1. Tableau Algorithm

2. Presentation Outline Description Logics Reasoning Tasks
Structural Subsumption
Tableau Algorithm
Examples
Q & A

3. DL Basics Concepts (unary predicates/formulae with one free variable)
E.g., Person, Doctor, HappyParent
Roles (binary predicates/formulae with two free variables)
E.g., hasChild, loves, hasBrother, hasDaughter
Individuals (constants)
E.g., John, Mary, Italy
Operators (for forming concepts and roles) restricted so that:
Satisfiability/subsumption is decidable and, if possible, of low complexity

4. DL Semantics Interpretation function I Interpretation domain I
Individuals iI 2 I
John
Mary
Concepts CI µ I
Lawyer
Doctor
Vehicle
Roles rI µ I £ I
hasChild
owns
(Lawyer u Doctor)

5. DL Knowledge Base
A TBox is a set of “schema” axioms (sentences), e.g.:
{Doctor v Person,
HappyParent ´ Person u 8hasChild.(Doctor t 9hasChild.Doctor)}
An ABox is a set of “data” axioms (ground facts), e.g.:
{John:HappyParent,
John hasChild Mary}
A Knowledge Base (KB) is just a TBox plus an ABox

6. Example of TBox
Woman ≡ Person Female Man ≡ Person ¬Woman Mother ≡ Woman hasChild.Person Father ≡ Man hasChild.Person Parent ≡ Father Mother Grandmother ≡ Mother hasChild.Parent MotherWithManyChildren ≡ Mother  3 hasChild MotherWithoutDaughter ≡ Mother hasChild.¬Woman Wife ≡ Woman hasHusband.Man

7. Example of ABox
MotherWithoutDaughter(mary) Father(peter) hasChild(mary, peter) hasChild(peter, harry) hasChild(mary, paul)

8. Reasoning Tasks
Whether a TBox description is satisfiable (i.e., non-contradictory)
Whether one description subsumes another one in a TBox - organize the concepts of a terminology into a hierarchy according to their generality
Find out whether the set of assertions in a ABox is consistent (has a model)
Whether the assertions in the ABox entail that a particular individual is an instance of a given concept description
A concept description can also be conceived as a query - retrieve the individuals that satisfy the query.

9. Types of Reasoning
The simplest form of reasoning involves computing the subsumption relation between two concept expressions, i.e., verifying whether one expression always denotes a subset of the objects denoted by another expression.
Parent is a specialization of Person, i.e., Person subsumes Parent A B

10. Types of Reasoning
A more complex reasoning task consists in checking whether a certain assertion is logically implied by a knowledge base.
For example, Bill is an instance of Parent

11. Structural Subsumption
Normalize descriptions
Compare syntactical structure of normal forms
Normal form of C: A1 ⊓ …⊓ Am ⊓ ∀R1.C1⊓…⊓∀Rn.Cn
Normal form of D: B1 ⊓ …⊓ Bk ⊓ ∀S1.D1⊓…⊓∀Sl.Dl
C ⊑ D iff:
For all i, 1 <= i <= k, there exists j, 1<=j<=m, such that Bi=Aj
For all i, 1 <= i <= l, there exists j, 1<=j<=n, such that Si=Rj and Ci ⊑ Dj

12. Structural Subsumption
C ≡ Person Person hasChild.(Lawyer Doctor Rockstar)
D ≡ Person hasChild.Doctor hasChild.Lawyer
C ⊑ D? Yes
C: Person Person hasChild.Lawyer hasChild.Doctor hasChild.Rockstar
D: Person hasChild.Doctor hasChild.Lawyer

13. Structural Subsumption
What if we introduce disjunction
C: A ⊔ (B ⊓ E)
D: A ⊔ E
C ⊑ D?
Cannot be handled with structural subsumption
Solution: Tableau

