An Infornation-Theoretic Analysis of Relational Databases—Part I: Data Dependencies and Information Metric

Database design is based on the concept of data dependency, which is the interrelationship between data contained in various sets of attributes. In particular, functional, multivalued and acyclic join, dependencies play an essential role in the design of database schemas. The basic definition of an information metric and how this notion can be used in relational database are discussed in this paper. We use Shannon entropy as an information metric to quantify the information associated with a set of attributes. Thus, we prove that data dependencies can be formulated in terms of entropies. These formulas make the numerical computation and testing of data dependencies feasible. Among the different types of data dependencies, the acyclic join dependency is most important to the design of a relational database schema. The acyclic join dependency, with multivalued dependency as a special case, impose a constraint on the information-preserving decomposition of a relation. It is interesting that this constraint on a relation is similar to Gibbs' condition for separating physical systems in statistical mechanics. They both assert that entropy is preserved during the decomposition process. That is, the entropies of the corresponding set of attributes must satisfy the inclusion–exclusion identity.     

Relational database schema design for uncertain data

Driven by the dominance of the relational model, we investigate how the requirements of applications on the certainty of functional dependencies can improve the outcomes of relational database schema design. For that purpose, we assume that tuples are assigned a degree of possibility with which they occur in a relation, and that functional dependencies are assigned a dual degree of certainty which says to which tuples they apply. A design theory is developed for functional dependencies with degrees of certainty, including efficient axiomatic and algorithmic characterizations of their implication problem. Naturally, the possibility degrees of tuples bring forward different degrees of data redundancy, caused by functional dependencies with the dual degree of certainty. Variants of the classical syntactic Boyce–Codd and Third Normal Forms are established. They are justified semantically in terms of eliminating data redundancy and update anomalies of given degrees, and minimizing data redundancy of given degrees across all dependency-preserving decompositions, respectively. As a practical outcome of our results, designers can simply fix the degree of certainty they target, and then apply classical decomposition and synthesis to the set of functional dependencies whose associated degree of certainty meets the target. Hence, by fixing the certainty degree a designer controls which integrity requirements will be enforced for the application and which data will be processed by the application. The choice of the certainty degree also balances the classical trade-off between query and update efficiency on future database instances. Our experiments confirm the effectiveness of our control parameter, and provide original insight into classical normalization strategies and their implementations.     

XML DTD的一种范式

研究了XML DTD的规范化问题。由于DTD在设计上存在不足之处，DTD中可能包含类似于关系数据库模式中存在的异常依赖，从而导致XML文档包含冗余的数据和各种操作异常。提出了关于DTD的多值依赖的概念，然后基于多值依赖的概念，提出了一种XML的范式XNF，并且用DTD的关系表示形式给出了关于DTD的无损联接分解的概念。最后给出了把DTD无损联接地分解成XNF的一种算法。     

NF-NR: A practical normal form for nested relations

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.     

具有多时间粒度的时态多值依赖及时态模式分解方法研究

一个好的数据库逻辑设计目标是消除数据冗余以及插入、删除和更新异常.对于时态数据库,通过具有多时间粒度的时态函数依赖约束对时态数据库进行规范化已有大量研究.基于时态函数依赖和多值依赖理论提出了多时间粒度约束的时态多值依赖(TMVD)等概念,并给出了时态多值依赖的推理规则,对其有效性、完备性进行了证明.由于包含有限个TMVD的TMVD集通常蕴含着无限个TMVD,给出了TMVD的有限推理规则,对其有效性、完备性进行了证明.最后,基于时态多值依赖集提出了时态第四范式,并给出了时态模式的T4NF的无损分解算法,对算法的可终止性、正确性进行了证明, 并对时间复杂度进行了分析.     

具有无损联结性且保持依赖性关系模式的BCNF完备分解算法

本文给出一种具有完备性的合成方法，来把一个关系模式分解成具有无损性和保持依赖性的ＢＣ范式关系数据库模式，只要这个模式“本质上能作这种分解的话”；同时对这种“本质上能分解为保持某些性质的某一范式”提法进行了形式化描述；最后，讨论了这种合成法的固有复杂度．     

An Information-Theoretic Analysis of Relational Databases—Part II: Information Structures of Database Schemas

The structural properties of acyclic database schemas, especially the interrelationships between acyclic join dependencies and multivalued dependencies, are examined in this paper. The intersection closure of a database schema is a semilattice, which is usually represented by Hasse diagram in algebra. The Hasse diagram of a schema is analogous to the Bachman diagram of a network model. Based on the topological structures of Hasse diagrams, the acyclic join dependency is proved to be equivalent to a set of conflict-free multivalued dependencies. Furthermore, we show that if the Hasse diagram is loop free, then the corresponding set of multivalued dependencies is also contention-free. This result is consistent with previous results derived from the Bachman diagram.     

