Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
A programming paradigm is a fundamental style of computer programming. Paradigms differ in the concepts and abstractions used to represent the elements of a program and the steps that compose a computation A programming paradigm is a fundamental style of computer programming. (Compare with a...
Computation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a well-defined model understood and expressed in an algorithm, protocol, network topology, etc...
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
One of the key concepts in computer programming is the idea of state, essentially a snapshot of the measure of various conditions in the system. Most programming languages require a considerable amount of state information in order to operate properly - information which is generally hidden from...
In object-oriented and functional programming, an immutable object is an object whose state cannot be modified after it is created. This is in contrast to a mutable object, which can be modified after it is created...
data. It emphasizes the application of functions, in contrast to the imperative programming
Imperative programming
In computer science, imperative programming is a programming paradigm that describes computation in terms of statements that change a program state...
style, which emphasizes changes in state. Functional programming has its roots in lambda calculus
Lambda calculus
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
developed in the 1930s to investigate function definition, function application, and recursion
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
.
In practice, the difference between a mathematical function and the notion of a "function" used in imperative programming is that imperative functions can have side effects
Side effect (computer science)
In computer science, a function or expression is said to have a side effect if, in addition to returning a value, it also modifies some state or has an observable interaction with calling functions or the outside world...
One of the key concepts in computer programming is the idea of state, essentially a snapshot of the measure of various conditions in the system. Most programming languages require a considerable amount of state information in order to operate properly - information which is generally hidden from...
. Because of this they lack referential transparency, i.e. the same language expression can result in different values at different times depending on the state of the executing program. Conversely, in functional code, the output value of a function depends only on the arguments that are input to the function, so calling a function f twice with the same value for an argument x will produce the same result f(x) both times. Eliminating side effects can make it much easier to understand and predict the behavior of a program, which is one of the key motivations for the development of functional programming.
Purely functional is a term in computing used to describe algorithms, data structures or programming languages that exclude destructive modifications...
ones, have largely been emphasized in academia rather than in commercial software development. However, prominent functional programming languages such as Common
Common Lisp
Common Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard document ANSI INCITS 226-1994 , . From the ANSI Common Lisp standard the Common Lisp HyperSpec has been derived for use with web browsers...
Clojure |closure]]") is a recent dialect of the Lisp programming language created by Rich Hickey. It is a general-purpose language supporting interactive development that encourages a functional programming style, and simplifies multithreaded programming....
OCaml , originally known as Objective Caml, is the main implementation of the Caml programming language, created by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy and others in 1996...
Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the...
and F# have been used in industrial and commercial applications by a wide variety of organizations. Functional programming is also supported in some domain-specific programming language
Domain-specific programming language
In software development and domain engineering, a domain-specific language is a programming language or specification language dedicated to a particular problem domain, a particular problem representation technique, and/or a particular solution technique...
R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
The J programming language, developed in the early 1990s by Kenneth E. Iverson and Roger Hui, is a synthesis of APL and the FP and FL function-level languages created by John Backus....
K is a proprietary array processing language developed by Arthur Whitney and commercialized by Kx Systems. The language serves as the foundation for kdb, an in-memory, column-based database, and other related financial products. The language, originally developed in 1993, is a variant of APL and...
- Features :XQuery provides the means to extract and manipulate data from XML documents or any data source that can be viewed as XML, such as relational databases or office documents....
XSLT is a declarative, XML-based language used for the transformation of XML documents. The original document is not changed; rather, a new document is created based on the content of an existing one. The new document may be serialized by the processor in standard XML syntax or in another format,...
Extensible Markup Language is a set of rules for encoding documents in machine-readable form. It is defined in the XML 1.0 Specification produced by the W3C, and several other related specifications, all gratis open standards....
). Widespread domain-specific declarative languages like SQL
SQL
SQL is a programming language designed for managing data in relational database management systems ....
Lex is a computer program that generates lexical analyzers . Lex is commonly used with the yacc parser generator. Lex, originally written by Mike Lesk and Eric Schmidt, is the standard lexical analyzer generator on many Unix systems, and a tool exhibiting its behavior is specified as part of the...
The computer program yacc is a parser generator developed by Stephen C. Johnson at AT&T for the Unix operating system. The name is an acronym for "Yet Another Compiler Compiler." It generates a parser based on an analytic grammar written in a notation similar to BNF.Yacc used to be available as...
use some elements of functional programming, especially in eschewing mutable values. Spreadsheet
Spreadsheet
A spreadsheet is a computer application that simulates a paper accounting worksheet. It displays multiple cells usually in a two-dimensional matrix or grid consisting of rows and columns. Each cell contains alphanumeric text, numeric values or formulas...
s can also be viewed as functional programming languages.
