Equivalence and Normal Forms for the Restricted and Bounded Fixpoint in the Nested Algebra

The nested model is an extension of the traditional, “flat” relational model in which relations can also have relation-valued entries. Its “default” query language, the nested algebra, is rather weak, unfortunately, since it is only a conservative extension of the traditional, flat relational algebra, and thus can express only a small fraction of the polynomial-time queries. Therefore, it was proposed to extend the nested algebra with a fixpoint construct, but the resulting language turned out to be too powerful: many inherently exponential queries could also be expressed. Two polynomial-time restrictions of the fixpoint closure of the nested algebra were proposed: the restricted fixpoint closure (by Gyssens and Van Gucht) and the bounded fixpoint closure (by Suciu). Here, we prove two results. First we show that both restrictions are equivalent in expressive power. The proof technique relies on known encodings of nested relations into flat ones, and on a novel technique, called type substitution, by which we reduce the equivalence of the two restrictions to its obvious counterpart in the flat relational model. Second we prove that both the bounded fixpoint queries and the restricted fixpoint queries admit normal forms, in which the fixpoint occurs exactly once. The proof technique relies on a novel encoding method of nested relations into flat ones.     

基于嵌套语义的约束代数

1 引言约束数据库近期被Kanellakis等提出作为处理空间数据的一般性框架。约束数据库用约束来建模和检索数据。在数据层,约束能用有限的形式来表示可能是无限的关系元组集。例如,约束x~2 y~2≤9表示中心在点(0,0)处,半径为3的圆。在查询语言层,约束通过允许数学计算而增强了简单关系语言的表达能力,同时约束查询语言保留了关系查询语言的所有特征,如封闭性和自底向上求值。关系代数能被扩充来处理约束关系,这个新的代数叫做约束代数CALG。     

The Relationlog system prototype

The Relationlog system is a novel persistent deductive database system for advanced data and knowledge‐based applications. It directly supports the storage and inference of data with complex structures, especially data supported in nested relational and complex‐object models. The Relationlog system supports the Relationlog query language, which is a typed extension of Datalog with tuples and sets and stands in the same relationship to the nested relational and complex‐object models as Datalog stands to the relational model. It also supports an SQL‐like data definition language and a declarative data manipulation language. This article introduces the Relationlog language, discusses the system architecture, the design decisions incorporated within its implementation, and our experience in developing the system. Copyright © 2001 John Wiley & Sons, Ltd.     

Temporal relational data model

This paper incorporates a temporal dimension to nested relations. It combines research in temporal databases and nested relations for managing the temporal data in nontraditional database applications. A temporal data value is represented as a temporal atom; a temporal atom consists of two parts: a temporal set and a value. The temporal atom asserts that the value is valid over the time duration represented by its temporal set. The data model allows relations with arbitrary levels of nesting and can represent the histories of objects and their relationships. Temporal relational algebra and calculus languages are formulated and their equivalence is proved. Temporal relational algebra includes operations to manipulate temporal data and to restructure nested temporal relations. Additionally, we define operations to generate a power set of a relation, a set membership test, and a set inclusion test, which are all derived from the other operations of temporal relational algebra. To obtain a concise representation of temporal data (temporal reduction), collapsed versions of the set-theoretic operations are defined. Procedures to express collapsed operations by the regular operations of temporal relational algebra are included. The paper also develops procedures to completely flatten a nested temporal relation into an equivalent 1 NF relation and back to its original form, thus providing a basis for the semantics of the collapsed operations by the traditional operations on 1 NF relations     

An extended algebra for constraint databases

Constraint relational databases use constraints to both model and query data. A constraint relation contains a finite set of generalized tuples. Each generalized tuple is represented by a conjunction of constraints on a given logical theory and, depending on the logical theory and the specific conjunction of constraints, it may possibly represent an infinite set of relational tuples. For their characteristics, constraint databases are well suited to model multidimensional and structured data, like spatial and temporal data. The definition of an algebra for constraint relational databases is important in order to make constraint databases a practical technology. We extend the previously defined constraint algebra (called generalized relational algebra). First, we show that the relational model is not the only possible semantic reference model for constraint relational databases and we show how constraint relations can be interpreted under the nested relational model. Then, we introduce two distinct classes of constraint algebras, one based on the relational algebra, and one based on the nested relational algebra, and we present an algebra of the latter type. The algebra is proved equivalent to the generalized relational algebra when input relations are modified by introducing generalized tuple identifiers. However, from a user point of view, it is more suitable. Thus, the difference existing between such algebras is similar to the difference existing between the relational algebra and the nested relational algebra, dealing with only one level of nesting. We also show how external functions can be added to the proposed algebra     

