Solving satisfiability, implication, and equivalence problems in database systems

Satisfiability, implication and equivalence problems are important and widely-encountered database problems that need to be efficiently and effectively solved. We provide a comprehensive and systematic study of these problems. We consider three popular types of arithmetic inequalities, (X op C), (X op Y), and (X op Y + C), where X and Y are attributes, C is a constant of the domain of X, and op $\in\ \{{\u3c},\ {\le},\ {=},\ {\not=},\ {\u3e},\ {\ge}\}.$ These inequalities are most frequently used in a database system, since the first type of inequalities represents $\theta$-join, the second type represents selection, and the third type is popular in deductive databases. We study the problems under the integer domain and the real domain, as well as under two different operator sets. Our results show that solutions under different domains and/or different operator sets are quite different. In this dissertation, we either report the first necessary and sufficient conditions as well as their efficient algorithms with complexity analysis, or provide improved algorithms