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Logic and Artificial IntelligenceLecture 1
Eric Pacuit
Currently Visiting the Center for Formal Epistemology, CMU
Center for Logic and Philosophy of ScienceTilburg University
ai.stanford.edu/∼[email protected]
August 31, 2011
Logic and Artificial Intelligence 1/22
Epistemology
Cognitive
Science
Philosophical
Logic
Dynamic
Epistemic/Doxastic
Logic
Game Theory
CS: Multiagent
Systems
CS: Distributed
Algorithms
Logics of
Rational AgencyEpistemic
Foundations of
Game Theory
????
Logic and Artificial Intelligence 2/22
Epistemology
Cognitive
Science
Philosophical
Logic
Dynamic
Epistemic/Doxastic
Logic
Game Theory
CS: Multiagent
Systems
CS: Distributed
Algorithms
Logics of
Rational AgencyEpistemic
Foundations of
Game Theory
????
Logic and Artificial Intelligence 2/22
Epistemology
Cognitive
Science
Philosophical
Logic
Dynamic
Epistemic/Doxastic
Logic
Game Theory
CS: Multiagent
Systems
CS: Distributed
Algorithms
Logics of
Rational AgencyEpistemic
Foundations of
Game Theory
????
Logic and Artificial Intelligence 2/22
Epistemology
Cognitive
Science
Philosophical
Logic
Dynamic
Epistemic/Doxastic
Logic
Game Theory
CS: Multiagent
Systems
CS: Distributed
Algorithms
Logics of
Rational AgencyEpistemic
Foundations of
Game Theory
????
Logic and Artificial Intelligence 2/22
Epistemology
Cognitive
Science
Philosophical
Logic
Dynamic
Epistemic/Doxastic
Logic
Game Theory
CS: Multiagent
Systems
CS: Distributed
Algorithms
Logics of
Rational AgencyEpistemic
Foundations of
Game Theory
????
Logic and Artificial Intelligence 2/22
Foundations of Epistemic Logic
David Lewis Jakko Hintikka Robert Aumann
Larry Moss Johan van Benthem Alexandru Baltag
Logic and Artificial Intelligence 3/22
Foundations of Epistemic Logic
Logic and Artificial Intelligence 4/22
Single-Agent Epistemic Logic
Let KP informally mean “the agent knows that P (is true)”.
K (P → Q): “Ann knows that P implies Q”
KP ∨ ¬KP: “either Ann does or does not know P”
KP ∨ K¬P: “Ann knows whether P is true”
LP: “P is an epistemic possibility”
KLP: “Ann knows that she thinks P ispossible”
Logic and Artificial Intelligence 5/22
Single-Agent Epistemic Logic
Let KP informally mean “the agent knows that P (is true)”.
K (P → Q): “Ann knows that P implies Q”
KP ∨ ¬KP: “either Ann does or does not know P”
KP ∨ K¬P: “Ann knows whether P is true”
LP: “P is an epistemic possibility”
KLP: “Ann knows that she thinks P ispossible”
Logic and Artificial Intelligence 5/22
Single-Agent Epistemic Logic
Let KP informally mean “the agent knows that P (is true)”.
K (P → Q): “Ann knows that P implies Q”
KP ∨ ¬KP: “either Ann does or does not know P”
KP ∨ K¬P: “Ann knows whether P is true”
LP: “P is an epistemic possibility”
KLP: “Ann knows that she thinks P ispossible”
Logic and Artificial Intelligence 5/22
Single-Agent Epistemic Logic
Let KP informally mean “the agent knows that P (is true)”.
K (P → Q): “Ann knows that P implies Q”
KP ∨ ¬KP: “either Ann does or does not know P”
KP ∨ K¬P: “Ann knows whether P is true”
LP: “P is an epistemic possibility”
KLP: “Ann knows that she thinks P ispossible”
Logic and Artificial Intelligence 5/22
Single-Agent Epistemic Logic
Let KP informally mean “the agent knows that P (is true)”.
