Resiliency to Multiple Nucleation in Temperature-1 Self-Assembly

  • Matthew J. PatitzEmail author
  • Trent A. Rogers
  • Robert T. Schweller
  • Scott M. Summers
  • Andrew Winslow
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9818)


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 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 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.


Turing Machine Tile Type Input Tape Tile Assembly Model Tape Cell 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Matthew J. Patitz
    • 1
    Email author
  • Trent A. Rogers
    • 1
  • Robert T. Schweller
    • 2
  • Scott M. Summers
    • 3
  • Andrew Winslow
    • 4
  1. 1.Department of Computer Science and Computer EngineeringUniversity of ArkansasFayettevilleUSA
  2. 2.Department of Computer ScienceUniversity of Texas–Rio Grande ValleyEdinburgUSA
  3. 3.Computer Science DepartmentUniversity of Wisconsin–OshkoshOshkoshUSA
  4. 4.Département d’InformatiqueUniversité libre de BruxellesBrusselsBelgium

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