The rules for the Turing machine and the cellular automaton are shown above. In the rules for the cellular automaton, indicates a cell of any color.

And one can then consider the axioms of a system as defining possible transformations from one sequence of these elements to another—just like the rules in the multiway systems we discussed in Chapter 5 . … The rules on the left in effect correspond to axioms that specify valid transformations between strings of black and white elements.

For it implies that all the wonders of our universe can in effect be captured by simple rules, yet it shows that there can be no way to know all the consequences of these rules, except in effect just to watch and see how they unfold.

The basic action of Mathematica is then to transform such expressions according to whatever rules it knows. Most often these rules are specified in terms of Mathematica patterns—expressions in which _ can stand for any expression.

Pure equational logic Proofs in operator systems always rely on certain underlying rules about equality, such as the equivalence of u  v and v  u , and of u  v and u  v/. a  b . And as Garrett Birkhoff showed in 1935, any equivalence between expressions that holds for all possible forms of operator must have a finite proof using just these rules.

And if one always does computations using systems that have only nearest-neighbor rules then just combining 2t + 1 bits of information can take up to t steps—even if the bits are combined in a way that is not computationally irreducible. … But I strongly suspect that computational irreducibility prevents outcomes in systems like rule 30 and rule 110 from being found by computations that are in NC—implying in effect that allowing arbitrary connections does not help much in computing the evolution of such systems. … A notable example due to Cristopher Moore from 1996 is the 3D majority cellular automaton with rule UnitStep[a + AxesTotal[a, 3] - 4] (see page 927 ); another example is the Ising model cellular automaton from page 982 .

The pictures below show the behavior of several sequential cellular automata with k = 2 , r = 1 elementary rules. … Additive rules. … Even though the basic rule is additive, there seems to be no simple traditional mathematical description of the results.

With the particular rule shown, the behavior always eventually stabilizes—though sometimes only after an astronomically long time.

[No text on this page] Examples of the evolution of networks in which a single cluster of nodes is replaced at each step according to the rules shown.

The cellular automaton rule used is a 4-color totalistic one with code 1004600.