Finding Faithful Boyce-Codd Normal Form Decompositions

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.

Boyce–Codd normal form and object normal forms

Some results about normal forms for functional dependency in the relational datamodel

Characterizations for functional dependency and Boyce-Codd normal form families

Several anonymization techniques, such as generalization and bucketization, have been designed for privacy preserving microdata publishing. Recently it has become major importance in research areas and numerous approaches have been proposed to preserve privacy in literature. In order to solve this issue, recent work study the problem of privacy preservation hazard for mined Conditional Functional Dependency (CFD) against (d,l) inference model using Compact frequent pattern growth branch sort algorithm (CFPGBS) methods. Boyce–Codd normal Form (BCNF) method is presented for publishing multiple subtables and it is anonymized through (d, l)- inference model. But the major problem of the work is that initial partitioning is done without considering the importance of the feature, so it contains all quasi identifier features. It becomes more complex to perform initial partitioning if the data samples becomes large,thus may also reduce the privacy results. To overcome this issue in this paper presents a hybrid feature selection method which removes unimportant quasi identifier in the final Boyce–Codd normal Form (BCNF) table. The proposed hybrid feature selection method follows the procedure of hybrid firefly algorithm with harmony search (HFAHS) algorithm to select optimal feature in the table for initial partition of top down approach. For selected features expectation Maximization (EM) based Gaussian mixture distribution (EM-GMM) based top down initial partition is done to group the partition data from the result d,l -inference model along with changed publishing data from BCNF. Experimental results show that the proposed BCNF- d,l -inference model can adeptly anonymize the microdata with less information loss when compared to existing methods against CFD. The effectiveness and privacy results of proposed methods are also compared with existing conventional methods. Our workload experiments confirm that proposed preserves better utility, privacy than d,l -inference model against CFD attack and is more effective in workloads with selection of optimal quasi identifiers.

The design of the logical structure of databases using the entity-relationship (E-R) model is considered. The goal is to focus on E-R conditions for desirable logical design, using an extended translation of the full E-R model into the more easily implementable relational database model; all entity sets in a regular E-R diagram will map to relation schemes that are in the very desirable Boyce-Codd normal form (BCNF). Conditions under which extended E-R relationship sets translate to BCNF schemes are found. These results extend to both types of weak entity sets, subset and dependent entity sets, and contribute to the logical design of robust database systems. An E-R-diagram-based design system currently under development enforces the conditions that ensure BCNF by ensuring extended regularity and other measures of well-behaved E-R models. >

The relational database design approach requires the process of normalization in order to minimize data redundancy and update anomalies in the relational schema. Algorithms defined in normalization theory depend upon various dependencies namely functional, multivalued, join and inclusion dependencies that should be carefully defined for a database application. Identification of these dependencies and a minimal cover is a complex and time consuming task for almost all practical problems. This work discusses how the normalization process can be eliminated from the required steps of database design. It explores various constructs of entity relationship diagram (ERD) and their transformation to relational schema. This work elaborates how un-normalized relations are created during the entity relationship (ER) model to relational schema transformation. A set of rules is presented which if followed at the stage of conceptual modeling would always generate a relational schema that satisfies normal forms up to Boyce-Codd normal form (BCNF), thus eliminating the need for normalization. The motivation behind this paper is to save the database designer's valuable time and effort otherwise required in defining dependencies, in finding a minimal cover and in normalizing a given relational schema.

The problem of designing database schemes satisfying the three design properties lossless join, Boyce-Codd normal form and α-acyclicity is considered. First, it is shown by a sample relation scheme, that these three properties are not always obtainable. This result is independent of any design method. Secondly, the problem of using the BCNF-decomposition algorithm for designing database schemes, which are additionally α-acyclic, is investigated. A characterization of those decomposotion processes yielding α-acyclic, lossless database schemes is given.The main result of the paper is, that for each relation scheme an α-acyclic, lossless BCNF-database scheme can be constructed. This goal is reached by using the concept of lattice compatible augmentation, cf. /KaMa 80/, i.e. the original relation scheme is expanded by additional surrogate attributes (and related functional dependencies). Moreover, this result yields a modified two phase BCNF-decomposition algorithm.

The analysis of a relational database to reduce its component schemes into fourth normal form is not trivial. Decompositions into Boyce–Codd normal form is not difficult, but the concepts and analysis needed to further decompose into fourth normal form are usually difficult to comprehend and use. A diagrammatic representation of dependencies is described which provides an informal aid to the construction of relational schemes in fourth normal form.

9 - Normalization

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 \(O(\frac{{\sqrt {\ln \tfrac{1}{\delta }} }}{\varepsilon }\sqrt {\left| T \right|} )\).

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.

6 - Normalization

In this paper we present a theory of relational database systems based on the partition lattice, which represents a new mathematical approach to the structure of relational database systems. A partition lattice can be defined for any given relation. This partition lattice is shown to be a meet-morphic image of the Boolean algebra of subsets of the attribute set. The partial ordering in the lattice is proved to be equivalent to the concept of functional dependency, and thus Armstrong's axioms for functional dependencies are proved. We solve the problem of finding the list of all keys by seeking the prime implicants of the Boolean function associated with the principal ideals generated by the attributes. We demonstrate the properties of the Boyce-Codd Normal Form (BCNF), and give a modified algorithm for synthesizing an information-lossless BCNF based on the principal filter. The necessary and sufficient conditions for multivalued dependency (MVD) are given in terms of a lattice equation, and the inference rules of MVD are proved. The necessary and sufficient conditions for join dependency (JD) are given; consequently, we can prove the known result that acyclic join dependency (AJD) is equivalent to a set of MVDs. The concept of data independence is introduced, and is extended to conditional independence and mutual independence. We established this algebraic theory of relational databases in the same spirit that the theory of probability was constructed. We present a comparison that demonstrates the similarities.

Managing inconsistency in databases has long been recognized as an important problem. One of the most promising approaches to coping with inconsistency in databases is the framework of database repairs, which has been the topic of an extensive investigation over the past several years. Intuitively, a repair of an inconsistent database is a consistent database that differs from the given inconsistent database in a minimal way. So far, most of the work in this area has addressed the problem of obtaining the consistent answers to a query posed on an inconsistent database. Repair checking is the following decision problem: given two databases r and r′, is r′ a repair of r? Although repair checking is a fundamental algorithmic problem about inconsistent databases, it has not received as much attention as consistent query answering. In this paper, we give a polynomial-time algorithm for subset-repair checking under integrity constraints that are the union of a weakly acyclic set of local-as-view (LAV) tuple-generating dependencies and a set of equality-generating dependencies. This result significantly generalizes earlier work for subset-repair checking when the integrity constraints are the union of an acyclic set of inclusion dependencies and a set of functional dependencies. We also give a polynomial-time algorithm for symmetric-difference repair checking, when the integrity constraints form a weakly acyclic set of LAV tgds. After this, we establish a number of complexity-theoretic results that delineate the boundary between tractability and intractability for the repair-checking problem. Specifically, we show that the aforementioned tractability results are optimal; in particular, subset-repair checking for arbitrary weakly acyclic sets of tuple-generating dependencies is a coNP-complete problem. We also study cardinality-based repairs and show that cardinality-repair checking is coNP-complete for various classes of integrity constraints encountered in database design and data exchange. Copyright 2009 ACM.