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

This article focuses on defining rule semantics and database semantics for deductive databases within the framework of four-valued logic, which accommodates true, false, unknown, and inconsistent information. A database is modeled as a pair consisting of a finite set of ground facts with associated truth values and a set of generalized rules whose heads may be positive or negative literals and whose bodies use connectors from four-valued logic. The semantics is defined as the least fixed point of a monotonic operator and is shown to correspond to the unique minimal model of the database formulas under the knowledge ordering of four-valued logic. The paper characterizes safe rules that guarantee finite semantics and introduces a novel approach to database updating, where updates depend not only on the new fact but also on its current truth value, allowing for flexible update policies. Comparisons with related semantics by Fitting and Arieli highlight differences in rule interpretation and handling of inconsistency, while the approach is motivated by applications in data integration where contradictory and incomplete information naturally arise.