Signal transmission across tile assemblies: 3D static tiles simulate active self-assembly by 2D signal-passing tiles

Abstract

The 2-handed assembly model (2HAM) is a tile-based self-assembly model in which, typically beginning from single tiles, arbitrarily large aggregations of static tiles combine in pairs to form structures. The signal-passing tile assembly model (STAM) is an extension of the 2HAM in which the tiles are dynamically changing components which are able to alter their binding domains as they bind together. In this paper, we examine the $\hbox {STAM}^+$, a restriction of the STAM that does not allow glues to be turned “off”, and prove that there exists a 3D tile set at temperature $\tau >1$ in the 2HAM which is intrinsically universal for the class of all 2D $\hbox {STAM}^+$ systems at temperature $\tau$ for each $\tau$ (where the $\hbox {STAM}^+$ does not make use of the STAM’s power of glue deactivation and assembly breaking, as the tile components of the 2HAM are static and unable to change or break bonds). This means that there is a single tile set $U$ in the 3D 2HAM which can, for an arbitrarily complex $\hbox {STAM}^+$ system $S$, be configured with a single input configuration which causes $U$ to exactly simulate $S$ at a scale factor dependent upon $S$. Furthermore, this simulation uses only two planes of the third dimension. This implies that there exists a 3D tile set at temperature $2$ in the 2HAM which is intrinsically universal for the class of all 2D $\hbox {STAM}^+$ systems at temperature $1$. Moreover, we also show that there exists an $\hbox {STAM}^+$ tile set for temperature $\tau$ which is intrinsically universal for the class of all 2D $\hbox {STAM}^+$ systems at temperature $\tau$, including the case where $\tau = 1$. To achieve these results, we also demonstrate useful techniques and transformations for converting an arbitrarily complex $\hbox {STAM}^+$ tile set into an $\hbox {STAM}^+$ tile set where every tile has a constant, low amount of complexity, in terms of the number and types of “signals” they can send, with a trade off in scale factor. While the first result is of more theoretical interest, showing the power of static tiles to simulate dynamic tiles when given one extra plane in 3D, the second result is of more practical interest for the experimental implementation of STAM tiles, since it provides potentially useful strategies for developing powerful STAM systems while keeping the complexity of individual tiles low, thus making them easier to physically implement.