Axiomatic Specification of Database Domain Statics

Roelf J. Wieringa

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    In the past ten years, much work has been done to add more structure to database models 1 than what is represented by a mere collection of flat relations (Albano & Cardelli [1985], Albano et al. [1986], Borgida eta. [1984], Brodie [1984], Brodie & Ridjanovic [1984], Brodie & Silva (1982], Codd (1979], Hammer & McLeod (1981], King (1984], King & McLeod [1984], [1985], Mylopoulos et al. [1980], Smith & Smith 1977a & b). 2 The informal approach which most of these studies advocate has a number of disadvantages. First, a recent survey of some of the pro­ posed models by Urban & Delcambre [1986] reveals a wide divergence in terminology and con­ cepts, making comparison of the expressive power of these models difficult. Second, undefined or even ill-defined concepts are a hindrance, not an aid, for the analysis of the Universe of Discourse (UoD). Third, informal treatment 9f such complex structures as set hierarchies, gen­ eralization hierarchies and aggregation hierarchies all in one model, with some dynamics thrown in for good measure, bodes ill for the consistency of these theories. The first goal of the research reported on is to integrate the static structures which these models propose in one coherent, axiomatic framework. It will be shown in chapter 7 that the theory presented here provides the needed conceptual foundations for these models. A second aim is to provide a possible worlds framework onto which to graft theories of the dynamics of the UoD. The third aim is to provide clear concepts which can aid the database model designer in his or her thinking about the UoD. In this report we concentrate on the first goal only, leav­ ing the formulation of theories of domain dynamics and the application to system development as research goals for the near future. The structure of this report is as follows. Chapter 1 provides the necessary context for the theory by defining a four level structure for information systems (IS's). The goal of this and of forthcoming reports can then be stated in terms of this IS structure. Chapter 2 formalizes the concept of object by combining ideas from database theory (sur­ rogates and identities), philosophical logic (identity and rigid designation), axiomatic set theory (the hierarchy of sets) and systems theory (state spaces and state transition functions). Chapter 3 defines attributes and introduces an example database domain. It also draws an important distinction between attributes and operations. I . This is one of the three meanings i n which "model" is used , the other two being that of absrracrion and of model for a jonnal language. See section 7.2 for an explication of these three meanings. 2. In the literature, the term "semantic model" is often used to indicate a model which has one or more of structures li ke generalization and aggregation and grouping. It is not clear to me why the possesion of one or more of these structures makes a model more semantic than others. In whatever sense the word "model" is used, a model has semantics, i.e. it has an intuitive meaning or it serves as an interpretation strucrure for a form al language. A related misnomer is the term "conceptual model" for such models, for there is no model which is not conceptua l. Borgida et al. [19841 motivate the term -conceptual model" by saying that "such a model consists of symbol structures and symbol structure manipulators which. according to a rather naive men talistic philosophy. are supposed to correspond to t he conceptua lizations of the world by human ob­ servers" ( p. 89). Given the fact that human beings often have quite mistaken ideas about what is going on around them , trying to represent these ideas cannot be the goal of the development of an DB. Rather. the DB is built. among other reasons, so that human beings can form ideas about the multitude of events occur­ ring around them which deviate less from the truth than would have been the case without t he aid of an DB. Chapter 4 investigates the lattice structure of the specialization/generalization hierarchy. The concepts of kind and type are defined, and criteria are given for a kind (type) to be natural. The results of chapters 2-4 are distilled in a number of axioms about the UoD, some of which are common to all UoD's and others which are specific to individual UoD's. For the example UoD, these a xioms are summarized in appendix 3. Appendix 4 gives a formal model for these axioms. Chapter 5 defines the concepts of state and combines it with the concept of identity to detine objects. A possible world is then defined as a function which assigns one state to each different object identity. Different possible definitions of object existence are discussed. Chapter 6 treats the set of possible worlds as a set of possible models for static integrity constraints. Worlds which satisfy the static constraints are called admissible. Chapter 7 uses the conceptual apparatus developed in chapter 2-6 to analyze the structures of ,.semantic,. data models like TAXIS, RM/T, SDM and others. Two other models which stand in the relational tradition, the universal relation model and several proposals for non­ first-normal-form models, are investigated as well. Also, a brief comparison with some prob­ lems and their solutions i n philosophical logic, relevant to the specification of data models, is given. Chapter 8, finally, summarizes the main results and lists topics for future research. The appendices contain a list of notational conventions, an overview of ZF, a summary of the axioms for the example UoD used in this report and a description of a formal model for these axioms. Thanks are due to Jan Willem Klop, Reind van de Riet and Hans Weigand for their care­ ful reading of different versions of this report.
    Original languageUndefined
    Place of PublicationAmsterdam
    PublisherVrije Universiteit
    Number of pages103
    Publication statusPublished - Oct 1987

    Publication series

    NameTechnical Report / Faculty of Mathematics and Computer Science
    PublisherFree University, Department of Mathematics and Computer Science


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