At 1/17/04, Eric Hawthorne wrote:
1. All cellular automata which are computationally universal are reducible to each other, by the definition of universality, so it doesn't matter which D the automaton program itself is. The subject matter that they can represent and compute is equivalent.

That's correct on one level, but what we're really interested in is the dimensionality of space that SASs within the computation would perceive their world as having. For instance, we know that there is a very simple 1D CA that's computation universal (Wolfram's rule 110), so we know that we can implement any higher-dimensional cellular automata in rule 110. However, if we implement in rule 110 some 3D CA which contains SASs, these SASs would go right on moving around in their 3D world and perceiving their space as 3D. In an important sense, it would be incorrect to say that those SASs live in a 1D world, even though ultimately their "substrate" is 1D. This is really just another example of the familiar concept of "substrate neutrality".

So at the least the 2D or 4 or 5D sentient creatures would be frustrated
(remember, they are SUBSTRUCTURES, they're not computing the space itself, they're
part of the space and perceiving and acting on other parts of it).

But what possible reason do we have for believing that 4D or 5D cellular automata (or, to be more careful, cellular automata which would be perceived as 4D or 5D by the SASs within them) are somehow hostile to the existence of SASs? The arguments in Tegmark's paper about how universes with more than 3 spacial dimensions can't support stable structures like atoms simply don't apply to 4D cellular automata. Those arguments are very specific, applying only to quantum-physical and string-theory models.

-- Kory

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