Normalization and lossless join decomposition of similarity‐based fuzzy relational databases

Fuzzy relational database models generalize the classical relational database model by allowing uncertain and imprecise information to be represented and manipulated. In this article, we introduce fuzzy extensions of the normal forms for the similarity‐based fuzzy relational database model. Within this framework of fuzzy data representation, similarity, conformance of tuples, the concept of fuzzy functional dependencies, and partial fuzzy functional dependencies are utilized to define the fuzzy key notion, transitive closures, and the fuzzy normal forms. Algorithms for dependency preserving and lossless join decompositions of fuzzy relations are also given. We include examples to show how normalization, dependency preserving, and lossless join decomposition based on the fuzzy functional dependencies of fuzzy relation are done and applied to some real‐life applications. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 885–917, 2004.     

On interpreting and inferring propositional formulas of data dependencies in a relational database

This paper concerns generally the satisfaction and the inference problem involving functional and/or multivalued dependencies in a relational database. In particular, two independent aids in solving an inference problem, concerning the logical counterparts of functional as well as multivalued dependencies, are introduced. The first aid is provided by establishing a pair of complementary inequivalence and equivalence theorems between the propositional formula corresponding to the difference, U-X, in set theory and the propositional formula not(X) where U is a relation scheme and X is a subset of U. By applying these theorems, correctness of solving an inference problem is assured. The second aid is the application of a Venn diagram for simplifying a propositional formula involving conjunctions, differences, etc., for solving an inference problem. A guideline for constructing simplified Venn diagrams is also given and discussed.     

样本依赖与连接依赖的等价性

以抽象代数为工具,探索了全样本依赖与全连接依赖之间的关系.首先,分别在全样本依赖集和全连接依赖集上建立等价关系,这两种等价关系都将作用相同的依赖视为等价依赖. 然后证明了在这两个等价关系下的商集分别构成么半群,并且这两个么半群是同构的.这就等于证明了全样本依赖类本质上等同于全连接依赖类.最后给出了一个关于全无环连接依赖的有趣结果.有关结果可以在关系数据库的设计中发挥积极作用.     

用于XML模式和DTD规范化设计的层次模式设计

XML模式和DTD(document type definition)规范化设计是给出一个很好地表示数据间依赖关系并消除了冗余的XML模式或DTD的集合.目前在这一方面开展的研究还不多,而且才刚起步.Provost提出将关系数据库理论应用于XML模式规范化设计的思想,这一思想还没有付诸实施.在Provost思想的基础上给出用于XML模式和DTD规范化设计的层次模式设计的算法.首先分析了基于Provost思想的层次分解;然后给出用于消除冗余模式的分解树设计算法;最后给出用于XML模式和DTD规范化设计的层次     

Learning Sets of Antecedent-restricted Functional and Multivalued Dependencies with Queries

Identifying dependencies that hold in relational databases is essential to produce good databases designs. In particular, functional and multivalued dependencies are used to obtain relation schemes that satisfy the 4th normal form, a property that is considered satisfactory for most applications. It is known that the class of sets of functional dependencies is learnable in the exact model of learning with queries. Also a subclass of multivalued dependencies, the class of consequent-restricted multivalued dependencies, has been shown to be learnable in this model. In this paper, we present an algorithm that learns a class that contains sets of both functional and multivalued dependencies under some restrictions imposed on the antecedents of dependencies. We also show, as a by-product, an algorithm that learns a non-trivial subclass of 2-quasi Horn formulas, closely related to the class just mentioned.     

不完全信息环境下存在XML强多值依赖的XML文档规范化研究

不完全信息环境下XML文档中的数据存在多值依赖时,为了避免在没有约束条件下XML文档数据出现冗余及更新异常,引入XML强多值依赖的概念和理论对XML文档的规范化进行了系统研究.基于节点信息等价、节点信息相容的概念给出了XML强多值依赖的定义;基于层次化的XML强多值依赖,提出了不完全XML文档树满足XML强多值依赖范式的条件;给出了满足该条件的不完全XML文档树无数据冗余的判定定理;提出了不完全XML文档树的规范化算法,对其时间复杂性进行了分析.理论研究和实例分析表明：研究成果较好地解决了在不完全信息环境下XML文档中存在层次化的XML强多值依赖引起的数据冗余问题.     

The Relational Structure of Belief Networks

This paper demonstrates the relational structure of belief networks by establishing an extended relational data model which can be applied to both belief networks and relational applications. It is demonstrated that a Markov network can be represented as a generalized acyclic join dependency (GAJD) which is equivalent to a set of conflict-free generalized multivalued dependencies (GMVDs). A Markov network can also be characterized by an entropy function, which greatly facilitates the manipulation of GMVDs. These results are extensions of results established in relational theory. It is shown that there exists a complete set of inference rules for the GMVDs. This result is important from a probabilistic perspective. All the above results explicitly demonstrate that there is a unified model for relational database and probabilistic reasoning systems. This is not only important from a theoretical point of view in that one model has been developed for a number of domains, but also from a practical point of view in that one system can be implemented for both domains. This implemented system can take advantage of the performance enhancing techniques developed in both fields. Thereby, this paper serves as a theoretical foundation for harmonizing these two important information domains.     

