One of the main things I have done in this book is in a sense to introduce a new approach to generalization in which one considers systems that have simple but completely arbitrary rules—and that are not set up with any constraint about what theorems they should satisfy.

I argued above that if the rules for a system are as simple as they can be, then this may suggest the presence of purpose.

For while this is undoubtedly very common say in cellular automata, the most immediate suggestions of it are in class 4 systems like rule 110 that in effect happen to do their computations in a way that looks at least somewhat similar to the way we as humans are used to doing them.

The rules for such register machines are then idealizations of practical programs, and are taken to consist of fixed sequences of instructions, to be executed in turn.

Multiplication systems [from cellular automata] The rules for the cellular automaton shown here are {{_, 0, 3 | 8}  5, {_, 0, 2 | 7}  8, {_, 1, 4 | 9}  9, {_, 1, 3 | 8}  4, {_, 1, 2 | 7}  8, {_, 10, 4 | 9}  3, {_, 10, 3 | 8}  7, {_, 10, 2 | 7}  2, {5 | 6, 1, 0}  9, {5 | 6, 10, 0}  3, {5 | 6, 1, _}  6, {5 | 6, 10, _}  5, {_, 2 | 3 | 4 | 5, _}  10, {_, 6 | 7 | 8 | 9, _}  1, {_, x_, _}  x} and the initial condition consists of a single 3 surrounded by 0 's.

RAM [emulated with cellular automata] The rules for the cellular automaton shown here are {{2, 4 | 8, 2 | 11, _, _}  2, {11 | 10, 4 | 8, 2 | 11, _, _}  11, {2, 4 | 8, _, _, _}  10, {11 | 10, 4 | 8, _, _, _}  2, {2, 0, _, _, _}  2, {11, 0, _, _, _}  11, {3 | 7 | 6, _, 10, _, _}  1, {x : (3 | 7 | 6), _, _, _, _}  x, {_, _, 6, 4, 10}  5, {_, _, 6, 8, 10}  9, {_, 3, _, 10, _}  4, {_, 7, _, 10, _}  8, {_, _, 1, _, x : (5 | 9)}  x, {1, _, _, _, _}  1, {_, _, 1, _, _}  1, {_, _, _, _, 1}  1, {_, _, x : (4 | 8 | 0), _, _}  x} The initial conditions are divided into two parts: instructions on the left and memory on the right.

Mathematica works by taking its input and repeatedly applying transformation rules—a process which normally reaches a fixed point that is returned as the answer, but with definitions like x = x + 1 ( x having no value) formally does not.

In models based both on equations and other kinds of rules the existence of formulas for conserved quantities is in general undecidable.

As a reduced analog of algorithmic information theory one can for example ask what the simplest cellular automaton rule is that will generate a given sequence if started from a single black cell. Page 1186 gives some results, and suggests that sequences which require more complicated cellular automaton rules do tend to look to us more complicated and more random.

But as the pictures on page 761 suggest, arbitrary computational systems—even Turing machines with very simple rules—can exhibit much more complicated behavior with no clear asymptotic growth rate.