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Consistency, Acyclicity, and Positive Semirings

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.