Map (higher-order function) - AbsoluteAstronomy.com

Programming language

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....

s, map is the name of a higher-order function

Higher-order function

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...

that applies a given function

Procedural parameter

In computing, a procedural parameter is a parameter of a procedure that is itself a procedure.This concept is an extremely powerful and versatile programming tool, because it allows programmers to modify certain steps of a library procedure in arbitrarily complicated ways, without having to...

to each element of a list, returning a list of results. They are examples of both 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...

s and anamorphism

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. This is often called apply-to-all when considered a functional form

Functional form

In programming and mathematics, a functional form is an operator or function that can either be applied to other operators or yield operators as result, or both...

.

For example, if we define a function square as follows:

square x = x * x

Then calling map square [1,2,3,4,5] will return [1,4,9,16,25], as map will go through the list, and apply the function square to each element.

Language comparison

The map function originated in functional programming

Functional programming

In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state...

languages but is today supported (or may be defined) in many procedural

Procedural programming

Procedural programming can sometimes be used as a synonym for imperative programming , but can also refer to a programming paradigm, derived from structured programming, based upon the concept of the procedure call...

, object oriented, and multi-paradigm languages as well: In C++

C++

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...

's Standard Template Library

Standard Template Library

The Standard Template Library is a C++ software library which later evolved into the C++ Standard Library. It provides four components called algorithms, containers, functors, and iterators. More specifically, the C++ Standard Library is based on the STL published by SGI. Both include some...

, it is called transform, in C# (3.0)'s LINQ library, it is provided as an extension method called Select. Map is also a frequently used operation in high level languages such as Perl

Perl

Perl is a high-level, general-purpose, interpreted, dynamic programming language. Perl was originally developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions and become widely popular...

, 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...

and Ruby

Ruby (programming language)

Ruby is a dynamic, reflective, general-purpose object-oriented programming language that combines syntax inspired by Perl with Smalltalk-like features. Ruby originated in Japan during the mid-1990s and was first developed and designed by Yukihiro "Matz" Matsumoto...

; the operation is called map in all three of these languages. A collect alias for map is also provided in Ruby (from Smalltalk

Smalltalk

Smalltalk is an object-oriented, dynamically typed, reflective programming language. Smalltalk was created as the language to underpin the "new world" of computing exemplified by "human–computer symbiosis." It was designed and created in part for educational use, more so for constructionist...

). Common Lisp

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...

provides a family of map-like functions; the one corresponding to the behavior described here is called mapcar (-car indicating access using the CAR operation

Car and cdr

car and cdr are primitive operations on cons cells introduced in the Lisp programming language. A cons cell is composed of two pointers; the car operation extracts the first pointer, and the cdr operation extracts the second.Thus, the expression evaluates to x, and evaluates to...

). There are also languages with syntactic constructs providing the same functionality as the map function.

Map is sometimes generalized to accept dyadic

Dyadic

Dyadic may refer to:*Adicity of a mathematical relation or function *Dyadic communication* Dyadic counterpoint, the voice-against-voice conception of polyphony...

(2-argument) functions that can apply a user-supplied function to corresponding elements from two lists; some languages use special names for this, such as map2 or zipWith. Languages using explicit variadic function

Variadic function

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....

s may have versions of map with variable arity

Arity

In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...

to support variable-arity functions. Map with 2 or more lists encounters the issue of handling when the lists are of different lengths. Various languages differ on this; some raise an exception, some stop after the length of the shortest list and ignore extra items on the other lists; some continue on to the length of the longest list, and for the lists that have already ended, pass some placeholder value to the function indicating no value.

In languages which support first-class function

First-class function

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, map may be partially applied

Partial application

In computer science, partial application refers to the process of fixing a number of arguments to a function, producing another function of smaller arity...

to "lift" functions to element-wise versions; for instance, map square is a Haskell function which squares lists element-wise.

