Extended ontologies: a cognitively inspired approach

Abstract

Within the Knowledge representation community, in general, an ontology 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 concepts in a logical way, adopting the so-called classical theory of representation. Due to this, ontologies can support classification, based on necessary and sufficient conditions, and rule-based reasoning. In this work, we discuss a cognitively 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 reasoning, besides the usual logical reasoning.

1. Introduction

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 theory, 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 exemplar 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 prototypes 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 representation 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 sufficient 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 [ Ga¨rdenfors 2004 ]. Besides that, according to evidences taken from the research within the Cognitive Sciences [ Ga¨rdenfors 2004 ], for most of the concepts, humans can perform similarity-based classifications, and can consider the typical features of the concepts during the classification process. In this work, we assume that a knowledge representation framework that preserves 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 ( O), which incorporates the conventional features and capabilities of the classical ontologies with the possibility 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 2015 a, Carbonera and Abel 2015 b].

Definition 1. An extended ontology ( O) is a tuple

O = (C; ; R; A; ,!; D; d; I; v; ext; E ; ex; P ; prot) (1) , where:

C is a set C = fc1 ; c2 ; :::; cn g 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 reflexive, transitive, and anti-symmetric. Thus, represents a relation of subsumption between two concepts.

R is a set R = fr1 ; r2 ; :::; rm g of m symbols that represents relations, where each ri is a symbolic representation of a given relation.

A is a set A = fa1 ; a2 ; :::; al g 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 2 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 2 A. d : A ! 2D is a function that maps a given attribute ai 2 A to a set Dai D, which is its domain of values. Notice that D = Sli=1 d(al ).

I is a set I = fi1 ; i2 ; :::; ip g 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 2 I and a given attribute ai 2 A to the specific value v 2 D that the attribute ai assumes in ij . ext : C ! 2I is a function that maps a given concept ci 2 C to a set Ici I, which is its extension (the set of individuals that it classifies).

E is a set E = fe1 ; e2 ; :::; en g of n sets of individuals, where each ei 2 E represents 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 2 C to its set of exemplars ei 2 E .

P is a set P = fp1 ; p2 ; :::; pn g of n prototypes, where each pi 2 P represents the prototype of a given concept ci 2 C. prot : C ! P is a function that maps a given concept ci 2 C to its prototype pi 2 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 2 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 2 P of a given concept ci 2 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 2 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 individuals 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 2 C is defined as a percentage ep (defined by the user) of jext(ci )j (where jSj 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 2 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 similarity 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 exemplariness index. The exemplariness index is computed using the notion of density of a given individual. Regarding some concept ci 2 C, the density of some individual ij 2 ext(ci ), is computed by the function density : I C ! R, such that, density(ij ; ci ) =

1 jexXt(ci )j jext(ci )j p=1 d(ip ; ij ) (2) , where d is some dissimilarity (or distance) function (a function that measures the dissimilarity 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 1; imax null; foreach individual 2 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 [ findividualg; return exemplars;

Notice that Algorithm 1 basically selects from ext(c), the individuals that maximize 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 component 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 providing 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 classifications, 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 classificationset of concepts representing the classifications of i. begin classificationset ?; Perform the logical reasoning for interpreting i, and include the concepts of the resulting classification in classificationset ; if the concepts in classificationset are not specific then hypset ?; foreach c 2 classificationset do

Find the leaves in the taxonomy, whose root is c, and include them in hypset ; classificationset ?; MAX 1; foreach c 2 hypset do app applicability(c; i); if app > MAX then

MAX app; classificationset fcg; else if app = MAX then

classificationset classificationset [ fcg; return classificationset ;

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 2 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 2 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 common 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 flexibility 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-Lo´ pez 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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