The paper investigates the decidability of role mappings between modular ontologies represented using distributed description logics (DDL) and package-based description logics (P-DL). It is shown that extending ALC with both concept and role mappings in DDL results in undecidability if one of negated roles, cardinality restrictions in bridge rules, or inverse bridge rules is allowed. However, a decidable P-DL family ALCHIO(¬)P is identified that extends ALC with concept/role importing and various other features while retaining decidability. The results help characterize the decidability boundary of modular ontology languages.

1. On the Decidability of Role Mappings Between Modular
Ontologies
Jie Bao1
baojie@cs.rpi.edu
George Voutsadakis2 , Giora Slutzki2 , Vasant Honavar2
{gvoutsad,slutzki,honavar}@cs.iastate.edu
1 Department of Computer Science,
Rensselaer Polytechnic Institute, Troy, NY 12180, USA.
2 Department of Computer Science,
Iowa State University, Ames, IA 50011, USA.
The 23rd Conference on Artificial Intelligence
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 1 / 19

2. Outline
1 Background
Modular Ontology
Decidability
Contributions
2 Preliminary
Notations
Semantics
3 (Un)decidability Results
Basic Approach
Undecidable DDLs
Decidable P-DL Family ALCHIO(¬)P
4 Conclusion
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 2 / 19

3. Background Modular Ontology
Modular Ontology and Ontology Mapping
1 On Semantic Web: Single Ontology or Modular Ontologies?
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 3 / 19

4. Background Modular Ontology
Modular Ontology and Ontology Mapping
1 On Semantic Web: Single Ontology or Modular Ontologies?
2 Mappings between modular ontologies are required.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 3 / 19

5. Background Decidability
Decidability
Important for automated reasoning
Decidability (in general)
Given an ontology language, can we always finish inference with any
ontology in that language in finite time?
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 4 / 19

6. Background Decidability
Decidability
Important for automated reasoning
Decidability (in general)
Given an ontology language, can we always finish inference with any
ontology in that language in finite time?
Decidability in modular ontologies
Given a modular ontology language, can we always finish inference with
any ontology in that language in finite time?
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 4 / 19

7. Background Decidability
Decidability
Important for automated reasoning
Decidability (in general)
Given an ontology language, can we always finish inference with any
ontology in that language in finite time?
Decidability in modular ontologies
Given a modular ontology language, can we always finish inference with
any ontology in that language in finite time?
What language features are allowed within each module?
What language features (i.e., mapping) are allowed between
modules?
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 4 / 19

8. Background Decidability
Decidability w.r.t. Language
Language features within each module
E.g., negated role? (does not know=¬knows)
E.g., inverse role? (is known by=knows− )
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 5 / 19

9. Background Decidability
Decidability w.r.t. Language
Language features within each module
E.g., negated role? (does not know=¬knows)
E.g., inverse role? (is known by=knows− )
Mappings between modules
E.g., concept mapping? (a Husband is a Man)
E.g., role mapping? (marriedTo implies knows)
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 5 / 19

10. Background Decidability
Decidability w.r.t. Language
Language features within each module
E.g., negated role? (does not know=¬knows)
E.g., inverse role? (is known by=knows− )
Mappings between modules
E.g., concept mapping? (a Husband is a Man)
E.g., role mapping? (marriedTo implies knows)
Research focus of the paper
Where is the decidability frontier of some modular ontology languages?
Distributed Description Logics (DDL) [2, 3]
Package-based Description Logics (P-DL) [1]
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 5 / 19

11. Background Contributions
Known Results
For the Decidability of DDL
Extending SHIQ with concept mappings is decidable [2];
Extending SHIQ with concept mappings and role mapping is
decidable [3].
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 6 / 19

12. Background Contributions
Known Results
For the Decidability of DDL
Extending SHIQ with concept mappings is decidable [2];
Extending SHIQ with concept mappings and role mapping is
decidable [3].
For the Decidability of P-DL
Extending DL SHOIQ with concept, role and nominal importing, and
disallowing imported role being used in role inclusions, is decidable [1].
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 6 / 19

13. Background Contributions
New Results of the Paper
For the Decidability of DDL
Extending ALC with concept and role mappings, and one of i) negated
roles, ii) cardinality restrictions in bridge rules or iii) inverse bridge rules,
yields undecidable DDLs.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 7 / 19

