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In this paper, we investigate rule semantics for deductive databases in the context of four-valued logic. In our approach, a database is a pair   $\varDelta =(E,R)$  , where   $E$   is a set of pairs, each pair associating a ground fact with a truth value (thus, allowing to store true, false or inconsistent facts) and   $R$   is a set of rules generalizing standard Datalog rules in the following sense: (i) the head of a rule can be a positive or a negative atom and (ii) the body can involve any among the connectors of four-valued logic. We define the database semantics as the least fixed point of a monotonic operator and we compare this semantics with that of k-existential programs defined by Fitting and paraconsistent extended logic programs defined by Arieli. Our main contribution is to show that, if we consider rules as implications (i.e. if we view the database as a set of formulas) then the semantics of the database is the unique minimal model of the set of database formulas. Here, minimality is understood with respect to the knowledge ordering of four-valued logic satisfying a monotonicity property whereby the truth value of the head of an instantiated rule is greater than that of the body. Moreover, we characterize databases having finite semantics and then, we address the issue of database updating. We argue that our approach allows for a new kind of updates, in which the update result depends not only on the fact involved in the update but also on its current truth value in the database.

Deductive databases in four-valued logic: rule semantics and models