Satisfiability and Query Answering in Description Logics with Global - - PowerPoint PPT Presentation

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Satisfiability and Query Answering in Description Logics with Global - - PowerPoint PPT Presentation

Sep 23, 2022 178 likes •918 views

Satisfiability and Query Answering in Description Logics with Global and Local Cardinality Constraints Bartosz Bednarczyk Franz Baader, , Sebastian Rudolph TU Dresden & University of Wrocaw Running example: Greek mythology ALCQ knowledge

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SLIDE 1

Satisfiability and Query Answering in Description Logics

with Global and Local Cardinality Constraints

Bartosz Bednarczyk

Franz Baader, , Sebastian Rudolph

TU Dresden & University of Wrocław

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SLIDE 2

Running example: Greek mythology ALCQ knowledge base

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-3
SLIDE 3

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-4
SLIDE 4

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus)

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

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SLIDE 5

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus)

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-6
SLIDE 6

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea)

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-7
SLIDE 7

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene)

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-8
SLIDE 8

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

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SLIDE 9

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety ⊤ ⊑ ∃hasParent.Male ⊓ ∃hasParent.Female

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

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SLIDE 10

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety ⊤ ⊑ ∃hasParent.Male ⊓ ∃hasParent.Female KingOfGods ⊑ (≥ 13 hasChildren).⊤

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-11
SLIDE 11

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety ⊤ ⊑ ∃hasParent.Male ⊓ ∃hasParent.Female KingOfGods ⊑ (≥ 13 hasChildren).⊤ Main problem: Counting

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-12
SLIDE 12

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety ⊤ ⊑ ∃hasParent.Male ⊓ ∃hasParent.Female KingOfGods ⊑ (≥ 13 hasChildren).⊤ Main problem: Counting is very-limited and local.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-13
SLIDE 13

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety ⊤ ⊑ ∃hasParent.Male ⊓ ∃hasParent.Female KingOfGods ⊑ (≥ 13 hasChildren).⊤ Main problem: Counting is very-limited and local. How to express:

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-14
SLIDE 14

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety ⊤ ⊑ ∃hasParent.Male ⊓ ∃hasParent.Female KingOfGods ⊑ (≥ 13 hasChildren).⊤ Main problem: Counting is very-limited and local. How to express:

  • Zeus is the only KingOfGods?
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-15
SLIDE 15

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety ⊤ ⊑ ∃hasParent.Male ⊓ ∃hasParent.Female KingOfGods ⊑ (≥ 13 hasChildren).⊤ Main problem: Counting is very-limited and local. How to express:

  • Zeus is the only KingOfGods? there are exactly 12 Titans?
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-16
SLIDE 16

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety ⊤ ⊑ ∃hasParent.Male ⊓ ∃hasParent.Female KingOfGods ⊑ (≥ 13 hasChildren).⊤ Main problem: Counting is very-limited and local. How to express:

  • Zeus is the only KingOfGods? there are exactly 12 Titans?
  • No more than 40% of Zeus’ children are Male?
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

slide-17
SLIDE 17

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox)

Icons downloaded from icon-icons.com by c Rena Xiao and c Eucalyp Studio (both under CC BY 4.0). No changes has been made.

hasParent(Heracles, Zeus) Diety(Zeus), KingOfGods(Zeus) Titan(Rhea), Female(Rhea) Mortal(Alcmene) Mortal ⊑ ¬Diety ⊤ ⊑ ∃hasParent.Male ⊓ ∃hasParent.Female KingOfGods ⊑ (≥ 13 hasChildren).⊤ Main problem: Counting is very-limited and local. How to express:

  • Zeus is the only KingOfGods? there are exactly 12 Titans?
  • No more than 40% of Zeus’ children are Male?
  • Most of people stored in DB are Zeus’ children?
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 1 / 5

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SLIDE 18

QFBAPA

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

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SLIDE 19

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

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SLIDE 20

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics counting toolkit

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-21
SLIDE 21

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-22
SLIDE 22

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-23
SLIDE 23

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-24
SLIDE 24

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-25
SLIDE 25

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-26
SLIDE 26

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-27
SLIDE 27

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c)

