For all one need do is to repeat the process that was used for encryption, and reverse the color of every square in (c) for which the corresponding square in (b) is black. … But perhaps not surprisingly it is fairly easy to do cryptanalysis in such a case.

But that does not mean that there cannot exist higher forms of perception and analysis that succeed in recognizing at least some regularities that our existing methods do not.

A crucial feature of cyclic tag systems is that the choice of what block of elements can be added does not depend in any way on the form of the sequence. … But while picture (c) shows the effects of various lines carrying information around the system, it gives no indication of why the lines should behave in the way they do.

In general what one needs to do in order to prove universality is to find a procedure for setting up initial conditions in one system so as to make it emulate some general class of other systems. … Typically what we are used to doing is constructing things in stages.

But when one examines the known examples of such systems—all of which have very intricate underlying rules—one finds that even though the particular part of their behavior that is identified as output is sufficiently restricted to avoid universality, almost every other part of their behavior nevertheless does exhibit universality—just as one would expect from the Principle of Computational Equivalence. … When we look at the patterns produced by such systems they certainly do not seem to have any great regularity; indeed in most

So where in this distribution do the typical axiom systems of ordinary mathematics lie? … I suspect that it has to do with the fact that in mathematics one usually wants axiom systems that one can think of as somehow describing definite kinds of objects—about which one then expects to be able to establish all sorts of definite statements.

And if one does this it is indeed perfectly possible, say, to program arithmetic to reproduce any proof in set theory. In fact, all one need do is to encode the axioms of set theory in something like the arithmetic equation system of page 786 .

But while traditional engineering has usually ended up finding ways to avoid searches for the limited kinds of systems it considers, the phenomenon of computational irreducibility makes it inevitable that if one considers all possible simple programs then finding particular forms of behavior can require doing searches that involve irreducibly large amounts of computational work. … So in practice a better approach will often be in effect just to do basic science—and much as I have done in this book to try to build up a body of abstract knowledge about how all sorts of simple programs behave.

[Number representations in] practical computing Numbers used for arithmetic in practical computing are usually assumed to have a fixed length of, say, 32 bits, and thus do not need to be self-delimiting.

[Systems based on] undirected networks Networks with connections that do not have definite directions are discussed at length in Chapter 9 , mainly as potential models for space in the universe.