Introducing fuzzy quanti cation in OWL 2 ontologies

Abstract

In this paper, we brie y report our latest achievements in fuzzy granulation of OWL 2 ontologies. More precisely, we extend a previously presented method in order to address a new class of sentences with fuzzy quanti ers. 1 http://www.w3.org/TR/2009/REC-owl2-overview-20091027/

Introduction

Humans continually acquire, manipulate and communicate imprecise knowledge: therefore any knowledge base capable of dealing with imprecision, when a precise alternative leads to an exceedingly complex representation, could be more interpretable by human users, i.e.easier to read and understand. Moreover, tolerance to imprecision may lead to concrete bene ts such as compact knowledge representation, e cient and robust reasoning, etc.

In this work we focus on imprecision due to the lack of boundaries in concepts (usually of perceptual nature), which is well modeled through fuzzy set theory [ 8 ]. In [ 5 ] we proposed a data-driven approach for generating linguistic descriptions from data available in OWL 2 1 ontologies, which uses Fuzzy OWL 2 (see [ 1 ] for details about the language) as a formalism to represent the semantics of fuzzy sets through annotations. The proposed approach can be synthesized in the following main steps: (i) individuals of speci ed (numerical) data properties are granulated into fuzzy sets through clustering ; (ii) the prototypes of these clusters are used to granulate the numerical domain into a number of information granules that are represented as new individuals in the ontology; (iii) the cardinalities of such information granules are compared against a number of fuzzy quanti ers and the corresponding results are integrated in the ontology through new axioms.

The proposed approach has been successively re ned in [ 4 ] and a software prototype called GranulO has been implemented and presented in [ 3 ]. What emerged from these later studies was the ability of representing information granules that have a coarser granularity than our original idea in [ 5 ]. In this study, we present a further development of the proposed method, which enables the representation and integration of fuzzy information granules with ner granularity, so as to fully comply with our initial proposal.

Fuzzy sets and fuzzy quanti ers

From the modeling viewpoint, fuzzy sets are very useful to represent perceptionbased information granules, which are characterized by both granularity (i.e. concepts that refer to a multiplicity of objects) and graduality (i.e. the reference of concepts to objects is a matter of degree) [ 7 ]. In the case of numerical domains, a simple yet e ective way to de ne such fuzzy information granules is through a so-called Strong Fuzzy Partition (SFP). A SFP can be easily de ned by trapezoidal fuzzy sets by properly constraining the characterizing parameters.

Fuzzy sets, like crisp sets, can be quanti ed in terms of their cardinality. Several de nitions of cardinality of fuzzy sets have been proposed [ 2 ], although in this paper we consider only relative scalar cardinalities. Since the range of a scalar cardinality is always the unitary interval, a number of fuzzy sets can be de ned to represent granular concepts about cardinalities, such as Many, Most, etc. These concepts are called fuzzy quanti ers [ 6 ]. As usual, they can be de ned so as to form a SFP; in this way linguistic labels can be easily attached, as illustrated in Fig. 1. Fuzzy quanti ers are used to express imprecise properties on fuzzy information granules. More speci cally, given a quanti er Q labeled with Q and a fuzzy set F labeled with F, the membership degree Q ( (F )) quanti es the truth degree of the proposition

Q x are F For example, the fuzzy proposition Many x are Low asserts that many data points have a low value. Its truth degree is quanti ed by QMany ( (FLow)). 3

Representing fuzzy quanti ed sentences in OWL 2

In [ 5 ] we discussed several cases of application of our fuzzy granulation method for OWL 2 ontologies. A case of particular interest to the present paper is given by OWL 2 schemes modeling ternary relations. A ternary relation is a subset of the Cartesian product involving three domains C D N (for our purposes, we will assume N a numerical domain). Ternary relations are not directly representable in OWL 2. However, they can be indirectly represented through an auxiliary class E, two object properties R1 and R2, and one datatype property T , as depicted in Figure 2.

As an illustrative example, let us consider the distances between hotels and attractions in the touristic domain. This is clearly a case of a ternary relation that needs to be modeled through an auxiliary class Distance, which is connected to the classes Hotel and Attraction by means of the object properties hasDistance and isDistanceFor, respectively, and plays the role of domain for a datatype property hasValue with range xsd:double. The knowledge that \Hotel Verdi has a distance of 100 meters from Corso Italia" can be therefore represented with the following three triples: hverdi hasDistance d1 i, hd1 isDistanceFor corso italiai, and hd1 hasValue 100i.

The purpose of our study is to de ne a method for representing granular information in OWL 2 which can be linguistically described as \Hotel Verdi has a low distance from many attractions" [ 5 ]. However, the method we have developed so far enables the representation of sentences with a coarser level of granulation, such as \Many hotels have a low distance from attractions". In the following we present a re nement of the method in order to represent granular information of the former type, with ner granularity.

The structure shown in Figure 2 corresponds to a tabular representation with three columns, and as many rows as the number of elements of the relation, as in Table 1. Data in the third column is numerical, therefore it can be clustered in order to obtain a number of prototypes which are used to generate fuzzy information granules characterized by trapezoidal and triangular membership functions (see [ 5 ] for details).

Given an individual ai of the class C and an information granule Fj , the relative -count of Fj w.r.t. ai is given by i(Fj ) =

Pkn=i1 Fj (vk) ni (1) which represents the relative cardinality of Fj over all tuples of the relation as in Table 1 corresponding to the individual ai. The cardinality i(Fj ) can be quanti ed according to the fuzzy quanti ers Q1; : : : ; Qm as in Fig. 1.

In order to integrate the cardinality information in the ontology, we introduce a number of new axioms. (We follow the Hotel example for the sake of clarity.) After fuzzy granulation, d1 is member (to some degree) of the class LowDistance

Distance u 9isDistanceFor:Attraction u 9hasValue:Low A new individual, ld verdi, is introduced as an instance of a new class Granule with data property hasCardinality. The punctual distance d1 then maps to the granular distance ld verdi. Finally, hasDistance is replaced by its granular counterpart g hasDistance so that hverdi g hasDistance ld verdi i holds.

Let us suppose that hld verdi hasCardinality 0:5i and that, after quanti cation of the cardinality, ld verdi is member to some degree of the class 9hasCardinality:Many. This corresponds to the truth degree that Hotel Verdi has low distance from many attractions.

A possible arrangement of the granulated ontology is illustrated in Figure 3. 4

Conclusions

In this paper we have brie y reported our latest achievements on fuzzy granulation of OWL 2 ontologies. In particular, we have addressed a new class of fuzzy quanti ed sentences involving ternary relations. This has required an extension of our former method originally presented in [ 5 ]. An implementation of this extension within the GranulO system is ongoing.

Acknowledgments This work was partially funded by the Universita degli Studi di Bari \Aldo Moro" under the IDEA Giovani Ricercatori 2011 grant \Dealing with Vague Knowledge in Ontology Re nement".

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