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Logics for Data and Knowledge Representation

Logics for Data and Knowledge Representation. Exercises: ClassL. Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese. Outline. Syntax Symbols and formation rules Reasoning with a TBox Reasoning with an ABox. 2. Symbols in ClassL. SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX.

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Logics for Data and Knowledge Representation

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  1. Logics for Data and KnowledgeRepresentation Exercises: ClassL Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

  2. Outline • Syntax • Symbols and formation rules • Reasoning with a TBox • Reasoning with an ABox 2

  3. Symbols in ClassL SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX • Which of the following symbols are used in ClassL? ⊓  ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨ • Which of the following symbols are in well formed formulas? ⊓  ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨ 3

  4. Symbols in ClassL (solution) SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX • Which of the following symbols are used in ClassL? ⊓ ⊤∨ ≡⊔ ⊑→ ↔ ⊥ ∧ ⊨ • Which of the following symbols are in well formed formulas? ⊓ ⊤ ∨≡ ⊔ ⊑→ ↔ ⊥ ∧ ⊨ Remember the BNF grammar: <Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ <wff> ::= <Atomic Formula> | ¬<wff> |<wff> ⊓<wff> | <wff> ⊔ <wff> | <Atomic Formula> ⊑ <wff> | <Atomic Formula> ≡ <wff> 4

  5. Formation rules SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX • Which of the following is not a wff in ClassL? •  MonkeyLow⊔ BananaHigh •   MonkeyLow⊓ BananaHigh ⊑  GetBanana • MonkeyLow  ⊓ BananaHigh • MonkeyLow GetBanana NUM 2, 3, 4 ! 5

  6. Satisfiability A B SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX • Given the TBox T={A⊑B, B⊑A}, is (A⊓B) satisfiable in ClassL? RECALL: to prove satisfiability it is enough to find one model. We can use Venn Diagrams. In alternative, this can be proved with entailment + DPLL: P: A⊑B Q: B⊑A S: (A⊓B) {P, Q} ⊨ S DPLL(RewriteInPL(P)  RewriteInPL(Q)  RewriteInPL(S)) DPLL(((A B)  (B A))  (A  B)) 6

  7. ClassL Exercises 2 A B C SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX • Given the TBox T={C⊑A, C⊑B} is (A⊓B) satisfiable in ClassL? 7

  8. Proprieties of the ∧ and ⊓ (I) SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX • Suppose that A and B are satisfiable. Is A ∧ B always satisfiable in PL? We can observe that the fact that A and B are satisfiable (alone) does not necessarily imply that A ∧ B is also satisfiable. It is instead the case when they are satisfiable by the same model. Think for instance to the case B =  A. MODEL for A MODEL for B MODEL for A MODEL for B 8

  9. Proprieties of the ∧ and ⊓ (II) SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX • Suppose that A and B are satisfiable. Is A ⊓ B always satisfiable in ClassL? We can easily observe that the fact that A and B are satisfiable does not imply that A ⊓ B is also satisfiable. Think to the case in which their extensions are disjoint. Differently from PL, this might not be the case even when they are satisfiable by the same model. A A B B A B 9

  10. Satisfiability with respect to a TBox T SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX RECALL: A concept P is satisfiable w.r.t. a terminology T, if there exists an interpretationI with I⊨θ for all θ∈T, and such that I⊨P, namely I(P) is not empty 10

  11. Satisfiability with respect to a TBox T Suppose we model the Monkey-Banana problem as follows: “If the monkey is low in position then it cannot get the banana. If the monkey gets the banana it survives”. TBox T MonkeyLow ⊑  GetBanana GetBanana ⊑ Survive Is T satisfiable? Survive GetBanana MonkeyLow SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX YES! Look at the Venn diagram 11

  12. Satisfiability with respect to a TBox T Suppose we model the Monkey-Banana problem as follows: TBox T MonkeyLow ⊑  GetBanana GetBanana ⊑ Survive Is it possible for a monkey to survive even if it does not get the banana? We can restate the problem as follow: does T ⊨  GetBanana ⊓ Survive ? Survive GetBanana MonkeyLow SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX YES! Look at the Venn diagram 12