A frame-based knowledge representation model and its mapping to nested relations

The first part of this paper introduces the basic constructs of a frame representation model and gives a formal definition for them. Subsequently the nested relational model (NF²) is described as an extension of the classical relational model to support relation-valued attributes. In the second part of this paper both models are compared with each other and a mapping of frame representation structures to NF² structures is specified. The structural similarities between frames and NF² relations are made clear and it is shown that their main difference is due to the type polymorphism introduced by the concept hierarchy of the frame model. This causes type collisions to occur in the strictly typed NF² model when frames are directly mapped to NF² structures. Two solutions to this problem are suggested and compared. The paper concludes with a reformulation of query operations of the frame model in terms of NF² algebra operations.     

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.     

微机关系数据库的递归查询接口

为满足大型企业中产品零部件数据管理的需要，解决关系数据库所不能解决的递归查询问题，为微机关系数据库设计并实现了一个递归查询接口，该接口能够计算了Ｄａｔａｌｏｇ逻辑程序，通过允许规划头部包含函数符号，使规则增加了数值计算能力。     

Typed query languages for databases containing queries

This paper introduces and studies the relational meta algebra, a statically typed extension of the relational algebra to allow for meta programming in databases. In this meta algebra one can manipulate database relations involving not only stored data values (as in classical relational databases) but also stored relational algebra expressions. Topics discussed include modeling of advanced database applications involving “procedural data” ; desirability as well as limitations of a strict typing discipline in this context; equivalence with a first-order calculus; and global expressive power and non-redundancy of the proposed formalism.     

基于嵌入式SQL的Datalog演绎规则解释器的设计

文章提出了一个建立在传统关系数据库基础上的能支持ANSISQL与嵌入式SQL的演绎规则解释器。利用这个解释器,用户能够定义一个蕴含关系并可以像在演绎数据库中使用Datalog规则一样来提出查询。其方法是把演绎规则和查询翻译成嵌入式SQL程序,该程序在执行查询时能被调用。这个解释器可以被认为是扩充RDBMS演绎查询功能的一个前端工具。     

Near-optimum storage models for nested relations based on workloadinformation

The problem of choosing a storage model for a nested relation (i.e., a relation containing relations) is considered. A technique is introduced that uses the workload information of the database system under consideration to obtain a better storage model (i.e., one with a lower query cost) for a given nested relation. The nested relation scheme is first represented as a tree called the scheme tree. By using the workload information and by performing a series of merges in the nodes of the scheme tree, a near-optimum scheme tree is produced, and file organization types are assigned to each node (file) in the scheme tree. The authors' methodology is applied by using a specific nested relational algebra and three file organization types, namely, sequential, heap, and dense index files. The proposed methodology locates the optimum storage model and the optimum file organization techniques for the external and internal relations of the nested relations tested     

On the expressive power of the relational algebra with partially ordered domains

Assuming data domains are partially ordered, we define the partially ordered relational algebra (PORA) by allowing the ordering predicate ? to be used in formulae of the selection operator σ. We apply Paredaens and Bancilhon's Theorem to examine the expressiveness of the PORA, and show that the PORA expresses exactly the set of all possible relations which are invariant under order-preserving automorphisms of databases. The extension is consistent with the two important extreme cases of unordered and linearly ordered domains. We also investigate the three hierarchies of: (1) computable queries, (2) query languages and (3) partially ordered domains, and show that there is a one-to-one correspondence between them.     

Unifying Concurrent and Relational Refinement

Refinement in a concurrent context, as typified by a process algebra, takes a number of different forms depending on what is considered observable, where observations record, for example, which events a system is prepared to accept or refuse. Examples of concurrent refinement relations include trace refinement, failures-divergences refinement and bisimulation.Refinement in a state-based language such as Z, on the other hand, is defined using a relational model in terms of input/output behaviour of abstract programs. These refinements are verified by using two simulation rules which help make the verification tractable.The purpose of this paper is to unify these two standpoints, and we do so by generalising the standard relational model to include additional observable aspects. The central result of the paper is then to develop simulation rules to verify relations such as failures-divergences refinement in a relational setting, and show how these might be applied in a specification language such as Z.     

