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Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Rewriting in DL-Lite(HN )horn
Elena Botoeva, Alessandro Artale, and Diego Calvanese
KRDB Research CentreFree University of Bozen-Bolzano
I-39100 Bolzano, Italy
Description Logics Workshop, 2010
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
1/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Outline
1 Motivation
2 The DL DL-Lite(HN)horn
3 Knowledge Base Satisfiability
4 Query Answering
5 Conclusions
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
2/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Motivation: Ontology-Based Data Access
• Ontologies are used for accessing data
query
Application
response
Application 1 Application 2
query
response
Data source 1
Data source 2
Data source 3
• An ontology provides a high-level conceptual view of informationsources
• Data sources can be queried through ontologies
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
3/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Motivation: Ontology-Based Data Access
• Ontologies are used for accessing data
Data source 1
Data source 2
Data source 3
C1
A2
R1
C2
C3
A1
query
Ontology
Application
response
Application
C1
A2
R1
C2
C3
A1
Ontology
• An ontology provides a high-level conceptual view of informationsources
• Data sources can be queried through ontologies
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
3/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering by Rewriting
• We want to compute certain answers to a query
• Rewriting approach:
1 Rewrite the query using the constraints in the ontology2 Evaluate the rewritten query over the database
Reasoning
Rewritten Query
Query Result
Reasoning
DataSource
LogicalSchema
Schema /Ontology
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
4/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering by Rewriting
• We want to compute certain answers to a query
• Rewriting approach:
1 Rewrite the query using the constraints in the ontology2 Evaluate the rewritten query over the database
Reasoning
Rewritten Query
Query Result
Reasoning
DataSource
LogicalSchema
Schema /Ontology
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
4/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering by Rewriting: Example
Ontology: O = {PhDStudent v Student}Database: DBA = {PhDStudent(john)}Query: q(x)← Student(x)
• The rewriting of q:qucq(x)← Student(x)qucq(x)← PhDStudent(x)
• By evaluating the rewriting over the ABox viewed as a DB :eval(qucq, DBA) = {john} = ans(q, 〈O,DBA〉)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
5/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering by Rewriting: Example
Ontology: O = {PhDStudent v Student}Database: DBA = {PhDStudent(john)}Query: q(x)← Student(x)
• The rewriting of q:qucq(x)← Student(x)qucq(x)← PhDStudent(x)
• By evaluating the rewriting over the ABox viewed as a DB :eval(qucq, DBA) = {john} = ans(q, 〈O,DBA〉)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
5/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
FOL Rewritable Logics
• Such a rewriting approach can be applied only to FOL rewritablelogics.
• DL-Lite is a family of logics that has been shown to enjoy FOLrewritability:
I DL-LiteR, DL-LiteF , DL-LiteA
• Extended DL-Lite family: additional constructs have been proposed
I DL-Lite(HN )horn is the most interesting logic
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
6/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
FOL Rewritable Logics
• Such a rewriting approach can be applied only to FOL rewritablelogics.
• DL-Lite is a family of logics that has been shown to enjoy FOLrewritability:
I DL-LiteR, DL-LiteF , DL-LiteA
• Extended DL-Lite family: additional constructs have been proposed
I DL-Lite(HN )horn is the most interesting logic
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
6/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
FOL Rewritable Logics
• Such a rewriting approach can be applied only to FOL rewritablelogics.
• DL-Lite is a family of logics that has been shown to enjoy FOLrewritability:
I DL-LiteR, DL-LiteF , DL-LiteA
• Extended DL-Lite family: additional constructs have been proposed
I DL-Lite(HN )horn is the most interesting logic
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
6/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Outline
1 Motivation
2 The DL DL-Lite(HN)horn
3 Knowledge Base Satisfiability
4 Query Answering
5 Conclusions
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
7/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
DL-Lite(HN )horn
In this work we consider the logic DL-Lite(HN )horn :
• The most expressive tractable variant of DL-Lite [ACKZ09].
