The pictures below show what happens if one takes various patterns, arranges their rows one after another in a long line, and then applies pointer-based encoding to the resulting sequences. … So how can one do better? The basic answer is that one needs to take account of the two-dimensional nature of the patterns.

So if the encrypting sequence corresponds to a single column, how can one find an adjacent column? … One picks some sequence of cells for the right half of the top row, then evolves down the page. … (a) shows ordinary evolution from one row to the next.

To compare the general computational capabilities of continuous and discrete systems one needs to find some basic scheme for constructing inputs and decoding outputs that one can use in both types of systems. … And the same is presumably true if one works with essentially any of what are normally considered standard mathematical functions. But what happens if one assumes that one can set up a system that not only finds values of such functions but also finds solutions to arbitrary equations involving them?

In a multiway system, one can imagine identifying "true" with a string consisting of a single black element. … One can take negation to be the operation that interchanges black and white. … The third one, however, is incomplete, since for example it never generates either or its negation .

For their results have the kind of precision and clarity that one expects of theoretical or mathematical statements. … And this principle applies both to the structure of the actual systems one studies—and to the procedures that one uses for studying them. … In addition, looking at systems with simpler underlying structures gives one a better chance of being able to tell what is really responsible for any phenomenon one sees—for there are fewer features that have been put into the system and that could lead one astray.

For in making a summary one inevitably has to pick out only certain features, and in doing this one can remove or obscure the most interesting effects. But one of the problems with very direct experiments is that they often generate huge amounts of raw data. Yet what I have typically found is that if one manages to present this data in the form of pictures then it effectively becomes possible to analyze very quickly just with one's eyes.

If there are almost no similarities then one can reasonably conclude that the model is wrong. … Quite often this will depend, at least in part, on how one intends to use the model. … One might perhaps think that in the end one could always tell whether a model was correct by explicitly looking at sufficiently low-level underlying elements in a system and comparing them with elements in the model.

One feature of causal networks is that they tell one not only what the consequences of a particular event will be, but also in a sense what its causes were. Thus, for example, if one starts, say, with event 17 in the first causal network below, then to find out that its causes were events 11 and 16 one simply has to trace backwards along the connections which lead to it. … And at least when there is overall apparent randomness, the networks that one gets by going forwards and backwards from a particular node will look very similar.

If one starts from a given initial string, then typically one will generate different strings by applying different replacements. But if one is going to get the same causal network, then it must always be the case that there are replacements one can apply to the strings one has generated that yield the same final string. So what this means is that any pair of paths in the multiway system that diverge must be able to converge again within just one step—so that all the arrows in pictures like the ones below must lie on the edges of quadrilaterals.

But it also means that if one once discovers a rule that reproduces sufficiently many features of the universe, then it becomes extremely likely that this rule is indeed the final and correct one for the whole universe. … And as a result, if these basic processes are reproduced correctly, then I believe that one can have considerable confidence that one in fact has the complete rule for the universe. … For one might expect that there would always be some other esoteric phenomenon, say in particle physics, that would be discovered and would show that whatever rule one has found is somehow incomplete.