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
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