=Paper=
{{Paper
|id=Vol-1442/paper_29
|storemode=property
|title=Extended Ontologies: a Cognitively Inspired Approach
|pdfUrl=https://ceur-ws.org/Vol-1442/paper_29.pdf
|volume=Vol-1442
|dblpUrl=https://dblp.org/rec/conf/ontobras/CarboneraA15
}}
==Extended Ontologies: a Cognitively Inspired Approach==
https://ceur-ws.org/Vol-1442/paper_29.pdf
Extended ontologies: a cognitively inspired approach
Joel Luis Carbonera1 , Mara Abel1
1
Instituto de Informática – Universidade Federal do Rio Grande do Sul (UFRGS)
Caixa Postal 15.064 – 91.501-970 – Porto Alegre – RS – Brazil
{jlcarbonera,marabel}@inf.ufrgs.br
Abstract. Within the Knowledge representation community, in general, an on-
tology is considered as a formal specification of a shared conceptualization. In
this sense, ontologies would be constituted of concepts and could be understood
as an approach of representing knowledge. In general, ontologies represent con-
cepts in a logical way, adopting the so-called classical theory of representation.
Due to this, ontologies can support classification, based on necessary and suf-
ficient conditions, and rule-based reasoning. In this work, we discuss a cogni-
tively inspired approach for extending the knowledge representation capabilities
of ontologies. We propose an extended notion of ontologies which incorporates
other cognitively plausible representations, such as prototypes and exemplars.
The extended ontology has the advantage of supporting similarity-based reason-
ing, besides the usual logical reasoning.
1. Introduction
Nowadays, ontologies are widely adopted for knowledge reusing and for promoting the
semantic interoperability among different systems (and humans). Within the knowledge
representation community, in general, ontologies are considered as formal and explicit
specifications of a shared conceptualization in a given domain [Studer et al. 1998]. It
is important to notice that, according to this perspective, ontologies would be consti-
tuted of concepts. In this work, following other works in the field of Artificial Intelli-
gence [Oltramari and Lebiere 2011, Carbonera et al. 2015], we adopt this conceptualist
[Smith 2004] view about ontologies.
In general, ontologies represent concepts in a logical way, assuming the so-called
classical theory of representation [Murphy 2002], where the concepts are represented
by sets of features that are shared by all the entities that are abstracted by the concept.
Due to this, ontologies are well suited for supporting classification based on necessary
and sufficient conditions and for supporting rule-based reasoning. However, in general,
ontologies cannot deal naturally with typical features of the concepts [Gärdenfors 2004];
that is, the features that are common to the entities abstracted by the concepts, but that are
neither necessary nor sufficient. In this paper, we propose the notion of extended ontology,
which incorporates other cognitively plausible representations, such as prototypes and
exemplars, and that can support similarity-based reasoning (dealing with prototypical
effects), besides the usual rule-based reasoning.
2. Theories of knowledge representation
Within the Cognitive Sciences there is an ongoing debate concerning how the knowledge
is represented in the human mind. According to [Murphy 2002] in this debate there are
three main theories. The classical theory assumes that each concept is represented by a
set of features that are shared by all the entities that are abstracted by the concept. In this
way, this set of features can be viewed as the necessary and sufficient conditions for a
given entity to be considered an instance of a given concept. Thus, according to this the-
ory, concepts are viewed as rules for classifying objects based on features. The prototype
theory, on the other hand, states that concepts are represented through a typical instance,
which has the typical features of the instances of the represented concept. Finally, the ex-
emplar theory assumes that each concept is represented by a set of exemplars of it, which
are explicitly represented in the memory. In theories based on prototypes or exemplars,
the categorization of a given entity is performed according to its similarity with proto-
types or exemplars; the instance is categorized by the category that has a prototype (or
exemplar) that is more similar to it. There are some works that apply these alternative
theories in computer applications [Fiorini et al. 2014].
3. Extended ontologies
As previously discussed, ontologies can be viewed as a paradigm of knowledge repre-
sentation that adopts the classical theory of knowledge representation. In this sense, the
classification of instances is performed by checking if they meet the necessary and suffi-
cient conditions of the considered concepts. However, it is well known in the knowledge
representation community that, for most of the common sense concepts, finding their
necessary and sufficient conditions can be a challenging task [Gärdenfors 2004]. Be-
sides that, according to evidences taken from the research within the Cognitive Sciences
[Gärdenfors 2004], for most of the concepts, humans can perform similarity-based clas-
sifications, and can consider the typical features of the concepts during the classification
process. In this work, we assume that a knowledge representation framework that pre-
serves the flexibility of the human cognition can provide advantages for knowledge-based
systems. For example, a system with this capability could classify some individual i as c
(where c is some concept) if it is sufficiently similar to a given prototype of c, even when
it does not present all the logically necessary features for being considered an instance of
c.
