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Tableau Algorithm
Presentation Outline
• Description Logics• Reasoning Tasks• Structural Subsumption• Tableau Algorithm• Examples• Q & A
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
DL Semantics
Interpretation domain IInterpretation function I
Individuals iI 2 I
John
Mary
Concepts CI µ I
Lawyer
Doctor
Vehicle
Roles rI µ I £ I
hasChild
owns
(Lawyer u Doctor)
• 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
DL Knowledge Base
Example of TBoxWoman ≡ 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
Example of ABoxMotherWithoutDaughter(mary)
Father(peter)
hasChild(mary, peter)
hasChild(peter, harry)
hasChild(mary, paul)
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.
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
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
Structural Subsumption• Normalize descriptions
• Compare syntactical structure of normal forms
• Normal form of C: A1 … A⊓ ⊓ m R⊓ ∀ 1.C1 … R⊓ ⊓∀ n.Cn
• Normal form of D: B1 … B⊓ ⊓ k S⊓ ∀ 1.D1 … S⊓ ⊓∀ l.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 D⊑ j
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
Structural Subsumption• What if we introduce disjunction
• C: A (⊔ B ⊓ E)
• D: A ⊔ E
• C ⊑ D?
• Cannot be handled with structural subsumption
• Solution: Tableau
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
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)
• 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)} ∪ ∃ ∃
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={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.∃
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)}∪ ∪
Transformation rules• ∃-rule
Condition: A contains ( R.C)(x), ∃ but there is no z such that both C(z) and R(x,z) are in AAction: 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) …
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)
Examples• Example 1
– DL knowledge base
• vegan ≐ person ⊓ ∀eats.plant
• vegetarian ≐ person ⊓ ∀eats.(plants ⊔ dairy)
– Query: vegan ⊑ vegetarian
• Example 2– Query: vegetarian ⋢ vegan
Example-1• DL knowledge base
– vegan ≐ person ⊓ ∀eats.plant
– vegetarian ≐ person ⊓ ∀eats.(plants ⊔ dairy)
• Query: vegan ⊑ vegetarian
• Convert to
– vegan ⊓ ¬ vegetarian is unsatisfiable
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)}
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
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
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)
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)}
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