The algorithm for the integration of classes/concepts which results in an ontology suitable for the preservation of database classes/concepts is described. It has been designed by means of original research at the boundary between mathematics, computer science and concept theory. The input conceptual graph of this algorithm is determined as a solution of a data mining problem, which has been approached as inverse mapping of conceptual database design. Similarities and differences are discussed between an algorithm formalized in concept theory, which is not suitable for implementation, and the algorithm for the integration of classes/concepts, which exploits the benefits of database notations.
TopIntroduction
In concept theory (Kauppi, 1997; Palomäki, 1994), it is possible to make a distinction between the intensional/concept level, which is the level of human thinking, and the extensional/set-theoretical level, which is the level of computer science. At the intensional level, concepts are dealt with, whereas at the extensional level, set of objects/classes are considered (Locuratolo & Palomäki, 2008). The world of concepts is large and yet unexplored; on the contrary, the world of object classes supported by computer systems is well known. In order to evidence the differences, let us observe that:
The distinction between the two worlds and their relationships is significant, because a principle of duality holds between intensions and extensions of concepts; thus, at the concept level the situation is in a way opposite to that of the extensional level. Relationships between concept theory and computer science have been established through ontology for database preservation (Locuratolo & Palomäki, 2013), a concept structure which satisfies the following properties:
- •
Encloses all and only the concepts related to an initial concept structure.
- •
Encloses all and only the intensional relations among concepts.
- •
Results in leaves which are mapped to a Universe of discourse and to database models.
Ontology for database preservation results from an algorithm, called algorithm of concept construction (Locuratolo & Palomäki, 2014), which is complete with respect to both concepts and classes; however, concept completeness is distinguished from completeness of database classes. Although ontology for database preservation is able to keep all the database concepts and their logical relationships, it is a structure defined at an abstraction level not suitable for implementations. Further, it can enclose concepts corresponding to necessarily empty database classes, which can be interesting in concept theory but not in computer science.
To overcome these drawbacks, a relationship has been introduced at the boundary between concept theory and computer science, and the algorithmic property of class/concept completeness has been proposed (Locuratolo, 2015, in press). An algorithm resulting into a structure, called structure for classes/concepts preservation, has been designed to achieve the class/concept completeness property. This algorithm, called algorithm for the integration of classes/concepts, is defined at an abstraction level suitable for implementation. Further, only classes/concepts useful in computer science belong to the structure for classes/concepts preservation. The algorithm for the integration of classes/concepts is applied to a specialization hierarchy of database classes determined as a solution of the following data mining problem (Locuratolo, 2006):
Starting from an object database model, where each object instance belongs to one and only one class, determine a formal mapping from the object database model to a conceptual database model, where each object instance can belong to any class of the model.