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Outline 1 Motivation 2 The DL DL - Lite ( HN ) horn 3 Knowledge Base - - PowerPoint PPT Presentation

Sep 14, 2022 272 likes •1.03k views

The DL DL - Lite ( HN ) Motivation Knowledge Base Satisfiability Query Answering Conclusions horn Query Rewriting in DL-Lite ( HN ) horn Elena Botoeva, Alessandro Artale, and Diego Calvanese KRDB Research Centre Free University of

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SLIDE 1

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Rewriting in DL-Lite(HN)

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Elena Botoeva, Alessandro Artale, and Diego Calvanese

KRDB Research Centre Free University of Bozen-Bolzano I-39100 Bolzano, Italy

Description Logics Workshop, 2010

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SLIDE 2

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation 2 The DL DL-Lite(HN)

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3 Knowledge Base Satisfiability 4 Query Answering 5 Conclusions

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SLIDE 3

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Motivation: Ontology-Based Data Access

query

Application

response

Application 1 Application 2

query response

Data source 1 Data source 2 Data source 3 Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 4

Motivation The DL DL-Lite(HN)

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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 information

sources

  • Data sources can be queried through ontologies

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SLIDE 5

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering by Rewriting

  • We want to compute certain answers to a query

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SLIDE 6

Motivation The DL DL-Lite(HN)

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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 ontology 2 Evaluate the rewritten query over the database

Reasoning Rewritten Query Query

Result

Reasoning

Data Source

Logical Schema Schema / Ontology

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SLIDE 7

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering by Rewriting: Example

Ontology: O = {PhDStudent ⊑ Student} Database: DBA = {PhDStudent(john)} Query: q(x) ← Student(x)

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SLIDE 8

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering by Rewriting: Example

Ontology: O = {PhDStudent ⊑ 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)

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SLIDE 9

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

FOL Rewritable Logics

  • Such a rewriting approach can be applied only to FOL rewritable

logics.

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SLIDE 10

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

FOL Rewritable Logics

  • Such a rewriting approach can be applied only to FOL rewritable

logics.

  • DL-Lite is a family of logics that has been shown to enjoy FOL

rewritability:

◮ DL-LiteR, DL-LiteF, DL-LiteA Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 11

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

FOL Rewritable Logics

  • Such a rewriting approach can be applied only to FOL rewritable

logics.

  • DL-Lite is a family of logics that has been shown to enjoy FOL

rewritability:

◮ DL-LiteR, DL-LiteF, DL-LiteA

  • Extended DL-Lite family: additional constructs have been proposed

◮ DL-Lite(HN )

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is the most interesting logic

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SLIDE 12

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation 2 The DL DL-Lite(HN)

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3 Knowledge Base Satisfiability 4 Query Answering 5 Conclusions

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SLIDE 13

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN)

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In this work we consider the logic DL-Lite(HN)

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:

  • The most expressive tractable variant of DL-Lite [ACKZ09].

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SLIDE 14

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN)

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In this work we consider the logic DL-Lite(HN)

horn

:

  • The most expressive tractable variant of DL-Lite [ACKZ09].
  • Extends DL-Lite with

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SLIDE 15

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN)

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In this work we consider the logic DL-Lite(HN)

horn

:

  • The most expressive tractable variant of DL-Lite [ACKZ09].
  • Extends DL-Lite with

◮ role inclusions H

  • hasConfPaper ⊑ hasPublication

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SLIDE 16

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

DL-Lite(HN)

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In this work we consider the logic DL-Lite(HN)

horn

:

  • The most expressive tractable variant of DL-Lite [ACKZ09].
  • Extends DL-Lite with

◮ role inclusions H

  • hasConfPaper ⊑ hasPublication

◮ number restrictions N

  • PhDStudent ⊑ ≥2 hasConfPaper
  • ≥2 teaches− ⊑ ⊥

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SLIDE 17

Motivation The DL DL-Lite(HN)

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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

◮ role inclusions H

  • hasConfPaper ⊑ hasPublication

◮ number restrictions N

  • PhDStudent ⊑ ≥2 hasConfPaper
  • ≥2 teaches− ⊑ ⊥

◮ horn inclusions horn

  • Student ⊓ ≥1 teaches ⊑ PhDStudent

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SLIDE 18

Motivation The DL DL-Lite(HN)

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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 for

the simpler DL-Lite logics?

  • What is the complexity of such algorithms?

