An algebraic approach to the information-lossless decomposition of relational databases.

摘要

Relational information systems, systems that can be represented by tables of finite states, are widely used in many areas such as logic circuits, finite state machines, and relational databases. Decomposition is a natural method to remove redundancy of complex systems. It divides a system into a network of simpler components. In order to preserve the original functionalities of the system, any valid decomposition has to be lossless. This work is divided into two parts. In the first part, we develop a mathematical model for lossless decompositions of relational information systems. Commutative partitions play an important role in decompositions. The commutativity is essentially a general algebraic formulation of independency of two partitions. We express the interdependency of two commutative partitions by a bipartite graph, and classify the hierarchical independency structures by the topological property of bipartite graphs. In particular, we show that two partitions are decomposable, the strongest kind of independency, if and only if the associated bipartite graph is uniform. Moreover, we adopt Shannon's entropy to quantify the amount of information contained in each partition, and formulate information-lossless decompositions by entropy equalities. Under the assumption of running intersection property, we show that the general formulation of information-lossless decompositions of relational information systems is given by the entropy inclusion-exclusion equality. We also present the applications of these formulations to the above engineering systems to manifest the information-lossless decomposition processes.;In the second part, we further investigate algebraic structure of relational databases. The decomposition theory for relational databases is based on data dependencies. Nevertheless, the set-theoretic representations of data dependencies in terms of the attributes of relation schemes are incompatible with partial ordering operations. This brings a gap between the database decomposition theory and our theory. We identify the unique component constraint as a necessary condition for binary decomposition of a relation, i.e. there is a unique component for every join key value in the bipartite graph. We generalize the running intersection property as the partial ordering counterpart under the unique component constraint. It follows that we characterize the multivalued and acyclic join dependencies in terms of commutativity and unique component constraint. This shows the decompositions specified by these dependencies are special cases of our theory. Furthermore, we propose a lossless decomposition method for the class of data dependencies that is based on commutativity, and demonstrate that existing relational operations are sufficient for this method.