K (P → Q): “Ann knows that P implies Q”
KP ∨ ¬KP: “either Ann does or does not know P”
KP ∨ K¬P: “Ann knows whether P is true”
LP: “P is an epistemic possibility”
KLP: “Ann knows that she thinks P ispossible”
Logic and Artificial Intelligence 5/22
Single-Agent Epistemic Logic
Let KP informally mean “the agent knows that P (is true)”.
K (P → Q): “Ann knows that P implies Q”
KP ∨ ¬KP: “either Ann does or does not know P”
KP ∨ K¬P: “Ann knows whether P is true”
LP: “P is an epistemic possibility”
KLP: “Ann knows that she thinks P ispossible”
Logic and Artificial Intelligence 5/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
(1, 2)
w1
(1, 3)
w2
(2, 3)
w3
(2, 1)
w4
(3, 1)
w5
(3, 2)
w6
Logic and Artificial Intelligence 6/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
What are the relevant states?
(1, 2)
w1
(1, 3)
w2
(2, 3)
w3
(2, 1)
w4
(3, 1)
w5
(3, 2)
w6
Logic and Artificial Intelligence 6/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
What are the relevant states?
(1, 2)
w1
(1, 3)
w2
(2, 3)
w3
(2, 1)
w4
(3, 1)
w5
(3, 2)
w6
Logic and Artificial Intelligence 6/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
Ann receives card 3 and card 1is put on the table
(1, 2)
w1
(1, 3)
w2
(2, 3)
w3
(2, 1)
w4
(3, 1)
w5
(3, 2)
w6
Logic and Artificial Intelligence 6/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
What information does Annhave?
(1, 2)
w1
(1, 3)
w2
(2, 3)
w3
(2, 1)
w4
(3, 1)
w5
(3, 2)
w6
Logic and Artificial Intelligence 6/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
What information does Annhave?
(1, 2)
w1
(1, 3)
w2
(2, 3)
w3
(2, 1)
w4
(3, 1)
w5
(3, 2)
w6
Logic and Artificial Intelligence 6/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
What information does Annhave?
(1, 2)
w1
(1, 3)
w2
(2, 3)
w3
(2, 1)
w4
(3, 1)
w5
(3, 2)
w6
Logic and Artificial Intelligence 6/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
Suppose Hi is intended tomean “Ann has card i”
Ti is intended to mean “card iis on the table”
Eg., V (H1) = {w1,w2}
(1, 2)
w1
(1, 3)
w2
(2, 3)
w3
(2, 1)
w4
(3, 1)
w5
(3, 2)
w6
Logic and Artificial Intelligence 6/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
Suppose Hi is intended tomean “Ann has card i”
Ti is intended to mean “card iis on the table”
Eg., V (H1) = {w1,w2}
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 6/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 7/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
Suppose that Ann receives card1 and card 2 is on the table.
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 7/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
Suppose that Ann receives card1 and card 2 is on the table.
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 7/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
M,w1 |= KH1
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 7/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
M,w1 |= KH1
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 7/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
M,w1 |= KH1
M,w1 |= K¬T1
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 7/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
M,w1 |= LT2
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 7/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
M,w1 |= K (T2 ∨ T3)
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 7/22
Multiagent Epistemic Logic
Many of the examples we are interested in involve more than oneagent!
KAP means “Ann knows P”
KBP means “Bob knows P”
I KAKBϕ: “Ann knows that Bob knows ϕ”
I KA(KBϕ ∨ KB¬ϕ): “Ann knows that Bob knows whether ϕ
I ¬KBKAKB(ϕ): “Bob does not know that Ann knows thatBob knows that ϕ”
Logic and Artificial Intelligence 8/22
Multiagent Epistemic Logic
Many of the examples we are interested in involve more than oneagent!
KAP means “Ann knows P”
KBP means “Bob knows P”
I KAKBϕ: “Ann knows that Bob knows ϕ”
I KA(KBϕ ∨ KB¬ϕ): “Ann knows that Bob knows whether ϕ
I ¬KBKAKB(ϕ): “Bob does not know that Ann knows thatBob knows that ϕ”
Logic and Artificial Intelligence 8/22
Multiagent Epistemic Logic
Many of the examples we are interested in involve more than oneagent!
