Tanya writes "Eric Drexler has submitted the following critique:
Wolfram's argument that the properties of certain automata "must almost inevitably succeed in reproducing the fundamental features of relativity theory" (A New Kind of Science, p.520) appears to be misconceived. This may explain why he offers no examples of automata in which particle-like features move, as physical particles do, at a range of relative speeds.
He describes a broad class of automata (yielding "causal invariant" networks) that are insensitive to update order. In these, one can generate an invariant structure by adding nodes in any of many orders — for example, adding parallel layers at one or another angle. Differently angled slices of this sort look a bit like spacelike surfaces in moving frames of reference. Wolfram states (and makes central to his argument) that "one can interpret slices at different angles as corresponding to motion at different speeds" (p.521).
One must ask, however, motion of what? Since these networks are insensitive to update order, differing update patterns can make no difference to structures within a network, hence tell us nothing about the relative motion of any particle-like features it may contain. Angled update slices do not necessarily correspond to motion of anything within the model world.
If causal invariant networks naturally modeled motion, then one would expect Wolfram's book to include pictures showing streaks corresponding to variable-speed particles, but it does not. Offered neither a coherent argument nor a concrete example, we are left with a model of physics that lacks a model of relative motion."