The document summarizes Vlad Ryzhikov's talk on exchanging OWL 2 QL knowledge bases. It discusses knowledge base exchange (KBE) where the source knowledge base (KB), mapping, and target KB use constraints, unlike data exchange (DE) where only mappings use constraints. It focuses on using DL-LiteR as the language for constraints, the formal counterpart of OWL 2 QL. The document defines solutions in KBE and introduces three notions - universal solution, universal UCQ-solution, and representation - and discusses their properties and issues with universal solutions in KBE.

1. Exchanging OWL 2 QL Knowledge Bases
Vlad Ryzhikov
joint work with E. Botoeva, D. Calvanese and M. Arenas
KRDB Research Centre, Free University of Bozen-Bolzano, Italy
ryzhikov@inf.unibz.it
Talk at University of KwaZulu-Natal, Durban, South Africa
Vlad Ryzhikov
Free University of Bozen-Bolzano
1/16

2. Knowledge Base Exchange
Problem
given a mapping M between the disjoint signatures Σ and Σ and a source
knowledge base (KB) K, find a target KB K that is a solution for K under
M.
M
Σ
Σ1
Σ2
target signature
source signature
A
T
D
B
T
A
C
B
C
solution
A
source KB K
Vlad Ryzhikov
A
target KB K
Free University of Bozen-Bolzano
2/16

3. Data Exchange vs. Knowledge Base Exchange
• In Data Exchange (DE) only mappings M (in some scenarios, also
solutions K ) use constraints – in Knowledge Base Exchange (KBE)
source KB K, M, and K use constraints.
Vlad Ryzhikov
Free University of Bozen-Bolzano
3/16

4. Data Exchange vs. Knowledge Base Exchange
• In Data Exchange (DE) only mappings M (in some scenarios, also
solutions K ) use constraints – in Knowledge Base Exchange (KBE)
source KB K, M, and K use constraints.
• We consider DL-LiteR as the language for the constraints; it is a
formal counterpart of OWL 2 QL standard.
Vlad Ryzhikov
Free University of Bozen-Bolzano
3/16

5. Data Exchange vs. Knowledge Base Exchange
• In Data Exchange (DE) only mappings M (in some scenarios, also
solutions K ) use constraints – in Knowledge Base Exchange (KBE)
source KB K, M, and K use constraints.
• We consider DL-LiteR as the language for the constraints; it is a
formal counterpart of OWL 2 QL standard.
• Some definitions of solutions in DE apply to KBE, however, KBE
allows for other natural definitions, which are easier to compute.
Vlad Ryzhikov
Free University of Bozen-Bolzano
3/16

6. Solutions
Solution is a target KB K that “at best” preserves the meaning of a source
KB K w.r.t. a mapping M.
Vlad Ryzhikov
Free University of Bozen-Bolzano
4/16

7. Solutions
Solution is a target KB K that “at best” preserves the meaning of a source
KB K w.r.t. a mapping M.
Example:
K = {A(a)}, M = {A
Vlad Ryzhikov
A },
Free University of Bozen-Bolzano
K = {A (a)} − solution?
4/16

8. Solutions
Solution is a target KB K that “at best” preserves the meaning of a source
KB K w.r.t. a mapping M.
Example:
K = {A(a)}, M = {A
K = {A(a), A
B}, M = {A
A },
A ,B
K = {A (a)} − solution?
B }, K = {A (a), B (a)} − solution?
K = {A (a), A
Vlad Ryzhikov
Free University of Bozen-Bolzano
B } − solution?
4/16

9. Solutions
Solution is a target KB K that “at best” preserves the meaning of a source
KB K w.r.t. a mapping M.
Example:
K = {A(a)}, M = {A
K = {A(a), A
B}, M = {A
A },
A ,B
K = {A (a)} − solution?
B }, K = {A (a), B (a)} − solution?
K = {A (a), A
K = {A
Vlad Ryzhikov
B}, M = {A
A ,B
B },
K = {A
Free University of Bozen-Bolzano
B } − solution?
B } − solution?
4/16

10. Solutions
Solution is a target KB K that “at best” preserves the meaning of a source
KB K w.r.t. a mapping M.
Example:
K = {A(a)}, M = {A
K = {A(a), A
B}, M = {A
A },
A ,B
K = {A (a)} − solution?
B }, K = {A (a), B (a)} − solution?
K = {A (a), A
K = {A
B}, M = {A
A ,B
B },
K = {A
B } − solution?
B } − solution?
Different definitions make different K above solutions!
Vlad Ryzhikov
Free University of Bozen-Bolzano
4/16

