  Previous work by Demaine et al. (2012) developed a strong connection between
smallest context-free grammars and staged self-assembly systems for
one-dimensional strings and assemblies. We extend this work to two-dimensional
polyominoes and assemblies, comparing staged self-assembly systems to a natural
generalization of context-free grammars we call polyomino context-free grammars
(PCFGs). We achieve nearly optimal bounds on the largest ratios of the smallest
PCFG and staged self-assembly system for a given polyomino with n cells. For
the ratio of PCFGs over assembly systems, we show the smallest PCFG can be an
Omega(n/(log(n))^3)-factor larger than the smallest staged assembly system,
even when restricted to square polyominoes. For the ratio of assembly systems
over PCFGs, we show that the smallest staged assembly system is never more than
a O(log(n))-factor larger than the smallest PCFG and is sometimes an
Omega(log(n)/loglog(n))-factor larger.
