1. A Kernel Revision Operator for Terminologies Algorithms and Evaluation
Guilin Qi1, Peter Haase1, Zhisheng Huang2, Qiu Ji1, Jeff Z. Pan3, Johanna Voelker1
1University of Karlsruhe, GE
2Vrije University Amsterdam
3The University of Aberdeen

2. Outline Motivation Preliminaries on Debugging Terminologies
Kernel Revision Operator for Terminologies
Algorithms for Specific Operators
Evaluation Results
Conclusion and Future Work

3. Outline Motivation Preliminaries on Debugging Terminologies
Kernel Revision Operator for Terminologies
Algorithms for Specific Operators
Evaluation Results
Conclusion and Future Work

4. Example scenario where we need to revise TBoxes
Motivation
Revision operator for terminologies: mapping from two Description Logic TBoxes T and T0 to a set of TBoxes or a single TBox which infer(s) every axiom in T0
Example scenario where we need to revise TBoxes
Ontology learning:
Starting with an initial empty TBox T
We generate a set of terminological axioms T0 from Text and add them to T
Result: a TBox without logical contradiction
Ontology mapping:
Integrate two heterogeneous source ontologies via mappings
The source ontologies are fixed and the set of generated mappings T0 is revised by their union T
Result: a merged ontology without logical contradiction

5. Problem: deal with logical contradictions
Motivation (Cont.)
Problem: deal with logical contradictions
Ontology learning: contradictions occur when expressive ontologies are learned
Ontology mapping: erroneous mappings are generated
Our revision operator
Is inspired by the kernel revision operator in propositional logic
Is based on the notion of minimal incoherence-preserving sub-terminologies (MIPS)
Is shown to satisfy some important logical properties
Has been instantiated by two algorithms which were implemented

6. Outline Motivation Preliminaries on Debugging Terminologies
Kernel Revision Operator for Terminologies
Algorithms for Specific Operators
Evaluation Results
Conclusion and Future Work

7. Debugging Terminologies
MUPS for A w.r.t. T: a subset T' of TBox T such that
A is unsatisfiable in T'
A is satisfiable in any T'' where T'' ½ T'
Example: T={Manager v Employee, Employee v JobPosition,
JobPosition v :Employee, Leader v JobPosition}
Manager is unsatisfiable
MUPS: {Manager v Employee, Employee v JobPosition, JobPosition v :Employee}
Incoherence: a concept in T is unsatisfiable
MIPS for T: a subset T' of TBox T such that
T' is incoherent
any T'' with T'' ½ T' is coherent
Example (cont.): One MIPS
{Employee v JobPosition, JobPosition v :Employee}
Minimal sub-TBox of
T in which A is unsatisfiable
Minimal sub-TBox of
T which is incoherent

8. Outline Motivation Preliminaries on Debugging Terminologies
Kernel Revision Operator for Terminologies
Algorithms for Specific Operators
Evaluation Results
Conclusion and Future Work

9. A Kernel Revision Operator
Idea: based on MIPS
step 1: find MIPS of T w.r.t. T0
step 2: remove some axioms in these MIPS
MIPS of T w.r.t. T0: a subset T' of TBox T s.t.
T'[T0 is incoherent (incoherence)
any T'' with T'' ½ T' is coherent with T0 (minimalism)
Example: T={Manager v Employee, Employee v JobPosition} and
T0={JobPosition v :Employee, Leader v JobPosition}
A MIPS of T w.r.t. T0:
{Manager v Employee, Employee v JobPosition}

10. A Kernel Revision Operator (Cont.)
Question: which axioms should be removed from MIPS?
Solution: an incision function
Incision function  for T: for each TBox T0 and the set MIPST0(T) of all MIPS of T w.r.t. T0
(MIPST0(T)) µ [Ti 2 MIPST0(T) Ti (axioms selected belong to some MIPS)
T’ Å (MIPST0(T)) ;, for any T’ 2 MIPST0(T) (each MIPS has at least one axiom selected)
Naïve incision function: (MIPST0(T))= [Ti 2 MIPST0(T) Ti
Principle: minimal change, i.e., select minimal number or set of axioms

