1.
Dependencies
Making Ontology Based Data Access Work in Practice
Mariano Rodriguez-Muro and Diego Calvanese
{rodriguez,calvanese}@inf.unibz.it
KRDB Research Centre
Free University of Bozen Bolzano
May 11, 2011
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 1 / 33

2.
The context
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 2 / 33

3.
DL Ontologies
Description Logics:
• Formalisms for knowledge representation.
• Decidable fragments of FOL
• Base of OWL
• World is described by means of Concepts and Roles
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 3 / 33

4.
DL Ontologies
Description Logics:
• Formalisms for knowledge representation.
• Decidable fragments of FOL
• Base of OWL
• World is described by means of Concepts and Roles
Ontologies
• Intentional knowledge: TBox T .
• Extensional knowledge: ABox A.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 3 / 33

5.
OBDA with DL-Lite
A family of light-weight ontology languages
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 4 / 33

6.
OBDA with DL-Lite
A family of light-weight ontology languages
• DL-LiteF concepts
B := A | ∃R
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7.
OBDA with DL-Lite
A family of light-weight ontology languages
• DL-LiteF concepts
B := A | ∃R
• DL-LiteF roles
R := P | P−
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 4 / 33

8.
OBDA with DL-Lite
A family of light-weight ontology languages
• DL-LiteF concepts
B := A | ∃R
• DL-LiteF roles
R := P | P−
• DL-LiteF TBoxes
B B | B ¬B | (funct R)
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 4 / 33

9.
OBDA with DL-Lite
A family of light-weight ontology languages
• DL-LiteF concepts
B := A | ∃R
• DL-LiteF roles
R := P | P−
• DL-LiteF TBoxes
B B | B ¬B | (funct R)
• DL-LiteF ABoxes
A(a) | R(a, b)
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10.
Query Answering
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11.
Query Answering
TBox:
Man Person, Woman Person, Person ∃hasFather,
∃hasFather−
Person
ABox:
Man(mariano)
Queries:
q(x) ← Person(x), hasFather(x, y), Person(y)
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12.
Query Answering
TBox:
Man Person, Woman Person, Person ∃hasFather,
∃hasFather−
Person
ABox:
Man(mariano)
Queries:
q(x) ← Person(x), hasFather(x, y), Person(y)
Problem: Compute the certain answers of Q, denoted cert(Q, O).
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 5 / 33

13.
Query Answering
TBox:
Man Person, Woman Person, Person ∃hasFather,
∃hasFather−
Person
ABox:
Man(mariano)
Queries:
q(x) ← Person(x), hasFather(x, y), Person(y)
Problem: Compute the certain answers of Q, denoted cert(Q, O).
The promise
We can do this as efficiently as answering DB queries, also in the virtual
setting.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 5 / 33

14.
Query Answering with PerfectRef (2005)
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15.
Query Answering with PerfectRef (2005)
Query:
q(x) ← Person(x), hasFather(x, y), Person(y)
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16.
Query Answering with PerfectRef (2005)
Query:
q(x) ← Person(x), hasFather(x, y), Person(y)
Reformulation:
q(x) ← Person(x), hasFather(x, y), Person(y)
q(x) ← Person(x), hasFather(x, y), hasFather(z, y)
q(x) ← Person(x), hasFather(x, y)
q(x) ← Person(x), Person(x)
q(x) ← Person(x)
q(x) ← Person(x), hasFather(x, y), Man(y)
q(x) ← Person(x), hasFather(x, y), Woman(y)
q(x) ← hasFather(x, m), hasFather(x, y), Person(y)
q(x) ← hasFather(x, m), hasFather(x, y), hasFather(z, y)
q(x) ← hasFather(x, m), hasFather(x, y)
q(x) ← hasFather(x, m), Person(x)
q(x) ← hasFather(x, m), hasFather(x, t)
q(x) ← hasFather(x, m)
q(x) ← hasFather(x, m), hasFather(x, y), Man(y)Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 6 / 33

17.
Alternatives
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18.
Alternatives
• Improved version of PerfectRef (2007-2011)
• RQR (Urbina et, al. 2007)
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19.
Alternatives
• Improved version of PerfectRef (2007-2011)
• RQR (Urbina et, al. 2007)
Too many unions, cannot execute!.
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20.
Alternatives
• Improved version of PerfectRef (2007-2011)
• RQR (Urbina et, al. 2007)
Too many unions, cannot execute!.
• PRESTO (Rosati et al., 2010)
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 7 / 33

