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

Next, 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

Next, 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

… 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
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PS
Or in terms of adjacency matrices: