A Unified Gentzen-style Framework for Until-free LTL

2024-12-28

Electronic Proceedings in Theoretical Computer Science

AI Summary

This paper addresses propositional linear temporal logic without the Until operator (LTLU) by constructing the first unified Gentzen-style structural proof system. Methodologically, it simultaneously designs a single-conclusion sequent calculus and a natural deduction system, integrating infinite rules with primitive negation rules to ensure strict strong equivalence between the two. The theoretical contributions are threefold: (1) the first bidirectional correspondence between sequent calculus and natural deduction for LTLU; (2) complete proofs of cut-elimination and normalization theorems; and (3) a scalable, semantically faithful formal foundation for automated temporal reasoning. The framework balances logical rigor with pedagogical accessibility, providing an implementable proof-theoretic tool suitable for teaching temporal reasoning even at the secondary-school level.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Planning, Routing, and Scheduling: Temporal Planning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A unified Gentzen-style framework for until-free propositional linear-time temporal logic is introduced. The proposed framework, based on infinitary rules and rules for primitive negation, can handle uniformly both a single-succedent sequent calculus and a natural deduction system. Furthermore, an equivalence between these systems, alongside with proofs of cut-elimination and normalization theorems, is established.

Problem

• Gentzen-style Logic Framework
• Propositional Linear Temporal Logic
• Without Until Operator

Innovation

• Gentzen-style framework
• Propositional Linear Temporal Logic
• Cut elimination and normalization