Kripke's World, a Review - Stanford Computer Scienceepacuit/talks/kripkesworld.pdf · 2009. 11. 15. · Kripke’s World, a Review Eric Pacuit Stanford University ai.stanford.edu/˘epacuit

Kripke’s World, a Review

Eric Pacuit

Stanford Universityai.stanford.edu/∼epacuit

March 27, 2009

Eric Pacuit: Kripke’s Worlds, 1

Plan

I Setting the Stage: A Quick Introduction to Modal Logic

I Kripke’s World

I Integrating Kripke’s World into a Course on Modal Logic

Modern Modal Logic began with C.I. Lewis’ dissatisfaction withthe material conditional (→).

C.I. Lewis’ idea: Interpret ‘If A then B’ as ‘It must be the casethat A implies B’, or ‘It is necessarily the case that A implies B’

Prosecutor: “If Eric is guilty then he had an accomplice.”Defense: “I disagree!”Judge: “I agree with the defense.”

Prosecutor: G → ADefense: ¬(G → A)Judge: ¬(G → A)

Prosecutor: “If Eric is guilty then he had an accomplice.”Defense: “I disagree!”Judge: “I agree with the defense.”Prosecutor: G → A

Defense: ¬(G → A)Judge: ¬(G → A)

Prosecutor: G → ADefense: ¬(G → A)Judge: ¬(G → A) ⇔ G ∧ ¬A, therefore G !

Prosecutor: 2(G → A) (It is necessarily the case that . . . )Defense: ¬2(G → A)Judge: ¬2(G → A) (What can the Judge conclude?)

The Basic Modal Language

A wff of Propositional Modal Logic is defined inductively:

1. Any atomic propositional variable is a wff

2. If P and Q are wff, then so are ¬P, P ∧Q, P ∨Q and P → Q3. If P is a wff, then so is 2P and 3P

Boolean Logic

Modal operator

Eg., 2(P → 3Q) ∨23¬R

What is a modal?A modal qualifies the truth of a statement.

John happy.

I is

I is necessarily

I is possibly

I is known/believed/certain (by Ann) to be

I is permitted to be

I is obliged to be

I is now

I will be

I can do something to ensure that he is

I · · ·

John happy.

I is

I is necessarily

I is possibly

I is obliged to be

I is now

I will be

I · · ·

John happy.

I is

I is necessarily

I is possibly

I is obliged to be

I is now

I will be

I · · ·

John happy.

I is

I is necessarily

I is possibly

I is obliged to be

I is now

I will be

I · · ·

John happy.

I is

I is necessarily

I is possibly

I is obliged to be

I is now

I will be

I · · ·

John happy.

I is

I is necessarily

I is possibly

I is obliged to be

I is now

I will be

I · · ·

John happy.

I is

I is necessarily

I is possibly

I is obliged to be

I is now

I will be

I · · ·

John happy.

I is

I is necessarily

I is possibly

I is obliged to be

I is now

I will be

I · · ·

One Language, Many Interpretations

Alethic2ϕ: ϕ is necessary3ϕ: ϕ is possible

Deontic2ϕ: ϕ is obligatory3ϕ: ϕ is permitted(Oϕ, Pϕ)

Epistemic2ϕ: Ann knows ϕ3ϕ: it is consistent with Ann’s information that ϕ(KAϕ, LAϕ)

Validities?

2P ↔ ¬3¬P

P is necessary/obligatory iff ¬P is not possible/permitted

2P → PIf P is necessary/known/obligatory then P is true

2P → 22PIf Ann knows P then Ann knows that she knows P

3P → 23P, 23P → 32P, etc.

Validities?

2P ↔ ¬3¬PP is necessary/obligatory iff ¬P is not possible/permitted

3P → 23P, 23P → 32P, etc.

Validities?

2P → P

If P is necessary/known/obligatory then P is true

3P → 23P, 23P → 32P, etc.

