A finer reparameterisation theorem for MSO and FO queries on strings

2025-12-06

AI Summary

This paper investigates the fine-grained parameterization of monadic second-order (MSO) and first-order (FO) queries over finite binary strings. Specifically, it addresses the setting where the output size is bounded by the product of the numbers of 0s and 1s—i.e., $O(n_0 n_1)$. The method integrates monadic second-order logic, aperiodic monoid theory, and automata-theoretic semantics to construct logically definable mappings between string positions and finite data structures. Key contributions include: (i) the first formal proof that FO-definable string-to-string interpretations admit a dimension minimization property; (ii) the establishment of full parameterized expressiveness of MSO for $O(n_0 n_1)$-bounded outputs, extended to FO; and (iii) a unifying repolarization theorem that precisely characterizes the tight correspondence between logical expressiveness and output size bounds.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Description LogicsMachine Learning: Statistical Relational/Logic Learning

Application Category

Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsGraph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $ imes$ 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.

Problem

• Reparameterises MSO and FO queries on strings
• Identifies query results using positions and finite data
• Proves dimension minimisation for first-order string interpretations

Innovation

• Reparameterisation theorem for MSO and FO queries
• Identifies results using positions and finite data
• Handles first-order logic and aperiodic monoids cases