic finite automaton State diagrams is a Diagram used in the field of Computer science, representing the behavior of a system which is composed of a finite number of states Machine code or machine language is a system of instructions and data executed directly by a Computer 's Central processing unit. See the terminology section below for information regarding inconsistent use of the terms assembly and assembler Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm

Sometimes it is helpful in the description of an algorithm to supplement small "flow charts" (state diagrams) with natural-language and/or arithmetic expressions written inside "block diagrams" to summarize what the "flow charts" are accomplishing. Block diagram is a Diagram of a System, in which the principal parts or functions are represented by blocks connected by lines that show the relationships of

Representations of algorithms are generally classed into three accepted levels of Turing machine description (Sipser 2006:157):

". . . 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"
". . . 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"
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. This article Algorithm examples supplements Algorithm and Algorithm characterizations.

Implementation

Most algorithms are intended to be implemented as computer programs. Computer programs (also software programs, or just programs) are instructions for a Computer. 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. Traditionally the term neural network had been used to refer to a network or circuit of biological neurons. The human brain controls the Central nervous system (CNS by way of the Cranial nerves and Spinal cord, the Peripheral nervous system (PNS Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone An electrical network is an interconnection of Electrical elements such as Resistors Inductors Capacitors Transmission lines Voltage

Example

One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm, which can be stated in a high-level description English prose, as:

High-level description:

  1. Assume the first item is largest. English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States
  2. Look at each of the remaining items in the list and if it is larger than the largest item so far, make a note of it.
  3. The last noted item is the largest in the list when the process is complete.

(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 non-empty list of numbers L. Pseudocode is a compact and informal high-level description of a Computer programming Algorithm that uses the structural conventions of some Programming language In Computer programming, pidgin code is a mixture of several Programming languages in the same program or Pseudocode that is a mixture of a programming 
  Output: The largest number in the list L. 

  largest ← L0
  for each item in the list L≥1, do
    if the item > largest, then
      largest ← the item
  return largest

For a more complex example of an algorithm, see Euclid's algorithm for the greatest common divisor, one of the earliest algorithms known. In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero

Algorithm analysis

As it happens, it is important to know how much of a particular resource (such as time or storage) is required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers; for example, the algorithm above has a time requirement of O(n), using the big O notation with n as the length of the list. To analyze an Algorithm is to determine the amount of resources (such as time and storage necessary to execute it In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments 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 (log 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, given two different recipes for making potato salad, one may have peel the potato before boil the potato while the other presents the steps in the reverse order, yet they both call for these steps to be repeated for all potatoes and end when the potato salad is ready to be eaten.

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. To analyze an Algorithm is to determine the amount of resources (such as time and storage necessary to execute it Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. 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. Pseudocode is a compact and informal high-level description of a Computer programming Algorithm that uses the structural conventions of some Programming language

Classes

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

Classification by implementation

One way to classify algorithms is by implementation means.

Classification 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 will include many different types of algorithms. Some commonly found paradigms include:

  1. Probabilistic algorithms are those that make some choices randomly (or pseudo-randomly); for some problems, it can in fact be proven that the fastest solutions must involve some randomness. A randomized algorithm or probabilistic algorithm is an Algorithm which employs a degree of randomness as part of its logic Randomness is a lack of order Purpose, cause, or predictability
  2. Genetic algorithms attempt to find solutions to problems by mimicking biological evolutionary processes, with a cycle of random mutations yielding successive generations of "solutions". A genetic algorithm (GA is a Search technique used in Computing to find exact or Approximate solutions to optimization and Search eVolution is the third Album by eLDee, it was due to be released in 2008 Thus, they emulate reproduction and "survival of the fittest". In genetic programming, this approach is extended to algorithms, by regarding the algorithm itself as a "solution" to a problem. In Artificial intelligence, genetic programming (GP is an Evolutionary algorithm based methodology inspired by Biological evolution to find
  3. Heuristic algorithms, whose general purpose is not to find an optimal solution, but an approximate solution where the time or resources are limited. heuristic (hyu̇-ˈris-tik is a method to help solve a problem commonly an informal method They are not practical to find perfect solutions. An example of this would be local search, tabu search, or simulated annealing algorithms, a class of heuristic probabilistic algorithms that vary the solution of a problem by a random amount. In Computer science, local search is a Metaheuristic for solving computationally hard optimization problems Tabu search is a mathematical optimization method belonging to the class of local search techniques Simulated annealing (SA is a generic probabilistic Meta-algorithm for the Global optimization problem namely locating a good approximation to the The name "simulated annealing" alludes to the metallurgic term meaning the heating and cooling of metal to achieve freedom from defects. Simulated annealing (SA is a generic probabilistic Meta-algorithm for the Global optimization problem namely locating a good approximation to the The purpose of the random variance is to find close to globally optimal solutions rather than simply locally optimal ones, the idea being that the random element will be decreased as the algorithm settles down to a solution.

