Conditional independence plays a foundational role in database theory, probability theory, information theory, and graphical models. In databases, conditional independence appears in database normalization and is known as the (embedded) multivalued dependency. Many properties of conditional independence are shared across various domains, and to some extent these commonalities can be studied through a measure-theoretic approach. The present paper proposes an alternative approach via semiring relations, defined by extending database relations with tuple annotations from some commutative semiring. Integrating various interpretations of conditional independence in this context, we investigate how the choice of the underlying semiring impacts the corresponding axiomatic and decomposition properties. We specifically identify positivity and multiplicative cancellativity as the key semiring properties that enable extending results from the relational context to the broader semiring framework. Additionally, we explore the relationships between different conditional independence notions through model theory, and consider how methods to test logical consequence and validity generalize from database theory and information theory to semiring relations.

翻译：条件独立性在数据库理论、概率论、信息论和图模型领域中具有基础性作用。在数据库中，条件独立性表现为（嵌入）多值依赖，是数据库规范化理论的核心概念。条件独立性的诸多性质在不同领域间具有共通性，这些共性在一定程度上可通过测度论方法进行研究。本文提出一种基于半环关系的替代方法——通过引入某个交换半环的元组标注来扩展数据库关系。在统一该框架下条件独立性的多种解释后，我们探究底层半环的选择如何影响相应的公理化性质与分解特性。研究特别指出，正性与乘法可消去性是使关系语境结果能推广至广义半环框架的关键半环性质。此外，我们通过模型理论研究不同条件独立性概念间的关联，并探讨从数据库理论与信息论中发展出的逻辑推论与有效性检验方法如何推广至半环关系。