shamebear (![[info]](http://l-stat.livejournal.com/img/userinfo.gif){kind=small} shamebear) wrote in alexanderwait, |
Re: Dimensionality and universality
I agree that Wolfram isn't much help here. I must also admit to know too little about graphs resembling euclidean spaces to make use of that approach.
However I liked the approach taken by for instance Kauffman in his book "at home in the universe" Consider a grid of "lightbulbs" where the input to each lightbulb is the state of the other lightbulbs it is connected to. Each set of inputs produce either light on or light off as output. Kauffman shows the "interesting" range to be at the "edge of chaos" for certain settings. A setting of K = 2 connections to other lightbulbs appear good. If K is increased, another parameter P should be adjusted to compensate.
P can be defined as (Number of input-combinations that give ON as output)/(Total number of input-combinations) and is a measure of the bias the lightbulb has towards one setting. If this bias is strong (P near 0 or near 1), the network may still be in the interesting range if K is high.
In a 2D cellular automata, each cell usually have four neighbours. in 3D it has six. That means our "lightbulb" should show a strong bias to compensate. (Kauffman also advices to stick to so-called "canalyzing" boolean functions)
These seem like good practical advices. While Kauffman talks of abstract spaces, not real-life euclidean, that's not too bad! Kauffman claims that life started as an "autocatalytic process", a certain web of chemical reactions in the primordial soup. Each cell then get to symbolise a certain molecule and the flow of nutrients could symbolise the relative amounts of molecules as one is spent to produce another. When it's all jumbled in a soup anyway, the elements exact position in the bowl is secondary.
"interesting" behaviour could be several instances of the same nearly closed webs of reactions (different individuals of the same species), showing homeostasis yet never stopping their evolution.
If these webs could be identified and displayed properly, very interesting behaviour would be allowed to occur on the screen without basing it in a physical space.
I agree that Wolfram isn't much help here. I must also admit to know too little about graphs resembling euclidean spaces to make use of that approach.
However I liked the approach taken by for instance Kauffman in his book "at home in the universe" Consider a grid of "lightbulbs" where the input to each lightbulb is the state of the other lightbulbs it is connected to. Each set of inputs produce either light on or light off as output. Kauffman shows the "interesting" range to be at the "edge of chaos" for certain settings. A setting of K = 2 connections to other lightbulbs appear good. If K is increased, another parameter P should be adjusted to compensate.
P can be defined as (Number of input-combinations that give ON as output)/(Total number of input-combinations) and is a measure of the bias the lightbulb has towards one setting. If this bias is strong (P near 0 or near 1), the network may still be in the interesting range if K is high.
In a 2D cellular automata, each cell usually have four neighbours. in 3D it has six. That means our "lightbulb" should show a strong bias to compensate. (Kauffman also advices to stick to so-called "canalyzing" boolean functions)
These seem like good practical advices. While Kauffman talks of abstract spaces, not real-life euclidean, that's not too bad! Kauffman claims that life started as an "autocatalytic process", a certain web of chemical reactions in the primordial soup. Each cell then get to symbolise a certain molecule and the flow of nutrients could symbolise the relative amounts of molecules as one is spent to produce another. When it's all jumbled in a soup anyway, the elements exact position in the bowl is secondary.
"interesting" behaviour could be several instances of the same nearly closed webs of reactions (different individuals of the same species), showing homeostasis yet never stopping their evolution.
If these webs could be identified and displayed properly, very interesting behaviour would be allowed to occur on the screen without basing it in a physical space.
