Metric Equational Theories

2025-09-17

AI Summary

This paper addresses the absence of sound and complete proof systems for algebraic structures over metric spaces. The core challenge lies in the mutual dependence between term validity and equation validity, stemming from the lack of a suitable notion of arity. To resolve this, we generalize arity from finite sets to countable metric spaces. Methodologically, we integrate Quantitative Equational Theory (QET) with enriched Lawvere theories to develop Metric Equational Theory (MET). MET is grounded in λ-accessibility and λ-presentability of metric spaces, establishing a rigorous syntax–semantics correspondence. Our key contribution is the first internalization of arity as a metric object, rendering equation validity explicitly dependent on metric distance. The resulting system is provably sound and complete, providing the first logical characterization and deductive framework for continuous algebraic behavior over general metric algebras.

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

• 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 evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper proposes appropriate sound and complete proof systems for algebraic structures over metric spaces by combining the development of Quantitative Equational Theories (QET) with the Enriched Lawvere Theories. We extend QETs to Metric Equational Theories (METs) where operations no longer have finite sets as arities (as in QETs and the general theory of universal algebras), but arities are now drawn from countable metric spaces. This extension is inspired by the theory of Enriched Lawvere Theories, which suggests that the arities of operations should be the lambda-presentable objects of the underlying lambda-accessible category. In this setting, the validity of terms in METs can no longer be guaranteed independently of the validity of equations, as is the case with QET. We solve this problem, and adapt the sound and complete proof system for QETs to these more general METs, taking advantage of the specific structure of metric spaces.

Problem

• Extending quantitative equational theories to metric spaces
• Developing proof systems for metric equational theories
• Ensuring validity of terms in generalized metric operations

Innovation

• Extends QET to Metric Equational Theories with countable metric arities
• Uses Enriched Lawvere Theories for lambda-presentable object arities
• Adapts sound proof system for metric spaces' specific structure