Re: computer-go: Go and chaos theory

[Ray Easton <ray.easton@xxxxxxxx>]

Thomas Johnson wrote:

> On 26-Oct-99 Ray Easton wrote:
> > Systems with a finite number of states are by definition not chaotic.
>
> On the contrary, chaos theory says nothing about whether or not a chaotic system
> has a finite set of states.

The whole concept of "sensitive dependence upon initial conditions" requires that
there exist states arbitrarily close to any given state, and this is false if there
are only a finite number of states.

> - it just says that the state one arrives at after
> some interval (be it real time, number moves, whatever) is highly dependent on
> the initial state down to it's final detail.

Then, according to your definition, *every* determinsitic system is chaotic!

> With 361 parameters on the go board at any one turn, a change in just one can
> have dramatic effects on the result of the game.  I would say go is a highly
> chaotic system.

In the loose sense of the term you employ here, perhaps so.  But it will be diffcult
to apply the results of chaos theory to Go -- an idea I find somewhat bizarre in any
case -- when Go does not satisfy the standard definition of a chaotic system.

> A nice definition of chaos theory is : "the qualitative study of unstable
> aperiodic behavior in deterministic nonlinear dynamical systems".

I agree -- but no system with a finite number of states can exhibit unstable
aperiodic behavior. The very definiton of such implies that there are not a finite
number states in the system.  If I am mistaken in this, perhaps you could clarify my
error for me by providing a definition of "unstable aperiodic behavior" that does
not assume that the system in question contains at least countably many distinct
states.

--

ray

    -- je suis marxiste, tendance groucho