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Currying
In mathematics and computer science, currying is the technique of translating a function that takes multiple arguments into a sequence of families of functions, each taking a single argument. In the prototypical example, one begins with a function f:(X\times Y)\to Z that takes two arguments, one from X and one from Y, and produces objects in Z. The curried form of this function treats the first argument as a parameter, so as to create a family of functions f_x :Y\to Z. The family is arranged so that for each object x in X, there is exactly one function f_x. In this example, \mbox itself becomes a function that takes f as an argument, and returns a function that maps each x to f_x. The proper notation for expressing this is verbose. The function f belongs to the set of functions (X\times Y)\to Z. Meanwhile, f_x belongs to the set of functions Y\to Z. Thus, something that maps x to f_x will be of the type X\to \to Z With this notation, \mbox is a function that takes objects from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Curry–Howard Correspondence
In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs. It is also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation. It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and the logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moses Schönfinkel
Moses Ilyich Schönfinkel (; 29 September 1888 – ) was a logician and mathematician, known for the invention of combinatory logic. Life Moses Schönfinkel was born on in Ekaterinoslav, Russian Empire (now Dnipro, Ukraine). He was born to a Jewish family. His father was Ilya Girshevich Schönfinkel, a merchant of first guild, who was in а grocery store trade, and his mother, Maria “Masha” Gertsovna Schönfinkel (née Lurie) came from a prominent Lurie family. Moses had siblings named Deborah, Natan, Israel and Grigoriy. Schönfinkel attended the Novorossiysk University of Odessa, studying mathematics under Samuil Osipovich Shatunovskii (1859–1929), who worked in geometry and the foundations of mathematics. From 1914 to 1924, Schönfinkel was a member of David Hilbert's group at the University of Göttingen in Germany. On 7 December 1920 he delivered a talk entitled ''Elemente der Logik'' ("Elements of Logic") to the group where he outlined the concept of combinatory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ML (programming Language)
ML (Meta Language) is a general-purpose, high-level, functional programming language. It is known for its use of the polymorphic Hindley–Milner type system, which automatically assigns the data types of most expressions without requiring explicit type annotations ( type inference), and ensures type safety; there is a formal proof that a well-typed ML program does not cause runtime type errors. ML provides pattern matching for function arguments, garbage collection, imperative programming, call-by-value and currying. While a general-purpose programming language, ML is used heavily in programming language research and is one of the few languages to be completely specified and verified using formal semantics. Its types and pattern matching make it well-suited and commonly used to operate on other formal languages, such as in compiler writing, automated theorem proving, and formal verification. Overview Features of ML include a call-by-value evaluation strategy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameter (computer Science)
In computer programming, a parameter, a.k.a. formal argument, is a variable that represents an argument, a.k.a. actual argument, a.k.a. actual parameter, to a subroutine call.. A function's signature defines its parameters. A call invocation involves evaluating each argument expression of a call and associating the result with the corresponding parameter. For example, consider subroutine def add(x, y): return x + y. Variables x and y are parameters. For call add(2, 3), the expressions 2 and 3 are arguments. For call add(a+1, b+2), the arguments are a+1 and b+2. Parameter passing is defined by a programming language. Evaluation strategy defines the semantics for how parameters can be declared and how arguments are passed to a subroutine. Generally, with call by value, a parameter acts like a new, local variable initialized to the value of the argument. If the argument is a variable, the subroutine cannot modify the argument state because the parameter is a copy. With call by re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Programming
In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declarative programming paradigm in which function definitions are Tree (data structure), trees of Expression (computer science), expressions that map Value (computer science), values to other values, rather than a sequence of Imperative programming, imperative Statement (computer science), statements which update the State (computer science), running state of the program. In functional programming, functions are treated as first-class citizens, meaning that they can be bound to names (including local Identifier (computer languages), identifiers), passed as Parameter (computer programming), arguments, and Return value, returned from other functions, just as any other data type can. This allows programs to be written in a Declarative programming, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haskell Curry
Haskell Brooks Curry ( ; September 12, 1900 – September 1, 1982) was an American mathematician, logician and computer scientist. Curry is best known for his work in combinatory logic, whose initial concept is based on a paper by Moses Schönfinkel, for which Curry did much of the development. Curry is also known for Curry's paradox and the Curry–Howard correspondence. Named for him are three programming languages: Haskell, Brook, and Curry, and the concept of ''currying'', a method to transform functions, used in mathematics and computer science. Life Curry was born on in Millis, Massachusetts, to Samuel Silas Curry and Anna Baright Curry, who ran a school for elocution. He entered Harvard University in 1916 to study medicine but switched to mathematics before graduating in 1920. After two years of graduate work in electrical engineering at Massachusetts Institute of Technology (MIT), he returned to Harvard to study physics, earning a Master of Arts (M.A.) in 1924. Cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher Strachey
Christopher S. Strachey (; 16 November 1916 – 18 May 1975) was a British computer scientist. He was one of the founders of denotational semantics, and a pioneer in programming language design and computer time-sharing.F. J. Corbató, et al., The Compatible Time-Sharing System A Programmer's Guide' (MIT Press, 1963) . "the first paper on time-shared computers by C. Strachey at the June 1959 UNESCO Information Processing conference" He has also been credited as possibly being the first developer of a video game and for coining terms such as polymorphism and referential transparency that are still widely used by developers today. He was a member of the Strachey family, prominent in government, arts, administration, and academia. Early life and education Christopher Strachey was born on 16 November 1916 to Oliver Strachey and Rachel (Ray) Costelloe in Hampstead, England. Oliver Strachey was the son of Richard Strachey and the great-grandson of Sir Henry Strachey, 1st Baronet. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Monoidal Category
In mathematics, especially in category theory, a closed monoidal category (or a ''monoidal closed category'') is a category that is both a monoidal category and a closed category in such a way that the structures are compatible. A classic example is the category of sets, Set, where the monoidal product of sets A and B is the usual cartesian product A \times B, and the internal Hom B^A is the set of functions from A to B. A non- cartesian example is the category of vector spaces, ''K''-Vect, over a field K. Here the monoidal product is the usual tensor product of vector spaces, and the internal Hom is the vector space of linear maps from one vector space to another. The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. Many examples of closed monoidal categories are symmetric. However, this need not always be the case, as non-symmetric monoidal categories can be encountered in category-theoret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haskell
Haskell () is a general-purpose, statically typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research, and industrial applications, Haskell pioneered several programming language features such as type classes, which enable type-safe operator overloading, and monadic input/output (IO). It is named after logician Haskell Curry. Haskell's main implementation is the Glasgow Haskell Compiler (GHC). Haskell's semantics are historically based on those of the Miranda programming language, which served to focus the efforts of the initial Haskell working group. The last formal specification of the language was made in July 2010, while the development of GHC continues to expand Haskell via language extensions. Haskell is used in academia and industry. , Haskell was the 28th most popular programming language by Google searches for tutorials, and made up less than 1% of active users on the GitHub source code repository ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (computer Programming)
In programming languages, a closure, also lexical closure or function closure, is a technique for implementing lexically scoped name binding in a language with first-class functions. Operationally, a closure is a record storing a function together with an environment. The environment is a mapping associating each free variable of the function (variables that are used locally, but defined in an enclosing scope) with the value or reference to which the name was bound when the closure was created. Unlike a plain function, a closure allows the function to access those ''captured variables'' through the closure's copies of their values or references, even when the function is invoked outside their scope. History and etymology The concept of closures was developed in the 1960s for the mechanical evaluation of expressions in the λ-calculus and was first fully implemented in 1970 as a language feature in the PAL programming language to support lexically scoped first-class functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