• Extends DL-Lite withI role inclusions H
• hasConfPaper v hasPublication
I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥
I horn inclusions horn• Student u ≥1 teaches v PhDStudent
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
8/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
DL-Lite(HN )horn
In this work we consider the logic DL-Lite(HN )horn :
• The most expressive tractable variant of DL-Lite [ACKZ09].
• Extends DL-Lite with
I role inclusions H• hasConfPaper v hasPublication
I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥
I horn inclusions horn• Student u ≥1 teaches v PhDStudent
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
8/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
DL-Lite(HN )horn
In this work we consider the logic DL-Lite(HN )horn :
• The most expressive tractable variant of DL-Lite [ACKZ09].
• Extends DL-Lite withI role inclusions H
• hasConfPaper v hasPublication
I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥
I horn inclusions horn• Student u ≥1 teaches v PhDStudent
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
8/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
DL-Lite(HN )horn
In this work we consider the logic DL-Lite(HN )horn :
• The most expressive tractable variant of DL-Lite [ACKZ09].
• Extends DL-Lite withI role inclusions H
• hasConfPaper v hasPublication
I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥
I horn inclusions horn• Student u ≥1 teaches v PhDStudent
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
8/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
DL-Lite(HN )horn
In this work we consider the logic DL-Lite(HN )horn :
• The most expressive tractable variant of DL-Lite [ACKZ09].
• Extends DL-Lite withI role inclusions H
• hasConfPaper v hasPublication
I number restrictions N• PhDStudent v ≥2 hasConfPaper• ≥2 teaches− v ⊥
I horn inclusions horn• Student u ≥1 teaches v PhDStudent
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
8/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Questions Addressed by Our Work
For the logic DL-Lite(HN )horn :
• Can we check ontology satisfiability by relying on RDB technology?
• Can we answer queries by relying on RDB technology?
• Can we extend the practical algorithms developed forthe simpler DL-Lite logics?
• What is the complexity of such algorithms?
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
9/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
DL-Lite(HN )horn : Syntax
Concept and role expressions
B ::= ⊥ | A | ≥k R
R ::= P | P−
TBox assertions
B1 u · · · u Bn v BR1 v R2
Dis(R1,R2)
Restriction to ensure FOL rewritability:
if R has a proper sub-role, then ≥k R with k ≥ 2does not occur in the lhs of concept inclusions.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
10/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
DL-Lite(HN )horn : Syntax
Concept and role expressions
B ::= ⊥ | A | ≥k R
R ::= P | P−
TBox assertions
B1 u · · · u Bn v BR1 v R2
Dis(R1,R2)
Restriction to ensure FOL rewritability:
if R has a proper sub-role, then ≥k R with k ≥ 2does not occur in the lhs of concept inclusions.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
10/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
A DL-Lite(HN )horn TBox
• basic concept inclusion
≥1 hasPublication− v Publication
• role inclusion
hasConfPaper v hasPublication
• number restrictions
PhDStudent v ≥2 hasConfPaper
• horn inclusion
Student u ≥1 teaches v PhDStudent
• local functionality assertion
PhDStudent u ≥2 teaches v ⊥
• global functionality assertion
≥2 teaches− v ⊥
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
11/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
A DL-Lite(HN )horn TBox
• basic concept inclusion
≥1 hasPublication− v Publication
• role inclusion
hasConfPaper v hasPublication
• number restrictions
PhDStudent v ≥2 hasConfPaper
• horn inclusion
Student u ≥1 teaches v PhDStudent
• local functionality assertion
PhDStudent u ≥2 teaches v ⊥
• global functionality assertion
≥2 teaches− v ⊥
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
11/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
A DL-Lite(HN )horn TBox
• basic concept inclusion
≥1 hasPublication− v Publication
• role inclusion
hasConfPaper v hasPublication