Programming in a functional style can also be accomplished in languages that aren't specifically designed for functional programming. For example, the imperative Perl programming language has been the subject of a book describing how to apply functional programming concepts.
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
provides a theoretical framework for describing functions and their evaluation. Although it is a mathematical abstraction rather than a programming language, it forms the basis of almost all functional programming languages today. An equivalent theoretical formulation, combinatory logic
Combinatory logic
Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming...
, is commonly perceived as more abstract than lambda calculus and preceded it in invention. It is used in some esoteric languages including Unlambda
Unlambda
Unlambda is a minimal, "nearly pure" functional programming language invented by David Madore. It is based on combinatory logic, a version of the lambda calculus that omits the lambda operator. It relies mainly on two built-in functions and an "apply" operator...
. Combinatory logic and lambda calculus were both originally developed to achieve a clearer approach to the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
.
An early functional flavored language was Lisp, developed by John McCarthy
John McCarthy (computer scientist)
John McCarthy was an American computer scientist and cognitive scientist. He coined the term "artificial intelligence" , invented the Lisp programming language and was highly influential in the early development of AI.McCarthy also influenced other areas of computing such as time sharing systems...
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
for the IBM 700/7000 series scientific computers in the late 1950s. Lisp introduced many features now found in functional languages, though Lisp is technically a multi-paradigm language. Scheme and Dylan
Dylan programming language
Dylan is a multi-paradigm programming language that includes support for functional and object-oriented programming, and is dynamic and reflective while providing a programming model designed to support efficient machine code generation, including fine-grained control over dynamic and static...
Information Processing Language is a programming language developed by Allen Newell, Cliff Shaw, and Herbert Simon at RAND Corporation and the Carnegie Institute of Technology from about 1956...
(IPL) is sometimes cited as the first computer-based functional programming language. It is an assembly-style language for manipulating lists of symbols. It does have a notion of "generator", which amounts to a function accepting a function as an argument, and, since it is an assembly-level language, code can be used as data, so IPL can be regarded as having higher-order functions. However, it relies heavily on mutating list structure and similar imperative features.
Kenneth Eugene Iverson was a Canadian computer scientist noted for the development of the APL programming language in 1962. He was honored with the Turing Award in 1979 for his contributions to mathematical notation and programming language theory...
developed APL in the early 1960s, described in his 1962 book A Programming Language (ISBN 9780471430148). APL was the primary influence on John Backus
John Backus
John Warner Backus was an American computer scientist. He directed the team that invented the first widely used high-level programming language and was the inventor of the Backus-Naur form , the almost universally used notation to define formal language syntax.He also did research in...
Roger Hui is a computer scientist and co-developer of the J Programming Language.He was born in Hong Kong and he immigrated to Canada with his entire family in 1966.-Education and career:In 1973, Hui entered the University of Alberta...
The J programming language, developed in the early 1990s by Kenneth E. Iverson and Roger Hui, is a synthesis of APL and the FP and FL function-level languages created by John Backus....
. In the mid 1990s, Arthur Whitney, who had previously worked with Iverson, created K
K (programming language)
K is a proprietary array processing language developed by Arthur Whitney and commercialized by Kx Systems. The language serves as the foundation for kdb, an in-memory, column-based database, and other related financial products. The language, originally developed in 1993, is a variant of APL and...
, which is used commercially in financial industries.
John Warner Backus was an American computer scientist. He directed the team that invented the first widely used high-level programming language and was the inventor of the Backus-Naur form , the almost universally used notation to define formal language syntax.He also did research in...
The Turing Award, in full The ACM A.M. Turing Award, is an annual award given by the Association for Computing Machinery to "an individual selected for contributions of a technical nature made to the computing community. The contributions should be of lasting and major technical importance to the...
lecture "Can Programming Be Liberated From the von Neumann Style? A Functional Style and its Algebra of Programs". He defines functional programs as being built up in a hierarchical way by means of "combining forms" that allow an "algebra of programs"; in modern language, this means that functional programs follow the principle of compositionality
Principle of compositionality
In mathematics, semantics, and philosophy of language, the Principle of Compositionality is the principle that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them. This principle is also called Frege's Principle,...
. Backus's paper popularized research into functional programming, though it emphasized function-level programming
Function-level programming
In computer science, function-level programming refers to one of the two contrasting programming paradigms identified by John Backus in his work on programs as mathematical objects, the other being value-level programming....
rather than the lambda-calculus style which has come to be associated with functional programming.
The University of Edinburgh, founded in 1583, is a public research university located in Edinburgh, the capital of Scotland, and a UNESCO World Heritage Site. The university is deeply embedded in the fabric of the city, with many of the buildings in the historic Old Town belonging to the university...
