Throughout, fix a first order language $\mathscr L$.
As we all know, $\mathscr L$-formulas are built-up inductively and it's precisely this inductive nature what allows us to define the complexity of an $\mathscr L$-formula. A bit more in detail, we can define the set of $\mathscr L$-formulas as $$\mathscr L\textit{-Fml} = \bigcup_{k \in \mathbb N_0}\mathscr L\textit{-Fml}_k,$$ where $\mathscr L\textit{-Fml}_0$ is the set of all atomic $\mathscr L$-formulas and the construction of the other sets $\mathscr L\textit{-Fml}_k$ is carried out as usual, so that for example $(\forall x(x = x) ) \to (\forall x(x = x) ) \in \mathscr L\textit{-Fml}_2$. This set-up naturally gives rise to a function $$c : \mathscr L\textit{-Fml}\longrightarrow \mathbb N_0$$ given by $c(\varphi) = \min\{k \in \mathbb N_0 : \varphi \in \mathscr L\textit{-Fml}_k\}$, and we call $c(\varphi)$ the complexity of $\varphi$. Having this settled, my question now is the following:
Are there any model-theoretic results which make use of the complexity function $c:\mathscr L\textit{-Fml}\longrightarrow \mathbb N_0$ in order to define or characterize certain properties of $\mathscr L$-structures or their definable sets?
If you are now thinking to yourself something like "Why on Earth would this guy come up with such silly question?", let me explain briefly some background motivation for it. A couple of weeks ago I came up with the notion of quantifier rank and I found the idea of introducing the $\equiv_k$ relation on $\mathscr L$-structures (where $\mathscr M \equiv_k \mathscr N$ if and only if $\mathscr M$ and $\mathscr N$ satisfy the same sentences of quantifier rank at most $k$) quite sexy, mainly due to its connection with the Ehrenfeucht-Fraïssé games and the usual elementary equivalence of structures. After working a bit with the notion of quantifier rank I thought to myself that it would be quite nice if there was some other syntactical property of formulas which translates to some property of models of such formulas, and the first thing I could think of is precisely the complexity of a formula.
On one hand something tells me that the question is indeed silly since I haven't seen any evidence which might lead me to think that such results using the complexity function $c$ exist, but on the other hand I'm still learning the subject so I'm aware that my view of it is quite limited.
Any comment is more than welcome!