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

TL;DR

This paper investigates the minimal number of states needed for self-assembling systems to form an n x n square, providing optimal bounds for various classes of seeded Tile Automata systems with different transition complexities.

Contribution

It establishes the first optimal bounds on state complexity for assembling squares in seeded Tile Automata, considering different restrictions on transition rules.

Findings

Seeded Tile Automata require Θ(log^{1/4} n) states in general.

Single-transition systems require Θ(log^{1/3} n) states.

Deterministic systems require Θ((log n / log log n)^{1/2}) states.

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 $n \times n$ 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 $\Theta( (\frac{\log n}{\log \log n})^\frac{1}{2} )$.