14. Background Contributions
New Results of the Paper
For the Decidability of DDL
Extending ALC with concept and role mappings, and one of i) negated
roles, ii) cardinality restrictions in bridge rules or iii) inverse bridge rules,
yields undecidable DDLs.
For the Decidability of P-DL
Extending ALC with concept, role and nominal importing, along with
unrestricted role inclusions, inverse roles, nominals and negated roles,
yields a decidable P-DL.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 7 / 19

15. Preliminary Notations
Notations
Module Language
Recall that the basic description logic ALC supports
Concept conjunction (⊓), concept disjunction (⊔), concept negation
(¬);
Existential restriction (∃), value restriction (∀)
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 8 / 19

16. Preliminary Notations
Notations
Module Language
Recall that the basic description logic ALC supports
Concept conjunction (⊓), concept disjunction (⊔), concept negation
(¬);
Existential restriction (∃), value restriction (∀)
Other DL features considered
Negated role (¬), Inverse role I, Role inclusion H, Nominal O
e.g., ALCHIO(¬)
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 8 / 19

17. Preliminary Notations
Notations
Module Language
Recall that the basic description logic ALC supports
Concept conjunction (⊓), concept disjunction (⊔), concept negation
(¬);
Existential restriction (∃), value restriction (∀)
Other DL features considered
Negated role (¬), Inverse role I, Role inclusion H, Nominal O
e.g., ALCHIO(¬)
Mapping Language (more details next)
Distributed Description Logics (DDL): D
Package-based Description Logics (P-DL): P
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 8 / 19

18. Preliminary Notations
DDL
In DDL, local vocabularies of modules are disjoint, mapping is by
directional bridge rules.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 9 / 19

19. Preliminary Notations
DDL
In DDL, local vocabularies of modules are disjoint, mapping is by
directional bridge rules.
Concept bridge rules DC
⊑
Into: 1 : Husband − 2 : Man (Husband is less general than Man seen
→
by O2 )
⊒
Onto: 1 : Human − 2 : Man (Human is more general than Man seen
→
by O2 )
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 9 / 19

20. Preliminary Notations
DDL
In DDL, local vocabularies of modules are disjoint, mapping is by
directional bridge rules.
Concept bridge rules DC
⊑
Into: 1 : Husband − 2 : Man (Husband is less general than Man seen
→
by O2 )
⊒
Onto: 1 : Human − 2 : Man (Human is more general than Man seen
→
by O2 )
Role bridge rules DR
⊑
Into: 1 : marriedTo − 2 : knows
→
⊒
Onto: 2 : familyOf − 1 : marriedTo
→
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 9 / 19

21. Preliminary Notations
P-DL
In P-DL, local vocabularies of modules are not disjoint, mapping is
by name importing and local inclusion.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 10 / 19

22. Preliminary Notations
P-DL
In P-DL, local vocabularies of modules are not disjoint, mapping is
by name importing and local inclusion.
Both concept mapping and role mapping are supported.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 10 / 19

23. Preliminary Notations
P-DL
In P-DL, local vocabularies of modules are not disjoint, mapping is
by name importing and local inclusion.
Both concept mapping and role mapping are supported.
Importing offers strong modeling ability.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 10 / 19

24. Preliminary Semantics
Semantics
Local Domains and Domain Relations
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 11 / 19

25. Preliminary Semantics
Semantics
Local Domains and Domain Relations
DDL: r12 (HusbandI1 ) ⊆ ManI2 [into rule; onto rules similar]
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 11 / 19

26. Preliminary Semantics
Semantics
Local Domains and Domain Relations
DDL: r12 (HusbandI1 ) ⊆ ManI2 [into rule; onto rules similar]
P-DL: HusbandI2 = r12 (HusbandI1 ); domain relations are one-to-one
and compositional consistent.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 11 / 19

27. (Un)decidability Results Basic Approach
Basic Approach
Undecidability Proof
Decidability Proof
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 12 / 19

28. (Un)decidability Results Basic Approach
Basic Approach
Undecidability Proof
Decidability Proof
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 13 / 19

29. (Un)decidability Results Undecidable DDLs
Undecidable DDLs
Undecidable extensions of ALCDCR
ALC(¬)DCR : allows negated roles, e.g., notKnow = ¬knows
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 14 / 19

