On Using Sub jective Logic to Build Consistent Merged Ontologies

Abstract

Ontologies encode a community's understanding of a domain and are thus subjective. Di erent communities may create di erent models of overlapping domains. If they need to be integrated, this may cause problems: combining subjective knowledge from diverse ontologies into a merged model may make it inconsistent. We provide a Subjective Logic-based approach to support users in creating a consistent merged ontology re ecting a commonly trusted view of the domain.

Keywords

• Ontology merging
• Consistency
• Subjective Logic

Introduction

Ontologies represent domain knowledge. They are subjective with regards to the creators' view of the domain. This can cause problems, when ontologies need to be merged: while each of them models a consistent view of the world, their combination may be inconsistent. Whereas inconsistencies in single ontologies that result from modeling errors are comparatively easy to resolve automatically [ 8,10 ], inconsistencies that arise due to di ering views of the world are di cult to deal with. In order to resolve them, the subjective beliefs about the world and how trustworthy they are need to be taken into consideration.

A formal approach to capture beliefs and trustworthiness is Subjective Logic [ 6 ], which consists of a belief model called opinion and a set of operations for combining opinions. It is applicable in situations with considerable uncertainty and incomplete knowledge. This formalism has been successfully applied to a number of Semantic Web related tasks, e.g., for ontology alignment [ 7 ], recommendation systems [ 9 ], and inconsistency handling in single ontology development environments [ 10 ].

We propose to use Subjective Logic theory for inconsistency resolution in ontology merging. More precisely, the problem that we are addressing is the following: Given an inconsistent ontology that is the result of merging several consistent source ontologies, determine which changes should be made to make the ontology consistent. These changes should preserve the most trusted point of view on the shared domain. A rst evaluation shows that this approach is promising.

Applying Subjective Logic to Achieve Consistency

We call an ontology O inconsistent i there is no model of O, i.e., O is unsatis able [ 2 ]. A rst step to resolving inconsistency in a merged ontology is to pinpoint its origin. To do so, the ontology is evaluated using an o -the-shelf reasoner. If the consistency test fails, the con icting axioms set that cause inconsistencies are extracted using the reasoner. In the next step, the trustworthiness of each axiom is computed using our Subjective Logic-based approach. For the least trustworthy axioms, a suggested revised axioms' set (following the approach from [ 8 ]) will be represented to the user.

Subjective Logic. Subjective opinions express beliefs of agents about the truth of propositions with degrees of uncertainty [ 6 ]. In our approach we use binary subjective logic and regard the source ontologies as agents, as they represent their communities' beliefs about the domain. Let P be a proposition such as \Axiom x is trustworthy in the merged ontology OM ". Following [ 6 ], the opinion w of agent Oi about the proposition P is equivalent to a beta distribution for the information source x as the combination of belief bxOi , disbelief dxOi , uncertainty uxOi , and base rate (atomicity) aOi with a tuple x wxOi = (bxOi ; dxOi ; uxOi ; axOi ), where bOi + dOi + uOi = 1.

x x x

Opinions are formed on the basis of positive r and negative s evidence about x available to agent Oi. Then, bxOi ; dxOi , and uxOi are: bOi = x rOi x rxOi + sxOi + W ; dOi = x sOi x rxOi + sxOi + W ; uOi = x

W rxOi + sxOi + W (1) where, W is the default non-informative prior weight that in binomial opinions is de ned as W = 2. Thus, the opinion's probability expectation value is computed by txOi = bxOi + axOi uxOi as the trustworthiness of x by agent Oi. This requires that we determine rxOi , sOi (to calculate bxOi ; dxOi ; uxOi ) and axOi .

x

Positive and negative evidence. To determine the positive evidence r of an axiom xj , we use provenance information. Each axiom in OM is derived from one or several input ontologies Oi. Therefore, r for xj from agent Oi is calculated in Eq. 2 as (i) the existence of axiom xj in Oi (provenance information), and (ii) the impact of the axiom's elements (e ect ), to re ect how much the ontology gets a ected if axiom xj is altered. By axiom's elements, we mean those elements which are involved in an axiom, e.g., A and B in the axiom x1 : A v B. To this end, we determine how often elements of axiom xj have been referenced in other axioms in the ontology. Let c(Oi) be the total number of axioms in Oi and cxj (Oi) be the number of axioms in Oi that contain elements of xj . Then fxj (Oi) = cxcj( O(Oi)i) is the fraction of axioms in Oi that contain elements of xj . The provenance of the axioms is represented by the and parameters; if xj 2= Oi, but the elements of the axioms exist in Oi, then fxj (Oi) multiplies with , otherwise, it multiplies with .

rxOji = ( fxj (Oi) if xj 2 Oi fxj (Oi) if xj 2= Oi (2)

A justi cation is a minimal subset of an ontology that causes it to be inconsistent. Let O be an ontology entailing axiom x ( O j= x). J is a justi cation for x in O if J O, and J j= , and for all J 0 ( J J 0 2 [ 4 ]. The ontology justi cation set J is the set of all justi cations, J = fJ1; J2; :::; Jlg, where there may be multiple, potentially overlapping justi cations in J . Each justi cation Jk 2 J includes several axioms, denoted by Jk = (x1; x2; :::; xz). In this follow, to determine the negative observations s, we use the axiom frequency cxj (J ) in the justi cation set J divided by the number of the con icting axioms set that cause inconsistencies c(X) which belong to Oi in Eq. 3 (to re ect the view of Oi). This metric is already used in the [ 8 ], to accelerate the process of getting rid of unsatis able concepts.

