Reasoning with Inconsistent Ontologies

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Reasoning with Inconsistent Ontologies????????
Zhisheng Huang, Frank van Harmelen, and Annette
ten Teije Vrije University Amsterdam (IJCAI2005
paper)
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Outline of This Talk

• Inconsistency in the Semantic Web
• General Framework
• Strategies and Algorithms
• Implementation
• Tests and Evaluation
• Future work and Conclusion

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Inconsistency and the Semantic Web

• The Semantic Web is characterized by
• scalability,
• distribution, and
• multi-authorship
• All these may introduce inconsistencies.

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Ontologies will be inconsistent

• Because of
• mistreatment of defaults
• polysemy
• migration from another formalism
• integration of multiple sources
• (Semantic Web as a wake-up call for KR)

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Example Inconsistency by mistreatment of
default rules

• MadCow Ontology
• Cow ? Vegetarian
• MadCow ? Cow
• MadCow ? ? Eat.BrainofSheep
• Sheep ? Animal
• Vegetarian ? ? Eat. ? (Animal? PartofAnimal)
• Brain ? PartofAnimal
• ......
• theMadCow ?MadCow
• ...

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Example Inconsistency through imigration from
other formalism

• DICE Ontology
• Brain ? CentralNervousSystem
• Brain ? BodyPart
• CentralNervousSystem ? NervousSystem
• BodyPart ? ?NervousSystem

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Inconsistency and Explosion

• The classical entailment is explosive P, P
  Q Any formula is a logical  consequence of a
  contradiction.
• The conclusions derived from an inconsistent
  ontology using the standard reasoning may be
  completely meaningless

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Why DL reasoning cannot escape the explosion

• The derivation checking is usually achieved by
  the satisfiability checking.
• ? ? ? ? ?? is not satisfiable.
• Tableau algorithms are approaches based on the
  satisfiability checking
• ? is inconsistent gt ? is not satisfiable gt ?
  ?? is not satisfiable.

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Two main approaches to deal with inconsistency

• Inconsistency Diagnosis and Repair
• Ontology Diagnosis(Schlobach and Cornet 2003)
• Reasoning with Inconsistency
• Paraconsistent logics
• Limited inference (Levesque 1989)
• Approximate reasoning(Schaerf and Cadoli 1995)
• Resource-bounded inferences(Marquis et al.2003)
• Belief revision on relevance (Chopra et al. 2000)

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What an inconsistency reasoner is expected

• Given an inconsistent ontology, return meaningful
  answers to queries.
• General solution Use non-standard reasoning to
  deal with inconsistency
• ? ? the standard inference relations
• ? ?? nonstandard inference relations

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Reasoning with inconsistent ontologies Main Idea

• Starting from the query,
• select consistent sub-theory by using a
  relevance-based selection function.
• apply standard reasoning on the selected
  sub-theory to find meaningful answers.
• If it cannot give a satisfying answer, the
  selection function would relax the relevance
  degree to extend consistent sub-theory for
  further reasoning.

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New formal notions are needed

• New notions
• Accepted
• Rejected
• Overdetermined
• Undetermined
• Soundness (only classically justified results)
• Meaningfulness (sound never overdetermined)sou
  ndness

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Some Formal Definitions

• Soundness ? ?? gt???? (? consistent and
  ??).
• Meaningfulness sound and consistent (? ?? gt ?
  ?).
• Local Completeness w.r.t a consistent ? ??
  (?? gt ? ??).
• Maximality locally complete w.r.t a maximal
  consistent set ?.
• Local Soundness w.r.t.a consistent set ? ? ??
  gt ??).

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Selection Functions

• Given an ontology T and a query ?, a selection
  function s(T,?,k)returns a subset of the
  ontology at each step kgt0.

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General framework

• Use selection function s(T,?,k),with s(T,?,k) ?
  s(T,?,k1)
• Start with k0 s(T,?,0) ? or s(T,?,0) ??
  ?
• Increase k, untils(T,?,k) ? or s(T,?,k) ??
• Abort when
• undetermined at maximal k
• overdetermined at some k

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Inconsistency Reasoning Processing Linear
Extension
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Proposition Linear Extension

• Never over-determined
• May undetermined
• Always sound
• Always meaningful
• Always locally complete
• May not maximal
• Always locally sound

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Direct Relevance and K Relevance

• Direct relevance (0-relevance).
• there is a common name in two formulas C(?) ?
  C(?)?? ? R(?) ? R(?)?? ? I(?)? I(?)??.
• K-relevance there exist formulas ?0, ?1,, ?k
  such that ? and ?0, ?0 and ?1 , , ?k and
  ? are directly relevant.

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Relevance-based Selection Functions

• s(T,?,0)?
• s(T,?,1) ?? T ? is directly relevant to ?.
• s(T,?,k) ?? T ? is directly relevant to
  s(T,?,k-1).

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PION Prototype
PION Processing Inconsistent ONtologies http//wa
sp.cs.vu.nl/sekt/pion
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An Extended DIG Description Logic Interface for
Prolog (XDIG)

• A logic programming infrastructure for the
  Semantic Web
• Similar to SOAP
• Application independent, platform independent
• Support for DIG clients and DIG servers.

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XDIG

• As a DIG client, the Prolog programs can call any
  external DL reasoner which supports the DIG DL
  interface.
• As a DIG server, the Prolog programs can serve as
  a DL reasoner, which can be used to support
  additional reasoning processing, like
  inconsistency reasoning multi-version reasoning,
  and inconsistency diagnosis and repair.

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XDIG package

• The XDIG package and the source code are now
  available for public download at the website
  http//wasp.cs.vu.nl/sekt/dig/
• In the package, we offer five examples how XDIG
  can be used to develop extended DL reasoners.

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PION Testbed

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Answer Evaluation

• Intended Answer (IA) PION answer Intuitive
  Answer
• Cautious Answer (CA) PION answer is
  undetermined, but intuitive answer is
  accepted or rejected.
• Reckless Answer (RA) PION answer is accepted
  or rejected, but intuitive answer is
  undetermined.
• Counter Intuitive Answer (CIA) PION answer is
  accepted but intuitive answer is rejected,
  or vice verse.

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Preliminary Tests with Syntactic-relevance
Selection Function
Ontology Queries IA CA RA CIA IA () ICR ()
Bird 50 50 0 0 0 100 100
Brain (DICE) 42 36 4 2 0 85.7 100
MarriedWoman 50 48 0 2 0 96 100
MadCow 254 236 16 0 2 92.9 99
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Observation

• Intended answers include many undetermined
  answers.
• Some counter-intuitive answers
• Reasonably good performance

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Intensive Tests on PION

• Evaluation and test on PION with several
  realistic ontologies
• Communication Ontology
• Transportation Ontology
• MadCow Ontology
• Each ontology has been tested by thousands of
  queries with different selection functions.

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Conclusions

• we proposed a general framework for reasoning
  with inconsistent ontologies
• based on selecting ever increasing consistent
  subsets
• choice of selection function is crucial
• query-based selection functions are flexible to
  find intended answers
• simple syntactic selection works surprisingly
  well

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Future Work

• understand better why simple selection functions
  work so well
• consider other selection functions(e.g. exploit
  more the structure of the ontology)
• Variants of strategies
• More tests on realistic ontologies
• Integrating with the diagnosis approach