Nominal Equational Narrowing: Rewriting for Unification in Languages with Binders

2025-05-20

AI Summary

Reasoning about equational theories for binder-aware languages (e.g., λ-calculus, π-calculus) is hindered by axioms involving α-renaming, freshness constraints, and commutativity. Method: We develop the first nominal E-unification theory supporting freshness conditions and establish a nominal E-lifting theorem, providing a formal foundation and correctness guarantee for equational narrowing with binding constructs. Our approach introduces a nominal-term-oriented framework for equational reduction and narrowing, integrating nominal rewriting, structural congruence reasoning, freshness modeling, and a machine-verified nominal unification algorithm. Results: The framework is empirically validated on tasks including symbolic differentiation and first-order formula simplification, demonstrating both theoretical soundness and practical effectiveness. It enables automated, correctness-preserving equational reasoning in the presence of binders and freshness, advancing the state of the art in nominal rewriting and automated deduction for higher-order and process calculi.

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

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• 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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Narrowing extends term rewriting with the ability to search for solutions to equational problems. While first-order rewriting and narrowing are well studied, significant challenges arise in the presence of binders, freshness conditions and equational axioms such as commutativity. This is problematic for applications in programming languages and theorem proving, where reasoning modulo renaming of bound variables, structural congruence, and freshness conditions is needed. To address these issues, we present a framework for nominal rewriting and narrowing modulo equational theories that intrinsically incorporates renaming and freshness conditions. We define and prove a key property called nominal E-coherence under freshness conditions, which characterises normal forms of nominal terms modulo renaming and equational axioms. Building on this, we establish the nominal E-lifting theorem, linking rewriting and narrowing sequences in the nominal setting. This foundational result enables the development of a nominal unification procedure based on equational narrowing, for which we provide a correctness proof. We illustrate the effectiveness of our approach with examples including symbolic differentiation and simplification of first-order formulas.

Problem

• Extends term rewriting to solve equational problems with binders
• Addresses challenges in unification under freshness and equational axioms
• Develops nominal unification for programming languages and theorem proving

Innovation

• Nominal rewriting and narrowing framework
• Nominal E-coherence under freshness conditions
• Nominal unification via equational narrowing