Nested Sequents for Horn-Characterizable Quantified Modal Logics with Equality via Reachability Rules

2026-04-20

AI Summary

This work addresses quantified modal logic featuring an inner–outer domain structure by constructing the first cut-free nested sequent calculus that is both sound and complete. The system uniformly handles diverse domain variations and path constraints through reachability rules parameterized by semi-graph grammars, enabling precise semantic characterization by propagating or consuming formulas and terms along paths within the nested structure. Key contributions include establishing a sound and complete proof system, revealing that standard universal quantifier rules implicitly impose restrictions equivalent to the converse Barcan formula, proving the invertibility of all inference rules, demonstrating a non-trivial cut-elimination theorem, and identifying several admissible structural rules.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Qualitative ReasoningMachine Learning: Calibration & Uncertainty Quantification

Application Category

Search and Retrieval-Augmented AI: Vertical and domain-specific searchSemantics 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: Foundation models and LLMs for Web-related graphs

Abstract

We introduce cut-free nested sequent systems for a broad class of quantified modal logics (QMLs). The QMLs we consider are semantically defined using relational models that assign both an inner and outer domain to each world. This rich model structure enables the specification of various QMLs by enforcing different frame conditions, including increasing, decreasing, constant, and empty domains, as well as general path conditions and seriality. We extend the usual notion of nested sequent to include signatures, i.e., multisets of terms, which let us naturally define rules capturing the aforementioned domain conditions. A distinctive feature of our nested sequent systems is the use of reachability rules--inference rules parameterized by formal grammars (viz., semi-Thue systems). These rules operate by propagating or consuming formulae or terms along certain paths within a nested sequent, where paths are encoded as strings generated by a parameterizing grammar. This paper is the first to provide sound and complete nested systems for QMLs semantically characterized by models using both inner and outer domains. We analyze the proof-theoretic properties of these systems, identify a number of admissible structural rules, establish the invertibility of all rules, and prove a non-trivial syntactic cut-elimination theorem. We also observe that the standard universal quantifier rule used in nested systems subsumes the Extended Barcan Rule, which forces nested systems to capture QMLs with constant outer domains.

Problem

• quantified modal logics
• nested sequents
• cut-elimination
• domain conditions
• equality

Innovation

• nested sequents
• quantified modal logics
• reachability rules
• cut-elimination
• domain conditions