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Examples include the Four-Color Theorem (coloring of maps), the optimality of the Kepler packing (see page 986 ), the completeness of the Robbins axiom system (see page 1151 ) and the universality of rule 110 (see page 678 ).
But on its own this would do me little good—for I need to represent not only traditional mathematics, but also more general rules and programs, as well as procedures and algorithms.
(If one considers for example theorems about computational issues such as whether Turing machines halt, then it becomes inevitable that to cover more Turing machines one needs more axioms—and to cover all possible machines one needs an infinite set of axioms, that cannot even be generated by any finite set of rules.)
As discussed on page 885 , the sequence appears in a vertical column of cellular automaton rule 150.
Schemes for such hash codes can fairly easily be constructed using rule 30 and other cellular automata.
In addition, my 1984 identification of rule 30 as a randomness generator was the result of a small-scale systematic search.
(My discussion of fundamental physics in Chapter 9 also suggests that no separate entities beyond simple rules are needed to capture space, time or matter.)
For while it emphasizes calculation rather than proof its symbolic expressions and transformation rules provide an extremely general way to represent mathematical objects and operations—as for example the notes to this book illustrate.
The ordinary axioms of arithmetic do not apply, but there are still fairly straightforward rules for manipulating such expressions.
Rule 30
The left-hand side of the pattern shown has an obvious repetitive character.