Reflections on Abstractions: Roaming the Subsumption Continuum

Abstraction Awareness is about deeper understanding of abstraction, a concept so basic to human thinking. Subsequently abstraction is discussed by basic concepts of graph theory.

In the recent post we defined 4 cases of subsumption. In order to get from these basic cases to real world situations, we see the 4 cases as the corners of a ‘continuum’ A, B, C, D, and thus we get figure (subsumption continuum).

continuum of subsumption

Here the following areas – well known from graph theory – can be identified:

CC Connected Component

A connected component is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph.* This is what we get when we reduce the density of A, i.e. components become less obvious to abstract/model just by their internal composition alone.

CY Connectivity

The connectivity asks for the minimum number of elements (nodes or edges) which need to be removed to disconnect the remaining nodes from each other. A higher degree of connectivity is what we get when we increase the strength of A, i.e. components become less obvious to abstract/model just by their external relationships alone.

MO Module

In a module all members have the same set of neighbors among nodes not in the module. This is what we get when we add edges to C, i.e. the abstraction/model of the class structure becomes a multi-level tree.

CL Concept Lattice

All formal concepts of a bi-partite graph constitute a lattice. This is what we get when we reduce edges from C, i.e. the abstraction/model of the class structure becomes a multi-level lattice.

EX Expander

An expander is a sparse graph that has strong connectivity properties. It is ‘quite’ hard to decompose by subsumption into classes or components. For this reason expanders are used to design robust IT networks etc.

Finally, we get E by reducing the edges in A and C and then joining them to a ‘cube’ that can not reasonably be structured by classes or components.

Thus …

we can take parts of graph theory for describing typical problem situations of abstraction. However, most of these parts of graph theory* do not address the issues of the actual abstracting mapping from original to model, as required for ‘modelling for the purpose of understanding’. This is what a theory of modelling should be about.

So long
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* apart from formal concept analysis

About modelpractice

Modeling Theory and Abstraction Awareness in strive for scientific rigour and relevance to information systems engineering.
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