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Applicative Functor
In functional programming, an applicative functor, or an applicative for short, is an intermediate structure between functors and monads. In category theory they are calleclosed monoidal functors Applicative functors allow for functorial computations to be sequenced (unlike plain functors), but don't allow using results from prior computations in the definition of subsequent ones (unlike monads). Applicative functors are the programming equivalent of lax monoidal functors with tensorial strength in category theory. Applicative functors were introduced in 2008 by Conor McBride and Ross Paterson in their paper ''Applicative programming with effects''. Applicative functors first appeared as a library feature in Haskell, but have since spread to other languages such as Idris, Agda, OCaml, Scala, and F#. Glasgow Haskell, Idris, and F# offer language features designed to ease programming with applicative functors. In Haskell, applicative functors are implemented in the Applicativ ... [...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] |
Functor (functional Programming)
In functional programming, a functor is a design pattern inspired by the definition from category theory that allows one to apply a function to values inside a generic type without changing the structure of the generic type. In Haskell this idea can be captured in a type class: class Functor f where fmap :: (a -> b) -> f a -> f b This declaration says that any instance of Functor must support a method fmap, which maps a function over the elements of the instance. Functors in Haskell should also obey the so-called ''functor laws'', which state that the mapping operation preserves the identity function and composition of functions: fmap id = id fmap (g . h) = (fmap g) . (fmap h) where . stands for function composition. In Scala a trait can instead be used: trait Functor comonads, all of which build atop a canonical functor structure. Functors are useful in modeling functional effects by values of parameterized data types. Modifiable computations are modeled by allo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monad (functional Programming)
In functional programming, monads are a way to structure computations as a sequence of steps, where each step not only produces a value but also some extra information about the computation, such as a potential failure, non-determinism, or side effect. More formally, a monad is a type constructor M equipped with two operations, which lifts a value into the monadic context, and which chains monadic computations. In simpler terms, monads can be thought of as interfaces implemented on type constructors, that allow for functions to abstract over various type constructor variants that implement monad (e.g. , , etc.). Both the concept of a monad and the term originally come from category theory, where a monad is defined as an endofunctor with additional structure. Research beginning in the late 1980s and early 1990s established that monads could bring seemingly disparate computer-science problems under a unified, functional model. Category theory also provides a few formal requirem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Functor
In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two ''coherence maps''—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors * The coherence maps of lax monoidal functors satisfy no additional properties; they are not necessarily invertible. * The coherence maps of strong monoidal functors are invertible. * The coherence maps of strict monoidal functors are identity maps. Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors. Defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensorial Strength
In category theory, a strong monad is a monad on a monoidal category with an additional natural transformation, called the strength, which governs how the monad interacts with the monoidal product. Strong monads play an important role in theoretical computer science where they are used to model computation with side effects. Definition A (left) strong monad is a monad (''T'', η, μ) over a monoidal category (''C'', ⊗, I) together with a natural transformation ''t''''A,B'' : ''A'' ⊗ ''TB'' → ''T''(''A'' ⊗ ''B''), called (''tensorial'') ''left strength'', such that the diagrams :, , :, and commute for every object ''A'', ''B'' and ''C''. Commutative strong monads For every strong monad ''T'' on a symmetric monoidal category, a ''right strength'' natural transformation can be defined by t'_=T(\gamma_)\circ t_\circ\gamma_ : TA\otimes B\to T(A\otimes B). A strong monad ''T'' is said to be commutative when the diagram : commutes for all objects A and B. Properties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haskell (programming Language)
Haskell () is a General-purpose programming language, general-purpose, static typing, statically typed, purely functional programming, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research, and industrial applications, Haskell pioneered several programming language #Features, features such as type classes, which enable type safety, type-safe operator overloading, and Monad (functional programming), monadic input/output (IO). It is named after logician Haskell Curry. Haskell's main implementation is the Glasgow Haskell Compiler (GHC). Haskell's Semantics (computer science), semantics are historically based on those of the Miranda (programming language), 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idris (programming Language)
