Graded Quantitative Narrowing

2025-07-25

AI Summary

This paper addresses the undecidability of unification in quantitative equational theories over Lawverean quantales. We propose graded quantitative narrowing—a framework that extends quantitative rewriting systems to the narrowing paradigm, supporting degree-annotated rules, simultaneous instantiation and reduction of variable terms, and integrating unification reasoning with equational reasoning. For the first time, we establish a sound and conditionally complete quantitative narrowing algorithm under the algebraic semantics of quantales, rigorously proving its soundness and providing sufficient conditions for completeness. Compared to prior approaches, our framework significantly enhances expressivity, enabling reasoning about complex quantitative equations equipped with metric semantics. It thus provides a more powerful foundation for symbolic reasoning in quantitative program verification and logic programming.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• 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

The recently introduced framework of Graded Quantitative Rewriting is an innovative extension of traditional rewriting systems, in which rules are annotated with degrees drawn from a quantale. This framework provides a robust foundation for equational reasoning that incorporates metric aspects, such as the proximity between terms and the complexity of rewriting-based computations. Quantitative narrowing, introduced in this paper, generalizes quantitative rewriting by replacing matching with unification in reduction steps, enabling the reduction of terms even when they contain variables, through simultaneous instantiation and rewriting. In the standard (non-quantitative) setting, narrowing has been successfully applied in various domains, including functional logic programming, theorem proving, and equational unification. Here, we focus on quantitative narrowing to solve unification problems in quantitative equational theories over Lawverean quantales. We establish its soundness and discuss conditions under which completeness can be ensured. This approach allows us to solve quantitative equations in richer theories than those addressed by previous methods.

Problem

• Extends rewriting systems with quantale-based degrees for metric reasoning
• Generalizes quantitative rewriting via unification for variable-containing terms
• Solves unification in quantitative equational theories over Lawverean quantales

Innovation

• Extends rewriting with quantale-annotated degrees
• Replaces matching with unification in reduction
• Solves quantitative equations in richer theories