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

Autor:	Alaniz, Robert M., Caballero, David, Cirlos, Sonya C., Gomez, Timothy, Grizzell, Elise, Rodriguez, Andrew, Schweller, Robert, Tenorio, Armando, Wylie, Tim
Rok vydání:	2022
Předmět:	Computer Science - Formal Languages and Automata Theory Computer Science - Distributed Parallel and Cluster Computing Computer Science - Emerging Technologies Nonlinear Sciences - Adaptation and Self-Organizing Systems Nonlinear Sciences - Cellular Automata and Lattice Gases
Druh dokumentu:	Working Paper
Popis:	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 $n \times n$ 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 $\Theta{(\log^{\frac{1}{4}} n)}$ states. For single-transition systems, where only one state may change in a transition rule, we show a bound of $\Theta{(\log^{\frac{1}{3}} n)}$, and for deterministic systems, where each pair of states may only have one associated transition rule, a bound of $\Theta( (\frac{\log n}{\log \log n})^\frac{1}{2} )$. Comment: An earlier version was published in the 2022 Symposium on Algorithmic Foundations of Dynamic Networks (SAND)
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2211.12589 Zobrazit plný text záznamu View this record from Arxiv