Reflections on Tiles (in Self-Assembly)

TL;DR

This paper introduces the Reflexive Tile Assembly Model (RTAM), exploring its computational capabilities and limitations, including shape assembly and tile complexity, revealing differences from the traditional aTAM model.

Contribution

It defines RTAM, analyzes its computational power at different temperatures, and classifies shape assembly possibilities, especially for odd and even squares, advancing understanding of reflective tile systems.

Findings

RTAM at temperature 1 can assemble odd-sized squares with n tiles

RTAM at temperature 1 cannot assemble even-sized squares

RTAM is computationally universal at temperature 2

Abstract

We define the Reflexive Tile Assembly Model (RTAM), which is obtained from the abstract Tile Assembly Model (aTAM) by allowing tiles to reflect across their horizontal and/or vertical axes. We show that the class of directed temperature-1 RTAM systems is not computationally universal, which is conjectured but unproven for the aTAM, and like the aTAM, the RTAM is computationally universal at temperature 2. We then show that at temperature 1, when starting from a single tile seed, the RTAM is capable of assembling n x n squares for n odd using only n tile types, but incapable of assembling n x n squares for n even. Moreover, we show that n is a lower bound on the number of tile types needed to assemble n x n squares for n odd in the temperature-1 RTAM. The conjectured lower bound for temperature-1 aTAM systems is 2n-1. Finally, we give preliminary results toward the classification of which finite connected shapes in Z^2 can be assembled (strictly or weakly) by a singly seeded (i.e. seed of size 1) RTAM system, including a complete classification of which finite connected shapes be strictly assembled by a "mismatch-free" singly seeded RTAM system.