|
Calculus Of Constructions
In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants. Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity). General traits The CoC is a higher-order typed lambda calculus, initially developed by Thierry Coquand. It is well known for being at the top of Barendregt's lambda cube. It is possible within CoC to define functions from terms to terms, as well as terms to types, types to types, and types to terms. The CoC is strongly normalizing, and hence consistent. Usage The CoC has been developed alongs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uzbekistan
Uzbekistan (, ; uz, Ozbekiston, italic=yes / , ; russian: Узбекистан), officially the Republic of Uzbekistan ( uz, Ozbekiston Respublikasi, italic=yes / ; russian: Республика Узбекистан), is a doubly landlocked country located in Central Asia. It is surrounded by five landlocked countries: Kazakhstan to the north; Kyrgyzstan to the northeast; Tajikistan to the southeast; Afghanistan to the south; and Turkmenistan to the southwest. Its capital and largest city is Tashkent. Uzbekistan is part of the Turkic world, as well as a member of the Organization of Turkic States. The Uzbek language is the majority-spoken language in Uzbekistan, while Russian is widely spoken and understood throughout the country. Tajik is also spoken as a minority language, predominantly in Samarkand and Bukhara. Islam is the predominant religion in Uzbekistan, most Uzbeks being Sunni Muslims. The first recorded settlers in what is now Uzbekistan were Easter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matita
Matita is an experimental proof assistant under development at the Computer Science Department of the University of Bologna. It is a tool aiding the development of formal proofs by man-machine collaboration, providing a programming environment where formal specifications, executable algorithms and automatically verifiable correctness certificates naturally coexist. Matita is based on a dependent type system known as the Calculus of (Co)Inductive Constructions (a derivative of Calculus of Constructions), and is compatible, to some extent, with Coq. The word "matita" means "pencil" in Italian (a simple and widespread editing tool). It is a reasonably small and simple application, whose architectural and software complexity is meant to be mastered by students, providing a tool particularly suited for testing innovative ideas and solutions. Matita adopts a tactic-based editing mode; (XML-encoded) proof objects are produced for storage and exchange. Main features Existential variabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Type Theory
In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies. This includes, among other lines of work, the construction of homotopical and Higher category theory, higher-categorical Model (mathematical logic), models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and higher category theory; the development of mathematics within a type-theoretic foundation of mathematics, foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the Formal proof, formalization of each of these in computer proof assistants. There is a large overlap between the work referred to as homotopy type theory, and as the univalent foundations project. Although neither is precisely delinea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dependent Type
In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". In functional programming languages like Agda, ATS, Coq, F*, Epigram, and Idris, dependent types help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations. Two common examples of dependent types are ''dependent functions'' and ''dependent pairs''. The return type of a dependent function may depend on the ''value'' (not just type) of one of its arguments. For instance, a function that takes a positive integer n may return an array of length n, where the array length is part of the type of the array. (Note that this is different from polymorphism and generic programming, both of which include the type as an argument.) A dependent pair may have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 bind a term or type. The respective dimensions of the λ-cube correspond to: * x-axis (\rightarrow): types that can bind terms, corresponding to dependent types. * y-axis (\uparrow): terms that can bind types, corresponding to polymorphism. * z-axis (\nearrow): types that can bind 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 generalized into the concept of a pure type system. Examples of Systems (λ� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Type System
__NOTOC__ 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 sorts and dependencies between any of these. The framework can be seen as a generalisation of 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 subsequent papers. In his PhD thesis, Berardi defined a cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Church Encoding
In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded data in the lambda calculus this way. Terms that are usually considered primitive in other notations (such as integers, booleans, pairs, lists, and tagged unions) are mapped to higher-order functions under Church encoding. The Church-Turing thesis asserts that any computable operator (and its operands) can be represented under Church encoding. In the untyped lambda calculus the only primitive data type is the function. Use A straightforward implementation of Church encoding slows some access operations from O(1) to O(n), where n is the size of the data structure, making Church encoding impractical. Research has shown that this can be addressed by targeted optimizations, but most functional programming languages instead expand their in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backus–Naur Form
In computer science, Backus–Naur form () or Backus normal form (BNF) is a metasyntax notation for context-free grammars, often used to describe the syntax of languages used in computing, such as computer programming languages, document formats, instruction sets and communication protocols. It is applied wherever exact descriptions of languages are needed: for instance, in official language specifications, in manuals, and in textbooks on programming language theory. Many extensions and variants of the original Backus–Naur notation are used; some are exactly defined, including extended Backus–Naur form (EBNF) and augmented Backus–Naur form (ABNF). Overview A BNF specification is a set of derivation rules, written as ::= __expression__ where: * is a '' nonterminal'' (variable) and the __expression__ consists of one or more sequences of either terminal or nonterminal symbols; * means that the symbol on the left must be replaced with the expression on the right. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpreta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Typed Lambda Calculus
A typed lambda calculus is a typed formalism that uses the lambda-symbol (\lambda) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered (see kinds below). From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus, but from another point of view, they can also be considered the more fundamental theory and ''untyped lambda calculus'' a special case with only one type. Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages. Typed lambda calculi play an important role in the design of type systems for programming languages; here, typability usually captures desirable properties of the program (e.g., the program will not cause a memory a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
