may boil down to issues analogous to whether it is possible to construct a configuration that has a certain property in, say, the rule 110 cellular automaton.

Whimsy Cellular automata and most of the other systems in this book readily admit various kinds of whimsical descriptions. The rule 30 cellular automaton, for example, can be described as follows.

Local conservation laws Whenever a system like a cellular automaton (or PDE) has a global conserved quantity there must always be a local conservation law which expresses the fact that every point in the system the total flux of the conserved quantity into a particular region must equal the rate of increase of the quantity inside it. … In any 1D k = 2 , r = 1 cellular automaton, it follows from the basic structure of the rule that one can tell what the difference in values of a particular cell on two successive steps will be just by looking at the cell and its immediate neighbor on each side.

Bitwise optimizations [of cellular automata] The C program above stores each cell value in a separate element of an integer array. … In general, bitwise optimizations require representing cellular automaton rules not by simple look-up tables but rather by Boolean expressions, which must be derived for each rule and can be quite complicated (see page 869 ). … The idea is to store the cellular automaton configuration in, say, m variables w[i] whose bits correspond respectively to the cell values {a 1 , a m + 1 , a 2m + 1 , …} , {a 2 , a m + 2 , a 2m + 2 , …} , {a 3 , …} , etc.

[Computational complexity of] finding outcomes If one sets up a function to compute the outcome after t steps of evolution from some fixed initial condition—say a single black cell in a cellular automaton—then the input to this function need contain only Log[2, t] digits.

These can be modelled by having a cellular automaton in which one starts from several separated seeds.

Fairly accurate cellular automaton models of this phenomenon were developed in the early 1990s.

One might expect, however, that it should be possible to construct a PDE that quite directly emulates a system like a cellular automaton. … For as suggested by the bottom row of pictures on page 732 one can imagine having localized structures whose interactions emulate the rules of the cellular automaton. … And this means that the overall state of the system will not be properly prepared for the next step of cellular automaton evolution.

Indeed, it is a general feature of class 4 cellular automata that with appropriate initial conditions they can mimic the behavior of all sorts of other systems. … But for now the main point is just how diverse and complex the behavior of class 4 cellular automata can be—even when their underlying rules are very simple. And perhaps the most striking example is the rule 110 cellular automaton that we first saw on page 32 .

The approach to equilibrium in a reversible cellular automaton with a variety of different initial conditions.