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Word (formal Language Theory)
In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also called "words"). Words that belong to a particular formal language are sometimes called ''well-formed words''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar. 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 that are associated with meanings or semantics. In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that can be parsed by machines with limited computational power. In l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Mathematics
Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theorems, proof (mathematics), proofs, algorithms, etc. in particular. This may also include the philosophy of mathematics, philosophical study of the relation of this framework with reality. The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancient Greek philosophers under the name of Aristotle's logic and systematically applied in Euclid's Elements, Euclid's ''Elements''. A mathematical assertion is considered as truth (mathematics), truth only if it is a theorem that is proved from true premises by means of a sequence of syllogisms (inference rules), the premises being either already proved theorems or self-evident assertions called axioms or postulat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Governorate, Congress Poland, Russian Empire (now Poland) into a Polish Jews, Polish-Jewish family that immigrated to New York City in May 1904. His parents were Arnold and Pearl Post. Post had been interested in astronomy, but at the age of twelve lost his left arm in a car accident. This loss was a significant obstacle to being a professional astronomer, leading to his decision to pursue mathematics rather than astronomy. Post attended the Townsend Harris High School and continued on to graduate from City College of New York in 1917 with a B.S. in mathematics. After completing his Doctor of philosophy, Ph.D. in mathematics in 1920 at Columbia University, supervised by Cassius Jackson Keyser, he did a post-doctorate at Princeton University in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axel Thue
Axel Thue (; 19 February 1863 – 7 March 1922) was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics. Work Thue published his first important paper in 1909. He stated in 1914 the so-called word problem for semigroups or Thue problem, closely related to the halting problem In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for .... Ronald V. Book and Friedrich Otto, ''String-rewriting Systems'', Springer, 1993, , p. 36. His only known PhD student was Thoralf Skolem. The esoteric programming language Thue is named after him. Publications * * See also * * * * * * * * References External links Axel Thue private archiveexists at NTNU University LibrarDorabiblioteket 1863 births 1922 deaths 20th-century Norwegian mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Begriffsschrift
''Begriffsschrift'' (German for, roughly, "concept-writing") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notation''; the full title of the book identifies it as "a formula language, modeled on that of arithmetic, for pure thought." Frege's motivation for developing his formal approach to logic resembled Gottfried Wilhelm Leibniz, Leibniz's motivation for his ''calculus ratiocinator'' (despite that, in the foreword Frege clearly denies that he achieved this aim, and also that his main aim would be constructing an ideal language like Leibniz's, which Frege declares to be a quite hard and idealistic—though not impossible—task). Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter-century. This is the first work in Analytical Philosophy, a field that later British and Anglo ... [...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, philosophy of logic, logic, and Philosophy of mathematics, 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 History of logic#Rise of modern logic, 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 ci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Notation
Gauss notation (also known as a Gauss code or Gauss words) 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 after the German mathematician Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ... (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 labelled 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 poi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855. While studying at the University of Göttingen, he propounded several mathematical theorems. As an independent scholar, he wrote the masterpieces '' Disquisitiones Arithmeticae'' and ''Theoria motus corporum coelestium''. Gauss produced the second and third complete proofs of the fundamental theorem of algebra. In number theory, he made numerous contributions, such as the composition law, the law of quadratic reciprocity and the Fermat polygonal number theorem. He also contributed to the theory of binary and ternary quadratic forms, the construction of the heptadecagon, and the theory of hypergeometric series. Due to Gauss' extensive and fundamental contributions to science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pictographs
A pictogram (also pictogramme, pictograph, or simply picto) is a graphical symbol that conveys meaning through its visual resemblance to a physical object. Pictograms are used in systems of writing and visual communication. A pictography is a writing system which uses pictograms. Some pictograms, such as hazard pictograms, may be elements of formal languages. In the field of prehistoric art, the term "pictograph" has a different definition, and specifically refers to art painted on rock surfaces. Pictographs are contrasted with petroglyphs, which are carved or incised. Small pictograms displayed on a computer screen in order to help the user navigate are called '' icons''. Historical Early written symbols were based on pictograms (pictures which resemble what they signify) and ideograms (symbols which represent ideas). Ancient Sumerian, Egyptian, and Chinese civilizations began to adapt such symbols to represent concepts, developing them into logographic writing systems. Pic ... [...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. Uses International communication Many Leibniz scholars writing in English seem to agree that he intended his ''characteristica universalis'' or "universal character" to be a form of pasigraphy, or ideographic language. This was to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics. Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. 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. Leibniz contributed to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Language
A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and change. It can take different forms, typically either a spoken language or a sign language. Natural languages are distinguished from constructed and formal languages such as those used to program computers or to study logic. Defining natural language Natural languages include ones that are associated with linguistic prescriptivism or language regulation. ( Nonstandard dialects can be viewed as a wild type in comparison with standard languages.) An official language with a regulating academy such as Standard French, overseen by the , is classified as a natural language (e.g. in the field of natural language processing), as its prescriptive aspects do not make it constructed enough to be a constructed language or controlled enough to be a controlled natural language. Natural language are different from: * artificial and constructed la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