30. (Un)decidability Results Undecidable DDLs
Undecidable DDLs
Undecidable extensions of ALCDCR
ALC(¬)DCR : allows negated roles, e.g., notKnow = ¬knows
ALCDCRI : allows inverse bridge rules,
⊑
e.g., 1 : Husband ← 2 : Man (a Husband is a Man seen by O1 )
−
Can avoid empty domain relations [6].
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 14 / 19

31. (Un)decidability Results Undecidable DDLs
Undecidable DDLs
Undecidable extensions of ALCDCR
ALC(¬)DCR : allows negated roles, e.g., notKnow = ¬knows
ALCDCRI : allows inverse bridge rules,
⊑
e.g., 1 : Husband ← 2 : Man (a Husband is a Man seen by O1 )
−
Can avoid empty domain relations [6].
ALCDCRN : allows cardinality restrictions on domain relations,
≥1
e.g., ⊤2 −→ 1 : Husband
−
Can help enforcing enforcing the partial injectivity of domain
relations [5].
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 14 / 19

32. (Un)decidability Results Undecidable DDLs
Domino Tiling Problem
Tiling of an infinite plane is known to be undecidable
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 15 / 19

33. (Un)decidability Results Undecidable DDLs
Reducing a Domino System into an DDL ontology
Main Idea
Use the ontology to enforce the existence of a grid-shaped solution
Two local domains, vertically alternating.
Vertical matching conditions: enforced by DL axioms
Horizontal matching conditions: enforced by DDL bridge rules.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 16 / 19

34. (Un)decidability Results Decidable P-DL Family ALCHIO(¬)P
ALCHIO(¬)P & ALBO
ALCHIO(¬)P
Concept grammar: C := A|o|C ⊓ C |¬k C |∃R.C
Role grammar: R := p|R − |¬k R
Importing: concept, role and nominal names
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 17 / 19

35. (Un)decidability Results Decidable P-DL Family ALCHIO(¬)P
ALCHIO(¬)P & ALBO
ALCHIO(¬)P
Concept grammar: C := A|o|C ⊓ C |¬k C |∃R.C
Role grammar: R := p|R − |¬k R
Importing: concept, role and nominal names
Reduction ⇓
ALBO
Concept grammar: C := A|o|C ⊓ C |¬C |∃R.C
Role grammar: R := p|R − |¬R|R ⊓ R|(R ⇂ C )|(R ↿ C )
Known to be decidable [4]
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 17 / 19

36. (Un)decidability Results Decidable P-DL Family ALCHIO(¬)P
ALCHIO(¬)P ⇒ ALBO
Main idea
Combine all ontology modules in ALCHIO(¬)P into one DL
ontology in ALBO
Based on the reduction P-DL SHOIQP to DL SHOIQ [1]
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 18 / 19

37. (Un)decidability Results Decidable P-DL Family ALCHIO(¬)P
ALCHIO(¬)P ⇒ ALBO
Main idea
Combine all ontology modules in ALCHIO(¬)P into one DL
ontology in ALBO
Based on the reduction P-DL SHOIQP to DL SHOIQ [1]
Some details about negated roles
Introduce new role names for imported roles
Preserve context of roles using (R ⇂ C ) and (R ↿ C )
Details in the paper
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 18 / 19

38. Conclusion
Conclusion
Results
Undecidability: DDL ALC(¬)D CR , ALCD CRI , ALCDCRN
Decidability: P-DL ALCHIO(¬)P
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 19 / 19

39. Conclusion
Conclusion
Results
Undecidability: DDL ALC(¬)D CR , ALCD CRI , ALCDCRN
Decidability: P-DL ALCHIO(¬)P
Future Work
DDL with nominals?
P-DL with Transitive roles, number restrictions?
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 19 / 19

40. References
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Mapping properties of heterogeneous ontologies.
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http://ftp.informatik.rwth–aachen.de/Publications/CEUR–WS/Vol–
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R. Schmidt and D. Tishkovsky.
Deciding ALBO with tableau.
Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 19 / 19

41. References
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Jie Bao et al. (RPI & ISU) Decidability of Role Mappings July 2008, AAAI 2008 19 / 19