Explanations for
DL-Lite
Alexander Borgida 2
Diego Calvanese 1
Mariano Rodríguez-Muro 1
1
1 Free University of Bozen Bolzano
2 Rutgers University
ODBASE 08 - Monterrey, México
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Outline
• Explanations
• DL-Lite and Explanations for DL-Lite
• Explanations for traditional services
• Explanations for conjunctive queries
• Conclusions and future work
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Explanations
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Explanations
Why explanations?
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Explanations
Why explanations?
What are explanations?
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Explanations
Why explanations?
What are explanations?
• Explanations are formal proofs, constructed
from premises using rules of inference.
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Explanations
Why explanations?
What are explanations?
• Explanations are formal proofs, constructed
from premises using rules of inference.
3
Features of an explanation:
ODBASE 08 - Monterrey, México
Thursday, August 8, 13

Explanations
Why explanations?
What are explanations?
• Explanations are formal proofs, constructed
from premises using rules of inference.
3
Features of an explanation:
• Style
• Length
• Presentation
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Explanations (cont.)
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Explanations (cont.)
Audience: KB developer, End user
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Explanations (cont.)
Audience: KB developer, End user
4
1. Style. Understandable inference rules (NOT
refutation or resolution)
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Explanations (cont.)
Audience: KB developer, End user
4
1. Style. Understandable inference rules (NOT
refutation or resolution)
2. Length: 'shorter' preferred
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Explanations (cont.)
Audience: KB developer, End user
4
1. Style. Understandable inference rules (NOT
refutation or resolution)
2. Length: 'shorter' preferred
3. Presentation: Complete proof vs Iterative process. as
indicated by the user. Possibly eliminating 'obvious'
parts (not addressed here)
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DL-Litef
Concept constructs
B ::= A | ƎP | ƎP-
C::= B | ¬B | C1 ⊓ C2
TBox Assertions
B ⊑ C
(funct P) (funt P-)
ABox Assertions
A(a) R(a,b)
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DL-Litef
Is-A Hierarchies
Class disjointness
Role-typing
Participation constraints
Non-participation constraints
Functionality restrictions
A1 ⊑ A2
A1 ⊑ ¬A2
ƎP ⊑ A1
ƎP- ⊑ A1
A1 ⊑ ƎP
A1 ⊑ ƎP-
A1 ⊑ ¬ƎP
A1 ⊑ ¬ƎP-
(funct P) (funct P-)
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DL-Litef
A1 ⊑ A2
A1 ⊑ ¬A2
ƎP ⊑ A1
ƎP- ⊑ A1
A1 ⊑ ƎP
A1 ⊑ ƎP-
A1 ⊑ ¬ƎP
A1 ⊑ ¬ƎP-
(funct P) (funct P-)
A1 ⊑ A2
disjoint(A1,A2)
domain(P) ⊑ A1
range(P) ⊑ A1
A1 ⊑ domain(P)
A1 ⊑ range(P)
disjoint(A1, domain(P))
disjoint(A1, range(P))
(funct P) (funct P-)
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Reasoning services
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Reasoning services
• Standard Inferences
• TBox reasoning (concept consistency, subsumption)
• ABox reasoning (KB satisfiability, Instance checking*)
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Reasoning services
• Standard Inferences
• TBox reasoning (concept consistency, subsumption)
• ABox reasoning (KB satisfiability, Instance checking*)
✦ Finite model reasoning
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Reasoning services
• Standard Inferences
• TBox reasoning (concept consistency, subsumption)
• ABox reasoning (KB satisfiability, Instance checking*)
✦ Finite model reasoning
• Conjunctive Query Answering
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Reasoning services
• Standard Inferences
• TBox reasoning (concept consistency, subsumption)
• ABox reasoning (KB satisfiability, Instance checking*)
✦ Finite model reasoning
• Conjunctive Query Answering
• Successful queries
• Failed queries
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B C
A
D
Concept Subsumption
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B C