14. Tableau Algorithm
Instead of directly testing subsumption of concept descriptions, these algorithms use negation to reduce subsumption to (un)satisfiability of concept descriptions.
Steps
Check unsatisfiability of the concept (C ⊑ D -> C ⊓ ¬D)
Check whether you can construct an instance b of this concept
Try to build a tree like model of the input concept
Concept in Negation Normal Form
Decomposition using Tableau rules
Stop when clash occurs or no more rules are applicable
If each branch in tableau contains a clash, the concept is inconsistent

15. Negation Normal Form
Rewrite Description such that only atomic roles and concepts are negated
Rewrite Rules
¬(C ⊓ D) -> ¬C ⊔ ¬D
¬(C ⊔ D) -> ¬C ⊓ ¬D
∀R.C -> ∃R.¬C
∃R.C -> ∀R. ¬C
Example
¬∃R.A ⊓ ∃R.B ⊓ ¬ (A ⊓ B) ⊓ ¬∀R.(A ⊔ B)
∀R.¬A ⊓ ∃R.B ⊓ (¬A ⊔ ¬ B) ⊓ ∃R. ¬ (A ⊔ B) then (¬A ⊓ ¬ B)

16. Transformation rules ⊓-rule
Condition: A contains (C1 ⊓ C2)(x), but not both C1(x) and C2(x)
Action: A’ = A ∪ {C1(x), C2(x)}
T={Mother ≡ Female ⊓ ∃hasChild.Person} A={Mother(Anna)}
Is ¬(∃hasChild.Person ⊓ ¬∃hasParent.Person)(Anna) satisfiable?
Expand A w.r.t. T
Mother(Anna)  (Female ⊓ ∃hasChild.Person)(Anna) 
A’ = A ∪ {Female(Anna), (∃hasChild.Person)(Anna)}
(¬∃hasChild.Person ⊓ ¬∃hasParent.Person)(Anna) 
A’ = A ∪ {¬∃hasChild.Person)(Anna), ¬∃hasParent.Person)(Anna)}

17. Transformation rules ⊔-rule
Condition: A contains (C1 ⊔ C2)(x), but neither C1(x) or C2(x)
Action: A’ = A ∪ {C1(x)} and A’’ = A ∪ {C2(x)}
T={Parent≡∃hasChild.Female⊔∃hasChild.Male,
Person≡Male⊔Female, Mother≡Parent ⊓Female}
A={Mother(Anna)}
Is ¬∃hasChild.Person(Anna) satisfiable?
Expand A w.r.t. T
A = {Mother(Anna)}  A’ = A ∪ {Parent(Anna), Female(Anna)}
Parent(Anna)  (∃hasChild.Female⊔∃hasChild.Male)(Anna) 
(∃hasChild.Female)(Anna) or (∃hasChild.Male)(Anna)
Both are in contradiction with ¬∃hasChild.Person, not satisfiable.

18. Transformation rules ∃-rule Action: A’ = A ∪ {C(z), R(x,z)}
Condition: A contains (∃R.C)(x), but there is no z such that both C(z) and R(x,z) are in A
Action: A’ = A ∪ {C(z), R(x,z)}
T={Parent≡∃hasChild.Female⊔∃hasChild.Male,
Person≡Male⊔Female, Mother≡Parent⊓Female}
A={Mother(Anna), hasChild(Anna,Bob), ¬Female(Bob)}
Is ¬∃hasChild.Person(Anna) satisfiable?
Expand A w.r.t. T
Mother(Anna)  Parent(Anna) 
(∃hasChild.Female⊔∃hasChild.Male)(Anna)
take (∃hasChild.Male)(Anna)  hasChild(Anna,Bob), Male(Bob) …

19. Transformation rules ∀-rule
Condition: A contains (∀R.C)(x) and R(x,z), but not C(z)
Action: A’ = A ∪ {C(z)}
T={DaughterParent≡∀hasChild.Female, Male⊓Female⊑⊥}
A={hasChild(Anna,Bob), ¬Female(Bob)}
Is DaughterParent(Anna) satisfiable?
Expand A w.r.t. T
DaughterParent(x)  ∀hasChild.Female(x) 
Given that hasChild(Anna,Bob)  A’ = A ∪ {Female(Bob)}
but this in contradiction with ¬Female(Bob)

