DISKIN, Z., CADISH, B.: Databases as graphical algebras: Algebraic graph-based approach to data modeling and database design. Tech. rep., Frame Inform Systems, Riga, Latvia, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....can be actually used in any of these approaches with the advantage of objectiveness and high abstraction that helps to describe the object oriented paradigm in a clear and unified way. In this paper, we will build on the results of the categorical approach to data specifications as presented in [5], 7] 17] 18] 19] and [21] In these works, classes and database objects are basically represented as categorical objects, operations as arrows, generalization and specialization constructions as limits. For our purpose , we will use just the basic notions of CT as presented in [1] The ....

....is not defined directly, but generated by the typegraph, and the instances are functors from this category to an extension category. The category Set (of sets and maps between them) is used only as one possibility to model the actual values stored in the database (this approach is followed also in [5] and [17] Data are represented in the typegraph model in a similar way as in [5] The type hierarchy is built on so called simple types which are the nodes of a directed acyclic graph that represents the inheritance hierarchy. The instances of these types (or classes) in the typegraph are sets. ....

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DISKIN, Z., CADISH, B.: Databases as graphical algebras: Algebraic graph-based approach to data modeling and database design. Tech. rep., Frame Inform Systems, Riga, Latvia, 1996.

....the development of the object oriented database system at the author s department. The main reason for using CT as a formal tool for the object oriented modeling is that it offers a formal, graphical and powerful abstract language for the modeling of objects (this idea is developed especially in [4]) This property gives a new view and helps to see nontrivial properties which would be difficult to discover otherwise (examples can be found in [5, 10, 11, 13] Moreover, all categorical constructions specify objects in terms of their relationship to other objects, and this specification ....

CADIS, B., DISKIN, Z.: Databases as graphical algebras: Algebraic graph-based approach to data modeling and database design. Tech. rep., Frame Inform Systems, Riga, Latvia, 1996.

....CT can be actually used in any of these approaches with the advantage of objectiveness and high abstraction that helps to describe the object oriented paradigm in a clear and unified way. In this paper, we will build on the results of the categorical approach to data specifications as presented in [5], 7] 17] 18] 19] and [21] In these works, classes and database objects are basically represented as categorical objects, operations as arrows, generalization and specialization constructions as limits. For our purpose , we will use just the basic notions of CT as presented in [1] The ....

....not defined directly, but generated by the typegraph, and the instances are functors from this category to an extension category. The category Set (of sets and maps between them) is used only as one possibility to model the actual values stored in the database (this approach is followed also in [5] and [17] Data are represented in the typegraph model in a similar way as in [5] The type hierarchy is built on so called simple types which are the nodes of a directed acyclic graph that represents the inheritance hierarchy. The instances of these types (or classes) in the typegraph are sets. ....

[Article contains additional citation context not shown here]

DISKIN, Z., CADISH, B.: Databases as graphical algebras: Algebraic graph-based approach to data modeling and database design. Tech. rep., Frame Inform Systems, Riga, Latvia, 1996.

....that is required is to bring more mathematical order in the subject, and organize a weakly connected collection of constructs into an integrated consistent framework. At the same time, the DB perspective on categorial concepts can lead to new notions and problems in the very CT (partly outlined in [16]) Thus, in our opinion, the exciting peculiarity of the situation is in extremely short theoretical distance between the abstract nonsense of CT and applied problems of DB theory. Nevertheless, development of this theoretical adjunction into a practically fruitful interaction is a difficult task ....

....discussion by Meseguer in [34] However, standard institutions do not capture the notion of query which is central for the DB theory. Several attempts to incorporate the notion into the institution framework were made but turned out to be not very successful. 6 Another approach was proposed in [8, 13, 16] where query languages are modeled via monads over categories of DB schemas. Indeed, it is well known in the DB theory that queries are operations enjoying some equational constraints, hence query languages, being collections of such operations, give rise to algebraic theories which can be modeled ....

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Z. Diskin and B. Cadish. Databases as graphical algebras: Algebraic graph-based approach to data modeling and database design. Submitted for Algebraic Methodology and Software Technology, AMAST'96 (On ftp: //ftp.cs.chalmers.se/pub/users/diskin/amast96.*), 1995.

....suitable for any data model. One must only define what are schemas and their mappings, and set up the 3 a standard reference suitable for computer science is [2] augmentation mechanism. In particular, the framework can be applied to the sketch data model introduced in [12, 13] and developed in [9, 11] (to make the paper selfcontained, a brief outline of sketches is presented in Appendix A) In the paper we will use sketches to exemplify and demonstrate the essence of our framework but we emphasize once more its suitability for any data model. On the other hand, sketches are important ....

Z. Diskin. Databases as graphical algebras: Algebraic graph-based approach to data modeling and database design. Technical Report 9503, Frame Inform Systems/LDBD, 1995. (On ftp: //ftp.cs.chalmers.se/pub/users/diskin/amast96.*).

....the signature. In other words, we have a unifying graphical mechanism for denoting both basic data and queries against them. Thus, general graph based logic can be developed as a logic of generalized sketches, and corresponding algebras are algebras of diagram operations over sketches ( Dis95d, Dis95b, Dis95c, Dis96a] In addition, if one extends (closes) a given semantic schema with items (nodes and arrows) denoting derived data which can be retrieved from the data instances over the schema, then one will come to a kind of another familiar structure of categorial logic called topos which is ....

.... machinery for such a strange thing are already developed in the mathematical category theory (which was even entitled by abstract nonsense during its early days) So, there emerges an important task of categorial formulating the DB theory conceptual framework (an initial step was made in [Dis95a, Dis95b, Dis96b] Quite briefly, the essence of category theory consists in characterizing internal structures of objects through mappings between them, thus constituting some universe of discourse as a collection of objects nodes, together with a collection of inter object mappings arrows closed ....

[Article contains additional citation context not shown here]

Z. Diskin. Databases as graphical algebras: Algebraic graphbased approach to data modeling and database design. Technical Report 9503, Frame Inform Systems/LDBD, 1995. (On ftp: //ftp.cs.chalmers.se/pub/users/diskin/amast96.*).

....as well. In addition, by designating among all formalizable objects and manipulations those which satisfy one or another special criteria one will get special kinds of toposes. For example, by considering only constructively feasible operations one gets a so called bounded topos described in [9]. A remarkable fact about sketches is that there is a signature of diagram constraints and operations, Sigma 0 , such that for any signature Sigma oe Sigma 0 the category of Sigma sketches, SKE( Sigma) is a bounded topos (see [9] The category of nested relational schemas and their ....

.... operations one gets a so called bounded topos described in [9] A remarkable fact about sketches is that there is a signature of diagram constraints and operations, Sigma 0 , such that for any signature Sigma oe Sigma 0 the category of Sigma sketches, SKE( Sigma) is a bounded topos (see [9]) The category of nested relational schemas and their mappings is also a bounded topos while the category of ordinary relational schemas is not because of the absense of the power object operation. So, for practical applications of definition 2.1.6 it is reasonable to require additionally that ....

[Article contains additional citation context not shown here]

Z. Diskin. Databases as graphical algebras: Algebraic graph-based approach to data modeling and database design. Technical Report 9503, Frame Inform Systems/LDBD, 1995. (On ftp: //ftp.cs.chalmers.se/pub/users/diskin/amast96.*).

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