  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.
