Section: Research Program

Ongoing work

Recent theoretical work (Meunier, Woods “The non-cooperative tile assembly model is not intrinsically universal or capable of bounded Turing machine simulation”) to be published in 2017 by has centered on the power of a model of self-assembly. In this model, called the noncooperative (or temperature 1) abstract Tile Assembly Model, square tiles assemble structures, called assemblies, in the discrete plane where each tile binds to a growing structure if one of its 4 coloured edges matches the colour of some available site on a growing assembly. It has been conjectured since 2000 that this model is not capable of computation or other sophisticated forms of growth. We show two results. One of our results states that time-bounded Turing machine computation is impossible in this model if we require the simulation to occur in a bounded rectangle in the plane. This result has a short proof that essentially follows from our other main result which states that this model is not “intrinsically universal”. This latter result means that there is no single tileset in this model that can simulate any instance of the model, answering a question from and contrasting a result for the more general cooperative (temperature 2) model.

Other work by Woods has focused on experimentally implementing a wide class of Boolean circuits of a certain form. Experiments were mostly carried out at Caltech, and the work is in collaboration with colleagues at Caltech, UC Davis, Harvard and Cambridge and a publication is in preparation with [Woods, Doty, Myhrvold, Hui, Zhou, Yin, Winfree]. Details will be described in a future report subsequent to publication.

Work published earlier in 2016 (Erik D Demaine, Matthew J Patitz, Trent A Rogers, Robert T Schweller Scott M Summers and Damien Woods, “The two-handed tile assembly model is not intrinsically universal”, Algorithmica 74:2, pages 812–850 (2016). not on HAL) shows results on a hierarchal model of algorithmic self-assembly called the two-handed self-assembly model (2HAM). Specifically, that the model is not intrinsically universal. In fact, we show that for all ${\tau }^{\text{'}}<\tau$, each temperature-${\tau }^{\text{'}}$ 2HAM tile system does not simulate at least one temperature-$\tau$ 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal and stands in contrast to the fact that the (single-tile addition) abstract Tile Assembly Model is intrinsically universal. On the positive side, we prove that, for every fixed temperature $\tau \ge 2$ , temperature-$\tau$ 2HAM tile systems are indeed intrinsically universal. In other words, for each $\tau$ there is a single intrinsically universal 2HAM tile set $U_{\tau }$ that, when appropriately initialized, is capable of simulating the behavior of any temperature-$\tau$ 2HAM tile system. As a corollary, we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. Finally, we show that for each $\tau$ , there is a temperature-$\tau$ 2HAM system that simultaneously simulates all temperature-$\tau$ 2HAM systems.

There are a number of projects being designed along the lines of topics above in Overall Objectives.