Unique complements and decompositions of database schemata

Abstract

In earlier work, Bancilhon and Spyratos introduced the concept of a complement to a database schema, and showed how this notion could be used in theories of decomposition and update semantics. However, they also showed that, except in trivial cases, even minimal complements are never unique, so that many desirable results, such as canonical decompositions, cannot be realized. Their work dealt with database schemata which are sets and database mappings which are functions, without further structure. In this work, we show that by adding a modest amount of additional structure, many important uniqueness results may be obtained. Specifically, we work with database schemata whose legal states form partially ordered sets (posets) with least elements, and with database mappings which are isotonic and which preserve this least element. This is a natural algebraic structure which is inherent in many important examples, including relational schemata constrained by data dependencies, with views constructed by composition of projection, restriction, and selection. Other examples include deductive database schemata in which views are defined by rules, and general first-order logic databases. Within this context of posets, we show that direct (i.e., independent) complements must be unique, and that in fact the directly complementable views have the structure, in a very natural sense, of a Boolean algebra. Decompositions of the schema then become identifiable with finite subalgebras of this Boolean algebra. To demonstrate the utility of our approach, we examine in some detail its applicability to the relational model. Particularly, we establish that under the condition that the schema is constrained by universal Horn sentences, there is a unique ultimate decomposition into a finite set of type restrictions. The latter are a special class of views which includes classical projections which occur in direct decompositions. In particular, classical join-based decomposition is completely recovered within a framework which explicitly axiomatizes independence via null values.

The Relative Complexity of Updates for a Class of Database Views

“…Modulo these modification, it is fair to say, at least in a general way, that [2] and [3] In [9], a theory of direct decomposition (i.e., situations in which the views are independent and so the meet is trivial) of order-based schemata is presented.…”

The complexity of embedded axiomatization for a class of closed database views

“…In [Heg94], a theory of direct decomposition (i.e., situations in which the views are independent and so the meet is trivial) of order-based schemata is presented.…”

Characterization of desirable properties of general database decompositions

“…However, the topic of decomposition in a set-based framework has fruitfully been pursued by Bancilhon and Spyratos [3], [2], who examined both update semantics and view independence in this context. This paper shares a common setting with the paper [17]. While here we study properties of decompositions in which the component schemata have nontrivial constraints connecting them, [17] examines direct decompositions, in which all views are independent.…”

“…This paper shares a common setting with the paper [17]. While here we study properties of decompositions in which the component schemata have nontrivial constraints connecting them, [17] examines direct decompositions, in which all views are independent. The paper [18] also deals with decompositions in a general context, but focuses upon the problem of updates rather than schema desirability.…”

“…Much of the material in this subsection can also be found in [17] and [18], with a somewhat different emphasis.…”