Using a variant of Lehmann's Default Logics and Probabilistic Description Logics we recently presented a framework that invalidates those unwanted inferences that cause concept unsatisfiability without the need to remove explicitly stated axioms. The solutions of this methods were shown to outperform classical ontology repair w.r.t. the number of inferences invalidated. However, conflicts may still exist in the knowledge base and can make reasoning ambiguous. Furthermore, solutions with a minimal number of inferences invalidated do not necessarily minimize the number of conflicts. In this paper we provide an overview over finding solutions that have a minimal number of conflicts while invalidating as few inferences as possible. Specifically, we propose to evaluate solutions w.r.t. the quantity of information they convey by recurring to the notion of entropy and discuss a possible approach towards computing the entropy w.r.t. an ABox

Bernstein, A

d'Amato, C

Fanizzi, N

Grütter, R

Scharrenbach, T

Waldvogel, B

ZORA

University of ZurichZurich Open Repository and ArchiveWinterthurerstr. 190CH-8057 Zurichhttp://www.zora.uzh.chYear: 2010Default logics for plausible reasoning with controversial axiomsScharrenbach, T; d'Amato, C; Fanizzi, N; Grütter, R; Waldvogel, B; Bernstein, AScharrenbach, T; d'Amato, C; Fanizzi, N; Grütter, R; Waldvogel, B; Bernstein, A (2010). Default logics forplausible reasoning with controversial axioms. In: 6th International Workshop on Uncertainty Reasoning for theSemantic Web (URSW-2010), Shanghai, China, 07 November 2010 - 07 November 2010.Postprint available at:http://www.zora.uzh.chPosted at the Zurich Open Repository and Archive, University of Zurich.http://www.zora.uzh.chOriginally published at:Scharrenbach, T; d'Amato, C; Fanizzi, N; Grütter, R; Waldvogel, B; Bernstein, A (2010). Default logics forplausible reasoning with controversial axioms. In: 6th International Workshop on Uncertainty Reasoning for theSemantic Web (URSW-2010), Shanghai, China, 07 November 2010 - 07 November 2010.Scharrenbach, T; d'Amato, C; Fanizzi, N; Grütter, R; Waldvogel, B; Bernstein, A (2010). Default logics forplausible reasoning with controversial axioms. In: 6th International Workshop on Uncertainty Reasoning for theSemantic Web (URSW-2010), Shanghai, China, 07 November 2010 - 07 November 2010.Postprint available at:http://www.zora.uzh.chPosted at the Zurich Open Repository and Archive, University of Zurich.http://www.zora.uzh.chOriginally published at:Scharrenbach, T; d'Amato, C; Fanizzi, N; Grütter, R; Waldvogel, B; Bernstein, A (2010). Default logics forplausible reasoning with controversial axioms. In: 6th International Workshop on Uncertainty Reasoning for theSemantic Web (URSW-2010), Shanghai, China, 07 November 2010 - 07 November 2010.Default logics for plausible reasoning with controversial axiomsAbstractUsing a variant of Lehmann's Default Logics and Probabilistic Description Logics we recently presenteda framework that invalidates those unwanted inferences that cause concept unsatisfiability without theneed to remove explicitly stated axioms. The solutions of this methods were shown to outperformclassical ontology repair w.r.t. the number of inferences invalidated. However, conflicts may still existin the knowledge base and can make reasoning ambiguous. Furthermore, solutions with a minimalnumber of inferences invalidated do not necessarily minimize the number of conflicts. In this paper weprovide an overview over finding solutions that have a minimal number of conflicts while invalidatingas few inferences as possible. Specifically, we propose to evaluate solutions w.r.t. the quantity ofinformation they convey by recurring to the notion of entropy and discuss a possible approach towardscomputing the entropy w.r.t. an ABox.Default Logics for Plausible Reasoning withControversial AxiomsThomas Scharrenbach1, Claudia d’Amato2, Nicola Fanizzi2, Rolf Gru¨tter1,Bettina Waldvogel1, and Abraham Bernstein31 Swiss Federal Institute for Forest, Snow and Landscape Research WSLBirmensdorf, Switzerland{thomas.scharrenbach, rolf.gruetter, bettina.waldvogel}@wsl.ch2 Universita` degli Studi di Bari Bari, Italy {claudia.damato, fanizzi}@di.uniba.it3 University of Zurich, Department of Informatics Zurich, Switzerland{bernstein}@ifi.uzh.chAbstract. Using a variant of Lehmann’s Default Logics and Probabilis-tic Description Logics we recently presented a framework that invalidatesthose unwanted inferences that cause concept unsatisfiability without theneed to remove explicitly stated axioms. The solutions of this methodswere shown to outperform classical ontology repair w.r.t. the number ofinferences invalidated. However, conflicts may still exist in the knowledgebase and can make reasoning ambiguous. Furthermore, solutions with aminimal number of inferences invalidated do not necessarily minimize thenumber of conflicts. In this paper we provide an overview over findingsolutions that have a minimal number of conflicts while invalidating asfew inferences as