logics of rational agencyepacuit/talks/lograt-tutorial.pdfbasic ingredients i what are the basic...
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![Page 1: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/1.jpg)
Logics of Rational Agency
Eric Pacuit
Tilburg University
October 6, 2009
Eric Pacuit 1
![Page 2: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/2.jpg)
Introduction and Motivation
We are interested in reasoning about rational agents interacting insocial situations.
Eric Pacuit 2
![Page 3: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/3.jpg)
Introduction and Motivation
We are interested in reasoning about rational agents interacting insocial situations.
I Philosophy (social philosophy, epistemology)
I Game Theory
I Social Choice Theory
I AI (multiagent systems)
Eric Pacuit 3
![Page 4: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/4.jpg)
Introduction and Motivation
We are interested in reasoning about rational agents interacting insocial situations.
What is a rational agent?
I maximize expected utility (instrumentally rational)
I react to observations
I revise beliefs when learning a surprising piece of information
I understand higher-order information
I plans for the future
I asks questions
I ????
Eric Pacuit 4
![Page 5: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/5.jpg)
Introduction and Motivation
We are interested in reasoning about rational agents interacting insocial situations.
There is a jungle of formal systems!
I logics of informational attitudes (knowledge, beliefs,certainty)
I logics of action & agency
I temporal logics/dynamic logics
I logics of motivational attitudes (preferences, intentions)
(Not to mention various game-theoretic/social choice modelsand logical languages for reasoning about them)
Eric Pacuit 5
![Page 6: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/6.jpg)
Introduction and Motivation
We are interested in reasoning about rational agents interacting insocial situations.
There is a jungle of formal systems!
I logics of informational attitudes (knowledge, beliefs,certainty)
I logics of action & agency
I temporal logics/dynamic logics
I logics of motivational attitudes (preferences, intentions)
(Not to mention various game-theoretic/social choice modelsand logical languages for reasoning about them)
I How do we compare different logical systems studyingthe same phenomena?
I How complex is it to reason about rational agents?
I (How) should we merge the various logical systems?
I What do the logical frameworks contribute to thediscussion on rational agency?
Eric Pacuit 6
![Page 7: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/7.jpg)
Introduction and Motivation
We are interested in reasoning about rational agents interacting insocial situations.
I playing a game (eg. a card game)
I having a conversation
I executing a social procedure
I ....
Eric Pacuit 7
![Page 8: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/8.jpg)
Introduction and Motivation
What about game-theoretic analyses?
Goal: incorporate/extend existing game-theoretic/social choiceanalyses
Eric Pacuit 8
![Page 9: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/9.jpg)
Introduction and Motivation
What about game-theoretic analyses?
Goal: incorporate/extend existing game-theoretic/social choiceanalyses
Eric Pacuit 9
![Page 10: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/10.jpg)
Introduction and Motivation
Formally, a game is described by its strategy sets and payofffunctions.
But in real life, may other parameters are relevant; thereis a lot more going on. Situations that substantively are vastlydifferent may nevertheless correspond to precisely the samestrategic game. For example, in a parliamentary democracy withthree parties, the winning coalitions are the same whether theparties hold a third of the seats, or, say, 49%, 39%, and 12%respectively. But the political situations are quite different. Thedifference lies in the attitudes of the players, in their expectationsabout each other, in custom, and in history, though the rules ofthe game do not distinguish between the two situations.
R. Aumann and J. H. Dreze. Rational Expectation in Games. American Eco-nomic Review (2008).
Eric Pacuit 10
![Page 11: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/11.jpg)
Introduction and Motivation
Formally, a game is described by its strategy sets and payofffunctions. But in real life, may other parameters are relevant; thereis a lot more going on. Situations that substantively are vastlydifferent may nevertheless correspond to precisely the samestrategic game.
For example, in a parliamentary democracy withthree parties, the winning coalitions are the same whether theparties hold a third of the seats, or, say, 49%, 39%, and 12%respectively. But the political situations are quite different. Thedifference lies in the attitudes of the players, in their expectationsabout each other, in custom, and in history, though the rules ofthe game do not distinguish between the two situations.
R. Aumann and J. H. Dreze. Rational Expectation in Games. American Eco-nomic Review (2008).
Eric Pacuit 11
![Page 12: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/12.jpg)
Introduction and Motivation
Formally, a game is described by its strategy sets and payofffunctions. But in real life, may other parameters are relevant; thereis a lot more going on. Situations that substantively are vastlydifferent may nevertheless correspond to precisely the samestrategic game. For example, in a parliamentary democracy withthree parties, the winning coalitions are the same whether theparties hold a third of the seats, or, say, 49%, 39%, and 12%respectively.
But the political situations are quite different. Thedifference lies in the attitudes of the players, in their expectationsabout each other, in custom, and in history, though the rules ofthe game do not distinguish between the two situations.
R. Aumann and J. H. Dreze. Rational Expectation in Games. American Eco-nomic Review (2008).
Eric Pacuit 12
![Page 13: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/13.jpg)
Introduction and Motivation
Formally, a game is described by its strategy sets and payofffunctions. But in real life, may other parameters are relevant; thereis a lot more going on. Situations that substantively are vastlydifferent may nevertheless correspond to precisely the samestrategic game. For example, in a parliamentary democracy withthree parties, the winning coalitions are the same whether theparties hold a third of the seats, or, say, 49%, 39%, and 12%respectively. But the political situations are quite different.
Thedifference lies in the attitudes of the players, in their expectationsabout each other, in custom, and in history, though the rules ofthe game do not distinguish between the two situations.
