A Kernel Revision Operator for Terminologies Algorithms and Evaluation

A Kernel Revision Operator for TerminologiesAlgorithms 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

Motivation • Revision operator for terminologies: mapping from two Description Logic TBoxesT andT0to a set of TBoxes or a single TBox which infer(s) every axiom inT0 • Example scenario where we need to revise TBoxes • Ontologylearning: • 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

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

Debugging Terminologies • MUPS for Aw.r.t.T: a subsetT'of TBoxTsuch that • A is unsatisfiable inT' • A is satisfiable in any T'' whereT'' ½T' • Example:T={ManagervEmployee, EmployeevJobPosition, JobPosition v:Employee, LeadervJobPosition} • Manager is unsatisfiable • MUPS: {ManagervEmployee, EmployeevJobPosition, JobPosition v:Employee} • Incoherence: a concept inTis unsatisfiable • MIPS forT: a subsetT'of TBoxTsuch that • T'is incoherent • anyT'' withT'' ½T'is coherent • Example (cont.): One MIPS • {EmployeevJobPosition, JobPosition v:Employee} Minimal sub-TBox of T in which A is unsatisfiable Minimal sub-TBox of T which is incoherent

A Kernel Revision Operator • Idea: based on MIPS • step 1: find MIPS ofTw.r.t.T0 • step 2: remove some axioms in these MIPS • MIPS ofTw.r.t.T0: a subsetT'of TBoxTs.t. • T'[T0is incoherent (incoherence) • anyT'' withT'' ½T'is coherent withT0 (minimalism) • Example:T={ManagervEmployee, EmployeevJobPosition} and T0={JobPosition v:Employee, LeadervJobPosition} • A MIPS ofTw.r.t.T0: • {ManagervEmployee, EmployeevJobPosition}

A Kernel Revision Operator (Cont.) • Question: which axioms should be removed from MIPS? • Solution: an incision function • Incision functionforT: for each TBoxT0and the set MIPST0(T) of all MIPS ofTw.r.t.T0 • (MIPST0(T)) µ[Ti2 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))= [Ti2 MIPST0(T) Ti • Principle: minimal change, i.e., select minimal number or set of axioms

A Kernel Revision Operator (Cont.) • Kernel revision operator: GivenTand  forT • T¤T0= (Tn(MIPST0(T))) [T0 • The result of revision is always a coherent TBox • Logical properties: • (R1) T0 µT¤T0 (success) • (R2) IfT [T0is coherent, thenT¤T0= T [ T0 • (R3) If T0is coherent thenT¤T0is coherent (coherence preserve) • (R4) IfT0,T'0, thenT¤T0,T¤T'0 (syntax independence) • (R5) If2Tand∉T¤T0, then there is a subsetSofTand a subset S0ofT0such that S[S0is coherent, but S[ S0[{} is not. (relevance)

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

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 1 algorithms can handle real life ontologies 2 Algorithm_MUPS is more scalable than others

Experimental Evaluation • Results for the ontology mapping scenario Algorithm_MUPS computes less unsat. Concepts and MUPS than others

Experimental Evaluation • Results for the ontology mapping scenario Algorithm_score bests complies the requirement of minimal change

Experimental Evaluation • Analysis of Meaningfulness correctness rate is considerably higher than error rate

Experimental Evaluation • Analysis of Meaningfulness

Conclusion • Problem addressed: • 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

A Kernel Revision Operator for Terminologies Algorithms and Evaluation