  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 coarse-grained 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.
