Suppose we have a (first-order) sentence, having quantifier rank $q$.
How can we prove that it is equivalent or not equivalent to a logical sentence of rank $p$, with $p \lt q$?
Say for example:
$$ F := \; \forall x \exists y \,(R(x,y) \lor R(y,x)\;)\lor \exists x R(x,x) $$
How can we prove that it can be, or cannot be, equivalent to any sentence (same language) of rank $1$?