A
D
Concept Subsumption
• Hierarchy Traversing
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B C
A
D
Concept Subsumption
• Hierarchy Traversing
A ⊑ D ⊓ C
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B C
A
D
Concept Subsumption
• Hierarchy Traversing
A ⊑ D ⊓ C
A ⊑ C
A ⊑ B
B ⊑ D
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B C
A
D
Concept Subsumption
• Hierarchy Traversing
A ⊑ D ⊓ C
A ⊑ C
A ⊑ B
B ⊑ D
• Minimal size explanations
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B1
C2
C1
A
C3
B2
B3
disjoint
n j
9
A is unsatisfiable
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B1
C2
C1
A
C3
B2
B3
disjoint
n j
9
Explanation
length
n + j = 6
A is unsatisfiable
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B1
C2
C1
A
E3
E2
E1D1
C3
E4
B2
B3
disjoint
disjoint
n j k l
10
A is unsatisfiable
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B1
C2
C1
A
E3
E2
E1D1
C3
E4
B2
B3
disjoint
disjoint
Explanation
length
k + l = 5
Explanation
length
n + j = 6
n j k l
10
A is unsatisfiable
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KB is inconsistent
11
1. PhD ⊑ Student
2. disjoint(Professor, Student)
3. range(supervisedBy) ⊑ Professor
4. PhD(al)
5. supevisedBy(tim, al)
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KB is inconsistent
11
1. PhD ⊑ Student
2. disjoint(Professor, Student)
3. range(supervisedBy) ⊑ Professor
4. PhD(al)
5. supevisedBy(tim, al)
al is a Student
al is a Professor
no Student can be a Professor ➔ 2
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KB is inconsistent
11
1. PhD ⊑ Student
2. disjoint(Professor, Student)
3. range(supervisedBy) ⊑ Professor
4. PhD(al)
5. supevisedBy(tim, al)
al is a Student
!every PhD is also a Student ➔ 1
!al is a PhD ➔ 4
al is a Professor
no Student can be a Professor ➔ 2
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KB is inconsistent
11
1. PhD ⊑ Student
2. disjoint(Professor, Student)
3. range(supervisedBy) ⊑ Professor
4. PhD(al)
5. supevisedBy(tim, al)
al is a Student
!every PhD is also a Student ➔ 1
!al is a PhD ➔ 4
al is a Professor
everything in the range of supervisedBy is also
a Professor ➔ 3
al is in range of supervisedBy
no Student can be a Professor ➔ 2
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KB is inconsistent
11
1. PhD ⊑ Student
2. disjoint(Professor, Student)
3. range(supervisedBy) ⊑ Professor
4. PhD(al)
5. supevisedBy(tim, al)
al is a Student
!every PhD is also a Student ➔ 1
!al is a PhD ➔ 4
al is a Professor
everything in the range of supervisedBy is also
a Professor ➔ 3
al is in range of supervisedBy
tim supervisedBy al ➔ 5
no Student can be a Professor ➔ 2
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Query Answering
• Successful queries
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⋮
Student(bob)
Student(juan)
supervisedBy(tom, bob)
Student(al)
supervisedBy(tim, al)
teaches(al, ben)
teaches(sam, ben)
teaches(sam, john)
teaches(sam, karl)
⋮
CQs in regular DBs
q(x) :- Student(x), supervisedBy(y,x), teaches(x,z)
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⋮
Student(bob)
Student(juan)
supervisedBy(tom, bob)
Student(al)
supervisedBy(tim, al)
teaches(al, ben)
teaches(sam, ben)
teaches(sam, john)
teaches(sam, karl)
⋮
CQs in regular DBs
q(x) :- Student(x), supervisedBy(y,x), teaches(x,z)
q(al)
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⋮
Student(bob)
Student(juan)
supervisedBy(tom, bob)
Student(al)
supervisedBy(tim, al)
teaches(al, ben)
teaches(sam, ben)
teaches(sam, john)
teaches(sam, karl)
⋮
CQs in regular DBs
q(x) :- Student(x), supervisedBy(y,x), teaches(x,z)
q(al)
Student(al), x=al
supervisedBy(tim, al), y=tim
teaches(al, ben), z = ben
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1.PhD ⊑ Student
2.PhD ⊑ dom(supervisedBy)
3.range(supervisedBy) ⊑ Professor
4.Professor ⊑ domain(teaches)
5.PhD(al)
q(x) :- Student(x), supervisedBy(x,y),
teaches(y,z)
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1.PhD ⊑ Student
2.PhD ⊑ dom(supervisedBy)
3.range(supervisedBy) ⊑ Professor
4.Professor ⊑ domain(teaches)
5.PhD(al)
q(x) :- Student(x), supervisedBy(x,y),
teaches(y,z)
q(al)
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al supervisedBy @1!!
