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Pregroup Grammar
Pregroup grammar (PG) is a grammar formalism intimately related to categorial grammars. Much like categorial grammar (CG), PG is a kind of type logical grammar. Unlike CG, however, PG does not have a distinguished function type. Rather, PG uses inverse types combined with its monoidal operation. Definition of a pregroup A pregroup is a partially ordered algebra (A, 1, \cdot, -^l, -^r, \leq) such that (A, 1, \cdot) is a monoid, satisfying the following relations: * x^l \cdot x \leq 1 \qquad x \cdot x^r \leq 1 (contraction) * 1 \leq x \cdot x^l \qquad 1 \leq x^r \cdot x (expansion) The contraction and expansion relations are sometimes called Ajdukiewicz laws. From this, it can be proven that the following equations hold: * 1^l = 1 = 1^r * x^ = x = x^ * (x\cdot y)^l = y^l \cdot x^l \qquad (x\cdot y)^r = y^r \cdot x^r x^l and x^r are called the left and right adjoints of ''x'', respectively. The symbol \cdot and \leq are also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Grammar
In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form. A formal grammar is defined as a set of production rules for such strings in a formal language. Formal language theory, the discipline that studies formal grammars and languages, is a branch of applied mathematics. Its applications are found in theoretical computer science, theoretical linguistics, formal semantics, mathematical logic, and other areas. A formal grammar is a set of rules for rewriting strings, along with a "start symbol" from which rewriting starts. Therefore, a grammar is usually thought of as a language generator. However, it can also sometimes be used as the basis for a " recognizer"—a function in computing that det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joachim Lambek
Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a German-born Canadian mathematician. He was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD degree in 1950 with Hans Zassenhaus as advisor. Biography Lambek was born in Leipzig, Germany, where he attended a Gymnasium. He came to England in 1938 as a refugee on the ''Kindertransport''. From there he was interned as an enemy alien and deported to a prison work camp in New Brunswick, Canada. There, he began in his spare time a mathematical apprenticeship with Fritz Rothberger, also interned, and wrote the McGill Junior Matriculation in fall of 1941. In the spring of 1942, he was released and settled in Montreal, where he entered studies at McGill University, graduating with an honours mathematics degree in 1945 and an MSc a year later. In 1950, he completed his doctorate under Hans Zassenhaus becoming McGill's first PhD in mathematics. Lambek became assistant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves ( wave–particle duality); and there ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sanskrit
Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late Bronze Age. Sanskrit is the sacred language of Hinduism, the language of classical Hindu philosophy, and of historical texts of Buddhism and Jainism. It was a link language in ancient and medieval South Asia, and upon transmission of Hindu and Buddhist culture to Southeast Asia, East Asia and Central Asia in the early medieval era, it became a language of religion and high culture, and of the political elites in some of these regions. As a result, Sanskrit had a lasting impact on the languages of South Asia, Southeast Asia and East Asia, especially in their formal and learned vocabularies. Sanskrit generally connotes several Old Indo-Aryan language varieties. The most archaic of these is the Vedic Sanskrit found in the Rig Veda, a colle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Persian Language
Persian (), also known by its endonym Farsi (, ', ), is a Western Iranian language belonging to the Iranian branch of the Indo-Iranian subdivision of the Indo-European languages. Persian is a pluricentric language predominantly spoken and used officially within Iran, Afghanistan, and Tajikistan in three mutually intelligible standard varieties, namely Iranian Persian (officially known as ''Persian''), Dari Persian (officially known as ''Dari'' since 1964) and Tajiki Persian (officially known as ''Tajik'' since 1999).Siddikzoda, S. "Tajik Language: Farsi or not Farsi?" in ''Media Insight Central Asia #27'', August 2002. It is also spoken natively in the Tajik variety by a significant population within Uzbekistan, as well as within other regions with a Persianate history in the cultural sphere of Greater Iran. It is written officially within Iran and Afghanistan in the Persian alphabet, a derivation of the Arabic script, and within Tajikistan in the Tajik alphabet, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
French Language
French ( or ) is a Romance languages, Romance language of the Indo-European languages, Indo-European family. It descended from the Vulgar Latin of the Roman Empire, as did all Romance languages. French evolved from Gallo-Romance, the Latin spoken in Gaul, and more specifically in Northern Gaul. Its closest relatives are the other langues d'oïl—languages historically spoken in northern France and in southern Belgium, which French (Francien) largely supplanted. French was also substratum, influenced by native Celtic languages of Northern Roman Gaul like Gallia Belgica and by the (Germanic languages, Germanic) Frankish language of the post-Roman Franks, Frankish invaders. Today, owing to France's French colonial empire, past overseas expansion, there are numerous French-based creole languages, most notably Haitian Creole language, Haitian Creole. A French-speaking person or nation may be referred to as Francophone in both English and French. French is an official language in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Italian Language
Italian (''italiano'' or ) is a Romance language of the Indo-European language family that evolved from the Vulgar Latin of the Roman Empire. Together with Sardinian, Italian is the least divergent language from Latin. Spoken by about 85 million people (2022), Italian is an official language in Italy, Switzerland ( Ticino and the Grisons), San Marino, and Vatican City. It has an official minority status in western Istria (Croatia and Slovenia). Italian is also spoken by large immigrant and expatriate communities in the Americas and Australia.Ethnologue report for language code:ita (Italy) – Gordon, Raymond G., Jr. (ed.), 2005. Ethnologue: Languages of the World, Fifteenth edition. Dallas, Tex.: SIL International. Online version ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
English Language
English is a West Germanic language of the Indo-European language family, with its earliest forms spoken by the inhabitants of early medieval England. It is named after the Angles, one of the ancient Germanic peoples that migrated to the island of Great Britain. Existing on a dialect continuum with Scots, and then closest related to the Low Saxon and Frisian languages, English is genealogically West Germanic. However, its vocabulary is also distinctively influenced by dialects of France (about 29% of Modern English words) and Latin (also about 29%), plus some grammar and a small amount of core vocabulary influenced by Old Norse (a North Germanic language). Speakers of English are called Anglophones. The earliest forms of English, collectively known as Old English, evolved from a group of West Germanic (Ingvaeonic) dialects brought to Great Britain by Anglo-Saxon settlers in the 5th century and further mutated by Norse-speaking Viking settlers starting in the 8 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Group
In mathematics, the free group ''F''''S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1''t'', but ''s'' ≠ ''t''−1 for ''s'',''t'',''u'' ∈ ''S''). The members of ''S'' are called generators of ''F''''S'', and the number of generators is the rank of the free group. An arbitrary group ''G'' is called free if it is isomorphic to ''F''''S'' for some subset ''S'' of ''G'', that is, if there is a subset ''S'' of ''G'' such that every element of ''G'' can be written in exactly one way as a product of finitely many elements of ''S'' and their inverses (disregarding trivial variations such as ''st'' = ''suu''−1''t''). A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property. History Free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
