Intrinsic and relative characterization results for logics with negative modalities

2025-12-17

AI Summary

This paper addresses the characterization problem for modal logics featuring non-classical negation and recovery modalities. To tackle their semantic complexity, we first introduce a unified bisimulation framework tailored to sub-classical negation and recovery operators. Building upon this, we establish a simulation adequacy theorem and derive both Hennessy–Milner-style (intrinsic) and van Benthem-style (relative) characterization results. Our main contribution is a rigorous proof that each recovery-enhanced modal language precisely corresponds to the fragment of first-order logic invariant under the associated simulation relation—thereby pinpointing its exact expressive power within first-order logic. This work achieves a two-dimensional semantic characterization of modal logics with non-classical negation, filling a key theoretical gap by providing both a unifying semantic framework and a fine-grained expressivity hierarchy for such logics.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We introduce simulations for modal logics with subclassical negations and restoration modalities, establish an adequacy theorem, and prove intrinsic (Hennessy-Milner-type) and relative (Van Benthem-type) characterization results. These results identify each restorative language with the fragment of first-order logic invariant under its simulations and delineate the expressive profile of modal logics with non-classical negations.

Problem

• Characterizing modal logics with subclassical negations and restoration modalities
• Establishing adequacy theorems and Hennessy-Milner-type intrinsic results
• Identifying expressive profiles via simulation-invariant first-order logic fragments

Innovation

• Simulations for modal logics with subclassical negations
• Adequacy theorem and characterization results established
• Identifies restorative languages with first-order logic fragments