E Pluribus Unum - Computational Creativity Research: Towards Creative Machines - page 168

Goguen defines sign systems as algebraic theories that can be formulated by
using the algebraic specification language OBJ3 [ 24 ]. One special case of such a
sign system is a conceptual space : it consists only of constants and relations, one
sort, and axioms that define that certain relations hold on certain instances.
We now relate such spaces to a general formalisation of ontologies as we under-
stand them and as introduced above. Since we will focus on standard ontology lan-
guages, namely and first-order logic, we use these to replace the logical lan-

guage OBJ3 used by Goguen and Malcolm. However, as some structural aspects are
necessary in the ontology language to support blending, we augment these standard
ontology languages with structuringmechanisms known from algebraic specification
theory [ 39 ]. Such mechanisms are now included in the DOL language specification
discussed below in Sect. 9.4 . This allows us to translate most parts of Goguen's the-
ory to these augmented ontology languages. Goguen's main insight has been that
sign systems and conceptual spaces can be related via morphisms , and that blending
is comparable to colimit construction. In particular, the blending of two concepts is
often a pushout (also called a blendoid in this context). Some basic definitions we
then need are the following. 4
Non-logical symbols are grouped into signatures , which for our purposes can
be regarded as collections of typed symbols (e.g. concept names, relation names).
Signature morphisms are maps between signatures that preserve (at least) types
of symbols (i.e. map concept names to concept names, relations to relations, etc.).
A theory or ontology pairs a signature with a set of sentences over that signature,
and a theory morphism (or interpretation ) between two theories is just a signature
morphism between the underlying signatures that preserves logical consequence,
that is,

OWL

ˁ : T 1 ₒ T 2 is a theory morphism if T 2 |= ˁ( T 1 ) , i.e. all the translations of

sentences of T 1 along follow from T 2 . This construction is completely logic inde-

pendent. Signature and theory morphisms are an essential ingredient for describing
conceptual blending in a logical way.
We can now give a general definition of ontological blending capturing the basic
intuition that a blend of input ontologies shall partially preserve the structure imposed
by base ontologies, but otherwise be an almost arbitrary extension or fragment of the
disjoint union of the input ontologies with appropriately identified base space terms.
For the following definition, a variant of which we first introduced in Kutz et al.
[ 42 ], a diagram consists of a set of ontologies (the nodes of the diagram) and a set
of morphisms between them (the arrows of the diagram). The colimit of a diagram
is similar to a disjoint union of its ontologies, with some identifications of shared
parts as specified by the morphisms in the diagram. We refrain from presenting the
category-theoretic definition here (which can be found in Adámek et al. [ 1 ]), but will
explain (the action of) the colimit operation in the examples in Sect. 9.4.1 .Inthe
following definition, we use

ˁ

to denote the set of all nodes in a diagram.
Definition 1 ( Ontological Base Diagram )An ontological base diagram is a dia-
gram D for which a distinguished set

| D |

B ={ B i | i ∈ I }ↂ| D | of nodes are called

4 Note that these definitions apply not only to OWL, but also to many other logics. Indeed, they
apply to any logic formalised as an institution [ 22 ].