Local consistency and axioms of functional dependence

Timon Barlag, Miika Hannula, Juha Kontinen, Nina Pardal, Jonni Virtema

Abstract: Local consistency arises in diverse areas, including Bayesian statistics, relational databases, and quantum foundations. Likewise, the notion of functional dependence arises in all of these areas. 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 schemes. We provide a complete axiomatisation and a PTIME algorithm for the entailment problem of unary FDs. In addition, we explore when contextual families over the Booleans have realisations as contextual families over various monoids.