Some computational problems related to normal forms

In the relational database theory the most desirable  normal form is the Boyce-Codd normal form (BCNF). This paper investigates some computational problems concerning BCNF relation scheme and BCNF relations. We give an effective algorithm finding a BCNF relation r such that r represents a given BCNF relation scheme s  (i.e., Kr=Ks, where Kr and Ks are  sets of all minimal keys of  r and s). This paper also gives an effective algorithm which  from a given  BCNF relation finds a BCNF relation scheme such that Kr=Ks. Based on these algorithms we prove that  the time  complexity of the  problem that  finds a BCNF relation r  representing a given BCNF relation scheme s is exponential in the size of s and conversely, the complexity of finding a BCNF relation scheme s from a given BCNF relation r such that r represents s also is exponential in the number of attributes. We give a new characterization of the relations and the relation scheme that are uniquely determined by their minimal keys. It is known that these relations and the relation schemes are in the BCNF class. From this characterization we give a polynomial time algorithm deciding whether an arbitrary relation is uniquely determined by its set of all  minimal keys. In the rest if this paper some new bounds of the  size of minimal Armstrong relations for  BCNF relation scheme are given. We show that given a Sperner system K and BCNF relation scheme s a set of minimal keys of which is K, the number of antikeys (maximal nonkeys) of K is polynomial in the number of attributes iff so is the size of minimal Armstrong relation of s.