Staged Self-Assembly and Polyomino Context-Free Grammars

TL;DR

This paper extends the connection between context-free grammars and self-assembly systems from strings to two-dimensional polyominoes, establishing bounds on their relative efficiencies and demonstrating the computational complexity involved.

Contribution

It introduces polyomino context-free grammars (PCFGs) and compares their efficiency to staged self-assembly systems for 2D shapes, providing nearly optimal bounds.

Findings

Smallest PCFG can be Omega(n/(log(n))^3)-times larger than staged assembly systems.

Smallest staged assembly system is at most O(log(n))-times larger than smallest PCFG.

Bounds are tight even for square polyominoes.

Abstract

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.