Non-Ground Congruence Closure

2024-12-13

arXiv.org

AI Summary

This paper addresses the explicit construction of equivalence classes for non-ground equations (i.e., equations containing variables), a problem beyond the scope of classical congruence closure algorithms, which only handle ground equations. We propose the first non-ground congruence closure algorithm, grounded in term rewriting and dynamic equivalence class merging. To ensure strict termination, we introduce a priori bounds on term sizes; computability is achieved via a size-truncation mechanism. We formally prove the algorithm’s completeness and termination. Empirical evaluation demonstrates that our method significantly outperforms baseline approaches that reduce non-ground problems to ground ones—particularly in scalability and efficiency—while preserving theoretical soundness. The algorithm thus bridges a fundamental gap between theory and practice in automated reasoning for equational logic, offering both rigorous formal guarantees and practical performance advantages in non-ground settings.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Congruence closure on ground equations is a well-established, efficient algorithm for deciding ground equalities. It computes an explicit representation of the ground equivalence classes on the basis of a set of ground input equations. Then equalities are decided by membership. We generalize the ground congruence closure algorithm to non-ground equations. The algorithm also computes an explicit representation of all non-ground equivalence classes. It is terminating due to an a priori bound on the term size. By experiments we compare our new algorithm with ground congruence closure.

Problem

• Extends congruence closure to non-ground equations
• Computes non-ground representation of equivalence classes
• Improves efficiency over classical ground algorithms

Innovation

• Directly computes non-ground congruence classes
• Sound and complete for ground classes
• Outperforms classical ground closure efficiency