• number restrictions
PhDStudent v ≥2 hasConfPaper
• horn inclusion
Student u ≥1 teaches v PhDStudent
• local functionality assertion
PhDStudent u ≥2 teaches v ⊥
• global functionality assertion
≥2 teaches− v ⊥
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
11/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
A DL-Lite(HN )horn TBox
• basic concept inclusion
≥1 hasPublication− v Publication
• role inclusion
hasConfPaper v hasPublication
• number restrictions
PhDStudent v ≥2 hasConfPaper
• horn inclusion
Student u ≥1 teaches v PhDStudent
• local functionality assertion
PhDStudent u ≥2 teaches v ⊥
• global functionality assertion
≥2 teaches− v ⊥
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
11/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
A DL-Lite(HN )horn TBox
• basic concept inclusion
≥1 hasPublication− v Publication
• role inclusion
hasConfPaper v hasPublication
• number restrictions
PhDStudent v ≥2 hasConfPaper
• horn inclusion
Student u ≥1 teaches v PhDStudent
• local functionality assertion
PhDStudent u ≥2 teaches v ⊥
• global functionality assertion
≥2 teaches− v ⊥
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
11/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
A DL-Lite(HN )horn TBox
• basic concept inclusion
≥1 hasPublication− v Publication
• role inclusion
hasConfPaper v hasPublication
• number restrictions
PhDStudent v ≥2 hasConfPaper
• horn inclusion
Student u ≥1 teaches v PhDStudent
• local functionality assertion
PhDStudent u ≥2 teaches v ⊥
• global functionality assertion
≥2 teaches− v ⊥
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
11/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Outline
1 Motivation
2 The DL DL-Lite(HN)horn
3 Knowledge Base Satisfiability
4 Query Answering
5 Conclusions
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
12/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Knowledge Base Satisfiability
Negative inclusions may lead to unsatisfiability:
• T : Student u Professor v ⊥, PhDStudent v Student
A : PhDStudent(john), Professor(john)
• T : Dis(teaches, attends)
A : teaches(john, cl), attends(john, cl)
• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)
⇒ We need to calculate closure of NIs w.r.t. PIs
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
13/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Knowledge Base Satisfiability
Negative inclusions may lead to unsatisfiability:
• T : Student u Professor v ⊥, PhDStudent v Student
A : PhDStudent(john), Professor(john)
• T : Dis(teaches, attends)
A : teaches(john, cl), attends(john, cl)
• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)
⇒ We need to calculate closure of NIs w.r.t. PIs
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
13/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Knowledge Base Satisfiability
Negative inclusions may lead to unsatisfiability:
• T : Student u Professor v ⊥, PhDStudent v Student
A : PhDStudent(john), Professor(john)
• T : Dis(teaches, attends)
A : teaches(john, cl), attends(john, cl)
• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)
⇒ We need to calculate closure of NIs w.r.t. PIs
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
13/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Knowledge Base Satisfiability
Negative inclusions may lead to unsatisfiability:
• T : Student u Professor v ⊥, PhDStudent v Student
A : PhDStudent(john), Professor(john)
• T : Dis(teaches, attends)
A : teaches(john, cl), attends(john, cl)
• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)
⇒ We need to calculate closure of NIs w.r.t. PIs
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
13/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Knowledge Base Satisfiability
Negative inclusions may lead to unsatisfiability:
• T : Student u Professor v ⊥, PhDStudent v Student
A : PhDStudent(john), Professor(john)
• T : Dis(teaches, attends)
A : teaches(john, cl), attends(john, cl)
• T : PhDStudent u ≥2 teaches v ⊥A : PhDStudent(john), teaches(john, cl), teaches(john, db)
⇒ We need to calculate closure of NIs w.r.t. PIs
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
13/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Knowledge Base Satisfiability Algorithm
We reduce the problem to FOL query evaluation.
