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The top row shows how a particular non-deterministic Turing machine behaves with successive sequences of choices for rules to apply.
And insofar as there are general principles for simple programs, these principles should also apply to biological organisms—making it possible to imagine constructing new kinds of general abstract theories in biology.
It is also possible to construct symbolic systems with the so-called confluence property, in which results from any fixed number of steps of cellular automaton evolution can be found by applying rules in any possible order (see page 1036 ).
And these kinds of expressions often arise in Mathematica when one manipulates functions as a whole before applying them to arguments. ( ∂ xx f[x] for example gives f''[x] which is Derivative[2][f][x] .)
In the most direct approach, one takes a string and at each step just applies the underlying rules or axioms of the multiway system. But as soon as one knows that there is a path from a string u to a string v , one can also imagine applying the rule u v to any string—in effect like a lemma.
[Computing] square roots
A standard way to compute √ n is Newton's method (actually used already in 2000 BC by the Babylonians), in which one takes an estimate of the value x and then successively applies the rule x 1/2 (x + n/x) .
And this principle applies both to the structure of the actual systems one studies—and to the procedures that one uses for studying them.
And in fact, to do so requires absolutely no sophisticated ideas from mathematics or elsewhere: all it takes is an understanding of how to apply simple rules repeatedly.
But in 1981 it so happened that I had for some years been deeply involved in both practical computing and basic science, and I was therefore in an almost unique position to apply ideas derived from practical computing to basic science.
And in particular—just as in a system like a cellular automaton—the network can be built up incrementally by starting with certain initial conditions and then applying appropriate underlying rules over and over again.