On Nov 5, 2005, at 2:22 AM, Russell Standish wrote:

Game of Life is an example 2D system capable of universal
computation. I'm not sure this implies consciousness is possible in
2D, but it needs to be considered.

It does imply that if the Game of Life is the laws of physics of your universe, then consciousness is possible, because at the very least a 3D physics could be simulated. Whether that should be interpreted as consciousness in 2D may be a subtle issue, because the perceptual world of the conscious entities would be 3D - perhaps that was your point?

However, one can easily imagine a perceptual 2D world existing for conscious entities. Even if there is no self-consistent 2D physics leading to atoms, planets, etc., one can computationally simulate Flatland (a la Abbott) or a Planiverse (a la Dewdney) in a 3D universe, with no requirement for a consistent micro-physics. (In fact the Planiverse is my simulation domain for my AI work.)

So, whether it's the base physical reality you care about, or the perceived reality of the conscious entities, I would say 2D consciousness is possible. (Admittedly, in the latter case, one has to consider whether the 2D creatures could at some point develop science sufficient to prove that they must be simulated in some higher-dimensional physics!)

I think Turing machines are impossible in 1D, however...

No, there are 1D cellular automata that are computation universal. Here's an abstract from a paper showing it; I don't seem to be able to find the paper online. The paper is from 1990. However, there are references to earlier constructions, e.g. here: http://www.stephenwolfram.com/publications/articles/mathematics/85-twenty/18/text.html . Again, I can't find the cited paper online.

The relevant Mathworld page is rather confused and misleading: http://mathworld.wolfram.com/UniversalCellularAutomaton.html

There it seems that by "universal" they mean that there is a certain class of 1D CA that can simulate any other 1D CA in that class. Hmm, so what? Cute, but hardly surprising. Mathworld is a great site, but it's too bad in some ways it's so tied in with the Wolfram mythos. There's a huge spin put on pages like the one above that you have to try to penetrate.



Robert A. Hearn



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