Normal forms and syntactic completeness proofs for functional independencies

Relational model is the most essential database model for decades and is examined in various ways. A very important concept in the relational modeling is the concept of dependencies. Particularly a set of functional dependencies and its equivalent minimal cover are discussed here for synthesizing third normal form relations. A traditional synthesis algorithm have been used to reduce the set of functional dependencies, F, into an equivalent minimal cover. This cover is then used to develop a relational schema; however, a given set of functional dependencies may have more than one minimal covers. In turn, different minimal covers may give rise to different relational schema. Database designers had already been agreed upon the use of popular inference rules of Armstrong to obtain minimal covers from the set F of functional dependencies. Despite the frequent use of the traditional synthesis algorithm, there is still a lack of formality and simplicity support in that algorithm to produce a minimum cover, especially; the step(s) of the algorithm in which the set of functional dependencies of F are used to make left-irreducible remain informal. In this paper, an extension is proposed to the synthesis algorithm that aids in to produce an optimal minimal cover more systematically from the given set of functional dependencies. The optimal minimal cover obtained by this extension would be appropriate for synthesizing third normal form relational schema from the universal relation scheme.

In the conceptual design of relational databases one of the main goals is to create a conceptual scheme, which minimize redundancies and eliminate deletion and addition anomalies, i.e. to create relation schemes in some good normal form. The study of relational databases has produced a host of normal forms: 2NF, 3NF, BCNF, Elementary-Key Normal Form, 4NF, Weak 4NF, PJ/NF, DK/NF, LTKNF, (3,3)NF, etc. There are two features which characterize these normal forms. First, they consider each relation separately. We believe that a normal form (which reflects the goodness of the conceptual design) should be related to the whole conceptual scheme. Second, the usefullness of all normal forms in relational database design have been based on the assumption that a data definition language (DDL) of a database management system (DBMS) is able to enforce key dependencies. However, different DDLs have different capabilities in defining constraints. In this paper we will discuss the design of conceptual relational schemes in general. We will also define a good normal form (GNF) which requires a minimally rich DDL; this normal form is based only on a primitive concept of constraints. We will not, however, discuss the normalization process itself – how one might, if possible, convert a relation scheme that is not in some normal form into a collection of relation schemes each of which is in that normal form.

A key is simple if it consists of a single attribute. It is shown that if a relation schema is in third normal form and every key is simple, then it is in projection-join normal form (sometimes called fifth normal form), the ultimate normal form with respect to projections and joins. Furthermore, it is shown that if a relation schema is in Boyce-Codd normal form and some key is simple, then it is in fourth normal form (but not necessarily projection-join normal form). These results give the database designer simple sufficient conditions, defined in terms of functional dependencies alone, that guarantee that the schema being designed is automatically in higher normal forms.

A new normal form, namely, object-normal form (ONF), is introduced. It is shown that the existing definitions of the fifth normal form (5NF) are unsatisfactory. The correct definition is given for the first time. The importance of the 5NF is demonstrated. For improving data representation in a database with a 5NF schema, the notion of negating relation is introduced. It serves as a basis for the new normal form, namely, the sixth normal form. Combining requirements of the ONF and other normal forms, new normal forms are defined: the fourth object-normal form, the fifth object-normal form, and the sixth object-normal form. It is shown that the standard order of steps of the normalization procedure should be changed: first, the specific requirements of the fourth normal form should be satisfied, and only then the requirements of the second normal form, and so forth.

This paper presents a PAC learning framework for inferring a set of functional dependencies from example tuples. A simple algorithm which outputs a set of all the functional dependencies consistent with example tuples is considered. Let T be a set of the whole tuples. The error is defined as the minimum \(\sum\limits_{t \in V} {p\left( t \right)}\) where V is a set such that a set of functional dependencies is consistent with T−V and p(t) denotes the probability of a tuple t. The number of examples required to infer a set of functional dependencies whose error does not exceed e with probability at least 1−δ under an arbitrary probability distribution is shown to be \(O(\frac{{\sqrt {\ln \tfrac{1}{\delta }} }}{\varepsilon }\sqrt {\left| T \right|} )\).

In this paper, we consider functional dependencies among Boolean dependencies (BDs, for short). Armstrong relations are defined for BDs (called BD-Armstrong relations). For BDs, two necessary and sufficient conditions for the existence of BD-Armstrong relations are given. A necessary and sufficient condition for the existence of Armstrong relations for functional dependencies (FDs, for short) is given, which in some sense is more convenient than the condition given in [3]. We give an algorithm that solves the problem of deciding if two BDs imply the same set of functional dependencies. If the BDs are given in perfect disjunctive normal form, then the algorithm requires only polynomial time. Although Mannila and Raiha have shown that for some relations exponential time is needed for computing any cover of the set of FDs defined in this relation, as a consequence, we show that the problem of deciding if two relations satisfy the same set of FDs can be solved in polynomial time. Another consequence is a new correspondence of the families of functional dependencies to the families of Sperner systems. By this correspondence, the estimate of the number of databases given previously in [6] is improved. It is shown that there is a one-to-one correspondence between the closure of the FDs that hold in a BD and its so-calledbasic cover. As applications of basic covers, we obtain a representation of a key, the family of minimal keys and a representation of canonical covers.

