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Lambda Cube
In mathematical logic and type theory, the λ-cube (also written lambda cube) is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency between terms and types. Here, "dependency" refers to the capacity of a term or type to Free variables and bound variables, bind a term or type. The respective dimensions of the λ-cube correspond to: * x-axis (\rightarrow): types that can depend on terms, corresponding to dependent types. * y-axis (\uparrow): terms that can depend on types, corresponding to Parametric polymorphism, polymorphism. * z-axis (\nearrow): types that can depend on other types, corresponding to (binding) type operators. The different ways to combine these three dimensions yield the 8 vertices of the cube, each corresponding to a different kind of typed system. The λ-cube can be generali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Cube Img
Lambda (; uppercase , lowercase ; , ''lám(b)da'') is the eleventh letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed. Lambda gave rise to the Latin L and the Cyrillic El (Л). The ancient grammarians and dramatists give evidence to the pronunciation as () in Classical Greek times. In Modern Greek, the name of the letter, Λάμδα, is pronounced . In early Greek alphabets, the shape and orientation of lambda varied. Most variants consisted of two straight strokes, one longer than the other, connected at their ends. The angle might be in the upper-left, lower-left ("Western" alphabets) or top ("Eastern" alphabets). Other variants had a vertical line with a horizontal or sloped stroke running to the right. With the general adoption of the Ionic alphabet, Greek settled on an angle at the top; the Romans put the angle at the lower-left. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Constructor
In the area of mathematical logic and computer science known as type theory, a type constructor is a feature of a typed formal language that builds new types from old ones. Basic types are considered to be built using nullary type constructors. Some type constructors take another type as an argument, e.g., the constructors for product types, function types, power types and list types. New types can be defined by recursively composing type constructors. For example, simply typed lambda calculus can be seen as a language with a single non-basic type constructor—the function type constructor. Product types can generally be considered "built-in" in typed lambda calculi via currying. Abstractly, a type constructor is an ''n''-ary type operator taking as argument zero or more types, and returning another type. Making use of currying, ''n''-ary type operators can be (re)written as a sequence of applications of unary type operators. Therefore, we can view the type operators as a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Type System
In the branches of mathematical logic known as proof theory and type theory, a pure type system (PTS), previously known as a generalized type system (GTS), is a form of typed lambda calculus that allows an arbitrary number of Structure (mathematical logic)#Many-sorted structures, sorts and dependencies between any of these. The framework can be seen as a generalisation of Henk Barendregt, Barendregt's lambda cube, in the sense that all corners of the cube can be represented as instances of a PTS with just two sorts. In fact, Barendregt (1991) framed his cube in this setting. Pure type systems may obscure the distinction between ''types'' and ''terms'' and collapse the type hierarchy, as is the case with the calculus of constructions, but this is not generally the case, e.g. the simply typed lambda calculus allows only terms to depend on terms. Pure type systems were independently introduced by Stefano Berardi (1988) and Jan Terlouw (1989). Barendregt discussed them at length in his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automath
Automath ("automating mathematics") is a formal language, devised by Nicolaas Govert de Bruijn starting in 1967, for expressing complete mathematical theories in such a way that an included automated proof checker can verify their correctness. Overview The Automath system included many novel notions that were later adopted and/or reinvented in areas such as typed lambda calculus and explicit substitution. Dependent types is one outstanding example. Automath was also the first practical system that exploited the Curry–Howard correspondence. Propositions were represented as sets (called "categories") of their proofs, and the question of provability became a question of non-emptiness (type inhabitation); de Bruijn was unaware of Howard's work, and stated the correspondence independently. L. S. van Benthem Jutting, as part of this Ph.D. thesis in 1976, translated Edmund Landau's ''Foundations of Analysis'' into Automath and checked its correctness. Automath was never widely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Arithmetic
In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic. The importance of formalizing arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Impredicativity
In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions. The opposite of impredicativity is predicativity, which essentially entails building stratified (or ramified) theories where quantification over a type at one 'level' results in types at a new, higher, level. A prototypical example is intuitionistic type theory, which retains ramification (without the explicit levels) so as to discard impredicativity. The 'levels' here correspond to the number of layers of dependency in a term definition. Russell's paradox is a famous example of an impredicative construction—namely the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archiv Für Mathematische Logik Und Grundlagenforschung
'' Archive for Mathematical Logic'' is a peer-reviewed mathematics journal published by Springer Science+Business Media. It was established in 1950 and publishes articles on mathematical logic. Abstracting and indexing The journal is abstracted and indexed in: Springer. 2022 * * * * According t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Transitive Closure
In mathematics, a subset of a given set is closed under an operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest superset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set equipped with one or several methods for producing elements of from other elements of . Operations and (partial) multivariate function are examples of such methods. If is a topological space, the limit of a sequence of eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
β-reduction
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computation based on function abstraction and application using variable binding and 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 logically consistent, and documented it in 1940. Lambda calculus consists of constructing lambda terms and performing 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 variable is a character or string representing a parameter. # (\lambda x.M): A lambda abstraction is a function definition, taking as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System F-omega
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-case \l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curry Howard Isomorphism
Curry is a dish with a sauce or gravy seasoned with spices, mainly derived from the interchange of Indian cuisine with European taste in food, starting with the Portuguese, followed by the Dutch and British, and then thoroughly internationalised. Many dishes that would be described as curries in English are found in the native cuisines of countries in Southeast Asia and East Asia. The English word is derived indirectly from some combination of Dravidian words. A first step in the creation of curry was the arrival in India of spicy hot chili peppers, along with other ingredients such as tomatoes and potatoes, part of the Columbian exchange of plants between the Old World and the New World. During the British Raj, Anglo-Indian cuisine developed, leading to Hannah Glasse's 18th century recipe for "currey the India way" in England. Curry was then spread in the 19th century by indentured Indian sugar workers to the Caribbean, and by British traders to Japan. Further exchanges ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