Algorithm for checking KB satisfiability
1 Calculate the closure of NIs.
2 Translate the closure into a UCQ qunsat asking for violation of some NI.
3 Evaluate qunsat over the ABox (viewed as a DB).
I if eval(qunsat ,DBA) = ∅, then the KB is satisfiable;I otherwise the KB is unsatisfiable.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
14/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Knowledge Base Satisfiability Algorithm
We reduce the problem to FOL query evaluation.
Algorithm for checking KB satisfiability
1 Calculate the closure of NIs.
2 Translate the closure into a UCQ qunsat asking for violation of some NI.
3 Evaluate qunsat over the ABox (viewed as a DB).
I if eval(qunsat ,DBA) = ∅, then the KB is satisfiable;I otherwise the KB is unsatisfiable.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
14/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs
• every NI is in cln(T ).
• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent
}⇒
add to cln(T ) : Professor u Student u ≥1 teaches v ⊥
• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches
}⇒
add to cln(T ): PhDStudent u FullProfessor v ⊥
• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends
}⇒
add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
15/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs
• every NI is in cln(T ).
• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent
}⇒
add to cln(T ) : Professor u Student u ≥1 teaches v ⊥
• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches
}⇒
add to cln(T ): PhDStudent u FullProfessor v ⊥
• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends
}⇒
add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
15/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs
• every NI is in cln(T ).
• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent
}⇒
add to cln(T ) : Professor u Student u ≥1 teaches v ⊥
• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches
}⇒
add to cln(T ): PhDStudent u FullProfessor v ⊥
• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends
}⇒
add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
15/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs
• every NI is in cln(T ).
• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent
}⇒
add to cln(T ) : Professor u Student u ≥1 teaches v ⊥
• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches
}⇒
add to cln(T ): PhDStudent u FullProfessor v ⊥
• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends
}⇒
add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
15/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs
• every NI is in cln(T ).
• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent
}⇒
add to cln(T ) : Professor u Student u ≥1 teaches v ⊥
• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches
}⇒
add to cln(T ): PhDStudent u FullProfessor v ⊥
• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends
}⇒
add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
15/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs
• every NI is in cln(T ).
• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent
}⇒
add to cln(T ) : Professor u Student u ≥1 teaches v ⊥
• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches
}⇒
add to cln(T ): PhDStudent u FullProfessor v ⊥
• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends
}⇒
add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
15/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs
• every NI is in cln(T ).
• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent
}⇒
add to cln(T ) : Professor u Student u ≥1 teaches v ⊥
• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches
}⇒
add to cln(T ): PhDStudent u FullProfessor v ⊥
• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends
}⇒
add to cln(T ): Professor u ≥1 registeredTo v ⊥
• · · ·
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
15/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Closure of Negative InclusionsClosure of NIs cln(T ) w.r.t. PIs
• every NI is in cln(T ).
• cln(T ) : Professor u PhDStudent v ⊥T : Student u ≥1 teaches v PhDStudent
}⇒
add to cln(T ) : Professor u Student u ≥1 teaches v ⊥
• cln(T ) : PhDStudent u ≥2 teaches v ⊥T : FullProfessor v ≥3 teaches
}⇒
add to cln(T ): PhDStudent u FullProfessor v ⊥
• cln(T ) : Professor u ≥1 attends v ⊥T : registeredTo v attends
}⇒
add to cln(T ): Professor u ≥1 registeredTo v ⊥• · · ·
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
15/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Translation to FOL Queries
• Professor u Student v ⊥ ⇒∃x .Professor(x) ∧ Student(x).
• ≥2 teaches− v ⊥ ⇒∃x1, x2, y .teaches(x1, y) ∧ teaches(x2, y) ∧ x1 6= x2.
• Dis(attends, teaches) ⇒∃x , y .attends(x , y) ∧ teaches(x , y).
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
16/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Translation to FOL Queries
• Professor u Student v ⊥ ⇒∃x .Professor(x) ∧ Student(x).
