A finer reparameterisation theorem for MSO and FO queries on strings

Lê Thành Dũng Nguyên; Paweł Parys

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arXiv · 2512.06466

A finer reparameterisation theorem for MSO and FO queries on strings

Lê Thành Dũng Nguyên·Paweł Parys

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Abstract

We show a theorem on monadic second-order k-ary queries on finite words. It may be illustrated by the following example: if the number of results of a query on binary strings is O(number of 0s $\times$ number of 1s), then each result can be MSO-definably identified from a 0-position, a 1-position and some finite data. Our proofs also handle the case of first-order logic / aperiodic monoids. Thus we can state and prove the folklore theorem that dimension minimisation holds for first-order string-to-string interpretations.

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Lê Thành Dũng Nguyên, Paweł Parys. 2025-12-06. A finer reparameterisation theorem for MSO and FO queries on strings. https://arxiv.org/abs/2512.06466

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