mud on forehead amp common knowledge Jeff Edmonds York University COSC 6111 Knowing Mud Problem Boys Learn Common Knowledge Formal Definition of Knowing Applying Formal Definition to Mud Problem ID: 287947
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Slide1
Distributed Systems mud on forehead & common knowledge
Jeff Edmonds York University
COSC 6111
Knowing Mud Problem
Boys Learn Common Knowledge
Formal Definition of "Knowing"Applying Formal Definition to Mud Problem
Grad AlgorithmsSlide2Slide3
Common KnowledgeTo be happy, it is important
For you to be a good personFor you to know you are a good personFor your mom to know you are a good personFor you to know your mom knows you are a good person.
For your dad to know your mom knows you are a good person.For you to know your dad knows your mom knows you are a good person.For it to be common knowledge
that you are a good person.Slide4
Knowing Mud
There is a group of boys on an island.
Each very smart and knows other boys very well.
Each either has mud on his forehead or not.
ie can see whether others have mud,
but not whether they themselves have mud.Slide5
Some of you have
mud on your foreheads
Knowing Mud
Man arrives and announces,
The boys, seeing other boys
with mud, are not impressed.Slide6
Some of you have
mud on your foreheads
Knowing Mud
Man arrives and announces,
Do you know whether you have mud
No
Each boy in unison
Do you know whether you have mud
Do you know whether you have mud
And so on ….
Does a boy ever know, or do they say “No” forever?
No
Each boy in unison
No
Each boy in unisonSlide7
Knowing Mud
And so on ….
Does a boy ever know, or do they say “No” forever?
Do you want to bet?
When might a boy know?Slide8
Some of you have
mud on your foreheads
Knowing Mud
Yes, I have mud
I see no mud.
Someone has mud.
It must be me
Do you know whether you have mud
Don’t know
Yes, I have no mud
Do you know whether you have mud
k = # of boys
with mud =1
k
1
B
1Slide9
Some of you have
mud on your foreheads
Knowing Mud
Yes, I have mud
I see no mud.
Someone has mud.
It must be me
k = # of boys
with mud =1
k=1
Boy with mud says yes first round.
Do you know whether you have mud
First round boys all say no k
2
k
1
B
2
B
1Slide10
Some of you have
mud on your foreheads
Knowing Mud
Do you know whether you have mud
No
Each boy in unison
k
1
I see k=1 mud.
k
2
has mud.
I must have mud
k
2
k = 2
B
2
B
1Slide11
Some of you have
mud on your foreheads
Knowing Mud
Do you know whether you have mud
No
Each boy in unison
k
2
k = 2
B
2
If I had no mud,
then B
2
would see nobody else with mud.
I know that he heard that someone has mud.
I know he is smart.
He would have known that he had mud.
But he did not know.
Hence, I must have mud
B
1Slide12
Some of you have
mud on your foreheads
Knowing Mud
Do you know whether you have mud
No
Each boy in unison
I must have mud
k
2
k = 2
B
2
B
1
Yes, I have mud
Do you know whether you have mudSlide13
Some of you have
mud on your foreheads
Knowing Mud
Do you know whether you have mud
No
Each boy in unison
I must have mud
k
2
k = 2
B
2
B
1
k=2
Boy with mud says yes second round.
Second round boys all say no k
3
t
th
round boys all say no k
t+1
by induction:Slide14
Knowing Mud
Do you know whether you have mud
No
Each boy in unison
Yes, I have mud
I see k=t mud.
k
t+1
has mud.
I must have mud
Do you know whether you have mud
k
t+1
t
th
round
…
t
th
round boys all say no k
t+1
k = t+1Slide15
Knowing Mud
Do you know whether you have mud
No
Each boy in unison
I see k=t mud.
k
t+1
has mud.
I must have mud
t
th
round
…
k = t+1
k=t+1
Boy with mud says yes t+1
st
round.
t+1
st
round boys all say no k
t+2
Induction complete
k
t+1 Slide16
The sky is blue
Knowing Mud
Man arrives and announces,
Do you know whether you have mud
No
Each boy in unison
Do you know whether you have mud
And so on ….
Does a boy ever know, or do they say “No” forever?
No
Each boy in unison
ForeverSlide17
Some of you have
mud on your foreheads
Knowing Mud
Man arrives and whispers in each boy ear
Do you know whether you have mud
No
Each boy in unison
Do you know whether you have mud
And so on ….
Does a boy ever know, or do they say “No” forever?
No
Each boy in unison
ForeverSlide18
Some of you have
mud on your foreheads
Knowing Mud
Man arrives and announces,
The boys, seeing other boys
with mud, are not impressed.
