The relational data model as proposed by Codd is a well-established method for data abstraction. Two essential aspects in this model are the definition of the data structure via the relation scheme and the data semantics via data dependencies. Various classes of data dependencies have been studied in the past. In the presence of data dependencies "update dependencies" (or anomalies) and "redundancy" may occur as first observed by Codd. Normal forms have been proposed as a means to control update anomalies and redundancy. But as the notion of redundancy has never been formally defined, one cannot make any precise statement concerning the presence or absence of redundancy for a given design. In this paper we attempt to provide a formal definition of the notion of redundancy for the case of a single relation respectively relation scheme. We first give a static semantic definition of redundancy and then present an operational analogue. Intuitively speaking a relation r contains redundancy, if some "part" of the information given in r can be "determined" from the "rest" of r. And a relation scheme with a given set of data dependencies admits redundancy if there is a relation belonging to this scheme that contains redundancy. The paper is organized in six sections. Section 1 contains the definition of the relational model that we use. We make use of partial "relations" that are built from constants and variables. In section 2 we present the semantic definition of redundancy. Section 3 introduces a class of data dependencies, i.e. implicational dependencies and a chase procedure for partial relations. Section 4 gives an operational characterization of redundancy. The main theorem in this section is theorem 4.3. It states that a relation r in a class of relations sat(D) contains redundancy if there exists a partial relation q that "contains less information" than rand for which chase D(q)
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