The logic of Dependence and Independence Bunched Implications (DIBI) is a logic to reason about conditional independence (CI); for instance, DIBI formulas can characterise CI in probability distributions and relational databases, using the probabilistic and relational DIBI models, respectively. Despite the similarity of the probabilistic and relational models, a uniform, more abstract account remains unsolved. The laborious case-by-case verification of the frame conditions required for constructing new models also calls for such a treatment. In this paper, we develop an abstract framework for systematically constructing DIBI models, using category theory as the unifying mathematical language. In particular, we use string diagrams -- a graphical presentation of monoidal categories -- to give a uniform definition of the parallel composition and subkernel relation in DIBI models. Our approach not only generalises known models, but also yields new models of interest and reduces properties of DIBI models to structures in the underlying categories. Furthermore, our categorical framework enables a logical notion of CI, in terms of the satisfaction of specific DIBI formulas. We compare it with string diagrammatic approaches to CI and show that it is an extension of string diagrammatic CI under reasonable conditions.

翻译：依赖与独立丛合蕴含逻辑（DIBI）是一种用于推理条件独立性（CI）的逻辑体系；例如，通过概率模型和关系模型，DIBI公式可分别刻画概率分布与关系数据库中的条件独立性。尽管概率模型与关系模型具有相似性，但尚未形成统一且更抽象的理论框架。构建新模型时需逐例验证框架条件的繁琐过程，也亟需此类系统性处理。本文利用范畴论作为统一的数学语言，发展出用于系统构建DIBI模型的抽象框架。特别地，我们采用弦图（幺半范畴的图形化表示）对DIBI模型中的并行组合与子核关系给出统一定义。该方法不仅推广了已知模型，还生成了有价值的新模型，并将DIBI模型的性质归结为底层范畴的结构特征。此外，基于特定DIBI公式的满足关系，我们的范畴框架阐明了CI的逻辑概念。通过与弦图式CI方法比较，证明在合理条件下该框架是弦图式CI的扩展。