But they must be learning something!
But what??Slide19
I give you common knowledge that some of you have
mud on your foreheads
Knowing Mud
I see no mud.
Someone has mud.
It must be me
The boy who answers in 1
st
round when sees 0 mud learns that
“Someone has mud”
The boy who answers in 2
nd
round when sees 1 mud learns that
The boy who answers in 3
rd
round when sees 2 mud learns that
The boy who answers in
t+1
st
round when sees t mud learns that
…
I see k=1 mud.
k
2
has mud.
I must have mud
I see k=2 mud.
k
3
has mud.
I must have mud
I see k=t mud.
k
t+1
has mud.
I must have mudSlide20
Knowing
Node for
Universe in which
v=1
sky is blue
If I don’t know the value of
v
,
it is because I can imagine a universe
in which it has one value
and another in which it has another value
and I can’t differentiate between these universes.
Node for
Universe in which
v=2
sky is blue
Edge labeled
A
between universes
indicates player
A
can’t differentiate
between them.A
v?Slide21
Knowing
If I don’t know the value of
v
,
it is because I can imagine a universe
in which it has one valueand another in which it has another valueand I can’t differentiate between these universes.Edge labeled A between universes indicates player A can’t differentiate between them.
v=1
v=2
A
v=1
v=1
v=1
v=1
v=3
v=2
v=2
A
A
A
A
A
A
A
A
ASlide22
Knowing
v=1
v=2
v=1
v=1
v=1
v=1
v=3
v=2
v=2
Equivalently, we can circle the undifferentiable universes.
One of the universe is the
Actual Universe.
Player
A
considers a universe possible
if he cannot differentiate it from the actual universe.
If the set is bigger, Player
A
has more
uncertainty
.
A
A
ASlide23
Knowing
v=1
v=2
v=1
v=1
v=1
v=1
v=3
v=2
v=2
We say “Player
A
knows
fact
F
”
if
F
is true in every universe that
A
considers possible.
We write
K
A
F
A
A
A
If this is the actual universe,
then
K
A
v=1Slide24
Knowing
v=1
v=2
v=1
v=1
v=1
v=1
v=3
v=2
v=2
A
A
A
If this is the actual universe,
then
K
A
v=2
We say “Player
A
knows
fact
F
”
if
F
is true in every universe that
A
considers possible.
We write
K
A
FSlide25
Knowing
v=1
v=2
v=1
v=1
v=1
v=1
v=3
v=2
v=2
A
A
A
If this is the actual universe,
then
K
A
v
We say “Player
A
knows
fact
F
”
if
F
is true in every universe that
A
considers possible.
We write
K
A
FSlide26
Knowing
v=1
v=2
v=1
v=1
v=1
v=1
v=3
v=2
v=2
A
A
A
We say “Player
A
knows
fact
F
”
if
F
is true in every universe that
A
considers possible.
We write
K
A
F
We say “Fact F is
true
”
if F is true in the actual
Universe.
Lemma:
K
A
F
F
Proof: A considers the actual universe possibleand hence F is true in it.Slide27
Knowing
v=1
v=2
v=1
v=1
v=1
v=1
v=3
v=2
v=2
A
A
A
Note that
knowing
fact
F
is different than
believing
fact
F
Or feeling you
should
do action A
Or
ought
to do action A.
(Ought has more of a moral obligation.)Slide28
Knowing
v=1
v=2
v=1
v=1
v=1
v=1
v=3
v=1
v=1
A
A
A
Player
B
can't differentiate
between these universes.
Assume that the players know each other so well
that they know which universes they can distinguish
and hence know the “graph”.
BSlide29
Knowing
v=1
v=2
v=1
v=1
v=1
v=1
v=3
v=1
v=1
A
A
A
In all of the universes that B
can't differentiate between,
the fact
K
A
v=1
is true.
Therefore,
K
B
K
A
v=1
i.e. Player
B
knows Player
A
knows
v=1
.
Lemma:
K
B
K
AF KB F
BProof: In every universe B considers possible,A
knows F and hence
F must be true.Slide30
Knowing
v=1
v=2
v=1
v=1
v=1
v=1
v=3
v=2
v=2
A
A
A
What do players know in this case?
K
A
v=1
K
B
v=1
In every universe
B
considers possible,
A
knows the value of
v
(be it 1 or 2)
Hence,
B
knows
A
knows the value of
v
,
K
B KA vDoes KB KA v and KB v contradict Lemma? No, because v is not a “fact”.