Map in various languages
Language	Map	Map 2 lists	Map n lists	Notes	Handling lists of different lengths
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...	map func list	zipWith func list1 list2	zipWithn func list1 list2 ...	n corresponds to the number of lists; predefined up to zipWith7	stops after the shortest list ends
haXe HaXe haXe is a versatile open-source high-level multiplatform programming language described on its website as a "universal language".It can produce:* Flash applications and games* Multi-platform client-side web applications* Apache CGI web applications...	Lambda.map(iterable, func)
J J (programming language) 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....	func list	list func list	func/ list1 , list2 , list3 ,: list4	J's array processing capabilities make operations like map implicit	length error if lists not equal
OCaml	List.map func list Array.map func array	List.map2 func list1 list2			raises Invalid_argument exception
Standard ML Standard ML 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...	map func list	ListPair.map func (list1, list2) ListPair.mapEq func (list1, list2)		For 2-argument map, func takes its arguments in a tuple	ListPair.map stops after the shortest list ends, whereas ListPair.mapEq raises UnequalLengths exception
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...	map(func, list)	map(func, list1, list2)	map(func, list1, list2, ...)		zip and map (3.x) stops after the shortest list ends, whereas map (2.x) and itertools.zip_longest (3.x) extends the shorter lists with None items
Ruby Ruby (programming language) Ruby is a dynamic, reflective, general-purpose object-oriented programming language that combines syntax inspired by Perl with Smalltalk-like features. Ruby originated in Japan during the mid-1990s and was first developed and designed by Yukihiro "Matz" Matsumoto...	enum.collect {block} enum.map {block}	enum1.zip(enum2).map {block}	enum1.zip(enum2, ...).map {block} [enum1, enum2, ...].transpose.map {block}	enum is an Enumeration	stops at the end of the object it is called on (the first list); if any other list is shorter, it is extended with nil items
C++ C++ 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...	std::transform(begin, end, result, func)	std::transform(begin1, end1, begin2, result, func)		in header begin, end, & result are iterators result is written starting at result
Perl Perl Perl is a high-level, general-purpose, interpreted, dynamic programming language. Perl was originally developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions and become widely popular...	map block list map expr, list			In block or expr special variable $_ holds each value from list in turn.	N/A
C# 3.0	ienum.Select(func)			Select is an extension method ienum is an IEnumerable Similarly in all .NET languages
C# 4.0	ienum.Select(func)	ienum1.Zip(ienum2, func)		Select is an extension method ienum is an IEnumerable Similarly in all .NET languages	stops after the shortest list ends
JavaScript JavaScript JavaScript is a prototype-based scripting language that is dynamic, weakly typed and has first-class functions. It is a multi-paradigm language, supporting object-oriented, imperative, and functional programming styles.... 1.6	array.map(func)
Common Lisp 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...	(mapcar func list)	(mapcar func list1 list2)	(mapcar func list1 list2 ...)		stops after the length of the shortest list
Scheme, 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....	(map func list)	(map func list1 list2)	(map func list1 list2 ...)		Stops after the shortest list ends
Smalltalk Smalltalk Smalltalk is an object-oriented, dynamically typed, reflective programming language. Smalltalk was created as the language to underpin the "new world" of computing exemplified by "human–computer symbiosis." It was designed and created in part for educational use, more so for constructionist...	aCollection collect: aBlock	aCollection1 with: aCollection2 collect: aBlock			Fails
Erlang Erlang Erlang may refer to:* Agner Krarup Erlang , a mathematician and engineer after whom several concepts are named** Erlang , a unit to measure traffic in telecommunications or other domains...	lists:map(Fun, List)	lists:zipwith(Fun, List1, List2)		zipwith3 also available	Lists must be equal length
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...	array_map(callback, array)	array_map(callback, array1,array2)	array_map(callback, array1,array2, ...)	The number of parameters for callback should match the number of arrays.	extends the shorter lists with NULL items
Prolog Prolog Prolog is a general purpose logic programming language associated with artificial intelligence and computational linguistics.Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is declarative: the program logic is expressed in terms of...	maplist(Cont, List1, List2).	maplist(Cont, List1, List2, List3).	maplist(Cont, List1, ...).	List arguments are input, output or both. Subsumes unzip, all	Failure
Mathematica Mathematica Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...	func /@ list Map[func, list]	MapThread[func, {list1, list2}]	MapThread[func, {list1, list2, ...}]		Lists must be same length
Maxima	map(f, expr₁, ..., expr_n) maplist(f, expr₁, ..., expr_n)			map returns an expression whose leading operator is the same as that of the expressions; maplist returns a list
S/R R (programming language) 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....	lapply(list,func)	mapply(func,list1,list2)	mapply(func,list1,list2,...)		Shorter lists are cycled
Scala	list.map(func)	(list1, list2).zipped.map(func)	(list1, list2,"list3").zipped.map(func)	note: more than 3 not possible.	stops after the shorter list ends

