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

Abstraction Awareness is about deeper understanding of abstraction, a concept so basic to human thinking. Subsequently abstraction is discussed by the means of basic graph theory and formal concept analysis.

Lets bring the definition of abstraction by subsumption, i.e. of not always lhs-unique mappings, to life. We want to detect complete subgraphs in graphs and build an abstract graph (model) from them. Think of this as building components, where their elements are extremely tightly bound. Notice that this is stronger than ‘connected components’ of graph theory.

Abstraction by ‘Siamese’ Edge

roa complete graph rhs unique not rhs uniqueFor example

(rhs-unique) gives a pretty nice example of abstraction by subsumption, where rhs-uniqueness holds. We get two rhs nodes with an edge amongst them, that ‘survived’ the mapping.

(not-rhs-unique) – a kind of ‘siamese graph’ – leaves us with the choice to simply map the middle node to two different nodes, and then introduce an additional type of edge for ‘having common node’ (e.g. meaning ‘these two rooms have a common wall’) or to introduce any other kind of construct.

Abstraction by Concept Lattice

There is an elegant construct for bi-partite graphs (that correspond to an object – attribute, or extension – intension situation) from formal concept analysis:

roa complete graph bi-partite concept lattice(bi-partite) shows an example where nodes 1 and 2 together with a and b are ‘as complete as possible’ since they have all connections allowed amongst them in a bi-partite graph. This kind of completeness corresponds to a ‘concept’ in formal concept analysis.  Thus we can draw a concept lattice as in (bi-partite), with 1-3 as objects and a-c as attributes. Notice that the edges here are read from bottom to top as generalisations, i.e. we can completely refrain from object-attribute edges here (no mesh up as by the ‘having common node’ edge above).

So long
|=

About modelpractice

Modeling Theory and Abstraction Awareness in strive for scientific rigour and relevance to information systems engineering.
This entry was posted in Mathematics, Reflections on Abstractions and tagged , , , , , , , , , , , , , , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s