Charting the completeness frontier of inference systems for multivalued dependencies

The implication of multivalued dependencies in relational databases has originally been defined in the context of some fixed finite universe. While axiomatisability and implication problems have been intensely studied with respect to this notion almost no research has been devoted towards the alternative notion of implication in which the underlying universe of attributes is left undetermined. Based on a set of common inference rules we establish all axiomatisations in undetermined universes, and all axiomatisations in fixed universes that indicate the role of the complementation rule as a means of database normalisation. This characterises the expressiveness of several incomplete sets of inference rules. We also establish relationships between axiomatisations in fixed and undetermined universes, and study the time complexity of the implication problem in undetermined universes. The results of this paper establish a foundation for reasoning about multivalued dependencies without the assumption of a fixed underlying universe.     

Appropriate inferences of data dependencies in relational databases

We study inference systems for the combined class of functional and full hierarchical dependencies in relational databases. Two notions of implication are considered: the original notion in which a dependency is implied by a given set of dependencies and the underlying set of attributes, and the alternative notion in which a dependency is implied by a given set of dependencies alone. The first main result establishes a finite axiomatisation for the original notion of implication which clarifies the role of the complementation rule in the combined setting. In fact, we identify inference systems that are appropriate in the following sense: full hierarchical dependencies can be inferred without use of the complementation rule at all or with a single application of the complementation rule at the final step of the inference only; and functional dependencies can be inferred without any application of the complementation rule. The second main result establishes a finite axiomatisation for the alternative notion of implication. We further show how inferences of full hierarchical dependencies can be simulated by inferences of multivalued dependencies, and vice versa. This enables us to apply both of our main results to the combined class of functional and multivalued dependencies. Furthermore, we establish a novel axiomatisation for the class of non-trivial functional dependencies.     

Complexity of dimension three and some related edge-covering characteristics of graphs

We study systems of inference rules for multivalued dependencies in database relations. For such systems we define a new notion of completeness in which the underlying universe of attributes is left undetermined, whereas the earlier studied concept of completeness refers to a fixed finite universe. We introduce a new inference rule, the subset rule, and using this rule we prove that a certain system is complete. Furthermore we clarify the role of the so-called complementation rule.     

Some results on the deductive structure of join dependencies

Among the many different data dependencies defined, the so-called join dependencies play a central role, since they explicitly capture lossless join properties for relation schemes.In this paper we state some inference rules for join dependencies and embedded join dependencies. A set of two rules is shown to be complete for monadic join dependency inferences (inferences from a single dependency). Furthermore, it is shown that there is no finite set of inference rules that is complete for embedded join dependencies.     

Characterisations of multivalued dependency implication over undetermined universes

In relational databases the original definition of a multivalued dependency is dependent on the underlying relation schema. In this context, the implication of multivalued dependencies has been characterised from multiple perspectives. Logically, it is equivalent to the logical implication of certain material implications in Boolean propositional logic. Proof-theoretically, the Chase procedure offers a convenient tool to decide implication. And algebraically, the implication can be characterised by the notion of closed attribute sets with respect to multivalued dependencies. The assumption of having a fixed underlying relation schema is not always feasible in practice, and also distinguishes multivalued dependencies from other classes of data dependencies. In this paper, we establish logical, proof-theoretical and algebraic characterisations for Biskup?s notion of multivalued dependency implication over undetermined universes. That is, we unburden the current theory of the assumption of having a fixed underlying relation schema. From the perspective of probability theory this means that is unnecessary to fix the set of discrete probabilistic variables in order to utilise conditional independencies.     

On the complementation rule for multivalued dependencies in database relations

Summary We study the interrelation between various versions of the complementation rule and other inference rules for multivalued dependencies in database relations. In particular we settle two open questions of [1] concerning the derivability of inference rules for Boolean operations on the right side of multivalued dependencies. Furthermore we prove that there is a trade-off between the complementation rule and the augmentation rule.     

An algebraic theory of functional and multivalued dependencies in relational databases

Computation of the dependency basis is the fundamental step in solving the membership problem for functional dependencies (FDs) and multivalued dependencies (MVDs) in relational database theory. We examine this problem from an algebraic perspective. We introduce the notion of the inference basis of a set M of MVDs and show that it contains the maximum information about the logical consequences of M. We propose the notion of a dependency-lattice and develop an algebraic characterization of inference basis using simple notions from lattice theory. We also establish several interesting properties of dependency-lattices related to the implication problem. Founded on our characterization, we synthesize efficient algorithms for (a): computing the inference basis of a given set M of MVDs; (b): computing the dependency basis of a given attribute set w.r.t. M; and (c): solving the membership problem for MVDs. We also show that our results naturally extend to incorporate FDs also in a way that enables the solution of the membership problem for both FDs and MVDs put together. We finally show that our algorithms are more efficient than existing ones, when used to solve what we term the ‘generalized membership problem’.     

