Updating TBoxes in DL-Lite D. Zheleznyakov

Outline I. Introduction II. Requirements And Principles of TBox 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 Concepts: PermStaff Manager AreaManager TopManager TBox: Manager ⊑ PermStaff Manager AreaManager ⊑ Manager ABox: ∅ We considered TBox updates only for KBs with empty ABoxes AreaManager TopManager

Updating Ontologies O: Mod(O): U: ✓ ✓ ✓ ✓ PermStaff Manager AreaManager TopManager U: ✓ ✓ ✓ ✓

Outline I. Introduction II. Requirements And Principles of TBox 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 ⊨ AreaManager ⊑ PermStaff PermStaff U: AreaManager ⊑ ¬ PermStaff IF new TBox ⊨ AreaManager ⊑ PermStaff ⊨ AreaManager ⊑ ¬ PermStaff Manager 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) AreaManager TopManager

Principles of TBox Updates Manager ⊑ PermSatff AreaManager ⊑ Manager PermStaff U: AreaManager ⊑ ¬ PermStaff Assume it is forbidden to change some parts of TBox.  There is a protected fragment Tpr ⊆ TBox E.g., Tpr = {Manager ⊑ PermSatff}. Manager Protection: We accept update iff Tpr and U together are fully satisfiable AreaManager TopManager

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 of TBox Updates III. Review of Model-Based Semantics IV. Review of Formula-Based Semantics V. Bold semantics VI. Conclusion

Model-Based Semantics (MBS) Mod(O): PermStaff Manager AreaManager TopManager Minimal distance U: Mod(U): ✓ ✓ ✓ ✓

Model-Based Semantics (MBS) Mod(O): PermStaff Manager AreaManager TopManager Employee Manager AreaManager TopManager Project 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 Semantics When distance(I, J) < distance(I, K) ? I: AI={ John, Frank } BI={ Mary } distance(I, J) distance(I, K) J: AJ={ John } BJ={ Mary } K: AK={ John } BK=∅

Winslett's Semantics When distance(I, J) < distance(I, K) ? I: AI={ John, Frank } BI={ Mary } diff(I, J) = { {Frank}, ∅ } distance(I, J) distance(I, K) J: AJ={ John } BJ={ Mary } K: AK={ John } BK=∅

Winslett's Semantics When distance(I, J) < distance(I, K) ? I: AI={ John, Frank } BI={ Mary } 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) distance(I, J) distance(I, K) J: AJ={ John } BJ={ Mary } K: AK={ John } BK=∅

Winslett's Semantics. Example PermStaff U: TopManager ⊑ Manager What should the updated result be? The expectation: like in the picture Is it so under Winslett’s semantics? Manager AreaManager TopManager

Winslett's Semantics. Example Frank PermStaff U: TopManager ⊑ Manager Winslett’s semantics: new TBox ⊨ U Mimimal change in models Frank Manager What is a new TBox here? new TBox: ⊨ TopManager ⊑ Manager ✓ ⊨ Manager ⊑ PermStuff ? ✗ ⊨ AreaManager ⊑ Manager ? ✓ ⊨ AreaManager ⊑ PermStaff ✓ ? John John AreaManager TopManager Anything else?

Winslett's Semantics. Example PermStaff This TBox has irrelevant models that cannot be obtained from any model of the old TBox. We should add something into the new TBox to cut off them Manager We cannot add any other DL-Lite assertion into the new TBox, otherwise, we cut off some relevant models AreaManager TopManager

Winslett's Semantics We have to drop important assertions (Manager ⊑ PermStuff) Every MBS has such a problem Result of update under Winslett’s semantics is inexpressible in DL-Lite.  Consider Formula-Based semantics

Outline I. Introduction II. Requirements And Principles of TBox Updates III. Review of Model-Based Semantics IV. Review of Formula-Based Semantics V. Bold semantics VI. Conclusion

Formula-Based Semantics (FBS) FBS: closeness is measured btw set of formulas How? O1: O: Satisfiable PermStaff Manager AreaManager TopManager PermStaff Manager ✓ Manager AreaManager TopManager We take such a subset Omax ⊆ O, which is maximal by: cardinality, or set inclusion, or some preferences AreaManager TopManager O2: Unsatisfiable ✗ The result is: Omax ∪ U O3: Satisfiable U: Omax is not unique! There are: O1max, O2max, … ✓ 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 j U: AreaManager ⊑ ¬ PermStaff Manager TBox: AreaManager ⊑ PermStaff Manager ⊑ PermStaff ⊈ O1max AreaManager ⊑ Manager ⊈ O2max AreaManager TopManager

Cross-Product Approach. Example The output is a disjunction of KBs, one KB for each Omax: The result is: U ∪ {∨ Ojmax} PermStaff j U: AreaManager ⊑ ¬ PermStaff Manager TBox: AreaManager ⊑ PermStaff OR Manager ⊑ PermStaff ⊈ O1max AreaManager ⊑ Manager ⊈ O2max AreaManager TopManager

Cross-Product Approach. Example The output is a disjunction of KBs, one KB for each Omax: The result is: U ∪ {∨ Ojmax} PermStaff Manager AreaManager TopManager PermStaff Manager AreaManager TopManager j U: AreaManager ⊑ ¬ PermStaff TBox: AreaManager ⊑ PermStaff OR 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 of TBox Updates III. Review of Model-Based Semantics IV. Review of Formula-Based Semantics V. Bold semantics VI. Conclusion

Bold Semantics Bold approach: – Takes on board only one Omax 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 Semantics. Example Start with empty TBox Add assertions from U Add assertions from TBox one by one, if no unsatisfiability appears PermStaff U: AreaManager ⊑ ¬ PermStaff ✓ Manager TBox: AreaManager ⊑ Manager ? ✓ Manager ⊑ PermStaff ? ✗ AreaManager ⊑ PermStaff ? ✗ The result is not unique AreaManager TopManager

Bold Semantics. Example Start with empty TBox Add assertions from U Add assertions from TBox one by one, if no unsatisfiability appears PermStaff U: AreaManager ⊑ ¬ PermStaff ✓ Manager TBox: AreaManager ⊑ Manager ✓ Manager ⊑ PermStaff ✗ AreaManager ⊑ PermStaff ✗ The result is not unique U: AreaManager ⊑ ¬ PermStaff ✓ AreaManager TopManager TBox: AreaManager ⊑ Manager ? ✗ Manager ⊑ PermStaff ✓ ? AreaManager ⊑ PermStaff ✗ ?

Checking Full Satisfiability

Outline I. Introduction II. Requirements And Principles of TBox 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 applicable to DL-Lite KBs

Conclusion We proposed new semantics: Bold Semantics We proposed polynomial time algorithm to compute update under Bold semantics

Thank you!

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