In real life it is very often the case that the available knowledge is imperfect in the sense that it represents multiple possible states of the external world, yet it is unknown which state corresponds to the actual situation of the world. Imperfect knowledge can be of two different categories. Knowledge is incomplete if it represents different states, one of which is true in the external world. On the contrary, knowledge is uncertain if it represents different states which may be satisfied or are likely to be true in the external world. Imperfect knowledge can be considered under two different perspectives: using either an algebraic or a logical approach. We present both approaches in relation with the standard relational model, providing the necessary background for the subsequent development. The study of imperfect knowledge has been an active area of research, in particular in the context of relational databases. However, due to the complexity of manipulating imperfect knowledge, little practical results have been obtained so far. In this thesis we provide a survey of the field of incompleteness and uncertainty in relational databases;it can be used also as an introductory tutorial for understanding the intuitive semantics and the problems encountered when representing and manipulating such imperfect knowledge. The survey concentrates in giving an unifying presentation of the different approaches and results found in the literature, thus providing a state of the art in the field. The rest of the thesis studies in detail the manipulation of one type of incomplete knowledge, namely disjunctive information, and one type of uncertain knowledge, namely probabilistic information. We study both types of imperfect knowledge using similar approaches, that is through an algebraic and a logical framework. The relational algebra operators are generalized for disjunctive and probabilistic relations, and we prove the correctness of these generalizations. In addition, disjunctive and probabilistic databases are formalized using appropriate logical theories and we give sound and complete query evaluation algorithms. A major implication of these studies is the conviction that viewing incompleteness and uncertainty as different facets of the same problem would allow to achieve a deeper understanding of imperfect knowledge, which is absolutely necessary for building information systems capable of modeling complex real-life situations. |