possible. Specifically, we propose to evaluate solutionsw.r.t. the quantity of information they convey by recurring to the no-tion of entropy and discuss a possible approach towards computing theentropy w.r.t. an ABox.1 IntroductionIn the Semantic Web, knowledge is represented by ontologies expressed in theWeb Ontology Language OWL. The current standard, OWL2 [1], defines differ-ent profiles all of which have some Description Logics as a rough syntactic vari-ant. These Description Logics (DL) are decidable fragments of first-order logicswhere knowledge is explicitly expressed in axioms and assertions. DL knowledgebases have well-defined model-theoretic semantics. They allow to express knowl-edge on different levels of expressivity and enable to infer new conclusions fromexisting knowledge.When ontologies evolve or one ontology is mapped to another, contradictionsmay be introduced that cause the knowledge base as a whole to be inconsis-tent. Yet, for an inconsistent knowledge base any conclusion—even meaninglessones—becomes trivially true. One cause of inconsistency is given by assertionsof concepts that are inferred to be unsatisfiable. Hence, it is desirable to preventconcepts from being inferred unsatisfiable. A knowledge base can become incon-sistent for other reasons, but we propose to start off with conflict-free conceptu-alizations and apply a method that never infers any concept to be unsatisfiable.In the Semantic Web, agents interacting with an ontology assume that boththe query and the answer are expressible in OWL2. Furthermore, the answershould have meaningful semantics but not infer conflicts. We therefore demandany formalism allowing for plausible reasoning on controversial information tofulfill the following properties:1. Permanence: The formalism for knowledge representation is not changed.2. Coherency: No concept is inferred to be unsatisfiable3. Autonomy: The procedure shall work automatically.4. Originality: The original information should be kept.5. Conservation: As little inferred information as possible shall be lost.We presented a method for solving unsatisfiable concepts [2] using a combina-tion of Lehmann’s Default Logics [3] and Lukasiewicz’ Probabilistic DescriptionLogics [4]. Instead of removing (explicit) axioms, we propose to invalidate thoseinferences that cause concepts to be inferred unsatisfiable [5]. While it is possi-ble to reason with all information provided, we may still produce contradictinginferences. In this paper we show that minimizing the number of inferences inval-idated does not necessarily minimize the number of those conflicts. For findingoptimal solutions we propose to evaluate these w.r.t. their information contentwhich requires the definition of the entropy of a solution. We discuss a possibleapproach towards computing the entropy w.r.t. an ABox and give an outlook onfuture work.2 ProcedureFor each unsatisfiable concept U of an ontology, its justifications JkUv⊥ [6], i.e.the minimal sets of axioms explaining the conflict, are determined in a firststep. Each of these justifications is split up into two sets: one that contains allaxioms which contain the unsatisfiable concept, Γ kUv⊥ and one that containsall other axioms of that very justification, ΘkUv⊥ [2]. Afterwards, the root unsatjustifications are determined, which are those justifications that do not dependon any other justification [7].According to the partition scheme of Lehmann’s Default Logics, the axiomsof the root justifications are put into partitions U0, . . . ,UN and a separate TBoxT∆ such that all concepts in T∆∪Un are satisfiable for n = 0, . . . N . Thanks to thesplitting, we do not have to perform additional satisfiability checks for computingthe partition. The resulting Default TBox is a family of (classical) TBoxes:DT = (T∆ ∪ U0, . . . , T∆ ∪ UN ). For such a Default TBox we may either use theinference methods provided by Probabilistic Description Logics [4] or stick toclassical reasoning on the single partitions, separately. Either approach defines adeductive closure of the Default TBox as a set of OWL2 axioms, but we prefer thelatter approach to change the formalism for reasoning only as little as possible.Instead of putting all axioms of the root unsat justifications into the parti-tions, we showed in [5] that we indeed have to put only two axioms of each rootunsat justification into the partitions—one of eachΘkUv⊥ and one of each ΓkUv⊥—while we may put the remaining axioms into T∆. While potentially invalidatingless inferences, however, finding partitions may become non-deterministic.We propose to approximate an optimal solution by a (stochastic) search pro-cess: On the one hand, the number of possible solutions is exponential in thenumber of axioms in the justifications. On the other hand, once the justifica-tions are known, finding a single valid solution can be performed efficiently, be-cause the complexity of the approach is dominated by the complexity of findingjustifications—a task which has to be performed anyhow.3 Minimizing