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Functions

Functions are a central concept in both mathematics and systems theory that capture how a system converts an input into an output

Functions are a central concept in both mathematics and systems theory that capture how a system converts an input into an output

A function is a very broad and fundamental concept that is central to systems theory. It is also used in many different domains, being particularly important in mathematics and engineering. A function may be defined as a process that transforms energy or resources from one state to another. There are three key things to note in this definition. We have a set of things that are the input, we have a process that changes these things in some way, and we have an output. Systems are essentially the global functionality that emerges out of the interaction and arrangement of a set of elements. Thus, systems are defined by the function that they perform.

Inputs

Firstly, inputs involve the capturing and assembling of elements that enter the system to be processed. Putting fuel into your car is an example of an input to a mechanical system. The water taken in by the roots of a plant is an example of an input to a biological system. An important thing to note is that any given system can only process a specific range of inputs. A car can process a certain type of fuel but not all fuels. What a system can and cannot process is a defining feature to its boundary that functions to filter inputs to the system. For example, an electrical power socket is designed in a particular shape to ensure that only the right plug is inputted to it, thus functioning as a boundary filter to accessing the system.

Processing

Secondly, resources that are successfully inputted are processed within the system. A process is a series of actions performed upon the input in order to achieve a particular end result. Processing is often understood in terms of information, i.e. an algorithm or set of instructions that are performed upon the input in order to produce the output. Baking a cake is an example of a process. It takes in a set of ingredients, such as eggs, water, flour etc. The cook then has a recipe to follow that represents the set of instruction to be performed upon these raw ingredients, such as chopping, mixing, baking and so on. If these stages in the process are correctly performed, the result should be the desired output. This same process is true for the internal workings of a computer, biological cell or a financial transaction. Processes are not necessarily linear in nature. They may be cyclical, feeding back on each other with the output from one process being the input for another. They may also be nested within larger processes or run parallel to them.

Output

Computer languages use black boxing in order to encapsulate the internal workings of a function, thus helping to reduce the overall complexity that the developer is exposed to

Computer languages use black boxing in order to encapsulate the internal workings of a function, thus helping to reduce the overall complexity that the developer is exposed to

But ultimately there will be some energy or resources produced by this process that travels across the systems boundary to be returned to its environment, and this is what is called the system’s output. One way of understanding a function then is simply as the difference between what goes into the system and what comes out. We call a system whose internal functioning we do not know a black box. In science, computing, and engineering, a black box is a device, system or object which can be viewed in terms of its inputs and outputs, without any knowledge of its internal workings. This can be of great value to us as it helps to hide away the complexity of the internal workings to the system.

Bidirectional Functions

A function is often symbolically denoted as an arrow from one element or set of elements to another, and in the language of mathematics may be called a mapping or a transformation. Functions may be unidirectional, meaning the function only maps an input to an output, or the function may be bi-directional, meaning it can also invert this process to transform the output back to the input through what is called an inverse function. Many processes are essentially unidirectional, requiring vastly or even infinitely more energy to invert the function than was required to perform it in the first place. Aging within the human body may be cited as a good example of a unidirectional process whereas the building of a Lego brick house is an example of a function that can be easily inverted and thus bidirectional.

Application

We should note that the model of a function cannot be properly used to describe a set of things because sets do not perform a common function. If we take a group of nations at war with each other, because they are not working together to perform a function, we can only describe them by talking about their attributes and interactions. But the model of a function can be effectively used to describe any type of system. The concept of a function may appear to be very simple and intuitive to us. This is due to its high level of abstraction, which also makes it a powerful model and very important tool in the systems thinking tool box.

Cite this article as: Joss Colchester, "Functions," in Complexity Labs, October 17, 2016, http://complexitylabs.io/functions/.
2017-05-05T11:55:28+00:00