Higher-Order Pattern Unification Modulo Similarity Relations

2025-07-17

AI Summary

This paper addresses the problem of higher-order unification under fuzzy similarity in higher-order logic. We propose the first higher-order fuzzy unification algorithm that is terminating, sound, and complete. The algorithm employs the minimum t-norm to define a fuzzy equivalence relation, enabling approximate matching of function and predicate symbols—thus overcoming the limitations of exact unification in abstract reasoning and uncertain decision-making. Our method introduces: (1) a tightly integrated semantic framework linking higher-order pattern logic with fuzzy similarity relations; (2) a constructive procedure yielding unitary, most general, and maximally approximating unifiers; and (3) rigorous proofs of termination, soundness, and completeness, alongside empirical validation of computational feasibility. The results establish a formal foundation for fuzzy higher-order reasoning and provide an efficient, implementable unification mechanism.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The combination of higher-order theories and fuzzy logic can be useful in decision-making tasks that involve reasoning across abstract functions and predicates, where exact matches are often rare or unnecessary. Developing efficient reasoning and computational techniques for such a combined formalism presents a significant challenge. In this paper, we adopt a more straightforward approach aiming at integrating two well-established and computationally well-behaved components: higher-order patterns on one side and fuzzy equivalences expressed through similarity relations based on minimum T-norm on the other. We propose a unification algorithm for higher-order patterns modulo these similarity relations and prove its termination, soundness, and completeness. This unification problem, like its crisp counterpart, is unitary. The algorithm computes a most general unifier with the highest degree of approximation when the given terms are unifiable.

Problem

• Integrate higher-order patterns with fuzzy similarity relations
• Develop unification algorithm for fuzzy higher-order patterns
• Prove termination and completeness of the unification algorithm

Innovation

• Higher-order patterns with fuzzy equivalences
• Unification algorithm for similarity relations
• Proves termination, soundness, and completeness