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Logics for Data and Knowledge Representation

Dive into ClassL syntax symbols, formation rules, modeling, and semantics in this comprehensive study guide. Learn about TBox, ABox, satisfiability, and more!

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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. SYNTAX

  3. Symbols in ClassL • 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) • Which of the following symbols are used in ClassL? ⊓ ⊤∨ ≡⊔ ⊑→ ↔ ⊥ ∧ ⊨ • Which of the following symbols are in well formed formulas? ⊓ ⊤ ∨≡ ⊔ ⊑→ ↔ ⊥ ∧ ⊨ 4

  5. Extended formation rules The basic BNF grammar: <Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ <wff> ::= <Atomic Formula> | ¬<wff> |<wff> ⊓<wff> | <wff> ⊔ <wff> TBox: <definition> ::= <Atomic Formula> ≡<wff> <specialization> ::= <Atomic Formula> ⊑ <wff> ABox: <individual> ::= a | b | ... | <assertion> ::= <Atomic Formula> (<individual>) 5

  6. Formation rules • Which of the following is not a wff in ClassL? •  MonkeyLow⊔ BananaHigh •   MonkeyLow⊓ BananaHigh ⊑  GetBanana • MonkeyLow  ⊓ BananaHigh • MonkeyLow GetBanana NUM 2, 3, 4 ! 6

  7. MODELING

  8. Formalization of simple sentences Propositional DL (ClassL) has pretty poor expressiveness. For instance, we cannot represent attributes and relations effectively. 8

  9. Formalization of a problem in ClassL Unicorns are mythical horses having a horn. Pegasus is a unicorn while Mike is not. Unicorn ⊑ mythical ⊓ horse ⊓ hasHorn Unicorn(Pegasus), Unicorn(Mike) There are two kinds of students: master students and PhD students. All PhD students do research. Ronald is a master student that does research. MasterStudent ⊑ Student PhDStudent ⊑ Student ⊓ doResearch MasterStudent(Ronald) doResearch(Ronald) 9

  10. Formalization of a semantic network T = {BodyOfWater ⊑ Location, PopulatedPlace ⊑ Location, Lake ⊑ BodyOfWater, City ⊑ PopulatedPlace, Country ⊑ PopulatedPlace} A = {Person(GiorgioNapolitano), Lake(GardaLake), City(Trento), Country(Italy), Part(GardaLake,Trento), Part(Trento, Italy), PresidentOf(GiorgioNapolitano, Italy)} 10

  11. Defining the TBox and ABox: the LDKR Class • 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. 11

  12. SEMANTICS

  13. Proprieties of the ∧ and ⊓ (I) • 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 13

  14. Proprieties of the ∧ and ⊓ (II) • 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 14

  15. TBOX REASONING

  16. Satisfiability with respect to a TBox T RECALL: Satisfiability in one model 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 Satisfiability in all models (validity) A concept P is satisfiable w.r.t. a terminology T, if for all interpretationsI with I⊨θ for all θ∈T, and such that I⊨P, namely I(P) is not empty 16

  17. Satisfiability with respect to a TBox (I) A B • Given the TBox T={A⊑B, B⊑A}, is (A⊓B) satisfiable in ClassL? This corresponds to the problem: T ⊨ (A⊓B) To prove satisfiability in one model it is enough to find one model; we can use Venn Diagrams. To prove satisfiability in all models we need to prove validity; this can be proved with DPLL: DPLL( (RewriteInPL(A⊑B)  RewriteInPL(B⊑A)  RewriteInPL((A⊓B) ))) DPLL( ((A B)  (B A))  (A  B)) 17

  18. Satisfiability with respect to a TBox (II) A B C • Given the TBox T={C⊑A, C⊑B} is (A⊓B) satisfiable? (in one model) 18

  19. Satisfiability with respect to a TBox (III) 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 YES! Look at the Venn diagram 19

  20. Satisfiability with respect to a TBox (IV) 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 YES! Look at the Venn diagram 20

  21. Subsumption Suppose we describe the students/attendees in a course: • Are assistants undergraduates? • T ⊨ Assistant ⊑ Undergraduate • Assistant ≡ PhD ⊓ Teach ≡ Master ⊓ Research ⊓ Teach ≡ • Student ⊓  Undergraduate ⊓ Research ⊓ Teach • Assistants are actually students who are not undergraduate. Undergraduate ⊑  Teach Bachelor ≡ Student ⊓ Undergraduate Master ≡ Student ⊓  Undergraduate PhD ≡ Master ⊓ Research Assistant ≡ PhD ⊓ Teach TBox T

  22. Disjointness Suppose we describe the students/attendees in a course: • Are Bachelor and master disjoint? • T ⊨ Bachelor ⊓ Master ⊑ ⊥ • (Student ⊓ Undergraduate) ⊓ (Student ⊓  Undergraduate) • Student ⊓ (Undergraduate ⊓  Undergraduate) ≡ ⊥ Undergraduate ⊑  Teach Bachelor ≡ Student ⊓ Undergraduate Master ≡ Student ⊓  Undergraduate PhD ≡ Master ⊓ Research Assistant ≡ PhD ⊓ Teach TBox T

  23. 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. 23

  24. 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 Notice that the fact that a monkey climbs the box does not necessarily mean that it survives. 24

  25. ABOX REASONING

  26. ABox: Consistency Check the consistency of A w.r.t. T via expansion. 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) 26

  27. 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 T MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡ Survive A MonkeyLow(Cita) Survive(Cita) Expansion of A MonkeyLow(Cita)  GetBanana(Cita) • ClimbBox(Cita)  Survive(Cita) GetBanana(Cita) 27

  28. Instance Retrieval. Consider the following expansion… 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) 28

  29. Instance Retrieval. … find the instances of Master 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) 29

  30. 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. T MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡ Survive A MonkeyLow(Cita) Survive(Cita) 30

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