On the $n$-back-and-forth types of Boolean algebras

The objective of this paper is to uncover the structure of the back-and-forth equivalence classes at the finite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the back-and-forth equivalence classes of a Boolean algebra for finite levels. This result has implications for characterizing the relatively intrinsically <!-- MATH $\Sigma^0_n$ --> relations of Boolean algebras as existential formulas over a finite set of relations.