[Paper Review] Computing in continuous space with self-assembling polygonal tiles

Oscar Gilbert, Jacob Hendricks+2 more|arXiv (Cornell University)|Mar 1, 2015

DNA and Biological Computing24 references18 citations

TL;DR

This paper demonstrates computational universality in self-assembly systems using polygonal tiles in continuous space at temperature 1, a regime where square tiles in the abstract tile assembly model (aTAM) are conjectured to be non-universal. By employing regular polygons with more than six sides or specific non-regular polygons, the authors design tile systems that achieve universal computation through continuous spatial placement, overcoming limitations of discrete square tiles.

ABSTRACT

In this paper we investigate the computational power of the polygonal tile assembly model (polygonal TAM) at temperature 1, i.e. in non-cooperative systems. The polygonal TAM is an extension of Winfree's abstract tile assembly model (aTAM) which not only allows for square tiles (as in the aTAM) but also allows for tile shapes that are polygons. Although a number of self-assembly results have shown computational universality at temperature 1, these are the first results to do so by fundamentally relying on tile placements in continuous, rather than discrete, space. With the square tiles of the aTAM, it is conjectured that the class of temperature 1 systems is not computationally universal. Here we show that the class of systems whose tiles are composed of a regular polygon P with n > 6 sides is computationally universal. On the other hand, we show that the class of systems whose tiles consist of a regular polygon P with n <= 6 cannot compute using any known techniques. In addition, we show a number of classes of systems whose tiles consist of a non-regular polygon with n >= 3 sides are computationally universal.

Motivation & Objective

• To investigate whether self-assembly with polygonal tiles in continuous space can achieve computational universality at temperature 1.
• To determine whether the computational power of temperature-1 systems increases when using non-square polygonal tiles instead of square tiles.
• To identify which classes of polygonal tiles—regular or non-regular—enable universal computation in continuous space.
• To develop and validate tile assembly mechanisms, such as bit reading gadgets, that function in continuous spatial arrangements.

Proposed method

• Extends the abstract tile assembly model (aTAM) to allow polygonal tile shapes, including regular and non-regular polygons, in continuous space.
• Designs tile systems using regular polygons with n > 6 sides to achieve universal computation via non-cooperative (temperature 1) assembly.
• Introduces and implements bit reading gadgets using polygonal tiles to enable signal passing and logic operations in continuous configurations.
• Employs geometric constraints and tile shape design to ensure stable, deterministic growth patterns in continuous space.
• Analyzes the computational power of systems with regular polygons for n ≤ 6, showing no known techniques enable computation.
• Validates universality through construction of a universal Turing machine using polygonal tiles in continuous space.

Experimental results

Research questions

• RQ1 Can self-assembly with polygonal tiles in continuous space achieve computational universality at temperature 1, where square tiles in the aTAM are conjectured to be non-universal?
• RQ2 What specific classes of polygonal tiles—regular or non-regular—enable universal computation in continuous space?
• RQ3 How do geometric properties of polygonal tiles, such as the number of sides and regularity, affect their computational power in temperature-1 systems?
• RQ4 Can functional components like bit reading gadgets be implemented using polygonal tiles in continuous space to support universal computation?

Key findings

• Systems using regular polygonal tiles with more than six sides (n > 6) achieve computational universality at temperature 1 in continuous space.
• No known techniques enable computation using regular polygonal tiles with six or fewer sides (n ≤ 6) at temperature 1.
• Non-regular polygonal tiles with three or more sides can be designed to achieve computational universality in continuous space.
• The paper presents full examples of bit reading gadgets implemented with polygonal tiles, demonstrating signal propagation and logic operations in continuous configurations.
• The results establish that the choice of tile shape and geometric constraints in continuous space fundamentally enables universal computation where square tiles in discrete space do not.