Computing in continuous space with self-assembling polygonal tiles (extended abstract)

Oscar Gilbert , Jacob Hendricks , Matthew J. Patitz ? ? , and Trent A. Rogers † 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 which are arbitrary 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 for each n > 6, the class of systems whose tiles are the shape of the regular polygon P with n 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 sides 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. 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 which are arbitrary 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 for each n > 6, the class of systems whose tiles are the shape of the regular polygon P with n 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 sides 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. ? Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR, USA. [email protected]. This author’s research was supported in part by National Science Foundation Grants CCF-1117672 and CCF-1422152. ?? Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA. [email protected]. This author’s research was supported in part by National Science Foundation Grants CCF-1117672 and CCF1422152. ? ? ? Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA. [email protected]. This author’s research was supported in part by National Science Foundation Grants CCF-1117672 and CCF-1422152. † Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA. [email protected]. This author’s research was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1450079, and National Science Foundation grants CCF1117672 and CCF-1422152. ar X iv :1 50 3. 00 32 7v 2 [ cs .C G ] 1 8 A ug 2 01 5