• ≥2 teaches− v ⊥ ⇒∃x1, x2, y .teaches(x1, y) ∧ teaches(x2, y) ∧ x1 6= x2.
• Dis(attends, teaches) ⇒∃x , y .attends(x , y) ∧ teaches(x , y).
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
16/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Translation to FOL Queries
• Professor u Student v ⊥ ⇒∃x .Professor(x) ∧ Student(x).
• ≥2 teaches− v ⊥ ⇒∃x1, x2, y .teaches(x1, y) ∧ teaches(x2, y) ∧ x1 6= x2.
• Dis(attends, teaches) ⇒∃x , y .attends(x , y) ∧ teaches(x , y).
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
16/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
KB Satisfiability: Complexity of the Algorithm
• Optimal data complexity: in AC0 (follows from FOL rewritability)
• Combined complexity: exponentialI Worst case: the size of cln(T ) is exponential in the size of the TBoxT = { A′1 v A1, . . . , A′n v An, A1 u · · · u An v ⊥ }.
Notice, that the problem is PTime [ACKZ09].
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
17/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
KB Satisfiability: Complexity of the Algorithm
• Optimal data complexity: in AC0 (follows from FOL rewritability)
• Combined complexity: exponentialI Worst case: the size of cln(T ) is exponential in the size of the TBoxT = { A′1 v A1, . . . , A′n v An, A1 u · · · u An v ⊥ }.
Notice, that the problem is PTime [ACKZ09].
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
17/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
KB Satisfiability: Complexity of the Algorithm
• Optimal data complexity: in AC0 (follows from FOL rewritability)
• Combined complexity: exponentialI Worst case: the size of cln(T ) is exponential in the size of the TBoxT = { A′1 v A1, . . . , A′n v An, A1 u · · · u An v ⊥ }.
Notice, that the problem is PTime [ACKZ09].
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
17/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Outline
1 Motivation
2 The DL DL-Lite(HN)horn
3 Knowledge Base Satisfiability
4 Query Answering
5 Conclusions
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
18/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Example
q(x)← hasPublication(x , y) ∧ Publication(y)
TBox T = {≥1 hasPublication− v Publication
hasConfPaper v hasPublication
PhDStudent v ≥2 hasConfPaper
Student u ≥1 teaches v PhDStudent
}
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
19/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Example
q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication
hasConfPaper v hasPublication
PhDStudent v ≥2 hasConfPaper
Student u ≥1 teaches v PhDStudent
⇓ ≥1 hasPublication− v Publication
q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)
⇓ unify the atoms
q3(x)← hasPublication(x , y)⇓ remove unbound variables
q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication
q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper
q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper
q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent
q7(x)← Student(x) ∧ E1teaches(x)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
19/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Example
q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication
hasConfPaper v hasPublication
PhDStudent v ≥2 hasConfPaper
Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication
q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)
⇓ unify the atoms
q3(x)← hasPublication(x , y)⇓ remove unbound variables
q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication
q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper
q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper
q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent
q7(x)← Student(x) ∧ E1teaches(x)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
19/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Example
q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication
hasConfPaper v hasPublication
PhDStudent v ≥2 hasConfPaper
Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication
q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)
⇓ unify the atoms
q3(x)← hasPublication(x , y)
⇓ remove unbound variables
q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication
q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper
q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper
q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent
q7(x)← Student(x) ∧ E1teaches(x)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
19/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Example
q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication
hasConfPaper v hasPublication
PhDStudent v ≥2 hasConfPaper
Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication
q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)
⇓ unify the atoms
q3(x)← hasPublication(x , y)⇓ remove unbound variables
q3(x)← E1hasPublication(x)
⇓ hasConfPaper v hasPublication
q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper
q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper
q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent
q7(x)← Student(x) ∧ E1teaches(x)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
19/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Example