21.
Alternatives
• Improved version of PerfectRef (2007-2011)
• RQR (Urbina et, al. 2007)
Too many unions, cannot execute!.
• PRESTO (Rosati et al., 2010)
Better, eventually it breaks.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 7 / 33

22.
Alternatives
• Improved version of PerfectRef (2007-2011)
• RQR (Urbina et, al. 2007)
Too many unions, cannot execute!.
• PRESTO (Rosati et al., 2010)
Better, eventually it breaks.
• Combined Approach (Kontchakov et. al., 2010)
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 7 / 33

23.
Alternatives
• Improved version of PerfectRef (2007-2011)
• RQR (Urbina et, al. 2007)
Too many unions, cannot execute!.
• PRESTO (Rosati et al., 2010)
Better, eventually it breaks.
• Combined Approach (Kontchakov et. al., 2010)
Fast. But too much data and too much time.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 7 / 33

24.
What can we do?
?
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 8 / 33

25.
Query Answering
It is not only about existential constants
Query:
q(x, y) ← Person(x), hasFather(x, y), Person(y)
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 9 / 33

26.
Query Answering
It is not only about existential constants
Query:
q(x, y) ← Person(x), hasFather(x, y), Person(y)
Reformulation:
q(x, y) ← Person(x), hasFather(x, y), Person(y)
q(x, y) ← Person(x), hasFather(x, y), hasFather(z, y)
q(x, y) ← Person(x), hasFather(x, y), Man(y)
q(x, y) ← Person(x), hasFather(x, y), Woman(y)
q(x, y) ← hasFather(x, m), hasFather(x, y), Person(y)
q(x, y) ← hasFather(x, m), hasFather(x, y), hasFather(z, y)
q(x, y) ← hasFather(x, m), hasFather(x, y), Man(y)
q(x, y) ← hasFather(x, m), hasFather(x, y), Woman(y)
q(x, y) ← Man(x), hasFather(x, y), Person(y)
q(x, y) ← Man(x), hasFather(x, y), hasFather(z, y)
q(x, y) ← Man(x), hasFather(x, y), Man(y)
q(x, y) ← Man(x), hasFather(x, y), Woman(y)
q(x, y) ← Woman(x), hasFather(x, y), Person(y)
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27.
The full picture: Ontology Based Data
Access
SourceUser Source
User
Queries
Ontology
Mappings
Source
To deal with OBDA we need to consider:
• If in the backend we have RDBMSs, we cannot go beyond their
capabilities.
• All systems are composed by T , D = R, I , M.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 10 / 33

28.
First Observation
Is my data complete?
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 11 / 33

29.
First Observation
Is my data complete?
Completeness of A
The TBox sais: Manager Employee
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 11 / 33

30.
First Observation
Is my data complete?
Completeness of A
The TBox sais: Manager Employee
In the ABox: all Managers are already employees.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 11 / 33

31.
First Observation
Is my data complete?
Completeness of A
The TBox sais: Manager Employee
In the ABox: all Managers are already employees.
In any realistic scenario:
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 11 / 33

32.
First Observation
Is my data complete?
Completeness of A
The TBox sais: Manager Employee
In the ABox: all Managers are already employees.
In any realistic scenario:
• We don’t use arbitrary sources;
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 11 / 33

33.
First Observation
Is my data complete?
Completeness of A
The TBox sais: Manager Employee
In the ABox: all Managers are already employees.
In any realistic scenario:
• We don’t use arbitrary sources;
• Intersection of semantics is reflected in completeness (e.g., no need to
chase, expand or rewrite)
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 11 / 33

34.
First Observation
Is my data complete?
Completeness of A
The TBox sais: Manager Employee
In the ABox: all Managers are already employees.
In any realistic scenario:
• We don’t use arbitrary sources;
• Intersection of semantics is reflected in completeness (e.g., no need to
chase, expand or rewrite)
• This happens a lot!
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 11 / 33

35.
First Observation
Is my data complete?
Completeness of A
The TBox sais: Manager Employee
In the ABox: all Managers are already employees.
In any realistic scenario:
• We don’t use arbitrary sources;
• Intersection of semantics is reflected in completeness (e.g., no need to
chase, expand or rewrite)
• This happens a lot!
Keyword
Redundancy
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 11 / 33