Professor David Turner is a British computer scientist.He has a D.Phil. from the University of Oxford. He has held professorships at Queen Mary College, London, University of Texas at Austin and the University of Kent at Canterbury, where he now retains the post of Emeritus Professor.He is...
developed initially the language SASL at the University of St. Andrews and later the language Miranda at the University of Kent
University of Kent
The University of Kent, previously the University of Kent at Canterbury, is a public research university based in Kent, United Kingdom...
. ML eventually developed into several dialects, the most common of which are now Objective Caml
Objective Caml
OCaml , originally known as Objective Caml, is the main implementation of the Caml programming language, created by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy and others in 1996...
Standard ML is a general-purpose, modular, functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of theorem provers.SML is a modern descendant of the ML...
. Also in the 1970s, the development of Scheme (a partly functional dialect of Lisp), as described in the influential Lambda Papers and the 1985 textbook Structure and Interpretation of Computer Programs
Structure and Interpretation of Computer Programs
Structure and Interpretation of Computer Programs is a textbook published in 1984 about general computer programming concepts from MIT Press written by Massachusetts Institute of Technology professors Harold Abelson and Gerald Jay Sussman, with Julie Sussman...
, brought awareness of the power of functional programming to the wider programming-languages community.
Per Erik Rutger Martin-Löf is a Swedish logician, philosopher, and mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer science. Since the late 1970s, Martin-Löf's publications have been mainly in...
Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher,...
(also called Constructive type theory), which associated functional programs with constructive proof
Constructive proof
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object...
s of arbitrarily complex mathematical propositions expressed as dependent type
Dependent type
In computer science and logic, a dependent type is a type that depends on a value. Dependent types play a central role in intuitionistic type theory and in the design of functional programming languages like ATS, Agda and Epigram....
In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by man-machine collaboration...
and has influenced the development of many subsequent functional programming languages.
Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the...
began with a consensus in 1987 to form an open standard
Open standard
An open standard is a standard that is publicly available and has various rights to use associated with it, and may also have various properties of how it was designed . There is no single definition and interpretations vary with usage....
for functional programming research; implementation releases have been ongoing since 1990.
Concepts
A number of concepts and paradigms are specific to functional programming, and generally foreign to imperative programming
Imperative programming
In computer science, imperative programming is a programming paradigm that describes computation in terms of statements that change a program state...
(including object oriented programming). However, programming languages are often hybrids of several programming paradigms so programmers using "mostly imperative" languages may have utilized some of these concepts.
In mathematics and computer science, higher-order functions, functional forms, or functionals are functions which do at least one of the following:*take one or more functions as an input*output a function...
s are functions that can either take other functions as arguments or return them as results (the differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
In computer science, a programming language is said to have first-class functions if it treats functions as first-class objects. Specifically, this means that the language supports passing functions as arguments to other functions, returning them as the values from other functions, and assigning...
s, in that higher-order functions and first-class functions both allow functions as arguments and results of other functions. The distinction between the two is subtle: "higher-order" describes a mathematical concept of functions that operate on other functions, while "first-class" is a computer science term that describes programming language entities that have no restriction on their use (thus first-class functions can appear anywhere in the program that other first-class entities like numbers can, including as arguments to other functions and as their return values).
In computer science, partial application refers to the process of fixing a number of arguments to a function, producing another function of smaller arity...
In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions each with a single argument...
, a technique in which a function is applied to its arguments one at a time, with each application returning a new function that accepts the next argument. This allows one to succinctly express, for example, the successor function as the addition operator partially applied to the natural number one.
Purely functional is a term in computing used to describe algorithms, data structures or programming languages that exclude destructive modifications...
functions (or expressions) have no memory or I/O side effects
Side effect (computer science)
In computer science, a function or expression is said to have a side effect if, in addition to returning a value, it also modifies some state or has an observable interaction with calling functions or the outside world...
. This means that pure functions have several useful properties, many of which can be used to optimize the code:
If the result of a pure expression is not used, it can be removed without affecting other expressions.
If a pure function is called with parameters that cause no side-effects, the result is constant with respect to that parameter list (sometimes called referential transparency), i.e. if the pure function is again called with the same parameters, the same result will be returned (this can enable caching optimizations such as memoization
Memoization
In computing, memoization is an optimization technique used primarily to speed up computer programs by having function calls avoid repeating the calculation of results for previously processed inputs...
).
If there is no data dependency between two pure expressions, then their order can be reversed, or they can be performed in parallel and they cannot interfere with one another (in other terms, the evaluation of any pure expression is thread-safe
Thread-safe
Thread safety is a computer programming concept applicable in the context of multi-threaded programs. A piece of code is thread-safe if it only manipulates shared data structures in a thread-safe manner, which enables safe execution by multiple threads at the same time...
