In several different settings, one comes across situations in which the objects of study are locally consistent but globally inconsistent. Earlier work about probability distributions by Vorob'ev (1962) and about database relations by Beeri, Fagin, Maier, Yannakakis (1983) produced characterizations of when local consistency always implies global consistency. Towards a common generalization of these results, we consider K-relations, that is, relations over a set of attributes such that each tuple in the relation is associated with an element from an arbitrary, but fixed, positive semiring K. We introduce the notions of projection of a K-relation, consistency of two K-relations, and global consistency of a collection of K-relations; these notions are natural extensions of the corresponding notions about probability distributions and database relations. We then show that a collection of sets of attributes has the property that every pairwise consistent collection of K-relations over those attributes is globally consistent if and only if the sets of attributes form an acyclic hypergraph. This generalizes the aforementioned results by Vorob'ev and by Beeri et al., and demonstrates that K-relations over positive semirings constitute a natural framework for the study of the interplay between local and global consistency. In the course of the proof, we introduce a notion of join of two K-relations and argue that it is the "right" generalization of the join of two database relations. Furthermore, to show that non-acyclic hypergraphs yield pairwise consistent K-relations that are globally inconsistent, we generalize a construction by Tseitin (1968) in his study of hard-to-prove tautologies in propositional logic.

中文翻译：

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一致性、非循环性和正半环

在几种不同的环境中，人们会遇到研究对象局部一致但全局不一致的情况。Vorob'ev (1962) 关于概率分布的早期工作和 Beeri、Fagin、Maier、Yannakakis (1983) 关于数据库关系的工作产生了局部一致性何时总是意味着全局一致性的特征。为了对这些结果进行普遍的概括，我们考虑 K 关系，即一组属性上的关系，使得关系中的每个元组都与来自任意但固定的正半环 K 的元素相关联。 我们引入了概念K-关系的投影、两个 K-关系的一致性以及 K-关系集合的全局一致性；这些概念是关于概率分布和数据库关系的相应概念的自然扩展。然后，我们证明属性集的集合具有以下属性：当且仅当属性集形成非循环超图时，这些属性上的 K 关系的每个成对一致集合都是全局一致的。这概括了 Vorob'ev 和 Beeri 等人的上述结果，并证明了正半环上的 K 关系构成了研究局部和全局一致性之间相互作用的自然框架。在证明过程中，我们引入了两个K-关系连接的概念，并认为这是两个数据库关系连接的“正确”推广。此外，