Note that without further conditions the continuous forms cannot be considered unique extensions of the discrete ones.

For typically one imagines that if something is to be a genuine reflection of human will then there must be some purpose to it.

If one looks at simple instances of problems (say PCP with short strings) then my experience is that many are easy to solve.

Operator systems One can represent the possible values of expressions like f[f[p, q], p] by rule numbers analogous to those used for cellular automata.

The argument for this is similar to the one on pages 941 and 954 for 1D cellular automata.

But eventually what happened was that I tried a few other very simple models, and to my great surprise one of them ended up showing the behavior I wanted, even though I had in no way explicitly built it in.

For among the 29 possible states allowed for each cell were ones set up to behave quite directly like components for practical electronic computers like the EDVAC—as well as to grow new memory areas and so on. … When only one glider is present, a new spaceship emerges on the right as the output. … If one considers rules with more than two colors, it becomes straightforward to emulate standard logic circuits.

One can always just use n copies of the same symbol to represent an integer n —and indeed this idea seems historically to have arisen independently quite a few times. But as soon as one tries to set up a more compact notation there inevitably seem to be many possibilities. … And it is fairly easy to see how a different historical progression might have ended up making another one of these seem the most natural.

For example, rule 90 on page 25 corresponds to f[1, _, 1] = 0; f[0, _, 1] = 1; f[1, _, 0] =1; f[0, _, 0] = 0 One can specify initial conditions for example by a[0, 0] = 1; a[0, _] = 0 (the cell on step 0 at position 0 has value 1, but all other cells on that step have value 0). Then just asking for a[4, 0] one will immediately get the value after 4 steps of the cell at position 0. … Thus, for example, one can use initial conditions a[0, -1] = p; a[0, 0] = q; a[0, 1] = r; a[0, _] = 0 to generate a formula for the value of a cell that holds for any choice of values for the three initial center cells.

It turns out that in order to get complex behavior in such systems, one needs either to allow more than two possible colors for each element, or to remove more than two elements from the beginning of the sequence at each step. … With more than two colors, one finds that rules of Post's type which remove just two elements at each step can yield complex behavior, even starting from an initial condition such as {0, 0} .