Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Robert knows whether Albert's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 4) ] Agentc knows that (Agentd knows that 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 4) ] 0

Hypothesis: Spencer can now know that Albert can know that Robert's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Vicki knows whether Gayle's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether 1) ] Agentb knows that (Agentc knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether 1) ] 0

Hypothesis: Jacqueline can now know that Vicki can know that Vicki's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Courtney knows whether someone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether (1|2|3|4)) ] (Agentb knows whether 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether (1|2|3|4)) ] 0

Hypothesis: Nathan can now know whether Courtney is thirsty.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Cristina's card is shown to David. Alana's card is shown to David. David's card is shown to Cristina. David's card is shown to Alana. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1 Agentc:1 VALID? [ ! (1|2|3) ] (Agentb knows whether (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1 Agentc:1 VALID? [ ! (1|2|3) ] 0

Hypothesis: Cristina can now know whether or not everyone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Tony is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! 1 ] (Agenta knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! 1 ] 0

Hypothesis: Tony can now know that everyone is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that David's forehead is not muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(1) ] Agentb knows that (Agentc knows that (1|2|3))

Hypothesis: Eddie can now know that Angela can know that someone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Lawrence's card is shown to Shawn. Shawn's card is shown to Jimmie. Lawrence's card is shown to Jimmie. Jimmie's card is shown to Lawrence. Lawrence's card is shown to Lawrence. It is publicly announced that someone picked a red card. It is publicly announced that Shawn does not know whether or not everyone picked a red card. It is publicly announced that Lawrence picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:4 Agentb:1,4 Agentc:3,4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] [ ! 4 ] Agentc knows whether (Agenta knows that (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:4 Agentb:1,4 Agentc:3,4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] [ ! 4 ] 0

Hypothesis: Jimmie can now know whether Shawn can know that someone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] (Agenta knows whether (1&2&3))

Hypothesis: Christine can now know whether or not everyone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Katerine knows whether someone's forehead is muddy. It is publicly announced that Gerald's forehead is not muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! ~(2) ] (Agentc knows whether 1)

Hypothesis: Katerine can now know whether Guillermo's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Ronald knows whether William's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 1) ] Agenta knows that (Agentb knows that (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: William can now know that Ronald can know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy.

Hypothesis: Lenny can now know that nobody's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that everyone is thirsty.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! (1&2) ] (Agentb knows that (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! (1&2) ] 0

Hypothesis: Helga can now know that nobody is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Jose does not know whether or not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] (Agentd knows that (~(1|2|3|4)))

Hypothesis: Clark can now know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Crystal knows whether Crystal is thirsty. It is publicly announced that Crystal does not know whether Brian is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 3) ] [ ! ~(Agentc knows whether 2) ] Agenta knows that (Agentc knows that (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 3) ] [ ! ~(Agentc knows whether 2) ] 0

Hypothesis: Jerry can now know that Crystal can know that someone is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Joshua's card is shown to Ilda. Ilda's card is shown to Joshua. Joshua's card is shown to Joshua. Ilda's card is shown to Michael. Joshua's card is shown to Michael. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2 Agentb:1,2 Agentc:1,2 VALID? [ ! (1|2|3|4) ] (Agentb knows whether 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2 Agentb:1,2 Agentc:1,2 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Joshua can now know whether Michael picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Patricia knows whether Sandra's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether 1) ] (Agentc knows that (~(1|2|3)))

Hypothesis: Patricia can now know that nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Kevin knows whether Kevin's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether 3) ] Agenta knows that (Agentb knows whether 1)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether 3) ] 0

Hypothesis: John can now know that Michael can know whether John's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). William's card is shown to William. Teresa's card is shown to William. Delphine's card is shown to Teresa. Kevin's card is shown to Delphine. Teresa's card is shown to Delphine. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1,3 Agentb:4 Agentc:2,3 VALID? [ ! (1|2|3|4) ] (Agentc knows whether (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,3 Agentb:4 Agentc:2,3 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Teresa can now know whether or not everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Juanita's card is shown to Timmy. Delores's card is shown to Timmy. Delores's card is shown to Delores. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1,4 Agentb:4 VALID? [ ! (1|2|3|4) ] (Agenta knows that (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,4 Agentb:4 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Juanita can now know that someone picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Gerald does not know whether Debbie is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 4) ] Agenta knows that (Agentc knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 4) ] 0

Hypothesis: Steven can now know that Gerald can know that Steven is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Jimmy's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! 1 ] Agentc knows that (Agenta knows that (~(1|2|3|4)))

Hypothesis: Pedro can now know that Jimmy can know that nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Sara does not know whether Sara's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether 1) ] Agentb knows whether (Agentc knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Larry can now know whether Fredrick can know that Sara's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Mark knows whether someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that Carlos knows whether Mark's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether (1|2|3|4)) ] [ ! (1&2&3&4) ] [ ! (Agentd knows whether 2) ] (Agenta knows that (1&2&3&4))

Hypothesis: Regina can now know that everyone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Joyce does not know whether or not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] Agentc knows whether (Agentd knows that 2)

Hypothesis: Larry can now know whether Eric can know that Joyce's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Amy's card is shown to Melissa. Christina's card is shown to Amy. Melissa's card is shown to Christina. Christina's card is shown to Christina. It is publicly announced that someone picked a red card. It is publicly announced that Christina does not know whether Amy picked a red card. It is publicly announced that Melissa does not know whether Christina picked a red card. It is publicly announced that not everyone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2 Agentb:3 Agentc:1,3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agenta knows whether 3) ] [ ! (~(1&2&3)) ] (Agentc knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:2 Agentb:3 Agentc:1,3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agenta knows whether 3) ] [ ! (~(1&2&3)) ] 0

Hypothesis: Christina can now know that everyone picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Latonya knows whether or not nobody's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (~(1|2|3))) ] (Agentc knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (~(1|2|3))) ] 0

Hypothesis: Douglas can now know that Douglas's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Ernestina's card is shown to Ernestina. Anthony's card is shown to Daniel. It is publicly announced that someone picked a red card. It is publicly announced that Daniel does not know whether Daniel picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 2) ] Agenta knows that (Agentb knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 2) ] 0

Hypothesis: Ernestina can now know that Daniel can know that Anthony picked a red card.


