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Currying
In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f that takes three arguments creates a nested unary function g, so that the code :\textx=f(a,b,c) gives x the same value as the code : \begin \texth = g(a) \\ \texti = h(b) \\ \textx = i(c), \end or called in sequence, :\textx = g(a)(b)(c). In a more mathematical language, a function that takes two arguments, one from X and one from Y, and produces outputs in Z, by currying is translated into a function that takes a single argument from X and produces as outputs ''functions'' from Y to Z. This is a natural one-to-one correspondence between these two types of functions, so that the sets together with functions between them form a Cartesian closed category. The currying of a function with more than two arguments can then be defined by induction. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Application
In computer science, partial application (or partial function application) refers to the process of fixing a number of arguments to a function, producing another function of smaller arity. Given a function f \colon (X \times Y \times Z) \to N , we might fix (or 'bind') the first argument, producing a function of type \text(f) \colon (Y \times Z) \to N . Evaluation of this function might be represented as f_(2, 3). Note that the result of partial function application in this case is a function that takes two arguments. Partial application is sometimes incorrectly called currying, which is a related, but distinct concept. Motivation Intuitively, partial function application says "if you fix the first arguments of the function, you get a function of the remaining arguments". For example, if function ''div''(''x'',''y'') = ''x''/''y'', then ''div'' with the parameter ''x'' fixed at 1 is another function: ''div''1(''y'') = ''div''(1,''y'') = 1/''y''. This is the same as the funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Closed Category
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation. Etymology Named after (1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of Cartesian product, which was later generalized to the notion of categorical product. Definition The category ''C'' is called Cartesian closed if and only if it satisfies the following three properties: * It has a terminal object. * Any two objects ''X'' and ''Y'' of ''C'' have a product ''X'' ×''Y'' in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haskell Curry
Haskell Brooks Curry (; September 12, 1900 – September 1, 1982) was an American mathematician and logician. Curry is best known for his work in combinatory logic. While the initial concept of combinatory logic was based on a single paper by Moses Schönfinkel, Curry did much of the development. Curry is also known for Curry's paradox and the Curry–Howard correspondence. There are three programming languages named after him, Haskell, Brook and Curry, as well as the concept of ''currying'', a technique used for transforming functions in mathematics and computer science. Life Curry was born on September 12, 1900, 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 MIT, he returned to Harvard to study physics, earning an MA in 1924. Curry's interes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ML (programming Language)
ML (Meta Language) is a general-purpose functional programming language. It is known for its use of the polymorphic Hindley–Milner type system, which automatically assigns the types of most expressions without requiring explicit type annotations, and ensures type safetythere 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. It 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, first-class functions, automatic memory management through garbage collection, parametric po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moses Schönfinkel
Moses Ilyich Schönfinkel (russian: Моисей Исаевич Шейнфинкель, translit=Moisei Isai'evich Sheinfinkel; 29 September 1888 – 1942) was a logician and mathematician, known for the invention of combinatory logic. Life Moses Schönfinkel was born in 1888 in Ekaterinoslav, Russian Empire (now Dnipro, Ukraine). Moses Schönfinkel 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 deli ... [...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-theoretic formu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameter (computer Science)
In computer programming, a parameter or a formal argument is a special kind of variable used in a subroutine to refer to one of the pieces of data provided as input to the subroutine. These pieces of data are the values of the arguments (often called ''actual arguments'' or ''actual parameters'') with which the subroutine is going to be called/invoked. An ordered list of parameters is usually included in the definition of a subroutine, so that, each time the subroutine is called, its arguments for that call are evaluated, and the resulting values can be assigned to the corresponding parameters. Unlike ''argument'' in usual mathematical usage, the ''argument'' in computer science is the actual input expression passed/supplied to a function, procedure, or routine in the invocation/call statement, whereas the ''parameter'' is the variable inside the implementation of the subroutine. For example, if one defines the add subroutine as def add(x, y): return x + y, then x, y are parame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Programming Language
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that map values to other values, rather than a sequence of imperative statements which update the 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 identifiers), passed as arguments, and returned from other functions, just as any other data type can. This allows programs to be written in a declarative and composable style, where small functions are combined in a modular manner. Functional programming is sometimes treated as synonymous with purely functional programming, a subset of functional programming which treats all functions as deterministic mathematical functions, or pure functions. When a pure function is called w ... [...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. 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. His elder sister was the writer Barbara Strachey. In 1919, the family moved to 51 Gordon Square. The Stracheys b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limit (mathematics), limits, continuous function, continuity and derivatives. The set of real numbers is mathematical notation, denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers subset, include t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes).Lyons, John. "Semantics, vol. I." Cambridge: Cambridge (1977). p.158 and on. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. Basic terminology In modern usage, a definition is something, typically expressed in words, that at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