R. Aumann and J. H. Dreze. Rational Expectation in Games. American Eco-nomic Review (2008).
Eric Pacuit 13
![Page 14: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/14.jpg)
Introduction and Motivation
Formally, a game is described by its strategy sets and payofffunctions. But in real life, may other parameters are relevant; thereis a lot more going on. Situations that substantively are vastlydifferent may nevertheless correspond to precisely the samestrategic game. For example, in a parliamentary democracy withthree parties, the winning coalitions are the same whether theparties hold a third of the seats, or, say, 49%, 39%, and 12%respectively. But the political situations are quite different. Thedifference lies in the attitudes of the players, in their expectationsabout each other, in custom, and in history, though the rules ofthe game do not distinguish between the two situations.
R. Aumann and J. H. Dreze. Rational Expectation in Games. American Eco-nomic Review (2008).
Eric Pacuit 14
![Page 15: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/15.jpg)
Introduction and Motivation
Logics of Rational Agency
Eric Pacuit 15
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Basic Ingredients
I What are the basic building blocks? (the nature of time
(continuous or discrete/branching or linear), how (primitive) events
or actions are represented, how causal relationships are represented
and what constitutes a state of affairs.)
I Single agent vs. many agents.
I What are the primitive operators?
• Informational attitudes• Motivational attitudes• Normative attitudes
I Static vs. dynamic
Eric Pacuit 16
![Page 17: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/17.jpg)
Basic Ingredients
I What are the basic building blocks? (the nature of time
(continuous or discrete/branching or linear), how (primitive) events
or actions are represented, how causal relationships are represented
and what constitutes a state of affairs.)
I Single agent vs. many agents.
I What are the primitive operators?
• Informational attitudes• Motivational attitudes• Normative attitudes
I Static vs. dynamic
Eric Pacuit 17
![Page 18: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/18.jpg)
Basic Ingredients
I What are the basic building blocks? (the nature of time
(continuous or discrete/branching or linear), how (primitive) events
or actions are represented, how causal relationships are represented
and what constitutes a state of affairs.)
I Single agent vs. many agents.
I What are the primitive operators?
• Informational attitudes• Motivational attitudes• Normative attitudes
I Static vs. dynamic
Eric Pacuit 18
![Page 19: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/19.jpg)
Basic Ingredients
I What are the basic building blocks? (the nature of time
(continuous or discrete/branching or linear), how (primitive) events
or actions are represented, how causal relationships are represented
and what constitutes a state of affairs.)
I Single agent vs. many agents.
I What are the primitive operators?
• Informational attitudes• Motivational attitudes• Normative attitudes
I Static vs. dynamic
Eric Pacuit 19
![Page 20: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/20.jpg)
Basic Ingredients
Basic Ingredients
I informational attitudes
I time, actions and ability
I motivational attitudes
Eric Pacuit 20
![Page 21: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/21.jpg)
Basic Ingredients
Single-Agent Epistemic Logic
Typically, we write KP when the intended interpretation is “P isknown”
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”
Eric Pacuit 21
![Page 22: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/22.jpg)
Basic Ingredients
Single-Agent Epistemic Logic
Typically, we write KP when the intended interpretation is “P isknown”
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”
Eric Pacuit 22
![Page 23: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/23.jpg)
Basic Ingredients
Single-Agent Epistemic Logic
Typically, we write KP when the intended interpretation is “P isknown”
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”
Eric Pacuit 23
![Page 24: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/24.jpg)
Basic Ingredients
Single-Agent Epistemic Logic
Typically, we write KP when the intended interpretation is “P isknown”
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”
Eric Pacuit 24
![Page 25: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/25.jpg)
Basic Ingredients
Single-Agent Epistemic Logic
Typically, we write KP when the intended interpretation is “P isknown”
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”
Eric Pacuit 25
![Page 26: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/26.jpg)
Basic Ingredients
Single-Agent Epistemic Logic
Typically, we write KP when the intended interpretation is “P isknown”
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”
Eric Pacuit 26
![Page 27: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/27.jpg)
Basic Ingredients
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
Eric Pacuit 27
![Page 28: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/28.jpg)
Basic Ingredients
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
Eric Pacuit 28
![Page 29: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/29.jpg)
Basic Ingredients
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
Eric Pacuit 29
![Page 30: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/30.jpg)
Basic Ingredients
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
Eric Pacuit 30
![Page 31: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/31.jpg)
Basic Ingredients
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
Eric Pacuit 31
![Page 32: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/32.jpg)
Basic Ingredients
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
Eric Pacuit 32
![Page 33: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/33.jpg)
Basic Ingredients
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
Eric Pacuit 33
![Page 34: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/34.jpg)
Basic Ingredients
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
Eric Pacuit 34
![Page 35: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/35.jpg)
Basic Ingredients
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
Eric Pacuit 35
![Page 36: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/36.jpg)
Basic Ingredients
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
Eric Pacuit 36
![Page 37: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/37.jpg)
Basic Ingredients
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
Eric Pacuit 37
![Page 38: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/38.jpg)
Basic Ingredients
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
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Basic Ingredients
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
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Basic Ingredients
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 H1
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
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Basic Ingredients
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 H1
M,w1 |= K¬T1
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
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Basic Ingredients
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
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Basic Ingredients
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
Eric Pacuit 43
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Basic Ingredients
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
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Basic Ingredients
Some Questions
Should we make additional assumptions about R (i.e., reflexive,transitive, etc.)
What idealizations have we made?
Eric Pacuit 45
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Basic Ingredients
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
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Basic Ingredients
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
Eric Pacuit 47
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Basic Ingredients
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
Eric Pacuit 48
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Basic Ingredients
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
Eric Pacuit 49
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Basic Ingredients
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
Eric Pacuit 50
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Basic Ingredients
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
Eric Pacuit 51
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Basic Ingredients
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
Eric Pacuit 52
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Basic Ingredients
Multi-agent Epistemic Logic
The Language: ϕ := p | ¬ϕ | ϕ ∧ ψ | Kiϕ with i ∈ A
Kripke Models: M = 〈W , Rii∈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
Eric Pacuit 53
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Basic Ingredients
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”
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Basic Ingredients
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
Eric Pacuit 55
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Basic Ingredients
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.
