From looking at the brain one might guess that parallel or other non-standard hardware might be required to achieve efficient human-like thinking.

Note that as discussed on page 1159 how one uses algebraic axioms can affect issues of universality and undecidability.

And for k = 3 , r = 1 there are 7,625,597,484,987 rules in all, with 2187 totalistic ones.)

The distinction may seem more obvious if one considers, for example, sequential substitution systems or cyclic tag systems.

The idea of this model is to add cells to a cluster one at a time, and to determine where a cell will be added by seeing where a random walk that starts far from the cluster first lands on a square adjacent to the cluster.

(It is as if one is defining constraints on the initial conditions for a cellular automaton by looking at the pattern generated by the cellular automaton after a long time.)

One development in the 1990s is the generation of phyllotaxis-like patterns in superconductors, ferrofluids and other physical systems.

In mathematics, rather little is usually done with network substitutions, though the proof of the Four-Color Theorem in 1976 was for example based on showing that 300 or so possible replacement rules—if applied in an appropriate sequence—can transform any graph to have one of 1936 smaller subgraphs that require the same number of colors. (32 rules and 633 subgraphs are now known to be sufficient.)

One might think that a more efficient approach would be to start with the trivial length t digit sequence for c t in base c , then to find a particular base k digit just by converting to base k . … Note that even though one may only want to find a single digit in m t , I know of no way to do this without essentially computing all the other digits in m t as well.

Out of the many hundreds of times that I have used hashing in practice, I recall only a couple of cases where schemes like the one just described were not adequate, and in these cases the data always turned out to have quite dramatic regularities. In typical applications hash codes give locations in computer memory, from which actual data is found either by following a chain of pointers, or by probing successive locations until an empty one is reached.