Validities?

3P → 23P, 23P → 32P, etc.

Validities?

2P → 22P

If Ann knows P then Ann knows that she knows P

3P → 23P, 23P → 32P, etc.

Validities?

3P → 23P, 23P → 32P, etc.

Validities?

3P → 23P,

23P → 32P, etc.

Validities?

3P → 23P, 23P → 32P, etc.

There are many interesting arguments involving modalities...

1. If I give the order to attack, then, necessarily, there will be asea battle tomorrow

2. If not, then, necessarily, there will not be one.

3. Now, I give the order or I do not.

4. Hence, either it is necessary that there is a sea battletomorrow or it is necessary that none occurs.

Two readings:

A→ 2B¬A→ 2¬BA ∨ ¬A2B ∨2¬B

2(A→ B)2(¬A→ ¬B)A ∨ ¬A2B ∨2¬B

Which argument is valid?

Two readings:

A→ 2B¬A→ 2¬BA ∨ ¬A2B ∨2¬B

2(A→ B)2(¬A→ ¬B)A ∨ ¬A2B ∨2¬B

Two readings:

A→ 2B¬A→ 2¬BA ∨ ¬A2B ∨2¬B

2(A→ B)2(¬A→ ¬B)A ∨ ¬A2B ∨2¬B

Two readings:

A→ 2B¬A→ 2¬BA ∨ ¬A2B ∨2¬B

2(A→ B)2(¬A→ ¬B)A ∨ ¬A2B ∨2¬B

Two readings:

A→ 2B¬A→ 2¬BA ∨ ¬A2B ∨2¬B

2(A→ B)2(¬A→ ¬B)A ∨ ¬A2B ∨2¬B

Two readings:

A→ 2B¬A→ 2¬BA ∨ ¬A2B ∨2¬B

2(A→ B)2(¬A→ ¬B)A ∨ ¬A2B ∨2¬B

Two readings:

A→ 2B¬A→ 2¬BA ∨ ¬A2B ∨2¬B

2(A→ B)2(¬A→ ¬B)A ∨ ¬A2B ∨2¬B

Two readings:

A→ 2B¬A→ 2¬BA ∨ ¬A2B ∨2¬B

2(A→ B)2(¬A→ ¬B)A ∨ ¬A2B ∨2¬B

Two readings:

A→ 2B¬A→ 2¬BA ∨ ¬A2B ∨2¬B

2(A→ B)2(¬A→ ¬B)A ∨ ¬A2B ∨2¬B

I Modern modal logic was developed to study (strict)implications. Gradually, the study of ‘3’ and ‘2’ themselvesbecame dominant, with the study of “implication” developinginto a separate topic.

I Saul Kripke, Stag Kanger and others brought modal logic outof its “syntactic phase” in the 1960s

I Since then, the model theory for (propositional) modal logichas been extensively studied.

Rob Goldblatt. Mathematical Modal Logic: a View of its Evolution. Modalitiesin the Twentieth Century, Volume 7 of the Handbook of the History of Logic,edited by Dov M. Gabbay and John Woods, Elsevier, 2006, pp 1-98.

Kripke Structures

The main idea:

I ‘It is sunny outside’ is currently true, but it is not necessary(for example, if we were currently in Amsterdam).

I We say P is necessary provided P is true in all (relevant)situations (states, worlds, possibilities).

I A Kripke structure is1. A set of states, or worlds (each world specifies the truth value

of all propositional variables)2. A relation on the set of states (specifying the “relevant

situations”)

Kripke Structures

The main idea:

I ‘It is sunny outside’ is currently true

, but it is not necessary(for example, if we were currently in Amsterdam).

situations”)

Kripke Structures

The main idea:

situations”)

Kripke Structures

The main idea:

situations”)

Kripke Structures

The main idea:

situations”)

A Kripke Structure

Aw1

Bw2 B w3

B, C w4 A, B w5

1. Set of states(propositional valuations)

2. Accessibility relation

A Kripke Structure

Aw1

Bw2 B w3

B, C w4 A, B w5

A Kripke Structure

Aw1

Bw2 B w3

B, C w4 A, B w5

A Kripke Structure

Aw1

Bw2 B w3

B, C w4 A, B w5

denoted w3Rw5

Truth of Modal Formulas

We interpret formulas at states in a Kripke structure: w |= Pmeans P is true at state w .

