Then Ask What is Complex Behavior

It is one thing to describe as we have done informally above, even qualitatively, what a complex system is, and to conjure up myriad examples of complex systems. It is quite another to quantify the notion of complexity itself, to describe the relationship between complexity and information, and/or to understand the role that complexity plays in various physical and/or computational contexts. Each of these difficult problems in fact remains very much open. While we may find it easy enough to distinguish a complex object from a less complex object, it is far from trivial to furnish anything that goes beyond a vague characterization as to how we have done so. Some recent attempts at quantifying the notion of complexity are sketched below. 

To set up the problem and in order to appreciate more fully the difficulty in quantifying complexity, consider figure 12.1. The figure shows three patterns (a) an area of a regular two-dimensional Euclidean lattice, (b) a space-time view of the evolution of the nearest-neighbor one-dimensional cellular automata rule RllO, starting from a random initial state,f and (c) a completely random collection of dots. 

These patterns illustrate the incongruity that exists between mathematically precise notions of entropy (see below), or the amount of disorder in a system, and intuitive notions of complexity. 

Whereas pattern (b) is intuitively the most complex of the three patterns, it has neither the highest entropy (which belongs to pattern (c)) or the lowest (which belongs to pattern (a)). Indeed, were we to plot our intuitive sense of complexity as a function of the amount of order or disorder in a system, it would probably look something like that shown in figure 12.2. The problem is to find an objective measure of the complexity of a system that matches this intuition. 

---