11. Definitions
DL-LiteR syntax: let a, b, . . . and n, m, . . . be infinite sets of constants and
nulls; Σ be a set of DL-LiteR concept names A and role names P, then
define
basic concepts B ::= A | ∃P | ∃P −
concept inclusions B1
concept disjointness B1
B2
B2
basic roles R ::= P | P −
role inclusions R1
⊥ role disjointness R1
R2
R2
⊥
concept membership B(a), B(n) role membership R(a, b), R(n, m), . . . ,
Vlad Ryzhikov
Free University of Bozen-Bolzano
5/16

12. Definitions
DL-LiteR syntax: let a, b, . . . and n, m, . . . be infinite sets of constants and
nulls; Σ be a set of DL-LiteR concept names A and role names P, then
define
basic concepts B ::= A | ∃P | ∃P −
concept inclusions B1
concept disjointness B1
B2
B2
basic roles R ::= P | P −
role inclusions R1
⊥ role disjointness R1
R2
R2
⊥
concept membership B(a), B(n) role membership R(a, b), R(n, m), . . . ,
DL-LiteR knowledge base is a set of concept/role inclusions/disjointness
(called TBox) and concept/role membership assertions (called ABox if
without nulls, otherwise extended ABox).
Vlad Ryzhikov
Free University of Bozen-Bolzano
5/16

13. Definitions
DL-LiteR syntax: let a, b, . . . and n, m, . . . be infinite sets of constants and
nulls; Σ be a set of DL-LiteR concept names A and role names P, then
define
basic concepts B ::= A | ∃P | ∃P −
concept inclusions B1
concept disjointness B1
B2
B2
basic roles R ::= P | P −
role inclusions R1
⊥ role disjointness R1
R2
R2
⊥
concept membership B(a), B(n) role membership R(a, b), R(n, m), . . . ,
DL-LiteR knowledge base is a set of concept/role inclusions/disjointness
(called TBox) and concept/role membership assertions (called ABox if
without nulls, otherwise extended ABox).
Mapping: defined over a pair of disjoint signatures Σ, Σ as the set of
concept inclusions/disjointness, where B is over Σ and B over Σ .
Vlad Ryzhikov
Free University of Bozen-Bolzano
5/16

14. Definitions
DL-LiteR syntax: let a, b, . . . and n, m, . . . be infinite sets of constants and
nulls; Σ be a set of DL-LiteR concept names A and role names P, then
define
basic concepts B ::= A | ∃P | ∃P −
concept inclusions B1
concept disjointness B1
B2
B2
basic roles R ::= P | P −
role inclusions R1
⊥ role disjointness R1
R2
R2
⊥
concept membership B(a), B(n) role membership R(a, b), R(n, m), . . . ,
DL-LiteR knowledge base is a set of concept/role inclusions/disjointness
(called TBox) and concept/role membership assertions (called ABox if
without nulls, otherwise extended ABox).
Mapping: defined over a pair of disjoint signatures Σ, Σ as the set of
concept inclusions/disjointness, where B is over Σ and B over Σ .
Semantics: standard, no unique name assumption.
Vlad Ryzhikov
Free University of Bozen-Bolzano
5/16

15. Solutions
We consider three notions:
• Universal solution
Inherited from incomplete data exchage; analogious to model conservative
extentions or Σ-model inseparability
Vlad Ryzhikov
Free University of Bozen-Bolzano
6/16

16. Solutions
We consider three notions:
• Universal solution
Inherited from incomplete data exchage; analogious to model conservative
extentions or Σ-model inseparability
• Universal UCQ-solution (UCQ = Union of Conjunctive Queries)
Based on what can be extracted from source and target with unions of
conjunctive queries; analogious to query conservative extentions or Σ-query
inseparability
Vlad Ryzhikov
Free University of Bozen-Bolzano
6/16

17. Solutions
We consider three notions:
• Universal solution
Inherited from incomplete data exchage; analogious to model conservative
extentions or Σ-model inseparability
• Universal UCQ-solution (UCQ = Union of Conjunctive Queries)
Based on what can be extracted from source and target with unions of
conjunctive queries; analogious to query conservative extentions or Σ-query
inseparability
• Representation
Like Universal UCQ-solution, but defined w.r.t. K and K containing only TBox;
uses universal quantification over possible the source and target ABoxes
Vlad Ryzhikov
Free University of Bozen-Bolzano
6/16

