Data Compression One usually thinks of perception and analysis as being done mainly in order to provide material for direct human consumption. … One simple example of such a process is run-length encoding—a method widely used in practice to compress data that involves long sequences of identical elements, such as bitmap images of pages of text with large areas of white. … This means, for example, that instead of having to list explicitly all the cells in a run of, say, 53 identical cells, one instead just gives the number "53".

For if the numbers corresponding to the lengths of successive runs are given one after another then there is no way to tell where the digits of one number end and the next begin. … One, illustrated in picture (c) at the bottom of the facing page , is to insert at the beginning of each number a specification of how many digits the number contains.

The point is that the two-dimensional blocks that one forms always contain cells whose colors are connected by the cellular automaton rule—and this greatly reduces the number of different arrangements of colors that can occur. In cases (e) and (f), however, there is no simple rule for going from one row to the next, and two-dimensional block encoding—like all the other encoding schemes we have discussed so far—does not yield any substantial compression. … The basic idea is just to scan two-dimensional data looking for repeats not of one-dimensional sequences, but instead of two-dimensional regions.

The main reason is that it potentially allows one to discuss in a unified way systems that have completely different underlying rules. … If one starts this cellular automaton with an even number of black cells, then after a few steps of evolution, no black cells are left. But if instead one starts it with an odd number of black cells, then a single black cell survives forever.

But while one can emulate each step in the evolution of a mobile automaton or a Turing machine with a single step of cellular automaton evolution, this is no longer in general true for substitution systems. That this must ultimately be the case one can see from the fact that the total number of elements in a substitution system can be multiplied by a factor from one step to the next, while in a cellular automaton the size of a pattern can only ever increase by a fixed amount at each step.

Among such 2-state 4-color Turing machines perhaps one in 50,000 shows complex behavior when started from a blank tape. Among 4-state 2-color Turing machines the same kind of complex behavior is also seen—as discussed on page 81 —but now it occurs only in perhaps one out of 200,000 cases. … One of the 14 essentially equivalent 2-state 3-color Turing machines that yield complicated behavior when started from a blank tape.

At least with the strings one can see in the pictures there are no inconsistencies. … But can every possible statement that one might expect to be true or false actually in the end be proved either true or false? … Other simple axiom systems for logic like those on page 808 yield networks similar to the one shown.

But if one looks at blocks of width 41, then such structures do eventually show up, as the picture on page 293 demonstrates. … In some cases, one structure essentially just passes through another with a slight delay. … But quite often it takes a very large number of steps before one can tell for sure what is going to happen.

And instead of looking explicitly at the complete pattern of digits, one can consider just finding the size of the fractional part of each successive number. … But division by 2 just does the opposite of multiplication by 2, so in base 2 it simply shifts all digits one position to the right.

But one might imagine that perhaps this representation is somehow perverse, and that if we were just to choose another one, then numbers generated by simple mathematical operations would no longer seem complex.