Transitive Roles in Number Restrictions Yevgeny Kazakov, Ulrike Sattler, Evgeny Zolin The University of Manchester

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

Motivating examples A human has 2 hands, each with 5 fingers: Human v 6 2 hasPart.Hand Hand v 6 5 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 ( 6 4 hasPart.Movement ) A quartet consists of exactly 4 instruments Quartet v ( 6 4 hasPart. Instrument ) u ( > 4 hasPart. Instrument ) …any other ontologies involving partonomy

… 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 ( 6 1 isFollowedBy. :9 hasNext. > ) A protein sequence D contains 5 amino-acids of type B ProtSeqD v ( > 5 isFollowedBy. AminoAcidB )

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

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...

Safety of an RBox Theorem. The following problem is undecidable: given R 0, T, C  satisfiable? (Yes/No) where R 0 = Star 4 (even without inverses!) We call Star 4 unsafe 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? transitive non-transitive

Main results. Good news: Decidability Theorem 3 (without inverses). An RBox is safe, if for any transitive roles R and S, either R v S or S v R. Corollary: { } and { } are safe! All examples from the first slides are decidable. Moreover: Theorem 4 (Modularity). If two RBoxes R 1 and R 2 are safe and do not share any roles, then R 1 [ R 2 is safe.

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:

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

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 …

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