Database normalization is a process which eliminates redundancy, organizes data efficiently and improves data consistency. Functional, multivalued, and join dependencies (FDs, MVDs, and JDs) play fundamental roles in relational databases where they provide semantics for the data and at the same time are the foundations for database design. In this study we investigate the issue of defining functional, multivalued and join dependencies and their normal forms in XML database model. We show that, like relational databases, XML documents may contain redundant information, and this redundancy may cause update anomalies. Furthermore, such problems are caused by certain dependencies among paths in the document. Our goal is to find a way for converting an arbitrary XML Schema to a well-designed one that avoids these problems. We extend the notion of tuple for relational databases to the XML model. We show that an XML tree can be represented as a set of tree tuples. We introduce the definitions of FD, MVD, and JD and new Normal Forms of XML Schema that based on these dependencies (X-1NF, X-2NF, X-3NF, X-BCNF, X-4NF, and X-5NF). We show that our proposed normal forms are necessary and sufficient to ensure all conforming XML documents have no redundancies.

The traditional relational database model may be extended into a fuzzy database model based on the mathematical framework of fuzzy set theory to process imprecise or uncertain information. While designing such a fuzzy relational database model that does not suffer from data redundancy and anomalies, the present authors have defined several fuzzy normal forms in ref [1]. However, as one decomposes an unnormalized relation into a desired normal form, it should also satisfy the essential properties of dependency preservation and lossless join of relation schemes which take a significant role in the design theory of a relational database. We have concentrated on these two important issues in this paper and have designed algorithms that confirm dependency preservation and lossless join decomposition of an unnormalized relation into the fuzzy third normal form or fuzzy Boyce Codd normal form. The algorithms have been tested and validated with examples.

An Equivalence between Dependencies in Nested Databases and a Fragment of Propositional Logic

Characterizations for functional dependency and Boyce-Codd normal form families

It is well known that faithful (i.e. dependency preserving) decompositions of relational database schemas into Boyce-Codd Normal Form (BCNF) do not always exist, depending on the set of functional dependencies given, and that the corresponding decision problem is NP-hard. The only algorithm to guarantee both faithfulness and BCNF (if possible) proposed so far in [Os79] is a brute-force approach which always requires exponential time. To be useful in practice, e.g. in automated design tools, we require more efficient means.In this paper we present an algorithm which always finds a faithful BCNF decomposition if one exists, and which is usually efficient, and exponential only in notorious cases.

This paper is about designing a complete interactive tool, named JMathNorm, for relational database (RDB) normalization using Mathematica. It is an extension of the prototype developed by the same authors [1] with the inclusion of Second Normal Form (2NF), and Boyce-Codd Normal Form (BCNF) in addition to the existing Third normal Form (3NF) module. The tool developed in this study is complete and can be used for real-time database design as well as an aid in teaching fundamental concepts of DB normalization to students with limited mathematical background. JMathNorm also supports interactive use of modules for experimenting the fundamental set operations such as closure, and full closure together with modules to obtain the minimal cover of the functional dependency set and testing an attribute for a candidate key. JMathNorm's GUI interface is written in Java and utilizes Mathematica's JLink facility to drive the Mathematica kernel.

We propose a normal form for nested relations, called NF-NR, which removes undesirable anomalies from a nested relational database schema. Both functional dependencies and multivalued dependencies are considered. NF-NR reduces to 3NF/4NF if the nested relation considered is actually a flat relation. Especially, NF-NR removes global redundancies among a set of nested relations. Two approaches to NF-NR database design, namely the restructuring rules approach and the ER approach, are discussed. We relate NF-NR to ER-NF, a normal form of ER defined earlier, by defining a simple mapping from an ERD in ER-NF to a set of nested relations in NF-NR. This approach effectively removes ambiguitics and redundancies on a semantic level and hence gives a set of nested relations with clean semantics and yet in good normal form. A set of desirable properties for any normal form for nested relations are described and an evaluation of several existing normal forms is given based on this set of properties. The evaluation shows that NF-NR improves over previously proposed normal forms in various aspects and is a more practical normal form for nested relations.

There is an established equivalence between relational database Functional Dependencies (FDs) and a fragment of switching algebra that is built typically of Horn clauses. This equivalence pertains to both concepts and procedures of the FD relational database domain and the switching algebraic domain. This study is an exposition of the use of switching-algebraic tools in solving problems typically encountered in the analysis and design of relational databases. The switching-algebraic tools utilized include purely-algebraic techniques, purely-visual techniques employing the Karnaugh map and intermediary techniques employing the variable-entered Karnaugh map. The problems handled include; (a) the derivation of the closure of a Dependency Set (DS), (b) the derivation of the closure of a set of attributes, (c) the determination of all candidate keys and (d) the derivation of irredundant dependency sets equivalent to a given DS and consequently the determination of the minimal cover of such a set. A relatively large example illustrates the switching-algebraic approach and demonstrates its pedagogical and computational merits over the traditional approach.

Vague relational database models generalize the classical relational database model by allowing uncertain and imprecise information by means of vague set theory. In this paper, we have introduced vague extensions of the normal forms for a similarity based vague relational database model. Firstly, we have designed an algorithm to find the vague closure of an attribute set which can be utilized to find vague key. Next, with the concepts of α-vfd and partial α-vfd as introduced in [1], we have defined different vague normal forms in a vague relational database. We have also presented examples based on real life application to demonstrate how normalization based on α-vfds of vague relation may be done.KeywordsVague setsimilarity measure of vague sets α-vfd partial α-vfd vague keyvague closure of attribute setvague normal forms