|
Apply
In mathematics and computer science, apply is a function that applies a function to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has a role in the study of the denotational semantics of computer programs, because it is a continuous function on complete partial orders. Apply is also a continuous function in homotopy theory, and, indeed underpins the entire theory: it allows a homotopy deformation to be viewed as a continuous path in the space of functions. Likewise, valid mutations (refactorings) of computer programs can be seen as those that are "continuous" in the Scott topology. The most general setting for apply is in category theory, where it is right adjoint to currying in closed monoidal categories. A special case of this are the Cartesian closed categories, whose internal language is simply typed lambda calculus. Programming In computer programming, apply applies a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Application
In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abstraction Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or " concrete") signifiers, first principles, or other methods. "An abst .... Representation Function application is usually depicted by juxtaposing the variable representing the function with its argument encompassed in parentheses. For example, the following expression represents the application of the function ''ƒ'' to its argument ''x''. :f(x) In some instances, a different notation is used where the parentheses aren't required, and function application can be expressed just by juxtaposition. For example, the following expression can be considered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scott Continuity
In mathematics, given two partially ordered sets ''P'' and ''Q'', a function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset ''D'' of ''P'' with supremum in ''P'', its image has a supremum in ''Q'', and that supremum is the image of the supremum of ''D'', i.e. \sqcup f = f(\sqcup D), where \sqcup is the directed join. When Q is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets. A subset ''O'' of a partially ordered set ''P'' is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets ''D'' with supremum in ''O'' have non-empty intersection with ''O''. The Scott-open subsets of a partially ordered set ''P'' form a topology on ''P'', the Scott topology. A function b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Language
__NOTOC__ Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category (mathematics), category, and an Interpretation (logic), interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type theory, type-theoretic constructions. The subject has been recognisable in these terms since around 1970. Overview There are three important themes in the categorical approach to logic: ;Categorical semantics: Categorical logic introduces the notion of ''structure valued in a category'' C with the classical model theory, model theoretic notion of a structure appearing in the particular case where C is the Category of sets, category of sets and functions. This notion has proven useful when the set-theoretic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haskell (programming Language)
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 has pioneered a number of programming language features such as type classes, which enable type-safe operator overloading, and monadic IO. Haskell's main implementation is the Glasgow Haskell Compiler (GHC). It is named after logician Haskell Curry. 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. His ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Go (programming Language)
Go is a statically typed, compiled programming language designed at Google by Robert Griesemer, Rob Pike, and Ken Thompson. It is syntactically similar to C, but with memory safety, garbage collection, structural typing, and CSP-style concurrency. It is often referred to as Golang because of its former domain name, golang.org, but its proper name is Go. There are two major implementations: * Google's self-hosting "gc" compiler toolchain, targeting multiple operating systems and WebAssembly. * gofrontend, a frontend to other compilers, with the ''libgo'' library. With GCC the combination is gccgo; with LLVM the combination is gollvm. A third-party source-to-source compiler, GopherJS, compiles Go to JavaScript for front-end web development. History Go was designed at Google in 2007 to improve programming productivity in an era of multicore, networked machines and large codebases. The designers wanted to address criticism of other languages in use at Google, b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection (computer Science)
In computer science, reflective programming or reflection is the ability of a process to examine, introspect, and modify its own structure and behavior. Historical background The earliest computers were programmed in their native assembly languages, which were inherently reflective, as these original architectures could be programmed by defining instructions as data and using self-modifying code. As the bulk of programming moved to higher-level compiled languages such as Algol, Cobol, Fortran, Pascal, and C, this reflective ability largely disappeared until new programming languages with reflection built into their type systems appeared. Brian Cantwell Smith's 1982 doctoral dissertation introduced the notion of computational reflection in procedural programming languages and the notion of the meta-circular interpreter as a component of 3-Lisp. Uses Reflection helps programmers make generic software libraries to display data, process different formats of data, perfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Java (programming Language)
Java is a high-level, class-based, object-oriented programming language that is designed to have as few implementation dependencies as possible. It is a general-purpose programming language intended to let programmers ''write once, run anywhere'' ( WORA), meaning that compiled Java code can run on all platforms that support Java without the need to recompile. Java applications are typically compiled to bytecode that can run on any Java virtual machine (JVM) regardless of the underlying computer architecture. The syntax of Java is similar to C and C++, but has fewer low-level facilities than either of them. The Java runtime provides dynamic capabilities (such as reflection and runtime code modification) that are typically not available in traditional compiled languages. , Java was one of the most popular programming languages in use according to GitHub, particularly for client–server web applications, with a reported 9 million developers. Java was originally develo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C Sharp (programming Language)
C# (pronounced ) is a general-purpose, high-level multi-paradigm programming language. C# encompasses static typing, strong typing, lexically scoped, imperative, declarative, functional, generic, object-oriented (class-based), and component-oriented programming disciplines. The C# programming language was designed by Anders Hejlsberg from Microsoft in 2000 and was later approved as an international standard by Ecma (ECMA-334) in 2002 and ISO/IEC (ISO/IEC 23270) in 2003. Microsoft introduced C# along with .NET Framework and Visual Studio, both of which were closed-source. At the time, Microsoft had no open-source products. Four years later, in 2004, a free and open-source project called Mono began, providing a cross-platform compiler and runtime environment for the C# programming language. A decade later, Microsoft released Visual Studio Code (code editor), Roslyn (compiler), and the unified .NET platform (software framework), all of which support C# and are free, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Common Lisp
Common Lisp (CL) is a dialect of the Lisp programming language, published in ANSI standard document ''ANSI INCITS 226-1994 (S20018)'' (formerly ''X3.226-1994 (R1999)''). The Common Lisp HyperSpec, a hyperlinked HTML version, has been derived from the ANSI Common Lisp standard. The Common Lisp language was developed as a standardized and improved successor of Maclisp. By the early 1980s several groups were already at work on diverse successors to MacLisp: Lisp Machine Lisp (aka ZetaLisp), Spice Lisp, NIL and S-1 Lisp. Common Lisp sought to unify, standardise, and extend the features of these MacLisp dialects. Common Lisp is not an implementation, but rather a language specification. Several implementations of the Common Lisp standard are available, including free and open-source software and proprietary products. Common Lisp is a general-purpose, multi-paradigm programming language. It supports a combination of procedural, functional, and object-oriented programming paradig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variadic Function
In mathematics and in computer programming, a variadic function is a function of indefinite arity, i.e., one which accepts a variable number of arguments. Support for variadic functions differs widely among programming languages. The term ''variadic'' is a neologism, dating back to 1936–1937. The term was not widely used until the 1970s. Overview There are many mathematical and logical operations that come across naturally as variadic functions. For instance, the summing of numbers or the concatenation of strings or other sequences are operations that can be thought of as applicable to any number of operands (even though formally in these cases the associative property is applied). Another operation that has been implemented as a variadic function in many languages is output formatting. The C function and the Common Lisp function are two such examples. Both take one argument that specifies the formatting of the output, and ''any number'' of arguments that provide the val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Reduction
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(\lam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
_waving.svg "Click for more on -> Java (programming Language)")
