[Paper Review] The Boyer-Moore Waterfall Model Revisited

Petros Papapanagiotou, Jacques Fleuriot|arXiv (Cornell University)|Aug 11, 2018

Artificial Intelligence in Games12 references3 citations

TL;DR

This paper revisits the 30-year-old Boyer-Moore waterfall model for automated inductive theorem proving by integrating it into the HOL Light theorem prover as a fully automated tactic. It enhances the model with modern generalization techniques, proof feedback, and loop detection, achieving 47% success on a 150-theorem benchmark, demonstrating the model's enduring relevance and extensibility in interactive proof environments.

ABSTRACT

In this paper, we investigate the potential of the Boyer-Moore waterfall model for the automation of inductive proofs within a modern proof assistant. We analyze the basic concepts and methodology underlying this 30-year-old model and implement a new, fully integrated tool in the theorem prover HOL Light that can be invoked as a tactic. We also describe several extensions and enhancements to the model. These include the integration of existing HOL Light proof procedures and the addition of state-of-the-art generalization techniques into the waterfall. Various features, such as proof feedback and heuristics dealing with non-termination, that are needed to make this automated tool useful within our interactive setting are also discussed. Finally, we present a thorough evaluation of the approach using a set of 150 theorems, and discuss the effectiveness of our additions and relevance of the model in light of our results.

Motivation & Objective

• To evaluate the viability of the 30-year-old Boyer-Moore waterfall model for modern automated inductive theorem proving in a contemporary proof assistant.
• To integrate and extend the Boyer-Moore model within HOL Light, leveraging its flexible, low-level architecture for deep system interaction.
• To improve automation through enhancements like state-of-the-art generalization heuristics, proof verbosity, and non-termination mitigation.
• To assess the effectiveness of combining original Boyer-Moore generalization with Aderhold’s inverse weakening technique in a modern context.
• To demonstrate that the waterfall model remains a viable foundation for scalable, extensible inductive proof automation in interactive theorem proving.

Proposed method

• Implement a fully integrated Boyer-Moore waterfall tactic in HOL Light, using its low-level OCaml-based architecture for seamless interaction with internal data structures and tactics.
• Reconstruct Boulton’s HOL 90 implementation of the Boyer-Moore system as a faithful basis for the new tactic, preserving the original waterfall logic.
• Integrate existing HOL Light proof procedures—such as MESON and rewriting—into the waterfall pipeline to enhance simplification and proof search.
• Enhance the generalization heuristic by incorporating Aderhold’s inverse weakening technique for variable generalization and comparing it with the original Boyer-Moore subterm generalization.
• Implement loop detection heuristics based on clause pool size and induction count, with incremental depth control similar to MESON to prevent non-termination.
• Add proof feedback mechanisms to improve transparency and user guidance during automated proof attempts.

Experimental results

Research questions

• RQ1 Can the classic Boyer-Moore waterfall model be effectively revived and extended within a modern, interactive theorem prover like HOL Light?
• RQ2 How do state-of-the-art generalization techniques, such as Aderhold’s inverse weakening, compare with the original Boyer-Moore subterm generalization in this context?
• RQ3 To what extent can loop detection and heuristic tuning improve the robustness and usability of the waterfall model in practice?
• RQ4 What is the performance and success rate of the enhanced Boyer-Moore tactic on a representative benchmark of 150 inductive theorems?
• RQ5 Can the integration of existing HOL Light procedures (e.g., MESON) and rewrite rules significantly improve proof automation within the waterfall framework?

Key findings

• The enhanced Boyer-Moore tactic successfully proved 47% of the 150 theorems in the evaluation set, indicating strong potential despite the fully automated setup.
• The original Boyer-Moore subterm generalization outperformed Aderhold’s inverse weakening in the context of the system, particularly due to the equation criterion and constructor/accessor handling.
• Aderhold’s algorithm for variable generalization showed value in specific cases, suggesting that a hybrid approach combining both techniques yields optimal results.
• The integration of HOL Light’s native procedures, such as MESON and rewriting, significantly improved simplification and proof search efficiency.
• Loop detection heuristics, including clause pool size and induction count monitoring, effectively mitigated non-termination, enhancing system reliability.
• The flexible, programmable nature of HOL Light enabled seamless integration and extension of the waterfall model, supporting future evolution similar to ACL2’s development from the original Boyer-Moore prover.