Propositional and predicate logics of incomplete information

International audienceOne of the most common scenarios of handling incomplete information occurs in relational databases. They describe in-complete knowledge with three truth values, using Kleene’s logic for propositional formulae and a rather peculiar exten-sion to predicate calculus. This design by a committee from several decades ago is now part of the standard adopted by vendors of database management systems. But is it really the right way to handle incompleteness in propositional and pred-icate logics?Our goal is to answer this question. Using an epistemic ap-proach, we first characterize possible levels of partial knowl-edge about propositions, which leads to six truth values. We impose rationality conditions on the semantics of the connec-tives of the propositional logic, and prove that Kleene’s logic is the maximal sublogic to which the standard optimization rules apply, thereby justifying this design choice. For exten-sions to predicate logic, however, we show that the additional truth values are not necessary: every many-valued extension of first-order logic over databases with incomplete informa-tion represented by null values is no more powerful than the usual two-valued logic with the standard Boolean interpreta-tion of the connectives. We use this observation to analyze the logic underlying SQL query evaluation, and conclude that the many-valued extension for handling incompleteness does not add any expressiveness to it

Intuitionistic Databases and Cylindric Algebra

The goal of this thesis is to develop an intuitionistic relevance-logic based semantics that allows us to handle Full First Order queries similar monotone First Order queries. Next, we fully investigate the relational model and universal nulls, showing that they can be treated on par with the usual existential nulls. To do so, we show that a suitable finite representation mechanism, called Star-Cylinders, handling universal nulls can be developed based on the Cylindric Set Algebra. Moreover, we show that any First Order Relational Calculus query over databases containing universal nulls can be translated into an equivalent expression in our star cylindric algebra, and vice versa. Furthermore, the representation mechanism is then extended to Naive Star-Cylinders, which are star-cylinders allowing existential nulls in addition to universal nulls. Beside the theory part, we also provide a practical approach for four-valued databases. We show that the four-valued database instances can be stored as a pair of two-valued instances. These two-valued instances store positive and negative information independently, in the format of current databases. In a similar way, we show that four-valued queries can be decomposed to two-valued queries and can be executed against decomposed instances to obtain the four-valued the result, after merging them back. Later, we show how these results can be extended to Datalog and we show that there is no need for any syntactical notion of stratification or non-monotonic reasoning when the intuitionistic logic is implemented. This is followed by presenting the complexity results