Termination of Innermost-Terminating Right-Linear Overlay Term Rewrite Systems (Full Version)

2026-04-22

AI Summary

This study investigates whether termination of right-linear overlay term rewriting systems is equivalent to innermost termination. By leveraging the simulation property of innermost reductions with respect to normalizing reduction sequences, the work establishes an equivalence between general dependency pair chains and innermost dependency pair chains within this class of systems. It presents the first proof that termination and innermost termination coincide for right-linear overlay term rewriting systems, thereby reducing termination verification to checking only innermost dependency pair chains. This result substantially lowers the complexity of termination analysis and provides a solid theoretical foundation for the development of automated verification tools in this domain.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

It has been shown that, regarding a terminating right-linear overlay term rewrite system (TRS), any rewrite sequence ending with a normal form can be simulated by the innermost reduction. In this paper, using this simulation property, we show that for a right-linear overlay TRS, there is no infinite minimal dependency-pair chain if and only if there is no infinite innermost minimal dependency-pair chain. This implies that a right-linear overlay TRS is terminating if and only if it is innermost terminating.

Problem

• termination
• innermost termination
• right-linear overlay TRS
• dependency pairs

Innovation

• innermost termination
• right-linear overlay TRS
• dependency pairs
• termination equivalence
• simulation property