DL'12 dl-lite explanations

Explanations for
DL-Lite
Alexander Borgida 2
Diego Calvanese 1
Mariano Rodríguez-Muro 1
1
1 Free University of Bozen Bolzano
2 Rutgers University
DL Workshop - May 2008 - Dresden

Outline
• Explanations
• DL-Lite and Explanations for DL-Lite
• Explanations for traditional services
• Explanations for conjunctive queries
• Conclusions and future work
2 DL Workshop - May 2008 - Dresden

Explanations

Explanations
Why explanations?

Explanations
Why explanations?
What are explanations?

Explanations
Why explanations?
• Explanations are formal proofs, constructed
from premises using rules of inference.

Explanations
Why explanations?
3
Features of an explanation:

Explanations
Why explanations?
3
Features of an explanation:
• Style
• Length
• Presentation

Explanations (cont.)

Audience: KB developer, End user

4
1. Style. Understandable inference rules (NOT
refutation or resolution)

4
2. Length: 'shorter' preferred

4
2. Length: 'shorter' preferred
3. Presentation: Complete proof vs Iterative process. as
indicated by the user. Possibly eliminating 'obvious'
parts (not addressed here)

Explanations
1. Explanations as formal proofs
2. Proof content: understandable inference rules
(NOT refutation or resolution)
• In DL-Lite, mostly simple rules
• but want performance system to help ﬁnd
‘explanation proofs’
3. Proof size: 'shorter' preferred
X

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)

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-)

DL-Litef
A1 ⊑ A2
A1 ⊑ ¬A2
ƎP ⊑ A1
ƎP- ⊑ A1
A1 ⊑ ƎP
A1 ⊑ ƎP-
A1 ⊑ ¬ƎP
A1 ⊑ ¬ƎP-
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))

Reasoning services

Reasoning services
• Standard Inferences
• TBox reasoning (concept consistency, subsumption)
• ABox reasoning (KB satisﬁability, Instance checking*)

Reasoning services
✦ Finite model reasoning

Reasoning services
• Conjunctive Query Answering

Reasoning services
• Conjunctive Query Answering
• Successful queries
• Failed queries

Generalized Inferece
Rules
<same cardinality>
<rng compos>
<dom compos>
<func compos>
X

B C
A
D
Concept Subsumption

B C
A
D
Concept Subsumption
• Hierarchy Traversing

B C
A
D
Concept Subsumption
A ⊑ D ⊓ C

B C
A
D
Concept Subsumption
A ⊑ D ⊓ C
A ⊑ C
A ⊑ B
B ⊑ D

B C
A
D
Concept Subsumption
A ⊑ D ⊓ C
A ⊑ C
A ⊑ B
B ⊑ D
• Minimal size explanations

B1
C2
C1
A
C3
B2
B3
disjoint
n j
9
A is unsatisﬁable

B1
C2
C1
A
C3
B2
B3
disjoint
n j
9
Explanation
length
n + j = 6
A is unsatisﬁable

B1
C2
C1
A
E3
E2
E1D1
C3
E4
B2
B3
disjoint
disjoint
n j k l
10
A is unsatisﬁable

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

B1
C2
C1
A
E3
E2
E1D1
C3
E1
B2
B3
disjoint
disjoint
Explanation
length
k + l = 5
A is unsatisﬁable
Explanation
length
n + j = 6
n j k l
X

KB is inconsistent
11
1. PhD ⊑ Student
2. disjoint(Professor, Student)
3. range(supervisedBy) ⊑ Professor
4. PhD(al)
5. supevisedBy(tim, al)

KB is inconsistent
11
1. PhD ⊑ Student
4. PhD(al)
al is a Student
al is a Professor
no Student can be a Professor ➔ 2

KB is inconsistent
11
1. PhD ⊑ Student
4. PhD(al)
al is a Student

every PhD is also a Student ➔ 1

al is a PhD ➔ 4
al is a Professor

KB is inconsistent
11
1. PhD ⊑ Student
4. PhD(al)
al is a Student


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

KB is inconsistent
11
1. PhD ⊑ Student
4. PhD(al)
al is a Student


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

Query Answering

⋮
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)

⋮
Student(bob)
Student(juan)
Student(al)
teaches(al, ben)
teaches(sam, ben)
teaches(sam, john)
teaches(sam, karl)
⋮
CQs in regular DBs
q(al)

