Applying Symmetric Enumeration Method to One-Dimensional Assembly of Rotatable Tiles

Abstract

Motivated by the increasing importance of the analysis of a complicated reaction system where molecules are interacting in various ways to produce a huge number of compounds of molecules, the author’s previous work proposed a new approach, called Symmetric Enumeration Method (SEM), to the efficient analysis of such reaction systems. The proposed theory provided a general method for the efficient computation of equilibrium states. In this paper, we will review the results of the theory of SEM, and apply it to the equilibria analysis of a one-dimensional assembly system of tiles which can be rotated around three axes, i.e., the x-, y-, and z-axes. Although the growth of a tile assembly is restricted to only one-dimensional direction, the exhaustive method of generating all tile assemblies and obtaining equilibria is intractable since the number of assemblies could be exponential with respect to the number of tiles given to the system. We will show that this equilibria analysis can be transformed into a convex programming problem with a set of variables of size polynomial with respect to the number of input tiles.