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 IIndividuals iI 2 I

JohnMary

Concepts CI µ I

LawyerDoctorVehicle

Roles rI µ I £ I

hasChildowns

(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. ¬CExample¬∃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