18. Universal Solutions
• Let Mod(K) be the set of models of K.
• Let I, J be a pair of DL-LiteR interpretations over signatures,
respectively Σ and Γ, s.t. Σ ⊆ Γ, then I is said to agree with J on Σ, if
aI = aJ for all constants a;
AI = AJ and P I = P J for all concept and role names A and P from Σ.
Given a Γ interpretation J , denote by agrΣ (J ) the set of all Σ
interpretations that agree with J on Σ; we also use agrΣ (J ), where J
is a set of Σ interpretations.
Vlad Ryzhikov
Free University of Bozen-Bolzano
7/16

19. Universal Solutions
• Let Mod(K) be the set of models of K.
• Let I, J be a pair of DL-LiteR interpretations over signatures,
respectively Σ and Γ, s.t. Σ ⊆ Γ, then I is said to agree with J on Σ, if
aI = aJ for all constants a;
AI = AJ and P I = P J for all concept and role names A and P from Σ.
Given a Γ interpretation J , denote by agrΣ (J ) the set of all Σ
interpretations that agree with J on Σ; we also use agrΣ (J ), where J
is a set of Σ interpretations.
• Let the mapping M be between the signatures Σ and Σ ; a KB K over
Σ is said to be a universal solution (US) for a KB K over Σ under M if
Mod(K ) = agrΣ (Mod(K ∪ M)).
Vlad Ryzhikov
Free University of Bozen-Bolzano
7/16

20. Universal Solutions contd.
Mod(K ) = agrΣ (Mod(K ∪ M))
Examples:
K = {A(a)}, M = {A
Vlad Ryzhikov
A },
Free University of Bozen-Bolzano
K = {A (a)} − US
8/16

21. Universal Solutions contd.
Mod(K ) = agrΣ (Mod(K ∪ M))
Examples:
K = {A(a)}, M = {A
K = {A(a), A
Vlad Ryzhikov
B}, M = {A
A },
A ,B
K = {A (a)} − US
B },
K = {A (a), B (a)} − US
K = {A (a), A
B } − not US
Free University of Bozen-Bolzano
8/16

22. Universal Solutions contd.
Mod(K ) = agrΣ (Mod(K ∪ M))
Examples:
K = {A(a)}, M = {A
K = {A(a), A
B}, M = {A
K = {∃R(a)}, M = {R
Vlad Ryzhikov
A },
A ,B
R , ∃R −
K = {A (a)} − US
B },
K = {A (a), B (a)} − US
K = {A (a), A
B } − not US
B }, K = {R (a, n), B (n)} − US
Free University of Bozen-Bolzano
8/16

23. Universal Solutions contd.
Mod(K ) = agrΣ (Mod(K ∪ M))
Examples:
K = {A(a)}, M = {A
K = {A(a), A
B}, M = {A
K = {∃R(a)}, M = {R
K = {A B ⊥,
A(a), B(b)}, M = {A
Vlad Ryzhikov
A },
A ,B
R , ∃R −
A ,B
K = {A (a)} − US
B },
K = {A (a), B (a)} − US
K = {A (a), A
B } − not US
B }, K = {R (a, n), B (n)} − US
B}
Free University of Bozen-Bolzano
− no US exists
8/16

24. Universal Solutions contd.
US is a fundamental and well-behaved notion in DE, however, in KBE it
has a number of issues:
• Nulls required in ABox, which are not part of OWL 2 QL standard.
Vlad Ryzhikov
Free University of Bozen-Bolzano
9/16

25. Universal Solutions contd.
US is a fundamental and well-behaved notion in DE, however, in KBE it
has a number of issues:
• Nulls required in ABox, which are not part of OWL 2 QL standard.
• USs cannot contain any non-trivial TBox, i.e., only (extended) ABoxes
can be materialized as the target.
Vlad Ryzhikov
Free University of Bozen-Bolzano
9/16

26. Universal Solutions contd.
US is a fundamental and well-behaved notion in DE, however, in KBE it
has a number of issues:
• Nulls required in ABox, which are not part of OWL 2 QL standard.
• USs cannot contain any non-trivial TBox, i.e., only (extended) ABoxes
can be materialized as the target.
• USs “very often” do not exists, when the source KB K1 contains
disjointness assertions. Reason: no unique name assumption, as it is
the case in OWL 2 QL.
Vlad Ryzhikov
Free University of Bozen-Bolzano
9/16

