I'm re-posting this question from cstheory.SE hoping to have more luck.

I'm a computer scientist learning a bit about algebraic logic and I was wondering how knowing the algebraic semantics of a given logic might help the study of the logic itself from a computational point of view.

In particular, is there any example of a complexity (or decidability) result for the satisfiability problem for some logics that can be obtained by reasoning about its algebraic semantics?

For example, the semantics of propositional logic can be given in terms of boolean algebras. Is there any connection between them and the fact that SAT is decidable and $NP$-complete?