abstract terms about the computation that is performed, without necessarily looking at all the details of how it actually works.

Why is such an abstraction useful? The main reason is that it potentially allows one to discuss in a unified way systems that have completely different underlying rules. For even though the internal workings of two systems may have very little in common, the computations the systems perform may nevertheless be very similar.

And by thinking in terms of such computations, it then becomes possible to imagine formulating principles that apply to a very wide variety of different systems—quite independent of the detailed structure of their underlying rules.

Computations in Cellular Automata

I have said that the evolution of a system like a cellular automaton can be viewed as a computation. But what kind of computation is it, and how does it compare to computations that we typically do in practice?

The pictures below show an example of a cellular automaton whose evolution can be viewed as performing a particular simple computation.

If one starts this cellular automaton with an even number of black cells, then after a few steps of evolution, no black cells are left. But if instead one starts it with an odd number of black cells, then a single black cell survives forever. So in effect this cellular automaton can be viewed as computing whether a given number is even or odd.

A simple cellular automaton whose evolution effectively computes the remainder after division of a number by 2. Starting from a row of n black cells, 0 black cells survive if n is even, and 1 black cell survives if n is odd. The cellular automaton follows elementary rule 132, as shown on the bottom left.

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From Stephen Wolfram: A New Kind of Science [citation]