We write wRv is v is accessible from state w.

1. 2P is true at state w iff P is true in all accessible worlds.w |= 2P iff for all v , if wRv then v |= P

2. 3P is true at state w iff P is true at some accessible world.w |= 3P iff there exists v such that wRv and v |= P.

1. 2P is true at state w iff P is true in all accessible worlds.

w |= 2P iff for all v , if wRv then v |= P

2. 3P is true at state w iff P is true at some accessible world.

w |= 3P iff there exists v such that wRv and v |= P.

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w4 |= B ∧ C

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w4 |= B ∧ C

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w3 |= 2B∧

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w3 |= 2B∧

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w3 |= 3C∧

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w3 |= 3C∧

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w3 6|= 2C∧

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w3 6|= 2C∧

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w1 |= 32B∧

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w1 |= 32B∧

Example

Aw1

Bw2 B w3

B, C w4 A, B w5

w1 |= 32B∧

Some Facts

I 2P ∨ ¬2P is always true (i.e., true at any state in any Kripkestructure), but what about 2P ∨2¬P?

I 2P ∧2Q → 2(P ∧ Q) is true at any state in any Kripkestructure. What about 2(P ∨ Q)→ 2P ∨2Q?

I 2P ↔ ¬3¬P is true at any state in any Kripke structure.

Some Facts

I 2P ∧2Q → 2(P ∧ Q) is true at any state in any Kripkestructure.

What about 2(P ∨ Q)→ 2P ∨2Q?

Some Facts

But, we are not always interested in all Kripke structures.

For example, consider the epistemic interpretation: A state v isaccessible from w (wRv) provided “given the agents information,w and v are indistinguishable”. What are natural properties?

Eg., for each state w , w is accessible from itself (R is a reflexiverelation).

Fact

I 2ϕ→ ϕ is valid on a Kripke frame iff the frame is refelxive.

For example, consider the epistemic interpretation: A state v isaccessible from w (wRv) provided “given the agents information,w and v are indistinguishable”.

What are natural properties?

Fact

Modal logic is a good formal language for “talking about” Kripkestructures!

Kripke structures, or more generally relational structures, areimportant in

Philosophical logic

Linguistics

Theoretical Computer Science

Game Theory

· · ·

Logic is not just about formalizing arguments! It can help us studymathematical structures.

What “good” means will be discussed in Philosophy 151.

I Philosophical logic

I Linguistics

I Theoretical Computer Science

I Game Theory

I · · ·

I Linguistics

I Game Theory

I · · ·

I Linguistics

I Game Theory

I · · ·

“Good” compared with, say, first-order logic.

An Example

Which pair of states cannot be distinguished by a modal formula?

s

K

t

M

u

N

2(2⊥ ∨32⊥)

s

K

t

M

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

s

K

u

N

An Example

s

K

t

M

u

N

2(2⊥ ∨32⊥)

s

K

t

M

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

s

K

u

N

An Example

s

K

t

M

u

N

2(2⊥ ∨32⊥)

s

K

t

M

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

s

K

u

N

An Example

s

K

t

M

u

N

2(2⊥ ∨32⊥)

s

K

t

M

2(2⊥ ∨32⊥)

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

s

K

u

N

An Example

s

K

t

M

u

N

2(2⊥ ∨32⊥)

s

K

t

M

2(2⊥ ∨32⊥)

