  In this paper, we explore relationships between two models of systems which
are governed by only the local interactions of large collections of simple
components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM).
While sharing several similarities, the models have fundamental differences,
most notably the dynamic nature of CA (in which every cell location is allowed
to change state an infinite number of times) versus the static nature of the
aTAM (in which tiles are static components that can never change or be removed
once they attach to a growing assembly). We work with 2-dimensional systems in
both models, and for our results we first define what it means for CA systems
to simulate aTAM systems, and then for aTAM systems to simulate CA systems. We
use notions of simulate which are similar to those used in the study of
intrinsic universality since they are in some sense strict, but also
intuitively natural notions of simulation. We then demonstrate a particular
nondeterministic CA which can be configured so that it can simulate any
arbitrary aTAM system, and finally an aTAM tile set which can be configured so
that it can be used to simulate any arbitrary nondeterministic CA system which
begins with a finite initial configuration.