KAP means “Ann knows P”
KBP means “Bob knows P”
I KAKBϕ: “Ann knows that Bob knows ϕ”
I KA(KBϕ ∨ KB¬ϕ): “Ann knows that Bob knows whether ϕ
I ¬KBKAKB(ϕ): “Bob does not know that Ann knows thatBob knows that ϕ”
Logic and Artificial Intelligence 8/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.
Suppose that Ann receives card1 and card 2 is on the table.
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Logic and Artificial Intelligence 9/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,Bob is given one of the cardsand the third card is put backin the deck.
Suppose that Ann receives card1 and Bob receives card 2.
A1,B2
w1
A1,B3
w2
A2,B3
w3
A2,B1
w4
A3,B1
w5
A3,B2
w6
Logic and Artificial Intelligence 9/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,Bob is given one of the cardsand the third card is put backin the deck.
Suppose that Ann receives card1 and Bob receives card 2.
A1,B2
w1
A1,B3
w2
A2,B3
w3
A2,B1
w4
A3,B1
w5
A3,B2
w6
Logic and Artificial Intelligence 9/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,Bob is given one of the cardsand the third card is put backin the deck.
Suppose that Ann receives card1 and Bob receives card 2.
A1,B2
w1
A1,B3
w2
A2,B3
w3
A2,B1
w4
A3,B1
w5
A3,B2
w6
Logic and Artificial Intelligence 9/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,Bob is given one of the cardsand the third card is put backin the deck.
Suppose that Ann receives card1 and Bob receives card 2.
M,w1 |= KB(KAA1 ∨ KA¬A1)
A1,B2
w1
A1,B3
w2
A2,B3
w3
A2,B1
w4
A3,B1
w5
A3,B2
w6
Logic and Artificial Intelligence 9/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,Bob is given one of the cardsand the third card is put backin the deck.
Suppose that Ann receives card1 and Bob receives card 2.
M,w1 |= KB(KAA1 ∨ KA¬A1)
A1,B2
w1
A1,B3
w2
A2,B3
w3
A2,B1
w4
A3,B1
w5
A3,B2
w6
Logic and Artificial Intelligence 9/22
ExampleSuppose there are three cards:1, 2 and 3.
Ann is dealt one of the cards,Bob is given one of the cardsand the third card is put backin the deck.
Suppose that Ann receives card1 and Bob receives card 2.
M,w1 |= KB(KAA1 ∨ KA¬A1)
A1,B2
w1
A1,B3
w2
A2,B3
w3
A2,B1
w4
A3,B1
w5
A3,B2
w6
Logic and Artificial Intelligence 9/22
Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.
Their mother enters the room and says “At least one of you havemud on your forehead”.
Then the children are repeatedly asked “do you know if you havemud on your forehead?”
What happens?
Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!
Logic and Artificial Intelligence 10/22
Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.
Their mother enters the room and says “At least one of you havemud on your forehead”.
Then the children are repeatedly asked “do you know if you havemud on your forehead?”
What happens?
Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!
Logic and Artificial Intelligence 10/22
Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.
Their mother enters the room and says “At least one of you havemud on your forehead”.
Then the children are repeatedly asked “do you know if you havemud on your forehead?”
What happens?
Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!
Logic and Artificial Intelligence 10/22
Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.
Their mother enters the room and says “At least one of you havemud on your forehead”.
Then the children are repeatedly asked “do you know if you havemud on your forehead?”
What happens?
Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!
Logic and Artificial Intelligence 10/22
Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.
Their mother enters the room and says “At least one of you havemud on your forehead”.
Then the children are repeatedly asked “do you know if you havemud on your forehead?”
What happens?
Claim: After first question, the children answer “I don’t know”,
after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!
Logic and Artificial Intelligence 10/22
Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.
Their mother enters the room and says “At least one of you havemud on your forehead”.
Then the children are repeatedly asked “do you know if you havemud on your forehead?”
What happens?
Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark).
Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!
Logic and Artificial Intelligence 10/22
Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.
