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L ogics for D ata and K nowledge R epresentation

L ogics for D ata and K nowledge R epresentation. Modal Logic: exercises. Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese. Truth relation (true in a world).

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L ogics for D ata and K nowledge R epresentation

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  1. Logics for Data and KnowledgeRepresentation Modal Logic: exercises Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

  2. Truth relation (true in a world) • Given a Kripke Model M = <W, R, I>, a proposition P ∈ LML and a possible world w ∈ W, we say that “w satisfies P in M” or that “P is satisfied by w in M” or “P is true in M via w”, in symbols: M, w ⊨ P in the following cases: 1. P atomic w ∈ I(P) 2. P = Q M, w ⊭ Q 3. P = Q  T M, w ⊨ Q and M, w ⊨ T 4. P = Q  T M, w ⊨ Q or M, w ⊨ T 5. P = Q  T M, w ⊭ Q or M, w ⊨ T 6. P = □Q for every w’∈W such that wRw’ then M, w’ ⊨ Q 7. P = ◊Q for some w’∈W such that wRw’ then M, w’ ⊨ Q NOTE: wRw’ can be read as “w’ is accessible from w via R” 2

  3. Kinds of frames • Serial: for every w ∈ W, there exists w’ ∈ W s.t. wRw’ • Reflexive: for every w ∈ W, wRw • Symmetric: for every w, w’ ∈ W, if wRw’ then w’Rw 1 2 3 1 2 1 2 3 3

  4. Kinds of frames • Transitive: for every w, w’, w’’ ∈ W, if wRw’ and w’Rw’’ then wRw’’ • Euclidian: for every w, w’, w’’ ∈ W, if wRw’ and wRw’’ then w’Rw’’ 1 2 3 1 2 3 4

  5. Kripke Models (I) • Find a Kripke model M where: • the formula M, 1 ⊨ ◊A is true • the formula M, 1 ⊨ ◊A is true • the formula M, 1 ⊨ A is true • the formula M, 1 ⊨ A is true and M is reflexive • the formula M, 1 ⊨  A is true and M is serial A 3 1 2 3 1 2 1 A 1 A 1 2 3 5

  6. Kripke Models (II) • Find a Kripke model M where: • the formula M, 1 ⊨ ◊A  ◊Bis true • Both the formulas M, 1 ⊨ A andM, 2 ⊨ ◊ B are true, IR2 and M is symmetric B A 3 1 2 A 1 2 6

  7. Kripke Models (III) B A 3 1 2 7

  8. Modeling 1 • Consider the paths designed between cities in the map. • worlds = cities • relations = roads • M, w ⊨ □P = “P is true in all cities that can be reached from w” • M, w ⊨ ◊P = “P is true in some cities that can be reached from w” • Express in Modal logic that: • It rains in all cities that can be reached directly from Trento M, 1 ⊨ Rain • If it rains in Florence, it must rain in Naples as well M, 4 ⊨ Rain  Rain 2 3 4 5 6 8

  9. Semantics: Kripke Model A, B A 1 2 • Given the Kripke model M = <W, R, I> with: W = {1, 2}, R = {<1, 2>, <2, 2>}, I(A) = {1,2} and I(B) = {1} (a) Say whether the frame <W, R> is serial, reflexive, symmetric, transitive or Euclidian. It is serial, transitive and euclidian. (b) Is M, 1 ⊨ ◊B? Yes, because 2 is accessible from 1 and M, 2 ⊨ B (c) Prove that □A is satisfiable in M By definition, it must be M, w ⊨ □A for all w in W. It is satisfiable because M, 2 ⊨ A and for all worlds w in {1, 2}, 2 is accessible from w. 9

  10. Semantics: Kripke Model 3 1 2 A A, B B • Given the Kripke model M = <W, R, I> with: W = {1, 2, 3}, R = {<1, 2>, <2, 1>, <1, 3>, <3, 3>}, I(A) = {1, 2} and I(B) = {2, 3} (a) Say whether the frame <W, R> is serial, reflexive, symmetric, transitive or Euclidian. It is serial. (b) Is M, 1 ⊨ ◊(A  B)? By definition, there must be a world w accessible from 1 where A  B is true. Yes, because A  B is true in 2 and 2 is accessible from 1. 10

  11. Semantics: Kripke Model 3 1 2 A A, B B • Given the Kripke model M = <W, R, I> with: W = {1, 2, 3}, R = {<1, 2>, <2, 1>, <1, 3>, <3, 3>}, I(A) = {1, 2} and I(B) = {2, 3} (c) Is □A satisfiable in M? By definition, it must be M, w ⊨ □A for all worlds w in W. This means that for all worlds w there is a world w’ such that wRw’ and M, w’ ⊨ A. For w = 1 we have 1R3 and M, 3 ⊨ A. Therefore the response is NO. 11

  12. Semantics: Kripke Model 1 2 3 A, B A B • Given the Kripke model M = <W, R, I> with: W = {1, 2, 3} , R = {<1, 3>, <3, 2>, <2, 1>, <2, 2>} I(A) = {1, 2} and I(B) = {1, 3} (a) Say whether the frame <W, R> is serial, reflexive, symmetric, transitive or Euclidian. It is serial (b) Is M, 1 ⊨ ◊ A? By definition, there must be a world w accessible from 1 where  A is true. Yes, because A is false in 3 and 3 is accessible from 1. 12

  13. Semantics: Kripke Model 1 2 3 A, B A B • Given the Kripke model M = <W, R, I> with: W = {1, 2, 3} , R = {<1, 3>, <3, 2>, <2, 1>, <2, 2>} I(A) = {1, 2} and I(B) = {1, 3} (c) Is ◊B satisfiable in M? We should have that M, w ⊨ ◊B for all worlds w. This means that for all worlds w there is at least a w’ such that wRw’ and M, w’ ⊨ B. However for w = 3 we have only 3R2 and B is false in 2. Therefore the response is NO. 13

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