Structural Similarity in Expressive Description Logics: An Extended Family of Kernels for OWL
چکیده
In the context of the Semantic Web many applications of inductive inference ultimately rely on a notion of similarity for the standard knowledge representations of the ontologies. We have tackled the problem of statistical learning with ontologies proposing a family of structural kernels for ALCN that has been integrated with support vector machines. Here we extend the definition of the kernels to more expressive languages of the family, namely those backing OWL. 1 Concept Similarity in Ontologies Although machine learning techniques may have a great potential for the Semantic Web (SW) applications, ontology learning tasks have focused mainly on methods for text [4]. Much less effort has been devoted to the application of machine learning methods to knowledge bases described in formal concept representations of the SW (formal ontologies) ultimately based on Description Logics (DL) [1]. In order to apply statistical learning some notion of similarity for the specific representation is needed [8]. It is particularly appealing to work with kernel methods that allow for decoupling the final learning algorithm (e.g. perceptrons, support vector machines, etc.) from the notion of similarity which depends on the particular instance space which is encoded in the kernel function. We have been investigating on the definition of kernel functions for instance spaces represented in relational languages such as those based on clauses and lately also on DLs. One of the first works concerns kernels for the Feature Description Logic [6] which proved particularly effective for relational structures elicited from text. More recently further proposals for working with more complex DL and ontology languages have been made [10, 2, 13]. In this work, we propose a family of declarative kernel functions that can be applied to DLs representations with different degrees of expressiveness. The kernels encode a notion of similarity of individuals in this representation, based on structural and semantic aspects of the reference representation. Specifically, we extend the definition of a family of kernels for the ALCN logic [12]. These kernel functions are based on both structural and also semantic aspects, namely a normal form and the extension of the concepts involved in the description, as elicited from the knowledge base. As such the kernels are designed for comparing concept descriptions. In order to apply them to instance comparison w.r.t. real ontologies, that likely exploit the full extent of the most expressive DL languages, (upper) approximations of the most specific concepts that cover the single individuals had to be computed [5]. This exposes the method to a number of problems. Firstly, a (partially) structural normal form may fail to fully capture the semantic similarity between the individuals. Scaling up to more complex languages wold require specific normal forms. Moreover, the existence of a most specific concept is not guaranteed [1]. Coupling valid kernels with efficient algorithms such as the support vector machines, many tasks based on inductive classification can be tackled. Particularly, we demonstrate how to perform important inferences on semantic knowledge bases, namely concept retrieval and query answering. These tasks are generally grounded on merely deductive procedures which easily fail in case of (partially) inconsistent or incomplete knowledge. The methods has been shown to work comparably well w.r.t. standard reasoners [12, 13], allowing the suggestion of new knowledge that was not previously logically derivable. Moreover, these kernels naturally induce distance measures which allow for extensions to further metric-based learning methods such as nearest-neighbor classification [7] and conceptual clustering [11]. However these extensions are beyond the scope of this work. 2 Preliminaries on Representation and Inference The basics of the DLs will be briefly recalled. The reader may refer to the DL handbook [1] for a thorough reference. Such representations provide the basic constructors adopted by the standard ontology languages employed in the SW, such as the Ontology Markup Language OWL. We will focus on the ALCQ logic which may represent a tradeoff between expressiveness and reasoning efficiency. 