Klein Shmuel Tomi - Language and automata theory and applications: 12th international conference, LATA 2018, Ramat Gan, Israel, April 9-11, 2018: proceedings

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Invited Talks

Springer International Publishing AG 2017

Frank Drewes , Carlos Martn-Vide and Bianca Truthe (eds.) Language and Automata Theory and Applications Lecture Notes in Computer Science 10168 10.1007/978-3-319-53733-7_1

Approximation in Description Logics: How Weighted Tree Automata Can Help to Define the Required Concept Comparison Measures in

Franz Baader 1

(1)

Theoretical Computer Science, TU Dresden, Dresden, Germany

Franz Baader (Corresponding author)

Email:

Oliver Fernndez Gil

Email:

Pavlos Marantidis

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Abstract

Recently introduced approaches for relaxed query answering, approximately defining concepts, and approximately solving unification problems in Description Logics have in common that they are based on the use of concept comparison measures together with a threshold construction. In this paper, we will briefly review these approaches, and then show how weighted automata working on infinite trees can be used to construct computable concept comparison measures for that are equivalence invariant w.r.t. general TBoxes. This is a first step towards employing such measures in the mentioned approximation approaches.

P. MarantidisSupported by DFG Graduiertenkolleg 1763 (QuantLA).

Introduction

Description Logics (DLs) [, and define three DLs of different expressive power, namely the DLs , , and .

Since the semantics of traditional DLs is based on classical first-order logic, the interpretation of the properties required for a concept is strict in the sense that all these properties need to be satisfied for an individual to belong to a concept, and the same is true for answers to queries. In applications where exact definitions are hard to come by, it would be useful to relax this strict requirement and allow for approximate definitions of concepts, where most, but not all, of the stated properties are required to hold. Similarly, if a query has no exact answer, approximate answers that satisfy most of the features the query is looking for could be useful. For example, in clinical diagnosis, diseases are often linked to a long list of medical signs and symptoms, but patients that have a certain disease rarely show all these signs and symptoms. Instead, one looks for the occurrence of sufficiently many of them. Similarly, people looking for a flat to rent or a bicycle to buy may have a long list of desired properties, but will also be satisfied if many, but not all, of them are met.

In order to allow for approximate definitions of concepts, we have introduced the notion of a graded membership function in [] since they need not map into [0,1]. In the following, we will call functions that compare pairs of concepts by mapping them into a (usually numerical) domain equipped with a partial order concept comparison measures .

An indispensable requirement for the concept comparison measures used in the three approximation approaches mentioned above is that they respect the semantics of concepts in the sense that they are invariant under equivalence of concepts w.r.t. their definitions in the TBox. For the DL , a framework for defining concept similarity measures that are equivalence invariant w.r.t. acyclic TBoxes has been introduced in [], which then yield the output of the measure. We will show that, under certain conditions on the weighted tree automata, this approach indeed yields computable concept comparison measures.

Description Logics, Concept Comparison Measures, and Approximation

We start by recalling basic notions of Description Logics, and in particular the DLs , , and . Then, we introduce concept comparison measures, which generalize concept similarity measures, and finally we show how such measures can be used to relax query answering, approximately define concepts, and approximately solve unification problems.

2.1 Description Logics

In Description Logics, concept constructors are used to build complex concept descriptions out of concept names (unary predicates) and role names (binary predicates). A particular DL is determined by the available constructors. Given finite, disjoint sets and of concept names and role names, respectively, we denote the set of all concept descriptions that can be built from and using the constructors of with .

As an example, consider the constructors top concept ( ), bottom concept ( ), conjunction ( ), disjunction ( ), negation ( ), value restriction ( ), and existential restriction ( ), which determine the DL

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