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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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Logics for Data and KnowledgeRepresentation 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) • 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
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
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
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. 5
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. 6
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. 7
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. 8
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. 9