⋮
Student(bob)
Student(juan)
Student(al)
teaches(al, ben)
teaches(sam, ben)
teaches(sam, john)
teaches(sam, karl)
⋮
CQs in regular DBs
q(al)
Student(al), x=al
supervisedBy(tim, al), y=tim
teaches(al, ben), z = ben

CQs in DL-Lite
X

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)

1.PhD ⊑ Student
5.PhD(al)
teaches(y,z)
q(al)

al supervisedBy @1

1.PhD ⊑ Student
5.PhD(al)
@1 teaches @2
al is a Student
teaches(y,z)
q(al)

al supervisedBy @1

1.PhD ⊑ Student
5.PhD(al)
@1 teaches @2
al is a Student
al is a PhD ➔ 5
teaches(y,z)
q(al)

al supervisedBy @1

al is in the domain of supervisedBy
1.PhD ⊑ Student
5.PhD(al)
@1 teaches @2
al is a Student
al is a PhD ➔ 5
teaches(y,z)
q(al)

al supervisedBy @1

every PhD is also in the domain of supervisedBy ➔ 2
al is a PhD ➔ 5
1.PhD ⊑ Student
5.PhD(al)
@1 teaches @2
al is a Student
al is a PhD ➔ 5
teaches(y,z)
q(al)

al supervisedBy @1

al is a PhD ➔ 5
1.PhD ⊑ Student
5.PhD(al)
@1 teaches @2
every Professor is in the domain of teaches ➔ 4
@1 is a Professor
al is a Student
al is a PhD ➔ 5
teaches(y,z)
q(al)

al supervisedBy @1

al is a PhD ➔ 5
1.PhD ⊑ Student
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
al is a PhD ➔ 5
teaches(y,z)
q(al)

Query Answering

Query Answering
• Failed queries

Query Answering
• Failed queries
• Due to missing information

Query Answering
• Failed queries
• Due to missing information
• Due to unsatisﬁability

1.PhD ⊑ Student
3.disjoint(Professor,Student)
q(x):- PhD(x), supervisedBy(y, x)
q inconsistent

@1 is a Professor
@1 is a Student
1.PhD ⊑ Student
q inconsistent
q(@1) :- PhD(@1), supervisedBy(@2, @1)

@1 is a Professor
@1 is a Student
@1 is a PhD
1.PhD ⊑ Student
q inconsistent

@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
1.PhD ⊑ Student
q inconsistent

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 exploiting existing DL-Lite query
rewriting algorithm.
• We looked into the problem of explaining failed
answers to queries.

Future Work
• Implementation for the QuOnto system.
• Explanation of failed answers to CQs over
DL-Lite ontologies.
• Explanation in Ontology Based Data Access
(presence of mappings).

Finite Model Reasoning
the set of TA is contained in student (2)
the number of students < the number of TAs
the number of students < the number of objects
in the domain of tutors (3 + subset count inference rule)
number of objects in the domain of tutors < number of objects
in the range of tutors

tutors is a function (ax 1)
number of objects in the range of tutors < number of TAs
(ax 4 + subset count inference rule)
19
TA ≣ Student

Finite Model Reasoning
the set of TA is contained in student (2)
the number of students < the number of TAs
the number of students < the number of objects
in the domain of tutors (3 + subset count inference rule)
number of objects in the domain of tutors < number of objects
in the range of tutors

tutors is a function (ax 1)
number of objects in the range of tutors < number of TAs
(ax 4 + subset count inference rule)
19
1) funct(tutors)
2) TA ⊑ Student
3) Student ⊑ range(tutors)
4) domain(tutors) ⊑ TATA ≣ Student

Why not?
explaining directly why

db |= lnot(exists y, v . A(b),R(b,y), B(y), S(y,w),C(w)).
equiv to db |= forall y,v. not A(b) / not R(b,y) / B(y)

Need to iterate through all values not in B!
The following uses an "intensional" approach

~italian(b),friendof(b,y),woman(y),drives(y,z),ferrari(z)
decreasing order to preference/brevity

not italian(b)

b not in domain(friendOf)

no values in friendOf(b) are in Woman

no values in friendOf(b) that are women are in domain(drives)

no values in drives(friendOf(b)) are in ferrari
X

Failed answers
italian(bob),friendOf(bob,y),woman(y),drives(y,z),ferrari(z)
Why not? q(bob)

DL'12 dl-lite explanations

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