variable, material> based on the engineering knowl-
edge; for each category the classification is built on
a specialisation/generalisation relations (i.e. inclu-
sion/coverage relations), i.e. moving to the next
lower level of the directed graph each category
(<event, system, function, variable, material>) is
specified in subclasses (and over sub-sub classes etc
down to specific concepts; i.e. “system elements” if
we follow the “System” category) and moving to the
next higher level of the directed graph sub-classes
(or individual concepts) are generalised to the next
level of sub-classes (or a class).
Keywords are going to be used to quickly find
documents through queries of (very) large databases;
this should be possible by building keyword combi-
nations without following the predefined structure of
the classification but using the relations
We superimpose a partonomy on the keyword
classification, or more precisely a fuzzy partonomy;
this will allow us to find keywords which are partly
the same for a query regardless of where they are
defined in the underlying keyword classification (or
where they are located in the directed graph).
A partonomy that is built on part-of relationships
is a primitive of the formal theory of parthood rela-
tions; parthood relations specify part-of and overlap
within a whole; part-of is reflexive, anti-symmetric
and transitive (the transitivity is sometimes difficult
to justify) and overlap between x and y is defined as
O(x, y) := {z │ z
x and z y} where the symbol
“” now denotes part-of.
The fuzzy keyword classification and partonomy
are built on inclusion and coverage, which are un-
derstood to be relations between fuzzy subsets. The
classifications and part-of relations are collected in
matrices of coverage/inclusion of keywords; the
cells of the matrix are numbers [0, 1] which show
the degree of coverage and inclusion.
A fuzzy ontology is a relation on fuzzy sets, i.e. a
relation associated with a membership function; let
K
i
be a finite fuzzy set of keywords identified with a
level of the directed graph and a category <event,
system, function, variable, material>, hence i = 1,
…, 5; a membership function is a mapping of K
i
x
K
j
on L, a lattice or a partially ordered set; the set of
linguistic labels {negligible, weak, moderate, strong,
perfect} is a lattice which means that a relation be-
tween two sets of keywords can be stated and de-
scribed with a linguistic label.
4.2 Fuzzy Reasoners
We need to find a way to combine linguistic labels
and numbers for the following reasoning schemes so
that we can use them to get numbers for the inclu-
sion/coverage matrix; this can be done in the follow-
ing way (the linguistic labels can be defined accord-
ing to the context; the labels can also be overlap-
ping; cf. Carlsson et al (2010b) for details). Let us
consider a domain of keywords that have
been classified based on some property with real
numbers in [0, 1]; we will consider three fuzzy sub-
sets A, B and C of keywords (similar to K
i
) in the
domain D; we will first work with the fuzzy subsets
A and B. We say that A is a fuzzy subset of B (both
defined in the domain D) and write
(1)
We can then define the two concepts inclusion and
coverage in terms of these fuzzy subsets (as both are
defined in the same domain) by following the
intuitive understanding we have in Figure 3
; it
should be noted that the min-operator is one of a
class of t-norms that can be used to express the
combinations (cf. Carlsson et al (2010b)).
Degree of subsethood (inclusion) of in
,
min
,
/
(2)
Degree of supersethood (coverage)
,
min
,
/
(3)
Now we can combine the two concepts as a
categorisation of the two subsets which can be used
to order the subsets of keywords – for this we have
several possibilities but we can use the following
simple characterisation:
Degree of similarity
,
min
,
/max
,
(4)
It is clear that , ,.
We will get a similar representation of the fuzzy
subset C as it is fully a subset of A (cf. Figure 3).
We can now illustrate these concepts with some
numerical examples; the numbers would be similar
to those used in Figure 4.
Let
0.4,0.6,0.8,0.3 and
0.5,0.4,0.8,0.6.
Then A is almost a subset of B since
for 1,3,4,5 but not quite since
2
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