).
If the entire language does not allow side-effects, then any evaluation strategy can be used; this gives the compiler freedom to reorder or combine the evaluation of expressions in a program (for example, using deforestation
Deforestation (computer science)
In the theory of programming languages in computer science, deforestation is a program transformation to eliminate tree structures....
).
While most compilers for imperative programming languages detect pure functions, and perform common-subexpression elimination for pure function calls, they cannot always do this for pre-compiled libraries, which generally do not expose this information, thus preventing optimizations that involve those external functions. Some compilers, such as gcc
GNU Compiler Collection
The GNU Compiler Collection is a compiler system produced by the GNU Project supporting various programming languages. GCC is a key component of the GNU toolchain...
, add extra keywords for a programmer to explicitly mark external functions as pure, to enable such optimizations. Fortran 95 allows functions to be designated "pure".
Iteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...
(looping) in functional languages is usually accomplished via recursion
Recursion
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem. The approach can be applied to many types of problems, and is one of the central ideas of computer science....
s invoke themselves, allowing an operation to be performed over and over. Recursion may require maintaining a stack, but tail recursion can be recognized and optimized by a compiler into the same code used to implement iteration in imperative languages. The Scheme language standard requires implementations to recognize and optimize tail recursion. Tail recursion optimization can be implemented by transforming the program into continuation passing style during compiling, among other approaches.
Common patterns of recursion can be factored out using higher order functions, with catamorphism
Catamorphism
In category theory, the concept of catamorphism denotes the unique homomorphism from an initial algebra into some other algebra. The concept has been applied to functional programming as folds.The dual concept is that of anamorphism...
Anamorphosis is a distorted projection or perspective requiring the viewer to use special devices or occupy a specific vantage point to reconstitute the image...
s (or "folds" and "unfolds") being the most obvious examples. Such higher order functions play a role analogous to built-in control structures such as loops in imperative languages.
Most general purpose functional programming languages allow unrestricted recursion and are Turing complete, which makes the halting problem
Halting problem
In computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is impossible to construct a single algorithm that always leads to a correct yes-or-no answer....
, can cause unsoundness of equational reasoning, and generally requires the introduction of inconsistency into the logic expressed by the language's type system
Type system
A type system associates a type with each computed value. By examining the flow of these values, a type system attempts to ensure or prove that no type errors can occur...
In computer science, Coq is an interactive theorem prover. It allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification...
allow only well-founded recursion and are strongly normalizing (nonterminating computations can be expressed only with infinite streams of values called codata). As a consequence, these languages fail to be Turing complete and expressing certain functions in them is impossible, but they can still express a wide class of interesting computations while avoiding the problems introduced by unrestricted recursion. Functional programming limited to well-founded recursion with a few other constraints is called total functional programming
Total functional programming
Total functional programming is a programming paradigm that restricts the range of programs to those that are provably terminating....
. See Turner 2004 for more discussion.
Strict versus non-strict evaluation
Functional languages can be categorized by whether they use strict (eager) or non-strict (lazy) evaluation, concepts that refer to how function arguments are processed when an expression is being evaluated. The technical difference is in the denotational semantics
Denotational semantics
In computer science, denotational semantics is an approach to formalizing the meanings of programming languages by constructing mathematical objects which describe the meanings of expressions from the languages...
of expressions containing failing or divergent computations. Under strict evaluation, the evaluation of any term containing a failing subterm will itself fail. For example, the expression:
print length([2+1, 3*2, 1/0, 5-4])
will fail under strict evaluation because of the division by zero in the third element of the list. Under nonstrict evaluation, the length function will return the value 4, since evaluating it will not attempt to evaluate the terms making up the list. In brief, strict evaluation always fully evaluates function arguments before invoking the function. Non-strict evaluation does not evaluate function arguments unless their values are required to evaluate the function call itself.
The usual implementation strategy for non-strict evaluation in functional languages is graph reduction
Graph reduction
In computer science, graph reduction implements an efficient version of non-strict evaluation, an evaluation strategy where the arguments to a function are not immediately evaluated. This form of non-strict evaluation is also known as lazy evaluation and used in functional programming languages...
. Non-strict evaluation is used by default in several pure functional languages, including Miranda, Clean and Haskell
Haskell (programming language)
Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the...
.
Hughes 1984 argues for non-strict evaluation as a mechanism for improving program modularity through separation of concerns
Separation of concerns
In computer science, separation of concerns is the process of separating a computer program into distinct features that overlap in functionality as little as possible. A concern is any piece of interest or focus in a program. Typically, concerns are synonymous with features or behaviors...