11. A Kernel Revision Operator (Cont.)
Kernel revision operator: Given T and  for T
T¤T0= (Tn(MIPST0(T))) [ T0
The result of revision is always a coherent TBox
Logical properties:
(R1) T0 µ T¤T0 (success)
(R2) If T [ T0 is coherent, then T¤T0= T [ T0
(R3) If T0 is coherent then T¤T0 is coherent (coherence preserve)
(R4) If T0,T'0, then T¤T0 ,T¤T'0 (syntax independence)
(R5) If 2T and ∉T¤T0, then there is a subset S of T and a subset S0 of T0 such that S[S0 is coherent, but S[ S0[{} is not. (relevance)

12. A Kernel Revision Operator (Cont.)
Kernel revision operator: Given T and  for T
T¤T0= (Tn(MIPST0(T))) [ T0
The result of revision is always a coherent TBox
Logical properties:
(R1) T0 µ T¤T0 (success)
(R2) If T [ T0 is coherent, then T¤T0= T [ T0
(R3) If T0 is coherent then T¤T0 is coherent (coherence preserve)
(R4) If T0,T'0, then T¤T0 ,T¤T'0 (syntax independence)
(R5) If 2T and ∉T¤T0, then there is a subset S of T and a subset S0 of T0 such that S[S0 is coherent, but S[ S0[{} is not. (relevance)

13. Outline Motivation Preliminaries on Debugging Terminologies
Kernel Revision Operator for Terminologies
Algorithms for Specific Operators
Evaluation Results
Conclusion and Future Work

14. Algorithms
Different incision functions will result in different specific kernel revision operators
Incision functions can be computed by Reiter's hitting set tree (HST) algorithm
However, there are potentially exponential number of hitting sets computed by the algorithm
We reduce the search space by using scoring function or
confidence values

15. Algorithms (Cont.)
Algorithm_score: based on the scoring function and HST algorithm
The score of an axiom is the number of MIPS it belongs to
Algorithm_confidence: based on confidence value and the HST algorithm
Algorithm_MUPS: adapted algorithm for repair based on confidence values
We compute MUPS and apply HST algorithm to them

16. Outline Motivation Preliminaries on Debugging Terminologies
Kernel Revision Operator for Terminologies
Algorithms for Specific Operators
Evaluation Results
Conclusion and Future Work

17. Experimental Evaluation Data sets
Ontology mapping data sets
Source ontologies
CONFTOOL: 197 axioms
CMT: 246 axioms
EKAW: 248 axioms
CRS: 69 axioms
SIGKDD: 122 axioms
Mappings
CONFTOOL-CMT: 14 mapping axioms
EKAW-CMT: 46 mapping axioms
CRS-SIGKDD: 22 mapping axioms

18. Experimental Evaluation
Revision time (efficiency)
Time to check coherence
Time to debug and resolve incoherence
Number of axioms removed (effectiveness)
Meaningfulness: correctness rate, error rate and unknown rate
Four users were asked to decide whether removal (1) was correct (2) was incorrect (3) whether they are unsure
We can also define Error_rate and Unknown_rate

19. Experimental Evaluation
Results for the ontology mapping scenario
Mappings
Strategy
# of unsatisf. Concepts
# of MUPS (All)
# of MUPS (Avg)
MUPS_
Size (Avg)
# of Removal Axioms
Time (sec)
CONFTOOL-CMT
Algorithm_score
Algorithm_Confidence
Algorithm_MUPS 26 4
351 15 14 6 5 8
331
332 12
EKAW-CMT 18
372 62 21 16
867
863 51
CRS-SIGKDD
Algorithm_score Algorithm_Confidence 19 64 13 3 10 7
1 algorithms can handle real life ontologies
2 Algorithm_MUPS is more scalable than others