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SLIDE 19

Motivation The DL DL-Lite(HN)

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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 ⊓ · · · ⊓ Bn ⊑ B R1 ⊑ R2 Dis(R1, R2)

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SLIDE 20

Motivation The DL DL-Lite(HN)

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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 ⊓ · · · ⊓ Bn ⊑ B R1 ⊑ R2 Dis(R1, R2)

Restriction to ensure FOL rewritability:

if R has a proper sub-role, then ≥k R with k ≥ 2 does not occur in the lhs of concept inclusions.

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SLIDE 21

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN)

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TBox

  • basic concept inclusion

≥1 hasPublication− ⊑ Publication

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SLIDE 22

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN)

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TBox

  • basic concept inclusion

≥1 hasPublication− ⊑ Publication

  • role inclusion

hasConfPaper ⊑ hasPublication

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SLIDE 23

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN)

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TBox

  • basic concept inclusion

≥1 hasPublication− ⊑ Publication

  • role inclusion

hasConfPaper ⊑ hasPublication

  • number restrictions

PhDStudent ⊑ ≥2 hasConfPaper

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SLIDE 24

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN)

horn

TBox

  • basic concept inclusion

≥1 hasPublication− ⊑ Publication

  • role inclusion

hasConfPaper ⊑ hasPublication

  • number restrictions

PhDStudent ⊑ ≥2 hasConfPaper

  • horn inclusion

Student ⊓ ≥1 teaches ⊑ PhDStudent

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SLIDE 25

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN)

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TBox

  • basic concept inclusion

≥1 hasPublication− ⊑ Publication

  • role inclusion

hasConfPaper ⊑ hasPublication

  • number restrictions

PhDStudent ⊑ ≥2 hasConfPaper

  • horn inclusion

Student ⊓ ≥1 teaches ⊑ PhDStudent

  • local functionality assertion

PhDStudent ⊓ ≥2 teaches ⊑ ⊥

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SLIDE 26

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

A DL-Lite(HN)

horn

TBox

  • basic concept inclusion

≥1 hasPublication− ⊑ Publication

  • role inclusion

hasConfPaper ⊑ hasPublication

  • number restrictions

PhDStudent ⊑ ≥2 hasConfPaper

  • horn inclusion

Student ⊓ ≥1 teaches ⊑ PhDStudent

  • local functionality assertion

PhDStudent ⊓ ≥2 teaches ⊑ ⊥

  • global functionality assertion

≥2 teaches− ⊑ ⊥

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SLIDE 27

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation 2 The DL DL-Lite(HN)

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3 Knowledge Base Satisfiability 4 Query Answering 5 Conclusions

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SLIDE 28

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

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SLIDE 29

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

  • T : Student ⊓ Professor ⊑ ⊥, PhDStudent ⊑ Student

A : PhDStudent(john), Professor(john)

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SLIDE 30

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

  • T : Student ⊓ Professor ⊑ ⊥, PhDStudent ⊑ Student

A : PhDStudent(john), Professor(john)

  • T : Dis(teaches, attends)

A : teaches(john, cl), attends(john, cl)

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SLIDE 31

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

  • T : Student ⊓ Professor ⊑ ⊥, PhDStudent ⊑ Student

A : PhDStudent(john), Professor(john)

  • T : Dis(teaches, attends)

A : teaches(john, cl), attends(john, cl)

  • T : PhDStudent ⊓ ≥2 teaches ⊑ ⊥

A : PhDStudent(john), teaches(john, cl), teaches(john, db)

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SLIDE 32

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability

Negative inclusions may lead to unsatisfiability:

  • T : Student ⊓ Professor ⊑ ⊥, PhDStudent ⊑ Student

A : PhDStudent(john), Professor(john)

  • T : Dis(teaches, attends)

A : teaches(john, cl), attends(john, cl)

  • T : PhDStudent ⊓ ≥2 teaches ⊑ ⊥

A : PhDStudent(john), teaches(john, cl), teaches(john, db) ⇒ We need to calculate closure of NIs w.r.t. PIs

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SLIDE 33

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Knowledge Base Satisfiability Algorithm

We reduce the problem to FOL query evaluation.

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SLIDE 34

Motivation The DL DL-Lite(HN)

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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).

◮ if eval(qunsat, DBA) = ∅, then the KB is satisfiable; ◮ otherwise the KB is unsatisfiable. Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 35

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative Inclusions

Closure of NIs cln(T ) w.r.t. PIs

  • every NI is in cln(T ).

