Resiliency to multiple nucleation in temperature-1 self-assembly

We consider problems in variations of the two-handed abstract Tile Assembly Model (2HAM), a generalization of Erik Winfree’s abstract Tile Assembly Model (aTAM). In the latter, tiles attach one-at-a-time to a seed-containing assembly. In the former, tiles aggregate into supertiles that then further combine to form larger supertiles; hence, constructions must be robust to the choice of seed (nucleation) tiles. We obtain three distinct temperature-1 results in two 2HAM variants whose aTAM siblings are well-studied. In the first variant, called the restricted glue 2HAM (rg2HAM), glue strengths are restricted to \(-\,1\), 0, or 1. We prove this model is Turing universal, overcoming undesired growth by breaking apart undesired computation assembly via repulsive forces. In the second 2HAM variant, the 3D 2HAM (3D2HAM), tiles are (three-dimensional) cubes. We prove that assembling a (roughly two-layer) \(n \times n\) square in this model at temperature 1 is possible with \(O(\log ^2{n})\) tile types. The construction uses “cyclic, colliding” binary counters, and assembles the shape non-deterministically. Finally, we prove that there exist 3D2HAM systems that only assemble infinite aperiodic shapes.