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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 study the role that parallelism plays in time complexity of variants of Winfree's abstract Tile Assembly Model (aTAM), a model of molecular algorithmic self-assembly. In the " hierarchical " aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the " seeded " aTAM, tiles attach one at a time to a growing… (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 investigate the role of nondeterminism in Winfree's abstract tile assembly model, which was conceived to model artificial molecular self-assembling systems constructed from DNA. By nondeterminism we do not mean a magical ability such as that possessed by a nondeterministic algorithm to search an exponential-size space in polynomial time. Rather, we study… (More)

— Working in Winfree's abstract tile assembly model, we show that a constant-size tile assembly system can be programmed through relative tile concentrations to build an n × n square with high probability, for any sufficiently large n. This answers an open question of Kao and Schweller (Randomized Self-Assembly for Approximate Shapes, ICALP 2008), who… (More)

Chemical reaction networks (CRNs) formally model chemistry in a well-mixed solution. CRNs are widely used to describe information processing occurring in natural cellular regulatory networks, and with upcoming advances in synthetic biology, CRNs are a promising language for the design of artificial molecular control circuitry. Nonetheless, despite the… (More)

The challenge of programming molecules to manipulate themselves.

We investigate methods for exploiting nondeterminism inherent within the Tile Assembly Model in order to generate uniform random numbers. Namely, given an integer range {0,. .. , n − 1}, we exhibit methods for randomly selecting a number within that range. We present three constructions exhibiting a trade-off between space requirements and closeness to… (More)

This paper explores the use of negative (i.e., repulsive) interactions in the abstract Tile Assembly Model defined by Winfree. Winfree in his Ph.D. thesis postulated negative interactions to be physically plausible, and Reif, Sahu, and Yin studied them in the context of reversible attachment operations. We investigate the power of negative interactions with… (More)