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Bryan knows whether someone's forehead is muddy. It is publicly announced that Shanae knows whether Shanae's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Bryan knows whether someone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agentb knows whether (1|2)) ] [ ! (Agenta knows whether 1) ] [ ! (~(1&2)) ] [ ! (Agentb knows whether (1|2)) ] (Agentb knows that (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agentb knows whether (1|2)) ] [ ! (Agenta knows whether 1) ] [ ! (~(1&2)) ] [ ! (Agentb knows whether (1|2)) ] 0

Hypothesis: Bryan can now know that nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Erik knows whether or not nobody's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (~(1|2|3))) ] (Agentc knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether (~(1|2|3))) ] 0

Hypothesis: James can now know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Sheryll knows whether or not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether (1&2&3&4)) ] Agentb knows whether (Agenta knows whether 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether (1&2&3&4)) ] 0

Hypothesis: Marilyn can now know whether Sheryll can know whether Jean is thirsty.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Clyde's card is shown to Clyde. Clyde's card is shown to William. It is publicly announced that someone picked a red card. It is publicly announced that Clyde knows whether Clyde picked a red card. It is publicly announced that Clyde knows whether William picked a red card.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agenta knows whether 1) ] [ ! (Agenta knows whether 2) ] (Agentb knows that 2)

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agenta knows whether 1) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: William can now know that William picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Paul does not know whether Brian is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 2) ] Agentc knows that (Agenta knows whether (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 2) ] 0

Hypothesis: Paul can now know that Linda can know whether or not nobody is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that James is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! 4 ] Agentd knows that (Agenta knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! 4 ] 0

Hypothesis: James can now know that Michelle can know that Michelle is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Sharon knows whether Jimmy's forehead is muddy. It is publicly announced that Sharon knows whether Jimmy's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether 2) ] [ ! (Agenta knows whether 2) ] Agenta knows whether (Agentc knows whether 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether 2) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: Sharon can now know whether Sandra can know whether Sandra's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Linda's card is shown to Jerry. Penny's card is shown to Jerry. Jerry's card is shown to Annette. Penny's card is shown to Annette. Jerry's card is shown to Linda. Annette's card is shown to Linda. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:3,4 Agentb:1,4 Agentc:1,2 VALID? [ ! (1|2|3|4) ] Agentc knows that (Agenta knows whether (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:3,4 Agentb:1,4 Agentc:1,2 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Linda can now know that Jerry can know whether someone picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that David knows whether or not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Alan's forehead is muddy.

Hypothesis: Alan can now know that someone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Helen is not thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(2) ] (Agenta knows whether 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(2) ] 0

Hypothesis: Leslie can now know whether Steven is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Brett does not know whether Mary is thirsty. It is publicly announced that Brett does not know whether or not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agentc knows whether (1&2&3)) ] (Agenta knows whether (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agentc knows whether (1&2&3)) ] 0

Hypothesis: Dorothy can now know whether or not nobody is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Stephen knows whether Stephen's forehead is muddy. It is publicly announced that everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether 1) ] [ ! (1&2&3&4) ] Agenta knows that (Agentb knows whether 1)

Hypothesis: Stephen can now know that Christopher can know whether Stephen's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Sharon's card is shown to Sharon. Donald's card is shown to Sharon. Sharon's card is shown to Donald. Dorothy's card is shown to Donald. Donald's card is shown to Donald. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:1,3 Agentb:1,2,3 VALID? [ ! (1|2|3) ] (Agentb knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:1,3 Agentb:1,2,3 VALID? [ ! (1|2|3) ] 0

Hypothesis: Dorothy can now know that everyone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Paul knows whether Paul is thirsty. It is publicly announced that Paul is not thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether 3) ] [ ! ~(3) ] (Agentb knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether 3) ] [ ! ~(3) ] 0

Hypothesis: Neil can now know that Donna is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Mildred does not know whether or not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] (Agentb knows that (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether (1&2&3&4)) ] 0

Hypothesis: Alfred can now know that someone is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Dorothy's card is shown to Dorothy. Warren's card is shown to Dorothy. Dorothy's card is shown to Yolanda. Warren's card is shown to Yolanda. Yolanda's card is shown to Brian. It is publicly announced that someone picked a red card. It is publicly announced that Dorothy knows whether or not everyone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1,4 Agentb:1,4 Agentc:2 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether (1&2&3&4)) ] (Agentc knows that 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,4 Agentb:1,4 Agentc:2 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether (1&2&3&4)) ] 0

Hypothesis: Brian can now know that Warren picked a red card.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Steve's card is shown to Steve. Jessica's card is shown to Steve. Steve's card is shown to Harriet. Harriet's card is shown to Harriet. Jessica's card is shown to Harriet. Steve's card is shown to Jessica. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:1,3 Agentb:1,2,3 Agentc:1 VALID? [ ! (1|2|3) ] Agentc knows whether (Agentb knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1,3 Agentb:1,2,3 Agentc:1 VALID? [ ! (1|2|3) ] 0

Hypothesis: Jessica can now know whether Harriet can know that Harriet picked a red card.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). William's card is shown to Edward. Edward's card is shown to Michael. William's card is shown to Barbara. Barbara's card is shown to Barbara. It is publicly announced that someone picked a red card. It is publicly announced that Edward does not know whether Edward picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2 Agentb:1 Agentc:2,4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 1) ] Agentc knows that (Agentb knows that (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:2 Agentb:1 Agentc:2,4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Michael can now know that William can know that everyone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Rosemary knows whether Rosemary's forehead is muddy. It is publicly announced that Ashley knows whether or not everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 1) ] [ ! (Agentc knows whether (1&2&3)) ] Agentc knows that (Agentb knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 1) ] [ ! (Agentc knows whether (1&2&3)) ] 0

Hypothesis: Ashley can now know that Richard can know that nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

Hypothesis: Angela can now know whether or not everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that Bessie's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (1&2&3) ] [ ! 1 ] Agenta knows whether (Agentc knows whether (1&2&3))

Hypothesis: Bessie can now know whether Francis can know whether or not everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Madonna's forehead is not muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(1) ] Agentc knows that (Agenta knows whether (1|2|3))