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Eric Pacuit 56
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Basic Ingredients
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.
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
Eric Pacuit 57
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Basic Ingredients
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.
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
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Basic Ingredients
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,w |= KB(KAH1 ∨ KA¬H1)
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
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Basic Ingredients
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,w |= KB(KAH1 ∨ KA¬H1)
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
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Basic Ingredients
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,w |= KB(KAH1 ∨ KA¬H1)
H1,T2
w1
H1,T3
w2
H2,T3
w3
H2,T1
w4
H3,T1
w5
H3,T2
w6
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Basic Ingredients
Group Knowledge
KAP: “Ann knows that P”
KBP: “Bob knows that P”
KAKBP: “Ann knows that Bob knows that P”
KAP ∧ KBP: “Every one knows P”. let EP := KAP ∧ KBP
KAEP: “Ann knows that everyone knows that P”.
EEP: “Everyone knows that everyone knows that P”.
EEEP: “Everyone knows that everyone knows that everyone knowsthat P.”
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Basic Ingredients
Group Knowledge
KAP: “Ann knows that P”
KBP: “Bob knows that P”
KAKBP: “Ann knows that Bob knows that P”
KAP ∧ KBP: “Every one knows P”. let EP := KAP ∧ KBP
KAEP: “Ann knows that everyone knows that P”.
EEP: “Everyone knows that everyone knows that P”.
EEEP: “Everyone knows that everyone knows that everyone knowsthat P.”
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Basic Ingredients
Group Knowledge
KAP: “Ann knows that P”
KBP: “Bob knows that P”
KAKBP: “Ann knows that Bob knows that P”
KAP ∧ KBP: “Every one knows P”. let EP := KAP ∧ KBP
KAEP: “Ann knows that everyone knows that P”.
EEP: “Everyone knows that everyone knows that P”.
EEEP: “Everyone knows that everyone knows that everyone knowsthat P.”
Eric Pacuit 64
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Basic Ingredients
Group Knowledge
KAP: “Ann knows that P”
KBP: “Bob knows that P”
KAKBP: “Ann knows that Bob knows that P”
KAP ∧ KBP: “Every one knows P”.
let EP := KAP ∧ KBP
KAEP: “Ann knows that everyone knows that P”.
EEP: “Everyone knows that everyone knows that P”.
EEEP: “Everyone knows that everyone knows that everyone knowsthat P.”
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Basic Ingredients
Group Knowledge
KAP: “Ann knows that P”
KBP: “Bob knows that P”
KAKBP: “Ann knows that Bob knows that P”
KAP ∧ KBP: “Every one knows P”. let EP := KAP ∧ KBP
KAEP: “Ann knows that everyone knows that P”.
EEP: “Everyone knows that everyone knows that P”.
EEEP: “Everyone knows that everyone knows that everyone knowsthat P.”
Eric Pacuit 66
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Basic Ingredients
Group Knowledge
KAP: “Ann knows that P”
KBP: “Bob knows that P”
KAKBP: “Ann knows that Bob knows that P”
KAP ∧ KBP: “Every one knows P”. let EP := KAP ∧ KBP
KAEP: “Ann knows that everyone knows that P”.
EEP: “Everyone knows that everyone knows that P”.
EEEP: “Everyone knows that everyone knows that everyone knowsthat P.”
Eric Pacuit 67
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Basic Ingredients
Group Knowledge
KAP: “Ann knows that P”
KBP: “Bob knows that P”
KAKBP: “Ann knows that Bob knows that P”
KAP ∧ KBP: “Every one knows P”. let EP := KAP ∧ KBP
KAEP: “Ann knows that everyone knows that P”.
EEP: “Everyone knows that everyone knows that P”.
EEEP: “Everyone knows that everyone knows that everyone knowsthat P.”
Eric Pacuit 68
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Basic Ingredients
Group Knowledge
KAP: “Ann knows that P”
KBP: “Bob knows that P”
KAKBP: “Ann knows that Bob knows that P”
KAP ∧ KBP: “Every one knows P”. let EP := KAP ∧ KBP
KAEP: “Ann knows that everyone knows that P”.
EEP: “Everyone knows that everyone knows that P”.
EEEP: “Everyone knows that everyone knows that everyone knowsthat P.”
Eric Pacuit 69
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Basic Ingredients
Common Knowledge
CP: “It is common knowledge that P”
— “Everyone knows thateveryone knows that everyone knows that · · · P”.
Is common knowledge different from everyone knows?
P
w1
P
w2
¬P
w3
A
B
A,B A,B
A,B
w1 |= EP ∧ ¬CP
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Basic Ingredients
Common Knowledge
CP: “It is common knowledge that P” — “Everyone knows thateveryone knows that everyone knows that · · · P”.
Is common knowledge different from everyone knows?
P
w1
P
w2
¬P
w3
A
B
A,B A,B
A,B
w1 |= EP ∧ ¬CP
Eric Pacuit 71
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Basic Ingredients
Common Knowledge
CP: “It is common knowledge that P” — “Everyone knows thateveryone knows that everyone knows that · · · P”.
Is common knowledge different from everyone knows?
P
w1
P
w2
¬P
w3
A
B
A,B A,B
A,B
w1 |= EP ∧ ¬CP
Eric Pacuit 72
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Basic Ingredients
Common Knowledge
CP: “It is common knowledge that P” — “Everyone knows thateveryone knows that everyone knows that · · · P”.
Is common knowledge different from everyone knows?
P
w1
P
w2
¬P
w3
A
B
A,B A,B
A,B
w1 |= EP ∧ ¬CP
Eric Pacuit 73
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Basic Ingredients
Common Knowledge
CP: “It is common knowledge that P” — “Everyone knows thateveryone knows that everyone knows that · · · P”.
Is common knowledge different from everyone knows?