Classification by field of study

See also: List of algorithms

Every field of science has its own problems and needs efficient algorithms. The following is a list of the algorithms described in Wikipedia 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, machine learning, cryptography, data compression algorithms and parsing techniques. In Computer science, a search algorithm, broadly speaking is an Algorithm that takes a problem as Input and returns a solution to the problem usually In Computer science and Mathematics, a sorting algorithm is an Algorithm that puts elements of a list in a certain order. Merge algorithms are a family of Algorithms that run sequentially over multiple sorted lists typically producing more sorted lists as output Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Computer programming and some branches of Mathematics, a string is an ordered Sequence of Symbols. Computational geometry is a branch of Computer science devoted to the study of algorithms which can be stated in terms of Geometry. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Machine learning is a subfield of Artificial intelligence that is concerned with the design and development of Algorithms and techniques that allow computers to "learn" Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" In Computer science and Linguistics, parsing, or more formally syntactic analysis, is the process of analyzing a sequence of tokens to

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 originally invented for optimization of resource consumption in industry, but is now used in solving a broad range of problems in many fields.

Classification by complexity

See also: Complexity class

Algorithms can be classified by the amount of time they need to complete compared to their input size. In Computational complexity theory, a complexity class is a set of problems of related complexity There is a wide variety: some algorithms complete in linear time relative to input size, some do so in an exponential amount of time or even worse, and some never halt. Additionally, 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.

Classification by computing power

Another way to classify algorithms is by computing power. This is typically done by considering some collection (class) of algorithms. A recursive class of algorithms is one that includes algorithms for all Turing computable functions. Looking at classes of algorithms allows for the possibility of restricting the available computational resources (time and memory) used in a computation. A subrecursive class of algorithms is one in which not all Turing computable functions can be obtained. For example, the algorithms that run in polynomial time suffice for many important types of computation but do not exhaust all Turing computable functions. In Computational complexity theory, P, also known as PTIME or DTIME ( n O(1 is one of the most fundamental Complexity The class algorithms implemented by primitive recursive functions is another subrecursive class. The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict Subset of the recursive

Burgin (2005, p. 24) uses a generalized definition of algorithms that relaxes the common requirement that the output of the algorithm that computes a function must be determined after a finite number of steps. He defines a super-recursive class of algorithms as "a class of algorithms in which it is possible to compute functions not computable by any Turing machine" (Burgin 2005, p. 107). This is closely related to the study of methods of hypercomputation. Hypercomputation refers to various hypothetical methods for the Computation of non- Turing-computable functions (see also Supertask)

Legal issues

See also: Software patents for a general overview of the patentability of software, including computer-implemented algorithms. Software patent does not have a universally accepted definition

Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals do not constitute "processes" (USPTO 2006) and hence algorithms are not patentable (as in Gottschalk v. Benson). The United States of America —commonly referred to as the Gottschalk v Benson, was a United States Supreme Court case that ruled that a process involving a numerical Algorithm was not Patentable if 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. Diamond v Diehr,, was a US Supreme Court decision which held that the execution of a Process, controlled by running a Computer program was Feedback is a circular causal Process whereby some proportion of a system's output is returned (fed back to the Input. Elastomer Synthetic rubber is any type of artificially made Polymer material which acts as an Elastomer. The patenting of software is highly controversial, and there are highly criticized patents involving algorithms, especially data compression algorithms, such as Unisys' LZW patent. Software patent debate is the argument dealing with the extent to which it should be possible to patent software and computer-implemented inventions as a matter of Unisys Corporation ( based in Blue Bell, Pennsylvania, United States, and incorporated in Delaware, is a global provider of information technology

Additionally, some cryptographic algorithms have export restrictions (see export of cryptography). The export of cryptography is the transfer from one country to another of devices and technology related to Cryptography.

History: Development of the notion of "algorithm"

Origin of the word

See also: Timeline of algorithms

The word algorithm comes from the name of the 9th century Persian mathematician Abu Abdullah Muhammad ibn Musa al-Khwarizmi whose works introduced Indian numerals and algebraic concepts. The following timeline outlines the development of Algorithms (mainly "mathematical recipes" since their inception layout and formatting it should ensure no clashes with the top of the infobox He worked in Baghdad at the time when it was the centre of scientific studies and trade. Baghdad (بغداد) is the Capital of Iraq and of Baghdad Governorate, with which it is also coterminous The word algorism originally referred only to the rules of performing arithmetic using Arabic numerals but evolved via European Latin translation of al-Khwarizmi's name into algorithm by the 18th century. Algorism is the technique of performing basic Arithmetic by writing numbers in Place value form and applying a set of memorized rules and facts to the digits Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century The word evolved to include all definite procedures for solving problems or performing tasks.

