Outline. Structure-Based Partitioning of Large Concept Hierarchies. Ontologies and the Semantic Web. The Case for Partitioning

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1 Outline Structure-Based Partitioning of Large Concept Hierarchies Heiner Stuckenschmidt, Michel Klein Vrije Universiteit Amsterdam Motivation: The Case for Ontology Partitioning Lots of Pictures A Partitioning Method Create a dependency graph etermine the strength of dependencies Compute Partitioning Improve Partitioning Experiments Ontologies and the Semantic Web Ontologies are the backbone of semantic web applications Content-Based Retrieval Information Integration Web Service iscovery More and More Large Ontologies become available General Purpose: Open irectory, Yahoo!, Medicine: GALEN, UMLS, FMA, The Case for Partitioning istributed evelopment and Maintenance Experts can update their portion independently of other parts Selective Publication and Use of Terminologies Stable subsets can be published in the development phase Users can chose relevant subset of an ontology M l I i d V lid i An Abstract View of the Problem espite the standardization of Languages there is no agreement on the way ontologies are represented. All ontologies contain classes Most organize them in a hierarchy Many define relations between classes Some provide formal definitions of classes Overview of the Process 1.Create ependency Graph 2.etermine Strength of ependencies 3.Compute Partitions We concentrate on partitioning ontologies 1

2 ependencies I: Hierarchy There is a strong dependency between classes and their subclasses: ependencies I: Hierarchy We can include definitions by computing the subsumption hierarchy:?? ependencies II: Shared Relations Overview of the Process 1.Create ependency Graph 2.etermine Strength of ependencies 3.Compute Partitions Relative Strength Networks Result for the Example Compute relative strength [Burt 92] of dependencies Example:

3 Overview of the Process 1.Create ependency Graph 2.etermine Strength of ependencies 3.Compute Partitions Computing Islands We use maximal line islands [Batagejl 2000] to compute partitions in the dependency graph. A set of vertices is a line island in network if and only if it induces a connected subgraph and the lines inside the island are stronger related among them than with the neighboring vertices. In particular there is a maximal spanning tree T over nodes in the island such that: Understanding Islands Result for the Example 0.5 > 1 > < < 1 < 0.5 Overview of the Process 1.Create ependency Graph 2.etermine Strength of ependencies Improving Partitions (work in progress) Islands are often very small (2-4 nodes) resulting in unwanted partitions of the ontology. Observation: small islands almost always have a large height value (1 or 0.5): 3.Compute Partitions 3

4 Improving Partitions Ontology Partitioning Tool Islands are often very small (2-4 nodes) resulting in unwanted partitions of the ontology. Module M1 Module M2 Height 0.5 Height. c1 c2 c c4 Features: OWL and KIF Import Observation: small islands almost always have a large height value (1 or 0.5). Module M3 Height 1 Selection of Criteria Computation of line islands Approach: Merge G h E t Experiments The ACM Computer Science Classification 1000 concepts, fixed hierarchy The Standard Upper Model Ontology SUMO 600 concepts, fixed hierarchy The NCI cancer ontology Subset with 2400 concepts, fixed hierarchy The ICE ontology About 2000 concepts, classified hierarchy istributed Representation and Reasoning Based on Work of: Alex Borgida Paolo Bouquet Fausto Giunchiglia Frank van Harmelen Luciano Serafini Heiner Stuckenschmidt Andrei Tamilin Holger Wache f Ontologies istributed Representation and Reasoning Rentals Accommodation Apartment Accommodation Based on Work of: Apartment First-Class- 4Star Alex Borgida Paolo Bouquet Fausto Giunchiglia Frank van Harmelen Luciano Serafini Heiner Stuckenschmidt Andrei Tamilin Holger Wache Bungalow Congress- Bed & Breakfast mapping mapping mapping Hostel mapping 5Star 4

5 Ontology Mappings Problem: Semantics of Mappings Accommodation Accommodation IAO.APARTMENT is a synonym of SAO.apartment is a hyponym of SAO.hotel Rentals Apartment First-Class- Apartment 4Star <,80%> IAO.PRICE is a hypernym of is a hypernym of overlaps is a synonym of SAO.4Star SAO.5Stars <SAO.,50% > SAO.price Bungalow Bed & Breakfast IAO.APARTMENT Congress- is a synonym of is a hyponym of is a hypernym of is a hypernym of SAO.apartment SAO.hotel SAO.4Star SAO.5Stars Hostel 5Star What is the semantic of this mapping? Share different approaches the semantic? If not where are the differences? <IAO.ACC,80%> overlaps <SAO.Acc,50% > IAO.PRICE is a synonym of SAO.price The Logic of Mappings Global Interpretation Syntax = Accommodation Interpretation Rentals Hostel Bungalow Apartment Bed & Breakfast First-Class- Congress- 4Star 5Star omain An Alternative Semantics Local omain Semantics Syntax Interpretation Syntax Interpretation How to express some statements about r ij ij resp. r ji ji? omain relation omain omain 5

