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arXiv · 2610.22218

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

Klas Marström·Eero Räty
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Abstract

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.

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Klas Marström, Eero Räty. 2026-10-15. Logical isomorphism and logical limit laws for small permutation classes and regular languages. https://arxiv.org/abs/2610.22218

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