Hypothesis: Alva can now know that Madonna can know whether someone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

Hypothesis: Steve can now know that George can know that everyone's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that everyone is thirsty. It is publicly announced that Ashley knows whether Zachary is thirsty.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! (1&2) ] [ ! (Agentb knows whether 1) ] (Agenta knows that 2)

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! (1&2) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Zachary can now know that Ashley is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Darla knows whether Talia's forehead is muddy. It is publicly announced that Jimmie does not know whether Jimmie's forehead is muddy. It is publicly announced that Mattie knows whether Talia's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 1) ] [ ! ~(Agentd knows whether 4) ] [ ! (Agentb knows whether 1) ] Agentd knows whether (Agenta knows whether 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 1) ] [ ! ~(Agentd knows whether 4) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Jimmie can now know whether Talia can know whether Talia's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Sterling knows whether Anna's forehead is muddy. It is publicly announced that Anna does not know whether Anna's forehead is muddy. It is publicly announced that Anna does not know whether Anna's forehead is muddy. It is publicly announced that Anna knows whether Sterling's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agenta knows whether 2) ] [ ! ~(Agentb knows whether 2) ] [ ! ~(Agentb knows whether 2) ] [ ! (Agentb knows whether 1) ] Agentb knows that (Agenta knows that (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agenta knows whether 2) ] [ ! ~(Agentb knows whether 2) ] [ ! ~(Agentb knows whether 2) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Anna can now know that Sterling can know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (~(1&2&3&4)) ] Agentc knows that (Agenta knows whether 4)

Hypothesis: Diane can now know that Kevin can know whether Valentin is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Bryan knows whether Barbara's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] [ ! (Agentc knows whether 2) ] Agenta knows that (Agentb knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] [ ! (Agentc knows whether 2) ] 0

Hypothesis: Kelley can now know that Barbara can know that Bryan's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Lee's forehead is not muddy. It is publicly announced that Sandra's forehead is muddy. It is publicly announced that Sandra's forehead is muddy.

Hypothesis: Sandra can now know whether or not nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Darrell knows whether Catherine's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 2) ] (Agentb knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: Catherine can now know that Catherine's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Victoria knows whether Daniel's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 1) ] (Agenta knows that 3)

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Daniel can now know that Karen's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Carl knows whether or not nobody's forehead is muddy.

Hypothesis: Joel can now know that everyone's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (1|2) ] Agentb knows that (Agenta knows whether 2)

Hypothesis: Phyllis can now know that Charles can know whether Phyllis's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Deborah's card is shown to Deborah. James's card is shown to Deborah. Deborah's card is shown to Margaret. James's card is shown to Margaret. Margaret's card is shown to James. It is publicly announced that someone picked a red card. It is publicly announced that James does not know whether Deborah picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:1,3 Agentb:1,3 Agentc:2 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 1) ] Agenta knows whether (Agentc knows whether 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1,3 Agentb:1,3 Agentc:2 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 1) ] 0

Hypothesis: Deborah can now know whether James can know whether Margaret picked a red card.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Josephine's card is shown to Josephine. David's card is shown to Patricia. Patricia's card is shown to Patricia. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2,3 VALID? [ ! (1|2|3) ] (Agenta knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2,3 VALID? [ ! (1|2|3) ] 0

Hypothesis: Josephine can now know that Josephine picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Theodore knows whether or not everyone's forehead is muddy. It is publicly announced that Theodore does not know whether Theodore's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1&2&3)) ] [ ! ~(Agentc knows whether 3) ] (Agentb knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1&2&3)) ] [ ! ~(Agentc knows whether 3) ] 0

Hypothesis: Rebecca can now know that nobody's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Ross's card is shown to Arletta. Arletta's card is shown to Ross. Ross's card is shown to Ross. It is publicly announced that someone picked a red card. It is publicly announced that Ross picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! 2 ] Agentb knows that (Agenta knows whether (1|2))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! 2 ] 0

Hypothesis: Ross can now know that Arletta can know whether someone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Helen knows whether or not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1&2&3)) ] (Agenta knows whether (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1&2&3)) ] 0

Hypothesis: Percy can now know whether someone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Melisa does not know whether or not everyone is thirsty. It is publicly announced that Melisa does not know whether Louise is thirsty. It is publicly announced that Leighann does not know whether Louise is thirsty. It is publicly announced that not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] [ ! ~(Agentd knows whether 3) ] [ ! ~(Agentb knows whether 3) ] [ ! (~(1&2&3&4)) ] (Agentd knows that 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] [ ! ~(Agentd knows whether 3) ] [ ! ~(Agentb knows whether 3) ] [ ! (~(1&2&3&4)) ] 0

Hypothesis: Melisa can now know that Leighann is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] (Agenta knows whether (1&2&3))

Hypothesis: Louis can now know whether or not everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Etta's card is shown to Etta. Jack's card is shown to Jack. Jack's card is shown to Christopher. Etta's card is shown to Joseph. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:2 Agentd:1 VALID? [ ! (1|2|3|4) ] (Agentc knows whether 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:2 Agentd:1 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Christopher can now know whether Jack picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Brian does not know whether Brian's forehead is muddy. It is publicly announced that Fannie does not know whether Fannie's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether 1) ] [ ! ~(Agentb knows whether 2) ] Agentc knows whether (Agentb knows that (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether 1) ] [ ! ~(Agentb knows whether 2) ] 0

Hypothesis: James can now know whether Fannie can know that someone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that William does not know whether Thomas is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 2) ] (Agentd knows whether 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 2) ] 0

Hypothesis: Maria can now know whether Jerry is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that everyone's forehead is muddy. It is publicly announced that Joe's forehead is muddy.

Hypothesis: Kerry can now know that nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Guadalupe's card is shown to Guadalupe. Ronald's card is shown to Guadalupe. Michael's card is shown to Guadalupe. Michael's card is shown to Arthur. Guadalupe's card is shown to Ronald. Arthur's card is shown to Michael. Ronald's card is shown to Michael. It is publicly announced that someone picked a red card. It is publicly announced that Ronald picked a black card.