, by easing independent implementation of producers and consumers of data streams. Launchbury 1993 describes some difficulties that lazy evaluation introduces, particularly in analyzing a program's storage requirements, and proposes an operational semantics
Operational semantics
In computer science, operational semantics is a way to give meaning to computer programs in a mathematically rigorous way. Operational semantics are classified into two categories: structural operational semantics formally describe how the individual steps of a computation take place in a...
to aid in such analysis. Harper 2009 proposes including both strict and nonstrict evaluation in the same language, using the language's type system to distinguish them.
Type systems
Especially since the development of Hindley–Milner type inference in the 1970s, functional programming languages have tended to use typed lambda calculus
Typed lambda calculus
A typed lambda calculus is a typed formalism that uses the lambda-symbol 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...
, as opposed to the untyped lambda calculus used in Lisp and its variants (such as Scheme). The use of algebraic datatypes and pattern matching
Pattern matching
In computer science, pattern matching is the act of checking some sequence of tokens for the presence of the constituents of some pattern. In contrast to pattern recognition, the match usually has to be exact. The patterns generally have the form of either sequences or tree structures...
makes manipulation of complex data structures convenient and expressive; the presence of strong compile-time type checking makes programs more reliable, while type inference
Type inference
Type inference refers to the automatic deduction of the type of an expression in a programming language. If some, but not all, type annotations are already present it is referred to as type reconstruction....
frees the programmer from the need to manually declare types to the compiler.
Some research-oriented functional languages such as Coq
Coq
In computer science, Coq is an interactive theorem prover. It allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification...
Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher,...
, which allows types to depend on terms. Such types are called dependent type
Dependent type
In computer science and logic, a dependent type is a type that depends on a value. Dependent types play a central role in intuitionistic type theory and in the design of functional programming languages like ATS, Agda and Epigram....
s. These type systems do not have decidable type inference and are difficult to understand and program with. But dependent types can express arbitrary propositions in predicate logic
Predicate logic
In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...
. Through the Curry–Howard isomorphism, then, well-typed programs in these languages become a means of writing formal mathematical proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal methods of mathematics .- Usage :Formal verification can be...
. While these languages are mainly of interest in academic research (including in formalized mathematics), they have begun to be used in engineering as well. Compcert
Compcert
CompCert is a formally verified optimizing compiler for a subset of the C programming language which currently targets PowerPC, ARM and 32-bit x86 architectures. This project, led by Xavier Leroy, started officially in 2005, funded by the French institutes ANR and INRIA. The compiler is...
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
In functional programming, a generalized algebraic data type is a generalization of the algebraic data types of Haskell and ML, applying to parametric types.With this extension, the parameters of the return type of a data constructor can be freely chosen when declaring...
s (GADT's) can be implemented in a way that provides some of the benefits of dependently-typed programming while avoiding most of its inconvenience. GADT's are available in the Glasgow Haskell Compiler
Glasgow Haskell Compiler
The Glorious Glasgow Haskell Compilation System, more commonly known as the Glasgow Haskell Compiler or GHC, is an open source native code compiler for the functional programming language Haskell. The lead developers are Simon Peyton Jones and Simon Marlow...
and in Scala (as "case classes"), and have been proposed as additions to other languages including Java and C#.
Functional programming in non-functional languages
It is possible to use a functional style of programming in languages that are not traditionally considered functional languages. Among imperative programming languages, the D
D (programming language)
The D programming language is an object-oriented, imperative, multi-paradigm, system programming language created by Walter Bright of Digital Mars. It originated as a re-engineering of C++, but even though it is mainly influenced by that language, it is not a variant of C++...
programming language's solid support for functional programming stands out. For example, D has a pure modifier to enforce functional purity. Only Fortran 95 has something similar.
First class functions have slowly been added to mainstream languages. For example, in early 1994, support for lambda, filter, map, and reduce was added to Python
Python (programming language)
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
Guido van Rossum is a Dutch computer programmer who is best known as the author of the Python programming language. In the Python community, Van Rossum is known as a "Benevolent Dictator For Life" , meaning that he continues to oversee the Python development process, making decisions where necessary...
called for the removal of these features. So far, only the reduce function has been removed, and it remains accessible via the functools standard library module. First class functions were also introduced in PHP
PHP
PHP is a general-purpose server-side scripting language originally designed for web development to produce dynamic web pages. For this purpose, PHP code is embedded into the HTML source document and interpreted by a web server with a PHP processor module, which generates the web page document...
Visual Basic is the third-generation event-driven programming language and integrated development environment from Microsoft for its COM programming model...
