Distributed Systems mud on forehead & common knowledge

Grad Algorithms Distributed Systems mud on forehead & common knowledge Jeff Edmonds York University • Knowing Mud Problem • Boys Learn Common Knowledge • Formal Definition of "Knowing" • Applying Formal Definition to Mud Problem COSC 6111

Common Knowledge To be happy, it is important • For you to be a good person • For you to know you are a good person • For your mom to know you are a good person • For 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.

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.

Knowing Mud Man arrives and announces, Some of you have mud on your foreheads The boys, seeing other boys with mud, are not impressed.

Knowing Mud Each boy in unison Each boy in unison Each boy in unison No No No Man arrives and announces, Some of you have mud on your foreheads Do you know whether you have mud 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?

Knowing Mud Do you want to bet? When might a boy know? And so on …. Does a boy ever know, or do they say “No” forever?

Knowing Mud k = # of boys with mud =1 Some of you have mud on your foreheads I see no mud. Someone has mud. It must be me k1 B1 Do you know whether you have mud Don’t know Yes, I have mud Do you know whether you have mud Yes, I have no mud

Knowing Mud k = # of boys with mud =1 Some of you have mud on your foreheads I see no mud. Someone has mud. It must be me k1 B2 B1 Do you know whether you have mud Yes, I have mud k=1  Boy with mud says yes first round. First round boys all say no  k2

Knowing Mud Each boy in unison No k = 2 Some of you have mud on your foreheads Do you know whether you have mud B2 k2 k1 I see k=1 mud. k2 has mud. I must have mud B1

Knowing Mud Each boy in unison No k = 2 Some of you have mud on your foreheads Do you know whether you have mud B2 k2 B1 If I had no mud, then B2 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

Knowing Mud Each boy in unison No k = 2 Some of you have mud on your foreheads Do you know whether you have mud B2 k2 I must have mud B1 Do you know whether you have mud Yes, I have mud

Knowing Mud Each boy in unison No k = 2 Some of you have mud on your foreheads Do you know whether you have mud B2 k2 I must have mud B1 k=2  Boy with mud says yes second round. Second round boys all say no  k3 by induction: tth round boys all say no  kt+1

Knowing Mud … tth round Each boy in unison No k = t+1 Do you know whether you have mud kt+1 I see k=t mud. kt+1 has mud. I must have mud Do you know whether you have mud Yes, I have mud tth round boys all say no  kt+1

Knowing Mud … tth round Each boy in unison No k = t+1 Do you know whether you have mud kt+1 I see k=t mud. kt+1 has mud. I must have mud k=t+1  Boy with mud says yes t+1st round. t+1st round boys all say no  kt+2 Induction complete

Knowing Mud Each boy in unison Each boy in unison No No Forever Man arrives and announces, The sky is blue 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?

Knowing Mud Each boy in unison Each boy in unison No No Forever Man arrives and whispers in each boy ear Some of you have mud on your foreheads 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?

Knowing Mud Man arrives and announces, Some of you have mud on your foreheads The boys, seeing other boys with mud, are not impressed. But they must be learning something! But what??

Knowing Mud I give you common knowledge that some of you have mud on your foreheads I see no mud. Someone has mud. It must be me 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 mud The boy who answers in t+1stround when sees t mud learns that … The boy who answers in 3rd round when sees 2 mud learns that The boy who answers in 2nd round when sees 1 mud learns that The boy who answers in 1st round when sees 0 mud learns that “Someone has mud”

Knowing Edge labeled A between universes indicates player A can’t differentiate between them. Node for Universe in which v=2 sky is blue Node forUniverse in which v=1sky is blue A v? • 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.

Knowing • 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. Edge labeled A between universes indicates player Acan’t differentiate between them. v=2 A v=1 v=1 v=3 A A A A A A A v=1 v=2 v=2 v=1 v=1 A A

Knowing • Equivalently, we can circle the undifferentiable universes. • One of the universe is the Actual Universe. • Player Aconsiders a universe possibleif he cannot differentiate it from the actual universe. • If the set is bigger, Player A has more uncertainty. A A A v=2 v=1 v=1 v=3 v=1 v=2 v=2 v=1 v=1

Knowing We say “Player Aknows fact F” if F is true in every universe that A considers possible. We write KAF If this is the actual universe,then KA v=1 A A A v=2 v=1 v=1 v=3 v=1 v=2 v=2 v=1 v=1

Knowing We say “Player Aknows fact F” if F is true in every universe that A considers possible. We write KAF If this is the actual universe,then KA v=2 A A A v=2 v=1 v=1 v=3 v=1 v=2 v=2 v=1 v=1

Knowing We say “Player Aknows fact F” if F is true in every universe that A considers possible. We write KAF If this is the actual universe,then KA v A A A v=2 v=1 v=1 v=3 v=1 v=2 v=2 v=1 v=1

Knowing We say “Player Aknows fact F” if F is true in every universe that A considers possible. We write KAF We say “Fact F is true” if F is true in the actualUniverse. Proof: A considers the actual universe possibleand hence F is true in it. Lemma: KAF  F A A A v=2 v=1 v=1 v=3 v=1 v=2 v=2 v=1 v=1

Knowing 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.) A A A v=2 v=1 v=1 v=3 v=1 v=2 v=2 v=1 v=1

Knowing Player B can't differentiate between these universes. Assume that the players know each other so wellthat they know which universes they can distinguishand hence know the “graph”. B A A A v=1 v=1 v=1 v=3 v=1 v=1 v=2 v=1 v=1

