Reflections on Abstractions: Subsumption I

Abstraction Awareness is about deeper understanding of abstraction, a concept so basic to human thinking. Subsequently abstraction is discussed by the means of basic relational mathematics.

We’re going to look at a subsuming mapping from lhs original to rhs model. When we assume subsumption, i.e. non-lhs-uniqueness, a single rhs node in model ‘contains’ , i.e. is mapped from, 1..n nodes of original.

Two model nodes with no edge

subsumption 2 nodes 0 edgeNext, as in figure (no edge), we look at two rhs nodes a and b with no (rhs) edge amongst them. From no edge on the rhs we can derive that not all lhs nodes in node a have a (lhs) edge with all lhs nodes in b, else we would need a negation in the mapping rule to map from all-to-all edges on lhs to no edge on rhs. Among the remaining mapping options, the rule ‘no lhs edge iff no rhs edge’ is the strictest (the ‘opposite’ of all-to-all, so to say).

Moreover, in this case no edge among the lhs nodes of a and b would make no sense, since the abstraction could not be justified by the structure of the graph. Thus, the nodes of a and b must be connected amongst themselves. Again, we take the strictest case of all-to-all connection as in figure (no edge).

Two model nodes with one edge

subsumption 2 nodes 1 edgeNext, as in figure (one edge), we look at two rhs nodes a and b with one (rhs) edge amongst them. From one edge on the rhs we can derive that one lhs nodes in node a has a (lhs) edge with one lhs node in b, else we would need a negation in the mapping rule to map from no edges at all on lhs to one edge on rhs. Among the remaining mapping options, the rule ‘all-to-all lhs edges iff one rhs edge’ is the strictest  (the ‘opposite’ of no edge at all, so to say) .

Moreover, in this case all-to-all edges among the lhs nodes of a and b would make no sense, since the abstraction could not be justified by the structure of the graph. Thus, the nodes of a and b must not be connected completely amongst themselves. Again, we take the strictest case of no edge at all connection as in figure (one edge).

Thus

abstraction subsumption

… we get four cases of subsumption as in figure (strictest cases), which next can be developed into a continuum with the four cases as corner points.

So long
|=

PS
Or in terms of adjacency matrices:

4 cases abstraction by subsumption adjacency

About modelpractice

Modeling Theory and Abstraction Awareness in the gap btw rigour of science and relevance to engineering.
This entry was posted in Reflections on Abstractions and tagged , , , , , , , , , , , , . Bookmark the permalink.

One Response to Reflections on Abstractions: Subsumption I

  1. Pingback: Reflections on Abstractions: Roaming the Subsumption Continuum | modelpractice

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