An Efficient Method for Computing Alignment Diagnoses Christian Meilicke, Heiner Stuckenschmidt University of Mannheim Lehrstuhl für Künstliche Intelligenz {christian, heiner}@informatik.uni-mannheim.de

Computing a local optimal diagnosis Problem Statement Automatically and manually (!) generated ontology alignments are often incoherent See OAEI-2008 results of conference track => Incoherent alignments are a problem in many application scenarios* Instance migration results in inconsistent ontologies Query translation results in ‚a priori‘ empty result sets Find a way to automatically repair incoherent alignments in a very efficient way, because … ‚Agents on the web‘ require coherent alignments on the fly Large ontologies require efficient algorithms * C.Meilicke and H.Stuckenschmidt. Incoherence as a Basis for Measuring the Quality of Ontology Mappings. OM-08. Computing a local optimal diagnosis

Computing a local optimal diagnosis Outline Alignment Semantics Incoherence of an alignment, MIPS alignments Alignment Diagnosis Diagnosis, Minimal Hitting Set, Local Optimal Diagnosis Computing a Local Optimal Diagnosis (LOD) Brute-Force LOD and Efficient LOD Experimental Results Runtime, Quality of the Diagnosis Computing a local optimal diagnosis

"Natural" Semantics O1 ∪A O2 O2 O1 Merged Ontology <1#Person, 2#Person, =, 0.98> <1#hasName, 2#name, =, 0.87> <1#writtenBy, 2#docWrittenBy, = 0.7> <1#authorOf, 2#hasWritten, =, 0.56> <1#firstAuthor, 2#Author, ⊑ , 0.56> O1 ∪A O2 Correspondences An alignment A and two ontologies O1 and O2 O2 O1 1#firstAuthor ⊑ 2#Author Axioms 1#Person ≣ 2#Person … Computing a local optimal diagnosis

Incoherence of an Alignment Definition: Incoherence of an Alignment An alignment A between ontologies O1 and O2 is incoherent iff there exists an satisfiable concept i#C or property i#R in Oi  {1,2} that is unsatisfiable in O1 ∪A O2. can be reduced to the satisfiability of ∃i#R.⊤ Definition: MIPS Alignment (minimal conflict set) Given an incoherent alignment A between ontologies O1 and O2. A subalignment M ⊆ A is a MIPS alignment (= minimal incoherence preserving subalignment) iff M is incoherent and there exists no M‘ ⊂ M such that M‘ is incoherent. Computing a local optimal diagnosis

Computing a local optimal diagnosis "Terminology" Alignment Correspondence Alignment with MIPS shown as subsets Alignment in a sequence ordered by confidences MIPS depicted by red-dotted links Computing a local optimal diagnosis

Computing a local optimal diagnosis Alignment Diagnosis Definition: Alignment Diagnosis Alignment ∆ ⊆ A is an alignment diagnosis for O1 and O2 iff A \ ∆ is coherent with respect to O1 and O2 and for each ∆‘ ⊂ ∆ alignment A \ ∆‘ is incoherent with respect to O1 and O2. Proposition: Alignment Diagnosis and minimal Hitting Sets Alignment ∆ ⊆ A is an alignment diagnosis for O1 and O2 iff ∆ is a minimal hitting set over all MIPS in A. Computing a local optimal diagnosis

Local Optimal Diagnosis (LOD) high confidence Definition: Accused correspondence A correspondence c  A is accused by A iff there exists a MIPS in A with c  M such that for all c‘ ≠ c in M it holds that (1) conf(c‘) > conf(c) and (2) c‘ is not accused by A. Definition: Local optimal diagnosis (LOD) The set of all accussed correspondences is referred to as local optimal diagnosis (LOD). important! low confidence Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1 1 2 3 4 5 6 7 8 9 10 Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1 Coherent? YES! 1 2 3 4 5 6 7 8 9 10 Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1 Coherent? YES! 1 2 3 4 5 6 7 8 9 10 Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1 Coherent? NO! 1 2 3 4 5 6 7 8 9 10 Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1 Coherent? Now it is! 1 2 3 4 5 6 7 8 9 10 Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1 Coherent? YES! 1 2 3 4 5 6 7 8 9 10 Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1 Coherent? YES! 1 2 3 4 5 6 7 8 9 10 Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1 Coherent? NO! 1 2 3 4 5 6 7 8 9 10 Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1 Coherent? Now it is! 1 2 3 4 5 6 7 8 9 10 … continue the same way Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 1: Result … and after a few more slides we would end up like this: 1 2 3 4 5 6 7 8 9 10 Note: 10 times checking coherence for constructing a local optimal diagnosis, which is a minimal hitting set over all MIPS We have not computed a single MIPS alignment! First sketch: Meilicke,Völker, Stuckenschmidt. Learning Disjointness for Debugging Mappings between Lightweight Ontologies (EKAW-08) With focus on relation to belief revision discussed in: Qi, Ji, Haase: A Conflict-based Operator for Mapping Revision (ISWC-09) Computing a local optimal diagnosis

„Patternbased“ reasoning Idea: Use incomplete method for incoherence detection in A‘ ⊆A Classify O1 and O2 once, then check for each pair of correspondence in A‘ wether a certain pattern occurs If pattern occurs for some pair of an alignment A‘, then A‘ is incoherent If no pattern occurs A‘ can nevertheless be incoherent! Oj Oi Computing a local optimal diagnosis

Computing a local optimal diagnosis That doesn‘t work … Use the efficient coherence test instead of complete reasoning in algorithm described above Reasoning about A' ⊆ A does not require to reason in O1 ∪A' O2, but is replaced by iterating over all pairs in A' Hoewever: Resulting alignment might still be incoherent and ∆ is not a LOD Missing out one MIPS might result in a chain of incorrect follow-up decisions! Thus, afterwards removal of missed-out MIPS does not work! How to exploit the efficient method while still constructing a LOD? Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 2: Example 1 2 3 4 5 6 7 8 9 10 Detectable by efficient method Only detectable by complete method Resolved due to removal of correspondence Computing a local optimal diagnosis

Algorithm 2: Example Run the BF algorithm with efficient reasoning. Still incoherent? Verification Step: Use binary search to detect correspondence k such that A[0… k-1] is coherent and A[0 … k] is incoherent safe part, efficient reasoning did not fail up to k 1 2 3 4 5 6 7 8 9 10 k=8 incorrect part, recompute! Detectable by efficient method Only detectable by complete method Resolved due to removal of correspondence Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 2: Example Run the main algorithm again with efficient reasoning for A[k+1 … n] where ∆1-k ∪ A[k] for A[1… k] is a fixed part of the resulting diagnosis. Still incoherent? If yes, we have knew > kold repeat again the same verification step A[1…k] 1 2 3 4 5 6 7 8 9 10 A[k+1…n] Detectable by efficient method Only detectable by complete method Resolved due to removal of correspondence Computing a local optimal diagnosis

Computing a local optimal diagnosis Algorithm 2: Example Final result is a LOD. 1 2 3 4 5 6 7 8 9 10 Detectable by efficient method Only detectable by complete method Resolved due to removal of correspondence Computing a local optimal diagnosis

Runtime Considerations (Theory) n = size of alignment A m = number of times the binary search is applied The "more complete„ pattern-based reasoning is => the less verification steps/ iterations are necesarry Runtime of pattern based reasoning not really matters with respect to runtime! Runtime Comparison Brute Force LOD: O(n) Efficient LOD: O(log(n) * m) Do we have m << n ? Computing a local optimal diagnosis

Computing a local optimal diagnosis Results: Runtime Based on experiments with OAEI conference ontologies and submission from 2007/08 Expressivity SHIN(D), ELI(D), SIF(D), ALCIF(D) Four different state of the art matching systems n m Better results for benchmark datasets: 5 to 10 times faster Computing a local optimal diagnosis

Results: Quality of Diagnosis Removing the LOD results in an alignment with increased precision and slightly decreased recall => slightly increased f-measure For alignments with low precision positive effects are very strong. In rare cases an incorrect correspondences annotated with high confidence has negative effects Computing a local optimal diagnosis

Computing a local optimal diagnosis Summary Algorithm 1: Algorithm for computing a LOD Without computing MIPS or MUPS! Algorithm 2: General approach for improving the algorithms of type 1 Shown for natural interpretation of correspondences as axioms and a specific type of incomplete reasoning In principle applicable to each semantic for which we can find a similar efficient reasoning approach! Good results for natural interpretation + pattern based reasoning: between 2 and 10 times faster! Computing a local optimal diagnosis

Thanks for attention Questions? Computing a local optimal diagnosis

Computing a local optimal diagnosis Back-Up Slides Computing a local optimal diagnosis

Property Pattern Example ∃readPaper.⊤ ⊑ Reviewer Reviewer ⊑ Person Document ⊑ ¬Person O2 ∃readPaper.⊤ ∃reviewOfPaper.⊤ ≣ readPaper reviewOfPaper disjoint disjoint ≣ Document Document ∃reviewOfPaper.⊤ ⊑ Review ⊑ Document O1 Computing a local optimal diagnosis