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 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)

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

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

Example of ABox MotherWithoutDaughter(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 ⊓ …⊓ 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

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)

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)}

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.

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) …

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 Example 2 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