P
w1
P
w2
¬P
w3
A
B
A,B A,B
A,B
w1 |= EP ∧ ¬CP
Eric Pacuit 74
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Basic Ingredients
Common Knowledge
CP: “It is common knowledge that P” — “Everyone knows thateveryone knows that everyone knows that · · · P”.
Is common knowledge different from everyone knows?
P
w1
P
w2
¬P
w3
A
B
A,B A,B
A,B
w1 |= EP ∧ ¬CP
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Basic Ingredients
Common Knowledge
CP: “It is common knowledge that P” — “Everyone knows thateveryone knows that everyone knows that · · · P”.
Is common knowledge different from everyone knows?
P
w1
P
w2
¬P
w3
A
B
A,B A,B
A,B
w1 |= EP ∧ ¬CP
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Basic Ingredients
Common Knowledge
The operator “everyone knows P”, denoted EP, is defined asfollows
EP :=∧i∈A
KiP
w |= CP iff every finite path starting at w ends with a statesatisfying P.
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Basic Ingredients
CP → ECP
Suppose you are told “Ann and Bob are going together,”’and respond “sure, that’s common knowledge.” Whatyou mean is not only that everyone knows this, but alsothat the announcement is pointless, occasions nosurprise, reveals nothing new; in effect, that the situationafter the announcement does not differ from that before....the event “Ann and Bob are going together” — call itP — is common knowledge if and only if some event —call it Q — happened that entails P and also entails allplayers’ knowing Q (like all players met Ann and Bob atan intimate party). (Robert Aumann)
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Basic Ingredients
CP → ECP
Suppose you are told “Ann and Bob are going together,”’and respond “sure, that’s common knowledge.” Whatyou mean is not only that everyone knows this, but alsothat the announcement is pointless, occasions nosurprise, reveals nothing new; in effect, that the situationafter the announcement does not differ from that before....the event “Ann and Bob are going together” — call itP — is common knowledge if and only if some event —call it Q — happened that entails P and also entails allplayers’ knowing Q (like all players met Ann and Bob atan intimate party). (Robert Aumann)
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Basic Ingredients
P ∧ C (P → EP)→ CP
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Basic Ingredients
An Example
Two players Ann and Bob are told that the following will happen.Some positive integer n will be chosen and one of n, n + 1 will bewritten on Ann’s forehead, the other on Bob’s. Each will be ableto see the other’s forehead, but not his/her own.
Suppose the number are (2,3).
Do the agents know there numbers are less than 1000?
Is it common knowledge that their numbers are less than 1000?
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Basic Ingredients
An Example
Two players Ann and Bob are told that the following will happen.Some positive integer n will be chosen and one of n, n + 1 will bewritten on Ann’s forehead, the other on Bob’s. Each will be ableto see the other’s forehead, but not his/her own.
Suppose the number are (2,3).
Do the agents know there numbers are less than 1000?
Is it common knowledge that their numbers are less than 1000?
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Basic Ingredients
An Example
Two players Ann and Bob are told that the following will happen.Some positive integer n will be chosen and one of n, n + 1 will bewritten on Ann’s forehead, the other on Bob’s. Each will be ableto see the other’s forehead, but not his/her own.
Suppose the number are (2,3).
Do the agents know there numbers are less than 1000?
Is it common knowledge that their numbers are less than 1000?
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Basic Ingredients
An Example
Two players Ann and Bob are told that the following will happen.Some positive integer n will be chosen and one of n, n + 1 will bewritten on Ann’s forehead, the other on Bob’s. Each will be ableto see the other’s forehead, but not his/her own.
Suppose the number are (2,3).
Do the agents know there numbers are less than 1000?
Is it common knowledge that their numbers are less than 1000?
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Basic Ingredients
(0,1) (2,1)
(2,3) (4,3)
(4,5) (6,5)
(6,7)
A
B
A
B
A
B
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Basic Ingredients
Basic Ingredients
X informational attitudes (knowledge, group knowledge, belief,certainty, etc.)
I time, actions and ability
I motivational attitudes
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Basic Ingredients
Actions: Two Views
1. Actions transition between states, or situations
s t
a
2. Actions restrict the set of possible future histories
· · ·
· · ·
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Basic Ingredients
Actions: Two Views1. Actions transition between states, or situations
s t
a
2. Actions restrict the set of possible future histories
· · ·
· · ·
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Basic Ingredients
Actions: Two Views1. Actions transition between states, or situations
s t
a
2. Actions restrict the set of possible future histories
· · ·
· · ·
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Basic Ingredients
Propositional Dynamic Logic
Semantics: M = 〈W , Ra | a ∈ P,V 〉 where for each a ∈ P,Ra ⊆W ×W and V : At→ ℘(W )
I Rα∪β := Rα ∪ RβI Rα;β := Rα RβI Rα∗ := ∪n≥0Rn
α
I Rϕ? = (w ,w) | M,w |= ϕ
M,w |= [α]ϕ iff for each v , if wRαv then M, v |= ϕ
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Basic Ingredients
Background: Propositional Dynamic Logic
1. Axioms of propositional logic
2. [α](ϕ→ ψ)→ ([α]ϕ→ [α]ψ)
3. [α ∪ β]ϕ↔ [α]ϕ ∧ [β]ϕ
4. [α;β]ϕ↔ [α][β]ϕ
5. [ψ?]ϕ↔ (ψ → ϕ)
6. ϕ ∧ [α][α∗]ϕ↔ [α∗]ϕ
7. ϕ ∧ [α∗](ϕ→ [α]ϕ)→ [α∗]ϕ
8. Modus Ponens and Necessitation (for each program α)
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Basic Ingredients
Background: Propositional Dynamic Logic
1. Axioms of propositional logic
2. [α](ϕ→ ψ)→ ([α]ϕ→ [α]ψ)
3. [α ∪ β]ϕ↔ [α]ϕ ∧ [β]ϕ
4. [α;β]ϕ↔ [α][β]ϕ
5. [ψ?]ϕ↔ (ψ → ϕ)
6. ϕ ∧ [α][α∗]ϕ↔ [α∗]ϕ (Fixed-Point Axiom)
7. ϕ ∧ [α∗](ϕ→ [α]ϕ)→ [α∗]ϕ (Induction Axiom)
8. Modus Ponens and Necessitation (for each program α)
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Basic Ingredients
Propositional Dynamic Logic
Theorem PDL is sound and weakly complete with respect to theSegerberg Axioms.