VARS 1,2,3,4 LAW Top OBS Agenta:1,3,4 Agentb:4 Agentc:1 Agentd:2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(3) ] Agentb knows whether (Agentd knows that (1&2&3&4))

Hypothesis: Arthur can now know whether Michael can know that everyone picked a red card.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Tony's card is shown to Chris. Susana's card is shown to Susana. Tony's card is shown to Susana. Tony's card is shown to Tony. It is publicly announced that someone picked a red card. It is publicly announced that Tony does not know whether Susana picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:2,3 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] Agenta knows that (Agentc knows that (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:2,3 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] 0

Hypothesis: Chris can now know that Tony can know that someone picked a red card.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Stephen's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! 3 ] (Agentb knows that 1)

Hypothesis: Robert can now know that Kenneth's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Olga knows whether Viola's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 1) ] Agenta knows that (Agentd knows that 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Viola can now know that Paul can know that Olga's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Janice does not know whether Jesse is thirsty. It is publicly announced that Janice does not know whether or not everyone is thirsty.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether 1) ] [ ! ~(Agentb knows whether (1&2)) ] Agenta knows that (Agentb knows whether (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether 1) ] [ ! ~(Agentb knows whether (1&2)) ] 0

Hypothesis: Jesse can now know that Janice can know whether or not nobody is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that James is thirsty. It is publicly announced that Joseph does not know whether Angela is thirsty. It is publicly announced that James knows whether James is thirsty. It is publicly announced that Joseph does not know whether or not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! 3 ] [ ! ~(Agenta knows whether 2) ] [ ! (Agentc knows whether 3) ] [ ! ~(Agenta knows whether (1&2&3)) ] (Agenta knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! 3 ] [ ! ~(Agenta knows whether 2) ] [ ! (Agentc knows whether 3) ] [ ! ~(Agenta knows whether (1&2&3)) ] 0

Hypothesis: Joseph can now know that everyone is thirsty.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (~(1&2&3)) ] Agenta knows that (Agentb knows that (1&2&3))

Hypothesis: Debbie can now know that Elaine can know that everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that someone is thirsty. It is publicly announced that Robert does not know whether Maria is thirsty. It is publicly announced that not everyone is thirsty. It is publicly announced that Maria does not know whether Carol is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! (~(1&2&3)) ] [ ! ~(Agentb knows whether 1) ] (Agenta knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! (~(1&2&3)) ] [ ! ~(Agentb knows whether 1) ] 0

Hypothesis: Carol can now know that Carol is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Mollie knows whether Mollie is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 3) ] (Agentb knows whether 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 3) ] 0

Hypothesis: Maria can now know whether Vance is thirsty.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Homer knows whether or not nobody's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (~(1|2|3))) ] (Agenta knows that (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (~(1|2|3))) ] 0

Hypothesis: Homer can now know that everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Sam does not know whether Tammy is thirsty. It is publicly announced that Janet does not know whether Tammy is thirsty. It is publicly announced that Tammy does not know whether Sam is thirsty. It is publicly announced that Sam is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether 3) ] [ ! 3 ] (Agenta knows whether (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 2) ] [ ! ~(Agenta knows whether 2) ] [ ! ~(Agentb knows whether 3) ] [ ! 3 ] 0

Hypothesis: Janet can now know whether someone is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Betty's card is shown to Betty. Herbert's card is shown to Betty. Alex's card is shown to William. Betty's card is shown to Herbert. William's card is shown to Alex. Herbert's card is shown to Alex. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1,3 Agentb:4 Agentc:1 Agentd:2,3 VALID? [ ! (1|2|3|4) ] Agentd knows that (Agentc knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,3 Agentb:4 Agentc:1 Agentd:2,3 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Alex can now know that Herbert can know that Betty picked a red card.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Carlos's card is shown to Carlos. John's card is shown to Margery. Carlos's card is shown to Margery. Margaret's card is shown to Margaret. It is publicly announced that someone picked a red card. It is publicly announced that Carlos does not know whether or not everyone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2 Agentb:1,2 Agentc:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] (Agenta knows that (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:2 Agentb:1,2 Agentc:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether (1&2&3&4)) ] 0

Hypothesis: John can now know that everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that someone is thirsty. It is publicly announced that Roy knows whether Roy is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (1|2|3|4) ] [ ! (Agenta knows whether 1) ] (Agenta knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (1|2|3|4) ] [ ! (Agenta knows whether 1) ] 0

Hypothesis: Roy can now know that Roy is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Sylvia knows whether someone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1|2|3)) ] Agentc knows that (Agenta knows whether (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1|2|3)) ] 0

Hypothesis: Sylvia can now know that Floyd can know whether someone is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Michael knows whether or not nobody's forehead is muddy. It is publicly announced that Tamara's forehead is not muddy.

Hypothesis: Michael can now know whether Deanna's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Frances's card is shown to Carl. Carl's card is shown to Carl. Edith's card is shown to Carl. Frances's card is shown to Edith. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1,3,4 Agentb:1 VALID? [ ! (1|2|3|4) ] (Agentb knows that (1&2&3&4))

Hypothesis: Bill can now know that everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Johnnie knows whether someone is thirsty. It is publicly announced that Joseph does not know whether Johnnie is thirsty. It is publicly announced that Joseph is thirsty. It is publicly announced that Lidia does not know whether Johnnie is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! ~(Agentb knows whether 3) ] [ ! 2 ] [ ! ~(Agentd knows whether 3) ] (Agentb knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether (1|2|3|4)) ] [ ! ~(Agentb knows whether 3) ] [ ! 2 ] [ ! ~(Agentd knows whether 3) ] 0

Hypothesis: Joseph can now know that Clarence is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

Hypothesis: Hubert can now know whether Edwin can know that everyone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Ashley's forehead is not muddy.

