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