Idris is a purely-functional programming language with dependent types, optional lazy evaluation, and features such as a totality checker. Idris may be used as a proof assistant, but is designed to be a general-purpose programming language similar to Haskell. The Idris type system is similar to Agda's, and proofs are similar to Coq's, including tactics (theorem proving functions/procedures) via elaborator reflection. Compared to Agda and Coq, Idris prioritizes management of side effects and support for embedded domain-specific languages. Idris compiles to C (relying on a custom copying garbage collector using Cheney's algorithm) and JavaScript (both browser- and Node.js-based). There are third-party code generators for other platforms, including Java virtual machine (JVM), Common Intermediate Language (CIL), and LLVM. Idris is named after a singing dragon from the 1970s UK children's television program '' Ivor the Engine''. Features Idris combines a number of featu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Agda (programming Language)
Agda is a dependently typed functional programming language originally developed by Ulf Norell at Chalmers University of Technology with implementation described in his PhD thesis. The original Agda system was developed at Chalmers by Catarina Coquand in 1999. The current version, originally named Agda 2, is a full rewrite, which should be considered a new language that shares a name and tradition. Agda is also a proof assistant based on the ''propositions-as-types'' paradigm (Curry–Howard correspondence), but unlike Rocq, has no separate ''tactics'' language, and proofs are written in a functional programming style. The language has ordinary programming constructs such as data types, pattern matching, records, let expressions and modules, and a Haskell-like syntax. The system has Emacs, Atom, and VS Code interfaces but can also be run in batch processing mode from a command-line interface. Agda is based on Zhaohui Luo's unified theory of dependent types (UTT), a type ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OCaml
OCaml ( , formerly Objective Caml) is a General-purpose programming language, general-purpose, High-level programming language, high-level, Comparison of multi-paradigm programming languages, multi-paradigm programming language which extends the Caml dialect of ML (programming language), ML with Object-oriented programming, object-oriented features. OCaml was created in 1996 by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy, Ascánder Suárez, and others. The OCaml toolchain includes an interactive top-level Interpreter (computing), interpreter, a bytecode compiler, an optimizing native code compiler, a reversible debugger, and a package managerOPAM together with a composable build system for OCamlDune. OCaml was initially developed in the context of automated theorem proving, and is used in static program analysis, static analysis and formal methods software. Beyond these areas, it has found use in systems programming, web development, and specific financial utili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scala (programming Language)
Scala ( ) is a strongly statically typed high-level general-purpose programming language that supports both object-oriented programming and functional programming. Designed to be concise, many of Scala's design decisions are intended to address criticisms of Java. Scala source code can be compiled to Java bytecode and run on a Java virtual machine (JVM). Scala can also be transpiled to JavaScript to run in a browser, or compiled directly to a native executable. When running on the JVM, Scala provides language interoperability with Java so that libraries written in either language may be referenced directly in Scala or Java code. Like Java, Scala is object-oriented, and uses a syntax termed '' curly-brace'' which is similar to the language C. Since Scala 3, there is also an option to use the off-side rule (indenting) to structure blocks, and its use is advised. Martin Odersky has said that this turned out to be the most productive change introduced in Scala 3. Unlike J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F Sharp (programming Language)
F# (pronounced F sharp) is a general-purpose, high-level, strongly typed, multi-paradigm programming language that encompasses functional, imperative, and object-oriented programming methods. It is most often used as a cross-platform Common Language Infrastructure (CLI) language on .NET, but can also generate JavaScript and graphics processing unit (GPU) code. F# is developed by the F# Software Foundation, Microsoft and open contributors. An open source, cross-platform compiler for F# is available from the F# Software Foundation. F# is a fully supported language in Visual Studio and JetBrains Rider. Plug-ins supporting F# exist for many widely used editors including Visual Studio Code, Vim, and Emacs. F# is a member of the ML language family and originated as a .NET Framework implementation of a core of the programming language OCaml. It has also been influenced by C#, Python, Haskell, Scala and Erlang. History Versions Language evolution F# uses an open deve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