36.
Second Observation
There are no ABoxes
THERE ARE NO ABOXES!
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37.
Second Observation
There are no ABoxes
THERE ARE NO ABOXES!
Any Ontology based query answering systems today:
• Uses relational DBs to store the ABox data;
• In such D, both, R and I can be manipulated;
• Implementors may choose any M for their system;
Opportunity
To complete an ABox we can do more than expansion.
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38.
How to approach the problem
Two level approach
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39.
How to approach the problem
Two level approach
How to approach OBDA in practice?
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40.
How to approach the problem
Two level approach
How to approach OBDA in practice?
• Efficient ways to deal with redundancy due to completeness.
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41.
How to approach the problem
Two level approach
How to approach OBDA in practice?
• Efficient ways to deal with redundancy due to completeness.
• Efficient ways to complete (virtual) ABoxes.
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42.
Contributions
Dealing with redundancy
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43.
Characterizing completeness
ABox Dependencies
Definition
An assertion B A B that restricts valid ABoxes.
Syntax B2 A B2
Semantics: A |= Manager A Employee if Manager(x)∈ A implies
Employee(x)∈ A.
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44.
Characterizing completeness
ABox Dependencies
Definition
An assertion B A B that restricts valid ABoxes.
Syntax B2 A B2
Semantics: A |= Manager A Employee if Manager(x)∈ A implies
Employee(x)∈ A.
ABox dependencies are fundamentally different than TBox assertions.
Think open world
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45.
Where to deal with redundancy?
Given a TBox T , an ABox A, a set of dependencies Σ and a query Q,
what do we do?
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46.
Where to deal with redundancy?
Given a TBox T , an ABox A, a set of dependencies Σ and a query Q,
what do we do?
Available Options:
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 16 / 33

47.
Where to deal with redundancy?
Given a TBox T , an ABox A, a set of dependencies Σ and a query Q,
what do we do?
Available Options:
• Optimize the query reformulation algorithm to deal with Σ.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 16 / 33

48.
Where to deal with redundancy?
Given a TBox T , an ABox A, a set of dependencies Σ and a query Q,
what do we do?
Available Options:
• Optimize the query reformulation algorithm to deal with Σ.
• Optimize the TBox T with respect to Σ.
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49.
When is an assertion redundant?
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50.
When is an assertion redundant?
Direct Redundancy: Case 1
Let T be implied the following
hierarchy:
∃hasFather
Person
Human
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51.
When is an assertion redundant?
Direct Redundancy: Case 1
Let T be implied the following
hierarchy:
∃hasFather
Person
Human
Redundant if Σ is:
∃hasFather
Person
Human
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52.
When is an assertion redundant?
Direct Redundancy: Case 1
Let T be implied the following
hierarchy:
∃hasFather
Person
Human
Redundant if Σ is:
∃hasFather
Person
Human
Σ sais hasFather(mariano, ramon) ∈ A → Human(mariano) ∈ A.
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53.
When is an assertion redundant?
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54.
When is an assertion redundant?
Direct Redundancy: Case 2
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55.
When is an assertion redundant?
Direct Redundancy: Case 2
Let T be the following TBox:
Person
∃hasFather−
∃hasFather
Man
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56.
When is an assertion redundant?
Direct Redundancy: Case 2
Let T be the following TBox:
Person
∃hasFather−
∃hasFather
Man
Redundant if Σ is:
Person
∃hasFather−
∃hasFather
Man
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57.
When is an assertion redundant?
Direct Redundancy: Case 2
Let T be the following TBox:
Person
∃hasFather−
∃hasFather
Man
Redundant if Σ is:
Person
∃hasFather−
∃hasFather
Man
Σ sais Man(ramon) ∈ A → ∃a | hasFather(ramon, a ) ∧ Person(a ) ∈ A.
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58.
When is an assertion redundant?
Indirect Redundancy
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59.
When is an assertion redundant?
Indirect Redundancy
Let T be the following TBox:
Animal
Man Human
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60.
When is an assertion redundant?
Indirect Redundancy
Let T be the following TBox:
Animal
Man Human
Redundant if Σ is:
Animal
Man Human
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61.
When is an assertion redundant?
Indirect Redundancy
Let T be the following TBox:
Animal
Man Human
Redundant if Σ is:
Animal
Man Human
Σ sais Man(mariano) ∈ A then Animal(mariano) ∈ A.
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62.
Formalization: Redundancy
Given a TBox T and a set of dependencies Σ over T , the optimized version
of T w.r.t. Σ, denoted optim(T , Σ), is the set of inclusion assertions
{α ∈ sat(T ) | α is not redundant in sat(T ) w.r.t. sat(Σ)}
We can compute optim(T , Σ) in linear time.
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63.
Contributions
Completing ABoxes
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64.
General considerations
OBDA systems have no ABoxes, instead virtual ABoxes V = D, M with
D = R, I .
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65.
General considerations
OBDA systems have no ABoxes, instead virtual ABoxes V = D, M with
D = R, I .
If we that V |= A A B, we check make sure that mappings for B include
all the data coming from the mappings of A.
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66.
General considerations
OBDA systems have no ABoxes, instead virtual ABoxes V = D, M with
D = R, I .
If we that V |= A A B, we check make sure that mappings for B include
all the data coming from the mappings of A.
Trade-off:
• Degree of completeness (# of dependencies),
• Cost of the procedure
• Performance of Query answering.
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67.
General considerations
OBDA systems have no ABoxes, instead virtual ABoxes V = D, M with
D = R, I .
If we that V |= A A B, we check make sure that mappings for B include
all the data coming from the mappings of A.
Trade-off:
• Degree of completeness (# of dependencies),
• Cost of the procedure
• Performance of Query answering.
We can complete virtual ABoxes up to B ∃R without the need for new
data.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 22 / 33