q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication
hasConfPaper v hasPublication
PhDStudent v ≥2 hasConfPaper
Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication
q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)
⇓ unify the atoms
q3(x)← hasPublication(x , y)⇓ remove unbound variables
q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication
q4(x)← E1hasConfPaper(x)
⇓ ≥2 hasConfPaper v ≥1 hasConfPaper
q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper
q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent
q7(x)← Student(x) ∧ E1teaches(x)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
19/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Example
q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication
hasConfPaper v hasPublication
PhDStudent v ≥2 hasConfPaper
Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication
q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)
⇓ unify the atoms
q3(x)← hasPublication(x , y)⇓ remove unbound variables
q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication
q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper
q5(x)← E2hasConfPaper(x)
⇓ PhDStudent v ≥2 hasConfPaper
q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent
q7(x)← Student(x) ∧ E1teaches(x)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
19/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Example
q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication
hasConfPaper v hasPublication
PhDStudent v ≥2 hasConfPaper
Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication
q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)
⇓ unify the atoms
q3(x)← hasPublication(x , y)⇓ remove unbound variables
q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication
q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper
q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper
q6(x)← PhDStudent(x)
⇓ Student u ≥1 teaches v PhDStudent
q7(x)← Student(x) ∧ E1teaches(x)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
19/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Example
q(x)← hasPublication(x , y) ∧ Publication(y)≥1 hasPublication− v Publication
hasConfPaper v hasPublication
PhDStudent v ≥2 hasConfPaper
Student u ≥1 teaches v PhDStudent⇓ ≥1 hasPublication− v Publication
q2(x)← hasPublication(x , y) ∧ E1hasPublication−(y)
⇓ unify the atoms
q3(x)← hasPublication(x , y)⇓ remove unbound variables
q3(x)← E1hasPublication(x)⇓ hasConfPaper v hasPublication
q4(x)← E1hasConfPaper(x)⇓ ≥2 hasConfPaper v ≥1 hasConfPaper
q5(x)← E2hasConfPaper(x)⇓ PhDStudent v ≥2 hasConfPaper
q6(x)← PhDStudent(x)⇓ Student u ≥1 teaches v PhDStudent
q7(x)← Student(x) ∧ E1teaches(x)
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
19/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering Algorithm
1 Compute the rewriting of the initial query, a UCQ.I Application of PIs to query atoms.
q(x)← hasPublication(x , y) ∧ Publication(y)
⇓ ≥1 hasPublication− v Publicationq′(x)← hasPublication(x , y) ∧ E1hasPublication
−(y)
I Unification of query atoms.
q(x)← hasPublication(x , y) ∧ E1hasPublication−(y)
⇓ unifyq′(x)← hasPublication(x , y)
2 Evaluate the obtained UCQ over the ABox viewed as a DB.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
20/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite
• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′
I ≥k R v ≥k R ′ for each subrole R of R ′
• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R
• Unification for newly introduced predicates
I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)
I EkR(x) unifies with E1R−( )
Notice that E1R−( ) stands for R( , )
• Horn inclusions increase the length of the query
I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
21/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite
• Number restrictions imply new inclusions:
extend the TBoxI ≥k R v ≥k ′ R, where k > k ′
I ≥k R v ≥k R ′ for each subrole R of R ′
• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R
• Unification for newly introduced predicates
I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)
I EkR(x) unifies with E1R−( )
Notice that E1R−( ) stands for R( , )
• Horn inclusions increase the length of the query
I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
21/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite
• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′
I ≥k R v ≥k R ′ for each subrole R of R ′
• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R
• Unification for newly introduced predicates
I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)
I EkR(x) unifies with E1R−( )
Notice that E1R−( ) stands for R( , )
• Horn inclusions increase the length of the query
I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
21/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite
• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′
I ≥k R v ≥k R ′ for each subrole R of R ′
• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R
• Unification for newly introduced predicates
I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)
I EkR(x) unifies with E1R−( )
Notice that E1R−( ) stands for R( , )
