1 / 30

Logics for Data and Knowledge Representation

Logics for Data and Knowledge Representation. Exercises: ClassL. Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese. SYNTAX. Symbols in ClassL. Which of the following symbols are used in ClassL? ⊓  ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨ Which of the following symbols are in well formed formulas?

axel-stanley
Download Presentation

Logics for Data and Knowledge Representation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related