1.PhD ⊑ Student
2.PhD ⊑ dom(supervisedBy)
3.range(supervisedBy) ⊑ Professor
4.Professor ⊑ domain(teaches)
5.PhD(al)
@1 teaches @2
al is a Student
q(x) :- Student(x), supervisedBy(x,y),
teaches(y,z)
q(al)
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al supervisedBy @1!!
1.PhD ⊑ Student
2.PhD ⊑ dom(supervisedBy)
3.range(supervisedBy) ⊑ Professor
4.Professor ⊑ domain(teaches)
5.PhD(al)
@1 teaches @2
al is a Student
every PhD is also a Student ➔ 1
al is a PhD ➔ 5
q(x) :- Student(x), supervisedBy(x,y),
teaches(y,z)
q(al)
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al supervisedBy @1!!
al is in the domain of supervisedBy
1.PhD ⊑ Student
2.PhD ⊑ dom(supervisedBy)
3.range(supervisedBy) ⊑ Professor
4.Professor ⊑ domain(teaches)
5.PhD(al)
@1 teaches @2
al is a Student
every PhD is also a Student ➔ 1
al is a PhD ➔ 5
q(x) :- Student(x), supervisedBy(x,y),
teaches(y,z)
q(al)
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al supervisedBy @1!!
al is in the domain of supervisedBy
every PhD is also in the domain of supervisedBy ➔ 2
al is a PhD ➔ 5
1.PhD ⊑ Student
2.PhD ⊑ dom(supervisedBy)
3.range(supervisedBy) ⊑ Professor
4.Professor ⊑ domain(teaches)
5.PhD(al)
@1 teaches @2
al is a Student
every PhD is also a Student ➔ 1
al is a PhD ➔ 5
q(x) :- Student(x), supervisedBy(x,y),
teaches(y,z)
q(al)
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al supervisedBy @1!!
al is in the domain of supervisedBy
every PhD is also in the domain of supervisedBy ➔ 2
al is a PhD ➔ 5
1.PhD ⊑ Student
2.PhD ⊑ dom(supervisedBy)
3.range(supervisedBy) ⊑ Professor
4.Professor ⊑ domain(teaches)
5.PhD(al)
@1 teaches @2
every Professor is in the domain of teaches ➔ 4
@1 is a Professor
al is a Student
every PhD is also a Student ➔ 1
al is a PhD ➔ 5
q(x) :- Student(x), supervisedBy(x,y),
teaches(y,z)
q(al)
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al supervisedBy @1!!
al is in the domain of supervisedBy
every PhD is also in the domain of supervisedBy ➔ 2
al is a PhD ➔ 5
1.PhD ⊑ Student
2.PhD ⊑ dom(supervisedBy)
3.range(supervisedBy) ⊑ Professor
4.Professor ⊑ domain(teaches)
5.PhD(al)
@1 teaches @2
every Professor is in the domain of teaches ➔ 4
@1 is a Professor
@1 is in the range of supervisedBy
everything in the range of supervisedBy is a
Professor ➔ 3
al is a Student
every PhD is also a Student ➔ 1
al is a PhD ➔ 5
q(x) :- Student(x), supervisedBy(x,y),
teaches(y,z)
q(al)
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al supervisedBy @1!!
