1. An Efficient Method for Computing Alignment Diagnoses
Christian Meilicke, Heiner Stuckenschmidt University of Mannheim
Lehrstuhl für Künstliche Intelligenz
{christian,

2. 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

3. 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

4. "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

5. 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

6. 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

7. 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

8. 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

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

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

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

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

13. 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

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

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

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

17. 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

18. 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

19. „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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. Thanks for attention Questions?
Computing a local optimal diagnosis

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

31. 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