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-28
SLIDE 28

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-29
SLIDE 29

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T)

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-30
SLIDE 30

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U)

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-31
SLIDE 31

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-32
SLIDE 32

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-33
SLIDE 33

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-34
SLIDE 34

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-35
SLIDE 35

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-36
SLIDE 36

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are N · |S| (set term S, N ∈ Z) or
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-37
SLIDE 37

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are N · |S| (set term S, N ∈ Z) or E + E ′ or
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-38
SLIDE 38

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are N · |S| (set term S, N ∈ Z) or E + E ′ or

N

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-39
SLIDE 39

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are N · |S| (set term S, N ∈ Z) or E + E ′ or

N Example: 2 · |T c ∪ S| + (−3) · |U ∩ S|

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-40
SLIDE 40

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are N · |S| (set term S, N ∈ Z) or E + E ′ or

N Example: 2 · |T c ∪ S| + (−3) · |U ∩ S| Cardinality constraints: E = E ′, E < E ′, N dvd E ′

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-41
SLIDE 41

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are N · |S| (set term S, N ∈ Z) or E + E ′ or

N Example: 2 · |T c ∪ S| + (−3) · |U ∩ S| Cardinality constraints: E = E ′, E < E ′, N dvd E ′ 5 dvd

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-42
SLIDE 42

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are N · |S| (set term S, N ∈ Z) or E + E ′ or

N Example: 2 · |T c ∪ S| + (−3) · |U ∩ S| Cardinality constraints: E = E ′, E < E ′, N dvd E ′ 5 dvd QFBAPA formula =

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-43
SLIDE 43

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are N · |S| (set term S, N ∈ Z) or E + E ′ or

N Example: 2 · |T c ∪ S| + (−3) · |U ∩ S| Cardinality constraints: E = E ′, E < E ′, N dvd E ′ 5 dvd QFBAPA formula = boolean combination of

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-44
SLIDE 44

QFBAPA Quantifier-Free Boolean Algebra with Presburger Arithmetics Step 1. Boolean algebra of sets.

  • Set variables: S, T, . . .
  • Special set constants: ∅, U
  • Boolean operators: ∩, ∪, ·c
  • Set terms
  • apply operations on variables and constants

Example: ∅ ∪ (Sc ∩ (T ∪ R)c) Set constraints: S = T or S ⊆ T (for set terms S, T) ⊆ Rc ∩ (S ∪ U) Step 2. Presburger arithmetics (PA).

  • Integer constants: .., −1, 0, 1, 2, ..
  • Set cardinalities constants: |S|

(S is a set term)

  • PA expressions E, E ′ are N · |S| (set term S, N ∈ Z) or E + E ′ or

N Example: 2 · |T c ∪ S| + (−3) · |U ∩ S| Cardinality constraints: E = E ′, E < E ′, N dvd E ′ 5 dvd QFBAPA formula = boolean combination of set and cardinality constraints

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

slide-45
SLIDE 45

ALC extended with QFBAPA: ALCSCC++

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-46
SLIDE 46

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-47
SLIDE 47

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-48
SLIDE 48

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I

DietyI = {d1}, FemaleI = {d2},

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-49
SLIDE 49

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I

DietyI = {d1}, FemaleI = {d2},

  • 2. Roles: hasParent, hasChildren . . . ⊆ ∆I × ∆I
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-50
SLIDE 50

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I

DietyI = {d1}, FemaleI = {d2},

  • 2. Roles: hasParent, hasChildren . . . ⊆ ∆I × ∆I

hasParentI = {(d1, d2)}

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-51
SLIDE 51

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I

DietyI = {d1}, FemaleI = {d2},

  • 2. Roles: hasParent, hasChildren . . . ⊆ ∆I × ∆I

hasParentI = {(d1, d2)}

  • 3. Boolean combination of concepts: Diety ⊓ Male, Diety ⊔ Mortal, ¬Mortal
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-52
SLIDE 52

