Coarse-graining of cellular automata, emergence, and the predictability of complex systems

Abstract

We study the predictability of emergent phenomena in complex systems. Using nearest neighbor, one-dimensional Cellular Automata (CA) as an example, we show how to construct local coarsegrained descriptions of CA in all classes of Wolfram's classification. The resulting coarse-grained CA that we construct are capable of emulating the large-scale behavior of the original systems without accounting for small-scale details. Several CA that can be coarse-grained by this construction are known to be universal Turing machines; they can emulate any CA or other computing devices and are therefore undecidable. We thus show that because in practice one only seeks coarse-grained information, complex physical systems can be predictable and even decidable at some level of description. The renormalization group flows that we construct induce a hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity and is therefore a good candidate for a complexity measure and a classification method. Finally we argue that the large scale dynamics of CA can be very simple, at least when measured by the Kolmogorov complexity of the large scale update rule, and moreover exhibits a novel scaling law. We show that because of this large-scale simplicity, the probability of finding a coarse-grained description of CA approaches unity as one goes to increasingly coarser scales. We interpret this large scale simplicity as a pattern formation mechanism in which large scale patterns are forced upon the system by the simplicity of the rules that govern the large scale dynamics.

Neutral Emergence and Coarse Graining

“…In terms of emergence, the two languages L and S are different CA rules (except in a few cases), and information is lost in the coarse graining (in that low-level fine-grained detail is washed out). Interestingly, the figures illustrating coarse grainings in [7] seem to highlight some of the underlying L structure (for example, various propagating 'signals'), maybe because they have smoothed out other, irrelevant, structure.…”

“…In relation to levels, various research identifies 'natural' scales. Israeli and Goldenfeld [7] note that there is an emergent natural length scale for coarse graining (see section 6.1) cellular automata (CA). In the process of "reconstructing the attractor" from time-lagged observations [17] (also a form of coarse-graining), the correct time lag can be found using mutual information [12].…”

“…The higher level rule S may be a CA rule of lower complexity than the rule at level L: because "the system (the update rule, not the cell lattice) does not contain enough information to be complex at large scales" [7]. This complexity-reducing kind of coarse graining (of 'relevant degrees of freedom') removes information that is relevant to the dynamics at L, but is redundant under the coarse graining at S: it does not capture all of the microscopic dynamics.…”

“…In investigating neutral emergence, we start by recasting Israeli and Goldenfeld's [7] coarse-graining results in the information theoretic emergent framework described above. We then consider an extension of coarse graining that has greater robustness.…”

“…Israeli and Goldenfeld [7] coarse grain CAs. Their exact coarse graining captures the dynamics of a CA rule with another rule, expressed over a coarser grain of space and time.…”