Updating TBoxes in DL-Lite

D. Zheleznyakov

Outline

I. Introduction

II. Requirements And Principles ofTBox Updates

III. Review of Model-Based Semantics

IV. Review of Formula-Based Semantics

V. Bold semantics

VI. Conclusion

Description Logics (DLs)

Formalism to represent structered knowledge

Traditinal inference tasks for static DL KBs:– concept satisfiability– concept, role hierarchies

More recently – query answering

Web services are getting more important

Web Services (?)

There are many things that might be called Web Services

We use the following meaning:software system designed to support interoperable machine-to-machine interaction over a network

DLs for Web Services

Services access data through ontologies

Services can be specified using ontologies There are needs:• to enable services do data modification

ABox changes

• to modify web services

TBox changes

Ontology Changes

There are several types of ontology changes:– Revision– Update– Smth.– and such

Updating DL-Lite Ontologies

We study updates for DL-Lite KBs:it is the most tractable family of OWL 2

ABox updates:– Prelim./limited studied in [De Giacomo&al:2006] (?)– We revised and extended it [Calvanese&al:2010]

TBox updates:– Only TBox revision studied in [Qi,Du:2009]

– Topic of this talk is TBox updates

Ontologies

PermStaff

Manager

AreaManager TopManager

Concepts: PermStaffManagerAreaManagerTopManager

TBox: Manager ⊑ PermStaffAreaManager ⊑ Manager

ABox: ∅

We considered TBox updates only for KBs with empty ABoxes

Updating Ontologies

PermStaff

Manager

AreaManager TopManager

O:

U:

✓✓ ✓ ✓

Mod(O):

Outline

I. Introduction

II. Requirements And Principles ofTBox Updates

III. Review of Model-Based Semantics

IV. Review of Formula-Based Semantics

V. Bold semantics

VI. Conclusion

Tractable Closure under Updates

We want an update operator such that: Results are expressible in DL-Lite:

we require updated KBs to be expressible in DL-Lite

Results computation is tractable:we require PTIME complexity

Principles of TBox Updates

PermStaff

Manager

AreaManager TopManager

⊨ AreaManager PermStaff⊑TBox:

U: AreaManager ¬ PermStaff⊑

IF new TBox ⊨

AreaManager PermStaff⊑AreaManager ¬ PermStaff⊑

THEN AreaManagerM=∅ ∀M – model of the new TBox

Satisfiability Preservation:IF AM≠∅ before update,THEN AM≠∅ after update

(A is a atomic concept or role)

⊨

Principles of TBox Updates

PermStaff

Manager

AreaManager TopManager

TBox: Manager PermSatff⊑AreaManager Manager⊑

Assume it is forbidden to changesome parts of TBox. There is a protected fragmentTpr ⊆ TBox E.g., Tpr = {Manager PermSatff⊑ }.

U: AreaManager ¬ PermStaff⊑

Protection:We accept update iffTpr and U togetherare fully satisfiable

Principles of TBox Updates

Satisfiability Preservation:IF AM≠ ∅ before update,THEN AM≠ ∅ after update

Protection:We accept update iff protected part and U together are fully satisfiable

Moreover, we reject any update that enforces us to drop protected part

Outline

I. Introduction

II. Requirements And Principles ofTBox Updates

III. Review of Model-Based Semantics

IV. Review of Formula-Based Semantics

V. Bold semantics

VI. Conclusion

Model-Based Semantics (MBS)PermStaff

Manager

AreaManager TopManager

O:Mod(O):

Mod(U):U:

Minimaldistance

✓ ✓ ✓ ✓

Model-Based Semantics (MBS)PermStaff

Manager

AreaManager TopManager

Employee

Manager

AreaManager TopManager

Project?

O:

O’:

✓✓ ✓ ✓

Mod(O):

Mod(O’):

Winslett's Semantics

What does minimal distance mean?This depends on semantics.

Winslett’s semantics:• Well known

• There are works on ABox update under Winslett’s semantics

• Representative of MBS

Winslett's SemanticsI:

J: K:

distance(I, J) distance(I, K)

When distance(I, J) < distance(I, K) ?AI={ John, Frank }BI={ Mary }

AJ={ John }BJ={ Mary }

AK={ John }BK=∅

Winslett's SemanticsI:

J: K:

distance(I, J) distance(I, K)

When distance(I, J) < distance(I, K) ?AI={ John, Frank }BI={ Mary }

AJ={ John }BJ={ Mary }

AK={ John }BK=∅

diff(I, J) = { {Frank}, ∅ }

Winslett's SemanticsI:

J: K:

distance(I, J) distance(I, K)

When distance(I, J) < distance(I, K) ?AI={ John, Frank }BI={ Mary }

AJ={ John }BJ={ Mary }

AK={ John }BK=∅

diff(I, J) = { {Frank}, ∅ }diff(I, K) = { {Frank}, {Mary} }

diff(I, J) ⊂ diff(I, K)

So, distance(I, J) < distance(I, K) iff diff(I, J) ⊂ diff(I, K)

Winslett's Semantics. Example

PermStaff

Manager

AreaManager TopManager

U: TopManager Manager⊑

What should the updated result be?

The expectation: like in the picture

Is it so under Winslett’s semantics?