Knowing In all of the universes that B can't differentiate between, the fact KA v=1 is true. Therefore, KB KA v=1i.e. Player B knows Player A knows v=1. Lemma: KB KAF  KB F Proof: In every universe B considers possible,A knows F and hence F must be true. B A A A v=1 v=1 v=1 v=3 v=1 v=1 v=2 v=1 v=1

Knowing • What do players know in this case? • KAv=1  KBv=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, KB KAv • Does KB KAv and  KBv contradict Lemma? • No, because v is not a “fact”. B A A A v=2 v=1 v=1 v=3 v=1 v=2 v=2 v=1 v=1

Knowing • KAF: • if F is true in every universe reachable from the actual universe by an edge labeled A. A F F F F

Knowing • KBKAF: • 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. A A A F F F F F F F F F F F F B F

Knowing • KCKBKAF: • if KBKAF 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 CBA. A A A A A A F F F F F F F F F F F F F F F F F F F F F F F F B B C F F F

Knowing • Common Knowledge ofF: • if F is true in every universe reachable from the actual universe by a path with any labels. A A A A A A F F F F F F F F F F F F F F F F F F F F F F F F B B C F F F

Knowing Mud 1 2 8 7 3 4 6 5 Universe with mud on the foreheads of boys 1-4 and no mud on boys 5-8. 4 5 6 8 7 1 2 3

Knowing Mud 1 1 2 2 8 8 7 7 3 3 4 4 6 6 5 5 Boy1 can’t differentiate between these universes. 1

Knowing Mud 1 1 1 1 1 1 2 2 2 2 2 2 8 8 8 8 8 8 7 7 7 7 7 7 3 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 5 5 5 5 5 5 2 4 3 1 1

Knowing Mud 1 1 1 1 1 2 2 2 2 2 8 8 8 8 8 k=0 7 7 7 3 3 7 3 7 3 3 4 4 4 4 4 6 6 6 6 6 5 5 2 5 5 5 4 3 1 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.

Knowing Mud K4K3 K2K1k≥1 K2K1k≥1  K1 k≥1 K3 K2K1k≥1 1 1 1 1 1 2 2 2 2 2 8 8 8 8 8 k=0 7 7 7 3 3 7 3 7 3 3 4 4 4 4 4 6 6 6 6 6 5 5 2 5 5 5 4 3 1 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.

Knowing Mud 1 1 1 1 1 2 2 2 2 2 8 8 8 8 8 k=0 7 7 7 3 3 7 3 7 3 3 4 4 4 4 4 6 6 6 6 6 5 5 2 5 5 5 4 3 1 Some of you have mud on your foreheads It becomes common knowledge that k≥1.

Knowing Mud 1 1 1 1 1 2 2 2 2 2 8 8 8 8 8 k=0 7 7 7 3 3 7 3 7 3 3 4 4 4 4 4 6 6 6 6 6 5 5 2 5 5 5 4 3 1 If this were the actual universe, then boy1 learns that he has mud. Do you know whether you have mud Yes, I have mud

Knowing Mud 1 1 1 1 1 2 2 2 2 2 8 8 8 8 8 k=0 7 7 7 3 3 7 3 7 3 3 4 4 4 4 4 6 6 6 6 6 5 5 2 5 5 5 4 3 1 On the other hand, if boy1does not learn that he has mud. Do you know whether you have mud No, I don’t know. It becomes common knowledge that k≥2.

Knowing Mud To each boy, the man whispers either Some of you have mud on your foreheads Or The sky is blue 1 1 1 1 1 2 2 2 2 2 8 8 8 8 8 k=0 7 7 7 3 3 7 3 7 3 3 4 4 4 4 4 6 6 6 6 6 5 5 2 5 5 5 4 3 1

Knowing Mud 1 1 2 2 8 8 7 7 3 3 4 4 6 6 5 5 2 2 1 1 1 1 1 2 2 2 2 2 8 8 8 8 8 Boy1 toldno mud Boy1 toldhas mud 7 7 7 3 3 7 3 3 7 3 4 4 4 4 4 6 6 6 6 6 5 5 5 5 5 4 3 1 1 Boy1 toldsky blue Boy1 told sky blue If this were the actual universe, then boy1 learns that he has mud. Do you know whether you have mud Yes, I have mud

Knowing Mud 1 1 2 2 8 8 7 7 3 3 4 4 6 6 5 5 2 2 1 1 1 1 1 2 2 2 2 2 8 8 8 8 8 Boy1 toldno mud Boy1 toldhas mud 7 7 7 3 3 7 3 3 7 3 4 4 4 4 4 6 6 6 6 6 5 5 5 5 5 4 3 1 1 Boy1 toldsky blue Boy1 told sky blue Do you know whether you have mud No, I don’t know.

Knowing Mud 1 1 2 2 8 8 7 7 3 3 4 4 6 6 5 5 2 2 1 1 1 1 1 2 2 2 2 2 8 8 8 8 8 Boy1 toldno mud Boy1 toldhas mud 7 7 7 3 3 7 3 3 7 3 4 4 4 4 4 6 6 6 6 6 5 5 5 5 5 4 3 1 1 Boy1 toldsky blue Boy1 told sky blue If this were the actual universe, then boy2 does not learn that he has mud. Do you know whether you have mud No, I don’t knowand never will.

Distributed Systems mud on forehead & common knowledge