In this work, we propose the notion of extended ontology (χO), which incorpo-
rates the conventional features and capabilities of the classical ontologies with the possi-
bility of representing typical features of the concepts and of supporting similarity-based
reasoning. This proposal adopts some notions originally proposed in our previous works
[Carbonera and Abel 2015a, Carbonera and Abel 2015b].
Definition 1. An extended ontology (χO) is a tuple
χO = (C, ≤, R, A, ,→, D, d, I, v, ext, E, ex, P, prot) (1)
, where:
• C is a set C = {c1 , c2 , ..., cn } of n symbols that represents concepts (or classes),
where each ci is a symbolic representation of a given concept.
• ≤ is a partial order on C, that is, ≤ is a binary relation ≤⊆ C × C, which is reflex-
ive, transitive, and anti-symmetric. Thus, ≤ represents a relation of subsumption
between two concepts.
• R is a set R = {r1 , r2 , ..., rm } of m symbols that represents relations, where each
ri is a symbolic representation of a given relation.
• A is a set A = {a1 , a2 , ..., al } of l symbols that represents properties (or attributes
or features), where each ai is a symbolic representation of a given property.
• ,→ is a binary relation that relates properties in A to concepts in C, such that
,→⊆ A × C. Thus ai ,→ cj means that the attribute ai ∈ A is an attribute of the
concept cj , in the sense that ai characterizes cj .
• D is the set of every possible value of every attribute ai ∈ A.
• d : A → 2D is a function that maps a given attribute ai ∈ A to a set Dai ⊆ D,
which is its domain of values. Notice that D = li=1 d(al ).
S
• I is a set I = {i1 , i2 , ..., ip } of p symbols that represents individuals, where each
ij represents a given individual.
• v : I × A → D is a function that maps a given individual ij ∈ I and a given
attribute ai ∈ A to the specific value v ∈ D that the attribute ai assumes in ij .
• ext : C → 2I is a function that maps a given concept ci ∈ C to a set Ici ⊆ I,
which is its extension (the set of individuals that it classifies).
• E is a set E = {e1 , e2 , ..., en } of n sets of individuals, where each ei ∈ E repre-
sents the set of exemplars of a given concept ci . Notice that E ⊆ 2I .
• ex : C → E is a function that maps a given concept ci ∈ C to its set of exemplars
ei ∈ E.
• P is a set P = {p1 , p2 , ..., pn } of n prototypes, where each pi ∈ P represents the
prototype of a given concept ci ∈ C.
• prot : C → P is a function that maps a given concept ci ∈ C to its prototype
pi ∈ P.
Besides that, for our purposes, the individuals (members of I) are considered as
q − tuples, representing the respective values of the q attributes that characterize each
instance. Thus, each ij ∈ I = (v(ij , ah ), v(ij , al ), ..., v(ij , ap )), where ah , al and ap are
attributes of ij .
In our proposal, the sets E and P can be explicitly assigned to the members of
C, or can be automatically determined from the set I. As a basic strategy, a prototype
pi ∈ P of a given concept ci ∈ C, such that prot(ci ) = pi can be extracted by analyzing
the individuals in ext(ci ) and by determining the typical value of each attribute of the
individuals. If the attribute is numeric, the typical value can be the average; if the attribute
is categorical (or nominal or symbolic), the typical value can be the most frequent (the
mode).
Considering a given ci ∈ C, the set of its exemplars, ex(ci ), should be selected in
a way that, collectively, its members provide a good sample of the variability of the indi-
viduals in ext(ci ). Also, it is important to consider that the exemplars of a concept can
be used for supporting the classification of a given individual i and that, for performing
this process, it can be necessary to compare i with every exemplar of every concept of
the ontology. Thus, it is not desirable to consider all records in ext(ci ) as exemplars for
representing ci , since the computational cost of the classification process is proportional
to the number of exemplars that are selected for representing the concepts. Due to this, in
our approach we consider that the number of exemplars related to each concept ci ∈ C is
defined as a percentage ep (defined by the user) of |ext(ci )| (where |S| is the cardinality
of the set S). This raises the problem of how to select which individuals in ext(ci ) will
be consider as the exemplars in e(ci ). We select three main criteria that an individual
ij ∈ ext(ci ) should meet for being included in ex(ci ): (i) ij should have a high degree of
dissimilarity with the prototype given by prot(ci ); (ii) ij should have a high degree of sim-
ilarity with a big number of observations in ext(ci ); and (iii) ij should have a high degree
of dissimilarity with each exemplar already included in ex(ci ). This set of criteria was
developed for ensuring that the set of exemplars in ex(ci ) will cover in a reasonable way
the spectrum of variability of the individuals in ext(ci ). That is, our goal is to preserve in
ex(ci ) some uncommon individuals, which can be not well represented by prot(ci ), but
that represent the variability of the individuals. In our approach, we apply these criteria,
by including in ex(ci ) the k first individuals from ext(ci ) that maximize their exemplar-
iness index. The exemplariness index is computed using the notion of density of a given
individual. Regarding some concept ci ∈ C, the density of some individual ij ∈ ext(ci ),
is computed by the function density : I × C → R, such that,
|ext(ci )|
1 X
density(ij , ci ) = − d(ip , ij ) (2)
|ext(ci )| p=1
, where d is some dissimilarity (or distance) function (a function that measures the dis-
similarity between to entities). Considering this, the set ex(ci ) of some concept ci , with
k exemplars, can be computed by the Algorithm 1.