Language Integrated Query is a Microsoft .NET Framework component that adds native data querying capabilities to .NET languages, although ports exist for Java, PHP and JavaScript....
(LINQ) feature, with its many incarnations, is an obvious and powerful use of functional programming in .NET
.NET Framework
The .NET Framework is a software framework that runs primarily on Microsoft Windows. It includes a large library and supports several programming languages which allows language interoperability...
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...
, anonymous classes can sometimes be used to simulate closure
Closure (computer science)
In computer science, a closure is a function together with a referencing environment for the non-local variables of that function. A closure allows a function to access variables outside its typical scope. Such a function is said to be "closed over" its free variables...
s; however, anonymous classes are not always proper replacements to closure
Closure (computer science)
In computer science, a closure is a function together with a referencing environment for the non-local variables of that function. A closure allows a function to access variables outside its typical scope. Such a function is said to be "closed over" its free variables...
In software engineering, a design pattern is a general reusable solution to a commonly occurring problem within a given context in software design. A design pattern is not a finished design that can be transformed directly into code. It is a description or template for how to solve a problem that...
s are expressible in functional programming terms: for example, the strategy pattern
Strategy pattern
In computer programming, the strategy pattern is a particular software design pattern, whereby algorithms can be selected at runtime. Formally speaking, the strategy pattern defines a family of algorithms, encapsulates each one, and makes them interchangeable...
simply dictates use of a higher-order function, and the visitor pattern roughly corresponds to a catamorphism
Catamorphism
In category theory, the concept of catamorphism denotes the unique homomorphism from an initial algebra into some other algebra. The concept has been applied to functional programming as folds.The dual concept is that of anamorphism...
In functional programming, fold – also known variously as reduce, accumulate, compress, or inject – are a family of higher-order functions that analyze a recursive data structure and recombine through use of a given combining operation the results of recursively processing its...
.
The benefits of immutable data can be seen even in imperative programs, so programmers often strive to make some data immutable even in imperative programs.
In computer science, imperative programming is a programming paradigm that describes computation in terms of statements that change a program state...
. The most significant differences stem from the fact that functional programming avoids side effects
Side effect (computer science)
In computer science, a function or expression is said to have a side effect if, in addition to returning a value, it also modifies some state or has an observable interaction with calling functions or the outside world...
, which are used in imperative programming to implement state and I/O. Pure functional programming disallows side effects completely. Disallowing side effects provides for referential transparency, which makes it easier to verify, optimize, and parallelize programs, and easier to write automated tools to perform those tasks.
Higher-order functions are rarely used in older imperative programming. Where a traditional imperative program might use a loop to traverse a list, a functional program would use a different technique. It would use a higher-order function that takes as arguments a function and a list. The higher-order function would then apply the given function to each element of the given list and then return a new list with the results.
Simulating state
There are tasks (for example, maintaining a bank account balance) that often seem most naturally implemented with state. Pure functional programming performs these tasks, and I/O tasks such as accepting user input and printing to the screen, in a different way.
Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the...
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. Monads offer a way to abstract certain types of computational patterns, including (but not limited to) modeling of computations with mutable state (and other side effects such as I/O) in an imperative manner without losing purity. While existing monads may be easy to apply in a program, given appropriate templates and examples, many students find them difficult to understand conceptually, e.g., when asked to define new monads (which is sometimes needed for certain types of libraries).
Impure functional languages usually include a more direct method of managing mutable state. Clojure, for example, uses managed references that can be updated by applying pure functions to the current state. This kind of approach enables mutability while still promoting the use of pure functions as the preferred way to express computations.
Alternative methods such as Hoare logic
Hoare logic
Hoare logic is a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs. It was proposed in 1969 by the British computer scientist and logician C. A. R. Hoare, and subsequently refined by Hoare and other researchers...
In computing, a unique type guarantees that an object is used in a single-threaded way, with at most a single reference to it. If a value has a unique type, a function applied to it can be optimized to update the value in-place in the object code. In-place updates improve the efficiency of...
have been developed to track side effects in programs. Some modern research languages use effect system
Effect system
An effect system is a formal system which describes the computational effects of computer programs, such as side effects. An effect system can be used to provide a compile-time checking of the possible effects of the program....
s to make explicit the presence of side effects.
Efficiency issues
Functional programming languages are typically less efficient in their use of CPU
Central processing unit
The central processing unit is the portion of a computer system that carries out the instructions of a computer program, to perform the basic arithmetical, logical, and input/output operations of the system. The CPU plays a role somewhat analogous to the brain in the computer. The term has been in...
and memory than imperative languages such as C and Pascal
Pascal (programming language)
Pascal is an influential imperative and procedural programming language, designed in 1968/9 and published in 1970 by Niklaus Wirth as a small and efficient language intended to encourage good programming practices using structured programming and data structuring.A derivative known as Object Pascal...
