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