Noncooperative algorithms in self-assembly

We show the first non-trivial positive algorithmic results (i.e. programs whose output is larger than their size), in a model of self-assembly that has so far resisted many attempts of formal analysis or programming: the planar non-cooperative variant of Winfree's abstract Tile Assembly Model. This model has been the center of several open problems and conjectures in the last fifteen years, and the first fully general results on its computational power were only proven recently (SODA 2014). These results, as well as ours, exemplify the intricate connections between computation and geometry that can occur in self-assembly. In this model, tiles can stick to an existing assembly as soon as one of their sides matches the existing assembly. This feature contrasts with the general cooperative model, where it can be required that tiles match on \emph{several} of their sides in order to bind. In order to describe our algorithms, we also introduce a generalization of regular expressions called Baggins expressions. Finally, we compare this model to other automata-theoretic models.Comment: A few bug fixes and typo correction

Directed Non-Cooperative Tile Assembly Is Decidable

We provide a complete characterisation of all final states of a model called directed non-cooperative tile self-assembly, also called directed temperature 1 tile assembly, which proves that this model cannot possibly perform Turing computation. This model is a deterministic version of the more general undirected model, whose computational power is still open. Our result uses recent results in the domain, and solves a conjecture formalised in 2011. We believe that this is a major step towards understanding the full model. Temperature 1 tile assembly can be seen as a two-dimensional extension of finite automata, where geometry provides a form of memory and synchronisation, yet the full power of these "geometric blockings" was still largely unknown until recently (note that nontrivial algorithms which are able to build larger structures than the naive constructions have been found)

Intrinsic universality in tile self-assembly requires cooperation

We prove a negative result on the power of a model of algorithmic self-assembly for which it has been notoriously difficult to find general techniques and results. Specifically, we prove that Winfree's abstract Tile Assembly Model, when restricted to use noncooperative tile binding, is not intrinsically universal. This stands in stark contrast to the recent result that, via cooperative binding, the abstract Tile Assembly Model is indeed intrinsically universal. Noncooperative self-assembly, also known as "temperature 1", is where tiles bind to each other if they match on one or more sides, whereas cooperative binding requires binding on multiple sides. Our result shows that the change from single- to multi-sided binding qualitatively improves the kinds of dynamics and behavior that these models of nanoscale self-assembly are capable of. Our lower bound on simulation power holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. On the positive side, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. Our negative result can be interpreted to mean that Turing universal algorithmic behavior in self-assembly does not imply the ability to simulate arbitrary algorithmic self-assembly processes.Comment: Added references. Improved presentation of definitions and proofs. This article uses definitions from arXiv:1212.4756. arXiv admin note: text overlap with arXiv:1006.2897 by other author

The Need for Seed (in the abstract Tile Assembly Model)

In the abstract Tile Assembly Model (aTAM) square tiles self-assemble, autonomously binding via glues on their edges, to form structures. Algorithmic aTAM systems can be designed in which the patterns of tile attachments are forced to follow the execution of targeted algorithms. Such systems have been proven to be computationally universal as well as intrinsically universal (IU), a notion borrowed and adapted from cellular automata showing that a single tile set exists which is capable of simulating all aTAM systems (FOCS 2012). The input to an algorithmic aTAM system can be provided in a variety of ways, with a common method being via the "seed" assembly, which is a pre-formed assembly from which all growth propagates. In this paper we present a series of results which investigate the the trade-offs of using seeds consisting of a single tile, versus those containing multiple tiles. We show that arbitrary systems with multi-tile seeds cannot be converted to functionally equivalent systems with single-tile seeds without using a scale factor > 1. We prove tight bounds on the scale factor required, and also present a construction which uses a large scale factor but an optimal number of unique tile types. That construction is then used to develop a construction that performs simultaneous simulation of all aTAM systems in parallel, as well as to display a connection to other tile-based self-assembly models via the notion of intrinsic universality.Comment: To appear in the SODA 2023 proceeding