Transitive Roles inNumber 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

• Main results: (Un)decidability

• Open problems: towards a criterion

Motivating examples• A human has 2 hands, each with 5 fingers:

Human v 62 hasPart.Hand

Hand v 65 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

… 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 )

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

• Syntax for knowledge base:

The problem and the usual remedyTheorem. 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 RBoxTheorem. The following problem is undecidable:

given R0, T, C satisfiable? (Yes/No)

where R0 = Star4 (even without inverses!)

We call Star4 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 R1 and R2 are safe and do not share any roles, then R1 [ R2 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 safetyTheorem 5 (without inverses). These RBoxes are unsafe:

and any their extensions.

Conjecture (Criterion): No other unsafe RBoxes exist!

min!

R S

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!