Theorem The satisfiability problem for PDL is decidable(EXPTIME-Complete).
D. Kozen and R. Parikh. A Completeness proof for Propositional Dynamic Logic..
D. Harel, D. Kozen and Tiuryn. Dynamic Logic. 2001.
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Basic Ingredients
Actions and Ability
An early approach to interpret PDL as logic of actions was putforward by Krister Segerberg.
Segerberg adds an “agency” program to the PDL language δAwhere A is a formula.
K. Segerberg. Bringing it about. JPL, 1989.
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Basic Ingredients
Actions and Agency
The intended meaning of the program ‘δA’ is that the agent“brings it about that A’: formally, δA is the set of all paths p suchthat
1. p is the computation according to some program α, and
2. α only terminates at states in which it is true that A
Interestingly, Segerberg also briefly considers a third condition:
3. p is optimal (in some sense: shortest, maximally efficient,most convenient, etc.) in the set of computations satisfyingconditions (1) and (2).
The axioms:
1. [δA]A
2. [δA]B → ([δB]C → [δA]C )
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Basic Ingredients
Actions and Agency
The intended meaning of the program ‘δA’ is that the agent“brings it about that A’: formally, δA is the set of all paths p suchthat
1. p is the computation according to some program α, and
2. α only terminates at states in which it is true that A
Interestingly, Segerberg also briefly considers a third condition:
3. p is optimal (in some sense: shortest, maximally efficient,most convenient, etc.) in the set of computations satisfyingconditions (1) and (2).
The axioms:
1. [δA]A
2. [δA]B → ([δB]C → [δA]C )
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Basic Ingredients
Actions and Agency
The intended meaning of the program ‘δA’ is that the agent“brings it about that A’: formally, δA is the set of all paths p suchthat
1. p is the computation according to some program α, and
2. α only terminates at states in which it is true that A
Interestingly, Segerberg also briefly considers a third condition:
3. p is optimal (in some sense: shortest, maximally efficient,most convenient, etc.) in the set of computations satisfyingconditions (1) and (2).
The axioms:
1. [δA]A
2. [δA]B → ([δB]C → [δA]C )
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Basic Ingredients
Actions and Agency
The intended meaning of the program ‘δA’ is that the agent“brings it about that A’: formally, δA is the set of all paths p suchthat
1. p is the computation according to some program α, and
2. α only terminates at states in which it is true that A
Interestingly, Segerberg also briefly considers a third condition:
3. p is optimal (in some sense: shortest, maximally efficient,most convenient, etc.) in the set of computations satisfyingconditions (1) and (2).
The axioms:
1. [δA]A
2. [δA]B → ([δB]C → [δA]C )
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Basic Ingredients
Actions and Agency
J. Horty. Agency and Deontic Logic. 2001.
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Basic Ingredients
Logics of Action and AgencyAlternative accounts of agency do not include explicit descriptionof the actions:
t0 t1 t2 t3
· · ·
· · ·
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Basic Ingredients
STIT
I Each node represents a choice point for the agent.
I A history is a maximal branch in the above tree.
I Formulas are interpreted at history moment pairs.
I At each moment there is a choice available to the agent(partition of the histories through that moment)
I The key modality is [stit]ϕ which is intended to mean that theagent i can “see to it that ϕ is true”.
• [stit]ϕ is true at a history moment pair provided the agent canchoose a (set of) branch(es) such that every futurehistory-moment pair satisfies ϕ
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Basic Ingredients
STIT
We use the modality ‘♦’ to mean historic possibility.
♦[stit]ϕ: “the agent has the ability to bring about ϕ”.
Example Consider the example of an agent (call her Ann)throwing a dart. Suppose Ann is not a very good dart player, butshe just happens to throw a bull’s eye. Intuitively, we do not wantto say that Ann has the ability to throw a bull’s eye even though ithappens to be true. That is, the following principle should befalsifiable:
ϕ→ ♦[stit]ϕ
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Basic Ingredients
STIT
We use the modality ‘♦’ to mean historic possibility.
♦[stit]ϕ: “the agent has the ability to bring about ϕ”.
Example Consider the example of an agent (call her Ann)throwing a dart. Suppose Ann is not a very good dart player, butshe just happens to throw a bull’s eye.
Intuitively, we do not wantto say that Ann has the ability to throw a bull’s eye even though ithappens to be true. That is, the following principle should befalsifiable:
ϕ→ ♦[stit]ϕ
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Basic Ingredients
STIT
We use the modality ‘♦’ to mean historic possibility.
♦[stit]ϕ: “the agent has the ability to bring about ϕ”.
Example Consider the example of an agent (call her Ann)throwing a dart. Suppose Ann is not a very good dart player, butshe just happens to throw a bull’s eye. Intuitively, we do not wantto say that Ann has the ability to throw a bull’s eye even though ithappens to be true. That is, the following principle should befalsifiable:
ϕ→ ♦[stit]ϕ
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Basic Ingredients
STIT
Example Continuing with this example, suppose that Ann has theability to hit the dart board, but has no other control over theplacement of the dart. Now, when she throws the dart, as a matterof fact, it will either hit the top half of the board or the bottomhalf of the board. Since, Ann has the ability to hit the dart board,she has the ability to either hit the top half of the board or thebottom half of the board.
