View source for Function

Functions are abstract mathematical representation of [[relation]]s between the [[domain]] and [[codomain]]. It can be represented as a [[key-value pairs]], or in the [[lambda calculus]]<ref>{{:Book/Structure and Interpretation of Computer Programs}}</ref> form. 
=Tag Line=
[[Data]] with [[continuation]] together makes [[function]].

Function can be used as a generic building block to represent information in general. It can be thought of as a primitive kind of data structure. This argument can be substantiated by thinking of [[Lambda calculus]] as a generalized way of encoding functions in an explicit structured [[data]] format. 

==A Critical Observation==
In most cases, people think of MediaWiki as a generic note taking or a dictionary of relevant pieces of static information. However, knowing the capabilities of modern browsers, HTML 5, JavaScript, and the server side PHP + Lua (Scribuntu) extension of MediaWiki, not to mention Semantic MediaWiki and its MySQL/MariaDB data storage/retrieval capabilities, the overall functions of MediaWiki must not be just seen as a passive data browsing tool, but a full blown programming and data creating environment. In programmer's term, MediaWiki can be configured to be a full blown Integrated Development Environment, for any subject matter. This is what [[PKC]] strives to become, it tries to raise users' awareness of data elements as functions, so that all data are also functions in a [[hyperlinked]] context. In other words, hyplerink is a kind of data-oriented function, that allows users to incrementally construct small and large functions at many locations and across many time scales. On the Internet, everyone is constructing some piece of a bigger function that they may never be directly conscientiously aware, but never-the-less, all users, all computers are incrementally constructing functions of many kinds through accumulating data elements, all in the [[hyperlinked]] fashion as articulated by [[Ted Nelson]].

===Scale-Free, Time-like Structures, and Ordered Namespace of Functions===
Function as a data structure is not only [[Universal abstraction|universal]], but it can also be designed to be [[Scale-free]], meanningly that the formal properties of a specific '''Function''', don't have to be bounded to a specific [[spacetime physical scale]]. However, in the space of all functions, the containment relationship in terms of what functions are classified into what sub-class(es) of functions can be studied in terms of [[Category Theory]], and more specifically, in the field of [[Lattice Theory]] or just think of them in terms of [[Partially ordered set]]s. Thinking of named functions in terms of [[lattice]] or [[Partially ordered set]], a.k.a. [[Poset]]s, is an important mental model for both software engineering, and in everyday life. Thinking of functions as elements in [[Poset]]s will give them a consistent abstraction in an [[ordered]] or [[time-like]] structure, so that it will build a generically applicable mental model for [[causal relation]]s and in terms of an [[ordered namespace]].

===Function as Causal Relation===
Function can be represented as a [[key-value pair]] of [[inputs]] and [[outputs]], which defines a way to represent [[causal relation]]s.

{{:PKC/Design Principle}}

==Implementation==
Computationally, functions can almost always be thought of as a dictionary, or a [[lookup table]]/[[hashtable]] that relates a value of an input, to a value as the output of a function. This simplification is useful in the sense that all functions can be conceptualized as dictionaries/lookup tables, so that all functions can be treated as one type, which is a way to ensure structural purity, mathematically called a [[Monad]]. According the Beckman's talk on [[Don't fear the Monad]]<ref>{{:Video/Don't fear the Monad}}</ref>, this monadic approach will help construct complexity out of simplicity.
===Function as verb===
It would be rather easy to imagine why functions can be thought of as verbs. They can act on inputs and transform them into outputs. In MediaWiki, once [[Scribunto]] is installed, this can be realized rather easily. Otherwise, the [[JavaScript]] functional programming that comes with most modern browsers, can also demonstrate the nature of the effects of functions being verbs.
===Function as noun===
Function can also considered as nouns. Many function are associated with certain unique name in a defined functional namespace, functions can also be encoded in a formalized structure, that are called anonymous functions. In this case, each function's unique structural content, defines its identity. In this case, function can be thought of as names. In a computing system that treats computation as a pure structure, all functions must have their unique structures or unique names, so that their behavior can be consistently defined and asserted within the defined computing system. It is interesting to think of numbers as a kind of namespace, then, integer numbers, floating point numbers, are two kinds of namespaces. To study the nature of these namespaces in terms of their structures, such as [[IEEE 754]]<ref>{{:Video/IEEE 754}}</ref>, the functional nature becomes rather obvious.

===Function as Dictionary===
When thinking of function as a computational device, which needs to serve both the roles of verb and noun, it could be best thought of as a dictionary of [[key-value pair]]. Each instance of a function can be thought of as a [[dictionary]] or [[associative array]] containing many [[key-value pair]]s. In most cases, this tuple structure best capture the duality of function that must serve as noun and verb. To extend on this generalizable foundation, a function can be thought of as a special case of [[morphism]], which has two other types, called [[functor]] and [[natural transformation]].

====A piece of sample code====
[[PKC]] is based on [[MediaWiki]], which in term installs [[MW:Scribunto|Scribunto]] to enable [[Lua]] programming. [[Lua]] defines functions as below:
<syntaxhighlight lang=LUA>
f = function (op, a, b)
 if op == 'add' then
   return a + b
 else if op == 'sub' then
   return a - b
 end
 error("invalid operation")
end
</syntaxhighlight>

==Function represented in a Logic Model==
{{Template:LogicModel
|name=Function
}}
<noinclude>

=References=
<references/>

==Related Pages==
* [[Logically related::Data]]
* [[Logically related::Morphism]]
* [[Logically related::Functor]]
* [[Logically related::Natural transformation]]

[[Category:Morphism]]
[[Category:Data]]

</noinclude>