Normal Forms and Minimal Keys in the Relational Datamodel

  J. Demetrovics Vu Duc Thi

Abstract:

  The normalization of relations was introduced by E.F. Codd. The main purpose of normalization is to delete undesired redundancy and anormalies. The most desirable normal forms are second normal form (2NF), third normal form (3NF) and Boyce-Codd normal form (BCNF) that have been investigated in a lot of papers. The concepts of minimal key and prime attribute (recall that an attribute is prime if it belongs to a minimal key, and nonprime otherwise) directly concern 2NF, 3NF and BCNF. This paper investigates connections between these normal forms and sets of minimal keys. Lucchesi and Osborn showed [11] that the problem to decide if an arbitrary attribute is prime is NP-complete for relation scheme. We proved [9] that a set of all nonprime attributes is the intersection of all antikeys (maximal nonkeys) and this prime attribute problem can be solved by polynomial time algorithm for relation. From these results some problems are NP-complete for relation scheme, but for relation these problems are solved by polynomial time algorithms. It is known [5] hat a set of all minimal keys of a relation scheme (and a relation) is a Sperner System (sometimes it is called an antichain) and for an arbitrary Sperner system there exists a relation scheme the set of all minimal keys of which is exactly this Sperner system. In this paper the following concepts are introduced.

 A Sperner system is in 2NF (3NF, BCNF, respectively), where then is in 2NF (3NF, BCNF, respectively), where is a set of all minimal keys of . This paper gives necessary and sufficient conditions for an arbitrary Sperner system is in 2NF or 3NF or BCNF. We prove that problems of deciding whether is in 2NF (3NF, respectively) are NP-complete. However, we show that if a relation scheme is changed to a relation then these problems are solved by polynomial time algorithms. We give a new characterization of relations and relation schemes that are uniquely determined by their minimal keys. From this characterization we give a polynomial time algorithm deciding whether an arbitrary relation is uniquely determined by its set of all minimal keys. Osborn [14] gives a polynomial time algorithm testing BCNF property of a given relation scheme. This paper gives a polynomial time algorithm recognizing BCNF and finding a set of all minimal keys and a minimum cover if a given relation scheme is in BCNF.

 Key Words and Phrases: database, relation, relational datamodel, functional dependency, relation scheme, second normal form, third normal form, Boyce-Codd normal form, closure, closed set, minimal generator, key, minimal key, antikey.

 Gyenizse Pal 1996. Szeptember 4. Szerda 13:21:16 MET DST