27. Universal UCQ-solutions
Universal UCQ-solution is a “softer” notion of solution, that avoids the
above mentioned issues. It uses UCQs q and certains answers cert(q, K):
Vlad Ryzhikov
Free University of Bozen-Bolzano
10/16

28. Universal UCQ-solutions
Universal UCQ-solution is a “softer” notion of solution, that avoids the
above mentioned issues. It uses UCQs q and certains answers cert(q, K):
• Let the mapping M be between the signatures Σ and Σ ; a KB K over
Σ is said to be a universal UCQ-solution (UUCQS) for a KB K over Σ
under M if
cert(q, K ) = cert(q, K ∪ M)
for each UCQ q over Σ .
Vlad Ryzhikov
Free University of Bozen-Bolzano
10/16

29. Universal UCQ-solutions
Universal UCQ-solution is a “softer” notion of solution, that avoids the
above mentioned issues. It uses UCQs q and certains answers cert(q, K):
• Let the mapping M be between the signatures Σ and Σ ; a KB K over
Σ is said to be a universal UCQ-solution (UUCQS) for a KB K over Σ
under M if
cert(q, K ) = cert(q, K ∪ M)
for each UCQ q over Σ .
• if only the inclusion ⊇ in the equation above satisfied, K is called a
UCQ-solution
Vlad Ryzhikov
Free University of Bozen-Bolzano
10/16

30. Universal UCQ-solutions contd.
cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ
Examples:
K = {A(a)}, M = {A
Vlad Ryzhikov
A },
Free University of Bozen-Bolzano
K = {A (a)} − UUCQS
11/16

31. Universal UCQ-solutions contd.
cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ
Examples:
K = {A(a)}, M = {A
K = {A(a), A
Vlad Ryzhikov
B}, M = {A
A },
A ,B
K = {A (a)} − UUCQS
B },
K = {A (a), B (a)} − UUCQS
K = {A (a), A
B } − UUCQS
Free University of Bozen-Bolzano
11/16

32. Universal UCQ-solutions contd.
cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ
Examples:
K = {A(a)}, M = {A
K = {A(a), A
B}, M = {A
K = {∃R(a)}, M = {R
Vlad Ryzhikov
A },
A ,B
R , ∃R −
K = {A (a)} − UUCQS
B },
K = {A (a), B (a)} − UUCQS
K = {A (a), A
B } − UUCQS
B },
Free University of Bozen-Bolzano
K = {∃R (a), ∃R − B }
− UUCQS
11/16

33. Universal UCQ-solutions contd.
cert(q, K ) = cert(q, K ∪ M) for each UCQ q over Σ
Examples:
K = {A(a)}, M = {A
K = {A(a), A
B}, M = {A
K = {∃R(a)}, M = {R
K = {A B ⊥,
A(a), B(b)}, M = {A
Vlad Ryzhikov
A },
A ,B
R , ∃R −
A ,B
K = {A (a)} − UUCQS
B },
K = {A (a), B (a)} − UUCQS
K = {A (a), A
B } − UUCQS
B },
B}
Free University of Bozen-Bolzano
K = {∃R (a), ∃R − B }
− UUCQS
K = {A (a), B (b)}
−UUCQS
11/16

34. Universal UCQ-solutions contd.
• UUCQS is a notion of the solution, that is better suited for KBE.
Vlad Ryzhikov
Free University of Bozen-Bolzano
12/16

35. Universal UCQ-solutions contd.
• UUCQS is a notion of the solution, that is better suited for KBE.
• Still, this notion is dependent on data, i.e., ABox; computing UUCQS
requires processing big amounts of frequently changing data.
Vlad Ryzhikov
Free University of Bozen-Bolzano
12/16

36. Universal UCQ-solutions contd.
• UUCQS is a notion of the solution, that is better suited for KBE.
• Still, this notion is dependent on data, i.e., ABox; computing UUCQS
requires processing big amounts of frequently changing data.
• UCQ-representation is a notion of the solution, that is not dependent
on data.
Vlad Ryzhikov
Free University of Bozen-Bolzano
12/16

37. UCQ-representation
For the definition, we need to consider UCQ-solutions over KBs consisting
of only ABoxes. Cosider A = {A(a)} and M = {A A }, then
• A = {A (a), A (b)} - UCQ-solution;
• A = {A (b)} - not UCQ-solution.
Vlad Ryzhikov
Free University of Bozen-Bolzano
13/16

