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{{Wiktionary|apply}}
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In [[mathematics]] and [[computer science]], '''Apply''' is a function that applies functions to arguments. It is central to [[programming language]]s derived from [[lambda calculus]], such as [[LISP]] and [[Scheme (programming language)|Scheme]], and also in [[functional language]]s. In particular, it has a role in the study of the [[denotational semantics]] of computer programs, because it is a [[Scott continuity|continuous function]] on [[complete partial order]]s.
 
In [[category theory]], '''Apply''' is important in [[Cartesian closed categories]], (and thus, also in [[Topos theory]]), where it is a [[universal morphism]], [[right adjoint]] to [[currying]].
 
==Programming==
In computer programming, '''apply''' applies a function to a list of arguments. ''[[Eval]]'' and ''apply'' are the two interdependent components of the ''[[eval-apply cycle]]'', which is the essence of evaluating Lisp, described in [[Structure and Interpretation of Computer Programs|SICP]].<ref>Harold Abelson, Gerald Jay Sussman, Julie Sussman, ''Structure and Interpretation of Computer Programs'', (1996) MIT Press, ISBN 0-262-01153-0. ''See [http://mitpress.mit.edu/sicp/full-text/book/book-Z-H-26.html Section 4.1, The Metacircular Evaluator]''</ref>
 
===Apply function===
Apply is also the name of a special function in many languages, which takes a function and a list, and uses the list as the function's own argument list, as if the function were called with the elements of the list as the arguments.  This is important in languages with [[variadic function]]s, because this is the only way to call a function with an indeterminate (at compile time) number of arguments.
 
====Common Lisp and Scheme====
In [[Common Lisp]] '''apply''' is a function that applies a function to a list of arguments (note here that "+" is a variadic function that takes any number of arguments):<source lang="lisp">(apply #'+ (list 1 2))</source>
Similarly in Scheme:<source lang="scheme">(apply + (list 1 2))</source>
 
====C++====
In [[C++]], Bind <ref>http://www.boost.org/doc/libs/1_49_0/libs/bind/bind.html#with_functions</ref> is used either via the std namespace or via the boost namespace.
 
====C# and Java====
In [[C Sharp (programming language)|C#]] and [[Java (programming language)|Java]], variadic arguments are simply collected in an array. Caller can explicitly pass in an array in place of the variadic arguments. This can only be done for a variadic parameter. It is not possible to apply an array of arguments to non-variadic parameter without using [[reflection (computer science)|reflection]]. An ambiguous case arises should the caller want to pass an array itself as one of the arguments rather than using the array as a ''list'' of arguments. In this case, the caller should cast the array to <code>Object</code> to prevent the compiler from using the ''apply'' interpretation.<source lang="java">variadicFunc(arrayOfArgs);</source>
 
====Go====
In [[Go (programming language)|Go]], typed variadic arguments are simply collected in a slice. The caller can explicitly pass in a slice in place of the variadic arguments, by appending a <code>...</code> to the slice argument. This can only be done for a variadic parameter. The caller can not apply an array of arguments to non-variadic parameters, without using reflection..<pre>s := []string{"foo", "bar"}
variadicFunc(s...)</pre>
 
====Haskell====
In [[Haskell (programming language)|Haskell]], functions may be applied by simple juxtaposition:
<source lang="haskell">
func param1 param2 ...
</source>
 
In Haskell, the syntax may also be interpreted that each parameter curries its function in turn. In the above example, "func param1" returns another function accepting one fewer parameters, that is then applied to param2, and so on, until the function has no more parameters.
 
====JavaScript====
In [[JavaScript]], function objects have an <code>apply</code> method, the first argument is the value of the <code>this</code> keyword inside the function; the second is the list of arguments:<source lang="javascript">func.apply(null, args);</source>
 
====Lua====
In [[Lua (programming language)|Lua]], apply can be written this way: <source lang="lua">function apply(f,l)
  return f(unpack(l))
end </source>
 
====Perl====
In [[Perl]], arrays, hashes and expressions are automatically "flattened" into a single list when evaluated in a list context, such as in the argument list of a function:<source lang="perl">
# Equivalent subroutine calls:
@args = (@some_args, @more_args);
func(@args);
 
func(@some_args, @more_args);
</source>
 
====PHP====
In [[PHP]], <code>apply</code> is called <code>call_user_func_array</code>:<source lang="php">call_user_func_array('func_name', $args);</source>
 