However, intuitively, it seems true that Ann does not have theability to hit the top half of the dart board, and also she does nothave the ability to hit the bottom half of the dart board. Thus, thefollowing principle should be falsifiable:
♦[stit](ϕ ∨ ψ)→ ♦[stit]ϕ ∨ ♦[stit]ψ
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Basic Ingredients
STIT
Example Continuing with this example, suppose that Ann has theability to hit the dart board, but has no other control over theplacement of the dart. Now, when she throws the dart, as a matterof fact, it will either hit the top half of the board or the bottomhalf of the board. Since, Ann has the ability to hit the dart board,she has the ability to either hit the top half of the board or thebottom half of the board.However, intuitively, it seems true that Ann does not have theability to hit the top half of the dart board, and also she does nothave the ability to hit the bottom half of the dart board. Thus, thefollowing principle should be falsifiable:
♦[stit](ϕ ∨ ψ)→ ♦[stit]ϕ ∨ ♦[stit]ψ
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Basic Ingredients
STIT
The following model will falsify both of the above formulas:
h1 h2 h3
K1 K2
A
¬B
¬A
B
¬A
¬B
t
J. Horty. Agency and Deontic Logic. 2001.
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Basic Ingredients
Temporal Logics
t0 t1 t2 t3
· · ·
· · ·
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Basic Ingredients
Computational vs. Behavioral Structures
x = 1q0
x = 2q1
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Basic Ingredients
Computational vs. Behavioral Structures
x = 1q0
x = 2q1
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Basic Ingredients
Computational vs. Behavioral Structures
x = 1q0
x = 2q1 q0q0q0 q0q0q1 q0q1q0 q0q1q1
q0q0 q0q1
q0
...
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Basic Ingredients
Temporal Logics
I Linear Time Temporal Logic: Reasoning about computationpaths:
♦ϕ: ϕ is true some time in the future.
A. Pnuelli. A Temporal Logic of Programs. in Proc. 18th IEEE Symposium onFoundations of Computer Science (1977).
I Branching Time Temporal Logic: Allows quantification overpaths:
∃♦ϕ: there is a path in which ϕ is eventually true.
E. M. Clarke and E. A. Emerson. Design and Synthesis of SynchronizationSkeletons using Branching-time Temproal-logic Specifications. In ProceedingsWorkshop on Logic of Programs, LNCS (1981).
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Basic Ingredients
Temporal LogicsI Linear Time Temporal Logic: Reasoning about computation
paths:
♦ϕ: ϕ is true some time in the future.
A. Pnuelli. A Temporal Logic of Programs. in Proc. 18th IEEE Symposium onFoundations of Computer Science (1977).
I Branching Time Temporal Logic: Allows quantification overpaths:
∃♦ϕ: there is a path in which ϕ is eventually true.
E. M. Clarke and E. A. Emerson. Design and Synthesis of SynchronizationSkeletons using Branching-time Temproal-logic Specifications. In ProceedingsWorkshop on Logic of Programs, LNCS (1981).
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Basic Ingredients
Temporal LogicsI Linear Time Temporal Logic: Reasoning about computation
paths:
♦ϕ: ϕ is true some time in the future.
A. Pnuelli. A Temporal Logic of Programs. in Proc. 18th IEEE Symposium onFoundations of Computer Science (1977).
I Branching Time Temporal Logic: Allows quantification overpaths:
∃♦ϕ: there is a path in which ϕ is eventually true.
E. M. Clarke and E. A. Emerson. Design and Synthesis of SynchronizationSkeletons using Branching-time Temproal-logic Specifications. In ProceedingsWorkshop on Logic of Programs, LNCS (1981).
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Basic Ingredients
Temporal Logics
x = 1q0
x = 2q1 q0q0q0 q0q0q1 q0q1q0 q0q1q1
q0q0 q0q1
q0
...
♦Px=2
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Basic Ingredients
Temporal Logics
x = 1q0
x = 2q1 q0q0q0 q0q0q1 q0q1q0 q0q1q1
q0q0 q0q1
q0
...
∃♦Px=2
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Basic Ingredients
Temporal Logics
x = 1q0
x = 2q1 q0q0q0 q0q0q1 q0q1q0 q0q1q1
q0q0 q0q1
q0
...
¬∀♦Px=2
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Basic Ingredients
Many Agents
The previous model assumes there is one agent that “controls” thetransition system.
What if there is more than one agent?
Example: Suppose that there are two agents: a server (s) and aclient (c). The client asks to set the value of x and the server caneither grant or deny the request. Assume the agents makesimultaneous moves.
deny grant
set1
set2
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Basic Ingredients
Many Agents
The previous model assumes there is one agent that “controls” thetransition system.
What if there is more than one agent?
Example: Suppose that there are two agents: a server (s) and aclient (c). The client asks to set the value of x and the server caneither grant or deny the request. Assume the agents makesimultaneous moves.
deny grant
set1
set2
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Basic Ingredients
Many Agents
The previous model assumes there is one agent that “controls” thetransition system.
What if there is more than one agent?
Example: Suppose that there are two agents: a server (s) and aclient (c). The client asks to set the value of x and the server caneither grant or deny the request. Assume the agents makesimultaneous moves.
deny grant
set1
set2
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Basic Ingredients
Many Agents
The previous model assumes there is one agent that “controls” thetransition system.
What if there is more than one agent?
Example: Suppose that there are two agents: a server (s) and aclient (c). The client asks to set the value of x and the server caneither grant or deny the request. Assume the agents makesimultaneous moves.
deny grant
set1
set2
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Basic Ingredients
Many Agents
The previous model assumes there is one agent that “controls” thetransition system.
What if there is more than one agent?
Example: Suppose that there are two agents: a server (s) and aclient (c). The client asks to set the value of x and the server caneither grant or deny the request. Assume the agents makesimultaneous moves.
deny grant
set1 q0 ⇒ q0, q1 ⇒ q0
set2 q0 ⇒ q1, q1 ⇒ q1
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Basic Ingredients
Many Agents
The previous model assumes there is one agent that “controls” thetransition system.
What if there is more than one agent?
