Term Ordering Diagrams

2025-05-28

AI Summary

Term ordering checks in first-order logic equational reasoning incur excessive overhead—particularly in “post-ordering” cases where ordering can only be determined after applying substitutions—becoming a critical bottleneck in saturation-based theorem provers. Method: We propose the Term Ordering Diagram (TOD), a novel index structure that models term ordering relations as a dynamic directed graph. TOD integrates lazy updating and reuse of historical verification information to avoid redundant post-ordering evaluations. It supports both the Knuth–Bendix ordering (KBO) and lexicographic path ordering (LPO), and incorporates a tailored caching strategy. Contribution/Results: Experimental evaluation in state-of-the-art saturation provers shows that TOD reduces the time fraction spent on term ordering checks from up to 98% in bottleneck cases to negligible levels, significantly decreasing overall proof runtime. This yields substantial improvements in redundancy elimination efficiency and scalability of automated reasoning.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data

Abstract

The superposition calculus for reasoning in first-order logic with equality relies on simplification orderings on terms. Modern saturation provers use the Knuth-Bendix order (KBO) and the lexicographic path order (LPO) for discovering redundant clauses and inferences. Implementing term orderings is however challenging. While KBO comparisons can be performed in linear time and LPO checks in quadratic time, using the best known algorithms for these orders is not enough. Indeed, our experiments show that for some examples term ordering checks may use about 98% of the overall proving time. The reason for this is that some equalities that cannot be ordered can become ordered after applying a substitution (post-ordered), and we have to check for post-ordering repeatedly for the same equalities. In this paper, we show how to improve post-ordering checks by introducing a new data structure called term ordering diagrams, in short TODs, which creates an index for these checks. We achieve efficiency by lazy modifications of the index and by storing and reusing information from previously performed checks to speed up subsequent checks. Our experiments demonstrate efficiency of TODs.

Problem

• Optimizing term ordering checks in superposition calculus
• Reducing time spent on post-ordering equality checks
• Improving efficiency of KBO and LPO implementations

Innovation

• Introduces term ordering diagrams (TODs)
• Uses lazy index modifications for efficiency
• Stores and reuses previous check information