Let's take as an example the 3d → 2d reduction: What's the cost of simulating a 3d cellular automaton by a 2d cellular automaton?
Here is a bunch of more specific questions:
What kind of algorithms will have their time complexity changed, by how much?
What would be the basic idea for the encoding; how is a 3d grid efficiently (or not efficiently…) mapped to a 2d grid? (The challenge seems to achieve communication between two cells that where originally neighbors on the 3d grid, but are not neighbors anymore on the 2d grid).
In particular, I'm interested in the complexity drift for exponential complexity algorithms (which I guess remains exponential whatever the dimension, is it the case?)
Note: I'm not interested in low complexity classes for which the chosen I/O method has an influence on complexities. (Maybe the best is to assume that the I/O method is dimensionless: done locally on one specific cell during a variable amount of time steps.)
Some context: I'm interested in parallel local graph rewriting, but those graphs are closer to 3d (or maybe ωd…) grids than to 2d grids, I'd like to know what to expect of a hardware implementation on a 2-dimentional silicon chip.