Chapter 6: Starting from Randomness

Section 7: The Notion of Attractors

History [of dynamical systems approaches]

The analysis of cellular automata given in this section is largely as I worked it out in the early 1980s. Parts of it, however, are related to earlier investigations, particularly in dynamical systems theory. Starting in the 1930s the idea of symbolic dynamics began to emerge, in which one partitions continuous values in a system into bins represented by discrete symbols, and then looks at the sequences of such symbols that can be produced by the evolution of the system. In connection with early work on chaos theory, it was noted that there are some systems that act like "full shifts", in the sense that the set of sequences they generate includes all possibilities—and corresponds to what one would get by starting with any possible number, then successively shifting digits to the left, and at each step picking off the leading digit. It was noted that some systems could also yield various kinds of subshifts that are subsets of full shifts. But since—unlike in cellular automata—the symbol sequences being studied were obtained by rather arbitrary partitionings of continuous values, the question arose of what effect using different partitionings would have. One approach was to try to find invariants that would remain unchanged in different partitionings—and this is what led, for example, to the study of topological entropy in the 1960s. Another approach was to look at actual possible transformations between partitionings, and this led from the late 1950s to various studies of so-called shift-commuting block maps (or sliding-block codes)—which turn out to be exactly 1D cellular automata (see page 878). The locality of cellular automaton rules was thought of as making them the analog for symbol sequences of continuous functions for real numbers (compare page 869). Of particular interest were invertible (reversible) cellular automaton rules, since systems related by these were considered conjugate or topologically equivalent.

In the 1950s and 1960s—quite independent of symbolic dynamics—there was a certain amount of work done in connection with ideas about self-reproduction (see page 876) on the question of what configurations one could arrange to produce in 1D and 2D cellular automata. And this led for example to the study of so-called Garden of Eden states that can appear only in initial conditions—as well as to some general discussion of properties such as surjectivity.

When I started working on cellular automata in the early 1980s I wanted to see how far one could get by following ideas of statistical mechanics and dynamical systems theory and trying to find global characterizations of the possible behavior of individual cellular automata. In the traditional symbolic dynamics of continuous systems it had always been assumed that meaningful quantities must be invariant under continuous invertible transformations of symbol sequences. It turns out that the spacetime (or "invariant") entropy defined in the previous note has this property. But the spatial and temporal entropies that I introduced do not—and indeed in studying specific cellular automata there seems to be no particular reason why such a property would be useful.

From Stephen Wolfram: A New Kind of Science [citation]