含有全部值语义的SQL 查询

SQL语句在表达具有全部值语义时需要使用谓词NOT EXISTS构建复杂的多层嵌套结构,学习者学习起来难度大,易出错。元组关系演算可以充分表达全部值语义,通过应用实例探讨了元组关系演算表达式转换为SQL语句的转换规则,对学习SQL语句者非常有帮助。     

Relational Concurrent Refinement

Refinement in a concurrent context, as typified by a process algebra, takes a number of different forms depending on what is considered observable. Observations record, for example, which events a system is prepared to accept or refuse. Concurrent refinement relations include trace refinement, failures–divergences refinement, readiness refinement and bisimulation. Refinement in a state-based language such as Z, on the other hand, is defined using a relational model in terms of the input–output behaviour of abstract programs. These refinements are normally verified by using two simulation rules which help make the verification tractable. This paper unifies these two standpoints by generalising the standard relational model to include additional observable aspects. These are chosen in such a way that they represent exactly the notions of observation embedded in the various concurrent refinement relations. As a consequence, simulation rules for the tractable verification of concurrent refinement can be derived. We develop such simulation rules for failures–divergences refinement and readiness refinement in particular, using an alternative relational model in the latter case.     

A generalised temporal algebra

The algebra (TNA) of a generalised temporal database model supporting temporal relations nested to any finite depth is presented. The temporal nested relations consist of temporal nested attributes which are formed from temporal attributes together with the corresponding time-varying attributes. Therefore, the temporal dimension of the model is nested and is not integral with the corresponding time-dependent value. All the operations of the algebra are defined recursively and are proved to be closed. In particular, considering the natural join operation for temporal nested relations, different cases are presented, distinguished by the types and the nesting levels of the common attributes that participate in the natural join operation.     

The expressive power of temporal relational query languages

The authors consider the representation of temporal data based on tuple and attribute timestamping. They identify the requirements in modeling temporal data and elaborate on their implications in the expressive power of temporal query languages. They introduce a temporal relational data model where N1NF relations and attribute timestamping are used and one level of nesting is allowed. For this model, a nested relational tuple calculus (NTC) is defined. They follow a comparative approach in evaluating the expressive power of temporal query languages, using NTC as a metric and comparing it with the existing temporal query languages. They prove that NTC subsumes the expressive power of these query languages. They also demonstrate how various temporal relational models can be obtained from the temporal relations by NTC and give equivalent NTC expressions for their languages. Furthermore, they show the equivalence of intervals and temporal elements (sets) as timestamps in their model     

A Reasoning Mechanism for DeductiveObject-Oriented Databases

Based on the approach implementing a deductive object-oriented database system through the underlying relational database,this paper presents and object reasoning language O-Datalog,which is the extension of Datalog in form and can deal with object-oriented data.For any O-Datalog program,an dquivalent Datalog program can be built to help evaluate the original program.This paper focuses on the syntax,semantics and evaluation of O-Datalog.     

Extending the power of datalog recursion

Supporting aggregates in recursive logic rules represents a very important problem for Datalog. To solve this problem, we propose a simple extension, called Datalog $^{FS}\,$ (Datalog extended with frequency support goals), that supports queries and reasoning about the number of distinct variable assignments satisfying given goals, or conjunctions of goals, in rules. This monotonic extension greatly enhances the power of Datalog, while preserving (i) its declarative semantics and  (ii) its amenability to efficient implementation via differential fixpoint and other optimization techniques presented in the paper. Thus, Datalog $^{FS}\,$ enables the efficient formulation of queries that could not be expressed efficiently or could not be expressed at all in Datalog with stratified negation and aggregates. In fact, using a generalized notion of multiplicity called frequency, we show that diffusion models and page rank computations can be easily expressed and efficiently implemented using Datalog $^{FS}\,$ .     

A data model and algebra for probabilistic complex values

We present a probabilistic data model for complex values. More precisely, we introduce probabilistic complex value relations, which combine the concept of probabilistic relations with the idea of complex values in a uniform framework. We elaborate a model-theoretic definition of probabilistic combination strategies, which has a rigorous foundation on probability theory. We then define an algebra for querying database instances, which comprises the operations of selection, projection, renaming, join, Cartesian product, union, intersection, and difference. We prove that our data model and algebra for probabilistic complex values generalizes the classical relational data model and algebra. Moreover, we show that under certain assumptions, all our algebraic operations are tractable. We finally show that most of the query equivalences of classical relational algebra carry over to our algebra on probabilistic complex value relations. Hence, query optimization techniques for classical relational algebra can easily be applied to optimize queries on probabilistic complex value relations.