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. Roman numerals are a Numeral system originating in ancient Rome, adapted from Etruscan numerals. An abacus, also called a counting frame, is a calculating tool used primarily by Asians for performing arithmetic processes 16–41). Tally marks appear prominently in unary numeral system arithmetic used in Turing machine and Post-Turing machine computations. The unary numeral system is the bijective base - 1 Numeral system. Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm The article Turing machine gives a general introduction to Turing machines while this article covers a specific class of Turing machines

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

The work of the ancient Greek geometers, Persian mathematician Al-Khwarizmi (often considered as the "father of algebra"), and Western European mathematicians culminated in Leibniz's notion of the calculus ratiocinator (ca 1680):

"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" (Davis 2000:1)

Mechanical contrivances with discrete states

The clock: Bolter credits the invention of the weight-driven clock as “The key invention [of Europe in the Middle Ages]", in particular the verge escapement< (Bolter 1984:24) that provides us with the tick and tock of a mechanical clock. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. The Calculus Ratiocinator is a theoretical universal logical calculation framework a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently Circadian Locomotor Output Cycles Kaput, or Clock is a gene which encodes proteins regulating Circadian rhythm. The verge (or crown wheel) escapement is the earliest known type of mechanical Escapement, the mechanism in a mechanical Clock that controls “The accurate automatic machine” (Bolter 1984:26) led immediately to "mechanical automata" beginning in the thirteenth century and finally to “computational machines" – the difference engine and analytical engines of Charles Babbage and Countess Ada Lovelace (Bolter p. The Difference Engine was an automatic mechanical calculator designed to tabulate polynomial functions. The analytical engine, an important step in the History of computers, was the design of a mechanical general-purpose Computer by the British mathematician Charles Augusta Ada King Countess of Lovelace (10 December 1815 London England &ndash 27 November 1852 Marylebone, London England born Augusta Ada Byron, was the only 33–34, p. 204–206).

Jacquard loom, Hollerith punch cards, telegraphy and telephony — the electromechanical relay: Bell and Newell (1971) indicate that the Jacquard loom (1801), precursor to Hollerith cards (punch cards, 1887), and “telephone switching technologies” were the roots of a tree leading to the development of the first computers (Bell and Newell diagram p. The Jacquard Loom is a mechanical Loom, invented by Joseph Marie Jacquard in 1801, that has holes punched in pasteboard each row of which corresponds to 39, cf Davis 2000). By the mid-1800s the telegraph, the precursor of the telephone, was in use throughout the world, its discrete and distinguishable encoding of letters as “dots and dashes” a common sound. By the late 1800s the ticker tape (ca 1870s) was in use, as was the use of Hollerith cards in the 1890 U. Ticker tape was used by ticker tape machines, the Ticker tape timer stock ticker machines, or just stock tickers. S. census. Then came the Teletype (ca 1910) with its punched-paper use of Baudot code on tape. A teleprinter ( The Baudot code, invented by Émile Baudot, is a character set predating EBCDIC and ASCII, and the root predecessor to International Telegraph

Telephone-switching networks of electromechanical relays (invented 1835) was behind the work of George Stibitz (1937), the inventor of the digital adding device. A relay is an electrical Switch that opens and closes under the control of another Electrical circuit. George Robert Stibitz ( April 20, 1904 &ndash January 31, 1995) is internationally recognized as a father of the modern digital computer 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". (Valley News, p. 13).

Davis (2000) observes the particular importance of the electromechanical relay (with its two "binary states" open and closed):

It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had envisioned. " (Davis, p. 14).

Mathematics during the 1800s up to the mid-1900s

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. George Boole (buːl ( November 2, 1815 &ndash December 8, 1864) was a British Mathematician and Philosopher. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional 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:81ff).

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" (van Heijenoort:1). The work of Frege was further simplified and amplified by Alfred North Whitehead and Bertrand Russell in their Principia Mathematica (1910–1913). Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian The Principia Mathematica is a 3-volume work on the Foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell

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 (Dixon 1906, cf Kleene 1952:36–40). In Set theory, a field of Mathematics, the Burali-Forti paradox demonstrates that naively constructing "the set of all Ordinal numbers quot leads to Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the Richard's paradox is a fallacious Paradox of mathematical mapping first described by the French Mathematician Jules Richard in 1905 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. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition

