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Formal Languages
In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (formal languages), alphabet and are well-formedness, well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or ''well-formed formulas''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syntax Tree , used in linguistics
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Syntax tree may refer to: * Abstract syntax tree, used in computer science * Concrete syntax tree A parse tree or parsing tree or derivation tree or concrete syntax tree is an ordered, rooted tree that represents the syntactic structure of a string according to some context-free grammar. The term ''parse tree'' itself is used primarily in com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whether a given natural number is prime. Another is the problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?". The answer is either 'yes' or 'no' depending upon the values of ''x'' and ''y''. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" would give the steps for determining whether ''x'' evenly divides ''y''. One such algorithm is long division. If the remainder is zero the answer is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm is called ''decidable''. Decision problems typically appear in mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though he was largely ignored during his lifetime, Giuseppe Peano (1858–1932), Bertrand Russell (1872–1970), and, to some extent, Ludwig Wittgenstein (1889–1951) introduced his work to later generations of philosophers. Frege is widely considered to be the greatest logician since Aristotle, and one of the most profound philosophers of mathematics ever. His contributions include the development of modern logic in the ''Begriffsschrift'' and work in the foundations of mathematics. His book the ''Foundations of Arithmetic'' is the seminal text of the logicist project, and is cited by Michael Dummett as where to pinpoint the linguistic turn. His philosophical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Notation
Gauss notation (also known as a Gauss code or Gauss word) is a notation for mathematical knots. It is created by enumerating and classifying the crossings of an embedding of the knot in a plane. It is named for the mathematician Carl Friedrich Gauss (1777–1855). Gauss code represents a knot with a sequence of integers. However, rather than every crossing being represented by two different numbers, crossings are labeled with only one number. When the crossing is an overcrossing, a positive number is listed. At an undercrossing, a negative number. For example, the trefoil knot in Gauss code can be given as: 1,−2,3,−1,2,−3. Gauss code is limited in its ability to identify knots by a few problems. The starting point on the knot at which to begin tracing the crossings is arbitrary, and there is no way to determine which direction to trace in. Also, Gauss code is unable to indicate the handedness of each crossing, which is necessary to identify a knot versus its mirror. For exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referred to as the ''Princeps mathematicorum'' () and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and he is ranked among history's most influential mathematicians. Also available at Retrieved 23 February 2014. Comprehensive biographical article. Biography Early years Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pictographs
A pictogram, also called a pictogramme, pictograph, or simply picto, and in computer usage an icon, is a graphic symbol that conveys its meaning through its pictorial resemblance to a physical object. Pictographs are often used in writing and graphic systems in which the characters are to a considerable extent pictorial in appearance. A pictogram may also be used in subjects such as leisure, tourism, and geography. Pictography is a form of writing which uses representational, pictorial drawings, similarly to cuneiform and, to some extent, hieroglyphic writing, which also uses drawings as phonetic letters or determinative rhymes. Some pictograms, such as Hazards pictograms, are elements of formal languages. "Pictograph" has a different definition in the field of prehistoric art (which includes recent art by traditional societies), where it means art painted on rock surfaces. This is in comparison to petroglyphs, where the images are carved or incised. Such images may or may no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristica Universalis
The Latin term ''characteristica universalis'', commonly interpreted as ''universal characteristic'', or ''universal character'' in English, is a universal and formal language imagined by Gottfried Leibniz able to express mathematical, scientific, and metaphysical concepts. Leibniz thus hoped to create a language usable within the framework of a universal logical calculation or ''calculus ratiocinator''. The ''characteristica universalis'' is a recurring concept in the writings of Leibniz. When writing in French, he sometimes employed the phrase ''spécieuse générale'' to the same effect. The concept is sometimes paired with his notion of a ''calculus ratiocinator'' and with his plans for an encyclopaedia as a compendium of all human knowledge. A universal pictographic language: the key to all the sciences International communication Many Leibniz scholars writing in English seem to agree that he intended his ''characteristica universalis'' or "universal character" to be a fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history and philology. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. In addition, he contributed to the field of library science: while serving as overseer of the Wolfenbüttel library in Germany, he devised a cataloging system that would have served as a guide for many of Europe's largest libraries. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages can take different forms, such as speech or signing. They are distinguished from constructed and formal languages such as those used to program computers or to study logic. Defining natural language Natural language can be broadly defined as different from * artificial and constructed languages, e.g. computer programming languages * constructed international auxiliary languages * non-human communication systems in nature such as whale and other marine mammal vocalizations or honey bees' waggle dance. All varieties of world languages are natural languages, including those that are associated with linguistic prescriptivism or language regulation. ( Nonstandard dialects can be viewed as a wild type in comparison with standard l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syntax
In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituency), agreement, the nature of crosslinguistic variation, and the relationship between form and meaning (semantics). There are numerous approaches to syntax that differ in their central assumptions and goals. Etymology The word ''syntax'' comes from Ancient Greek roots: "coordination", which consists of ''syn'', "together", and ''táxis'', "ordering". Topics The field of syntax contains a number of various topics that a syntactic theory is often designed to handle. The relation between the topics is treated differently in different theories, and some of them may not be considered to be distinct but instead to be derived from one another (i.e. word order can be seen as the result of movement rules derived from grammatical relations). Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formalism (philosophy Of Mathematics)
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess." According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics). In contrast to mathematical realism, logicism, or intuitionism, formalism's contours are less ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic System
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system. Properties An axiomatic system is said to be ''consistent'' if it lacks contradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion). In an axiomatic syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
