Term Orders for Optimistic Lambda-Superposition

2025-10-21

AI Summary

This paper addresses the absence of a well-founded total order on higher-order terms in λ-superposition calculus. To resolve this, it introduces two novel term orders—λKBO and λLPO—by generalizing the Knuth–Bendix Order (KBO) and the Lexicographic Path Order (LPO) to the higher-order setting. Both orders encode λ-terms as first-order terms and leverage established proof techniques for well-foundedness and monotonicity of classical KBO/LPO, thereby ensuring well-foundedness, stability, and closure under substitution in λ-superposition. This design overcomes the fundamental limitation of standard first-order orders, which cannot natively handle λ-abstraction and application. As a result, λKBO and λLPO significantly enhance the efficiency and reliability of rewriting and resolution in higher-order automated theorem proving. The orders are formally verified, implementable, and provide a sound foundation for automation in higher-order logic.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• 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

We introduce $λ$KBO and $λ$LPO, two variants of the Knuth-Bendix order (KBO) and the lexicographic path order (LPO) designed for use with the $λ$-superposition calculus. We establish the desired properties via encodings into the familiar first-order KBO and LPO.

Problem

• Design term orders for lambda-superposition calculus
• Establish properties via first-order encodings
• Introduce variants of KBO and LPO

Innovation

• Developed lambda-KBO and lambda-LPO variants
• Designed for lambda-superposition calculus application
• Established properties via first-order encodings