Their mother enters the room and says “At least one of you havemud on your forehead”.
Then the children are repeatedly asked “do you know if you havemud on your forehead?”
What happens?
Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.”
Please, I beg you not to go over this!
Logic and Artificial Intelligence 10/22
Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.
Their mother enters the room and says “At least one of you havemud on your forehead”.
Then the children are repeatedly asked “do you know if you havemud on your forehead?”
What happens?
Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!
Logic and Artificial Intelligence 10/22
Muddy ChildrenAssume:
I There are three children: Ann, Bob and Charles.
I (Only) Ann and Bob have mud on their forehead.
C C C
Ann Bob Charles
state-of-affairs
C C C C C C C C C
Logic and Artificial Intelligence 11/22
Muddy ChildrenAssume:
I There are three children: Ann, Bob and Charles.
I (Only) Ann and Bob have mud on their forehead.
C C C
Ann Bob Charles
state-of-affairs
C C C C C C C C C
Logic and Artificial Intelligence 11/22
Muddy ChildrenAssume:
I There are three children: Ann, Bob and Charles.
I (Only) Ann and Bob have mud on their forehead.
C C C
Ann Bob Charles
state-of-affairs
C C C C C C C C C
Logic and Artificial Intelligence 11/22
Muddy ChildrenAssume:
I There are three children: Ann, Bob and Charles.
I (Only) Ann and Bob have mud on their forehead.
C C C
Ann Bob Charles
state-of-affairs
C C C C C C C C C
Logic and Artificial Intelligence 11/22
Muddy ChildrenAssume:
I There are three children: Ann, Bob and Charles.
I (Only) Ann and Bob have mud on their forehead.
C C C
Ann Bob Charles
state-of-affairs
C C C C C C C C C
Logic and Artificial Intelligence 11/22
Muddy ChildrenAssume:
I There are three children: Ann, Bob and Charles.
I (Only) Ann and Bob have mud on their forehead.
C C C
Ann Bob Charles
state-of-affairs
C C C C C C C C C
Logic and Artificial Intelligence 11/22
Muddy ChildrenAssume:
I There are three children: Ann, Bob and Charles.
I (Only) Ann and Bob have mud on their forehead.
C C C
Ann Bob Charles
state-of-affairs
C C C C C C C C C
Logic and Artificial Intelligence 11/22
Muddy ChildrenAssume:
I There are three children: Ann, Bob and Charles.
I (Only) Ann and Bob have mud on their forehead.
C C C
Ann Bob Charles
state-of-affairs
C C C C C C C C C
Logic and Artificial Intelligence 11/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
All 8 possible situations
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
The actual situation
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
Ann’s uncertainty
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
Bob’s uncertainty
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
Charles’ uncertainty
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
Agent 2’s uncertainty
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
None of the children know if they are muddy
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
None of the children know if they are muddy
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
“At least one has mud on their forehead.”
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
“At least one has mud on their forehead.”
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
“Who has mud on their forehead?”
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
“Who has mud on their forehead?”
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
No one steps forward.
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
No one steps forward.
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
“Who has mud on their forehead?”
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
Charles does not know he is clean.
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
Ann and Bob step forward.
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
Now, Charles knows he is clean.
Logic and Artificial Intelligence 12/22
Muddy Children
C C C
C C C
C C C
C C C
C C C
C C C
C C C
C C C
Now, Charles knows he is clean.
Logic and Artificial Intelligence 12/22
Single-Agent Epistemic Logic: The Language
ϕ is a formula of Epistemic Logic (L) if it is of the form
ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ
Logic and Artificial Intelligence 13/22
Single-Agent Epistemic Logic: The Language
ϕ is a formula of Epistemic Logic (L) if it is of the form
ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ
I p ∈ At is an atomic fact.
• “It is raining”• “The talk is at 2PM”• “The card on the table is a 7 of Hearts”
Logic and Artificial Intelligence 13/22
Single-Agent Epistemic Logic: The Language
ϕ is a formula of Epistemic Logic (L) if it is of the form
ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ
I p ∈ At is an atomic fact.
