On similarity in fuzzy description logics

Abstract

This paper is a contribution to the study of similarity relations between objects represented as attribute-value pairs in Fuzzy Description Logics. For this purpose we use concrete domains in the fuzzy description logic $\mathfrak{I}\mathcal{A}\mathcal{L}\mathcal{C}\mathcal{E}\mathcal{F}(\mathcal{D})$ associated either with a left-continuous or with a finite t-norm. We propose to expand this fuzzy description logic by adding a Similarity Box (SBox) including axioms expressing properties of fuzzy equalities. We also define a global similarity between objects from similarities between the values of each object attribute (local similarities) and we prove that the global similarity defined using a t-norm inherits the usual properties of the local similarities (reflexivity, symmetry or transitivity). We also prove a result relative to global similarities expressing that, in the context of the logic MTL∀, similar objects have similar properties, being these properties expressed by predicate formulas evaluated in these objects.

Introduction

Similarity has been a central issue for decades in different disciplines, ranging from philosophy (Leibniz's Principle of the Identity of Indiscernibles [33]) and psychology (Tversky's stimuli judged similarity [62]) to natural sciences (taxonomy [37]) and mathematics (geometric similarity [25, Chapter 4]).

In artificial intelligence (AI) similarity plays an important role because the analogy reasoning is behind some of the early machine learning methods. For instance, case-based reasoning methods (see [38]) are based on the principle that “similar problems have similar solutions”. In clustering [39] objects are grouped in clusters according to their similarities. The key point of these methods is to define a similarity metrics to express the similarity between objects. In AI, domain objects are commonly represented using attribute-value pairs. Metrics used to assess the similarity between two objects have to take into account such representation, in order to do that, it is usual to consider the number of similar attribute-values. The global similarity between two objects has to be seen as an aggregation of the local similarities of the attributes describing them (see [43] for a collection of similarity measures; for the item of aggregation operators see [50], [61]).

In the present paper we study how to deal with similarities when objects are represented by sets of attribute-value pairs in Fuzzy Description Logics. Description Logics (DLs) are knowledge representation languages built on the basis of classical logic. DLs allow the creation of knowledge bases and provide ways to reason on the contents of these bases. A full reference manual of the field can be found in [4]. Hirsh and Kudenko [42] proposed a way to apply feature-based learners to DL learning tasks by presenting a method to compute an attribute vector representation of DL instances. Although it is a work oriented to learning, the authors deal with the problem of how to represent attribute-value objects in DL, and they manage it by expressing each attribute as a concept.

Fuzzy Description Logics (FDLs) are natural extensions of DLs expressing vague concepts commonly present in real applications (see for instance [9], [45], [56], [57], [59], [60]). Hájek [36] proposed to deal with FDLs taking as basis t-norm based fuzzy logics. His aim was to enrich the expressive possibilities in FDLs and to capitalize on recent developments in the field of mathematical fuzzy logic. From this perspective, in [34] a family of FDL languages was defined. These languages include truth constants for representing truth degrees, thus allowing the definition of the axioms of the knowledge bases as sentences of a predicate fuzzy language in much the same way as in classical DLs.

In the fuzzy framework, the notion of similarity was introduced by Zadeh in [64] as a generalization of the notion of equivalence relation (see [53] for a historical overview on the notion of t-norm based similarity). As Zadeh pointed out, one of the possible semantics of fuzzy sets is in terms of similarity. Indeed, the membership degree of an object to a fuzzy set can be seen as the degree of resemblance between this object and prototypes of the fuzzy set. Ruspini suggests in [54] that the degree of similarity between two objects A and B may be regarded as the degree of truth of the vague proposition “A is similar to B”. Thus, similarity among objects can be seen as a phenomenon essentially fuzzy. Following this idea, we want to use the capabilities of languages of FDLS to express similarity degrees between objects.