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

s

K

u

N

An Example

s

K

t

M

u

N

2(2⊥ ∨32⊥)

s

K

t

M

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

2(2⊥ ∨32⊥)

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

s

K

u

N

An Example

s

K

t

M

u

N

2(2⊥ ∨32⊥)

s

K

t

M

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

2(2⊥ ∨32⊥)

s

K

u

N

An Example

s

K

t

M

u

N

2(2⊥ ∨32⊥)

s

K

t

M

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

2(2⊥ ∨32⊥)

s

K

u

N

An Example

s

K

t

M

u

N

2(2⊥ ∨32⊥)

s

K

t

M

2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)2(2⊥ ∨32⊥)

s

K

u

N

Defining States

w2

w1 w4

w3

I w4 |= 2⊥I w3 |= 32⊥ ∧22⊥I w2 |= 32⊥ ∧33>I w1 |= 3(32⊥ ∧22⊥)

Defining States

w2

w1 w4

w3

I w4 |= 2⊥I w3 |= 32⊥ ∧22⊥I w2 |= 32⊥ ∧33>I w1 |= 3(32⊥ ∧22⊥)

Defining States

w2

w1 w4

w3

I w4 |= 2⊥I w3 |= 32⊥ ∧22⊥I w2 |= 32⊥ ∧33>I w1 |= 3(32⊥ ∧22⊥)

Defining States

w2

w1 w4

w3

I w4 |= 2⊥I w3 |= 32⊥ ∧22⊥I w2 |= 33> ∧ ¬3ϕw3I w1 |= 3(32⊥ ∧22⊥)

Defining States

w2

w1 w4

w3

I w4 |= 2⊥I w3 |= 32⊥ ∧22⊥I w2 |= 33> ∧ ¬3ϕw3I w1 |= 3(32⊥ ∧22⊥)

Defining States

w2

w1 w4

w3

I w4 |= 2⊥I w3 |= 32⊥ ∧22⊥I w2 |= 33> ∧ ¬3ϕw3I w1 |= 3(32⊥ ∧22⊥)

Defining States

w2

w1 w4

w3

I w4 |= 2⊥I w3 |= 32⊥ ∧22⊥I w2 |= 33> ∧ ¬3ϕw3I w1 |= 3(32⊥ ∧22⊥)

Defining Classes of Kripke Structures

Fact A Kripke frame F is transitive iff F |= 2ϕ→ 22ϕ

Fact A Kripke structure F is symmetric iff F |= 3ϕ→ 23ϕ

Which properties are modally definable? (Goldblatt-ThomasonTheorem, Salqhvist Theorem)

Which properties are modally definable?

(Goldblatt-ThomasonTheorem, Salqhvist Theorem)

Which properties are modally definable? (Goldblatt-ThomasonTheorem, Salqhvist Theorem)

Comparing Modal Logic and First Order Logic

I share many meta-theoretic properties (eg., sound andcomplete axiomatization, compactness)

I there is an analogy between 2 and ∀ and 3 and ∃ that can bemade precise (the standard translation).

I valuation games are essentially the same

I Theorem Modal logic is the bisimulation-invariant fragmentof first-order logic (so, first order logic is more expressive, butthis comes with a price).

Kripke’s World

I Uses a first-order modal language. ∃x3Cube(x),3∃xCube(x), 2Tet(a), etc.

I States are blocks worlds

I Accessibility relations can be reflexive, symmetric, transitive,Euclidean, etc.

Kripke’s World: Rigid Designation

2Cube(a)

find the object named by a in the current world, look at that objectin all accessible worlds and check if they have the property Cube.

Kripke’s World: Rigid Designation

2Cube(a)

find the object named by a in the current world, look at that objectin all accessible worlds and check if they have the property Cube.

Demonstration

Comments

1. How should/will Kripke’s World be integrated in a logiccurriculum?

1. As a stand alone introduction to modal logic course (do formodal logic what Tarski’s world/LPL does for introduction tofirst-order logic courses)

2. Integrate it into the current logic sequence (i.e., integrate itinto an introduction to logic class, intermediate logic class,advanced logic class).

