|
System U
In mathematical logic, System U and System U− are pure type systems, i.e. special forms of a typed lambda calculus with an arbitrary number of sorts, axioms and rules (or dependencies between the sorts). System U was proved inconsistent by Jean-Yves Girard in 1972 (and the question of consistency of System U− was formulated). This result led to the realization that Martin-Löf's original 1971 type theory was inconsistent, as it allowed the same "Type in Type" behaviour that Girard's paradox exploits. Formal definition System U is defined as a pure type system with * three sorts \; * two axioms \; and * five rules \. System U− is defined the same with the exception of the (\triangle, \ast) rule. The sorts \ast and \square are conventionally called “Type” and “ Kind”, respectively; the sort \triangle doesn't have a specific name. The two axioms describe the containment of types in kinds (\ast:\square) and kinds in \triangle (\square:\triangle). Intuitively, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
System F
System F (also polymorphic lambda calculus or second-order lambda calculus) is a typed lambda calculus that introduces, to simply typed lambda calculus, a mechanism of universal quantification over types. System F formalizes parametric polymorphism in programming languages, thus forming a theoretical basis for languages such as Haskell and ML. It was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds. Whereas simply typed lambda calculus has variables ranging over terms, and binders for them, System F additionally has variables ranging over ''types'', and binders for them. As an example, the fact that the identity function can have any type of the form ''A'' → ''A'' would be formalized in System F as the judgement :\vdash \Lambda\alpha. \lambda x^\alpha.x: \forall\alpha.\alpha \to \alpha where \alpha is a type variable. The upper-case \Lambda is traditionally used to denote type-level functions, as opposed to the lower-ca ... [...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] |
IEEE Computer Society
IEEE Computer Society (commonly known as the Computer Society or CS) is a technical society of the Institute of Electrical and Electronics Engineers (IEEE) dedicated to computing, namely the major areas of hardware, software, standards and people, "advancing the theory, practice, and application of computer and information processing science and technology." It was founded in 1946 and is the largest of 39 technical societies organized under the IEEE Technical Activities Board with over 375,000 members in 150 countries, more than 100,000 being based in the United States alone. It operates as a "global, non-governmental, not-for-profit professional society" publishing 23 peer-reviewed journals, facilitating numerous technical committees, and developing IEEE computing standards, It maintains its headquarters in Washington, DC and additional offices in California, China, and Japan. History The IEEE Computer Society traces its origins to the Subcommittee on Large-Scale Computing, est ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