2.1 Knowledge Bases in Description Logics Let us consider a triple 〈NC , NR, NI〉 made up. respectively, by a set of primitive concept names NC , to be interpreted as sets of objects in a certain domain, a set of primitive role names NR, to be interpreted as binary relationships between the mentioned objects, and a set of individual names NI for the objects themselves. The semantics of the descriptions is defined by an interpretation I = (∆I , ·I), where ∆I is a non-empty set, the domain of the interpretation, and ·I is the interpretation function that maps each individual a ∈ NI to a domain object aI ∈ ∆I , each primitive concept name A ∈ NC to its extension AI ⊆ ∆I and for each R ∈ NR the extension is a binary relation RI ⊆ ∆I ×∆I . Complex descriptions can be built in ALCQ using the language constructors listed in Table 1, along with their semantics derived from the interpretation of atomic concepts and roles [1]. The top concept > is interpreted as the whole domain ∆I , while the bottom concept ⊥ corresponds to ∅. Complex descriptions can be built in ALCQ using the following constructors. The language supports Table 1. Syntax and semantics of concepts in the ALCQ logic. Name Syntax Semantics top concept > ∆I bottom concept ⊥ ∅ full concept negation ¬C ∆I \ CI concept conjunction C1 u C2 CI 1 ∩ CI 2 concept disjunction C1 t C2 CI 1 ∪ CI 2 existential restriction ∃R.C {x ∈ ∆I | ∃y ∈ ∆I((x, y) ∈ RI ∧ y ∈ CI)} universal restriction ∀R.C {x ∈ ∆I | ∀y ∈ ∆I((x, y) ∈ RI → y ∈ CI)} qual. at least restriction ≥ nR.C {x ∈ ∆I | |{y ∈ ∆I : (x, y) ∈ RI ∧ y ∈ CI |} ≥ n} qual. at most restriction ≤ nR.C {x ∈ ∆I | |{y ∈ ∆I : (x, y) ∈ RI ∧ y ∈ CI |} ≤ n} full negation: a concept negation ¬C has an extension that amounts to the complement of CI w.r.t. the domain. The conjunction and disjunction of two concepts are simply interpreted as the intersection and union of their extensions. Concepts can be also defined as restrictions on the roles. The existential restriction constrains the elements of its extension to be related via a certain role R to some instances of concept C while the value (or universal) restriction comprises those elements who are related through R only to instances of concept C (if any). Finally, qualified numeric restrictions define concepts by constraining its instances with the minimal or maximal number of instances of concept C related through R. OWL offers further constructors that extend the expressiveness of this language. Its DL equivalent is SHOIQ(D) [1], that extends ALCQ with individual classes, role hierarchies, transitive and inverse roles (I). Besides concrete domains (D), i.e. well-founded external data types (such as numerical types, tuples of the relational calculus, spatial regions, or time intervals), can be dealt with. The main inference employed with these representations is assessing whether a concept subsumes another concept based on their semantics: Definition 2.1 (subsumption). Given two descriptions C and D, C is subsumed by D, denoted by C v D, iff for every interpretation I it holds that CI ⊆ DI . When C v D and D v C then they are equivalent, denoted with C ≡ D. Note that this naturally induces a generality relationship on the space of concepts. Generally subsumption is not assessed in isolation but rather related to the models of a system of axioms representing the background knowledge and the world state: Definition 2.2 (knowledge base). A knowledge base K = 〈T ,A〉 contains a TBox T and an ABox A. T is the set of terminological axioms of concept descriptions C ≡ D, where C is the concept name and D is its description. A contains assertions on individuals C(a) and R(a, b). An interpretation that satisfies all its axioms is a model of the knowledge base. Note that defined concepts should have a unique definition. However defining concepts through inclusions axioms (C v D) is also generally admitted. Moreover, such definitions are assumed to be unfoldable. Example 2.1 (royal family). This example shows a knowledge base modeling concepts and roles related to the British royal family: T = { Male ≡ ¬Female, Woman ≡ Human u Female, Man ≡ Human u Male, Mother ≡ Woman u ∃hasChild.Human, Father ≡ Man u ∃hasChild.Human, Parent ≡ Father t Mother, Grandmother ≡ Mother u ∃hasChild.Parent, Mother-w/o-daughter ≡ Mother u ∀hasChild.¬Female, Super-mother ≡ Mother u ≥3 hasChild.Human } A = { Woman(elisabeth), Woman(diana), Man(charles), Man(edward), Man(andrew), Mother-w/o-daughter(diana), hasChild(elisabeth, charles), hasChild(elisabeth,edward), hasChild(elisabeth, andrew), hasChild(diana, william), hasChild(charles, william) } Note that the Open World Assumption (OWA) is made in the underlying semantics, since normally knowledge representation systems are applied in situations where one cannot assume that the knowledge in the base is complete. Although this is quite different from the standard settings considered in machine learning and knowledge discovery, it is convenient for the Semantic Web context where new resources (e.g. Web pages, Web services) may be continuously made available. 