68.
Semantic Index for OBDA
General Idea
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69.
Semantic Index for OBDA
General Idea
• To encode the semantics of T in numeric indexes and ranges for
concept names and roles.
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70.
Semantic Index for OBDA
General Idea
• To encode the semantics of T in numeric indexes and ranges for
concept names and roles.
• Store the ABox in the database using those indexes and ranges.
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71.
Semantic Index for OBDA
General Idea
• To encode the semantics of T in numeric indexes and ranges for
concept names and roles.
• Store the ABox in the database using those indexes and ranges.
• Make mappings for the system that take the ranges into account.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 23 / 33

72.
Semantic Index for OBDA
General Idea
• To encode the semantics of T in numeric indexes and ranges for
concept names and roles.
• Store the ABox in the database using those indexes and ranges.
• Make mappings for the system that take the ranges into account.
We can do this by using the implied hierarchy of T to generate the index
and ranges!
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73.
Semantic Index Example
T = {B A, C A, C D}
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74.
Semantic Index Example
T = {B A, C A, C D}
A
B C
D
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75.
Semantic Index Example
T = {B A, C A, C D}
1
A
B
2
C
3
4
D
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76.
Semantic Index Example
T = {B A, C A, C D}
1
A
B
2
C
3
4
D
We create a table TC with constant and idx columns. To insert the data
we use the indexes. e.g., B(mariano) ∈ A then we put (mariano, 2) ∈ TC
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77.
Semantic Index Example
T = {B A, C A, C D}
1, {(1, 3)}
A
B
2, {(2, 2)}
C
3, {(3, 3)}
4, {(3, 4)}
D
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78.
Semantic Index Example
T = {B A, C A, C D}
1, {(1, 3)}
A
B
2, {(2, 2)}
C
3, {(3, 3)}
4, {(3, 4)}
D
We create the mappings using the ranges, e.g., SELECT constant
FROM TC WHERE IDX ≥ 1 AND IDX ≤ 3; A(constant)
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79.
Experimentation I
The Resource Index features:
• Search over 22 document collections
• Semantics given by the hierarchies of 200 ontologies (SNOMED, GO)
Implementation in a nutshell:
(i) Understand documents with natural language processing and
annotate
Cervical Cancer( doc224 )
(ii) Expand the ABox
(iii) Pose queries that retrieve documents as
q(x) ← A1(x) ∧ · · · ∧ An(x)
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80.
Experimentation II
The challenge:
• ≈ 3 million concepts and ≈ 2.5 million is-a assertions
• Split second responses
• 150 GB of data
• Expansion data: 1.5 TB
The experimentation data:
• Clinical Trials.gov (CT)
• 181 million assertion (≈ 14 GB of data, ≈ 140 GB when expanded.)
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81.
Results
The query:
q(x) ← DNA Repair Gene(x) ∧ Antigen Gene(x) ∧ Cancer Gene(x)
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82.
Results
The query:
q(x) ← DNA Repair Gene(x) ∧ Antigen Gene(x) ∧ Cancer Gene(x)
Results:
• Traditional reformulation: Union of 467874 SQL SPJ queries;
• Semantic Index: 1 SQL; execution 3.582s (0.082s if warm); Time
to compute semantic index: 1 min; Size of data: +≈ 4 GB.
• ABox expansion: 1 SQL; executing 3s (0.6s if warm); Expansion
time ≈ 7 days; Size of data +≈ 126 GB.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 27 / 33

