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We prove that if a set X ⊆ Z 2 weakly self-assembles at temperature 1 in a deterministic (Winfree) tile assembly system satisfying a natural condition known as pumpability, then X is a semilinear set. This shows that only the most simple of infinite shapes and patterns can be constructed using pumpable temperature 1 tile assembly systems, and gives evidence… (More)

We prove that the abstract Tile Assembly Model (aTAM) of nanoscale self-assembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is " intrinsic " in the sense that the self-assembly process carried out by U is exactly that carried out… (More)

We consider the problem of fault-tolerance in nanoscale algorithmic self-assembly. We employ a standard variant of Winfree's abstract Tile Assembly Model (aTAM), the two-handed aTAM, in which square " tiles " – a model of molecules constructed from DNA for the purpose of engineering self-assembled nanostructures – aggregate according to specific binding… (More)

We study the difference between the standard seeded model of tile self-assembly, and the " seedless " two-handed model of tile self-assembly. Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit… (More)

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… (More)

In this paper, we explore relationships between two models of systems which are governed by only the local interactions of large collections of simple components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM). While sharing several similarities, the models have fundamental differences, most notably the dynamic nature of CA (in which… (More)

This paper explores the impact of geometry on computability and complexity in Winfree's model of nanoscale self-assembly. We work in the two-dimensional tile assembly model, i.e., in the discrete Euclidean plane Z × Z. Our first main theorem says that there is a roughly quadratic function f such that a set A ⊆ Z + is computably enumerable if and only if the… (More)

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… (More)

The Two-Handed Tile Assembly Model (2HAM) is a model of algorithmic self-assembly in which large structures, or assemblies of tiles, are grown by the binding of smaller assemblies. In order to bind, two assemblies must have matching glues that can simultaneously touch each other, and stick together with strength that is at least the temperature $$\tau $$ τ… (More)