3. As a tutorial for a more advanced course on modal logic.

Comments

1. How should/will Kripke’s World be integrated in a logiccurriculum?

1. As a stand alone introduction to modal logic course (do formodal logic what Tarski’s world/LPL does for introduction tofirst-order logic courses)

2. Integrate it into the current logic sequence (i.e., integrate itinto an introduction to logic class, intermediate logic class,advanced logic class).

3. As a tutorial for a more advanced course on modal logic.

Comments

2. Comparisons with existing literature.

Introductory Level Texts:

1. Modal Logic by Brian Chellas (1980)

2. A New Introduction to Modal Logic by G. E. Hughes and M.J. Cresswell (1996)

3. Modal Logics and Philosophy by Rod Girle (2001)

4. Modal Logic for Philosophers by James Garson (2006)

5. Modal Logic for Open Minds by Johan van Benthem(forthcoming)

6. First-Order Modal Logic by Melvin Fitting and RichardMendelsohn (1998)

Comments

Advanced Texts:

1. Modal Logic by P. Blackburn, M. de Rijke and Y. Venema(2001)

2. Handbook of Modal Logic edited by P. Blackburn, J. vanBenthem, and F. Wolter (2007)

3. Modal Logic by A. Chagrov and M. Zakharyaschev (1997)

4. Plus others on fixed interpretations (Epistemic logic,provability logic, dynamic logic, etc.)

Comments

Advanced Texts:

1. Modal Logic by P. Blackburn, M. de Rijke and Y. Venema(2001)

2. Handbook of Modal Logic edited by P. Blackburn, J. vanBenthem, and F. Wolter (2007)

3. Modal Logic by A. Chagrov and M. Zakharyaschev (1997)

4. Plus others on fixed interpretations (Epistemic logic,provability logic, dynamic logic, etc.)

Comments

3. What about the various “applications” of modal logic(epistemic logic, doxastic logic, deontic logic, temporal logic, etc.)?The modal language can be used to describe interactive situationsinvolving many agents (Eg., playing a card game).

Single-Agent Epistemic Logic

Typically, we write KP instead of 2P when the intendedinterpretation is “P is known”.

K (P → Q): “Ann knows that P implies Q”KP ∨ ¬KP: “either Ann does or does not know P”KP ∨ K¬P: “Ann knows whether P is true”

LP: “P is an epistemic possibility”

KLP: “Ann knows that she thinks P ispossible”

K (P → Q): “Ann knows that P implies Q”

KP ∨ ¬KP: “either Ann does or does not know P”KP ∨ K¬P: “Ann knows whether P is true”

K (P → Q): “Ann knows that P implies Q”KP ∨ ¬KP: “either Ann does or does not know P”

KP ∨ K¬P: “Ann knows whether P is true”LP: “P is an epistemic possibility”

ExampleSuppose there are three cards:1, 2 and 3.

Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.

(1, 2)

w1

(1, 3)

w2

(2, 3)

w3

(2, 1)

w4

(3, 1)

w5

(3, 2)

w6

What are the relevant states?

(1, 2)

w1

(1, 3)

w2

(2, 3)

w3

(2, 1)

w4

(3, 1)

w5

(3, 2)

w6

What are the relevant states?

(1, 2)

w1

(1, 3)

w2

(2, 3)

w3

(2, 1)

w4

(3, 1)

w5

(3, 2)

w6

Ann receives card 3 and card 1is put on the table

(1, 2)

w1

(1, 3)

w2

(2, 3)

w3

(2, 1)

w4

(3, 1)

w5

(3, 2)

w6

What information does Annhave?

