Asynchronous Signal Passing for Tile Self-Assembly: Fuel Efficient Computation and Efficient Assembly of Shapes

Abstract

In this paper we demonstrate the power of a model of tile self-assembly based on active glues which can dynamically change state. We formulate the Signal-passing Tile Assembly Model (STAM), based on the model of Padilla, et al. [20] to be asynchronous, allowing any action of turning a glue on or off, attaching a new tile, or breaking apart an assembly to happen in any order. Within this highly generalized model we provide three new solutions to tile self-assembly problems that have been addressed within the abstract Tile Assembly Model and its variants, showing that signal passing tiles allow for substantial improvement across multiple complexity metrics. Our first result utilizes a recursive assembly process to achieve tile-type efficient assembly of linear structures, using provably fewer tile types than what is possible in standard tile assembly models. Our second system of signal-passing tiles simulates any Turing machine with high fuel efficiency by using only a constant number of tiles per computation step. Our third system assembles the discrete Sierpinski triangle, demonstrating that this pattern can be strictly self-assembled within the STAM. This result is of particular interest in that it is known that this pattern cannot self-assemble within a number of well studied tile self-assembly models. Notably, all of our constructions are at temperature 1, further demonstrating that signal-passing confers the power to bypass many restrictions found in standard tile assembly models.

Dynamic Simulation of 1D Cellular Automata in the Active aTAM

“…Both Active aTAM 9) and STAM 13) can be seen as extensions of the conventional aTAM being equipped with a signaling mechanism to control the state of glue labels. Originally these models were defined as an extension of 2HAM (two handed assembly model), 4) where the assembly growth does not need a seed and it can be achieved by attachments of larger structures, not just a single tile.…”

“…6, 14,17) This has resulted in the expansion of the Winfree's theoretical model aTAM to the Active aTAM 9) and signal passing TAM (STAM). 13) Active aTAM was motivated by the idea of signals being transmitted across a building block similarly as in the use of strand displacement mechanism for the “walkers,” 6) while STAM was mainly motivated by the idea that activation of a signal at one side of the block initiates the activation of a signal on another side of the block. 14) With both of these models, it has been shown that universal computation can be achieved at temperature one, 9, 13) which has been suspected to be unachievable in standard aTAM.…”

“…13) Active aTAM was motivated by the idea of signals being transmitted across a building block similarly as in the use of strand displacement mechanism for the “walkers,” 6) while STAM was mainly motivated by the idea that activation of a signal at one side of the block initiates the activation of a signal on another side of the block. 14) With both of these models, it has been shown that universal computation can be achieved at temperature one, 9, 13) which has been suspected to be unachievable in standard aTAM. 12) The model STAM is more general than Active aTAM in the sense that it allows for signals to deliver not just activation, but also deactivation of the sticky ends.…”

“…12) The model STAM is more general than Active aTAM in the sense that it allows for signals to deliver not just activation, but also deactivation of the sticky ends. This deactivation mechanism in STAM was used to describe a strict assembly of the Sierpinski triangle 13) and in shape replication mechanisms. 10) …”

“…The reusable space in our simulator is similar to that of the STAM simulation of a Turing machine (TM) by Padilla et al, 13) which consumes 7 tiles for each step of the TM computation. The simulation presented here assumes a reuse of the tiles and there is presumably no tile consumption.…”