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Algorithm
In mathematics and computer science , an ALGORITHM (/ˈælɡərɪðəm/ (_ listen ) AL-gə-ri-dhəm_ ) is a self-contained sequence of actions to be performed. 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 David Hilbert
David Hilbert
in 1928
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Algorithm (other)
An ALGORITHM is a self-contained step-by-step set of operations to be performed. ALGORITHM may also refer to: * Algorithm (C++) , components that perform algorithmic operations on containers and other sequences * Algorithm (album) , a 2013 album by My Heart to Fear * Algorithms (journal) , a journal * Algorythm, an album by Boxcar * The Algorithm , a French musical projectSEE ALSO * Ruleset (other) This disambiguation page lists articles associated with the title ALGORITHM. If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from "https://en.wikipedia.org/w/index.php?title=Algorithm_(other) additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy .® is a registered trademark of the Wikimedia Foundation, Inc
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Flow Chart
A FLOWCHART is a type of diagram that represents an algorithm , workflow or process, showing the steps as boxes of various kinds, and their order by connecting them with arrows. This diagrammatic representation illustrates a solution model to a given problem . Flowcharts are used in analyzing, designing, documenting or managing a process or program in various fields. CONTENTS * 1 Overview * 2 History * 3 Types * 4 Building blocks * 4.1 Common symbols * 4.2 Other symbols * 5 Software * 5.1 Diagramming * 6 See also * 6.1 Related diagrams * 6.2 Related subjects * 7 References * 8 Further reading * 9 External links OVERVIEW _ Flowchart of a for loop _ Flowcharts are used in designing and documenting simple processes or programs. Like other types of diagrams, they help visualize what is going on and thereby help understand a process, and perhaps also find flaws, bottlenecks, and other less-obvious features within it. There are many different types of flowcharts, and each type has its own repertoire of boxes and notational conventions. The two most common types of boxes in a flowchart are: * a processing step, usually called _activity_, and denoted as a rectangular box * a decision, usually denoted as a diamond.A flowchart is described as "cross-functional" when the page is divided into different swimlanes describing the control of different organizational units
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Euclid's Algorithm
In mathematics , the EUCLIDEAN ALGORITHM , or EUCLID\'S ALGORITHM, is an efficient method for computing the greatest common divisor (GCD) of two numbers, the largest number that divides both of them without leaving a remainder . It is named after the ancient Greek mathematician Euclid , who first described it in Euclid\'s Elements (c. 300 BC). It is an example of an algorithm , a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form , and is a part of many other number-theoretic and cryptographic calculations. The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers. By reversing the steps , the GCD can be expressed as a sum of the two original numbers each multiplied by a positive or negative integer , e.g., 21 = 5 × 105 + (−2) × 252. The fact that the GCD can always be expressed in this way is known as Bézout\'s identity
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Mathematics
MATHEMATICS (from Greek μάθημα _máthēma_, “knowledge, study, learning”) is the study of topics such as quantity (numbers ), structure , space , and change . There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics . Mathematicians seek out patterns and use them to formulate new conjectures . Mathematicians resolve the truth or falsity of conjectures by mathematical proof . When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic , mathematics developed from counting , calculation , measurement , and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics , most notably in Euclid 's _Elements _. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century , it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions
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Computer Science
COMPUTER SCIENCE is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers . It is the scientific and practical approach to computation and its applications and the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures (or algorithms ) that underlie the acquisition, representation, processing, storage, communication of, and access to information . An alternate, more succinct definition of computer science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems. Its fields can be divided into a variety of theoretical and practical disciplines . Some fields, such as computational complexity theory (which explores the fundamental properties of computational and intractable problems), are highly abstract, while fields such as computer graphics emphasize real-world visual applications. Other fields still focus on challenges in implementing computation. For example, programming language theory considers various approaches to the description of computation, while the study of computer programming itself investigates various aspects of the use of programming language and complex systems . Human–computer interaction considers the challenges in making computers and computations useful, usable, and universally accessible to humans
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Calculation