• Horn inclusions increase the length of the query
I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
21/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite
• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′
I ≥k R v ≥k R ′ for each subrole R of R ′
• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R
• Unification for newly introduced predicates
I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)
I EkR(x) unifies with E1R−( )
Notice that E1R−( ) stands for R( , )
• Horn inclusions increase the length of the query
I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
21/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite
• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′
I ≥k R v ≥k R ′ for each subrole R of R ′
• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R
• Unification for newly introduced predicates
I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)
I EkR(x) unifies with E1R−( )
Notice that E1R−( ) stands for R( , )
• Horn inclusions increase the length of the query
I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
21/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite
• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′
I ≥k R v ≥k R ′ for each subrole R of R ′
• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R
• Unification for newly introduced predicates
I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)
I EkR(x) unifies with E1R−( )
Notice that E1R−( ) stands for R( , )
• Horn inclusions increase the length of the query
I remove duplicated atoms
I remove Ek′R(z), if EkR(x) occurs in the query and k > k ′
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
21/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
What Is New?Differences w.r.t. the algorithm for simpler variants of DL-Lite
• Number restrictions imply new inclusions: extend the TBoxI ≥k R v ≥k ′ R, where k > k ′
I ≥k R v ≥k R ′ for each subrole R of R ′
• Introduce new predicates EkR(x) to handle inequalitiesimplied by number restrictions ≥k R
• Unification for newly introduced predicates
I P(x,y) unifies with P(z,w), E1P(z), or E1P−(w)
I EkR(x) unifies with E1R−( )
Notice that E1R−( ) stands for R( , )
• Horn inclusions increase the length of the query
I remove duplicated atomsI remove Ek′R(z), if EkR(x) occurs in the query and k > k ′
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
21/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Complexity of the Algorithm
• Optimal data complexity: in AC0
• Combined complexity: in NP
Note that the size of the rewriting is exponentialalready w.r.t. the size of the TBox.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
22/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Query Answering: Complexity of the Algorithm
• Optimal data complexity: in AC0
• Combined complexity: in NP
Note that the size of the rewriting is exponentialalready w.r.t. the size of the TBox.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
22/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Outline
1 Motivation
2 The DL DL-Lite(HN)horn
3 Knowledge Base Satisfiability
4 Query Answering
5 Conclusions
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
23/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Conclusion
• We reduced knowledge satisfiability and query answering in
DL-Lite(HN )horn to FOL evaluation.
I Practically implementable algorithms.I We can rely on relational database technology for managing the data
and query evaluation.
• The computational complexity of the algorithms is optimal w.r.t.data complexity:
I in AC0.
• Future work:I Implement the developed algorithms.I Study optimization techniques for the algorithm.I Extend the practical algorithm to positive existential queries.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
24/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Conclusion
• We reduced knowledge satisfiability and query answering in
DL-Lite(HN )horn to FOL evaluation.
I Practically implementable algorithms.I We can rely on relational database technology for managing the data
and query evaluation.
• The computational complexity of the algorithms is optimal w.r.t.data complexity:
I in AC0.
• Future work:I Implement the developed algorithms.I Study optimization techniques for the algorithm.I Extend the practical algorithm to positive existential queries.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
24/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Conclusion
• We reduced knowledge satisfiability and query answering in
DL-Lite(HN )horn to FOL evaluation.
I Practically implementable algorithms.I We can rely on relational database technology for managing the data
and query evaluation.
• The computational complexity of the algorithms is optimal w.r.t.data complexity:
I in AC0.
• Future work:I Implement the developed algorithms.I Study optimization techniques for the algorithm.I Extend the practical algorithm to positive existential queries.
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
24/25
Motivation The DL DL-Lite(HN)horn
Knowledge Base Satisfiability Query Answering Conclusions
Thank youfor your attention!
Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )horn
25/25