al is in the domain of supervisedBy
every PhD is also in the domain of supervisedBy ➔ 2
al is a PhD ➔ 5
1.PhD ⊑ Student
2.PhD ⊑ dom(supervisedBy)
3.range(supervisedBy) ⊑ Professor
4.Professor ⊑ domain(teaches)
5.PhD(al)
@1 teaches @2
every Professor is in the domain of teaches ➔ 4
@1 is a Professor
@1 is in the range of supervisedBy
everything in the range of supervisedBy is a
Professor ➔ 3
al is a Student
every PhD is also a Student ➔ 1
al is a PhD ➔ 5
q(x) :- Student(x), supervisedBy(x,y),
teaches(y,z)
q(al)
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Query Answering
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Query Answering
• Successful queries
• Failed queries
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Query Answering
• Successful queries
• Failed queries
• Due to missing information
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Query Answering
• Successful queries
• Failed queries
• Due to missing information
• Due to unsatisfiability
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1.PhD ⊑ Student
2.range(supervisedBy) ⊑ Professor
3.disjoint(Professor,Student)
q(x):- PhD(x), supervisedBy(y, x)
q inconsistent
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@1 is a Professor
@1 is a Student
1.PhD ⊑ Student
2.range(supervisedBy) ⊑ Professor
3.disjoint(Professor,Student)
q(x):- PhD(x), supervisedBy(y, x)
q inconsistent
q(@1) :- PhD(@1), supervisedBy(@2, @1)
no Student can be a Professor ➔ 3
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@1 is a Professor
@1 is a Student
@1 is a PhD
every PhD is also a Student ➔ 1
1.PhD ⊑ Student
2.range(supervisedBy) ⊑ Professor
3.disjoint(Professor,Student)
q(x):- PhD(x), supervisedBy(y, x)
q inconsistent
q(@1) :- PhD(@1), supervisedBy(@2, @1)
no Student can be a Professor ➔ 3
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@1 is a Professor
@1 is a Student
@1 is a PhD
every PhD is also a Student ➔ 1
1.PhD ⊑ Student
2.range(supervisedBy) ⊑ Professor
3.disjoint(Professor,Student)
q(x):- PhD(x), supervisedBy(y, x)
q inconsistent
q(@1) :- PhD(@1), supervisedBy(@2, @1)
no Student can be a Professor ➔ 3
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@1 is a Professor
@2 supervisedBy @1
everything in the range of supervisedBy is a Professor ➔ 2
@1 is a Student
@1 is a PhD
every PhD is also a Student ➔ 1
1.PhD ⊑ Student
2.range(supervisedBy) ⊑ Professor
3.disjoint(Professor,Student)
q(x):- PhD(x), supervisedBy(y, x)
q inconsistent
q(@1) :- PhD(@1), supervisedBy(@2, @1)
no Student can be a Professor ➔ 3
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@1 is a Professor
@2 supervisedBy @1
everything in the range of supervisedBy is a Professor ➔ 2
@1 is a Student
@1 is a PhD
every PhD is also a Student ➔ 1
1.PhD ⊑ Student
2.range(supervisedBy) ⊑ Professor
3.disjoint(Professor,Student)
q(x):- PhD(x), supervisedBy(y, x)
q inconsistent
q(@1) :- PhD(@1), supervisedBy(@2, @1)
no Student can be a Professor ➔ 3
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Conclusions
• We addressed DL-Lite explanations can be given for
traditional reasoning services (shortness of proof).
• Finite model case.
• Explaining conjunctive queries when reasoning is
present by a) exploiting existing DL-Lite query
rewriting algorithm and b) Prolog-based program.
• We looked into the problem of explaining failed
answers to queries.
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Future Work
• Explanation of failed answers to CQs over
DL-Lite ontologies.
• Integration of prototype with the QuOnto
Reasoner.
• Explanation in Ontology Based Data Access
(presence of mappings).
• Field testing of the algorithms.
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