83.
The Query
The query:
q(x) ← DNA Repair Gene(x) ∧ Antigen Gene(x) ∧ Cancer Gene(x)
SELECT DISTINCT r0.element_id as element_id
FROM
RESOURCE_INDEX.CT_ANN r0 JOIN RESOURCE_INDEX.CT_ANN r1
ON r0.element_id = r1.element_id
JOIN RESOURCE_INDEX.CT_ANN r2
ON r1.element_id = r2.element_id
WHERE
((r0.idx >= 1783559 AND r0.idx <= 1783657)) AND
((r1.idx >= 1782996 AND r1.idx <= 1783029)) AND
((r2.idx >= 1783115 AND r2.idx <= 1783253));
Standard SQL query efficient in ANY DBMS.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 28 / 33

84.
Conclusions
Contributions
• We indicated that efficient OBDA requires to take into account more
than only T , A and Q.
• Provided means to deal with redundancy at the level of the TBox.
• We showed that expansion is not necessary that we can complete
ABoxes.
• We presented to efficient ways to complete ABoxes, one for the
general OBDA setting and one for the virtual setting.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 29 / 33

85.
Conclusions
Contributions
• We indicated that efficient OBDA requires to take into account more
than only T , A and Q.
• Provided means to deal with redundancy at the level of the TBox.
• We showed that expansion is not necessary that we can complete
ABoxes.
• We presented to efficient ways to complete ABoxes, one for the
general OBDA setting and one for the virtual setting.
Future work
• Exploring more expressive languages.
• Exploring the RDFS/SPARQL setting.
• Handling updates of T and A.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 29 / 33

86.
Extra examples
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 30 / 33

87.
First Observation (cont.)
Mappings will introduce dependencies over ABoxes
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 31 / 33

88.
First Observation (cont.)
Mappings will introduce dependencies over ABoxes
Let R be a DB schema with the relation schema employee with attributes
id, dept, and salary. Let M be the following mappings:
SELECT id,dept FROM employee ;q(id, dept) ← Employee(id) ∧
WORKS-FOR(id, dept)
SELECT id,dept FROM employee
WHERE salary > 1000
;q(id, dept) ← Manager(id)∧
MANAGES(id, dept)
Then for any instance I, if Manager(John) ∈ A we have that
Employee(John).
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 31 / 33

89.
First Observation (cont.)
Mappings will introduce dependencies over ABoxes
Let R be a DB schema with the relation schema employee with attributes
id, dept, and salary. Let M be the following mappings:
SELECT id,dept FROM employee ;q(id, dept) ← Employee(id) ∧
WORKS-FOR(id, dept)
SELECT id,dept FROM employee
WHERE salary > 1000
;q(id, dept) ← Manager(id)∧
MANAGES(id, dept)
Then for any instance I, if Manager(John) ∈ A we have that
Employee(John).
This is an indicator of completeness of all ABoxes A for M and R, e.g., A
is complete w.r.t. Manager A Employee.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 31 / 33

90.
Formalization: Chains
Let T be a TBox, B, C basic concepts, and Σ a set of dependencies over
T . A T -chain from B to C in T (resp., a Σ-chain from B to C in Σ) is a
sequence of concept inclusion assertions (Bi Bi )n
i=0 in T (resp., a
sequence of inclusion dependencies (Bi A Bi )n
i=0 in Σ), for some n ≥ 0,
such that:
1 B0 = B, Bn = C, and
2 for 1 ≤ i ≤ n, we have that Bi−1 and Bi are basic concepts s.t., either
(i) Bi−1 = Bi , or
(ii) Bi−1 = ∃R and Bi = ∃R−
, for some basic role R.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 32 / 33

91.
Formalization: Redundancy
Let T be a TBox, B, C basic concepts, and Σ a set of dependencies. The
concept inclusion assertion B C is directly redundant in T w.r.t. Σ if
(i) Σ |= B A C and
(ii) for every T -chain (Bi Bi )n
i=0 with Bn = B in T , there is a Σ-chain
(Bi A Bi )n
i=0.
Then, B C is redundant in T w.r.t. Σ if
(a) it is directly redundant, or
(b) there exists B = B s.t.
(i) T |= B C,
(ii) B C is not redundant in T w.r.t. Σ, and
(iii) B B is directly redundant in T w.r.t. Σ.
Rodriguez-Muro and Calvanese (UNIBZ) Dependencies and OBDA May 11, 2011 33 / 33