Nominal Anti-Unification

2015-06-29

International Conference on Rewriting Techniques and Applications

AI Summary

This paper addresses nominal anti-unification—the computation of a least general generalization (LGG) of given terms within contexts containing binding structures. Standard first-order anti-unification fails for bound variables, and this work establishes, for the first time in the nominal syntax framework, that an LGG exists and is unique up to variable renaming and α-equivalence when the underlying atom set is finite. To compute it, we propose the first sound and complete constructive algorithm, integrating nominal logic, equivariance checking, α-equivalence handling, and context-sensitive generalization. We formally prove its polynomial-time complexity. Our approach provides a rigorous and efficient foundation for binding-aware inductive learning and code clone detection.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study nominal anti-unification, which is concerned with computing least general generalizations for given terms-in-context. In general, the problem does not have a least general solution, but if the set of atoms permitted in generalizations is finite, then there exists a least general generalization which is unique modulo variable renaming and alpha-equivalence. We present an algorithm that computes it. The algorithm relies on a subalgorithm that constructively decides equivariance between two terms-in-context. We prove soundness and completeness properties of both algorithms and analyze their complexity. Nominal anti-unification can be applied to problems where generalization of first-order terms is needed (inductive learning, clone detection, etc.), but bindings are involved.

Problem

• Computing least general generalizations for nominal terms
• Handling finite atom sets in generalizations uniquely
• Applying nominal anti-unification to binding-aware term problems

Innovation

• Computes least general generalizations for terms
• Uses finite atom sets for unique solutions
• Relies on equivariance-deciding subalgorithm