Algorithm 1: extractExemplars
Input: A concept c and a number h of exemplars
Output: A set exemplars of h instances representing the exemplars of the concept c.
begin
exemplars ← ∅;
for j ← 1 to h do
eIndexmax ← −∞;
imax ← null;
foreach individual ∈ ext(c) do
density ← density(individual, c);
dp ← d(individual, prot(c));
med ← 0;
if exemplars is not empty then
Compute the distance between individual and each exemplar already included in exemplars and assign
to med the distance of the nearest exemplar from individual;
/* eIndex is the exemplariness index */
eIndex = dp + density + med;
if eIndex > eIndexmax then
eIndexmax ← eIndex;
imax ← individual;
exemplars ← exemplars ∪ {individual};
return exemplars;
Notice that Algorithm 1 basically selects from ext(c), the individuals that maxi-
mize the exemplariness index, which is the sum of: (i) distance (or dissimilarity) of the
individual from the prot(c); (ii) the density of the individual, considering the set ext(c);
and the distance (or dissimilarity) of the individual from its nearest exemplar, already
included in exemplars.
Once a given extended ontology has its concepts, prototypes and exemplars, they
can be used by a hybrid classification engine for classifying individuals. This compo-
nent takes as input an individual and provides its corresponding classifications (a set of
concepts classif ications ⊆ C). Firstly, the classification engine applies a conventional
logical reasoning procedure (using the classical part of the extended ontology) for pro-
viding a first set of classification hypothesis. Notice that this reasoning process can infer
more than one classification for the same individual. If this process provides, as clas-
sifications, concepts that are not specific (if they are not leaves of the taxonomy), the
similarity-based reasoning can be used for determining more specific interpretations. The
hybrid classification engine implements the Algorithm 2.
Algorithm 2: hybridClassification
Input: An individual i.
Output: A set classif icationset of concepts representing the classifications of i.
begin
classif icationset ← ∅;
Perform the logical reasoning for interpreting i, and include the concepts of the resulting classification in
classif icationset ;
if the concepts in classif icationset are not specific then
hypset ← ∅;
foreach c ∈ classif icationset do
Find the leaves in the taxonomy, whose root is c, and include them in hypset ;
classif icationset ← ∅;
M AX ← −∞;
foreach c ∈ hypset do
app ← applicability(c, i);
if app > M AX then
M AX ← app;
classif icationset ← {c};
else if app = M AX then
classif icationset ← classif icationset ∪ {c};
return classif icationset ;
Notice that the Algorithm 2 uses the notion of applicability, which, intuitively
measures the degree in that a given concept c can be applied as an interpretation for a
given observation individual. The applicability is computed by the Algorithm 3, using
the prototypes and exemplars of the concepts.
Algorithm 3: applicability
Input: A concept c and an instance i.
Output: A value r ∈ R, which is the degree in that c can be applied as a classification for i.
begin
app ← 0;
pSimilarity ← sim(i, prot(c));
eSimilarity ← 0;
Calculate the similarity sim(i, exi ) between i and each exi ∈ e(c), and assign to eSimilarity the similarity
value of the most similar exi ;
app ← pSimilarity + eSimilarity;
return app;
Notice that the Algorithm 3 uses the function sim for measuring the similarity.
Intuitively, the similarity is the inverse of the dissimilarity (or distance) between two
individuals. Thus, sim has values that are inversely proportional to the values obtained
by the function d. Here, we assume that sim(ij , il ) = exp(−d(ij , il )).
4. Conclusions and future works
In this paper, we propose the notion of extended ontology, which integrates the com-
mon features and capabilities of conventional ontologies (based on the classical paradigm
of knowledge representation) with the capability of dealing with typical features in
similarity-based reasoning processes. The extended ontologies can provide more flex-
ibility in classification processes, in the cases that do not have enough information for
being classified according to necessary and sufficient conditions.
In future works, we intend to investigate approaches of instance selection
[Olvera-López et al. 2010] for enhancing our approach for selecting exemplars. Also,
we intend to apply the notion of extended ontologies (as well as the algorithms proposed
here) for improving the results obtained in [Carbonera et al. 2011, Carbonera et al. 2013,
Carbonera et al. 2015] for the task of visual interpretation of depositional processes, in
the domain of Sedimentary Stratigraphy. We are also investigating how this approach can
be applied for solving other problems, such as ontology alignment. We hypothesize that
it is possible to take advantage of the information represented in the form of prototypes
and exemplars, as additional sources of evidences in the process of ontology alignment.
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