Locally Consistent K-relations: Entailment and Axioms of Functional Dependence

TL;DR

This paper develops a general framework for logical inference in systems with local consistency and global inconsistency, covering probabilistic and possibilistic relations, and introduces novel axioms for functional dependencies.

Contribution

It introduces a unified approach to analyze local consistency in K-relations, providing new axioms for functional dependencies and a complete axiomatisation for unary FDs.

Findings

Transitivity for FDs is not sound; replaced by two new axioms.

Complete axiomatisation for unary FDs in this setting.

NL-completeness of the entailment problem for unary FDs.

Abstract

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.