query answering for owl-dl with rules boris motik ulrike sattler rudi studer

Query Answering forOWL-DL with Rules

Boris MotikUlrike SattlerRudi Studer

2

Contents

•Introduction

•Preliminaries

•Decidability Problems

•DL-safe Rules

•Decidability of Query Answering

•Query Answering by Disjunctive Datalog

•Conclusion & Future Work

3

Introduction

• OWL-DL – a decidable fragment of FOL allows existential and universal quantifiers

quantifier usage restricted to make reasoning decidable only tree-like axioms allowed

expressivity not sufficient for certain practical problems

• Rule systems – a different set of choices decidability achieved by allowing universal quantifiers only

existential quantifiers possible (function symbols required;

easily lead to undecidability)

usually support non-monotonic reasoning

4

Goals

• Extending OWL-DL with rules is needed query answering should be decidable

SWRL approach is undecidable

• In this talk I… …explain why adding rules to DL leads to undecidability

…present DL-safe rules

…discuss the expressivity of the approach

…show that query answering is decidable

…give an algorithm for query answering

5

Contents

•Introduction

•Preliminaries


•DL-safe Rules




6

Preliminaries

Atomic concepts

:CC u D

C t D9 R.C

8 R.C

· n R.C (R is simple)

¸ n R.C (R is simple)

{ i1, …, in }

Concept Expressions

C v D

C ´ D

TBox Axioms

C(a)

R(a,b)

a ¼ b

a ¼ b

ABox Axioms

R v S

Trans(R)

RBox Axioms

Atomic roles

R– (inverse roles)

Roles

• OWL-DL is SHOIN(D)

• My algorithms support SHIQ(D)

• Main difference: nominals

• Semantics is (KB) by translating KB into FOL

7

Contents

•Introduction

•Preliminaries


•DL-safe Rules




8

Example for Termination Problems

• Question: does KB ² Grandchild(Jane)? this is the case iff KB [ { :Grandchild(Jane) } is unsatisfiable

show by trying to build a model

peter x1 x2

Person9 father.Person

father


father

Person

father

9 father.Person

• KB implies an infinite sequence of fathers enumerating all of them leads to non-termination advanced techniques needed to ensure termination

Grandchild

9

Termination in DL Algorithms

• Each father does not need to be distinct, so an infinite model can be wound up to a finite model (using blocking)

Peter x1 x2


father


father

Person

father

9 father.Person

Peter x1


father


father

x1 is blocked by Peter,so reuse successors of Peter.

Grandchild

Grandchild

10

Why is Blocking Possible?

9 S.(9 R.C u 9 R.D) v Q ,8 x:{[ 9 y: S(x,y) Æ (9 x: R(y,x) Æ C(x)) Æ (9 x: R(y,x) Æ D(x))] ! Q(x)} ,8 x,x1,x2,x3:{ S(x,x1) Æ R(x1,x2) Æ C(x2) Æ R(x1,x3) Æ D(x3) ! Q(x) }

• In OWL-DL only tree-like axioms are allowed (modulo technicalities concerning transitivity or nominals)

x

x1

S

x2 x3

R R

• This restriction ensures the tree-model property if there is a model, a tree-like model always exists as well

• Tree-like models can always be wound up into finite (representations of) models SHIQ models can be infinite trees, but can be finitely represented SHOIN models need not be trees, but can be finitely represented

11

Reasoning with (function-free) Rules

• No existential quantifiers limited to only explicitly introduced individuals

= a finite number for finite knowledge bases

• Can enforce arbitrary-shaped models for reasoning, examine all possible assignments of individuals to

variables (grounding) reasoning is reduced to propositional logic

hasAunt(x,y) Ã hasParent(x,z), hasSibling(y,z), Female(y)

hasAunt(Jane,Mary) Ã hasParent(Jane,Ann), hasSibling(Mary,Ann), Female(Mary)

hasAunt(Ann,Jane) Ã hasParent(Ann,Mary), hasSibling(Jane,Mary), Female(Jane)

…

JaneMaryAnn propositional clauses

12

Combining OWL-DL with Rules

OWL-DL + Rules

=Decidability due to tree-like axioms

+Decidability due to finite number of individuals

=

Trouble!

OWL-DL + Rules

=Decidability due to tree-like axioms

+Decidability due to finite number of individuals

=

Trouble!