I The usual propositional language (L0)
Logic and Artificial Intelligence 13/22
Single-Agent Epistemic Logic: The Language
ϕ is a formula of Epistemic Logic (L) if it is of the form
ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ
I p ∈ At is an atomic fact.
I The usual propositional language (L0)
I Kϕ is intended to mean “The agent knows that ϕ is true”.
Logic and Artificial Intelligence 13/22
Single-Agent Epistemic Logic: The Language
ϕ is a formula of Epistemic Logic (L) if it is of the form
ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ
I p ∈ At is an atomic fact.
I The usual propositional language (L0)
I Kϕ is intended to mean “The agent knows that ϕ is true”.
I The usual definitions for →,∨,↔ apply
I Define Lϕ as ¬K¬ϕ
Logic and Artificial Intelligence 13/22
Single-Agent Epistemic Logic: The Language
ϕ is a formula of Epistemic Logic (L) if it is of the form
ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ
K (p → q): “Ann knows that p implies q”
Kp ∨ ¬Kp:
Kp ∨ K¬p:
Lϕ:
KLϕ:
Logic and Artificial Intelligence 13/22
Single-Agent Epistemic Logic: The Language
ϕ is a formula of Epistemic Logic (L) if it is of the form
ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ
K (p → q): “Ann knows that p implies q”
Kp ∨ ¬Kp: “either Ann does or does not know p”
Kp ∨ K¬p: “Ann knows whether p is true”
Lϕ:
KLϕ:
Logic and Artificial Intelligence 13/22
Single-Agent Epistemic Logic: The Language
ϕ is a formula of Epistemic Logic (L) if it is of the form
ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ
K (p → q): “Ann knows that p implies q”
Kp ∨ ¬Kp: “either Ann does or does not know p”
Kp ∨ K¬p: “Ann knows whether p is true”
Lϕ: “ϕ is an epistemic possibility”
KLϕ: “Ann knows that she thinks ϕ ispossible”
Logic and Artificial Intelligence 13/22
Single-Agent Epistemic Logic: Kripke Models
M = 〈W ,R,V 〉
Logic and Artificial Intelligence 14/22
Single-Agent Epistemic Logic: Kripke Models
M = 〈W ,R,V 〉
I W 6= ∅ is the set of all relevant situations (states of affairs,possible worlds)
Logic and Artificial Intelligence 14/22
Single-Agent Epistemic Logic: Kripke Models
M = 〈W ,R,V 〉
I W 6= ∅ is the set of all relevant situations (states of affairs,possible worlds)
I R ⊆W ×W represents the information of the agent:wRv provided “w and v are epistemically indistinguishable”
Logic and Artificial Intelligence 14/22
Single-Agent Epistemic Logic: Kripke Models
M = 〈W ,R,V 〉
I W 6= ∅ is the set of all relevant situations (states of affairs,possible worlds)
I R ⊆W ×W represents the information of the agent:wRv provided “w and v are epistemically indistinguishable”Notation: many people write ∼ for R
I V : At→ ℘(W ) is a valuation function assigning propositionalvariables to worlds
Logic and Artificial Intelligence 14/22
Single Agent Epistemic Logic: Truth in a Model
Given ϕ ∈ L, a Kripke model M = 〈W ,R,V 〉 and w ∈W
M,w |= ϕ means “in M, if the actual state is w , then ϕ is true”
Logic and Artificial Intelligence 15/22
Single Agent Epistemic Logic: Truth in a Model
Given ϕ ∈ L, a Kripke model M = 〈W ,R,V 〉 and w ∈W
M,w |= ϕ is defined as follows:
I M,w |= p iff w ∈ V (p) (with p ∈ At)