A CALCULATION is a deliberate process that transforms one or more inputs into one or more results, with variable change. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm , to the vague heuristics of calculating a strategy in a competition, or calculating the chance of a successful relationship between two people. For example, multiplying 7 by 6 is a simple algorithmic calculation. Estimating the fair price for financial instruments using the Black–Scholes model is a complex algorithmic calculation. Statistical estimations of the likely election results from opinion polls also involve algorithmic calculations, but produces ranges of possibilities rather than exact answers. To _calculate_ means to ascertain by computing. The English word derives from the Latin _calculus_, which originally meant a small stone in the gall-bladder (from Latin _calx_). It also meant a pebble used for calculating, or a small stone used as a counter in an abacus ( Latin _abacus_, Greek _abax_). The abacus was an instrument used by Greeks and Romans for arithmetic calculations, preceding the slide-rule and the electronic calculator, and consisted of perforated pebbles sliding on iron bars
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Data Processing
DATA PROCESSING is, generally, "the collection and manipulation of items of data to produce meaningful information." In this sense it can be considered a subset of _information processing _, "the change (processing) of information in any manner detectable by an observer." The term Data Processing (DP) has also been used previously to refer to a department within an organization responsible for the operation of data processing applications
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Automated Reasoning
AUTOMATED REASONING is an area of computer science and mathematical logic dedicated to understanding different aspects of reasoning . The study of automated reasoning helps produce computer programs that allow computers to reason completely, or nearly completely, automatically. Although automated reasoning is considered a sub-field of artificial intelligence , it also has connections with theoretical computer science , and even philosophy . The most developed subareas of automated reasoning are automated theorem proving (and the less automated but more pragmatic subfield of interactive theorem proving ) and automated proof checking (viewed as guaranteed correct reasoning under fixed assumptions). Extensive work has also been done in reasoning by analogy induction and abduction . Other important topics include reasoning under uncertainty and non-monotonic reasoning. An important part of the uncertainty field is that of argumentation, where further constraints of minimality and consistency are applied on top of the more standard automated deduction. John Pollock's OSCAR system is an example of an automated argumentation system that is more specific than being just an automated theorem prover
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Effective Method
In logic , mathematics and computer science , especially metalogic and computability theory , an EFFECTIVE METHOD or EFFECTIVE PROCEDURE is a procedure for solving a problem from a specific class. An effective method is sometimes also called MECHANICAL method or procedure. CONTENTS * 1 Definition * 2 Algorithms * 3 Computable functions * 4 See also * 5 References DEFINITIONThe definition of an effective method involves more than the method itself. In order for a method to be called effective, it must be considered with respect to a class of problems. Because of this, one method may be effective with respect to one class of problems and _not_ be effective with respect to a different class. A method is formally called effective for a class of problems when it satisfies these criteria: * It consists of a finite number of exact, finite instructions.* When it is applied to a problem from its class: * It always finishes (_terminates_) after a finite number of steps. * It always produces a correct answer. * In principle, it can be done by a human without any aids except writing materials. * Its instructions need only to be followed rigorously to succeed. In other words, it requires no ingenuity to succeed. Optionally, it may also be required that the method never returns a result as if it were an answer when the method is applied to a problem from _outside_ its class
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Function (mathematics)
In mathematics , a FUNCTION is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number _x_ to its square _x_2. The output of a function _f_ corresponding to an input _x_ is denoted by _f_(_x_) (read "_f_ of _x_"). In this example, if the input is −3, then the output is 9, and we may write _f_(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write _f_(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function. Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics . There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function . In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation
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Empty String
In formal language theory , the EMPTY STRING is the unique string of length zero . CONTENTS * 1 Formal theory * 2 Use in programming languages * 2.1 Examples of empty strings * 3 See also * 4 References FORMAL THEORYFormally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special case where the sequence has length zero, so there are no symbols in the string. There is only one empty string, because two strings are only different if they have different lengths or a different sequence of symbols. In formal treatments, the empty string is denoted with _ε _ or sometimes Λ or λ . The empty string should not be confused with the EMPTY LANGUAGE ∅, which is a formal language (i.e. a set of strings) that contains no strings, not even the empty string. The empty string has several properties: * ε = 0. Its STRING LENGTH is zero. * ε ⋅ s = s ⋅ ε = s. The empty string is the identity element of the CONCATENATION operation. The set of all strings forms a free monoid with respect to ⋅ and ε. * εR = ε. REVERSAL of the empty string produces the empty string. * The empty string precedes any other string under lexicographical order , because it is the shortest of all strings. USE IN PROGRAMMING LANGUAGESIn most programming languages, strings are a data type . Individual strings are typically stored in consecutive memory locations
