Note that because of the probabilistic nature of the correlations it turns out to be impossible to use them to do anything that would normally be considered communicating information faster than the speed of light.) … The approach I discuss in the main text is quite different, in effect using the idea that in a network model of space there can be direct connections between particles that do not in a sense ever have to go through ordinary intermediate points in space.

But in practice it is difficult to do this, because without knowing the factors of m one cannot even readily tell whether a given x is a quadratic residue modulo m .

In 1979 Robert Axelrod tried setting up computer programs as players and found that in tournaments between them the winner was often a simple "tit-for-tat" program that cooperates on the first step, then on subsequent steps just does whatever its opponent did on the previous step.

For whether one does calculations by hand, by mechanical calculator or by electronic computer, one always needs an explicit representation for numbers, typically in terms of a sequence of digits of a certain length.

One approach to finding constraints that can be satisfied only by nested patterns is nevertheless to start from specific nested patterns, look at what templates occur, and then see whether these templates are such that they do not allow any purely repetitive patterns.

And from studying phase transitions in cellular automata, it does not seem that an interpretation in terms of symmetry is particularly useful.

But from looking at such simulations, as well as from my own experiments done from 1980 onwards, I increasingly came to believe that almost any complexity being generated had its origin in phenomena similar to those I had seen in cellular automata—and had essentially nothing to do with natural selection.

But by the end of the 1980s, the idea had emerged of doing explicit computer simulations with entities in the market represented by practical programs.

But showing that it is unprovable whether a Turing machine halts with every input (a Π 2 statement in the notation of page 1139 ) does not immediately imply anything about whether this is in fact true or false.

But while it was immediately clear that most cellular automata do not have the kind of reversible underlying rules assumed in traditional statistical mechanics, it still seemed initially very surprising that their overall behavior could be so elaborate—and so far from the complete orderlessness one might expect on the basis of traditional ideas of entropy maximization.