Logical isomorphism and logical limit laws for small permutation classes and regular languages

In this paper we show that for each small permutation class $\mathcal{F}$ there exists a regular language $L_{\mathcal{F}}$ such that the monadic second-order TOTO logic on $\mathcal{F}$ is isomorphic to classical monadic second-order logic on $L_{\mathcal{F}}$. Thanks to the fact that regular languages are closed under monadic second-order restrictions this shows that many subsets of $\mathcal{F}$ have a well-behaved structure. In particular, this allows us to show that random permutations from any small permutation class satisfies convergence laws for both first-order logic and monadic second-order logic. In the latter case, the law is modular. Our proofs introduce a way of using the automaton description of regular languages to prove properties of generating functions which may be of independent interest. Potential applications include first-order limit laws for random words in some regular languages.