Modelling & Abstraction of Relational Structures

This is a primer on the very rock bottom principles of modelling & abstraction from and to finite relational structures.

relational structureRelational Structures are ubiquitous. Relational structures appear in many forms and situations, like database models, all kinds of spreadsheet usage, state machines or flow diagrams, directory trees, all kinds of ontologies and taxonomies etc. Figure (simple) shows a very simple relation in x-matrix form.

When Relational Structures get harder, understanding helps. As the number of relation elements grows, as in figure (hard), the RS becomes harder to ‘deal with’. For this purpose a deeper understanding of the relation might be quite helpful. So, we look for ways how such an understanding can be provided.

Modelling provides understanding. In understanding, abstraction plays a central role as described in part one ‘Modelling for Understanding’. Performing an abstracting mapping from original to model is what we call modelling. This mapping is our object of study in part two ‘Understanding of Modelling’.

Modelling for Understanding

Modelling – with abstraction at its very heart – is seen here as a technique to gain understanding. As with reasoning, where we have practical common sense as well as the theory of formal logic, we consider it fruitful for the topic of modelling, too, to have beyond its everyday use a treatment by a rigorous theory.

  1. The role of abstraction in Understanding

Understanding of Modelling

How can we gain the deepest understanding of modelling? Of course by modelling it. Here, by the means of basic discrete mathematics like order theory, graph theory, formal concept analysis, or finite model theory.

  1. Reductional Mappings: Subsumption and Omittance
  2. Four Cases of Subsumption
  3. The Subsumption Continuum
  4. Omittance

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