  Modern applications combine information from a great variety of sources.
Oftentimes, some of these sources, like Machine-Learning systems, are not
strictly binary but associated with some degree of (lack of) confidence in the
observation. We propose MV-Datalog and MV-Datalog+- as extensions of Datalog
and Datalog+-, respectively, to the fuzzy semantics of infinite-valued
Lukasiewicz logic L as languages for effectively reasoning in scenarios where
such uncertain observations occur. We show that the semantics of MV-Datalog
exhibits similar model-theoretic properties as Datalog. In particular, we show
that (fuzzy) entailment can be decided via minimal fuzzy models. We show that
when they exist, such minimal fuzzy models are unique (when they exist) and can
be characterised in terms of a linear optimisation problem over the output of a
fixed-point procedure. On the basis of this characterisation, we propose
similar many-valued semantics for rules with existential quantification in the
head, extending Datalog+-. This paper is under consideration for acceptance in
TPLP.