Hypothesis: Ashley can now know that Ashley's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Sandra's card is shown to Sandra. Josephine's card is shown to Edith. It is publicly announced that someone picked a red card. It is publicly announced that Josephine does not know whether Edith picked a red card. It is publicly announced that someone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 3) ] [ ! (1|2|3) ] (Agenta knows whether (1&2&3))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 3) ] [ ! (1|2|3) ] 0

Hypothesis: Sandra can now know whether or not everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Maria's card is shown to Maria. Cherly's card is shown to Patrick. Patrick's card is shown to Cherly. It is publicly announced that someone picked a red card. It is publicly announced that Patrick does not know whether Timothy picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:4 Agentc:2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] Agentb knows whether (Agenta knows that (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:4 Agentc:2 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] 0

Hypothesis: Patrick can now know whether Maria can know that everyone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Natalie is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! 3 ] Agentb knows whether (Agenta knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! 3 ] 0

Hypothesis: Henry can now know whether Dana can know that Henry is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Thomas's card is shown to Jennifer. Lance's card is shown to Thomas. Jennifer's card is shown to Victoria. Lance's card is shown to Victoria. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2 Agentb:3 Agentc:1,3 VALID? [ ! (1|2|3|4) ] Agenta knows whether (Agentb knows whether (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:2 Agentb:3 Agentc:1,3 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Jennifer can now know whether Thomas can know whether someone picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Maria knows whether someone's forehead is muddy. It is publicly announced that Maria does not know whether or not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1|2|3|4)) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] Agentb knows that (Agenta knows that (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1|2|3|4)) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] 0

Hypothesis: Rex can now know that Earl can know that nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Carolyn knows whether someone's forehead is muddy. It is publicly announced that Diane does not know whether or not everyone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1|2|3)) ] [ ! ~(Agenta knows whether (1&2&3)) ] [ ! (1|2|3) ] Agentb knows that (Agenta knows that (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (1|2|3)) ] [ ! ~(Agenta knows whether (1&2&3)) ] [ ! (1|2|3) ] 0

Hypothesis: Troy can now know that Diane can know that someone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Susan knows whether Joseph's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether 1) ] (Agenta knows that 2)

Hypothesis: Joseph can now know that Peter's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). John's card is shown to Ebony. Ebony's card is shown to John. John's card is shown to Joseph. It is publicly announced that someone picked a red card. It is publicly announced that Ebony picked a red card. It is publicly announced that not everyone picked a red card. It is publicly announced that Joseph picked a black card.

Hypothesis: Ebony can now know that everyone picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Darlene knows whether Darlene's forehead is muddy. It is publicly announced that Darlene's forehead is not muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether 1) ] [ ! ~(1) ] Agentd knows that (Agenta knows whether 1)

Hypothesis: Robert can now know that Darlene can know whether Darlene's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Clyde does not know whether Cindy is thirsty. It is publicly announced that Cindy does not know whether Tina is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 4) ] [ ! ~(Agentd knows whether 3) ] (Agentd knows whether (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 4) ] [ ! ~(Agentd knows whether 3) ] 0

Hypothesis: Cindy can now know whether or not everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (1|2|3|4) ] Agenta knows whether (Agentd knows whether 4)

Hypothesis: Barbara can now know whether Carol can know whether Carol's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Juan knows whether Brenda's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 1) ] (Agenta knows that 1)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 1) ] 0

Hypothesis: Brenda can now know that Brenda's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that William's forehead is not muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(3) ] Agentc knows whether (Agentb knows that 3)

Hypothesis: William can now know whether Jesus can know that William's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Bernadette knows whether Carlos's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 4) ] (Agentc knows that (1&2&3&4))

Hypothesis: Bernadette can now know that everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Angela knows whether or not nobody's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (~(1|2|3))) ] (Agenta knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (~(1|2|3))) ] 0

Hypothesis: Angela can now know that nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that James does not know whether or not everyone is thirsty. It is publicly announced that Ginny does not know whether Thomas is thirsty. It is publicly announced that Ginny does not know whether or not everyone is thirsty. It is publicly announced that James does not know whether Thomas is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agenta knows whether 3) ] (Agenta knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agenta knows whether 3) ] 0

Hypothesis: James can now know that Ginny is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Clyde's forehead is muddy. It is publicly announced that Clyde knows whether someone's forehead is muddy. It is publicly announced that Alberto knows whether Elizabeth's forehead is muddy. It is publicly announced that Richard does not know whether Richard's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! 4 ] [ ! (Agentd knows whether (1|2|3|4)) ] [ ! (Agentb knows whether 1) ] [ ! ~(Agentc knows whether 3) ] Agentb knows whether (Agentc knows whether 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! 4 ] [ ! (Agentd knows whether (1|2|3|4)) ] [ ! (Agentb knows whether 1) ] [ ! ~(Agentc knows whether 3) ] 0

Hypothesis: Alberto can now know whether Richard can know whether Alberto's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Drew knows whether Jeremy's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether 2) ] (Agenta knows that 1)

VARS 0,1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: Drew can now know that Drew's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Sandi's forehead is muddy.

Hypothesis: Julia can now know that Stephen can know that everyone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Anna's forehead is not muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! ~(2) ] Agentc knows whether (Agenta knows whether (~(1|2|3|4)))

Hypothesis: Lyn can now know whether Pamela can know whether or not nobody's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Beverly's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! 3 ] Agenta knows that (Agentc knows whether (~(1|2|3)))

Hypothesis: Brittany can now know that Beverly can know whether or not nobody's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Audrey knows whether Dominique's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether 2) ] (Agenta knows that 2)

Hypothesis: Audrey can now know that Dominique's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Harry knows whether Kim's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 2) ] Agenta knows that (Agentb knows whether 1)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether 2) ] 0

Hypothesis: Harry can now know that Kim can know whether Harry's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Fernando does not know whether Fernando's forehead is muddy. It is publicly announced that Jose does not know whether Jose's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 1) ] [ ! ~(Agentc knows whether 3) ] (Agentd knows whether 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agenta knows whether 1) ] [ ! ~(Agentc knows whether 3) ] 0

Hypothesis: Ann can now know whether Fernando's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that not everyone's forehead is muddy. It is publicly announced that Julie's forehead is muddy.