XML多值依赖及其消除冗余模式的算法

从消除XML文档内数据冗余的角度出发研究了文档的规范化问题.首先引入XML上的数据冗余及其消除处理示例,同时基于函数依赖,提出了规范化的DTD概念和XML DTD 规范化处理规则;其次通过XML多值依赖的定义,给出用于消除冗余模式的算法;最后给出用于XML模式及其消除冗余模式的算法.该算法相应于其他XML模式的研究,在算法产生的层次模式中,完全MVD和嵌入MVD的集合由给出的MVD集合导出;并且产生的XML模式具有消除冗余模式和满足无损连接的特性.     

Losslessness and project-join constructibility in relational databases

Summary Checking a database scheme for the lossless join property with respect to a set, M, of multivalued dependencies (MVDs) is NP-hard. We prove that, for a class of MVDs that includes the set of projected full MVDs, this check can be performed in polynomial time. Even with a lossless database scheme and a consistent database, joining the set of relations in the database can take time and space that is exponential in the size of the relation finally obtained. Joining the set of relations of such a database can be performed in polynomial time if the database scheme is project-join constructible with respect to M. We prove that project-join constructibility, a stricter condition than the lossless join property, can be detected in a database scheme in polynomial time.     

Controlling FD and MVD inferences in multilevel relational databasesystems

The authors investigate the inference problems due to functional dependencies (FD) and multivalued dependencies (MVD) in a multilevel relational database (MDB) with attribute and record classification schemes, respectively. The set of functional dependencies to be taken into account in order to prevent FD-compromises is determined. It is proven that incurring minimum information loss to prevent compromises is an NP-complete problem. An exact algorithm to adjust the attribute levels so that no compromise due to functional dependencies occurs is given. Some necessary and sufficient conditions for MVD-compromises are presented. The set of MVDs to be taken into account for controlling inferences is determined. An algorithm to prevent MVD-compromises in a relation with conflict-free MVDs is given     

Adding inclusion dependencies to an object-oriented data model with uniqueness constraints

We study an object-oriented data model that allows to express both uniqueness constraints and inclusion dependencies as semantic constraints. The data model is based on a subset of F-logic. Uniqueness constraints comprise path functional dependencies which generalise functional dependencies and reflect the navigational power of object-oriented query languages. As inclusion dependencies, we consider explicit class inclusion constraints, besides inclusions required by class hierarchies, and onto constraints that enforce reachability of objects. For these classes of semantic constraints we present an axiomatisation and prove its inference rules to be correct and complete with respect to general logical implication, leaving the decision problem open. The completeness proof combines the known construction for path functional dependencies alone with a possibly infinite model generation process to enforce onto constraints. The results prepare the grounds for normal forms in object-oriented data models and subsequently for computer aided object-oriented database design, following the decomposition approach for the relational data model. Beyond the application for schema design, the achievements could also be exploited for related tasks like semantic query optimisation and mediated data integration within a variety of graph based data models. Received: 11 October 2000 / 27 January 2003     

XML多值依赖的推理规则集问题

XML多值依赖的推理规则集问题是解决XML数据依赖的蕴涵问题的基础,是XML规范化理论的关键问题之一。该文对XML树、树元组等进行了重新定义,与Vincent等人不同,提出了基于DTD的XML多值依赖的概念,通过对XML的关系化表示给出了其形式化定义,定义了XML多值依赖集的闭包、XML多值依赖路径依赖基以及XML多值依赖路径集的闭包等概念,给出了一个有效且完备的推理规则集,并对其有效性及完备性进行了证明。     

Reasoning with truth values on compacted fuzzy chained rules

In this paper, we consider the problem of executing a fuzzy knowledge base (FKB) with rule chaining. The inference process used as starting point is the one based on forward reasoning functions which, obtained from the compositional rule of inference, permits performing the execution of rules in the truth space. This way the process is totally independent from the universes of discourse in which the different variables are defined, allowing a homogeneous treatment for all the variables in the FKB. The execution of the rules is interpreted as the "propagation" of linguistic truth values of the linguistic truth variable that reflect the linguistic degree of fulfillment of each of the propositions in the rules. This execution process is analyzed in two fields of application: control systems, where it is customary to assume t-norm operators as implication functions, and the aggregation process is implemented through the maximum operator and expert systems applications, where other implication functions may be needed and t-norm operators are generally used as aggregation operators. For both of these situations, we present a compaction mechanism which allows a noticeable part of the operations to be performed a priori, thus achieving an important computation time saving.     

