Staged self-assembly and polyomino context-free grammars

Previous work by Demaine et al. (Nat Comput 6937:100–114, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} cells. For the ratio of PCFGs over assembly systems, we show that the smallest PCFG can be an Ω(n/log3n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (n/\log ^3{n})$$\end{document}-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(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log {n})$$\end{document}-factor larger than the smallest PCFG and is sometimes an Ω(logn/loglogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (\log {n}/\log \log {n})$$\end{document}-factor larger.