There are many authors that have focused on a fuzzy notion of similarity. In a more general context of predicate fuzzy logics, Hájek studied similarities and applied the obtained results to the analysis of fuzzy control [35]. Bělohlávek [19] presented a general theory of fuzzy relational systems. Model-theoretic properties of algebras with fuzzy equalities were studied in [19], [24]. Dubois and Prade pointed out in [28] that three main semantics for membership functions existed in the literature: similarity, preference and uncertainty. Each semantic underlying a particular class of applications. Similarity notions, for instance, have been exploited in clustering analysis and fuzzy controllers. The authors stated that the similarity semantics of fuzzy sets could serve as a basis for the estimation of preference and uncertainty.

In [20] Běhounek et al. studied fuzzy relations in the graded framework of Fuzzy Class Theory (FCT) generalizing existing crisp results on fuzzy relations to the graded framework. In FCT we can express the fact that a fuzzy relation is reflexive, symmetric or transitive up to a certain degree, and similarity is defined as a first-order sentence which is the fusion of three sentences corresponding to the graded notions of reflexivity, symmetry and transitivity. This allows to speak in a natural way of the degree of similarity of a relation. In [3] the relationship between global and local similarities in the graded framework of FCT was investigated.

Another interesting approach is the one taken by Bobillo and Straccia in [11]. The authors provide a simple solution to join two formalisms, fuzzy DLs and rough DLs, and define a fuzzy rough DL. This logic is more general than other related approaches, including tight and loose fuzzy rough approximations and being independent of the fuzzy logic operators considered. The key idea in rough set theory is the approximation of a vague concept by means of a pair of concepts, usually this approximation is based on an equivalence relation between elements of the domain. They extend this idea using fuzzy similarity relations instead of equivalence relations, giving raise to fuzzy rough sets. Bobillo and Straccia use a fixed set of similarities in order to introduce the semantics for the upper and lower approximation constructors. A revised and extended version of their work is [12].

Regarding the work done on distances defined from t-norms, in [2] Alsina introduced the idea of constructing distances from a t-norm and its dual. He proved that being a copula is a sufficient condition for the t-norm to induce a distance. Y. Ouyang in [52] gives an example of a continuous strict Archimedean t-norm that is not a copula and that generates a distance. An interesting problem recently solved in [1] (for the t-norms with the same zero region as Łukasiewicz) is the characterization of those t-norms that induce distances. The authors give a necessary and sufficient condition for a pair consisting of a t-norm and a t-conorm (not necessarily its dual) to generate a distance.

Our paper is a contribution to the study of similarity relations between objects represented as attribute-value pairs in Fuzzy Description Logics. It is organized as follows. In Section 2 we recall the notions and results from predicate fuzzy logics necessary to the understanding of the paper. In Sections 3 and 4 we recall the syntax and semantics of the classical description logic $\mathcal{ALC}$ and the fuzzy description logic $\mathfrak{I}\mathcal{ALCE}$. In Section 5 we expand the Fuzzy Description Language $\mathfrak{I}\mathcal{ALCE}$ by introducing a similarity box (SBox) including axioms expressing properties of fuzzy equalities, allowing models of the language with a non-geometrical interpretation of the similarity symbols. In the Similarity Box we express that a role is reflexive, symmetric, transitive or that it is a congruence. We also obtain some results stating the equivalence of these axioms with certain role inclusion axioms. In Section 6 we explain how the attribute-value representation can be captured in the framework of the classical description language $\mathcal{ALC}$ and of the fuzzy description language $\mathfrak{I}\mathcal{ALCEF}$ by means of the so-called concrete domains. In the same section we define a global similarity between objects from similarities between the values of each object's attribute (local similarities) and we show that the global similarity inherits the usual properties of the local similarities (reflexivity, symmetry or transitivity). In Section 7 we generalize a result of [35] to the logic MTL∀ relative to the global similarity. Finally there is a section devoted to concluding remarks and future work.