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SLIDE 36

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative Inclusions

Closure of NIs cln(T ) w.r.t. PIs

  • every NI is in cln(T ).
  • cln(T ) :

Professor ⊓ PhDStudent ⊑ ⊥

T :

Student ⊓ ≥1 teaches ⊑ PhDStudent

  • ⇒

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SLIDE 37

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative Inclusions

Closure of NIs cln(T ) w.r.t. PIs

  • every NI is in cln(T ).
  • cln(T ) :

Professor ⊓ PhDStudent ⊑ ⊥

T :

Student ⊓ ≥1 teaches ⊑ PhDStudent

  • ⇒

add to cln(T ) : Professor ⊓ Student ⊓ ≥1 teaches ⊑ ⊥

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SLIDE 38

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative Inclusions

Closure of NIs cln(T ) w.r.t. PIs

  • every NI is in cln(T ).
  • cln(T ) :

Professor ⊓ PhDStudent ⊑ ⊥

T :

Student ⊓ ≥1 teaches ⊑ PhDStudent

  • ⇒

add to cln(T ) : Professor ⊓ Student ⊓ ≥1 teaches ⊑ ⊥

  • cln(T ) :

PhDStudent ⊓ ≥2 teaches ⊑ ⊥

T :

FullProfessor ⊑ ≥3 teaches

  • ⇒

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SLIDE 39

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative Inclusions

Closure of NIs cln(T ) w.r.t. PIs

  • every NI is in cln(T ).
  • cln(T ) :

Professor ⊓ PhDStudent ⊑ ⊥

T :

Student ⊓ ≥1 teaches ⊑ PhDStudent

  • ⇒

add to cln(T ) : Professor ⊓ Student ⊓ ≥1 teaches ⊑ ⊥

  • cln(T ) :

PhDStudent ⊓ ≥2 teaches ⊑ ⊥

T :

FullProfessor ⊑ ≥3 teaches

  • ⇒

add to cln(T ): PhDStudent ⊓ FullProfessor ⊑ ⊥

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SLIDE 40

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative Inclusions

Closure of NIs cln(T ) w.r.t. PIs

  • every NI is in cln(T ).
  • cln(T ) :

Professor ⊓ PhDStudent ⊑ ⊥

T :

Student ⊓ ≥1 teaches ⊑ PhDStudent

  • ⇒

add to cln(T ) : Professor ⊓ Student ⊓ ≥1 teaches ⊑ ⊥

  • cln(T ) :

PhDStudent ⊓ ≥2 teaches ⊑ ⊥

T :

FullProfessor ⊑ ≥3 teaches

  • ⇒

add to cln(T ): PhDStudent ⊓ FullProfessor ⊑ ⊥

  • cln(T ) :

Professor ⊓ ≥1 attends ⊑ ⊥

T :

registeredTo ⊑ attends

  • ⇒

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SLIDE 41

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative Inclusions

Closure of NIs cln(T ) w.r.t. PIs

  • every NI is in cln(T ).
  • cln(T ) :

Professor ⊓ PhDStudent ⊑ ⊥

T :

Student ⊓ ≥1 teaches ⊑ PhDStudent

  • ⇒

add to cln(T ) : Professor ⊓ Student ⊓ ≥1 teaches ⊑ ⊥

  • cln(T ) :

PhDStudent ⊓ ≥2 teaches ⊑ ⊥

T :

FullProfessor ⊑ ≥3 teaches

  • ⇒

add to cln(T ): PhDStudent ⊓ FullProfessor ⊑ ⊥

  • cln(T ) :

Professor ⊓ ≥1 attends ⊑ ⊥

T :

registeredTo ⊑ attends

  • ⇒

add to cln(T ): Professor ⊓ ≥1 registeredTo ⊑ ⊥

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 42

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Closure of Negative Inclusions

Closure of NIs cln(T ) w.r.t. PIs

  • every NI is in cln(T ).
  • cln(T ) :

Professor ⊓ PhDStudent ⊑ ⊥

T :

Student ⊓ ≥1 teaches ⊑ PhDStudent

  • ⇒

add to cln(T ) : Professor ⊓ Student ⊓ ≥1 teaches ⊑ ⊥

  • cln(T ) :

PhDStudent ⊓ ≥2 teaches ⊑ ⊥

T :

FullProfessor ⊑ ≥3 teaches

  • ⇒

add to cln(T ): PhDStudent ⊓ FullProfessor ⊑ ⊥

  • cln(T ) :

Professor ⊓ ≥1 attends ⊑ ⊥

T :

registeredTo ⊑ attends

  • ⇒

add to cln(T ): Professor ⊓ ≥1 registeredTo ⊑ ⊥

  • · · ·

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SLIDE 43

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Translation to FOL Queries

  • Professor ⊓ Student ⊑ ⊥ ⇒

∃x.Professor(x) ∧ Student(x).