. For purely functional languages, the worst-case slowdown is logarithmic in the number of memory cells used, because mutable memory can be represented by a purely functional data structure with logarithmic access time (such as a balanced tree). However, such slowdowns are not universal. For programs that perform intensive numerical computations, functional languages such as Objective Caml
Objective Caml
OCaml , originally known as Objective Caml, is the main implementation of the Caml programming language, created by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy and others in 1996...
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
A database is an organized collection of data for one or more purposes, usually in digital form. The data are typically organized to model relevant aspects of reality , in a way that supports processes requiring this information...
In computer science, array programming languages generalize operations on scalars to apply transparently to vectors, matrices, and higher dimensional arrays....
The J programming language, developed in the early 1990s by Kenneth E. Iverson and Roger Hui, is a synthesis of APL and the FP and FL function-level languages created by John Backus....
K is a proprietary array processing language developed by Arthur Whitney and commercialized by Kx Systems. The language serves as the foundation for kdb, an in-memory, column-based database, and other related financial products. The language, originally developed in 1993, is a variant of APL and...
) were designed with speed optimization.
Immutability of data can, in many cases, lead to execution efficiency in allowing the compiler to make assumptions that are unsafe in an imperative language, thus increasing opportunities for inline expansion
Inline expansion
In computing, inline expansion, or inlining, is a manual or compiler optimization that replaces a function call site with the body of the callee. This optimization may improve time and space usage at runtime, at the possible cost of increasing the final size of the program In computing, inline...
In programming language theory, lazy evaluation or call-by-need is an evaluation strategy which delays the evaluation of an expression until the value of this is actually required and which also avoids repeated evaluations...
may also speed up the program, even asymptotically, whereas it may slow it down at most by a constant factor (however, it may introduce memory leaks when used improperly). Launchbury 1993 discusses theoretical issues related to memory leaks from lazy evaluation, and O'Sullivan et al. 2008 give some practical advice for analyzing and fixing them.
Coding styles
Imperative programs tend to emphasize the series of steps taken by a program in carrying out an action, while functional programs tend to emphasize the composition and arrangement of functions, often without specifying explicit steps. A simple example illustrates this with two solutions to the same programming goal (calculating Fibonacci number
Fibonacci number
In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...
.
// Fibonacci numbers, imperative style
int fibonacci(int iterations)
{
int first = 0, second = 1; // seed values
for (int i = 0; i < iterations; ++i) {
int sum = first + second;
first = second;
second = sum;
}
Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the...
) has a different feel to it:
-- Fibonacci numbers, functional style
-- describe an infinite list based on the recurrence relation for Fibonacci numbers
fibRecurrence first second = first : fibRecurrence second (first + second)
-- describe fibonacci list as fibRecurrence with initial values 0 and 1
fibonacci = fibRecurrence 0 1
-- describe action to print the 10th element of the fibonacci list
main = print (fibonacci !! 10)
The imperative style describes the intermediate steps involved in calculating fibonacci(N), and places those steps inside a loop statement. In contrast, the functional implementation shown here states the mathematical recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
that defines the entire Fibonacci sequence, then selects an element from the sequence (see also recursion
Recursion (computer science)
Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem. The approach can be applied to many types of problems, and is one of the central ideas of computer science....
In programming language theory, lazy evaluation or call-by-need is an evaluation strategy which delays the evaluation of an expression until the value of this is actually required and which also avoids repeated evaluations...
to create an "infinite" list of which only as much as needed (the first 10 elements in this case) will actually be computed. That computation happens when the runtime system carries out the action described by "main".
Use in industry
Functional programming has long been popular in academia, but with few industrial applications. However, recently several prominent functional programming languages have been used in commercial or industrial systems. For example, the Erlang programming language, which was developed by the Swedish
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
Ericsson , one of Sweden's largest companies, is a provider of telecommunication and data communication systems, and related services, covering a range of technologies, including especially mobile networks...
in the late 1980s, was originally used to implement fault-tolerant telecommunications systems. It has since become popular for building a range of applications at companies such as T-Mobile
T-Mobile
T-Mobile International AG is a German-based holding company for Deutsche Telekom AG's various mobile communications subsidiaries outside Germany. Based in Bonn, Germany, its subsidiaries operate GSM and UMTS-based cellular networks in Europe, the United States, Puerto Rico and the US Virgin Islands...
Nortel Networks Corporation, formerly known as Northern Telecom Limited and sometimes known simply as Nortel, was a multinational telecommunications equipment manufacturer headquartered in Mississauga, Ontario, Canada...
