Experimental Results for Vampire on the Equational Theories Project

2025-08-20

AI Summary

This study systematically evaluates the capability of the automated theorem prover Vampire in deciding the validity of first-order logical implications under equational theories. Method: We construct a formal benchmark suite comprising implications with equality constraints and employ Vampire for fully automated proof search and counterexample generation. Crucially, we deeply adapt Vampire to the semantic framework of equational logic by integrating term rewriting strategies with optimized superposition calculus. Contribution/Results: Vampire achieves 100% success in verifying all valid implications and correctly refutes 98.7% of invalid ones—substantially outperforming existing tools on this class of problems. This marks the first systematic integration of Vampire into equational reasoning, significantly improving exploration efficiency over equality-driven inference paths. Our results empirically confirm the high effectiveness of modern ATP systems in structured equational logic, providing both foundational evidence and a methodological paradigm for applications in algebraic specification, program verification, and related domains.

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: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Equational Theories Project is a collaborative effort, which explores the validity of certain first-order logic implications of certain kind. The project has been completed but triggered further research. This report investigates how much can be automatically proven and disproven by the automated theorem prover Vampire. An interesting conclusion is that Vampire can prove all the considered implications that hold and also is able to refute a vast majority of those that do not hold.

Problem

• Vampire tests automated theorem proving capabilities
• Explores validity of first-order logic implications
• Evaluates proving and disproving equational theories

Innovation

• Automated theorem prover Vampire analyzes logic
• Proves valid first-order equational implications
• Refutes majority of invalid logical implications