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Representation (b), for example, is more weighted towards large numbers, while (c) is more weighted towards small numbers.

Note (a) for The Threshold of Universality in Cellular Automata

Yet while practical computers have elaborate circuits containing huge numbers of Nand operations, we now know that for example simple cellular automata that can be implemented with just a few Nand operations (see page 619 ) are enough.

Note (b) for Intelligence in the Universe

But while it shows some of the essence of self-reproduction, it lacks many of the more elaborate features common in biological self-reproduction.

Note (a) for Symbolic Systems

[Enumerating] possible expressions LeafCount[expr] gives the number of symbols that appear anywhere in an expression, while Depth[expr] gives the number of closing brackets at the end of its functional representation—equal to the number of levels in the rightmost branch of the tree representation.

Note (c) for Register Machines

But while extending the instruction set can increase the speed of operations, it does not appear to yield a much larger density of machines with complex behavior.

Note (a) for Fluid Flow

A variety of tricks, however, maintain precision while in effect allowing a large number of molecules to be handled at the same time.

Note (d) for Ultimate Models for the Universe

But while I believe that every feature of our universe does indeed come from an ultimate discrete model, I would be very surprised if the values of constants which happen to be easy for us to measure in the end turn out to be given by simple traditional mathematical formulas.

Note (g) for Conserved Quantities and Continuum Phenomena

(With k = 2 , for b = 1 , {1, 1} represents conservation of the total number of cells, regardless of color, while for b = 2 , {1, 1, 1, 1} represents the same thing, while {0, 1, -1, 0} represents the fact that in going along in any state the number of black-to-white transitions must equal the number of white-to-black ones.)

Note (a) for Mathematical Constants

If h is rational, the sequence is repetitive, while if h is a quadratic irrational, it is nested.

Note (f) for The Intrinsic Generation of Randomness

In addition, one should recognize that while the complete evolution of the cellular automaton may effectively generate perfect randomness, there may be deviations from randomness introduced when one constructs a practical random number generator with a limited number of cells.