  13. Normalization of a TBox Normalize the TBox below: MonkeyLow ⊑  GetBanana GetBanana ≡Survive Possible solution: MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡Survive Note that, with this theory, the monkey necessarily needs to get the banana to survive. SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX 13

  14. Expansion of a TBox Expand the TBox below: MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡Survive T’, expansion of T (The Venn diagram gives a possible model): MonkeyLow ≡  Survive ⊓  ClimbBox GetBanana ≡Survive ClimbBox Survive GetBanana MonkeyLow SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX Notice that the fact that a monkey climbs the box does not necessarily mean that it survives. 14

  15. ABox: Consistency Check the consistency of A w.r.t. T via expansion. SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX T MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡ Survive A MonkeyLow(Cita) Survive(Cita) • Expansion of A is consistent: • MonkeyLow(Cita)  GetBanana(Cita) • ClimbBox(Cita)  Survive(Cita)  GetBanana(Cita) 15

  16. ABox: Instance checking Given T and A below Is Cita an instance of MonkeyLow? YES Is Cita an instance of ClimbBox? NO Is Cita an instance of GetBanana? NO SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX T MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡ Survive A MonkeyLow(Cita) Survive(Cita) • Expansion of A • MonkeyLow(Cita)  GetBanana(Cita) • ClimbBox(Cita)  Survive(Cita) GetBanana(Cita) 16

  17. Instance Retrieval. Consider the following expansion… SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX T Undergraduate ⊑  Teach Bachelor ≡ Student ⊓ Undergraduate Master ≡ Student ⊓  Undergraduate PhD ≡ Master ⊓ Research Assistant ≡ PhD ⊓ Teach A Master(Chen) PhD(Enzo) Assistant(Rui) The expansion of A • Master(Chen) • Student(Chen) • Undergraduate(Chen) • PhD(Enzo) • Master(Enzo) • Research(Enzo) • Student(Enzo) • Undergraduate(Enzo) • Assistant(Rui) • PhD(Rui) • Teach(Rui) • Master(Rui) • Research(Rui) • Student(Rui) • Undergraduate(Rui) 17

  18. Instance Retrieval. … find the instances of Master SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX T Undergraduate ⊑  Teach Bachelor ≡ Student ⊓ Undergraduate Master ≡ Student ⊓  Undergraduate PhD ≡ Master ⊓ Research Assistant ≡ PhD ⊓ Teach A Master(Chen) PhD(Enzo) Assistant(Rui) The expansion of A • Master(Chen) • Student(Chen) • Undergraduate(Chen) • PhD(Enzo) • Master(Enzo) • Research(Enzo) • Student(Enzo) • Undergraduate(Enzo) • Assistant(Rui) • PhD(Rui) • Teach(Rui) • Master(Rui) • Research(Rui) • Student(Rui) • Undergraduate(Rui) 18

  19. ABox: Concept realization Find the most specific concept C such that A ⊨ C(Cita) Notice that MonkeyLow directly uses GetBanana and ClimbBox, and it uses Survive. The most specific concept is therefore MonkeyLow. SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX T MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡ Survive A MonkeyLow(Cita) Survive(Cita) 19

  20. Defining the TBox and ABox: the LDKR Class SYNTAX :: REASONING WIN A TBOX :: REASONING WITH AN ABOX • Define a TBox and ABox for the following database: ABox = {Italian(Fausto), Italian(Enzo), Chinese(Rui), Indian(Bisu), BlackHair(Enzo), BlackHair(Rui), BlackHair(Bisu), WhiteHair(Fausto)} TBox = {Italian ⊑ LDKR, Indian ⊑ LDKR, Chinese ⊑ LDKR, BlackHair ⊑ LDKR, WhiteHair ⊑ LDKR} LDKR NOTE: ClassL is not expressive enough to represent database constrains such as keys involving two fields. 20

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