Studying Emergence

I previously defined the emergence ratio . It is a function of the size of the system, , together with some inherent level of emergence contained in the rules of interaction. It measures the relative efficiency of rule-based `understanding' versus simulation-based `understanding'. My results indicate that, in general, we cannot determine this ratio. Some interesting questions arise dealing with the behaviour of with . In very small systems we can usually perform some form of an analysis, so will be smaller than .

As systems become more emergent, and increases, the propagation of information through accumulated interaction will blur the boundaries of any analysis we try and perform. Trying to generalise in the initial-state space becomes more and more futile, until any previously useful outstrips . We gradually approach the worst case - that of being forced simply to classify points in the state-space purely by exhaustive enumeration, with each individual result being determined by a simulation. Generalisation has vanished.

Before moving on to discuss possible approaches to the study of emergent systems, it is important to point out that many systems are not emergent, and therefore amenable to some form of analysis. Such an analysis will certainly not generalise to the emergent complex systems, but is clearly an important and valuable contribution to understanding the context of our investigations - the continuum quantified by the emergence ratio.

For example, it has been demonstrated that in certain highly symmetric classes of one-dimensional cellular automata, the single cell at the bottom of a light-cone after time-steps can be predicted more quickly than the steps of a simulation:

Linear CA[10] have . Quasi-linear CA with radius 1/2 [11] have . The proof of the latter result is particularly informative in the direct manner in which it exploits the symmetry of, for example, the quaternion group.

• Research inside Emergent Systems
• Self-Organised Criticality