How to Contract Ontologies

The dynamic nature of knowledge development has motivated the formal study of ontology evolution problems. In this paper, we study ontology contraction—the problem of retracting information that is no longer considered to hold. Our prime interest is to focus on ontologies expressed in Description Logics DL-Lite and EL, which underpin the OWL 2 QL and OWL 2 EL profiles. Our goal is to understand how to compute contractions under different kinds of semantics. As we have already shown in [1, 2], ontology contraction is technically very challenging. 1 Importance of Ontology Contraction Ontologies written in the Web Ontology Language (OWL) [3] and its revision OWL 2 [4] are becoming increasingly important for a wide range of applications. The formal underpinning of OWL is based on Description Logics (DLs) – knowledge representation formalisms with well-understood computational properties [5]. A DL ontology K consists of a TBox T , describing general (i.e., schema-level) domain knowledge, and an ABox A, providing data about specific individuals. Ontologies are not static entities, but rather they are frequently modified when new information needs to be incorporated, or existing information is no longer considered valid. The impact of such changes on the semantics of the ontology, however, is difficult to predict and understand. This dynamic nature of ontologies motivates the study of ontology evolution problems from both foundational and practical perspectives [2, 6–15]. We are interested in a particular aspect of ontology evolution, namely contraction – the process of “retracting” information that is no longer considered to hold [16, 17]. This information is typically represented by a formula α. Two important forms of contraction are crucial for many ontology design and management tasks: – TBox contraction, where the axiom α to be retracted is a TBox axiom; and – ABox contraction, where α is an ABox assertion and the TBox of the original ontology should remain the same. 2 Bernardo Cuenca Grau, Evgeny Kharlamov, and Dmitriy Zheleznyakov Consider scenarios for TBox and ABox contraction. OWL TBoxes are extensively used in the clinical sciences, and clinical ontologies such as SNOMED CT and NCI are subject to frequent modifications that involve retracting unintended consequences from the TBox [18]. For example, the developers of NCI perform over 900 monthly changes [18]. Furthermore, ABox contraction is important for applications relying on widely-used reference TBoxes. For example, bio-informaticians working on gene extraction can describe the experimental results using an ABox according to standard gene TBoxes. New experiments may imply that some facts about specific individuals no longer hold, which should be reflected in the ABox; at the same time, TBoxes should clearly not be affected by these manipulations of the data. 2 Practical View of Contraction Approaches to ontology contraction, typically adopted in practice (especially when changes occur at the TBox-level), are essentially syntactic [9, 13, 19]. Many such approaches are based on the notion of a justification: a minimal subset of the ontology that entails a given consequence [20, 21]. For example, to retract an axiom α entailed by K, it suffices to compute all justifications for α in K, find a minimal subset R of K (a “repair”) with at least one axiom from each justification, and take Ko = K \R as the result of the contraction. Retracting α results in the deletion of a minimal set of axioms and hence the structure of K is maximally preserved. 3 Logic-Based View of Contraction From a logic-based perspective, the desirable properties of contraction should be dictated by the principle of minimal change [16], according to which the semantics of the ontology should change “as little as possible”, thus ensuring that the contraction has the least possible impact. Logic-based semantics derived from the principle of minimal change have been recently studied in the more general context of ontology evolution. These semantics are either model-based (MBS) or formula-based (FBS). Under both types of semantics, evolution of an ontology K written in a DL DL results in a DL-ontology Ko in which the required information is incorporated, retracted, or updated; the difference is in the way Ko is obtained. Under MBS the set of all models M of K evolves into a new set M′ of models that are “as close as possible” to those in M (w.r.t. some notion of distance between models); then, Ko is the ontology that axiomatises M′ [10–12, 22, 23]. Under FBS, Ko is an ontology defined in terms of the deductive closure of K that satisfies the evolution requirements. FBS, however, have been less studied in the context of ontologies [1, 2, 11, 24]. 3 http://www.nlm.nih.gov/research/umls/Snomed/snomed_main.html 4 National Cancer Institute Thesaurus How to Contract Ontologies 3 Under both MBS and FBS, we are interested in computing an “optimal” contraction —that is, a contraction that is as similar as possible to the original ontology. In particular, an optimal contraction of a DL-ontology K with an axiom α is a DL-ontology Ko such that (i) K entails Ko, (ii) Ko does not entail α, and (iii) Ko is “as similar as possible” to K according to the particular notion of minimal change adopted by the semantics under consideration. 4 Issues with Practical and Logic-Based Approaches Practical approaches suffer from intrinsic information loss. More precisely, by removing R from K, one may inadvertently retract consequences of K other than α, which are “intended”. Identifying and recovering such intended consequences is an important issue. For example, if K consists of two TBox axioms: β1 = “VW is-a car”, β2 = “car is-a vehicle”, with an implicit information β3 = “VW is-a vehicle”, and α = β1, then Ko = {β2}; thus, β3 is lost. Logic-based approaches limit information loss, but they suffer from the following two problems. First, given a DL-ontology K and an axiom α, an optimal contraction Ko may not exist in DL, that is DL may not be closed under contraction. It has been shown [11, 22] that DL-Lite is not closed under so-called update and revision on both TBox and ABox levels; in [1] these results were extended to TBox and ABox contraction. In [1, 2] it has been shown that EL is not closed under TBox and ABox contraction. The second problem is that logic-based approaches are also problematic from a modeling point of view. Indeed, in contrast to syntactic approaches, these semantics do not distinguish between the axioms in the deductive closure that are explicit in K, and those that are merely implied. Ontologies, however, are the result of a time-consuming modeling process, and thus contractions should also preserve as much as possible the structure of K. For example, consider a DL-Lite-ontology K with the following two axioms β2 = “car is-a vehicle” and β4 = “VW is-a car and golf”, and let α = “lecturer is-a professor”. Clearly α is unrelated to K, and the optimal contraction of K with α is expected to be K itself. At the same time, the ontology Ko consisting of the following axioms is a valid contraction too (under any FBS and MBS considered in the ontology evolution literature): β2 = “car is-a vehicle” and β5 = “VW is-a car”, β6 = “VW is-a golf”, β7 = “VW is-a vehicle”. Returning such Ko as the contraction result might be undesirable from a practical point of view. We believe that our results in [1] suggest that classical approaches to ontology contraction, which are well-understood and well-behaved for propositional logics, are intrinsically problematical in the context of ontology languages. 5 Bridging Logics and Practice In [2], these limitations of both syntactic and logic-based approaches were addressed (at least partly). On the one hand, the semantics in [2] provides a “bridge” 4 Bernardo Cuenca Grau, Evgeny Kharlamov, and Dmitriy Zheleznyakov between syntactic and formula-based approaches; on the other hand, this semantics provides a distinction between the languages DL in which the original ontology K and the resulting contraction are expressed, and the language LP (the preservation language), which expresses the entailments of K that must be maximally preserved. The principle of minimal change is reflected along two dimensions: (i) structural, where the explicit axioms in K are maximally preserved; (ii) deductive, where the consequences of K in LP are maximally preserved. More formally, the semantics of [2] can be defined as follows in the context of contraction. Definition 1. Let DL and LP be description logics with LP ⊆ DL. Let K be a DL-ontology, and let α be a DL-axiom such that K |= α. A DL-ontology Ko is an optimal contraction of K with α w.r.t. the preservation language LP if (i) K |= Ko; (ii) Ko 6|= α; (iii) Ko∪{β} |= α, for each β ∈ K\Ko; and (iv) Ko ∪ {γ} |= α, for each LP-axiom γ that is entailed by K but not by Ko.