It does not matter what initial conditions one starts from: one always reaches the same all-black attractor in the end. … In the first case, the sequences that can occur are ones that involve only black cells. In the second case, the sequences are ones in which every black cell is surrounded by white cells.

But what if one looks not at a complete organism but instead just at some part of an organism? … But what about the much more complicated forms that one sees in biological systems? … But still one might assume that to get significant complexity would require something more.

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. … One cannot then immediately use the method described above. … And one can then immediately An example of the basis for cryptanalysis of an additive cellular automaton.

For all one ever need do is to work out the remainder from dividing the position of a particular square by the size of the basic repeating block, and this then immediately tells one how to look up the color one wants. … So as the picture illustrates this means that new squares always have positions that involve numbers containing one extra digit. With the particular rules shown, the new squares always have the same color as the old one, except in one specific case: when a black square is replaced, the new square that appears in the upper right is always white.

For if one looks at the individual cells in the cellular automaton one can plainly see that they just follow definite rules, with absolutely no freedom at all. … For normally it has assumed that if one can only find the underlying rules for the components of a system then in a sense these tell one everything important about the system. … But at least with our normal methods of perception and analysis one

And one can see that in fact the third multiway system is incomplete, since by following its rules one can never for example generate either or its negation . … Can one always in the end make the system complete? If one is not quite careful, one will generate too many strings, and inevitably get inconsistencies where both a string and its negation appear, as in the second picture on the facing page .

But as soon as one also puts in an abstract function or relation with more than one argument, one gets universality. … If one puts enough constraints on the axioms one uses, one can eventually prevent universality—and in fact this happens for commutative group theory, and for the simplified versions of both real algebra and geometry on pages 773 and 774 . But of the axiom systems actually used in current mathematics research every single one is now known to be universal.

To test this explicitly one would have to look at an infinite number of possible integers. … So how can one see this? … If in general one was able to tell whether such a solution exists then it would mean that one could always answer the question of

For all one need do, as in the pictures at the top of the next page , is to evaluate the expressions for all possible values of each variable, and then to see whether the patterns of results one gets are the same. … And in principle one can use this method to establish any equivalence that exists between any expressions with an operator of any specific form. … And one advantage of this approach is that at least in principle it allows one to handle operators—like those found in many areas of mathematics—that are not based on finite tables.

one might wonder whether perhaps some other ordering would make it easier to compress more complex images. One simple approach is just to assemble a large collection of images typical of the ones that one wants to compress, and then to order the basic forms so that those which on average occur with larger weights in this collection appear first. The pictures on the next page show what happens if one does this first with images of cellular automata and then with images of letters.