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# Category Archives: Reflections on Abstractions

## Reflections on Abstractions: Subsumption I

We’re going to look at a subsumptional mapping from lhs original to rhs model. We will get four cases of subsumption as in figure (strictest cases), which now can be developed into a continuum with the four cases as corner points. Continue reading

## Reflections on Abstractions: From ‘Siamese’ Graphs to Concept Lattices

There is an elegant construct of dealing with ‘Siamese’ abstractions for object-attribute situations, from formal concept analysis. Where ‘Siamese’ means not-rhs-unique mapping of complete subgraphs. Continue reading

## Reflections on Abstractions: Subsumptions and Omissions

In addition to the recent posting ‘Abstractive and Functional Mappings’ we provide a simple visualisation of subsuming and omitting abstractions. Continue reading

Posted in Mathematics, Reflections on Abstractions
Tagged abstraction, Graph, left-total, left-unique, lhs, Mapping, model, modeling, omission, reflections on abstractions, Relation, rhs, ROA, subsumption
1 Comment

## Reflections on Abstractions: Abstractive vs Functional Mappings

We introduce the concepts of subsuming and omitting mappings, and see how they are better suited for abstraction and modelling than the classical mathematical concept of functions. Continue reading

Posted in Mathematics, Reflections on Abstractions
Tagged abstraction, bijection, function, Graph, left-total, left-unique, lhs, Mapping, model, modeling, reflections on abstractions, Relation, rhs, ROA, total, unique
3 Comments

## Reflections on Abstractions: The Use Case trade-off

Based on the elementary terms of relational structures, this little example shows the basic trade-off of Use Cases: understandability vs redundancy. Continue reading

## Reflections on Abstractions: Adjacent Rooms

Example of an abstraction by subsuming directly connected nodes in the original into a single node in the model. This corresponds to a situation where structures of wall are abstracted to rooms with the neighborhood relation. Continue reading

## Reflections on Abstractions: Joining Classification by Relationships and Properties

How does classification based on properties go together with relationship based classes? In addition to the former posting “Concepts vs Modules for Classification”, the fit of concept lattices and relationship graphs is examined in more detail. Continue reading

Posted in Mathematics, Reflections on Abstractions
Tagged abstraction, abstraction awareness, Class, classification, classifier, component, concept analysis, formal concept analysis, formal methods, Graph Theory, model, modeling, modeling theory, module, refinement, reflections on abstractions, ROA
1 Comment

## Reflections on Abstractions: Concepts vs Modules for Classification

A Concept (as in Formal Concept Analysis) and a Module (as in Graph Theory) both cover the notion of Classification. Although they share the same basic idea, they reveal differences in detail. Continue reading

## Reflections on Abstractions in Relational Structures. The very basic Setting.

Abstractional concepts can be found in the very basics of Graph Theory and Formal Concept Analysis. They provide the basic elements of Classification, Aggregation and Generalisation for a deeper rigorous analysis of Abstractions. Continue reading

Posted in Mathematics, Reflections on Abstractions
Tagged abstraction, abstraction awareness, aggregation, classification, component, concept analysis, connectivity, formal concept analysis, formal methods, generalisation, Graph Theory, model, modeling, modeling theory, module, refinement, reflections on abstractions, ROA
2 Comments