I M,w |= ¬ϕ if M,w 6|= ϕ
I M,w |= ϕ ∧ ψ if M,w |= ϕ and M,w |= ψ
I M,w |= Kϕ if for each v ∈W , if wRv , then M, v |= ϕ
Logic and Artificial Intelligence 15/22
Single Agent Epistemic Logic: Truth in a Model
Given ϕ ∈ L, a Kripke model M = 〈W ,R,V 〉 and w ∈W
M,w |= ϕ is defined as follows:
X M,w |= p iff w ∈ V (p) (with p ∈ At)
I M,w |= ¬ϕ if M,w 6|= ϕ
I M,w |= ϕ ∧ ψ if M,w |= ϕ and M,w |= ψ
I M,w |= Kϕ if for each v ∈W , if wRv , then M, v |= ϕ
Logic and Artificial Intelligence 15/22
Single Agent Epistemic Logic: Truth in a Model
Given ϕ ∈ L, a Kripke model M = 〈W ,R,V 〉 and w ∈W
M,w |= ϕ is defined as follows:
X M,w |= p iff w ∈ V (p) (with p ∈ At)
X M,w |= ¬ϕ if M,w 6|= ϕ
X M,w |= ϕ ∧ ψ if M,w |= ϕ and M,w |= ψ
I M,w |= Kϕ if for each v ∈W , if wRv , then M, v |= ϕ
Logic and Artificial Intelligence 15/22
Single Agent Epistemic Logic: Truth in a Model
Given ϕ ∈ L, a Kripke model M = 〈W ,R,V 〉 and w ∈W
M,w |= ϕ is defined as follows:
X M,w |= p iff w ∈ V (p) (with p ∈ At)
X M,w |= ¬ϕ if M,w 6|= ϕ
X M,w |= ϕ ∧ ψ if M,w |= ϕ and M,w |= ψ
X M,w |= Kϕ if for each v ∈W , if wRv , then M, v |= ϕ
Logic and Artificial Intelligence 15/22
Some Questions
Should we make additional assumptions about R (i.e., reflexive,transitive, etc.)
What idealizations have we made?
Logic and Artificial Intelligence 16/22
Some Notation
A Kripke Frame is a tuple 〈W ,R〉 where R ⊆W ×W .
ϕ is valid in a Kripke model M if M,w |= ϕ for all states w (wewrite M |= ϕ).
ϕ is valid on a Kripke frame F if M |= ϕ for all models M basedon F .
Logic and Artificial Intelligence 17/22
Logical Omniscience
Fact: ϕ is valid then Kϕ is valid
Logic and Artificial Intelligence 18/22
Logical Omniscience
Fact: Kϕ ∧ Kψ → K (ϕ ∧ ψ) is valid on all Kripke frames
Logic and Artificial Intelligence 18/22
Logical Omniscience
Fact: If ϕ→ ψ is valid then Kϕ→ Kψ is valid
Logic and Artificial Intelligence 18/22
Logical Omniscience
Fact: K (ϕ→ ψ)→ (Kϕ→ Kψ) is valid on all Kripke frames.
Logic and Artificial Intelligence 18/22
Logical Omniscience
Fact: ϕ↔ ψ is valid then Kϕ↔ Kψ is valid
Logic and Artificial Intelligence 18/22
Correspondence
DefinitionA model formula ϕ corresponds to a property P (of a relation in aKripke frame) provided
F |= ϕ iff F has P
Modal Formula Corresponding Property
Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric¬K⊥ Serial
Logic and Artificial Intelligence 19/22
Correspondence
DefinitionA model formula ϕ corresponds to a property P (of a relation in aKripke frame) provided
F |= ϕ iff F has P
Modal Formula Corresponding Property
Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric¬K⊥ Serial
Logic and Artificial Intelligence 19/22
Correspondence
DefinitionA model formula ϕ corresponds to a property P (of a relation in aKripke frame) provided
F |= ϕ iff F has P
Modal Formula Corresponding Property
Kϕ→ ϕ Reflexive
Kϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric¬K⊥ Serial
Logic and Artificial Intelligence 19/22
Correspondence
DefinitionA model formula ϕ corresponds to a property P (of a relation in aKripke frame) provided
F |= ϕ iff F has P
Modal Formula Corresponding Property
Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive
¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric¬K⊥ Serial
Logic and Artificial Intelligence 19/22
Correspondence
DefinitionA model formula ϕ corresponds to a property P (of a relation in aKripke frame) provided
F |= ϕ iff F has P
Modal Formula Corresponding Property
Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclidean
ϕ→ KLϕ Symmetric¬K⊥ Serial
Logic and Artificial Intelligence 19/22
Correspondence
DefinitionA model formula ϕ corresponds to a property P (of a relation in aKripke frame) provided
F |= ϕ iff F has P
Modal Formula Corresponding Property
Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric
¬K⊥ Serial
Logic and Artificial Intelligence 19/22
Correspondence
DefinitionA model formula ϕ corresponds to a property P (of a relation in aKripke frame) provided
F |= ϕ iff F has P
Modal Formula Corresponding Property
Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric¬K⊥ Serial
Logic and Artificial Intelligence 19/22
Modal Formula Property Philosophical Assumption
K (ϕ→ ψ)→ (Kϕ→ Kψ) — Logical OmniscienceKϕ→ ϕ Reflexive Truth
Kϕ→ KKϕ Transitive Positive Introspection¬Kϕ→ K¬Kϕ Euclidean Negative Introspection
¬K⊥ Serial Consistency
Logic and Artificial Intelligence 20/22
Modal Formula Property Philosophical Assumption
K (ϕ→ ψ)→ (Kϕ→ Kψ) — Logical Omniscience
Kϕ→ ϕ Reflexive TruthKϕ→ KKϕ Transitive Positive Introspection¬Kϕ→ K¬Kϕ Euclidean Negative Introspection
¬K⊥ Serial Consistency
Logic and Artificial Intelligence 20/22
Modal Formula Property Philosophical Assumption
K (ϕ→ ψ)→ (Kϕ→ Kψ) — Logical OmniscienceKϕ→ ϕ Reflexive Truth
Kϕ→ KKϕ Transitive Positive Introspection¬Kϕ→ K¬Kϕ Euclidean Negative Introspection
¬K⊥ Serial Consistency
Logic and Artificial Intelligence 20/22
Modal Formula Property Philosophical Assumption
K (ϕ→ ψ)→ (Kϕ→ Kψ) — Logical OmniscienceKϕ→ ϕ Reflexive Truth
Kϕ→ KKϕ Transitive Positive Introspection
¬Kϕ→ K¬Kϕ Euclidean Negative Introspection¬K⊥ Serial Consistency
Logic and Artificial Intelligence 20/22
Modal Formula Property Philosophical Assumption
K (ϕ→ ψ)→ (Kϕ→ Kψ) — Logical OmniscienceKϕ→ ϕ Reflexive Truth
Kϕ→ KKϕ Transitive Positive Introspection¬Kϕ→ K¬Kϕ Euclidean Negative Introspection
¬K⊥ Serial Consistency
Logic and Artificial Intelligence 20/22
Modal Formula Property Philosophical Assumption
K (ϕ→ ψ)→ (Kϕ→ Kψ) — Logical OmniscienceKϕ→ ϕ Reflexive Truth
Kϕ→ KKϕ Transitive Positive Introspection¬Kϕ→ K¬Kϕ Euclidean Negative Introspection
¬K⊥ Serial Consistency
Logic and Artificial Intelligence 20/22
The Logic S5
The logic S5 contains the following axiom schemes and rules:
Pc Axiomatization of Propositional CalculusK K (ϕ→ ψ)→ (Kϕ→ Kψ)T Kϕ→ ϕ4 Kϕ→ KKϕ5 ¬Kϕ→ K¬Kϕ
MPϕ ϕ→ ψ
ψ
Necϕ
Kψ
TheoremS5 is sound and strongly complete with respect to the class ofKripke frames with equivalence relations.
Logic and Artificial Intelligence 21/22
The Logic S5
The logic S5 contains the following axiom schemes and rules:
Pc Axiomatization of Propositional CalculusK K (ϕ→ ψ)→ (Kϕ→ Kψ)T Kϕ→ ϕ4 Kϕ→ KKϕ5 ¬Kϕ→ K¬Kϕ
MPϕ ϕ→ ψ
ψ
Necϕ
Kψ
TheoremS5 is sound and strongly complete with respect to the class ofKripke frames with equivalence relations.
