arXiv subjects

Phillip Drake

Publications and source records attributed to Phillip Drake.

3 recordsLinked to original sources

arXiv2026

Powers and Limitations of Synchronous Self-Assembly

In abstract models of algorithmic self-assembly, synchronization between attachments has emerged as a crucial distinction between the classical asynchronous model (aTAM) and a new synchronous model, the syncTAM. This paper presents recent advances in gauging the additional power afforded by the syncTAM. While it is known that the syncTAM and the aTAM are each unable to fully simulate the other, this paper offers evidence that the syncTAM is computationally significantly more powerful than the aTAM, especially in the non-cooperative setting. The additional power of the non-cooperative syncTAM is witnessed by the following constructions, all impossible in the non-cooperative aTAM: a flagpole, a strict self-assembly of a variant of the discrete Sierpinski triangle, and the ability to build the same assemblies (modulo scale factor) as directed aTAM systems. The second topic is that of limited synchronization, wherein, when the number of attachments is smaller than some threshold $l$, they happen synchronously, but attachments in excess of that number must wait. In that context, the precise value of $l$ is crucial, and changes to that value prevent simulation and can change which shapes can be obtained.

cs.CG↗

arXiv2024

Simulation of the abstract Tile Assembly Model Using Crisscross Slats

Tile assembly systems in the abstract Tile Assembly Model (aTAM) are computationally universal and capable of building complex shapes, but DNA-based implementations encounter formidable error rates that stifle this theoretical potential. Slat-based self-assembly is a recent development wherein DNA forms long slats that combine together in 2 layers, rather than the aTAM's square tiles in a plane. While tiles tend to bind to 2 neighboring tiles at a time, slats may bind to dozens of other slats. Large slat-based DNA constructions have been implemented in the lab with incredible resilience to many of the errors that plague tile-based constructions, but these come at a cost as slat-based systems are often more difficult to design and simulate. Also, it has not been clear if slats, with their larger sizes and different geometries, have the same theoretical capabilities as tiles. Here we show that slats do, at least at scale. We give constructions showing that any aTAM system may be simulated by a system of slats and that these can be made more efficiently, using shorter slats and a smaller scale factor, when simulating simpler classes of systems. We consider 5 classes of aTAM systems with increasing complexity, from zig-zag systems to the full class of all aTAM systems, and show how they can be converted to equivalent slat systems. Zig-zag systems can be simulated by slats at only a $2c \times 2c$ scale (where $c$ is the freely chosen cooperativity of the slats), the full class of aTAM systems at only a $5c \times 5c$ scale, and intermediate classes using scales between these. Together, these results prove that slats have the full theoretical power of aTAM tiles while providing constructions compact enough to potentially provide designs for DNA-based implementations of slat systems that are both capable of powerful algorithmic self-assembly and possessing the strong error resilience of slats.

cs.CG↗

arXiv2024

Self-Assembly of Patterns in the abstract Tile Assembly Model

In the abstract Tile Assembly Model, self-assembling systems consisting of tiles of different colors can form structures on which colored patterns are ``painted.'' We explore the complexity, in terms of the numbers of unique tile types required, of assembling various patterns. We first demonstrate how to efficiently self-assemble a set of simple patterns, then show tight bounds on the tile type complexity of self-assembling 2-colored patterns on the surfaces of square assemblies. Finally, we demonstrate an exponential gap in tile type complexity of self-assembling an infinite series of patterns between systems restricted to one plane versus those allowed two planes.

cs.ET↗