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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)

Abstract

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.

Keywords

Turing Machine Tile Type Input Tape Tile Assembly Model Tape Cell 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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