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Building Squares with Optimal State Complexity in Restricted Active Self-Assembly

Authors Robert M. Alaniz, David Caballero, Sonya C. Cirlos, Timothy Gomez, Elise Grizzell, Andrew Rodriguez, Robert Schweller, Armando Tenorio, Tim Wylie

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Author Details

Robert M. Alaniz
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
David Caballero
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Sonya C. Cirlos
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Timothy Gomez
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Elise Grizzell
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Andrew Rodriguez
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Robert Schweller
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Armando Tenorio
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Tim Wylie
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA

Funding

This research was supported in part by National Science Foundation Grant CCF-1817602.

Acknowledgements

We would like to thank the reviewers for their comments, specifically for pointing us toward relevant Cellular Automata Literature.

Cite As Get BibTex

Robert M. Alaniz, David Caballero, Sonya C. Cirlos, Timothy Gomez, Elise Grizzell, Andrew Rodriguez, Robert Schweller, Armando Tenorio, and Tim Wylie. Building Squares with Optimal State Complexity in Restricted Active Self-Assembly. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SAND.2022.6

Abstract

Tile Automata is a recently defined model of self-assembly that borrows many concepts from cellular automata to create active self-assembling systems where changes may be occurring within an assembly without requiring attachment. This model has been shown to be powerful, but many fundamental questions have yet to be explored. Here, we study the state complexity of assembling n × n squares in seeded Tile Automata systems where growth starts from a seed and tiles may attach one at a time, similar to the abstract Tile Assembly Model. We provide optimal bounds for three classes of seeded Tile Automata systems (all without detachment), which vary in the amount of complexity allowed in the transition rules. We show that, in general, seeded Tile Automata systems require Θ(log^{1/4} n) states. For Single-Transition systems, where only one state may change in a transition rule, we show a bound of Θ(log^{1/3} n), and for deterministic systems, where each pair of states may only have one associated transition rule, a bound of Θ(({log n}/{log log n})^{1/2}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Applied computing → Computational biology
  • Theory of computation → Self-organization
Keywords
  • Active Self-Assembly
  • State Complexity
  • Tile Automata

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