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So should one conclude from this that the universe is in fact a giant cellular automaton with rules like those of case (c)?
… For there are immediately simple issues like what one imagines happens at the edges of the cellular automaton array. But much more important is the fact that I do not believe in the distinction between space and its contents implied by the basic construction of a cellular automaton.

base 2, the presence of carry digits in the multiplication process makes the system not quite an ordinary cellular automaton. It turns out, however, that multiplication by 3 in base 6, or by 2 or 5 in base 10, never leads to carry digits, with the result that in such cases the system can be thought of as following a purely local cellular automaton rule of the kind illustrated in the second set of pictures.
… Cellular automaton rules equivalent to multiplication of digit sequences in various bases.

This is a consequence of the fact that the cellular automaton rule allows only certain blocks to appear in the pattern, as illustrated in the picture below. (e) is generated by a two-dimensional cellular automaton; (f) is the sequence that appears on the center column of rule 30.
Cellular automaton rule 30, and the 3×2 blocks which appear in large patterns generated by it.

But even though no simple formula may exist, it is still always in principle possible to represent the outcome of any process of cellular automaton evolution by at least some kind of formula.
… The picture at the top of the facing page shows what happens if one considers two steps of cellular automaton evolution. … The first row shows the original cellular automaton rules.

In each case the position of a cell is specified by a pair of numbers given as base 2 digit sequences in the initial conditions for a cellular automaton. The evolution of the cellular automaton then quickly determines what the color of the cell at that position in the pattern on the left will be. For rule 188 the cellular automaton that does this involves 12 colors; for rule 60 it involves 6.

Time and Causal Networks
I argued in the last section that the progress of time should be viewed at a fundamental level much like the evolution of a system like a cellular automaton. But one of the features of a cellular automaton is that it is set up to update all of its cells together, as if at each tick of some global clock. … At first it may seem bizarre, but one possibility that I believe is ultimately not too far from correct is that the universe might work not like a cellular automaton in which all cells get updated at once, but instead like a mobile automaton or Turing machine, in which just a single cell gets updated at each step.

And what this means is that if one has some segment of the encrypting sequence, corresponding to part of a column, then one can immediately use this to deduce the forms of a sequence of other columns, and thus to find the form of a row in the cellular automaton—and hence the original key.
… And one can then immediately
An example of the basis for cryptanalysis of an additive cellular automaton. The first set of pictures show the ordinary evolution of the rule 60 cellular automaton, in which each successive row is deduced from the one above.

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The behavior of a class 4 two-dimensional cellular automaton often known in recreational computing as the Game of Life. … The rule for this cellular automaton considers the 8 neighbors of a cell (including diagonals): if two of these neighbors are black, then the cell stays the same color as before; if three are black, then the cell becomes black; and if any other number of neighbors are black, then the cell becomes white.

The first picture below shows an extreme example of a class 1 cellular automaton in which after just one step the only sequences that can occur are those that contain only black cells.
The resulting configuration can be thought of as a so-called attractor for the cellular automaton evolution. … The second picture below shows a class 2 cellular automaton that once again evolves to an attractor after just one step.

Each page shows 700 steps in the evolution of the cellular automaton.