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SLIDE 44

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Translation to FOL Queries

  • Professor ⊓ Student ⊑ ⊥ ⇒

∃x.Professor(x) ∧ Student(x).

  • ≥2 teaches− ⊑ ⊥ ⇒

∃x1, x2, y.teaches(x1, y) ∧ teaches(x2, y) ∧ x1 = x2.

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SLIDE 45

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Translation to FOL Queries

  • Professor ⊓ Student ⊑ ⊥ ⇒

∃x.Professor(x) ∧ Student(x).

  • ≥2 teaches− ⊑ ⊥ ⇒

∃x1, x2, y.teaches(x1, y) ∧ teaches(x2, y) ∧ x1 = x2.

  • Dis(attends, teaches) ⇒

∃x, y.attends(x, y) ∧ teaches(x, y).

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SLIDE 46

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

KB Satisfiability: Complexity of the Algorithm

  • Optimal data complexity: in AC0 (follows from FOL rewritability)

Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 47

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

KB Satisfiability: Complexity of the Algorithm

  • Optimal data complexity: in AC0 (follows from FOL rewritability)
  • Combined complexity: exponential

◮ Worst case: the size of cln(T ) is exponential in the size of the TBox

T = { A′

1 ⊑ A1, . . . , A′ n ⊑ An, A1 ⊓ · · · ⊓ An ⊑ ⊥ }. Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 48

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

KB Satisfiability: Complexity of the Algorithm

  • Optimal data complexity: in AC0 (follows from FOL rewritability)
  • Combined complexity: exponential

◮ Worst case: the size of cln(T ) is exponential in the size of the TBox

T = { A′

1 ⊑ A1, . . . , A′ n ⊑ An, A1 ⊓ · · · ⊓ An ⊑ ⊥ }.

Notice, that the problem is PTime [ACKZ09].

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SLIDE 49

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation 2 The DL DL-Lite(HN)

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3 Knowledge Base Satisfiability 4 Query Answering 5 Conclusions

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SLIDE 50

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x) ← hasPublication(x, y) ∧ Publication(y) TBox T = {

≥1 hasPublication− ⊑ Publication hasConfPaper ⊑ hasPublication PhDStudent ⊑ ≥2 hasConfPaper Student ⊓ ≥1 teaches ⊑ PhDStudent

}

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SLIDE 51

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x) ← hasPublication(x, y) ∧ Publication(y)

≥1 hasPublication− ⊑ Publication hasConfPaper ⊑ hasPublication PhDStudent ⊑ ≥2 hasConfPaper Student ⊓ ≥1 teaches ⊑ PhDStudent Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 52

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x) ← hasPublication(x, y) ∧ Publication(y)

≥1 hasPublication− ⊑ Publication hasConfPaper ⊑ hasPublication PhDStudent ⊑ ≥2 hasConfPaper Student ⊓ ≥1 teaches ⊑ PhDStudent

⇓ ≥1 hasPublication− ⊑ Publication q2(x) ← hasPublication(x, y) ∧ E1hasPublication−(y)

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SLIDE 53

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x) ← hasPublication(x, y) ∧ Publication(y)

≥1 hasPublication− ⊑ Publication hasConfPaper ⊑ hasPublication PhDStudent ⊑ ≥2 hasConfPaper Student ⊓ ≥1 teaches ⊑ PhDStudent

⇓ ≥1 hasPublication− ⊑ Publication q2(x) ← hasPublication(x, y) ∧ E1hasPublication−(y) ⇓ unify the atoms q3(x) ← hasPublication(x, y)

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SLIDE 54

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x) ← hasPublication(x, y) ∧ Publication(y)

≥1 hasPublication− ⊑ Publication hasConfPaper ⊑ hasPublication PhDStudent ⊑ ≥2 hasConfPaper Student ⊓ ≥1 teaches ⊑ PhDStudent

