# Subsystems and Closure

Subsystems

With the basic definition of a system in hand, we can also define a subsystem. To anyone who has worked with mathematical sets before, the definition of a subsystem should be quite intuitive.

Given systems S and S’, then S’ is a subsystem of S if:

• They have the same property set.
• They have the same rule set.
• The element set of S’ is a subset of the element set of S.

We can likewise define a proper subsystem, when the element set of S’ is a proper subset of the element set of S.

Closure

The concept of closure should also be familiar to those with a mathematical background, though it’s application here probably won’t be quite as intuitive. We consider a system (or equivalently, a subsystem) closed if the application of its rules is determined only by information within the system or by true randomness.

This probably isn’t as obvious, but hopefully the example of Conway’s Game of Life will make it clearer. An instance of Life is a system, as we have already seen, and is in fact a closed system, since there is no information outside the set of elements that determines its behaviour at each step. We can, however, take a subset of the cells of an instance of Life, for example a 10×10 square of them. Together with the property set and rule set from normal Life, this forms a subsystem which is not closed.

To see why our subsystem is not closed, consider the cells around the edges of that 10×10 square. Their behaviour is determined by rules referring to their eight neighbours, but not all of their neighbours actually lie within that 10×10 square. Since the edge cells’ behaviours are determined in part by cells (elements) not in the 10×10 square (the element set of our subsystem), then the subsystem is not closed.