But so how do cellular automata with all these different rules behave? The next page shows a few examples in detail, while the following two pages [ 55 , 56 ] show what happens in all 256 possible cases. … In the very simplest cases, all the cells in the cellular automaton end up just having the same color after one step.

One can readily enumerate all 4096 possible Turing machines with 2 states and 2 colors. … Each of them works in a slightly different way, but all of them follow one of the three schemes shown at the bottom of the next page —and all of them end up exhibiting the same overall linear increase in number of steps with length of input.

In all cases, the systems are started from the initial string BAB .

And this means that it cannot successfully reproduce all the results of logic. … But as soon as one uses a total of just 6 Nand s, one suddenly finds that out of the 3402 possibilities with 3 variables 32 axiom systems equivalent to case (f) above all end up working all the way up to at least 4-value operators. … But it turns out that there are all sorts of other axioms that also do this.

Properties [of example multiway systems] The first multiway system here generates all strings that end in ; the third all strings that end in . The second system generates all strings where the second-to-last element is white, or the string ends with a run of black elements delimited by white ones.

In each case single cells are encoded as blocks of cells, and all distinct such encodings with blocks up to length 20 are shown.

But since there are only six possible positions in all, it is inevitable that after at most six steps the dot will always get to a position where it has been before. … In all cases, the behavior is repetitive, and the maximum repetition period is equal to the number of possible positions.

Yet the picture itself does not at first appear to be at all reversible. … Starting from an initial condition in which all black cells or particles lie at the center of a box, the distribution becomes progressively more random.

For in the course of that chapter it became clear that almost all the very varied systems in this book can actually be made to emulate each other in a quite comparable number of steps. … And similarly if one has a multiway system that yields exponentially many strings then to reproduce all these will again take exponentially many steps. … A Turing machine can quickly test the highlighted path but could take exponentially long to test all paths.

In all the cases shown, these results are quite simple, consisting of sequences that increase uniformly or fluctuate in a purely repetitive way. … With all the rules shown here, successive elements either increase smoothly or fluctuate in a purely repetitive way. … All rules of the kind shown here lead to sequences where f[n] can be expressed in terms of a simple sum of powers of the form a n .