2.2 Inference Services The basic inference services for DL knowledge bases can be viewed as entailments as they amount to verifying whether a generic relationship is a logical consequence of the knowledge bases axioms. Typical reasoning tasks for TBoxes regards the satisfiability of a concept, subsumption, equivalence or disjointness between two concepts. For example in the knowledge base reported in Ex. 2.1, Father is subsumed by Man and is disjoint with Woman. These inferences can be reduced to each of them, hence normally reasoners support only one of them. Reasoning is performed though tableau algorithms [9, 1]. Inference problems concerning ABoxes are mainly related to checking its consistency, that is, whether it has a model, especially w.r.t. those of the TBox. For example, an ABox like the one in Ex. 2.1 containing also the assertion Father(elisabeth) is not consistent. Since we aim at crafting inductive methods that manipulate individuals, a prototypical inference is instance checking [1], that amounts to checking whether an assertion α is entailed by the knowledge base (K |= α). If α is a role assertion then it is quite easy to perform the inference as normally roles do not have a definition in the TBox. However, we will focus on deciding whether an individual is an instance of a given concept. This can be easily reduced to a consistency problem: K |= C(a) iff K ∪ {¬C(a)} is inconsistent. The adopted open world semantics has important consequences on the way queries are answered. While the closed-world semantics identifies a database with a single model an ABox represents possibly infinitely many interpretations. Hence a reasoner might be unable to answer certain queries because it may be actually able to build models for both a positive (K ∪ {C(a)}) and a negative (K ∪ {¬C(a)}) answer. Example 2.2 (open world semantics). Given the ABox is Ex. 2.1, while it is explicitly stated that diana is a Mother-w/odaughter, it cannot be proven for elisabeth because she may have daughters that are simply not known in the knowledge base. Analogously, elisabeth is an instance of Super-mother while this cannot be concluded for diana as only two children are known in the current knowledge base. Although these instance checks fail because of the inherent incompleteness of the knowledge state, this does not imply that the individuals belong to the negated concepts, as could be inferred in case the CWA were made. Another related inference is retrieval which consists in querying the knowledge base to know the individuals that belong to a given concept: Definition 2.3 (retrieval). Given an knowledge base K and a concept C, find all individuals a such that K |= C(a). A straightforward algorithm for a retrieval query can be realized via instance checking on each individual occurring in the ABox, testing whether it is an instance of the concept. Dually, it may be necessary to find the (most specific) concepts which an individual belongs to. This is called a realization problem. One especially seeks for the most specific one (up to equivalence): Definition 2.4 (most specific concept). Given an ABox A and an individual a, the most specific concept of a w.r.t. A is the concept C, denoted MSCA(a), such that A |= C(a) and for any other concept D such that A |= D(a), it holds that C v D. This is a typical way to lift individuals to the conceptual level [5]. For some languages, the MSC may not be expressed by a finite description [1], yet it may be approximated by a more general concept [17]. Generally approximations up to a certain depth k of nested levels are considered, denoted MSC. We will generically indicate a maximal depth approximation with MSC∗. For further details see also [12]. 2.3 A Normal Form for ALCQ Many semantically equivalent (yet syntactically different) descriptions can be given for the same concept. Equivalent concepts can be reduced to a normal form by means of rewriting rules that preserve their equivalence [1]. We will adopt a normal form extending one given for ALC descriptions [3], for concepts that are already in negation normal normal form. Preliminarily, some notation is necessary for naming the various nested subparts (levels) of a concept description D: – prim(D) is the set of all the primitive concepts (or their negations) at the top-level of D; – valR(D) is the concept in ALCQ normal form in the scope of the value restriction (if any) at the top-level of D (otherwise valR(D) = >); – exR(D) is the set of the descriptions C ′ appearing in existential restrictions ∃R.C ′ at the top-level conjunction of D; – minR.C(D) = max{n ∈ IN | D v ≥nR.C} (always a finite number); – maxR.C(D) = min{n ∈ IN | D v ≤nR.C} (if unlimited, maxR.C(D) =∞). A normal form may be recursively defined as follows: Definition 2.5 (ALCQ normal form). A concept description C is in ALCQ normal form iff C = ⊥ or C = > or if C = C1 t · · · t Cn with