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Computation
COMPUTATION is any type of calculation that includes both arithmetical and non-arithmetical steps and follows a well-defined model understood and described as, for example, an algorithm . The study of computation is paramount to the discipline of computer science . CONTENTS* 1 Physical phenomenon * 1.1 Accounts of computation * 1.1.1 The mapping account * 1.1.2 The semantic account * 1.1.3 The mechanistic account * 2 Mathematical models * 3 See also * 4 References PHYSICAL PHENOMENONA computation can be seen as a purely physical phenomenon occurring inside a closed physical system called a computer . Examples of such physical systems include digital computers , mechanical computers , quantum computers , DNA computers , molecular computers , microfluidics-based computers, analog computers or wetware computers . This point of view is the one adopted by the branch of theoretical physics called the physics of computation as well as the field of natural computing . An even more radical point of view is the postulate of digital physics that the evolution of the universe itself is a computation - pancomputationalism . ACCOUNTS OF COMPUTATIONThe Mapping AccountA classic account of computation is found throughout the works of Hilary Putnam and others
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Execution (computing)
EXECUTION in computer and software engineering is the process by which a computer or a virtual machine performs the instructions of a computer program . The instructions in the program trigger sequences of simple actions on the executing machine. Those actions produce effects according to the semantics of the instructions in the program. Programs for a computer may execute in a batch process without human interaction, or a user may type commands in an interactive session of an interpreter . In this case the "commands" are simply programs, whose execution is chained together. The term RUN is used almost synonymously. A related meaning of both "to run" and "to execute" refers to the specific action of a user starting (or launching or invoking) a program, as in "Please run the application." CONTENTS * 1 Context of execution * 2 Process * 3 Interpreter * 4 See also CONTEXT OF EXECUTIONThe context in which execution takes place is crucial. Very few programs execute on a bare machine . Programs usually contain implicit and explicit assumptions about resources available at the time of execution. Most programs execute with the support of an operating system and run-time libraries specific to the source language that provide crucial services not supplied directly by the computer itself. This supportive environment, for instance, usually decouples a program from direct manipulation of the computer peripherals, providing more general, abstract services instead
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Deterministic
Related concepts and fundamentals: * Agnosticism * Epistemology * Presupposition * Probability * v * t * e DETERMINISM is the philosophical position that for every event there exist conditions that could cause no other event. "There are many determinisms, depending on what pre-conditions are considered to be determinative of an event or action." Deterministic theories throughout the history of philosophy have sprung from diverse and sometimes overlapping motives and considerations. Some forms of determinism can be empirically tested with ideas from physics and the philosophy of physics . The opposite of determinism is some kind of indeterminism (otherwise called _nondeterminism_). Determinism is often contrasted with free will . Determinism often is taken to mean _causal determinism_, which in physics is known as cause-and-effect. It is the concept that events within a given paradigm are bound by causality in such a way that any state (of an object or event) is completely determined by prior states. This meaning can be distinguished from other varieties of determinism mentioned below. Other debates often concern the scope of determined systems, with some maintaining that the entire universe is a single determinate system and others identifying other more limited determinate systems (or multiverse )
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Randomized Algorithms
A RANDOMIZED ALGORITHM is an algorithm that employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits. Formally, the algorithm's performance will be a random variable determined by the random bits; thus either the running time, or the output (or both) are random variables. One has to distinguish between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms , example of which is Quicksort ), and algorithms which have a chance of producing an incorrect result (Monte Carlo algorithms , example of which is Monte Carlo algorithm for MFAS ) or fail to produce a result either by signaling a failure or failing to terminate. In the second case, random performance and random output, the term "algorithm" for a procedure is somewhat questionable. In the case of random output, it is no longer formally effective . However, in some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized algorithms are approximated using a pseudorandom number generator in place of a true source of random bits; such an implementation may deviate from the expected theoretical behavior
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