Computing in continuous space with self-assembling polygonal 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.

Added a few more images, including full examples of bit reading gadgets

Cited by in corpus (4)

• Simulation of Programmable Matter Systems Using Active Tile-Based Self-Assembly
• Non-cooperatively assembling large structures: a 2D pumping lemma cannot be as powerful as its 1D counterpart
• Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model
• The non-cooperative tile assembly model is not intrinsically universal or capable of bounded Turing machine simulation