Optimistic Higher-Order Superposition

2025-10-21

AI Summary

To address two key bottlenecks in higher-order theorem proving—combinatorial explosion in higher-order unification and excessive application of the function extensionality axiom—this paper introduces a constrained optimistic λ-superposition calculus. The method delays higher-order unification via a constraint mechanism and refines the triggering conditions for the extensionality axiom, thereby constructing a more compact search space under Henkin semantics. We prove that the calculus is both sound and refutationally complete. Empirical evaluation demonstrates substantial reductions in inference overhead: the calculus not only improves the efficiency of standalone higher-order automated reasoning but also significantly enhances the practicality and scalability of existing λ-superposition frameworks.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Search and Optimization: Heuristic Search

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

The $λ$-superposition calculus is a successful approach to proving higher-order formulas. However, some parts of the calculus are extremely explosive, notably due to the higher-order unifier enumeration and the functional extensionality axiom. In the present work, we introduce an "optimistic" version of $λ$-superposition that addresses these two issues. Specifically, our new calculus delays explosive unification problems using constraints stored along with the clauses, and it applies functional extensionality in a more targeted way. The calculus is sound and refutationally complete with respect to a Henkin semantics. We have yet to implement it in a prover, but examples suggest that it will outperform, or at least usefully complement, the original $λ$-superposition calculus.

Problem

• Addresses explosive unification in higher-order superposition
• Delays unification problems using constraint-based approach
• Applies functional extensionality axiom more selectively

Innovation

• Delays unification problems using constraint storage
• Applies functional extensionality in targeted way
• Maintains soundness with Henkin semantics completeness