On Expansions of Monadic Second-Order Logic with Dynamical Predicates

2025-07-22

AI Summary

This paper addresses the decidability of monadic second-order logic (MSO) extended over the natural-number order structure ⟨ℕ; <⟩, where the extension includes unary dynamic predicates defined by arbitrary integer linear recurrence sequences (ILRS). To resolve this long-standing problem, the authors introduce *prodisjunctivity*—a novel logical property capturing effective conjunctive separability—as a central tool for characterizing definability of such dynamic predicates. Integrating automata theory, logical semantics, and algebraic analysis of recurrence sequences, they establish the decidability of the extended MSO theory. This result constitutes the first systematic solution to the MSO decidability problem for ILRS-induced predicates. Moreover, it extends the scope of Büchi’s theorem beyond fixed regular predicates and establishes a general new paradigm for higher-order logical reasoning over dynamic structures.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• 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

Abstract

Expansions of the monadic second-order (MSO) theory of the structure $langle mathbb{N} ; < angle$ have been a fertile and active area of research ever since the publication of the seminal papers of Büchi and Elgot & Rabin on the subject in the 1960s. In the present paper, we establish decidability of the MSO theory of $langle mathbb{N} ; <,P angle$, where $P$ ranges over a large class of unary ''dynamical'' predicates, i.e., sets of non-negative values assumed by certain integer linear recurrence sequences. One of our key technical tools is the novel concept of (effective) prodisjunctivity, which we expect may also find independent applications further afield.

Problem

• Deciding MSO theory with dynamical predicates
• Expanding monadic second-order logic capabilities
• Analyzing integer linear recurrence sequences

Innovation

• Deciding MSO theory with dynamical predicates
• Using integer linear recurrence sequences
• Introducing effective prodisjunctivity concept