38. UCQ-representation
For the definition, we need to consider UCQ-solutions over KBs consisting
of only ABoxes. Cosider A = {A(a)} and M = {A A }, then
• A = {A (a), A (b)} - UCQ-solution;
• A = {A (b)} - not UCQ-solution.
Let the mapping M be between the signatures Σ and Σ ; a TBox T over Σ
is said to be UCQ-representaton (UCQR) for a TBox T over Σ under M if
cert(q, T ∪ A ∪ M) =
cert(q, T ∪ A ).
A : A is an ABox over Σ that
is a UCQ-solution for A under M
for
• each UCQ q over Σ ,
• ABox A over Σ
such that T ∪ A is consistent.
Vlad Ryzhikov
Free University of Bozen-Bolzano
13/16

39. UCQ-representation cont.
cert(q, T ∪ A ∪ M) =
cert(q, T ∪ A )
A : A is an ABox over Σ that
is a UCQ-solution for A under M
for each UCQ q over Σ and ABox A over Σ, such that T ∪ A is consistent.
Examples:
T = {A
Vlad Ryzhikov
A}, M = {A
A },
T = {A
Free University of Bozen-Bolzano
A } − UCQR
14/16

40. UCQ-representation cont.
cert(q, T ∪ A ∪ M) =
cert(q, T ∪ A )
A : A is an ABox over Σ that
is a UCQ-solution for A under M
for each UCQ q over Σ and ABox A over Σ, such that T ∪ A is consistent.
Examples:
T = {A
A },
T = {A
Vlad Ryzhikov
A}, M = {A
B}, M = {A
A ,B
T = {A
B },
A } − UCQR
T = {A
A } − not UCQR
T = {A
B } − UCQR
Free University of Bozen-Bolzano
14/16

41. UCQ-representation cont.
cert(q, T ∪ A ∪ M) =
cert(q, T ∪ A )
A : A is an ABox over Σ that
is a UCQ-solution for A under M
for each UCQ q over Σ and ABox A over Σ, such that T ∪ A is consistent.
Examples:
T = {A
A },
T = {A
B}, M = {A
A ,B
T = {A
Vlad Ryzhikov
A}, M = {A
B}, M = {B
T = {A
B },
B },
A } − UCQR
T = {A
A } − not UCQR
T = {A
B } − UCQR
Free University of Bozen-Bolzano
− no UCQR exists
14/16

42. UCQ-representation cont.
cert(q, T ∪ A ∪ M) =
cert(q, T ∪ A )
A : A is an ABox over Σ that
is a UCQ-solution for A under M
for each UCQ q over Σ and ABox A over Σ, such that T ∪ A is consistent.
Examples:
T = {A
A}, M = {A
A },
T = {A
B}, M = {A
A ,B
T = {A
B}, M = {B
B },
⊥}, M = {A
A ,B
T = {A
Vlad Ryzhikov
B
T = {A
B },
A } − UCQR
T = {A
A } − not UCQR
T = {A
B } − UCQR
− no UCQR exists
B },
T = {A
Free University of Bozen-Bolzano
B
⊥} − UCQR
T = ∅ − UCQR
14/16

43. Summary of Complexity Results
Membership
Universal solutions
UCQ-representations
Non-emptiness
Universal solutions
UCQ-representations
ABoxes extended ABoxes
in NP
NP-complete
NLogSpace-complete
ABoxes
extended ABoxes
in NP
PSpace-hard, in ExpTime
NLogSpace-complete
• Membership problem: given source KB K1 , target KB K2 , and the
mapping M, decide, if K2 is correct.
• Non-emptyness problem: given source KB K1 and the mapping M,
decide, if there exists a target KB K2 , such that it is correct.
Vlad Ryzhikov
Free University of Bozen-Bolzano
15/16

44. Summary of Complexity Results
Membership
Universal solutions
UCQ-representations
Non-emptiness
Universal solutions
UCQ-representations
ABoxes extended ABoxes
in NP
NP-complete
NLogSpace-complete
ABoxes
extended ABoxes
in NP
PSpace-hard, in ExpTime
NLogSpace-complete
• Membership problem: given source KB K1 , target KB K2 , and the
mapping M, decide, if K2 is correct.
• Non-emptyness problem: given source KB K1 and the mapping M,
decide, if there exists a target KB K2 , such that it is correct.
• Universal UCQ-solution: membership is PSpace-hard, no other results
yet - future work.
Vlad Ryzhikov
Free University of Bozen-Bolzano
15/16

45. Thank you
for your attention!
Vlad Ryzhikov
Free University of Bozen-Bolzano
16/16