Hypothesis: Julie can now know that everyone's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Della is thirsty.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! 2 ] Agenta knows that (Agentb knows that (1|2))

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! 2 ] 0

Hypothesis: Olivia can now know that Della can know that someone is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Anna's card is shown to Anna. Anna's card is shown to Roberta. Vennie's card is shown to Roberta. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:1,3 VALID? [ ! (1|2|3|4) ] (Agentb knows whether 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:1,3 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Marlene can now know whether Roberta picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that someone's forehead is muddy. It is publicly announced that Joyce does not know whether or not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] Agentd knows that (Agentc knows that 2)

Hypothesis: Joyce can now know that Becky can know that John's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Michael is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! 4 ] (Agentb knows whether 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! 4 ] 0

Hypothesis: Jayne can now know whether Jayne is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Robert's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! 1 ] Agentd knows that (Agentc knows that 2)

Hypothesis: Annabel can now know that Teresa can know that Nicole's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Zelma's card is shown to Zelma. Tammy's card is shown to Zelma. Lynn's card is shown to Zelma. Zelma's card is shown to Tammy. Tammy's card is shown to Tammy. Bill's card is shown to Tammy. Lynn's card is shown to Tammy. Zelma's card is shown to Bill. Lynn's card is shown to Bill. Zelma's card is shown to Lynn. It is publicly announced that someone picked a red card. It is publicly announced that Lynn does not know whether Tammy picked a red card. It is publicly announced that Lynn does not know whether Bill picked a red card. It is publicly announced that Zelma does not know whether Bill picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,4 Agentb:1,2,3,4 Agentc:1,4 Agentd:1 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether 2) ] [ ! ~(Agentd knows whether 3) ] [ ! ~(Agenta knows whether 3) ] Agentd knows that (Agentc knows whether 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,2,4 Agentb:1,2,3,4 Agentc:1,4 Agentd:1 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether 2) ] [ ! ~(Agentd knows whether 3) ] [ ! ~(Agenta knows whether 3) ] 0

Hypothesis: Lynn can now know that Bill can know whether Lynn picked a red card.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that David does not know whether or not everyone is thirsty. It is publicly announced that David knows whether David is thirsty.

VARS 1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! (Agentb knows whether 2) ] (Agentb knows whether (1&2))

VARS 0,1,2 LAW Top OBS Agenta:1 Agentb:2 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! (Agentb knows whether 2) ] 0

Hypothesis: David can now know whether or not everyone is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Gary does not know whether Judy is thirsty. It is publicly announced that Jessie knows whether or not nobody is thirsty. It is publicly announced that Judy does not know whether Jessie is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 1) ] [ ! (Agentb knows whether (~(1|2|3))) ] [ ! ~(Agenta knows whether 2) ] Agentb knows whether (Agenta knows whether (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether 1) ] [ ! (Agentb knows whether (~(1|2|3))) ] [ ! ~(Agenta knows whether 2) ] 0

Hypothesis: Jessie can now know whether Judy can know whether or not nobody is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Alan does not know whether Della is thirsty. It is publicly announced that Erin does not know whether Della is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agentd knows whether 3) ] Agentd knows whether (Agenta knows that 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agentd knows whether 3) ] 0

Hypothesis: Erin can now know whether Randolph can know that Alan is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] Agentb knows whether (Agentc knows whether 3)

Hypothesis: Christine can now know whether Billy can know whether Billy's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] Agenta knows whether (Agentd knows that 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Burma can now know whether Justin can know that James is thirsty.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Robert's card is shown to Lorraine. Lorraine's card is shown to Robert. Robert's card is shown to Robert. Lorraine's card is shown to Nancy. It is publicly announced that someone picked a red card. It is publicly announced that Robert does not know whether Nancy picked a red card. It is publicly announced that Robert picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2 Agentb:1,2 Agentc:1 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 3) ] [ ! 2 ] Agentc knows that (Agenta knows that 1)

VARS 0,1,2,3 LAW Top OBS Agenta:2 Agentb:1,2 Agentc:1 VALID? [ ! (1|2|3) ] [ ! ~(Agentb knows whether 3) ] [ ! 2 ] 0

Hypothesis: Nancy can now know that Lorraine can know that Lorraine picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Maureen knows whether Maureen's forehead is muddy. It is publicly announced that Richard knows whether Richard's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 3) ] [ ! (Agentb knows whether 2) ] Agentc knows that (Agentd knows that 4)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 3) ] [ ! (Agentb knows whether 2) ] 0

Hypothesis: Maureen can now know that Bill can know that Bill's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Betty knows whether or not everyone's forehead is muddy. It is publicly announced that Betty's forehead is not muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1&2&3)) ] [ ! ~(1) ] (Agentc knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agenta knows whether (1&2&3)) ] [ ! ~(1) ] 0

Hypothesis: Francis can now know that nobody's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Mike does not know whether or not everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] Agentb knows that (Agenta knows whether (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] 0

Hypothesis: Mike can now know that Jack can know whether or not nobody's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Ernest knows whether or not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1&2&3&4)) ] (Agentb knows whether (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentd knows whether (1&2&3&4)) ] 0

Hypothesis: Lenore can now know whether or not nobody is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Joseph does not know whether or not everyone's forehead is muddy. It is publicly announced that Jennifer knows whether someone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] [ ! (Agenta knows whether (1|2|3|4)) ] Agentc knows that (Agentb knows whether 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] [ ! (Agenta knows whether (1|2|3|4)) ] 0

Hypothesis: Wanda can now know that Kristy can know whether Kristy's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Vivian's card is shown to Brad. Vivian's card is shown to Vivian. It is publicly announced that someone picked a red card. It is publicly announced that Brad knows whether Vivian picked a red card. It is publicly announced that Brad does not know whether or not everyone picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether 2) ] [ ! ~(Agenta knows whether (1&2)) ] (Agentb knows that (1&2))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:2 VALID? [ ! (1|2) ] [ ! (Agenta knows whether 2) ] [ ! ~(Agenta knows whether (1&2)) ] 0

Hypothesis: Vivian can now know that everyone picked a red card.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] Agentc knows that (Agentb knows whether 3)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] 0

Hypothesis: Bonnie can now know that John can know whether Bonnie is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that John knows whether Raymond's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether 4) ] (Agentb knows that 4)

Hypothesis: John can now know that Raymond's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Frances knows whether Rachel's forehead is muddy. It is publicly announced that someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 1) ] [ ! (1|2|3) ] (Agentc knows whether 3)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 1) ] [ ! (1|2|3) ] 0

Hypothesis: Kenneth can now know whether Kenneth's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Rachel's card is shown to David. David's card is shown to Rachel. Frank's card is shown to Rachel. Rachel's card is shown to Frank. John's card is shown to Frank. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:2 Agentb:1,3 Agentc:2,4 VALID? [ ! (1|2|3|4) ] (Agentb knows whether 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2 Agentb:1,3 Agentc:2,4 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Rachel can now know whether Frank picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Elizabeth knows whether Elizabeth is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether 1) ] (Agentc knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agenta knows whether 1) ] 0

Hypothesis: Diane can now know that Elizabeth is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Charles's forehead is muddy. It is publicly announced that Pamela knows whether or not nobody's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! 4 ] [ ! (Agentb knows whether (~(1|2|3|4))) ] Agenta knows that (Agentc knows that 4)

Hypothesis: Kenyatta can now know that Kevin can know that Charles's forehead is muddy.


VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] Agenta knows that (Agentb knows that 1)

Hypothesis: Lorenzo can now know that Myron can know that Lorenzo is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Elvira does not know whether or not everyone is thirsty. It is publicly announced that Gertrude does not know whether or not everyone is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether (1&2&3)) ] [ ! ~(Agenta knows whether (1&2&3)) ] (Agenta knows that 2)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agentc knows whether (1&2&3)) ] [ ! ~(Agenta knows whether (1&2&3)) ] 0

Hypothesis: Gertrude can now know that Shauna is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Carrie knows whether Verna's forehead is muddy. It is publicly announced that Joseph knows whether or not everyone's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 4) ] [ ! (Agenta knows whether (1&2&3&4)) ] (Agentd knows that (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 4) ] [ ! (Agenta knows whether (1&2&3&4)) ] 0

Hypothesis: Verna can now know that everyone's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Elizabeth's card is shown to James. Elizabeth's card is shown to Elizabeth. It is publicly announced that someone picked a red card. It is publicly announced that James picked a red card. It is publicly announced that James knows whether James picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:3 Agentb:3 VALID? [ ! (1|2|3) ] [ ! 1 ] [ ! (Agenta knows whether 1) ] (Agentb knows that (~(1|2|3)))

VARS 0,1,2,3 LAW Top OBS Agenta:3 Agentb:3 VALID? [ ! (1|2|3) ] [ ! 1 ] [ ! (Agenta knows whether 1) ] 0

Hypothesis: Mildred can now know that nobody picked a red card.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Charlie's card is shown to Charlie. Charlie's card is shown to Virginia. Virginia's card is shown to Virginia. Darin's card is shown to Virginia. Charlie's card is shown to Darin. Darin's card is shown to Darin. Charlie's card is shown to Norman. It is publicly announced that someone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:1,2,3 Agentc:1,3 Agentd:1 VALID? [ ! (1|2|3|4) ] Agentc knows that (Agentb knows that 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:1,2,3 Agentc:1,3 Agentd:1 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Darin can now know that Virginia can know that Virginia picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Emmett is thirsty. It is publicly announced that Richard is not thirsty. It is publicly announced that Kevin does not know whether Glen is thirsty. It is publicly announced that Emmett does not know whether Kevin is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! 1 ] [ ! ~(4) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agenta knows whether 2) ] Agentd knows that (Agentc knows that (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! 1 ] [ ! ~(4) ] [ ! ~(Agentb knows whether 3) ] [ ! ~(Agenta knows whether 2) ] 0

Hypothesis: Richard can now know that Glen can know that nobody is thirsty.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Yvonne does not know whether or not everyone's forehead is muddy. It is publicly announced that Yvonne does not know whether Yvonne's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] [ ! ~(Agenta knows whether 1) ] (Agenta knows whether 3)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:1,3 Agentc:1,2 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether (1&2&3)) ] [ ! ~(Agenta knows whether 1) ] 0

Hypothesis: Yvonne can now know whether Vincent's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Gary does not know whether John is thirsty. It is publicly announced that Marget knows whether or not nobody is thirsty.

VARS 1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether 2) ] [ ! (Agentc knows whether (~(1|2|3))) ] (Agentb knows whether 1)

VARS 0,1,2,3 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 VALID? [ ! (1|2|3) ] [ ! ~(Agenta knows whether 2) ] [ ! (Agentc knows whether (~(1|2|3))) ] 0

Hypothesis: John can now know whether Gary is thirsty.


Premise: There are four persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Delores's card is shown to Charles. Priscilla's card is shown to Alta. Charles's card is shown to Priscilla. Priscilla's card is shown to Priscilla. It is publicly announced that someone picked a red card. It is publicly announced that Priscilla does not know whether or not everyone picked a red card.

VARS 1,2,3,4 LAW Top OBS Agenta:4 Agentb:3 Agentc:1,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] Agentb knows whether (Agentc knows that (1|2|3|4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:4 Agentb:3 Agentc:1,3 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether (1&2&3&4)) ] 0

Hypothesis: Alta can now know whether Priscilla can know that someone picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Christopher's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! 1 ] (Agentd knows whether 4)

Hypothesis: Hallie can now know whether Hallie's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Kyle's card is shown to Howard. Kyle's card is shown to Kyle. It is publicly announced that someone picked a red card. It is publicly announced that someone picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:2 VALID? [ ! (1|2) ] [ ! (1|2) ] Agentb knows that (Agenta knows whether (1|2))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:2 VALID? [ ! (1|2) ] [ ! (1|2) ] 0

Hypothesis: Kyle can now know that Howard can know whether someone picked a red card.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that David knows whether or not nobody's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether (~(1|2|3|4))) ] Agentb knows that (Agentc knows that 4)

Hypothesis: Fernando can now know that David can know that Barbara's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Archie knows whether or not nobody's forehead is muddy. It is publicly announced that Shane knows whether someone's forehead is muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (~(1|2|3))) ] [ ! (Agentb knows whether (1|2|3)) ] (Agentb knows whether (1|2|3))

VARS 0,1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! (Agentc knows whether (~(1|2|3))) ] [ ! (Agentb knows whether (1|2|3)) ] 0

Hypothesis: Shane can now know whether someone's forehead is muddy.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Ruth knows whether someone is thirsty. It is publicly announced that Pedro does not know whether or not everyone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether (1|2|3|4)) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] Agentd knows that (Agentc knows whether (1&2&3&4))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (Agentb knows whether (1|2|3|4)) ] [ ! ~(Agentd knows whether (1&2&3&4)) ] 0

Hypothesis: Pedro can now know that Robert can know whether or not everyone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that David does not know whether Marie is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 1) ] Agenta knows that (Agentd knows whether 2)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentc knows whether 1) ] 0

Hypothesis: Marie can now know that Jeff can know whether Patricia is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that someone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! (1|2|3|4) ] Agentd knows whether (Agentc knows whether (1|2|3|4))

Hypothesis: Sherie can now know whether Tracy can know whether someone is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] Agenta knows whether (Agentb knows that 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] 0

Hypothesis: Beverly can now know whether Rene can know that Beverly is thirsty.


