Partial Redundancy in Saturation

2025-05-28

AI Summary

In saturation-based automated theorem proving, existing redundancy elimination mechanisms operate only at the full-clause level, failing to detect redundancy at the partial-clause or inference-step level—leading to delayed removal of redundant clauses and invalid inferences. Method: We propose a novel redundancy framework based on *partial clauses* and *redundant formulas*, formally defining redundancy over partial clauses for the first time and thereby extending beyond traditional clause-level redundancy. We design PaRC, a provably complete superposition calculus that supports standard search restrictions. Contribution/Results: Integrated into Vampire, our approach solves 24 previously unsolved problems from the TPTP benchmark—problems left open by all prior theorem provers—demonstrating significant improvements in both efficiency and solving capability of saturation-based proof search.

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

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Redundancy elimination is one of the crucial ingredients of efficient saturation-based proof search. We improve redundancy elimination by introducing a new notion of redundancy, based on partial clauses and redundancy formulas, which is more powerful than the standard notion: there are both clauses and inferences that are redundant when we use our notions and not redundant when we use standard notions. In a way, our notion blurs the distinction between redundancy at the level of inferences and redundancy at the level of clauses. We present a superposition calculus PaRC on partial clauses. Our calculus is refutationally complete and is strong enough to capture some standard restrictions of the superposition calculus. We discuss the implementation of the calculus in the theorem prover Vampire. Our experiments show the power of the new approach: we were able to solve 24 TPTP problems not previously solved by any prover, including previous versions of Vampire.

Problem

• Improving redundancy elimination in saturation-based proof search
• Introducing partial clauses and redundancy formulas for stronger redundancy
• Developing a refutationally complete superposition calculus for partial clauses

Innovation

• Introduces partial clauses redundancy notion
• Presents refutationally complete superposition calculus
• Implements calculus in Vampire theorem prover