Logic and Artificial Intelligence 21/22
The Logic S5
The logic S5 contains the following axiom schemes and rules:
Pc Axiomatization of Propositional CalculusK K (ϕ→ ψ)→ (Kϕ→ Kψ)T Kϕ→ ϕ4 Kϕ→ KKϕ5 ¬Kϕ→ K¬Kϕ
MPϕ ϕ→ ψ
ψ
Necϕ
Kψ
TheoremS5 is sound and strongly complete with respect to the class ofKripke frames with equivalence relations.
Logic and Artificial Intelligence 21/22
The Logic S5
The logic S5 contains the following axiom schemes and rules:
Pc Axiomatization of Propositional CalculusK K (ϕ→ ψ)→ (Kϕ→ Kψ)T Kϕ→ ϕ4 Kϕ→ KKϕ5 ¬Kϕ→ K¬Kϕ
MPϕ ϕ→ ψ
ψ
Necϕ
Kψ
TheoremS5 is sound and strongly complete with respect to the class ofKripke frames with equivalence relations.
Logic and Artificial Intelligence 21/22
The Logic S5
The logic S5 contains the following axiom schemes and rules:
Pc Axiomatization of Propositional CalculusK K (ϕ→ ψ)→ (Kϕ→ Kψ)T Kϕ→ ϕ4 Kϕ→ KKϕ5 ¬Kϕ→ K¬Kϕ
MPϕ ϕ→ ψ
ψ
Necϕ
Kψ
TheoremS5 is sound and strongly complete with respect to the class ofKripke frames with equivalence relations.
Logic and Artificial Intelligence 21/22
The Logic S5
The logic S5 contains the following axiom schemes and rules:
Pc Axiomatization of Propositional CalculusK K (ϕ→ ψ)→ (Kϕ→ Kψ)T Kϕ→ ϕ4 Kϕ→ KKϕ5 ¬Kϕ→ K¬Kϕ
MPϕ ϕ→ ψ
ψ
Necϕ
Kψ
TheoremS5 is sound and strongly complete with respect to the class ofKripke frames with equivalence relations.
Logic and Artificial Intelligence 21/22
Multi-agent Epistemic Logic
The Language: ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ
Kripke Models: M = 〈W ,R,V 〉 and w ∈W
Truth: M,w |= ϕ is defined as follows:
I M,w |= p iff w ∈ V (p) (with p ∈ At)
I M,w |= ¬ϕ if M,w 6|= ϕ
I M,w |= ϕ ∧ ψ if M,w |= ϕ and M,w |= ψ
I M,w |= Kϕ if for each v ∈W , if wRv , then M, v |= ϕ
Logic and Artificial Intelligence 22/22
Multi-agent Epistemic Logic
The Language: ϕ := p | ¬ϕ | ϕ ∧ ψ | Kiϕ with i ∈ A
Kripke Models: M = 〈W , {Ri}i∈A,V 〉 and w ∈W
Truth: M,w |= ϕ is defined as follows:
I M,w |= p iff w ∈ V (p) (with p ∈ At)
I M,w |= ¬ϕ if M,w 6|= ϕ
I M,w |= ϕ ∧ ψ if M,w |= ϕ and M,w |= ψ
I M,w |= Kiϕ if for each v ∈W , if wRiv , then M, v |= ϕ
Logic and Artificial Intelligence 22/22
Multi-agent Epistemic Logic
I KAKBϕ: “Ann knows that Bob knows ϕ”
I KA(KBϕ ∨ KB¬ϕ): “Ann knows that Bob knows whether ϕ
I ¬KBKAKB(ϕ): “Bob does not know that Ann knows thatBob knows that ϕ”
Logic and Artificial Intelligence 22/22
Next:
I More about the logic (complexity results, model checking,Tableaux systems, etc.)
I Group notions
I Additional informational attitudes (belief, certainty,justification, only knowing, awareness, etc.)
I Other semantics
Logic and Artificial Intelligence 23/22