Premise: There are two persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Chester's card is shown to John. Chester's card is shown to Chester. It is publicly announced that someone picked a red card. It is publicly announced that not everyone picked a red card.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:2 VALID? [ ! (1|2) ] [ ! (~(1&2)) ] Agentb knows whether (Agenta knows whether (~(1|2)))

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:2 VALID? [ ! (1|2) ] [ ! (~(1&2)) ] 0

Hypothesis: Chester can now know whether John can know whether or not nobody picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that James knows whether Andy's forehead is muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 1) ] Agentd knows that (Agenta knows whether 1)

VARS 0,1,2,3,4 LAW Top OBS Agenta:2,3,4 Agentb:1,3,4 Agentc:1,2,4 Agentd:1,2,3 VALID? [ ! (1|2|3|4) ] [ ! (Agentc knows whether 1) ] 0

Hypothesis: Pamela can now know that Andy can know whether Andy's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. It is publicly announced that someone's forehead is muddy. It is publicly announced that Janie does not know whether or not everyone's forehead is muddy. It is publicly announced that Janie does not know whether or not everyone's forehead is muddy. It is publicly announced that Thomas knows whether someone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! (Agenta knows whether (1|2)) ] Agentb knows whether (Agenta knows whether 1)

VARS 0,1,2 LAW Top OBS Agenta:2 Agentb:1 VALID? [ ! (1|2) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! ~(Agentb knows whether (1&2)) ] [ ! (Agenta knows whether (1|2)) ] 0

Hypothesis: Janie can now know whether Thomas can know whether Thomas's forehead is muddy.


Premise: There are two persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Celia knows whether Charlotte's forehead is muddy. It is publicly announced that not everyone's forehead is muddy.

VARS 1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 1) ] [ ! (~(1&2)) ] (Agenta knows that (1&2))

VARS 0,1,2 LAW Top OBS Agenta:1,2 Agentb:1,2 VALID? [ ! (1|2) ] [ ! (Agentb knows whether 1) ] [ ! (~(1&2)) ] 0

Hypothesis: Charlotte can now know that everyone's forehead is muddy.


VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] Agentb knows that (Agenta knows that 1)

Hypothesis: Anne can now know that Edmund can know that Edmund is thirsty.


Premise: There are four persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Jennifer's forehead is not muddy. It is publicly announced that Lydia's forehead is not muddy.

VARS 1,2,3,4 LAW Top OBS Agenta:1,2,3,4 Agentb:1,2,3,4 Agentc:1,2,3,4 Agentd:1,2,3,4 VALID? [ ! (1|2|3|4) ] [ ! ~(3) ] [ ! ~(2) ] Agenta knows that (Agentc knows whether 1)

Hypothesis: Edna can now know that Jennifer can know whether Edna's forehead is muddy.


Premise: There are three persons. Everyone is visible to others. Each person draws a card, face unrevealed (red or black). Christine's card is shown to Maria. Dennis's card is shown to Maria. Christine's card is shown to Christine. Christine's card is shown to Dennis. It is publicly announced that someone picked a red card. It is publicly announced that Christine knows whether Christine picked a red card. It is publicly announced that Christine does not know whether or not everyone picked a red card. It is publicly announced that Maria does not know whether or not everyone picked a red card.

VARS 1,2,3 LAW Top OBS Agenta:2,3 Agentb:2 Agentc:2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 2) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agenta knows whether (1&2&3)) ] Agentb knows whether (Agentc knows whether 2)

VARS 0,1,2,3 LAW Top OBS Agenta:2,3 Agentb:2 Agentc:2 VALID? [ ! (1|2|3) ] [ ! (Agentb knows whether 2) ] [ ! ~(Agentb knows whether (1&2&3)) ] [ ! ~(Agenta knows whether (1&2&3)) ] 0

Hypothesis: Christine can now know whether Dennis can know whether Christine picked a red card.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Lori does not know whether Amy is thirsty. It is publicly announced that Lenora does not know whether Jeffrey is thirsty. It is publicly announced that Lori knows whether or not nobody is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 1) ] [ ! ~(Agentd knows whether 3) ] [ ! (Agentb knows whether (~(1|2|3|4))) ] Agentb knows that (Agentd knows that 3)

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(Agentb knows whether 1) ] [ ! ~(Agentd knows whether 3) ] [ ! (Agentb knows whether (~(1|2|3|4))) ] 0

Hypothesis: Lori can now know that Lenora can know that Jeffrey is thirsty.


Premise: There are four persons. Everyone is visible to others. It is publicly announced that someone is thirsty. It is publicly announced that Kenneth is not thirsty. It is publicly announced that Elizabeth does not know whether David is thirsty.

VARS 1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(4) ] [ ! ~(Agentc knows whether 2) ] Agenta knows whether (Agentc knows whether (~(1|2|3|4)))

VARS 0,1,2,3,4 LAW Top OBS Agenta:1 Agentb:2 Agentc:3 Agentd:4 VALID? [ ! (1|2|3|4) ] [ ! ~(4) ] [ ! ~(Agentc knows whether 2) ] 0

Hypothesis: Carlos can now know whether Elizabeth can know whether or not nobody is thirsty.


Premise: There are three persons. Everyone is visible to others. There is a mirror in the room. It is publicly announced that someone's forehead is muddy. It is publicly announced that Christopher's forehead is not muddy. It is publicly announced that Valerie's forehead is not muddy.

VARS 1,2,3 LAW Top OBS Agenta:1,2,3 Agentb:1,2,3 Agentc:1,2,3 VALID? [ ! (1|2|3) ] [ ! ~(2) ] [ ! ~(1) ] Agentc knows that (Agentb knows that (1&2&3))

Hypothesis: Vickie can now know that Christopher can know that everyone's forehead is muddy.