sxOji = cxj (J ) c(X)

Base Rate. In the absence of evidence for belief, disbelief, and uncertainty, the base rate plays an important role. It re ects prior knowledge about the phenomenon at hand. In our case, using the centrality measure [ 1 ] of elements exj seems a suitable indicator. The base rate for axiom xj with t elements xj = fe1; e2; :::; etg is given by Eq. 4. This is determined by the number of super- and subclasses of the elements divided by the total number of elements jej in Oi. axOji =

1 jej 2 Oi

t X jSubClass(ek) [ SuperClass(ek)j; ek 2 xj k=1 (3) (4)

Combining Opinions and Conditional Opinions. The Subjective Logic operator consensus [ 6 ] combines the opinions in such a way, that the more trustworthiness opinions will be those that are agreed upon by multiple agents. Let wxO1 = (bxO1 ; dxO1 ; uxO1 ; axO1 ) and wxO2 = (bxO2 ; dxO2 ; uxO2 ; axO2 ) be opinions respectively held by O1 and O2 about the same proposition x. Then the consensus for these two opinions is wxO1O2 = wxO1 wxO2 [ 6 ], it re ects the opinion of an imaginary agent representing both O1 and O2. In principle, this operator can be used in our approach to combine opinions by di erent agents and reach a consensus.

However, this would have a drawback: it does not consider the e ect of the calculated ranked values for axioms in Jk on Jl, k 6= l. To overcome this issue, we use conditional theory of Subjective Logic [ 6 ], which re ects the e ect of the dependent opinions. Let us consider an example: J = fJ1; J2; J3g is a set of justi cations, where axioms are repeated in multiple J s, as J = f(x1; x2; x3); (x4; x5; x6; x7); (x3; x4; x5; x7; x8)g. The opinions for J1's axioms can be calculated as independent opinions. However, some elements of J3 have already obtained some ranked values from J1 and J2. Here, we can use the previous ranked values from J1 and J2 for x3; x4; x5; x7 in J3, but it might happen that these axioms compared to the remaining axioms in J3 get di erent ranked values. Therefore, in an incremental process, we calculate a new value in each J , but we also consider the e ect of the previous ranked values for axioms in other J s. Thus, we enrich our method by using the conditional deduction operator } introduced in [ 6 ] to express this. 3

Related Work

To handle inconsistencies in one ontology, various researches such as [ 8 ] have been done, where the authors ranked the justi cation with a single metric. In [ 10 ], Subjective Logic is used to solve ontology inconsistencies in a single ontology process, not applying Subjective Logic to ontologies merging process, where the agent's opinions from the input ontologies play a serious role. Moreover, no agent's opinion combination has been considered, and the authors only utilized the atomicity value and omitted the belief, disbelief and uncertainty values. To deal with multiple ontologies, the authors in [ 5 ] considered multiple ontologies that are networked via mappings for distributed environments, only. To the best of our knowledge, this is the rst work that by using the Subjective Logic theory considers the knowledge of input sources to handle the inconsistencies on a merged model. As a whole, we di er from other works in three key respects: we solve inconsistencies in the ontology merging process; we combine several criteria to rank the con icting axioms set that cause inconsistencies with belief, disbelief and atomicity values; we consider the combination of agents' opinions and we employ conditional ranking when the con icting axioms are dependent. 4

Preliminary Evaluation

As a preliminary evaluation, the proposed work ow has been implemented within our merge framework [ 3 ] 1. We conducted a series of tests 2 on the OAEI benchmark 3. Table 1 shows the TBox T and Abox A size of the merged ontologies. In a rst step, we use the reasoner to determine whether the merged ontology is consistent (PASSED) or not (FAILED). For the ontologies 1 http://comerger.uni-jena.de/ 2 on Intel Core i7 with 12 GB internal memory; Pellet reasoner; 3 http://oaei.ontologymatching.org/ = 1, = 0:5 that do not pass the consistency check, we then determine the unsatis able classes Cun and the justi cation set J . Afterwards, our method processes all axioms c(X) of the justi cation set, and ranks them. Table 1 shows the size of Cun; J and c(X) for each inconsistent merged ontology. The axioms with the lowest trustworthiness are presented to the user together with a suggested resolution. The rightmost columns of the table show the runtime (in millisecond) for detecting the inconsistencies (i.e., extracting Cun; J and c(X)), ranking and generating the resolution plan respectively. Given that ontology merging is a complex, time consuming task overall, they seem acceptable. 5

Conclusion and Future Work

We propose a novel approach using Subjective Logic to estimate the trustworthiness of axioms that cause inconsistencies within a merged ontology. We use provenance information and structural relevance to assess the opinions of the input ontologies. With the consensus operator, con icting opinions can be combined. Moreover, we adopted conditional theory in the Subjective Logic to re ect the opinion of an axiom which is dependent on another opinion. A rst evaluation shows that the approach is promising. A pretty straightforward extension of this work that we are pursuing already, is to determine root causes of inconsistencies and restrict the approach to those. Also, using domain knowledge might improve the estimation of an opinion's probability expectation.

Acknowledgments. S. Babalou is supported by a scholarship from German Academic Exchange Service (DAAD).

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