MV-Datalog+-: Effective Rule-based Reasoning with Uncertain Observations

Abstract 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 $\mathrm{MV-Datalog}^\pm$ as extensions of Datalog and $\mathrm{Datalog}^\pm$ , respectively, to the fuzzy semantics of infinite-valued Łukasiewicz logic $\mathbf{L}$ 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 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 $\mathrm{Datalog}^\pm$ .