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Mizar System
The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library of formalized mathematics, which can be used in the proof of new theorems. The system is maintained and developed by the Mizar Project, formerly under the direction of its founder Andrzej Trybulec. In 2009 the Mizar Mathematical Library was the largest coherent body of strictly formalized mathematics in existence. History The Mizar Project was started around 1973 by Andrzej Trybulec as an attempt to reconstruct mathematical vernacular so it can be checked by a computer. Its current goal, apart from the continual development of the Mizar System, is the collaborative creation of a large library of formally verified proofs, covering most of the core of modern mathematics. This is in line with the influential QED manifesto. Currently the project is developed and maintained by resear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mizar System Logo
Mizar is a second-magnitude star in the handle of the Big Dipper asterism in the constellation of Ursa Major. It has the Bayer designation ζ Ursae Majoris ( Latinised as Zeta Ursae Majoris). It forms a well-known naked eye double star with the fainter star Alcor, and is itself a quadruple star system. The Mizar and Alcor system lies about 83 light-years away from the Sun, as measured by the Hipparcos astrometry satellite, and is part of the Ursa Major Moving Group. Nomenclature ''ζ Ursae Majoris'' ( Latinised to ''Zeta Ursae Majoris'' and abbreviated to ζ UMa or Zeta UMa) is Mizar's Bayer designation. It also has the Flamsteed designation 79 Ursae Majoris. The traditional name ''Mizar'' derives from the Arabic ' ' meaning 'apron; wrapper, covering, cover'. In 2016, the International Astronomical Union organized a Working Group on Star Names (WGSN) to catalog and standardize proper names for stars. The WGSN's first bulletin of July 201 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shinshu University
, abbreviated to , is a Japanese national university located in Matsumoto, Nagano Prefecture, Japan. As the only university in Japan bearing the name of a former Japanese province, it bears the name "Shinshu" (alternative name for Shinano Province), and is firmly rooted in the many regions of Nagano Prefecture. It was the 18th ranked higher education institution in Japan. The University has several campuses; Matsumoto, Nagano, Ueda and Minami-Minowa. Nagano Prefecture is located in the center of Japan, a region blessed with traditional culture that is kept alive at many places throughout the prefecture such as Zenkoji, Matsumoto Castle, Ueda Castle. The school was founded in 1873 and it was established as a university in 1949, at which time the following institutions were subsumed into it; Nagano Normal School (established in 1873), Ueda Textile College (1910), Nagano Youths Normal School (1918), Matsumoto High School (1919), Nagano Technical College (1943), Matsumoto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kőnig's Lemma
Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives a sufficient condition for an infinite graph to have an infinitely long path. The computability aspects of this theorem have been thoroughly investigated by researchers in mathematical logic, especially in computability theory. This theorem also has important roles in constructive mathematics and proof theory. Statement of the lemma Let G be a connected, locally finite, infinite graph. This means that every two vertices can be connected by a finite path, the graph has infinitely many vertices, and each vertex is adjacent to only finitely many other vertices. Then G contains a ray: a simple path (a path with no repeated vertices) that starts at one vertex and continues from it through infinitely many vertices. A useful special case of the lemma is that every infinite tree contains either a vertex of infinite degree or an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hahn–Banach Theorem
The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry. History The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C , b/math> of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz. The first Hahn–Banach theorem was p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peer-review
Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work ( peers). It functions as a form of self-regulation by qualified members of a profession within the relevant field. Peer review methods are used to maintain quality standards, improve performance, and provide credibility. In academia, scholarly peer review is often used to determine an academic paper's suitability for publication. Peer review can be categorized by the type of activity and by the field or profession in which the activity occurs, e.g., medical peer review. It can also be used as a teaching tool to help students improve writing assignments. Henry Oldenburg (1619–1677) was a German-born British philosopher who is seen as the 'father' of modern scientific peer review. Professional Professional peer review focuses on the performance of professionals, with a view to improving quality, upholding standards, or providing certification. In academia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes what's viewed as basic elements in the economy, including individual agents and markets, their interactions, and the outcomes of interactions. Individual agents may include, for example, households, firms, buyers, and sellers. Macroeconomics analyzes the economy as a system where production, consumption, saving, and investment interact, and factors affecting it: employment of the resources of labour, capital, and land, currency inflation, economic growth, and public policies that have impact on these elements. Other broad distinctions within economics include those between positive economics, describing "what is", and normative economics, advocating "what ought to be"; between economic theory and applied economics; between rati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automated Proof Checking
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. System comparison * ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition. * Coq – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification. * HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ASCII
ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of technical limitations of computer systems at the time it was invented, ASCII has just 128 code points, of which only 95 are , which severely limited its scope. All modern computer systems instead use Unicode, which has millions of code points, but the first 128 of these are the same as the ASCII set. The Internet Assigned Numbers Authority (IANA) prefers the name US-ASCII for this character encoding. ASCII is one of the IEEE milestones. Overview ASCII was developed from telegraph code. Its first commercial use was as a seven- bit teleprinter code promoted by Bell data services. Work on the ASCII standard began in May 1961, with the first meeting of the American Standards Association's (ASA) (now the American National Standards ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Declarative Programming
In computer science, declarative programming is a programming paradigm—a style of building the structure and elements of computer programs—that expresses the logic of a computation without describing its control flow. Many languages that apply this style attempt to minimize or eliminate side effects by describing ''what'' the program must accomplish in terms of the problem domain, rather than describe ''how'' to accomplish it as a sequence of the programming language primitives (the ''how'' being left up to the language's implementation). This is in contrast with imperative programming, which implements algorithms in explicit steps. Declarative programming often considers programs as theories of a formal logic, and computations as deductions in that logic space. Declarative programming may greatly simplify writing parallel programs. Common declarative languages include those of database query languages (e.g., SQL, XQuery), regular expressions, logic programming, f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class shares characteristic properties: Gabbay, Dov, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press. # Law of excluded middle and double negation elimination # Law of noncontradiction, and the principle of explosion # Monotonicity of entailment and idempotency of entailment # Commutativity of conjunction # De Morgan duality: every logical operator is dual to another While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. Shapiro, Stewart (2000). Classical Logic. In Stanford En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Formalized Reasoning
The ''Journal of Formalized Reasoning'' is a peer-reviewed open access academic journal established in 2009. It publishes formalization efforts in any area, including classical mathematics, constructive mathematics, formal algorithms, and program verifications. It is maintained by ''AlmaDL'', the digital library A digital library, also called an online library, an internet library, a digital repository, or a digital collection is an online database of digital objects that can include text, still images, audio, video, digital documents, or other digital m ... of the University of Bologna. Abstracting and indexing The journal is abstracted and indexed in Scopus, MathSciNet, and '' Zentralblatt MATH''. External links * {{Official website, http://jfr.unibo.it/ Computer science journals Open access journals Reasoning Publications established in 2008 Biannual journals University of Bologna English-language journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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