Document Type
Article
Publication Date
12-2023
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 even with limited assembly size, 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 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(( logn loglogn ) 1 2 ) . Along the way, we provide optimal bounds for the subroutines of building binary strings and building O(logn)×n rectangles.
Recommended Citation
Alaniz, Robert M., 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." Journal of Computer and System Sciences 138 (2023): 103462. https://doi.org/10.1016/j.jcss.2023.103462
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Publication Title
Journal of Computer and System Sciences
DOI
https://doi.org/10.1016/j.jcss.2023.103462
Comments
Original published version available at https://doi.org/10.1016/j.jcss.2023.103462