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The standard mathematical framework of relativity theory implies that any massless particle must propagate at c in a vacuum—so that not only light but also gravitational waves presumably go at this speed (and the same is at least approximately true of neutrinos).
It implies that every block of length m that occurs at a particular step has exactly 4 immediate predecessor blocks of length m + 2 (see page 960 ). It also implies that all 2 t possible single columns of t cells can be generated from some initial condition.
Time reversal invariance
The reversibility of the laws of physics implies that given the state of a physical system at a particular time, it is always possible to work out uniquely both its future and its past. Time reversal invariance would further imply that the rules for going in each direction should be identical.
The Löwenheim–Skolem theorem from 1915 implies that any axiom system must always have a countable model.
Standard operations in logic can be generalized as Not[a_] = 1 - a , And[a_, b_] = Min[a, b] , Or[a_, b_] = Max[a, b] , Xor[a_, b_] = Abs[a - b] , Equal[a_, b_] = 1 - Abs[a - b] , Implies[a_, b_] = 1 - UnitStep[a - b](a - b) .
And this implies that in choosing initial conditions for a system like the shift map, one should therefore make no distinction between the exact number 1/2 and numbers that are sufficiently close in size to 1/2 .
It has then often been assumed that having maximal repetition period will somehow imply maximum randomness in all aspects of the sequence one gets.
But the crucial point is that this fact by no means implies that it is necessarily easy to go from the constraints to find a pattern that satisfies them.
And indeed in the literature of traditional science I have quite often seen models which were taken very seriously because they could be made to reproduce a few specific numbers, but which are shown up as completely wrong if one works out the overall behavior that they imply.
And the presence of such structures implies that at least with some ranges of parameters, even very small changes in underlying rules can lead to large changes in certain aspects of the patterns that are produced.