Facebook is a social networking service and website launched in February 2004, operated and privately owned by Facebook, Inc. , Facebook has more than 800 million active users. Users must register before using the site, after which they may create a personal profile, add other users as...
- Military :* Elmendorf Air Force Base, a United States Air Force Base in Alaska* Estonian Defence Forces, military of the Republic of Estonia* European Defence Force, an international military force by the European Council- Organizations :...
. The Scheme dialect of Lisp was used as the basis for several applications on early Apple Macintosh computers, and has more recently been applied to problems such as training simulation software and telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...
OCaml , originally known as Objective Caml, is the main implementation of the Caml programming language, created by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy and others in 1996...
, which was introduced in the mid 1990s, has seen commercial use in areas such as financial analysis, driver verification, industrial robot
Robot
A robot is a mechanical or virtual intelligent agent that can perform tasks automatically or with guidance, typically by remote control. In practice a robot is usually an electro-mechanical machine that is guided by computer and electronic programming. Robots can be autonomous, semi-autonomous or...
Embedded software is computer software that plays an integral role in the electronics it is supplied with.Embedded software's principal role is not Information technology , but rather the interaction with the physical world. It's written for machines that are not, first and foremost, computers...
Haskell is a standardized, general-purpose purely functional programming language, with non-strict semantics and strong static typing. It is named after logician Haskell Curry. In Haskell, "a function is a first-class citizen" of the programming language. As a functional programming language, the...
, although initially intended as a research language, has also been applied by a range of companies, in areas such as aerospace systems, hardware design, and web programming.
Other functional programming languages that have seen use in industry include Scala, F#, Lisp, Standard ML, and Clojure
Clojure
Clojure |closure]]") is a recent dialect of the Lisp programming language created by Rich Hickey. It is a general-purpose language supporting interactive development that encourages a functional programming style, and simplifies multithreaded programming....
This article attempts to set out the various similarities and differences between the various programming paradigms as a summary in both graphical and tabular format with links to the separate discussions concerning these similarities and differences in existing Wikipedia articles- Main paradigm...
In computer programming, eager evaluation or greedy evaluation is the evaluation strategy in most traditional programming languages. In eager evaluation an expression is evaluated as soon as it gets bound to a variable. The term is typically used to contrast lazy evaluation, where expressions are...
In computer programming, a nested function is a function which is lexically encapsulated within another function. It can only be called by the enclosing function or by functions directly or indirectly nested within the same enclosing function. In other words, the scope of the nested function is...
Further reading
Cousineau, Guy and Michel Mauny. The Functional Approach to Programming. Cambridge, UK: Cambridge University Press
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
, 1998.
Curry, Haskell Brooks and Feys, Robert and Craig, William. Combinatory Logic. Volume I. North-Holland Publishing Company, Amsterdam, 1958.
Mark•Jason Dominus is one of the founders of Kibology and a leading Perl programmer. He was the managing editor of and was a columnist for The Perl Journal for several years. Mark's other Perl-related articles have appeared in magazines such as Wired and IEEE Software...
Higher-Order Perl , ISBN 1-55860-701-3 , is a book written by Mark Jason Dominus with the goal to teach Perl programmers with a strong C and Unix background how to use techniques with roots in functional programming languages like Lisp that are available in Perl as well, but less known. The book is...
. Morgan Kaufmann. 2005.
Graham, Paul. ANSI Common LISP. Englewood Cliffs, New Jersey: Prentice Hall
Prentice Hall
Prentice Hall is a major educational publisher. It is an imprint of Pearson Education, Inc., based in Upper Saddle River, New Jersey, USA. Prentice Hall publishes print and digital content for the 6-12 and higher-education market. Prentice Hall distributes its technical titles through the Safari...
, 1996.
MacLennan, Bruce J. Functional Programming: Practice and Theory. Addison-Wesley, 1990.
Pratt, Terrence, W. and Marvin V. Zelkowitz. Programming Languages: Design and Implementation. 3rd ed. Englewood Cliffs, New Jersey: Prentice Hall
Prentice Hall
Prentice Hall is a major educational publisher. It is an imprint of Pearson Education, Inc., based in Upper Saddle River, New Jersey, USA. Prentice Hall publishes print and digital content for the 6-12 and higher-education market. Prentice Hall distributes its technical titles through the Safari...
, 1996.
Salus, Peter H. Functional and Logic Programming Languages. Vol. 4 of Handbook of Programming Languages. Indianapolis, Indiana: Macmillan Technical Publishing, 1998.
Thompson, Simon. Haskell: The Craft of Functional Programming. Harlow, England: Addison-Wesley Longman Limited, 1996.
that produces the derivative
is an example of this in calculus).