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.

Boyce–Codd normal form and object normal forms

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

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.

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.

9 - Normalization

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.

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

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. >

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.

On learning multivalued dependencies with queries

Let R be a γ-acyclic relational scheme, and let F be the set of functional dependencies (FD's) embodied in R. Given an existence constrained database r over R, it was shown in [1] that it is possible to connect tuples from different relations in r and construct a universal instance L, possibly containing null values δ, such that the total projection of L onto R yields exactly the set r. Moreover, conditions were given which guarantee that this L would satisfy the functional dependency with nulls (NFD) counterparts of FD's, in F. The purpose of this note is to generalize the latter result and show that under the same conditions, L actually satisfies NFD counterparts of FD's in the closure F+ of F.

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.

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.

Normal forms and syntactic completeness proofs for functional independencies