But exactly what this implies about the underlying physical processes is not clear.

But in most cases these details can be avoided—and in the end the universality of multiway systems implies that they can always be made to emulate any axiom system.

One can have a rule be applied only once using Module[{i = 1}, expr /. lhs  rhs /; i++  1] Many symbolic systems (including the one on page 103 ) have the so-called Church–Rosser property (see page 1036 ) which implies that if a fixed point is reached in the evolution of the system, this fixed point will be the same regardless of the order in which rules are applied.

One such principle implies that atoms in molecules will tend to arrange themselves so as to minimize their energy.

(For the cellular automaton on page 339 the simple condition for equilibrium is p  p 2 (3 - 2p) , which correctly implies that 0, 1/2 and 1 are possible equilibrium densities.)

[Models involving] non-local processes It follows from the fact that any path in a finite network must always eventually return to a node where it has been before that any Markov process must be fundamentally local, in the sense that the probabilities it implies for what happens at a given point in a sequence must be independent of those for points sufficiently far away.

Indeed, the phenomenon of computational irreducibility discussed in this chapter specifically implies that in many cases irreducible work has to be done in order to find out how any particular system will behave.

Rule 30 inversion The total numbers of sequences for t from 1 to 15 not yielding stripes of heights 1 and 2 are respectively {1, 2, 2, 3, 3, 6, 6, 10, 16, 31, 52, 99, 165, 260} {2, 5, 8, 14, 23, 40, 66, 111, 182, 316, 540, 921, 1530, 2543, 4122} The sideways evolution of rule 30 discussed on page 601 implies that if one fills cells from the left rather than the right then some sequence of length t + 1 will always yield any given stripe of height t .

For typically the continuity of such functions implies that only a limited number of shapes not related by limited variations in local magnification can occur at any scale.

In the limit of an infinite number of runs (or infinite number of cells), the behavior in the second case approaches the form implied by the continuum diffusion equation.