⇓ ≥1 hasPublication− ⊑ Publication q2(x) ← hasPublication(x, y) ∧ E1hasPublication−(y) ⇓ unify the atoms q3(x) ← hasPublication(x, y) ⇓ remove unbound variables q3(x) ← E1hasPublication(x)

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SLIDE 55

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x) ← hasPublication(x, y) ∧ Publication(y)

≥1 hasPublication− ⊑ Publication hasConfPaper ⊑ hasPublication PhDStudent ⊑ ≥2 hasConfPaper Student ⊓ ≥1 teaches ⊑ PhDStudent

⇓ ≥1 hasPublication− ⊑ Publication q2(x) ← hasPublication(x, y) ∧ E1hasPublication−(y) ⇓ unify the atoms q3(x) ← hasPublication(x, y) ⇓ remove unbound variables q3(x) ← E1hasPublication(x) ⇓ hasConfPaper ⊑ hasPublication q4(x) ← E1hasConfPaper(x)

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SLIDE 56

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x) ← hasPublication(x, y) ∧ Publication(y)

≥1 hasPublication− ⊑ Publication hasConfPaper ⊑ hasPublication PhDStudent ⊑ ≥2 hasConfPaper Student ⊓ ≥1 teaches ⊑ PhDStudent

⇓ ≥1 hasPublication− ⊑ Publication q2(x) ← hasPublication(x, y) ∧ E1hasPublication−(y) ⇓ unify the atoms q3(x) ← hasPublication(x, y) ⇓ remove unbound variables q3(x) ← E1hasPublication(x) ⇓ hasConfPaper ⊑ hasPublication q4(x) ← E1hasConfPaper(x) ⇓ ≥2 hasConfPaper ⊑ ≥1 hasConfPaper q5(x) ← E2hasConfPaper(x)

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SLIDE 57

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x) ← hasPublication(x, y) ∧ Publication(y)

≥1 hasPublication− ⊑ Publication hasConfPaper ⊑ hasPublication PhDStudent ⊑ ≥2 hasConfPaper Student ⊓ ≥1 teaches ⊑ PhDStudent

⇓ ≥1 hasPublication− ⊑ Publication q2(x) ← hasPublication(x, y) ∧ E1hasPublication−(y) ⇓ unify the atoms q3(x) ← hasPublication(x, y) ⇓ remove unbound variables q3(x) ← E1hasPublication(x) ⇓ hasConfPaper ⊑ hasPublication q4(x) ← E1hasConfPaper(x) ⇓ ≥2 hasConfPaper ⊑ ≥1 hasConfPaper q5(x) ← E2hasConfPaper(x) ⇓ PhDStudent ⊑ ≥2 hasConfPaper q6(x) ← PhDStudent(x)

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SLIDE 58

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Example

q(x) ← hasPublication(x, y) ∧ Publication(y)

≥1 hasPublication− ⊑ Publication hasConfPaper ⊑ hasPublication PhDStudent ⊑ ≥2 hasConfPaper Student ⊓ ≥1 teaches ⊑ PhDStudent

⇓ ≥1 hasPublication− ⊑ Publication q2(x) ← hasPublication(x, y) ∧ E1hasPublication−(y) ⇓ unify the atoms q3(x) ← hasPublication(x, y) ⇓ remove unbound variables q3(x) ← E1hasPublication(x) ⇓ hasConfPaper ⊑ hasPublication q4(x) ← E1hasConfPaper(x) ⇓ ≥2 hasConfPaper ⊑ ≥1 hasConfPaper q5(x) ← E2hasConfPaper(x) ⇓ PhDStudent ⊑ ≥2 hasConfPaper q6(x) ← PhDStudent(x) ⇓ Student ⊓ ≥1 teaches ⊑ PhDStudent q7(x) ← Student(x) ∧ E1teaches(x)

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SLIDE 59

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering Algorithm

1 Compute the rewriting of the initial query, a UCQ.

◮ Application of PIs to query atoms.

q(x) ← hasPublication(x, y) ∧ Publication(y) ⇓ ≥1 hasPublication− ⊑ Publication q′(x) ← hasPublication(x, y) ∧ E1hasPublication−(y)

◮ Unification of query atoms.

q(x) ← hasPublication(x, y) ∧ E1hasPublication−(y) ⇓ unify q′(x) ← hasPublication(x, y) 2 Evaluate the obtained UCQ over the ABox viewed as a DB.

