Generalised Quantifiers Based on Rabin-Mostowski Index

2026-01-08

arXiv.org

AI Summary

This study investigates how to characterize the Rabin–Mostowski index of automata within logical systems and its impact on expressive power and decidability. By embedding this index into a generalized quantifier framework for the first time, the work introduces novel game quantifiers and integrates them with an extended monadic second-order (MSO) logic to systematically analyze their behavior over ω-words and infinite trees. The main contributions include an effective reduction of the new quantifiers to pure MSO over ω-words, thereby preserving decidability, while demonstrating that their addition renders the logic undecidable over infinite trees—highlighting a fundamental logical distinction between these two structures. Additionally, the paper proposes a construction of ω-regular game strategy transducers with monadic parameters that supports index elimination.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• 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 semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this work we introduce new generalised quantifiers which allow us to express the Rabin-Mostowski index of automata. Our main results study expressive power and decidability of the monadic second-order (MSO) logic extended with these quantifiers. We study these problems in the realm of both $\omega$-words and infinite trees. As it turns out, the pictures in these two cases are very different. In the case of $\omega$-words the new quantifiers can be effectively expressed in pure MSO logic. In contrast, in the case of infinite trees, addition of these quantifiers leads to an undecidable formalism. To realise index-quantifier elimination, we consider the extension of MSO by game quantifiers. As a tool, we provide a specific quantifier-elimination procedure for them. Moreover, we introduce a novel construction of transducers realising strategies in $\omega$-regular games with monadic parameters.

Problem

• Generalised Quantifiers
• Rabin-Mostowski Index
• Monadic Second-Order Logic
• ω-words
• Infinite Trees

Innovation

• generalised quantifiers
• Rabin-Mostowski index
• monadic second-order logic
• quantifier elimination
• omega-regular games