Winslett's Semantics. Example

PermStaff

Manager

AreaManager TopManager

U: TopManager Manager⊑

John

Frank

Frank

Winslett’s semantics:• new TBox ⊨ U• Mimimal change in models

John

What is a new TBox here?

new TBox: ⊨ TopManager Manager⊑ ⊨ Manager PermStuff⊑ ⊨ AreaManager Manager⊑ ⊨ AreaManager PermStaff⊑

Anything else?

✓✗??✓?✓

Winslett's Semantics. Example

PermStaff

Manager

AreaManager TopManager

This TBox has irrelevant modelsthat cannot be obtainedfrom any model of the old TBox.

We cannot add any otherDL-Lite assertion into the new TBox,otherwise, we cut offsome relevant models

We should add something into the new TBoxto cut off them

Winslett's Semantics

Result of update under Winslett’s semantics is inexpressible in DL-Lite.

We have to drop important assertions(Manager PermStuff)⊑

Every MBS has such a problem

Consider Formula-Based semantics

Outline

I. Introduction

II. Requirements And Principles ofTBox Updates

III. Review of Model-Based Semantics

IV. Review of Formula-Based Semantics

V. Bold semantics

VI. Conclusion

Formula-Based Semantics (FBS)

PermStaff

Manager

AreaManager TopManager

O:

U:

PermStaff

Manager

AreaManager TopManager

Manager

AreaManager TopManager

O1:

O2:

O3:

Satisfiable

✓

Unsatisfiable

Satisfiable

✗

✓

FBS: closeness is measuredbtw set of formulas

How?

We take such a subset Omax ⊆ O, which is maximal by:• cardinality, or• set inclusion, or• some preferences

Omax is not unique!There are: O1

max, O2max, …

The result is: Omax ∪ U

What to do with all of them?Depends on an approach

WIDTIO Approach. Example

We take only those formulas that appear in every Omax:

The result is: U ∪ ∩ Ojmax

PermStaff

Manager

AreaManager TopManager

AreaManager ¬ PermStaff⊑U:TBox: AreaManager PermStaff⊑

Manager PermStaff⊑ ⊈ O1max

AreaManager Manager⊑ ⊈ O2max

j

Cross-Product Approach. Example

OR

The output is a disjunction of KBs,one KB for each Omax:

The result is: U ∪ {∨ Ojmax}

j

AreaManager ¬ PermStaff⊑U:TBox: AreaManager PermStaff⊑

PermStaff

Manager

AreaManager TopManager

Manager PermStaff⊑ ⊈ O1max

AreaManager Manager⊑ ⊈ O2max

Cross-Product Approach. Example

OR

The output is a disjunction of KBs,one KB for each Omax:

The result is: U ∪ {∨ Ojmax}

PermStaff

Manager

AreaManager TopManager

j

AreaManager ¬ PermStaff⊑U:TBox: AreaManager PermStaff⊑

PermStaff

Manager

AreaManager TopManager

Manager PermStaff⊑ ⊈ O1max

AreaManager Manager⊑ ⊈ O2max

Formula-Based Semantics

WIDTIO approach:– Loses too much information

Cross-product approach:– “Keeps” too much information– Inexpressible in DL-Lite

Outline

I. Introduction

II. Requirements And Principles ofTBox Updates

III. Review of Model-Based Semantics

IV. Review of Formula-Based Semantics

V. Bold semantics

VI. Conclusion

Bold Semantics

Which Omax to take?• A maximal one by cardinality.

NP-Hard• A maximal one by set inclusion.

Polynomial• A maximal one by some preferences

Bold approach:– Takes on board only one Omax

Bold Semantics. Example

PermStaff

Manager

AreaManager TopManager

• Start with empty TBox

• Add assertions from U• Add assertions from TBox

one by one,if no unsatisfiability appears

AreaManager ¬ PermStaff⊑U:TBox: AreaManager Manager⊑

The result is not unique

✓✓?

Manager PermStaff⊑ ✗?AreaManager PermStaff⊑ ?✗

Bold Semantics. Example

PermStaff

Manager

AreaManager TopManager

• Start with empty TBox

• Add assertions from U• Add assertions from TBox

one by one,if no unsatisfiability appears

AreaManager ¬ PermStaff⊑U:TBox: AreaManager Manager⊑

✓✓

Manager PermStaff⊑ ✗AreaManager PermStaff⊑ ✗

AreaManager ¬ PermStaff⊑U:TBox: AreaManager Manager⊑

✓✗

Manager PermStaff⊑ ✓AreaManager PermStaff⊑ ✗

??

?

The result is not unique

Checking Full Satisfiability

Outline

I. Introduction

II. Requirements And Principles ofTBox Updates

III. Review of Model-Based Semantics

IV. Review of Formula-Based Semantics

V. Bold semantics

VI. Conclusion

Conclusion

We proposed two principles for DL KB updates

Model-based approaches:not good for TBox updates

Formula-based approaches:WIDTIO and CP are not applicableto DL-Lite KBs

Conclusion

We proposed new semantics:Bold Semantics

We proposed polynomial time algorithmto compute update under Bold semantics

Thank you!

References

[De Giacomo&al:2006] [Calvanese&al:2010] [Qi,Du:2009]