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. In Mathematics, the Entscheidungsproblem ( German for ' Decision problem ' is a challenge posed by David Hilbert in 1928 e. , a calculation that would succeed). In rapid succession the following appeared: Alonzo Church, Stephen Kleene and J.B. Rosser's λ-calculus, (cf footnote in Alonzo Church 1936a:90, 1936b:110) a finely-honed definition of "general recursion" from the work of Gödel acting on suggestions of Jacques Herbrand (cf Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene (1935-6:237ff, 1943:255ff). Alonzo Church ( June 14, 1903 – August 11, 1995) was an American Mathematician and logician Stephen Cole Kleene ( January 5, 1909, Hartford Connecticut, USA &ndash January 25, 1994, Madison Wisconsin John Barkley Rosser Sr (1907–1989 was an American Logician, a student of Alonzo Church, and known for his part in the Church-Rosser theorem In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function Alonzo Church ( June 14, 1903 – August 11, 1995) was an American Mathematician and logician Jacques Herbrand ( February 12, 1908 - July 27, 1931) was a French Mathematician who was born in Paris, Church's proof (1936:88ff) 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 "Formulation I", Post 1936:289-290). In Mathematics, the Entscheidungsproblem ( German for ' Decision problem ' is a challenge posed by David Hilbert in 1928 Emil Leon Post, PhD, ( February 11 1897, Augustów – April 21 1954, New York City) was a Mathematician Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- [automatic-] machine"(Turing 1936-7:116ff) -- in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine" (Rosser 1939:226). Alan Mathison Turing, OBE, FRS (ˈt(jʊ(ərɪŋ (23 June 1912 &ndash 7 June 1954 was an English Mathematician John Barkley Rosser Sr (1907–1989 was an American Logician, a student of Alonzo Church, and known for his part in the Church-Rosser theorem S. C. Kleene's proposal of a precursor to "Church thesis" that he called "Thesis I" (Kleene 1943:273–274), 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). Stephen Cole Kleene ( January 5, 1909, Hartford Connecticut, USA &ndash January 25, 1994, Madison Wisconsin

Emil Post (1936) and Alan Turing (1936-7, 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.

Emil Post (1936) described the actions of a "computer" (human being) as follows:

". Emil Leon Post, PhD, ( February 11 1897, Augustów – April 21 1954, New York City) was a Mathematician . . 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 will terminate only when it comes to the direction of type (C ) [i. e. , STOP]. " (U p. 289–290) See more at Post-Turing machine

Alan Turing’s work (1936, 1939:160) preceded that of Stibitz (1937); it is unknown whether Stibitz knew of the work of Turing. The article Turing machine gives a general introduction to Turing machines while this article covers a specific class of Turing machines Alan Mathison Turing, OBE, FRS (ˈt(jʊ(ərɪŋ (23 June 1912 &ndash 7 June 1954 was an English Mathematician 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 Teletypes we might conjecture that all were influences.

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". Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm But he continues a step further and creates a machine as a model of computation of numbers (Turing 1936-7:116).

"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 1936-7:136).

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. " (Turing 1936-7:136)

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. Although it is fairly easy to get an intuitive grasp of this idea, it is neverthessless desirable to have some more definite, mathematical expressible definition . . . [he discusses the history of the definition pretty much as presented above with respect to Gödel, Herbrand, Kleene, Church, Turing and Post] . . . We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability † with effective calculability . . . .
"† We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculabile" refer to the intuitive idea without particular identification with any one of these definitions. "(Turing 1939:160)

J. B. Rosser (1939) and S. C. Kleene (1943)

J. Barkley Rosser boldly defined an ‘effective [mathematical] method’ in the following manner (boldface 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. John Barkley Rosser Sr (1907–1989 was an American Logician, a student of Alonzo Church, and known for his part in the Church-Rosser theorem With this special meaning, three different precise definitions have been given to date. [his footnote #5; see discussion immediately below]. The simplest of these to state (due to Post and Turing) says essentially that an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one. " (Rosser 1939:225–6)

Rosser's footnote #5 references the work of (1) Church and Kleene and their definition of λ-definability, in particular Church's use of it in his An Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand and Gödel and their use of recursion in particular Gödel's use in his famous paper On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931); and (3) Post (1936) and Turing (1936-7) in their mechanism-models of computation.

Stephen C. Kleene defined as his now-famous "Thesis I" known as "the Church-Turing Thesis". Stephen Cole Kleene ( January 5, 1909, Hartford Connecticut, USA &ndash January 25, 1994, Madison Wisconsin 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 Church-Turing Thesis) and philosophy of mind (especially arguments around artificial intelligence). Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory Philosophy of mind is the branch of Philosophy that studies the nature of the Mind, Mental events Mental functions mental properties For more, see Algorithm characterizations. The word Algorithm does not have a generally accepted definition

Algorithmic Repositories

See also

References

Secondary references

External links

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Dictionary

algorithm

-noun

  1. A precise step-by-step plan for a computational procedure that begins with an input value and yields an output value in a finite number of steps.
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