====Python and Ruby====
In [[Python (programming language)|Python]] and [[Ruby (programming language)|Ruby]], the same asterisk notation used in defining [[variadic function]]s is used for calling a function on a sequence and array respectively:<source lang="python">func(*args)</source>
Python originally had an apply function, but this was [[deprecated]] in favour of the asterisk in 2.3 and removed in 3.0.<ref>{{cite web | title = Non-essential built-in functions | work = Python Library Reference | date = 8 February 2005 | url = http://docs.python.org/release/2.3.5/lib/non-essential-built-in-funcs.html | accessdate = 19 May 2013}}</ref>
 
====R====
In [[R (programming language)|R]], <code lang="rsplus">do.call</code> constructs and executes a function call from a name or a function and a list of arguments to be passed to it:<source lang="rsplus">
f(x1, x2)
# can also be performed via
do.call(what = f, args = list(x1, x2))
</source>
 
====Smalltalk====
In [[Smalltalk]], block (function) objects have a <code>valueWithArguments:</code> method which takes an array of arguments: <source lang="smalltalk">aBlock valueWithArguments: args</source>
 
==Universal property==
Consider a [[Function (mathematics)|function]] <math>g:(X\times Y)\to Z</math>, that is, <math>g\isin [(X\times Y)\to Z]</math> where the bracket notation <math>[A\to B]</math> denotes the [[space of functions]] from ''A'' to ''B''. By means of [[currying]], there is a unique function <math>\mbox{curry}(g) :X\to [Y\to Z]</math>
<!-- something like this is needed, because not everybody knows what does currying means:  , where <math>curry (f)\stackrel{def}{=}\lambda x.(\lambda y.(g(x,y))</math> with type <math>curry:((\alpha\times\beta)\to\gamma)\to(\alpha\to(\beta\to\gamma))</math> here <math>\lambda</math> is the abstractor of the <math>\lambda</math>-calculus
The lambda may be confuse the reader not familiar with lambda calculus and its use in category theory as told below -->.
Then '''Apply''' provides the [[universal morphism]]
 
:<math>\mbox{Apply}:([Y\to Z]\times Y) \to Z</math>,
 
so that
 
:<math>\mbox {Apply}(f,y)=f(y)</math>
 
or, equivalently one has the [[commuting diagram]]
 
<!-- it is not clear what does this mean. can a pair of functions be applied to an object?, this needs an explanation -->
:<math>\mbox{Apply} \circ \left( \mbox{curry}(g) \times \mbox{id}_Y \right) = g</math>
 
<!-- Maybe something like this can improve reading
:<math>f(x)=g(f,x)</math>
:<math>= (\mbox{Apply} \circ (\mbox{curry}(g),\mbox{id}_Y))(f,x)</math>
:<math>= (\mbox{Apply} ((\mbox{curry}(g),\mbox{id}_Y ))(f,x))</math>
:<math>= (\mbox{Apply} ((\mbox{curry}(g))(f),\mbox{id}_Y x))
</math>
-->
 
The notation <math>[A\to B]</math> for the space of functions from ''A'' to ''B'' occurs more commonly in computer science. In [[category theory]], however, <math>[A\to B]</math> is known as the [[exponential object]], and is written as <math>B^A</math>. There are other common notational differences as well; for example ''Apply'' is often called ''Eval'',<ref>Saunders Mac Lane, ''Category Theory''</ref> even though in computer science, these are not the same thing, with [[eval]] distinguished from ''Apply'', as being the evaluation of the quoted string form of a function with its arguments, rather than the application of a function to some arguments. 
 
Also, in category theory, ''curry'' is commonly denoted by <math>\lambda</math>, so that <math>\lambda g</math> is written for ''curry''(''g''). This notation is in conflict with the use of <math>\lambda</math> in [[lambda calculus]], where lambda is used to denote free variables. With all of these notational changes accounted for, the adjointness of ''Apply'' and ''curry'' is then expressed in the commuting diagram
 
[[Image:ExponentialObject-01.png|center|Universal property of the exponential object]]
 
The articles on [[exponential object]] and [[Cartesian closed category]] provide a more precise discussion of the category-theoretic formulation of this idea. Thus use of lambda here is not accidental; Cartesian-closed categories provide the general, natural setting for lambda calculus.
 
==Topological properties==
In [[order theory]], in the category of [[complete partial order]]s endowed with the [[Scott topology]], both ''curry'' and ''apply'' are [[continuous function]]s (that is, they are [[Scott continuous]]).<ref>H.P. Barendregt, ''The Lambda Calculus'', (1984) North-Holland ISBN 0-444-87508-5</ref> This property helps establish the foundational validity of the study of the [[denotational semantics]] of computer programs.
 
==References==
<references/>
 
[[Category:Adjoint functors]]
[[Category:Lambda calculus]]
[[Category:Functional programming]]

Latest revision as of 12:57, 16 June 2014

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