E Pluribus Unum - Computational Creativity Research: Towards Creative Machines - page 179

Triple Blend with one Base Triple Blend with two Bases

C C

colimit morphisms colimit morphisms

Input 1

O1 O2

Input 2

O3

Input 3 Input 1

O1 O2 Input 2 O3

Input 3

base morphisms

base morphisms

Base Ontology Base 2

Base 1

Fig. 9.7 Blending three input spaces using one respectively two base ontologies

A blended theory through a bridge theory

C

O2

Input 1 O1 Input 2

B

Bridge

O1' O2'

Base 1 Base 2

Fig. 9.8 Blending two input spaces through two bases and a bridge theory, deviating from the
Goguen construction

trivial ones 11 —however, such a reduction loses the direct connection between the dia-
grammatic representation and the cognitive-conceptual processes that are being for-
malised here. In a similar vein, Definition 1 introducing the notion of an ontological
base diagram in Sect. 9.3 easily generalises to the case of partial base morphisms, i.e.
where only parts of the signature of an ontology are mapped. Such partial morphisms
can be coded as spans of two (total) theory morphisms B i

I j , where
the first morphism is the embedding of the domain (actually, the larger dom

ₐ dom (μ ij ) ₒ

is,
the more defined is the partial morphism), and the second action represents the
action of the partial morphism. 12 Similarly, arbitrary relations can be coded as spans
B i

(μ ij )

I j is a relation, and the arrows are the projections to
the first and second component. However, such complexities can be hidden from a
user by allowing partial morphisms to be used directly in the specification of a blend-
ing diagram, and by letting a tool handle the simulation through total morphisms as
discussed above.
Finally, a more severe deviation from the basic blending diagram is shown in
Fig. 9.8 . Here, we interpret Base1 into Input1, Base2 into Input2, and connect the

ₐ R ₒ I j . Here, R ↆ B i ×

11 A well-known theorem of category theory states that every finite colimit can be expressed by
pushouts and initial objects.
12 In this case, the base diagram becomes a bit more complex; in particular, there are minimal nodes
dom (μ ) which have only auxiliary purposes and do not belong to the base.

ij