6 Goal: Reasoning with and about mappings The Case Study: Select three topics with sufficient overlap Substances Structures Processes Analyse Subhierarchies in the different Terminologies efine partial ad-hoc mappings between individual concepts Use semantics to verify and complete mappings Example: Substances I Example: Substances II UMLS GALEN UMLS Tambis Verification of Mappings Infering new Mappings 6

7 Reasoning using Mappings The formal Basis: istributed escription Logics L syntax Bridge graph L is a family of description logics {L i } i I A bridge graph of a TB A bridge rule from i to j is an expression of the form: i:x j:y (into-bridge rule) T 1 B 12 T 2 T 7 i:x j:y (onto-bridge rule) T 6 B 23 where X and Y are concepts of L i and L j. B 56 B 64 T 3 B 34 A distributed T-box (TB) is a pair {T i } i I, {B ij } i j I where B ij is a collection of bridge rules from i to j T 5 B 54 T 4 L semantics A distributed interpretation (I) of a TB {I i } i I, {r ij } i j I I i is a local interpretation of T i on a local domain Δ I i T 1,T 2,T 3,T 4,T 5,T 6,T 7 I 1, I 2, I 3, I 4, I 5, I 6, I 7 r ij is a domain relations from I to j L satisfiability I= {I i } i I,{r ij } i j I satisfies TB= {T i } i I,{B ij } i j I I TB If all T i are satisfied all bridge rules B ij are satisfied r ij Δ I ix Δ I j 7

8 Into-bridge rule satisfiability Onto-bridge rule satisfiability I i:x j:y I i:x j:y r ij (x Ii ) Y Ij r ij (x Ii ) Y Ij Δ I i X r ij r 12 (X) Y Δ I j Δ I i X r ij r 12 (X) Y Δ I j Subsumption propagation in L TB = T 1, T 2, B 12 Generalized subsumption propagation TB= {T i } i I,{B ij } i j I T 1 T 2 A G T i A G T j isa TB isa B 1 H 1 B H B 2 H 2 B n H n G I2 r 12 (A I1 ) r 12 (B I1 ) H I2 irectionality property: Knowledge propagates ONLY along the direction of bridge rules! TB i:a B 1 B n TB j:g H 1 H n Soundness and completeness Let TB 12 = T 1, T 2, B 12 be a distributed T-box Bridge operator encapsulates propagated axioms Theorem n T 1 A B k n k=1 B 12 (T 1 ) = 1:A 2:G G Hk B 12 k=1 1:B k 2:H k B 12 for 1 k n, n 0 istributed tableau algorithm TB 12 2:X Y T 2 B 12 (T 1 ) X Y 8

9 Basic reasoning service of L TB i:c TB i:x i:x is satisfiable with respect to TB if there exist a I such that I TB and X I i 0 Restrictions: (1) bridge graph is cycle-free (2) bridge rules connect atomic concepts (3) no nominals istributed tableau intuition Tab 1 (X) Tab 6 (X) Tab 5 (X) Tab i (X) = Tab i (X) + lazy computation of bridge operator Tab 2 (X) Tab 4 (X) Tab 7 (X) Tab 3 (X) istributed tableau intuition Is j: is satisfiable wrt TB? Tab i (A (B 1 B 2 )) Tab j () L(z) = {A (B x L(x) = {} 1 B 2 )} z Algorithm formalization Tab j SHIQ-tableau expansion rules + bridge expansion rule: Clash i:a j:g i:b 1 j:h 1 B ij Standard tableau expansion rules y L(y) = {G, H} 1 H 2 } G H 1 H 2 A (B 1 B 2 ) If 1. G L(x), i:a j:g B ij, and 2. B {B i:b j:h B H {H ij, and Tab i (A B) = Unsatisfiable for some H L(x), then L(x) L(x) { H} i:b 2 j:h 2 Algorithm properties Theorem (Termination) For any acyclic distributed T-box and for any SHIQ concept X, Tab j (X) terminates. Theorem (Soundness and completeness) j:x is satisfiable in distributed T-box if and only if Tab j (X) can generate a complete and clash-free completion tree. RAGO reasoning architecture 9

10 istributed reasoning architecture RP 1 RP 3 URI 1 URI 3 URI 7 URI 2 RP 2 B 37 B 67 URI 4 URI 5 URI 6 Implementation OWL ontologies C-OWL semantic mappings istributed Reasoner is an extension to open source OWL Reasoner Pellet 10