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SLIDE 60

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

What Is New?

Differences w.r.t. the algorithm for simpler variants of DL-Lite

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SLIDE 61

Motivation The DL DL-Lite(HN)

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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:

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SLIDE 62

Motivation The DL DL-Lite(HN)

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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 TBox

◮ ≥k R ⊑ ≥k′ R, where k > k′ ◮ ≥k R ⊑ ≥k R′ for each subrole R of R′ Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 63

Motivation The DL DL-Lite(HN)

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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 TBox

◮ ≥k R ⊑ ≥k′ R, where k > k′ ◮ ≥k R ⊑ ≥k R′ for each subrole R of R′

  • Introduce new predicates EkR(x) to handle inequalities

implied by number restrictions ≥k R

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SLIDE 64

Motivation The DL DL-Lite(HN)

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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 TBox

◮ ≥k R ⊑ ≥k′ R, where k > k′ ◮ ≥k R ⊑ ≥k R′ for each subrole R of R′

  • Introduce new predicates EkR(x) to handle inequalities

implied by number restrictions ≥k R

  • Unification for newly introduced predicates

◮ P(x,y) unifies with P(z,w), E1P(z), or E1P−(w) ◮ EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

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SLIDE 65

Motivation The DL DL-Lite(HN)

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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 TBox

◮ ≥k R ⊑ ≥k′ R, where k > k′ ◮ ≥k R ⊑ ≥k R′ for each subrole R of R′

  • Introduce new predicates EkR(x) to handle inequalities

implied by number restrictions ≥k R

  • Unification for newly introduced predicates

◮ P(x,y) unifies with P(z,w), E1P(z), or E1P−(w) ◮ EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

  • Horn inclusions increase the length of the query

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SLIDE 66

Motivation The DL DL-Lite(HN)

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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 TBox

◮ ≥k R ⊑ ≥k′ R, where k > k′ ◮ ≥k R ⊑ ≥k R′ for each subrole R of R′

  • Introduce new predicates EkR(x) to handle inequalities

implied by number restrictions ≥k R

  • Unification for newly introduced predicates

◮ P(x,y) unifies with P(z,w), E1P(z), or E1P−(w) ◮ EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

  • Horn inclusions increase the length of the query

◮ remove duplicated atoms Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 67

Motivation The DL DL-Lite(HN)

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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 TBox

◮ ≥k R ⊑ ≥k′ R, where k > k′ ◮ ≥k R ⊑ ≥k R′ for each subrole R of R′

  • Introduce new predicates EkR(x) to handle inequalities

implied by number restrictions ≥k R

  • Unification for newly introduced predicates

◮ P(x,y) unifies with P(z,w), E1P(z), or E1P−(w) ◮ EkR(x) unifies with E1R−( )

Notice that E1R−( ) stands for R( , )

  • Horn inclusions increase the length of the query

◮ remove duplicated atoms ◮ remove Ek′R(z), if EkR(x) occurs in the query and k > k′ Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 68

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Query Answering: Complexity of the Algorithm

  • Optimal data complexity: in AC0

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SLIDE 69

Motivation The DL DL-Lite(HN)

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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 exponential already w.r.t. the size of the TBox.

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SLIDE 70

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Outline

1 Motivation 2 The DL DL-Lite(HN)

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3 Knowledge Base Satisfiability 4 Query Answering 5 Conclusions

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SLIDE 71

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Conclusion

  • We reduced knowledge satisfiability and query answering in

DL-Lite(HN)

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to FOL evaluation.

◮ Practically implementable algorithms. ◮ We can rely on relational database technology for managing the data

and query evaluation.

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SLIDE 72

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Conclusion

  • We reduced knowledge satisfiability and query answering in

DL-Lite(HN)

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to FOL evaluation.

◮ Practically implementable algorithms. ◮ 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:

◮ in AC0. Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 73

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Conclusion

  • We reduced knowledge satisfiability and query answering in

DL-Lite(HN)

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to FOL evaluation.

◮ Practically implementable algorithms. ◮ 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:

◮ in AC0.

  • Future work:

◮ Implement the developed algorithms. ◮ Study optimization techniques for the algorithm. ◮ Extend the practical algorithm to positive existential queries. Botoeva, Artale, Calvanese Query Rewriting in DL-Lite(HN )

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SLIDE 74

Motivation The DL DL-Lite(HN)

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Knowledge Base Satisfiability Query Answering Conclusions

Thank you for your attention!

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