LLogics for DData and KKnowledgeRRepresentation
Modal Logic: exercises
Originally by Alessandro Agostini and Fausto GiunchigliaModified 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
3
1 2 3
1 2
1 2 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’’
4
1 2 3
1 2
3
Semantics: Kripke Model 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
1 2
A, B A
Semantics: Kripke Model 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
1 2 3
A A, B B
Semantics: Kripke Model 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
1 2 3
A A, B B
Semantics: Kripke Model 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
1 2 3
A, B A B
Semantics: Kripke Model 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
1 2 3
A, B A B