An Efficient Method for

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• An Efficient Method for
• Computing Alignment Diagnoses
• Christian Meilicke, Heiner StuckenschmidtUniversi
  ty of Mannheim
• Lehrstuhl für Künstliche Intelligenz
• christian, heiner_at_informatik.uni-mannheim.de

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Problem Statement

• Automatically and manually (!) generated ontology
  alignments are often incoherent
• See OAEI-2008 results of conference track
• gt 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.
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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

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"Natural" Semantics
Merged Ontology
lt1Person, 2Person, , 0.98gt lt1hasName, 2name,
, 0.87gt lt1writtenBy, 2docWrittenBy,
0.7gt lt1authorOf, 2hasWritten, ,
0.56gt lt1firstAuthor, 2Author, ? , 0.56gt
O1 ?A O2
Correspondences
An alignment A and two ontologies O1 and O2
O2
O1
1firstAuthor ? 2Author
1Person ? 2Person
Axioms

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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 iC or property iR in Oi ? 1,2 that is
unsatisfiable in O1 ?A O2.
can be reduced to the satisfiability of ?iR.?
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.
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"Terminology"
Alignment
Correspondence
Alignmentwith MIPS shown as subsets
Alignmentin a sequence ordered by
confidencesMIPS depicted by red-dotted links
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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.
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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) gt 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
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Algorithm 1
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Algorithm 1
Coherent?YES!
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Algorithm 1
Coherent?YES!
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Algorithm 1
Coherent?NO!
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Algorithm 1
Coherent?Now it is!
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Algorithm 1
Coherent?YES!
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Algorithm 1
Coherent?YES!
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Algorithm 1
Coherent?NO!
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Algorithm 1
Coherent?Now it is!
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continue the same way
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Algorithm 1 Result

• and after a few more slides we would end up
  like this

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• 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)
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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
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That doesnt 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?

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Algorithm 2 Example
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Detectable by efficient method
Only detectable by complete method
Resolved due to removal of correspondence
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Algorithm 2 Example
Run the BF algorithm with efficient reasoning.
Still incoherent?
Verification Step Use binary search to detect
correspondence k such that A0 k-1 is coherent
and A0 k is incoherent
safe part, efficient reasoning did not fail up to
k
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k8
incorrect part,recompute!
Detectable by efficient method
Only detectable by complete method
Resolved due to removal of correspondence
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Algorithm 2 Example
Run the main algorithm again with efficient
reasoning for Ak1 n where ?1-k ? Ak for
A1 k is a fixed part of the resulting
diagnosis. Still incoherent?If yes, we have
knew gt kold repeat again the same verification
step
A1k
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Ak1n
Detectable by efficient method
Only detectable by complete method
Resolved due to removal of correspondence
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Algorithm 2 Example
Final result is a LOD.
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Detectable by efficient method
Only detectable by complete method
Resolved due to removal of correspondence
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Runtime Considerations (Theory)

• n size of alignment A
• m number of times the binary search is applied
• The "more complete pattern-based reasoning is gt
  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 ltlt n ?

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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

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Results Quality of Diagnosis

• Removing the LOD results in an alignment with
  increased precision and slightly decreased recall
  gt 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

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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!

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• Thanks for attention
• Questions?

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Back-Up Slides
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Property Pattern Example
?readPaper.? ? Reviewer Reviewer ? Person
Document ? Person
O2
?reviewOfPaper.?
?readPaper.?
?
readPaper
reviewOfPaper
disjoint
disjoint
?
Document
Document
?reviewOfPaper.? ? Review ? Document
O1