20. Examples Example 1 Example 2 DL knowledge base
vegan ≐ person ⊓ ∀eats.plant
vegetarian ≐ person ⊓ ∀eats.(plants ⊔ dairy)
Query: vegan ⊑ vegetarian
Example 2
Query: vegetarian ⋢ vegan

21. Example-1 DL knowledge base vegan ≐ person ⊓ ∀eats.plant
vegetarian ≐ person ⊓ ∀eats.(plants ⊔ dairy)
Query: vegan ⊑ vegetarian
Convert to
vegan ⊓ ¬ vegetarian is unsatisfiable

22. Example-1 Unfold and normalise vegan ⊓ ¬ vegetarian
A0 =( person ⊓ ∀eats.plant ⊓ (¬ person ⊔ ∃eats.(¬ plant ⊓ ¬ dairy))(x)
Apply ⊓-rule and add to C0:
A1 = A0 U {person(x), ∀eats.plant(x), (¬ person ⊔ ∃eats.(¬ plant ⊓ ¬ dairy))(x)}

23. Example-1 Apply ⊔-rule to ¬ person ⊔ ∃eats.(¬ plant ⊓ ¬ dairy):
A1 = {person(x), ∀eats.plant(x), (¬ person ⊔ ∃eats.(¬ plant ⊓ ¬ dairy))(x)}
Add ¬ person to A1: Clash
Go back and add ∃eats.(¬ plant ⊓ ¬ dairy) to A1
A2= A1 U {∃eats.(¬ plant ⊓ ¬ dairy) (x)}
Apply ∃-rule to ∃eats.(¬ plant ⊓ ¬ dairy):
A3= A2 U {(¬ plant ⊓ ¬ dairy) (y), eats(x,y)}
Apply ∀-rule to ∀eats.plant (x) in A1 and eats(x, y) in A3
Add plant(y) to A3

24. Example-1 Apply ⊓-rule to ¬ plant ⊓ ¬ dairy in A3
Add {¬ plant(y), ¬ dairy(y)} to A3: Clash
Conclusion
Both applications of the ⊔-rule have lead to clashes
So vegan ⊓ ¬ vegetarian is unsatisfiable
So vegan ⊑ vegetarian

25. Example-2 Query: vegetarian ⋢ vegan Convert to
vegetarian ⊓ ¬ vegan is satisfiable
Unfold and normalise vegetarian ⊓ ¬ vegan
person ⊓ ∀eats.(plant ⊔ dairy) ⊓ (¬ person ⊔ ∃eats. ¬ plant)
A0 = {person ⊓ ∀eats.(plant ⊔ dairy) ⊓ (¬ person ⊔ ∃eats. ¬ plant)} (x)

26. Example-2 Apply ⊓-rule and add to A0:
A1= A0 U {person(x), ∀eats.(plant ⊔ dairy) (x),(¬person ⊔ ∃eats.¬plant) (x)}
Apply ⊔-rule to ¬person ⊔ ∃eats.¬ plant:
Add ¬person (x) to A1: Clash
Go back and add ∃eats.¬plant to A1
Apply ∃-rule to ∃eats.¬ plant:
A2 = A1 U{¬plant(y), eats(x,y)}

27. Example-2 Apply ∀-rule to ∀eats.(plant ⊔ dairy) in A1
A2 = A1 U {(plant ⊔ dairy)(y)}
Apply ⊔-rule to plant ⊔ dairy in A2
Add plant (y) to A2: Clash
Go back and add dairy(y) to A2
Conclusion
No more rules are applicable
So vegetarian ⊓ ¬ vegan is satisfiable
So vegetarian ⋢ vegan