In the past it has normally been assumed that these phenomena occur only in quite special systems, and not, for example, in typical systems with simple rules or of the kind that might be seen in nature. … And indeed in example (a) this approach works well: in only 36 steps one finds that the pattern

And typically it has achieved this in one of two ways: either by considering only very specific cases that have been observed or constructed to be simple, or by looking at things in so much generality that only rather simple properties ever survive.

But since one can always just start by proving the new axioms, the change can only be by a fixed amount. … Indeed, it turns out that this must always be the case for any axiom system—like those on page 803 —that ends up allowing essentially only operators of a single form. … Statements like p  ¬ ¬ p do not hold in general with more than 2 truth values.

But in fact it turns out that if only one element is removed at each step, then a tag system always effectively acts just like a slow version of a neighbor-independent substitution system of the kind we discussed on page 83 . … Because only a single element is removed at each step, the systems effectively just cycle through all elements, replacing each one in turn.

Networks representing possible sequences that can occur in the evolution of the cellular automata at the top of the page , starting from initial conditions in which black cells are only allowed to appear in pairs.

With three possible states, only repetitive and nested patterns are ever ultimately produced, at least starting with all cells white.

The only requirement is that around each node the network must have the same structure (or at least a limited number of possible structures). … If the connections at each node are not labelled, then only totalistic cellular automaton rules can be implemented.

[Time compression in] ODEs The method of compressing time using algebraic transformations works not only in partial but also in ordinary differential equations.

Other [symbolic systems] rules If only a single variable appears in the rule, then typically only nested behavior can be generated—though in an example like ℯ [x_][_]  ℯ [x[ ℯ [ ℯ ][ ℯ ]][ ℯ ]] it can be quite complex.

The bottom groups together pairs of cells, and shows only every other step.