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I

DietyI = {d1}, FemaleI = {d2},

  • 2. Roles: hasParent, hasChildren . . . ⊆ ∆I × ∆I

hasParentI = {(d1, d2)}

  • 3. Boolean combination of concepts: Diety ⊓ Male, Diety ⊔ Mortal, ¬Mortal

(Male ⊓ Diety)I = {d1}, (Male ⊔ Female)I = {d1, d2}

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-53
SLIDE 53

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I

DietyI = {d1}, FemaleI = {d2},

  • 2. Roles: hasParent, hasChildren . . . ⊆ ∆I × ∆I

hasParentI = {(d1, d2)}

  • 3. Boolean combination of concepts: Diety ⊓ Male, Diety ⊔ Mortal, ¬Mortal

(Male ⊓ Diety)I = {d1}, (Male ⊔ Female)I = {d1, d2}

  • 4. Existential restriction: ∃hasParent.Female
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-54
SLIDE 54

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I

DietyI = {d1}, FemaleI = {d2},

  • 2. Roles: hasParent, hasChildren . . . ⊆ ∆I × ∆I

hasParentI = {(d1, d2)}

  • 3. Boolean combination of concepts: Diety ⊓ Male, Diety ⊔ Mortal, ¬Mortal

(Male ⊓ Diety)I = {d1}, (Male ⊔ Female)I = {d1, d2}

  • 4. Existential restriction: ∃hasParent.Female

(∃hasParent.Female)I = {d1}

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-55
SLIDE 55

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I

DietyI = {d1}, FemaleI = {d2},

  • 2. Roles: hasParent, hasChildren . . . ⊆ ∆I × ∆I

hasParentI = {(d1, d2)}

  • 3. Boolean combination of concepts: Diety ⊓ Male, Diety ⊔ Mortal, ¬Mortal

(Male ⊓ Diety)I = {d1}, (Male ⊔ Female)I = {d1, d2}

  • 4. Existential restriction: ∃hasParent.Female

(∃hasParent.Female)I = {d1}

  • 5. QFBAPA expressions: sat(Set/Cardinality Constraint)
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-56
SLIDE 56

ALC extended with QFBAPA: ALCSCC++

d1 d2

hasParent Male, Diety Female

I = ∆I = {d1, d2}

  • 1. Atomic concepts: Diety, Male, Female, Mortal . . . ⊆ ∆I

DietyI = {d1}, FemaleI = {d2},

  • 2. Roles: hasParent, hasChildren . . . ⊆ ∆I × ∆I

hasParentI = {(d1, d2)}

  • 3. Boolean combination of concepts: Diety ⊓ Male, Diety ⊔ Mortal, ¬Mortal

(Male ⊓ Diety)I = {d1}, (Male ⊔ Female)I = {d1, d2}

  • 4. Existential restriction: ∃hasParent.Female

(∃hasParent.Female)I = {d1}

  • 5. QFBAPA expressions: sat(Set/Cardinality Constraint)

We use role/concept names in place of set variables.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 3 / 5

slide-57
SLIDE 57

ALC extended with QFBAPA: examples

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-58
SLIDE 58

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-59
SLIDE 59

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-60
SLIDE 60

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • Number restrictions: ZeusI ∈ sat(|hasChildren| ≥ 13)
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-61
SLIDE 61

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • Number restrictions: ZeusI ∈ sat(|hasChildren| ≥ 13)

There is only one KingOfGods and there are exactly 12 Titans.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-62
SLIDE 62

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • Number restrictions: ZeusI ∈ sat(|hasChildren| ≥ 13)

There is only one KingOfGods and there are exactly 12 Titans.

  • Nominals: sat(|KingOfGods| = 1), sat(|Titan| = 12)
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-63
SLIDE 63

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • Number restrictions: ZeusI ∈ sat(|hasChildren| ≥ 13)

There is only one KingOfGods and there are exactly 12 Titans.

  • Nominals: sat(|KingOfGods| = 1), sat(|Titan| = 12)

No more than 40% of Zeus’ children are Male?

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-64
SLIDE 64

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • Number restrictions: ZeusI ∈ sat(|hasChildren| ≥ 13)

There is only one KingOfGods and there are exactly 12 Titans.

