Unification of Deterministic Higher-Order Patterns

2026-01-20

AI Summary

This study addresses the unification problem for deterministic higher-order patterns—a class of terms characterized by deterministic matching, yet whose unifiability has long remained undecidable. We propose a sound and complete unification procedure that extends the scope of tractable problems by relaxing the global restriction on variable arguments imposed in the function-as-constructor approach. The key innovation lies in applying a most general unifier strategy to flex-flex cases; although a most general unifier is not guaranteed to exist in such scenarios, our method reveals for the first time that solution sets may be infinite. This work not only achieves complete unification for deterministic higher-order patterns but also offers new insights into the theoretical boundaries of higher-order unification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present a sound and complete unification procedure for deterministic higher-order patterns, a class of simply-typed lambda terms introduced by Yokoyama et al. which comes with a deterministic matching problem. Our unification procedure can be seen as a special case of full higher-order unification where flex-flex pairs can be solved in a most general way. Moreover, our method generalizes Libal and Miller's recent functions-as-constructors higher-order unification (FCU) by dropping their global restriction on variable arguments, thereby losing the property that every solvable problem has a most general unifier. In fact, minimal complete sets of unifiers of deterministic higher-order patterns may be infinite, so decidability of the unification problem remains an open question. Nevertheless, our method can be more useful than FCU in practice.

Problem

• higher-order unification
• deterministic higher-order patterns
• lambda calculus
• unification problem

Innovation

• deterministic higher-order patterns
• higher-order unification
• flex-flex pairs
• functions-as-constructors
• most general unifier