20. Experimental Evaluation
Results for the ontology mapping scenario
Mappings
Strategy
# of unsatisf. Concepts
# of MUPS (All)
# of MUPS (Avg)
MUPS_
Size (Avg)
# of Removal Axioms
Time (sec)
CONFTOOL-CMT
Algorithm_score
Algorithm_Confidence
Algorithm_MUPS 26 4
351 15 14 6 5 8
331
332 12
EKAW-CMT 18
372 62 21 16
867
863 51
CRS-SIGKDD
Algorithm_score Algorithm_Confidence 19 64 13 3 10 7
Algorithm_MUPS computes less unsat. Concepts and MUPS than others

21. Experimental Evaluation
Results for the ontology mapping scenario
Mappings
Strategy
# of unsatisf. Concepts
# of MUPS (All)
# of MUPS (Avg)
MUPS_
Size (Avg)
# of Removal Axioms
Time (sec)
CONFTOOL-CMT
Algorithm_score
Algorithm_Confidence
Algorithm_MUPS 26 4
351 15 14 6 5 8
331
332 12
EKAW-CMT 18
372 62 21 16
867
863 51
CRS-SIGKDD
Algorithm_score Algorithm_Confidence 19 64 13 3 10 7
Algorithm_score bests complies the requirement of minimal change

22. Experimental Evaluation
Analysis of Meaningfulness
Data set
Algorithm
# of Removed Axioms
Correctness
Error_Rate
Unknown_Rate
bt_km
Algorithm_score
Algorithm_Confidence
Algorithm_MUPS 27 34 33
0.41
0.53
0.65
0.28
0.19
0.13
0.31
0.22
CONFTOOL-CMT 4 8
0.56
0.97
0.03
EKAW-CMT 16 15 14
0.68
0.64
0.84
0.11
0.05
0.07
0.21
0.09
CRS-SIGKDD 5 10 7
0.60
0.50
0.79
0.40
0.25
0.14
correctness rate is considerably higher than error rate

23. Experimental Evaluation
Analysis of Meaningfulness
Data set
Algorithm
# of Removed Axioms
Correctness
Error_Rate
Unknown_Rate
bt_km
Algorithm_score
Algorithm_Confidence
Algorithm_MUPS 27 34 33
0.41
0.53
0.65
0.28
0.19
0.13
0.31
0.22
CONFTOOL-CMT 4 8
0.56
0.97
0.03
EKAW-CMT 16 15 14
0.68
0.64
0.84
0.11
0.05
0.07
0.21
0.09
CRS-SIGKDD 5 10 7
0.60
0.50
0.79
0.40
0.25
0.14

24. Experimental Evaluation
Analysis of Meaningfulness
Data set
Algorithm
# of Removed Axioms
Correctness
Error_Rate
Unknown_Rate
bt_km
Algorithm_score
Algorithm_Confidence
Algorithm_MUPS 27 34 33
0.41
0.53
0.65
0.28
0.19
0.13
0.31
0.22
CONFTOOL-CMT 4 8
0.56
0.97
0.03
EKAW-CMT 16 15 14
0.68
0.64
0.84
0.11
0.05
0.07
0.21
0.09
CRS-SIGKDD 5 10 7
0.60
0.50
0.79
0.40
0.25
0.14

25. Outline Motivation Preliminaries on Debugging Terminologies
Kernel Revision Operator for Terminologies
Algorithms for Specific Operators
Evaluation Results
Conclusion and Future Work

26. Conclusion Problem addressed: Our approach: Evaluation results:
Revising terminologies by dealing with logical contradiction
Our approach:
A general revision operator was proposed using an incision function
Our operator satisfies desirable logical properties
Two algorithms were given to instantiate our revision operator
An algorithm based on computing MUPS was presented as an alternative
Evaluation results:
Our algorithms can handle real life ontologies
Algorithms based on confidence values lead to considerable more meaningful results
The algorithm based on computing MUPS shows good scalability
Application of our work: ontology learning, ontology matching, web syndication, ontology evolution

27. Future Work Explore efficient algorithms for computing MUPS or MIPS
Idea: extract modules which contains all the MUPS
Fine-grained approaches to resolving incoherence
Combine our tool with Cicero argumentation wiki to deal with collaborative ontology evolution