One Tile to Rule Them All: Simulating Any Turing Machine, Tile Assembly System, or Tiling System with a Single Puzzle Piece

TL;DR

This paper demonstrates that a single, shape-variant tile can simulate any tile assembly system or Turing machine, revealing a universal tile concept with implications for computational and tiling systems.

Contribution

It introduces a universal tile type capable of simulating any aTAM system and adapts this to various tiling systems, expanding the understanding of tile assembly universality.

Findings

A single shape can simulate any aTAM system with multiple tile types.

A universal tile can be used to encode and simulate any aTAM system.

Limited computational power for nonrotatable tiles without shape complexity.

Abstract

In this paper we explore the power of tile self-assembly models that extend the well-studied abstract Tile Assembly Model (aTAM) by permitting tiles of shapes beyond unit squares. Our main result shows the surprising fact that any aTAM system, consisting of many different tile types, can be simulated by a single tile type of a general shape. As a consequence, we obtain a single universal tile type of a single (constant-size) shape that serves as a "universal tile machine": the single universal tile type can simulate any desired aTAM system when given a single seed assembly that encodes the desired aTAM system. We also show how to adapt this result to convert any of a variety of plane tiling systems (such as Wang tiles) into a "nearly" plane tiling system with a single tile (but with small gaps between the tiles). All of these results rely on the ability to both rotate and translate tiles; by contrast, we show that a single nonrotatable tile, of arbitrary shape, can produce assemblies which either grow infinitely or cannot grow at all, implying drastically limited computational power. On the positive side, we show how to simulate arbitrary cellular automata for a limited number of steps using a single nonrotatable tile and a linear-size seed assembly.