Friday, July 21, 2006

EDGE OF CHAOS: Probability and Possibility Models

"In the sciences in general, the phrase [edge of chaos] has come to refer to a metaphor that some physical, biological, economic and social systems operate in a region between order and complete randomness or chaos, where the complexity is maximal." (Wikipedia: Edge of Chaos.)

The following figure represents what is referred to as the ‘edge of chaos’, which is a graphical model which illustrates, specifically, what is referred to as the ‘life-generation zone’ against what on the model constitutes the mathematical equivalent of the binary possibilities 0 and 1. Here is an example of this figure:

Copyright Mark Maxwell.

Two possible models of comprehension present themselves for interpreting the base figure, illustrated below:

1. Probability

-->(to infinity)Off|On/Off Possibilities(Limited)|On(to infinity)-->
-->(representing a linear progression baseline)-->

2. Possibility

|(1 instance)=Off --
oo(infinity)(1 instance)=On/Off --
|(1 instance)=On --
--- (representing a static per instance baseline) ---

The former model is one which represents linear progression of any given and the probability of that given falling within the mathematical range of infinite 0 (off) or the mathematical range of infinite 1 (on). The limited range in the middle is limited in terms of the probability of the given falling within any of an infinite combination of 0 and 1.
The latter model represents a static non-progression, in which the given’s position on the base figure is examined not by the probability of where it might fall but as a measurement of where it does fall per instance per occurence. The possibilities for the given’s location per instance are 0 (off), 1 (on), or an infinite possible chain combination of 0 and 1.
The model which asserts itself for the base figure is ultimately the former.
Another way of illustrating the base figure is demonstrated here:

Copyright ???.

Dynamics, Computation, and the "Edge of Chaos": A Re-Examination
Revisiting the Edge of Chaos: Evolving Cellular Automata to Perform Computations

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