Local consistency arises in diverse areas, including Bayesian statistics, relational databases, and quantum foundations, and so does the notion of functional dependence. We adopt a general approach to study logical inference in a setting that enables both global inconsistency and local consistency. Our approach builds upon pairwise consistent families of K-relations, i.e, relations with tuples annotated with elements of some positive commutative monoid. The framework covers, e.g., families of probability distributions arising from quantum experiments and their possibilistic counterparts. As a first step, we investigate the entailment problem for functional dependencies (FDs) in this setting. Notably, the transitivity rule for FDs is no longer sound, but can be replaced by two novel axiom schemas. We provide a complete axiomatisation for, and establish NL-completeness of, the entailment problem of unary FDs, and demonstrate that even this restricted case exhibits context-dependent subtleties. In addition, we explore when contextual families over the Booleans have realisations as contextual families over various monoids.

翻译：局部一致性广泛存在于贝叶斯统计、关系数据库与量子基础理论等多个领域，函数依赖的概念同样如此。本文采用一种通用方法研究在允许全局不一致性与局部一致性并存场景下的逻辑推理。该方法建立在成对一致的K-关系族之上，即元组由某个正交换幺半群元素标注的关系族。该框架涵盖例如量子实验产生的概率分布族及其可能性对应物。作为初步探索，我们研究了此背景下函数依赖的蕴涵问题。值得注意的是，函数依赖的传递规则在此不再可靠，但可被两条新颖的公理模式所替代。针对一元函数依赖的蕴涵问题，我们给出了完备的公理化体系并证明了其NL完全性，同时论证了即使在此受限情形下仍存在语境依赖的微妙性。此外，我们还探讨了布尔域上的语境族在何种条件下可实现为不同幺半群上的语境族。