Example: Suppose that there are two agents: a server (s) and aclient (c). The client asks to set the value of x and the server caneither grant or deny the request. Assume the agents makesimultaneous moves.
deny grant
set1 q ⇒ q q0 ⇒ q0, q1 ⇒ q0
set2 q ⇒ q q0 ⇒ q1, q1 ⇒ q1
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Basic Ingredients
From Temporal Logic to Strategy Logic
I Coalitional Logic: Reasoning about (local) group power.
[C ]ϕ: coalition C has a joint action to bring about ϕ.
M. Pauly. A Modal Logic for Coalition Powers in Games. Journal of Logic andComputation 12 (2002).
I Alternating-time Temporal Logic: Reasoning about (local andglobal) group power:
〈〈A〉〉ϕ: The coalition A has a joint action to ensure that ϕwill remain true.
R. Alur, T. Henzinger and O. Kupferman. Alternating-time Temproal Logic.Jouranl of the ACM (2002).
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Basic Ingredients
From Temporal Logic to Strategy Logic
I Coalitional Logic: Reasoning about (local) group power.
[C ]ϕ: coalition C has a joint action to bring about ϕ.
M. Pauly. A Modal Logic for Coalition Powers in Games. Journal of Logic andComputation 12 (2002).
I Alternating-time Temporal Logic: Reasoning about (local andglobal) group power:
〈〈A〉〉ϕ: The coalition A has a joint action to ensure that ϕwill remain true.
R. Alur, T. Henzinger and O. Kupferman. Alternating-time Temproal Logic.Jouranl of the ACM (2002).
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Basic Ingredients
Multi-agent Transition Systems
x = 1q0
x = 2q1
〈set2, grant〉 〈set1, grant〉
〈∗, deny〉
〈∗, deny〉
(Px=1 → [s]Px=1) ∧ (Px=2 → [s]Px=2)
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Basic Ingredients
Multi-agent Transition Systems
x = 1q0
x = 2q1
〈set2, grant〉 〈set1, grant〉
〈∗, deny〉
〈∗, deny〉
Px=1 → ¬[s]Px=2
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Basic Ingredients
Multi-agent Transition Systems
x = 1q0
x = 2q1
〈set2, grant〉 〈set1, grant〉
〈∗, deny〉
〈∗, deny〉
Px=1 → ¬[s, c]Px=2
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Basic Ingredients
Basic Ingredients
X informational attitudes (knowledge, group knowledge, belief,certainty, etc.)
X time, actions and ability (individual and coalitional ability)
I motivational attitudes
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Basic Ingredients
Preference (Modal) Logics
x , y objects
x y : x is at least as good as y
1. x y and y 6 x (x y)
2. x 6 y and y x (y x)
3. x y and y x (x ∼ y)
4. x 6 y and y 6 x (x ⊥ y)
Properties: transitivity, connectedness, etc.
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Basic Ingredients
Preference (Modal) Logics
x , y objects
x y : x is at least as good as y
1. x y and y 6 x (x y)
2. x 6 y and y x (y x)
3. x y and y x (x ∼ y)
4. x 6 y and y 6 x (x ⊥ y)
Properties: transitivity, connectedness, etc.
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Basic Ingredients
Preference (Modal) Logics
x , y objects
x y : x is at least as good as y
1. x y and y 6 x (x y)
2. x 6 y and y x (y x)
3. x y and y x (x ∼ y)
4. x 6 y and y 6 x (x ⊥ y)
Properties: transitivity, connectedness, etc.
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Basic Ingredients
Preference (Modal) Logics
Modal betterness model M = 〈W ,,V 〉
Preference Modalities 〈〉ϕ: “there is a world at least as good(as the current world) satisfying ϕ”
M,w |= 〈〉ϕ iff there is a v w such that M, v |= ϕ
M,w |= 〈〉ϕ iff there is v w and w 6 v such that M, v |= ϕ
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Basic Ingredients
Preference (Modal) Logics
Modal betterness model M = 〈W ,,V 〉
Preference Modalities 〈〉ϕ: “there is a world at least as good(as the current world) satisfying ϕ”
M,w |= 〈〉ϕ iff there is a v w such that M, v |= ϕ
M,w |= 〈〉ϕ iff there is v w and w 6 v such that M, v |= ϕ
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Basic Ingredients
Preference (Modal) Logics
Modal betterness model M = 〈W ,,V 〉
Preference Modalities 〈〉ϕ: “there is a world at least as good(as the current world) satisfying ϕ”
M,w |= 〈〉ϕ iff there is a v w such that M, v |= ϕ
M,w |= 〈〉ϕ iff there is v w and w 6 v such that M, v |= ϕ
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Basic Ingredients
Preference (Modal) Logics
1. 〈〉ϕ→ 〈〉ϕ2. 〈〉〈〉ϕ→ 〈〉ϕ3. ϕ ∧ 〈〉ψ → (〈〉ψ ∨ 〈〉(ψ ∧ 〈〉ϕ))
4. 〈〉〈〉ϕ→ 〈〉ϕ
Theorem The above logic (with Necessitation and Modus Ponens)is sound and complete with respect to the class of preferencemodels.
J. van Benthem, O. Roy and P. Girard. Everything else being equal: A modallogic approach to ceteris paribus preferences. JPL, 2008.