Optimizations

The mathematical basis of maps allow for a number of optimizations

Optimization (computer science)

In computer science, program optimization or software optimization is the process of modifying a software system to make some aspect of it work more efficiently or use fewer resources...

. If one has (map f . map g) xs ('.' is function composition

Function composition (computer science)

In computer science, function composition is an act or mechanism to combine simple functions to build more complicated ones...

) then it is the same as the simpler map (f . g) xs; that is,

. This particular optimization eliminates an expensive second map by fusing it with the first map; thus it is a "map fusion".

Map functions can be and often are defined in terms of a fold

Fold (higher-order function)

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...

such as foldr, which means one can do a "map-fold fusion": foldr f z . map g is equivalent to foldr (f . g) z.

The implementation of map above on singly linked lists is not tail-recursive, so may build up a lot of frames on the stack when called with a large list. Many languages alternately provide a "reverse map" function, which is equivalent to reversing a mapped list, but is tail-recursive. Here is an implementation which utilizes the fold

Fold (higher-order function)

-left function.

rev_map f = foldl (\ys x -> f x : ys) []

Since reversing a singly linked list is also tail-recursive, reverse and reverse-map can be composed to perform normal map in a tail-recursive way.

Generalization

In the Haskell programming language

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...

, the polymorphic

Type polymorphism

In computer science, polymorphism is a programming language feature that allows values of different data types to be handled using a uniform interface. The concept of parametric polymorphism applies to both data types and functions...

map :: (a -> b) -> ([a] -> [b]) is generalized to a polytypic

Polytypic

In zoology, polytypic refers to a taxonomic unit with more than one subgroup at the next lower level.-See also:*Linnaean taxonomy*monotypic*monotypic habitat...

function called fmap :: Functor f => (a -> b) -> (f a -> f b) which applies to any type in the Functor class.

map is used in Haskell's Prelude to define the list type constructor [] an instance of the Functor class as follows



   instance Functor [] where fmap = map

But trees may belong to Functor too, for example:



  data Tree a = Leaf a | Fork (Tree a) (Tree a)

  instance Functor Tree where

    fmap f (Leaf x) = Leaf (f x)

    fmap f (Fork l r) = Fork (fmap f l) (fmap f r)



  fmap (1+) (Fork(Fork(Leaf 0)(Leaf 1))(Fork(Leaf 2)(Leaf 3)))

evaluates to:



  Fork (Fork(Leaf 1)(Leaf 2))(Fork(Leaf 3)(Leaf 4))

For every instance of the Functor type class

Type class

In computer science, a type class is a type system construct that supports ad-hoc polymorphism. This is achieved by adding constraints to type variables in parametrically polymorphic types...

, fmap must be defined such that it obeys the functor laws:

fmap id = id -- identity
fmap (f . g) = fmap f . fmap g -- composition

Among other uses, this allows defining element-wise operations for other kind of collections.
Moreover, if

and

are two functors, a natural transformation

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

is a function of polymorphic type

which respects fmap:

for any function

.

If the h function is defined by parametric polymorphism

Parametric polymorphism

In programming languages and type theory, parametric polymorphism is a way to make a language more expressive, while still maintaining full static type-safety. Using parametric polymorphism, a function or a data type can be written generically so that it can handle values identically without...

as in the type definition above, this specification is always satisfied.