Functional dependencies with null markers
  Functional dependencies are an integral part of database design. However, they are only defined when we exclude null markers. Yet we commonly use null markers in practice. To bridge this gap between theory and practice, researchers have proposed definitions of functional dependencies over relations with null markers. Though sound, these definitions lack some qualities that we find desirable. For example, some fail to satisfy Armstrong's axioms---while these axioms are part of the foundation of common database methodologies. We propose a set of properties that any extension of functional dependencies over relations with null markers should possess. We then propose two new extensions having these properties. These extensions attempt to allow null markers where they make sense to practitioners.   They both support Armstrong's axioms and provide realizable null markers: at any time, some or all of the null markers can be replaced by actual values without causing an anomaly. Our proposals may improve database designs. 
Antonio Badia and Daniel Lemire
cs.DB