  • Nominals: sat(|KingOfGods| = 1), sat(|Titan| = 12)

No more than 40% of Zeus’ children are Male?

  • KingOfGods ⊑ sat(|hasChildren ⊓ Male| ≤ 0.4|hasChildren|)
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-65
SLIDE 65

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • Number restrictions: ZeusI ∈ sat(|hasChildren| ≥ 13)

There is only one KingOfGods and there are exactly 12 Titans.

  • Nominals: sat(|KingOfGods| = 1), sat(|Titan| = 12)

No more than 40% of Zeus’ children are Male?

  • KingOfGods ⊑ sat(|hasChildren ⊓ Male| ≤ 0.4|hasChildren|)

Roles hasParent and hasChildren are disjoint.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-66
SLIDE 66

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • Number restrictions: ZeusI ∈ sat(|hasChildren| ≥ 13)

There is only one KingOfGods and there are exactly 12 Titans.

  • Nominals: sat(|KingOfGods| = 1), sat(|Titan| = 12)

No more than 40% of Zeus’ children are Male?

  • KingOfGods ⊑ sat(|hasChildren ⊓ Male| ≤ 0.4|hasChildren|)

Roles hasParent and hasChildren are disjoint.

  • sat(⊤ ⊆ sat(hasParent ∩ hasChildren ⊆ ∅))
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-67
SLIDE 67

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • Number restrictions: ZeusI ∈ sat(|hasChildren| ≥ 13)

There is only one KingOfGods and there are exactly 12 Titans.

  • Nominals: sat(|KingOfGods| = 1), sat(|Titan| = 12)

No more than 40% of Zeus’ children are Male?

  • KingOfGods ⊑ sat(|hasChildren ⊓ Male| ≤ 0.4|hasChildren|)

Roles hasParent and hasChildren are disjoint.

  • sat(⊤ ⊆ sat(hasParent ∩ hasChildren ⊆ ∅))

A role r is universal, i.e. r I = ∆I × ∆I

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-68
SLIDE 68

ALC extended with QFBAPA: examples Roles are interpreted locally but concepts are interpreted globally. Zeus has at least 13 children

  • Number restrictions: ZeusI ∈ sat(|hasChildren| ≥ 13)

There is only one KingOfGods and there are exactly 12 Titans.

  • Nominals: sat(|KingOfGods| = 1), sat(|Titan| = 12)

No more than 40% of Zeus’ children are Male?

  • KingOfGods ⊑ sat(|hasChildren ⊓ Male| ≤ 0.4|hasChildren|)

Roles hasParent and hasChildren are disjoint.

  • sat(⊤ ⊆ sat(hasParent ∩ hasChildren ⊆ ∅))

A role r is universal, i.e. r I = ∆I × ∆I

  • sat(⊤ ⊆ sat(|r| = |U|))
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 4 / 5

slide-69
SLIDE 69

Our results

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 5 / 5

slide-70
SLIDE 70

Our results Theorem The satisfiability problem for ALCSCC++ is NExpTime-complete.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 5 / 5

slide-71
SLIDE 71

Our results Theorem The satisfiability problem for ALCSCC++ is NExpTime-complete. Theorem Conjunctive query answering is undecidable for ALCSCC++.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 5 / 5

slide-72
SLIDE 72

Our results Theorem The satisfiability problem for ALCSCC++ is NExpTime-complete. Theorem Conjunctive query answering is undecidable for ALCSCC++. Theorem The satisfiability problem for ALCSCC++ with inverses is undecidable.

  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 5 / 5

slide-73
SLIDE 73

Our results Theorem The satisfiability problem for ALCSCC++ is NExpTime-complete. Theorem Conjunctive query answering is undecidable for ALCSCC++. Theorem The satisfiability problem for ALCSCC++ with inverses is undecidable. Theorem If global constraints are properly restrained we obtain ExpTime-completeness

  • f both satisfiability problem and CQ entailment.
  • B. Bednarczyk, F. Baader, S. Rudolph

SAT and CQs for DLs with Global&Local Cardinality Constr. 5 / 5