Inclusion dependencies and their interaction with functional dependencies

Abstract

Inclusion dependencies, or INDs (which can say, for example, that every manager is an employee) are studied, including their interaction with functional dependencies, or FDs. A simple complete axiomatization for INDs is presented, and the decision problem for INDs is shown to be PSPACE-complete.(The decision problem for INDs is the problem of determining whether or not Z logically implies c, given a set Z of INDs and a single IND (T). It is shown that finite implication (implication over databases with a finite number of tuples) is the same as unrestricted implications for INDs, although finite implication and unrestricted implication are distinct for FDs and 1NDs taken together. It is shown that, although there are simple complete axiomatizations for FDs alone and for INDs alone, there is no complete axiomatization for FDs and INDs taken together, in which every rule is k-ary for some fixed k (and in particular, there is no finite complete axiomatization.) This is true whether we consider finite implication or unrestricted implication, and is true even if no relation scheme has more than three attributes. The nonexistence of a k-ary complete axiomatization for FDs and INDs taken together is proven by giving a condition which is necessary and sufficient in general for the existence of a k-ary complete axiomatization.

Algorithms for analysing related constraint business rules

“…So, in contrast to the majority of work on constraint analysis for relational database systems that assume a flat structure [7,23], we allow a structure to be nested. This is necessary because the nature of constraint evolution is such that we often need to work with constraints that are recovered from legacy systems.…”

“…ATMOST ðY ; nÞ, we must have the following: It is easy to see the two types of axiom are handled rather similarly: both involve essentially a reachability computation (from P 1 to P m ) [7,24] and they differ only in the actual generation of derived constraint. 2 For this reason, we will only describe how transitivity axioms are handled in this paper and the reader is referred to [13] for the handling of inheritance axioms.…”

“…The earlier research concentrated on deriving axiomatic properties for constraints. This has resulted in a rich collection of axioms for characterising, for example, inclusion dependency [7,24], functional dependency [7,24], numerical dependency [16,28] and multi-valued dependency [3]. These axioms have then been used in a variety of constraint analyses, including cardinality constraint satisfiability for ER models [21], predicate satisfiability and subsumption in FOL [20], schema equivalence in meta-level FOL [8], and constraint subsumption and satisfiability in Description Logic [1,4].…”

“…The majority of constraint analysis techniques assume a flat structure (such as normalised relations) and deal with the instances of such structures in their analysis [5,7,21]. Our algorithms, on the other hand, allow nested structures to be used in constraints.…”

A finite axiomatization of conditional independence and inclusion dependencies

“…Inclusion dependencies were axiomatized in [5], and an axiomatization for pure independence atoms is also known (see [27,13,21]). On the other hand, the implication problem for embedded multivalued dependencies (also embedded join dependencies), and for inclusion dependencies and functional dependencies together, are known to be undecidable [18,19,6].…”