BSlide31
Knowing
F
A
F
F
F
K
A
F
:
if
F
is true in every universe
reachable from the actual universe
by an edge labeled
A
.Slide32
Knowing
K
B
K
A
F: if KAF is true in every universe reachable from the actual universe by an edge labeled B. if F is true in every universe reachable from the actual universe by a path labeled BA.
B
F
F
A
F
F
F
F
A
F
F
F
F
A
F
F
FSlide33
Knowing
B
F
F
A
F
F
F
F
A
F
F
F
F
A
F
F
F
B
F
F
A
F
F
F
F
A
F
F
F
F
A
F
F
F
C
F
K
C
K
B
K
A
F
:
if
K
B
K
A
F
is true in every universe
reachable from the actual universe
by an edge labeled
C
.
if
F
is true in every universe
reachable from the actual universe
by a path labeled
C
B
A
.Slide34
Knowing
B
F
F
A
F
F
F
F
A
F
F
F
F
A
F
F
F
B
F
F
A
F
F
F
F
A
F
F
F
F
A
F
F
F
C
F
Common Knowledge
of
F
:
if
F
is true in every universe
reachable from the actual universe
by a path with any labels
.Slide35
2
Knowing Mud
1
5
Universe with mud on the foreheads of boys 1-4
and no mud on boys 5-8.
4
3
1
2
3
5
7
6
8
4
6
7
8Slide36
Knowing Mud
1
2
3
5
7
6
8
4
1
Boy
1
can’t differentiate
between these universes.
1
2
3
5
7
6
8
4Slide37
Knowing Mud
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3Slide38
Knowing Mud
Actual universe
Each boy knows that some boy has mud
i
Ki k≥1 Path from actual universe to one in which k=0.Hence, it is not common knowledge that k≥1.
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3Slide39
Knowing Mud
Actual universe
Each boy knows that some boy has mud
i
Ki k≥1 Path from actual universe to one in which k=0.Hence, it is not common knowledge that k≥1.
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3
K
1
k
≥1
K
2
K
1
k
≥1
K
3
K
2
K
1
k
≥1
K
4
K
3
K
2
K
1
k
≥1
Slide40
Knowing Mud
Some of you have
mud on your foreheads
It becomes common knowledge that
k
≥1
.
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3Slide41
Yes, I have mud
Knowing Mud
If this were the actual universe,
then boy
1
learns that he has mud.
Do you know whether you have mud
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3Slide42
No, I don’t know.
Knowing Mud
Do you know whether you have mud
It becomes common knowledge that
k
≥2
.
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3
On the other hand, if boy
1
does not learn that he has mud. Slide43
Yes, I have mud
Knowing Mud
If this were the actual universe,
then boy
2
learns that he has mud.
Do you know whether you have mud
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3Slide44
No, I don’t know.
Knowing Mud
Do you know whether you have mud
It becomes common knowledge that
k
≥3
.
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3
On the other hand, if boy
2
does not learn that he has mud. Slide45
Yes, I have mud
Knowing Mud
If this were the actual universe,
then boy
3
learns that he has mud.
Do you know whether you have mud
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3Slide46
No, I don’t know.
Knowing Mud
Do you know whether you have mud
It becomes common knowledge that
k
≥4
.
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3
On the other hand, if boy
3
does not learn that he has mud. Slide47
Yes, I have mud
Knowing Mud
If this were the actual universe,
then boy
4
learns that he has mud.
Do you know whether you have mud
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3Slide48
Knowing Mud
To each boy, the man whispers either
Some of you have
mud on your foreheads
The sky is blue
Or
k
=0
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3Slide49
Knowing Mud
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
3
Boy
1
told
has mud
Boy
1
told
no mud
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
1
Boy
1
told
sky blue
Boy
1
told
sky blue
2
Yes, I have mud
If this were the actual universe,
then boy
1
learns that he has mud.
Do you know whether you have mud
1Slide50
Knowing Mud
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
3
Boy
1
told
has mud
Boy
1
told
no mud
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
1
Boy
1
told
sky blue
Boy
1
told
sky blue
2
No, I don’t know.
Do you know whether you have mud
1Slide51
Knowing Mud
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
2
1
2
3
5
7
6
8
4
4
1
2
3
5
7
6
8
4
1
1
2
3
5
7
6
8
4
3
Boy
1
told
has mud
Boy
1
told
no mud
1
2
3
5
7
6
8
4
1
2
3
5
7
6
8
4
1
Boy
1
told
sky blue
Boy
1
told
sky blue
2
No, I don’t know
and never will.
Do you know whether you have mud
If this were the actual universe,
then boy
2
does
not
learn that he has mud. Slide52Slide53Slide54Slide55
Knowing Mud
Lots of fun!
The End