LDK R Logics for Data and Knowledge Representation 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 =, a proposition P ∈ L ML 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 atomicw ∈ I(P) 2. P =  QM, w ⊭ Q 3. P = Q  TM, w ⊨ Q and M, w ⊨ T 4. P = Q  TM, w ⊨ Q or M, w ⊨ T 5. P = Q  TM, 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

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’’

Kripke Models (I)  Find a Kripke model M where: 1. the formula M, 1 ⊨ ◊  A is true 2. the formula M, 1 ⊨  ◊A is true 3. the formula M, 1 ⊨ A is true 4. the formula M, 1 ⊨ A is true and M is reflexive 5. the formula M, 1 ⊨ A is true and M is serial 5 1 A A A

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

Kripke Models (III)  7 1 A 2 3 B

Modeling  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

Semantics: Kripke Model  Given the Kripke model M = with: W = {1, 2}, R = {, }, I(A) = {1,2} and I(B) = {1} (a) Say whether the frame 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 A, B A

Semantics: Kripke Model  Given the Kripke model M = with: W = {1, 2, 3}, R = {,,, }, I(A) = {1, 2} and I(B) = {2, 3} (a) Say whether the frame 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 AA, BB

Semantics: Kripke Model  Given the Kripke model M = with: W = {1, 2, 3}, R = {,,, }, 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 AA, BB

Semantics: Kripke Model  Given the Kripke model M = with: W = {1, 2, 3}, R = {,,, } I(A) = {1, 2} and I(B) = {1, 3} (a) Say whether the frame 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 A, BAB

Semantics: Kripke Model  Given the Kripke model M = with: W = {1, 2, 3}, R = {,,, } 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 A, BAB