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Transitive Roles in Number Restrictions

Transitive Roles in Number Restrictions. Yevgeny Kazakov, Ulrike Sattler, Evgeny Zolin The University of Manchester {kazakov,sattler,zolin}@cs.man.ac.uk. Talk Outline. Motivating examples Definitions: where the problem lies Solution: to give a good definition

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Transitive Roles in Number Restrictions

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  1. Transitive Roles inNumber Restrictions Yevgeny Kazakov, Ulrike Sattler, Evgeny Zolin The University of Manchester {kazakov,sattler,zolin}@cs.man.ac.uk

  2. Talk Outline • Motivating examples • Definitions: where the problem lies • Solution: to give a good definition • Main results: (Un)decidability • Open problems: towards a criterion

  3. Motivating examples • A human has 2 hands, each with 5 fingers: Human v62 hasPart.Hand Hand v65 hasPart.HandFinger Then the concept Human u ( >11 hasPart.HandFinger) is unsatisfiable, provoded (!) that hasPart is transitive • A symphony consists of (at most) 4 movements Symphony v MusicalComposition u ( 64 hasPart.Movement ) • A quartet consists of exactly 4 instruments Quartet v ( 64 hasPart. Instrument ) u ( >4 hasPart. Instrument ) • …any other ontologies involving partonomy

  4. … more examples • Lists are modeled in ontolodies using two roles: hasNext — to link elements in a chain isFollowedBy — to jump in few steps in a chain Axioms: hasNext v isFollowedBy,Trans(isFollowedBy) • A list has at most 1 ending element: List v ( 61 isFollowedBy.:9 hasNext.> ) • A protein sequence D contains 5 amino-acids of type B ProtSeqD v ( >5 isFollowedBy. AminoAcidB)

  5. What’s in the SHIQ? • Syntax for concepts of the Description Logic SHIQ : • Syntax for knowledge base:

  6. The problem and the usual remedy Theorem.The following problem is undecidable: given R, T, C satisfiable? (Yes/No) Patch: Then we regain decidability!... At the cost of loss of expressivity...

  7. transitive non-transitive Safety of an RBox Theorem.The following problem is undecidable: given R0, T, C satisfiable? (Yes/No) where R0 = Star4 (even without inverses!) We call Star4unsafe for ALCQ Q1. Which RBoxes are safe? (with or without inverses) Can we decide whether an RBox is safe? Q2. For safe RBoxes, what is the decision problem for satisfiability? Complexity? Algorithms? Implementation?

  8. Main results. Good news: Decidability Theorem 3 (without inverses). An RBox is safe, if for any transitive rolesRandS, either Rv Sor Sv R. Corollary: { } and { } are safe! All examples from the first slides are decidable. Moreover: Theorem 4 (Modularity).If two RBoxes R1 and R2 are safe and do not share any roles, then R1 [R2 is safe.

  9. Main results. Bad news: Undecidability Theorem 1 (with inverses). The following RBox is unsafe: R= { Trans (R) } So, is the hope for decidability lost? Conjecture. If we use number restrictions on R, but not on R–, i.e.: (?nR.C), but not (?nR–.C ), then it is safe. Theorem 2 (without inverses). These RBoxes are unsafe: Not yet: ?

  10. Q min! R S Towards a criterion of safety Theorem 5 (without inverses). These RBoxes are unsafe: and any their extensions. Conjecture (Criterion): No other unsafe RBoxes exist! A little nuance: Q = min (R,S)

  11. Conclusions and future directions • Borderline between safety and unsafety – almost done • A smarter notion of a simple role in SHIQ – to be found • “Semi-simple role”: number restrictions are allowed for the role, but not for the inverse – no results at all so far • Complexity • Practical algorithms • Implementations …

  12. The roads we take… The grids we tile… Thank you!

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