XML函数依赖及其推理规则

函数依赖在关系数据库和XML文档中都是一种重要的语义表达．通过分析函数依赖的表现形式在XML文档和关系数据库中的不同之处,提出了基于DTD中的路径表达式的XML函数依赖的概念．它不仅能表达元素的属性和元素的值之间的函数依赖,而且也能表达元素之间的函数依赖．给出了关于XML函数依赖的一组完备的推理规则集,这对解决XML函数依赖的蕴含问题具有重要的意义．     

Mining Short-Rule Covers in Relational Databases

An implication rule Q → R is a statement of the form "for all objects in the database, if an object has the attribute–value pairs Q then it has also the attribute–value pairs R ." This simple type of rule is theoretically interesting, because it supports reasoning, similar to functional dependencies in database theory, and it may be of practical significance because the size of the set of implication rules that hold in a relation can remain substantially high even when mining real data and considering only most general covers; i.e., covers containing rules with unredundant right and left sizes. Motivated by these observations, we focus on the extraction of short-rule covers, which cannot be efficiently mined by standard rule miners. We present an algorithm driven by "negative examples" (i.e., satisfy Q but not R ) to prune the rule-candidate lattice associated with each "positive example" (i.e., satisfies both Q and R ). The algorithm scales up quite well with respect to the number of objects and it is particularly suitable for databases with attributes described by large domains. Furthermore, a perfect hash function ensures extraction of short-rule covers even from databases containing a large number of attributes.     

Another view of functional and multivalued dependencies in the relational database model

In this paper, pseudo-functional and pseudo-multivalued dependencies are introduced. They are shown to be isomorphic with functional and multivalued dependencies, i.e., they behave in the same way with respect to implication. This formalism leads in a very natural way to a rather efficient algorithm for the inference of functional and multivalued dependencies. Some applications to acyclic join dependencies are discussed.     

关于传递闭包和函数依赖完备公理系统存在问题研究

数据依赖在数据库设计中起着十分重要的作用．自Ｃｏｄｄ提出函数依赖（ＦＤｓ）、Ｆａｇｉｎ引入多值依赖（ＭＶＤｓ）后，近几年来人们又根据设计中的需要引入多种新的依赖，如在工程数据库设计中所引进的传递闭包依赖（ＣＤｓ）等．对这些依赖一般是按其是否具有完备的公理系统而划分为两大类，因为完备性公理系统往往具有有效的判定算法为先决条件．本文对ＣＤｓ和ＦＤｓ的ｋ元完备公理系统存在问题进行了研究，证明了ＣＤｓ和ＦＤｓ不具有共同的ｋ元完备公理系统这一结论．     

基于XML Schema的XML强多值依赖的推理规则集

XML强多值依赖的推理规则集问题是解决不完全信息环境下XML数据依赖蕴涵问题的基础,是不完全信息环境下XML模式设计理论的关键问题之一。提出了XML Schema、符合XML Schema的不完全XML文档树等概念;基于子树信息等价和子树信息相容的概念提出了XML强多值依赖的定义及性质;给出了相应的推理规则集,并对其正确性和完备性进行了证明。研究成果为不完全信息环境下存在XSMVD的XML Schema设计奠定了基础。     

Characterizing the MVDs That Hold in a Connected Subtree of a Join Tree: A Correction

Given an acyclic database scheme R and a connected subtree T of a join tree for R, this paper provides a characterization of the set of multivalued dependencies (MVDs) that hold over T with respect to infinR, the corresponding join dependency (JD) of R. This result corrects an error in one of my previous papers "A Comparative Study of Various Nested Normal Forms," which was published in vol. 14, no. 2, pp. 369-385, Mar./Apr. 2002.     

一种基于粗集理论的动态近似规则挖掘推理方法

提出一种基于粗集理论的,把属性的重要性和属性值的出现频率综合起来进行规则推理的方法.分析了“激活一个,否则离开”原则的优缺点,指出在近似推理中,大前提中的规则数量应该可变.给出一种根据推理过程中规则的出现频率决定其是否保留,从而实现规则数量的动态变化的方法,证明了动态变化过程中规则的数量不会无限增加.实例表明此法是比较有效的.