13

Contents

•Introduction

•Preliminaries


•DL-safe Rules




14

Definition: DL-safe Rules• DL-safe program P contains rules with concepts and roles from

KB as unary resp. binary predicates in head or body (DL-atoms)

• Each variable occurs in a non-DL-atom in the body (DL-safety) makes rules applicable only to explicitly introduced individuals

• Semantics: (KB, P) is semantically equivalent to (KB) [ P rules interpreted as clauses (no non-monotonic reasoning)

Homeworker(x) Ã Person(x), livesAt(x,y), worksAt(x,y)

Not DL-safe, since x and y occur only in DL-atoms.

Homeworker(x) Ã Person(x), livesAt(x,y), worksAt(x,y), O(x), O(y)

We assume that there is a fact O() for each individual in the ABox.

(KB contains Homeworker, livesAt, Person, worksAt).

15

(KB, P) ² BadChild(Cain)

With normal (DL-unsafe) rules:

• Cain is a grandchild (as before)

• Cain has a father (Adam) and a sibling that he hates (Abel)

(KB, P) ² BadChild(Romulus)

• Romulus hates Remus

• We do not know who the father of Romulus and Remus is, but we know that he exists

Expressivity (I)

16

With DL-safe rules:

(KB, P) ² BadChild’(Cain)

• We know the identity of Cain’s father (Adam)

• We do not know the identity of Cain’s father, so O(y) cannot be matched to an individual

(KB, P) ² BadChild’(Romulus)

Expressivity (II)

Intuitive semantics: BadChild’ is a known child with a known father who hates some of his known siblings.

17

• Reasoning with DL-safe rules does not mean “derive DL consequences first, and then apply the rules.”

• common misconception; significantly changes semantics

Oedipus may be a good or a bad child.KB ² GoodChild(Oedipus)

Expressivity (III)

KB ² BadChild’(Oedipus)

KB ² Child(Oedipus)Either way, Oedipus is a child.This is not derived by applying the rules to consequences of the DL part.

18

Expressivity (IV)

• DL-safety does not reduce component languages

• DL-safety allows exchanging consequences between

components about explicit individuals only

• DL-safety does increase expressivity rules alone cannot derive KB ² BadChild’(Cain)

no existential quantifiers

DL alone cannot derive KB ² BadChild’(Cain) non-tree-like rules needed

19

Contents

•Introduction

•Preliminaries


•DL-safe Rules




20

Decidability of Query Answering

• Theorem: Query answering in (KB, P) is decidable.• Proof: By grounding P and reduction to DL satisfiability.

KB [ { BC’(x) Ã GC(x), par(x,y), par–(y,z), hates(x,z), O(x), O(y), O(z),… } is unsatisfiable

• grounding by explicit individuals { Cain, Abel and Adam }• possible since O contains only explicit individuals

KB [ { BC’(Cain) Ç :GC(Cain) Ç :par(Cain, Adam) Ç :par–(Adam, Abel) Ç :hates(Cain, Abel),

BC’(Abel) Ç :GC(Abel) Ç :par(Abel, Adam) Ç :par–(Adam, Cain) Ç :hates(Abel, Cain)…}

• select from each clause don’t-know non-deterministicallya literal and assume it is true

KB [ { BC’(Cain), :GC(Abel) …} = KB’ is unsatisfiable

KB’ is an OWL-DL knowledge base, so satisfiability can be decided by standard algorithms.

(KB, P) ² iff (KB) [ P [ {: } is unsatisfiable

21

Contents

•Introduction

•Preliminaries


•DL-safe Rules




22

Query Answering by Disjunctive Datalog

• Theorem: (KB, P) ² iff DD(KB) [ P ² .

• Proof: By adapting slightly the original correctness proof for the reduction.

• Expected to be practicable, since it allows reusing deductive database techniques

• The algorithm is inefficient due to non-determinism

• For SHIQ, query answering can be done by reduction to disjunctive datalog

SHIQKB

Elimination of Transitivity

Axioms

TranslationInto

Clauses

Saturationby BS

Elimination of Function Symbols

Conversion to DD

Disjunctive Program DD(KB)

23

Contents

•Introduction

•Preliminaries


•DL-safe Rules




24

Conclusion

• DL-safe rules: restrict application of rules to individuals explicitly

introduced in the ABox to achieve decidability do not restrict component languages increase expressivity of component languages …can be simply appended to the result of the reduction of

SHIQ to disjunctive datalog

• Future work: extend reduction to support nominals (to support OWL-DL) implement KAON2 – a new hybrid reasoner conduct a thorough performance evaluation support some kind of non-monotonic reasoning

query answering for owl-dl with rules boris motik ulrike sattler rudi studer

Documents