Let $L$ be the set of sentences in some logic. I am interested in cases where $L$ is the set of sentences in monadic second-order logic, or it is its $\Pi^1_1$ fragment. Let $K$ be a class of finite structures. The theory $\mathrm{Th}_L(K)$ of $K$ is the set of sentences $\phi$ such that $\forall M \in K$ $M \models \phi$. I am interested in decidability of $\mathrm{Th}_L(K)$.

For what kind of class $K$ is this known? What are important / deep techniques here? What is a good source to learn the results and the techniques?


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