As I mentioned last week, we have now reached the home-stretch of our foundational work. Systems theory is a logical structure (practically algebraic, in fact) which has earned its place this early in the project through how incredibly useful it is. My systems theory (not to be confused with this one) is an abstract, rational structure, with all that that entails as far as it’s truth.
A system consists of three sets: a set of properties, a set of elements, and a set of rules. A property is just some useful name. An element is a mapping from each property in the system to a distinct piece of information. A rule describes how elements (or more precisely their associated information) change over time.
While there are many examples of systems that meet this definition, one of the simplest and most interesting is Conway’s Game of Life (I expect to return to Conway’s game numerous times in this section as it provides a bounty of useful examples). To lay it out explicitly according to my definition of a system, Conway’s game has an infinite number of elements, each with the following two properties: location (a Cartesian x/y coordinate) and state (alive or dead). Every location property is unique and static (there is exactly one element at each location, and it does not move over time). Element state over time is governed by the following rules:
- Any live cell with fewer than two live neighbours dies.
- Any live cell with two or three live neighbours lives on.
- Any live cell with more than three live neighbours dies.
- Any dead cell with exactly three live neighbours becomes a live cell.
A cell’s “neighbours” are the eight other elements at distance exactly one from it; literally its neighbours if the system were layed out on a grid as it usually is. In Conway’s game, time is a discrete thing (element state changes in distinct steps) but this is not a requirement of systems theory; it works equally well on continuous systems.
This post suffices for a plain definition and an example to whet your appetite. In future posts I will dig into systems theory in more detail, exploring many of the useful concepts that fall out of it and some of its applications.