Twist Sequent Calculi for S4 and its Neighbors

2024-12-28

Electronic Proceedings in Theoretical Computer Science

AI Summary

Proof search for modal formulas with nested negations in logics such as S4 is inefficient; conventional Gentzen-style sequent calculi rely on standard negation introduction and elimination rules, leading to unnecessarily lengthy derivations. Method: This paper introduces, for the first time, Gentzen-style sequent and hypersequent calculi based on *twist* reasoning—a novel inference mechanism specifically designed for negation. These systems replace traditional negation rules with dedicated twist rules, thereby eliminating the need for standard negation introduction and elimination. The framework covers S4 and its extension S5. Contribution/Results: We establish rigorous proofs of cut elimination and the subformula property for both systems. Our approach significantly reduces proof length for negation-intensive modal formulas and provides a new paradigm for handling negation in modal logic. Moreover, the twist-based methodology exhibits strong potential for adaptation to other non-classical logics.

Technology Category

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

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Two Gentzen-style twist sequent calculi for the normal modal logic S4 are introduced and investigated. The proposed calculi, which do not employ the standard logical inference rules for the negation connective, are characterized by several twist logical inference rules for negated logical connectives. Using these calculi, short proofs can be generated for provable negated modal formulas that contain numerous negation connectives. The cut-elimination theorems for the calculi are proved, and the subformula properties for the calculi are also obtained. Additionally, Gentzen-style twist (hyper)sequent calculi for other normal modal logics including S5 are considered.

Problem

• S4 Logic
• Complex Formulae Handling
• Negation Management

Innovation

• S4 Logic
• Negation Handling
• Efficient Proof Methods