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

Satisfiability and Query Answering in Description Logics with Global and Local Cardinality Constraints Bartosz Bednarczyk Franz Baader, , Sebastian Rudolph TU Dresden & University of Wrocław

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

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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Diety ( Zeus ) , KingOfGods ( Zeus ) 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Diety ( Zeus ) , KingOfGods ( Zeus ) Titan ( Rhea ) , Female ( Rhea ) 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Diety ( Zeus ) , KingOfGods ( Zeus ) Titan ( Rhea ) , Female ( Rhea ) Mortal ( Alcmene ) 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) Titan ( Rhea ) , Female ( Rhea ) Mortal ( Alcmene ) 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) ⊤ ⊑ ∃ hasParent . Male ⊓ ∃ hasParent . Female Titan ( Rhea ) , Female ( Rhea ) Mortal ( Alcmene ) 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) ⊤ ⊑ ∃ hasParent . Male ⊓ ∃ hasParent . Female Titan ( Rhea ) , Female ( Rhea ) KingOfGods ⊑ ( ≥ 13 hasChildren ) . ⊤ Mortal ( Alcmene ) 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) ⊤ ⊑ ∃ hasParent . Male ⊓ ∃ hasParent . Female Titan ( Rhea ) , Female ( Rhea ) KingOfGods ⊑ ( ≥ 13 hasChildren ) . ⊤ Mortal ( Alcmene ) Main problem: Counting 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) ⊤ ⊑ ∃ hasParent . Male ⊓ ∃ hasParent . Female Titan ( Rhea ) , Female ( Rhea ) KingOfGods ⊑ ( ≥ 13 hasChildren ) . ⊤ Mortal ( Alcmene ) Main problem: Counting is very-limited and local. 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) ⊤ ⊑ ∃ hasParent . Male ⊓ ∃ hasParent . Female Titan ( Rhea ) , Female ( Rhea ) KingOfGods ⊑ ( ≥ 13 hasChildren ) . ⊤ Mortal ( Alcmene ) Main problem: Counting is very-limited and local. How to express: 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) ⊤ ⊑ ∃ hasParent . Male ⊓ ∃ hasParent . Female Titan ( Rhea ) , Female ( Rhea ) KingOfGods ⊑ ( ≥ 13 hasChildren ) . ⊤ Mortal ( Alcmene ) Main problem: Counting is very-limited and local. How to express: • Zeus is the only KingOfGods ? 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) ⊤ ⊑ ∃ hasParent . Male ⊓ ∃ hasParent . Female Titan ( Rhea ) , Female ( Rhea ) KingOfGods ⊑ ( ≥ 13 hasChildren ) . ⊤ Mortal ( Alcmene ) Main problem: Counting is very-limited and local. How to express: • Zeus is the only KingOfGods ? there are exactly 12 Titan s? 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) ⊤ ⊑ ∃ hasParent . Male ⊓ ∃ hasParent . Female Titan ( Rhea ) , Female ( Rhea ) KingOfGods ⊑ ( ≥ 13 hasChildren ) . ⊤ Mortal ( Alcmene ) Main problem: Counting is very-limited and local. How to express: • Zeus is the only KingOfGods ? there are exactly 12 Titan s? • No more than 40% of Zeus ’ children are Male ? 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

Running example: Greek mythology ALCQ knowledge base Database (ABox) Knowledge (TBox) hasParent ( Heracles , Zeus ) Mortal ⊑ ¬ Diety Diety ( Zeus ) , KingOfGods ( Zeus ) ⊤ ⊑ ∃ hasParent . Male ⊓ ∃ hasParent . Female Titan ( Rhea ) , Female ( Rhea ) KingOfGods ⊑ ( ≥ 13 hasChildren ) . ⊤ Mortal ( Alcmene ) Main problem: Counting is very-limited and local. How to express: • Zeus is the only KingOfGods ? there are exactly 12 Titan s? • No more than 40% of Zeus ’ children are Male ? • Most of people stored in DB are Zeus ’ children? 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

QFBAPA B. Bednarczyk, F. Baader, S. Rudolph SAT and CQs for DLs with Global&Local Cardinality Constr. 2 / 5

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

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

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

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