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

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

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

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

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

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

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

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

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}

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

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)

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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