Pictures (e) through (g) show how a network can be formed with nodes corresponding to updating events.

Regular languages The set of sequences obtained by following possible paths through a finite network is often called a regular language, and appears in studies of many kinds of systems.

Finding these effectively required searching through billions of possibilities.

Rules (f) through (i), however, produce exactly half the strings of any given length, and can be considered complete and consistent.

The three-body problem consists in determining the motion of three bodies—such as the Earth, Sun and Moon—that interact through gravitational attraction. … The pictures on the next page show a particular case of the three-body problem, in which there are two large masses in a simple elliptical orbit, together with an infinitesimally small mass moving up and down through the plane of this orbit.

And indeed the networks in pictures (g) through (i) on the previous page were specifically laid out so that successive rows of nodes going down the page would correspond, at least roughly, to events occurring at successively later times. As the numbering in pictures (e) through (g) illustrates, there is no direct correspondence between this notion of time and the sequence of updating events that occur in the underlying evolution of the mobile automaton.

And while they can get from one point to another through the large number of connections that define the background space, they cannot in a sense fit through a small number of connections in a thread.

So even though the systems themselves generate their behavior by going through a whole sequence of steps, we can readily shortcut this process and find the outcome with much less effort. … Indeed, whenever computational irreducibility exists in a system it means that in effect there can be no way to predict how the system will behave except by going through almost as many steps of computation as the evolution of the system itself.

And at least in the context of a highly regular causal network like the one in the picture on page 518 there is a simple interpretation to this: it just corresponds to looking at slices at different angles through the causal network. Successive parallel slices through the causal network in general correspond to successive states of the underlying system at successive moments in time.

In cases (a) through (f) the motion of the active cell is purely repetitive.