Negative results on learning multivalued dependencies with queries

Data dependencies are useful to design relational databases. There is a strong connection between dependencies and some fragments of the propositional logic. In particular, functional dependencies are closely related to Horn formulas. Also, multivalued dependencies are characterized in terms of multivalued formulas. It is known that both Horn formulas and sets of functional dependencies are learnable in the exact model of learning with queries. Here we proof that neither multivalued formulas nor multivalued dependencies can be learned using only membership queries or only equivalence queries.     

Full hierarchical dependencies in fixed and undetermined universes

Full hierarchical dependencies (FHDs) constitute a large class of relational dependencies. A relation exhibits an FHD precisely when it is the natural join over at least two of its projections that all share the same join attributes. Therefore, FHDs generalise multivalued dependencies (MVDs) in which case the number of these projections is precisely two. The implication of FHDs has originally been defined in the context of some fixed finite universe. This paper identifies a sound and complete set of inference rules for the implication of FHDs. This axiomatisation is very reminiscent of that for MVDs. Then, an alternative notion of FHD implication is introduced in which the underlying set of attributes is left undetermined. The first main result establishes a finite axiomatisation for FHD implication in undetermined universes. It is then formally clarified that the complementation rule is only a mere means for database normalisation. In fact, the second main result establishes a finite axiomatisation for FHD implication in fixed universes which allows to infer FHDs either without using the complementation rule at all or only in the very last step of the inference. This also characterises the expressiveness of an incomplete set of inference rules in fixed universes. The results extend previous work on MVDs by Biskup.     

Functional and multivalued dependencies in nested databases generated by record and list constructor

The impact of the list constructor on two important classes of relational dependencies is investigated. Lists represent an inevitable data structure whenever order matters and data is allowed to occur repeatedly. The list constructor is therefore supported by many advanced data models such as genomic sequence, deductive and object-oriented data models including XML. The article proposes finite axiomatisations of functional, multivalued and both functional and multivalued dependencies in nested databases supporting record and list constructor. In order to capture different data models at a time, an abstract algebraic approach based on nested attributes is taken. The presence of the list constructor calls for a new inference rule which allows to infer non-trivial functional dependencies from multivalued dependencies. Further differences to the relational theory become apparent when the independence of the inference rules is investigated. The extension of the relational theory to nested databases allows to specify more real-world constraints and increases therefore the number of application domains. * Results of this article were partly presented at the 3rd International Symposium on Foundations of Information and Knowledge Systems (FoIKS), Vienna, Austria, 2004 [58], and the 23rd International Symposium on Principles of Database Systems (PODS), Paris, France, 2004 [54]. ** Sebastian Link was supported by Marsden Funding, Royal Society of New Zealand.     

Constructing the Bayesian network structure from dependencies implied in multiple relational schemas

Relational models are the most common representation of structured data, and acyclic database theory is important in relational databases. In this paper, we propose the method for constructing the Bayesian network structure from dependencies implied in multiple relational schemas. Based on the acyclic database theory and its relationships with probabilistic networks, we are to construct the Bayesian network structure starting from implied independence information instead of mining database instances. We first give the method to find the maximum harmoniousness subset for the multi-valued dependencies on an acyclic schema, and thus the most information of conditional independencies can be retained. Further, aiming at multi-relational environments, we discuss the properties of join graphs of multiple 3NF database schemas, and thus the dependencies between separate relational schemas can be obtained. In addition, on the given cyclic join dependency, the transformation from cyclic to acyclic database schemas is proposed by virtue of finding a minimal acyclic augmentation. An applied example shows that our proposed methods are feasible.     

On characterizing boundedness of database schemes with bounded dependencies

We investigate the effect of bounded dependencies on the boundedness of database schemes. The following results are proved. A database scheme with only bounded equality-generating dependencies is always bounded with respect to dependencies; a lossless database scheme with bounded full implicational dependencies is bounded w.r.t. dependencies if and only if the implicational dependencies are equivalent to a single join dependency and some equality-generating dependencies. By a known method, this condition can be tested effectively. These results are relevant in database theory in that they determine in a rather general case whether queries under the representative instance approach can be expressed in relational algebra.     

对象类视图中蕴涵的依赖关系

本文研究关系模型中对象类视图关系模式的数据依赖问题。关系数据库中用一组表存放一个对象类的数据,并用视频实现对象类。本文分析该视图的列之间存在的函数依赖和多值依赖,并证明了类的所有单值和多值属性组成对象标识的多值依赖基。