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、信息论和图模型中具有基础性作用。在数据库中，条件独立性体现在数据库规范化中，被称为（嵌入）多值依赖。条件独立性的许多特性在不同领域中具有共性，且在一定程度上可以通过测度论方法研究这些共性。本文提出了一种基于半环关系的替代方法，该方法通过将数据库关系扩展为具有某个交换半环的元组标注来定义。在此框架下整合条件独立性的多种解释，我们研究了底层半环的选择如何影响相应的公理化与分解性质。我们特别指出，正性和乘法可消去性是使关系语境下的结果推广至更广泛半环框架的关键半环性质。此外，我们通过模型理论探讨了不同条件独立性概念之间的联系，并考虑了数据库理论和信息论中逻辑推论与有效性测试方法如何推广至半环关系。