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

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

1A

23

1 23

1

1A

1 2 3A

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

1A

23

B

1A

2

Kripke Models (III)

7

1A

23

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 Rain8

1

2

3

4

5

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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.

9

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.

10

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.

11

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.

12

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.

13

1 2 3

A, B A B