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Pregroup Grammar
Pregroup grammar (PG) is a Formal grammar, 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 partial order, partially ordered universal algebra, 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 adjoint, left and right adjoints ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Formal Grammar
A formal grammar is a set of Terminal and nonterminal symbols, symbols and the Production (computer science), production rules for rewriting some of them into every possible string of a formal language over an Alphabet (formal languages), alphabet. A grammar does not describe the semantics, meaning of the strings — only their form. In applied mathematics, formal language theory is the discipline that studies formal grammars and languages. Its applications are found in theoretical computer science, theoretical linguistics, Formal semantics (logic), formal semantics, mathematical logic, and other areas. A formal grammar is a Set_(mathematics), 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 determines whether a given string belongs to the language or is grammatical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Joachim Lambek
Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a 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 professor at Mc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quantum Mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lambda Calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using variable Name binding, binding and Substitution (algebra), substitution. Untyped lambda calculus, the topic of this article, is a universal machine, a model of computation that can be used to simulate any Turing machine (and vice versa). It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. In 1936, Church found a formulation which was #History, logically consistent, and documented it in 1940. Lambda calculus consists of constructing #Lambda terms, lambda terms and performing #Reduction, reduction operations on them. A term is defined as any valid lambda calculus expression. In the simplest form of lambda calculus, terms are built using only the following rules: # x: A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Sanskrit
Sanskrit (; stem form ; nominal singular , ,) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in northwest South Asia after its predecessor languages had Trans-cultural diffusion, diffused there from the northwest in the late Bronze Age#South Asia, 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 lingua franca, 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 effect on the languages of South Asia, Southeast Asia and East Asia, especially in their formal and learned vocabularies. Sanskrit generally connotes several Indo-Aryan languages# ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Persian Language
Persian ( ), also known by its endonym and exonym, endonym Farsi (, Fārsī ), is a Western Iranian languages, Western Iranian language belonging to the Iranian languages, Iranian branch of the Indo-Iranian languages, 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 mutual intelligibility, mutually intelligible standard language, standard varieties, respectively Iranian Persian (officially known as ''Persian''), Dari, Dari Persian (officially known as ''Dari'' since 1964), and Tajik language, 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 society, Persianate history in the cultural sphere o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
French Language
French ( or ) is a Romance languages, Romance language of the Indo-European languages, Indo-European family. Like all other Romance languages, it descended from the Vulgar Latin of the Roman Empire. French evolved from Northern Old Gallo-Romance, a descendant of the Latin spoken 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 language, Francien) largely supplanted. It was also substratum (linguistics), influenced by native Celtic languages of Northern Roman Gaul and by the Germanic languages, Germanic Frankish language of the post-Roman Franks, Frankish invaders. As a result of French and Belgian colonialism from the 16th century onward, it was introduced to new territories in the Americas, Africa, and Asia, and numerous French-based creole languages, most notably Haitian Creole, were established. A French-speaking person or nation may be referred to as Fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Italian Language
Italian (, , or , ) is a Romance language of the Indo-European language family. It evolved from the colloquial Latin of the Roman Empire. Italian is the least divergent language from Latin, together with Sardinian language, Sardinian. It is spoken by about 68 million people, including 64 million native speakers as of 2024. Italian is an official language in Languages of Italy, Italy, Languages of San Marino, San Marino, Languages of Switzerland, Switzerland (Ticino and the Grisons), and Languages of Vatican City, Vatican City; it has official Minority language, minority status in Minority languages of Croatia, Croatia, Slovene Istria, Romania, Bosnia and Herzegovina, and the municipalities of Santa Teresa, Espírito Santo, Santa Tereza, Encantado, Rio Grande do Sul, Encantado, and Venda Nova do Imigrante in Languages of Brazil#Language co-officialization, Brazil. Italian is also spoken by large Italian diaspora, immigrant and expatriate communities in the Americas and Austral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
English Language
English is a West Germanic language that developed in early medieval England and has since become a English as a lingua franca, global lingua franca. The namesake of the language is the Angles (tribe), Angles, one of the Germanic peoples that Anglo-Saxon settlement of Britain, migrated to Britain after its End of Roman rule in Britain, Roman occupiers left. English is the list of languages by total number of speakers, most spoken language in the world, primarily due to the global influences of the former British Empire (succeeded by the Commonwealth of Nations) and the United States. English is the list of languages by number of native speakers, third-most spoken native language, after Mandarin Chinese and Spanish language, Spanish; it is also the most widely learned second language in the world, with more second-language speakers than native speakers. English is either the official language or one of the official languages in list of countries and territories where English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