Conflicts by Minimizing the EntropyBy invalidating the inferences of the kind DT |= U v ⊥ we ignore the conflictsduring reasoning. Yet, inferences such as the co-occurrence of DT |= A andDT |= ¬A are still possible but not desired. Hence, a performance measure thatassesses the quality of a solution must not only take into account the numberof inferences invalidated but, even more important, the number of conflicts stillremaining.Assume the simple TBox T = {B v A,C v B,C v ¬A, } which has twoDefault TBoxes as potential solutions:DT 0 with T 0∆ = {C v B}, U00 = {B v A}, U01 = {C v ¬A}DT 1 with T 1∆ = {C v ¬A}, U10 = {B v A}, U11 = {C v B}In contrast to the latter, the first Default TBox DT 0 preserves the inferenceC v A. Yet, in the presence of an ABox that infers the assertion C(i), theassertion A(i) as well as its complement ¬A(i) can be inferred. The secondDefault TBox DT 1, in contrast, infers only ¬A(i). It is preferred over DT 0,because it contains fewer conflicts than DT 1.Conflicts potentially reduce the information content of a knowledge base. Forminimizing the number of conflicts as well as the number of inferences invalidatedwe are currently investigating qualitative measures based on the entropy of apossible solution. As opposed to methods based on the structure of an ontology[8], we propose that an entropy-measure should take into account the ambiguityof different ABoxes.In information theory, the entropy measures the average information contentof a random variable we are missing when the value of the random variable is notknown [9]. If we know the probability mass function p of the random variable X,we may explicitly denote the entropy byH(X) = −∑Nn=0 p(xn) log p(xn). In casep(xn) = 0, then p(xn) log p(xn) = 0. We propose to approximate the probabilitymass function pA for the axioms B v A ∈ DT by counting assertions for theconcept (¬BunionsqA) found by the instance retrieval service of the reasoning process:pA(B v A) = |{x ∈ AI | T ,A |= (¬B unionsqA)(x)}|∑DvC∈DT |{y ∈ AI | T ,A |= (¬D unionsq C)(y)}|The entropy of a Default TBox DT measures the information content of itsaxioms w.r.t. an ABoxA:H(DT ,A) = −∑BvA∈(DT ) pA(B v A) log pA(B v A).For the Default TBoxes in the example above, we obtain an entropy ofH(DT 0) = − log(1/3) and H(DT 1) = − log(1/2) which would make us chooseDT 1 rather than DT 0. Our current hypothesis is that a Default TBox with min-imal entropy also minimizes the number of explicit conflicts w.r.t. an ABox. Aprototype implementation is available 4.4 ConclusionWe recently introduced a framework that never infers any concept to be unsatis-fiable while keeping all originally provided information. This allows plausible rea-soning on ontologies that possibly contain controversial information—as it is thecase for mapped or dynamic ontologies. Finding solutions is non-deterministicand requires optimization techniques that, in turn, require a performance mea-sure for evaluating the quality of possible solutions.While reasoning ignores conflicts, they are still present in the knowledge baseand may lead to sub-optimal results. It was shown that solutions invalidating aminimal number of inferences do not necessarily minimize the number of con-flicts still present. For minimizing these we proposed to use an entropy-basedperformance measure. We provided a definition for the entropy of a solutionw.r.t an ABox which is currently being further investigated.References1. Hitzler, P., Kro¨tzsch, M., Parsia, B., Patel-Schneider, P.F., Rudolph, S.: OWL 2Web Ontology Language Primer. W3C Recommendation, W3C (2009)2. Scharrenbach, T., Gru¨tter, R., Waldvogel, B., Bernstein, A.: Structure PreservingTBox Repair using Defaults. In: 23rd Intl. Workshop on Description Logics. (2010)3. Lehmann, D.: Another perspective on default reasoning. Ann. Math. Artif. Intell15 (1995) 61–824. Lukasiewicz, T.: Expressive probabilistic description logics. Art. Intell. 172(6-7)(2008) 852–8835. Scharrenbach, T., d’Amato, C., Fanizzi, N., Gru¨tter, R., Waldvogel, B., Bernstein,A.: Unsupervised conflict-free ontology evolution without removing axioms. In: 4thInternational Workshop on Ontology Dynamics (IWOD-2010). (to appear).6. Schlobach, S., Cornet, R.: Non-standard reasoning services for the debugging ofdescription logic terminologies. In: Proc. of IJCAI 2003. (2003) 355–3627. Kalyanpur, A., Parsia, B., Sirin, E., Hendler, J.: Debugging unsatisfiable classes inowl ontologies. Journal of Web Semantics 3(4) (2005) 268–2938. Doran, P.S., Tamma, V., Payne, T.R., Palmisano, I.: An entropy inspired measurefor evaluating ontology modularization. In: Proc. of KCAP2009. (2009) 73–809. Shannon, C.E.: A mathematical theory of communication. Bell System TechnicalJournal 27 379–423, 623–6564 http://www.wsl.ch/info/mitarbeitende/scharren/owl-defaults/

Default logics for plausible reasoning with controversial axioms

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Default logics for plausible reasoning with controversial axioms

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