(1, 2)

w1

(1, 3)

w2

(2, 3)

w3

(2, 1)

w4

(3, 1)

w5

(3, 2)

w6

(1, 2)

w1

(1, 3)

w2

(2, 3)

w3

(2, 1)

w4

(3, 1)

w5

(3, 2)

w6

(1, 2)

w1

(1, 3)

w2

(2, 3)

w3

(2, 1)

w4

(3, 1)

w5

(3, 2)

w6

Suppose Hi is intended tomean “Ann has card i”

Ti is intended to mean “card iis on the table”

Eg., V (H1) = {w1, w2}

(1, 2)

w1

(1, 3)

w2

(2, 3)

w3

(2, 1)

w4

(3, 1)

w5

(3, 2)

w6

Suppose Hi is intended tomean “Ann has card i”

Ti is intended to mean “card iis on the table”

Eg., V (H1) = {w1, w2}

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

Suppose that Ann receives card1 and card 2 is on the table.

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

M, w1 |= KH1

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

M, w1 |= K H1

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

M, w1 |= K H1

M, w1 |= K¬T1

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

M, w1 |= LT2

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

M, w1 |= K (T2 ∨ T3)

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

Multiagent Epistemic Logic

Many of the examples we are interested in involve more than oneagent!

KAP means “Ann knows P”

KBP means “Bob knows P”

I KAKBϕ: “Ann knows that Bob knows ϕ”

I KA(KBϕ ∨ KB¬ϕ): “Ann knows that Bob knows whether ϕ

I ¬KBKAKB(ϕ): “Bob does not know that Ann knows thatBob knows that ϕ”

H1, T2

w1

H1, T3

w2

H2, T3

w3

H2, T1

w4

H3, T1

w5

H3, T2

w6

Ann is dealt one of the cards,Bob is given one of the cardsand the third card is put backin the deck.

Suppose that Ann receives card1 and Bob receives card 2.

A1, B2

w1

A1, B3

w2

A2, B3

w3

A2, B1

w4

A3, B1

w5

A3, B2

w6

A1, B2

w1

A1, B3

w2

A2, B3

w3

A2, B1

w4

A3, B1

w5

A3, B2

w6

A1, B2

w1

A1, B3

w2

A2, B3

w3

A2, B1

w4

A3, B1

w5

A3, B2

w6

M, w1 |= KB(KAA1 ∨ KA¬A1)

A1, B2

w1

A1, B3

w2

A2, B3

w3

A2, B1

w4

A3, B1

w5

A3, B2

w6

A1, B2

w1

A1, B3

w2

A2, B3

w3

A2, B1

w4

A3, B1

w5

A3, B2

w6

A1, B2

w1

A1, B3

w2

A2, B3

w3

A2, B1

w4

A3, B1

w5

A3, B2

w6

Comments

4. Three different levels of analysis:

1. With an interpreted (first-order) language such as the blockslanguage, we must describe how each object are related toeach other in different worlds.

2. Structural properties of Kripke structures viewed as graphs.

3. Dynamic operations that change the current model.

Comments

4. Three different levels of analysis:

1. With an interpreted (first-order) language such as the blockslanguage, we must describe how each object are related toeach other in different worlds.

2. Structural properties of Kripke structures viewed as graphs.

3. Dynamic operations that change the current model.

Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.

Their mother enters the room and says “At least one of you havemud on your forehead”.

Then the children are repeatedly asked “do you know if you havemud on your forehead?”

What happens?

Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Thenthe clean child says, “Oh, I must be clean.”

What happens?

Claim: After first question, the children answer “I don’t know”,

after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Thenthe clean child says, “Oh, I must be clean.”

What happens?

Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark).

Thenthe clean child says, “Oh, I must be clean.”

What happens?

Muddy ChildrenAssume:

I There are three children: Ann, Bob and Charles.

I (Only) Ann and Bob have mud on their forehead.

C C C

Ann Bob Charles

state-of-affairs

C C C C C C C C C

C C C

Ann Bob Charles

state-of-affairs