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Basic Ingredients
Preference Modalities
ϕ ≥ ψ: the state of affairs ϕ is at least as good as ψ(ceteris paribus)
G. von Wright. The logic of preference. Edinburgh University Press (1963).
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Basic Ingredients
From worlds to sets and back
Lifting
I X ≥∀∃ Y if ∀y ∈ Y ∃x ∈ X : x y
A(ϕ→ 〈〉ψ)
I X ≥∀∀ Y if ∀y ∈ Y ∀x ∈ X : x yA(ϕ→ []¬ψ)
DerivingP1 >> P2 >> P3 >> · · · >> Pn
x > y iff x and y differ in at least one Pi and the first Pi wherethis happens is one with Pix and ¬Piy
F. Liu and D. De Jongh. Optimality, belief and preference. 2006.
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Basic Ingredients
From worlds to sets and back
Lifting
I X ≥∀∃ Y if ∀y ∈ Y ∃x ∈ X : x yA(ϕ→ 〈〉ψ)
I X ≥∀∀ Y if ∀y ∈ Y ∀x ∈ X : x yA(ϕ→ []¬ψ)
DerivingP1 >> P2 >> P3 >> · · · >> Pn
x > y iff x and y differ in at least one Pi and the first Pi wherethis happens is one with Pix and ¬Piy
F. Liu and D. De Jongh. Optimality, belief and preference. 2006.
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Basic Ingredients
From worlds to sets and back
Lifting
I X ≥∀∃ Y if ∀y ∈ Y ∃x ∈ X : x yA(ϕ→ 〈〉ψ)
I X ≥∀∀ Y if ∀y ∈ Y ∀x ∈ X : x yA(ϕ→ []¬ψ)
DerivingP1 >> P2 >> P3 >> · · · >> Pn
x > y iff x and y differ in at least one Pi and the first Pi wherethis happens is one with Pix and ¬Piy
F. Liu and D. De Jongh. Optimality, belief and preference. 2006.
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Basic Ingredients
From worlds to sets and back
Lifting
I X ≥∀∃ Y if ∀y ∈ Y ∃x ∈ X : x yA(ϕ→ 〈〉ψ)
I X ≥∀∀ Y if ∀y ∈ Y ∀x ∈ X : x yA(ϕ→ []¬ψ)
DerivingP1 >> P2 >> P3 >> · · · >> Pn
x > y iff x and y differ in at least one Pi and the first Pi wherethis happens is one with Pix and ¬Piy
F. Liu and D. De Jongh. Optimality, belief and preference. 2006.
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Basic Ingredients
The Logic of Group Decisions
Fundamental Problem: groups are inconsistent!
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Basic Ingredients
The Logic of Group Decisions
Fundamental Problem: groups are inconsistent!
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Basic Ingredients
The Logic of Group Decisions: The Doctrinal “Paradox”(Kornhauser and Sager 1993)
p: a valid contract was in placeq: there was a breach of contractr : the court is required to find the defendant liable.
p q (p ∧ q)↔ r r
1 yes yes yes yes
2 yes no yes no
3 no yes yes no
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Basic Ingredients
The Logic of Group Decisions: The Doctrinal “Paradox”(Kornhauser and Sager 1993)
Should we accept r?
p q (p ∧ q)↔ r r
1 yes yes yes yes
2 yes no yes no
3 no yes yes no
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Basic Ingredients
The Logic of Group Decisions: The Doctrinal “Paradox”(Kornhauser and Sager 1993)
Should we accept r? No, a simple majority votes no.
p q (p ∧ q)↔ r r
1 yes yes yes yes
2 yes no yes no
3 no yes yes no
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Basic Ingredients
The Logic of Group Decisions: The Doctrinal “Paradox”(Kornhauser and Sager 1993)
Should we accept r? Yes, a majority votes yes for p and q and(p ∧ q)↔ r is a legal doctrine.
p q (p ∧ q)↔ r r
1 yes yes yes yes
2 yes no yes no
3 no yes yes no
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”
a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”
b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 151
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True
True
2 True False False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 152
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 153
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False
False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 154
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 155
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True
False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 156
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority
True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 157
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True
True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 158
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True True
False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 159
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 160
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Basic Ingredients
Discursive Dilemmaa: “Carbon dioxide emissions are above the threshold x”a→ b: “If carbon dioxide emissions are above the threshold x ,then there will be global warming”b “There will be global warming”
a a→ b b
1 True True True
2 True False False
3 False True False
Majority True True False
Conclusion: Groups are inconsistent, difference between‘premise-based’ and ‘conclusion-based’ decision making, ...
Eric Pacuit 161
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Basic Ingredients
Group Preference Logics
H. Andreka, M. Ryan and P Yves Schobbens. Operators and laws for combiningpreference relations. Journal of Logic and Computation, 2002.
P. Girard. Modal Logic for Lexicographic Preference Aggregation. Manuscript,2008.
Eric Pacuit 162
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Basic Ingredients
Basic Ingredients
X informational attitudes (knowledge, group knowledge, belief,certainty, etc.)
X time, actions and ability (individual and coalitional ability)
X motivational attitudes (individual preferences, grouppreferences)
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General Issues
General Issues
Once a semantics and language are fixed, then standard questionscan be asked: eg. develop a proof theory, completeness,decidability, model checking.
Eric Pacuit 164
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General Issues
General Issues
How should we compare the different logical systems?
I Embedding one logic in another:
coalition logic is a fragmentof ATL (tr([C ]ϕ) = 〈〈C 〉〉 © ϕ)
I Compare different models for a fixed language:
• Alternating-Time Temporal Logics: Three different semanticsfor the ATL language.
V. Goranko and W. Jamroga. Comparing Semantics of Logics for MultiagentSystems. KRA, 2004.
I Comparing different frameworks: eg. PDL vs. TemporalLogic, PDL vs. STIT, STIT vs. ATL, etc.
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General Issues
General Issues
How should we compare the different logical systems?
I Embedding one logic in another: coalition logic is a fragmentof ATL (tr([C ]ϕ) = 〈〈C 〉〉 © ϕ)
I Compare different models for a fixed language:
• Alternating-Time Temporal Logics: Three different semanticsfor the ATL language.
V. Goranko and W. Jamroga. Comparing Semantics of Logics for MultiagentSystems. KRA, 2004.
I Comparing different frameworks: eg. PDL vs. TemporalLogic, PDL vs. STIT, STIT vs. ATL, etc.
Eric Pacuit 166
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General Issues
General Issues
How should we compare the different logical systems?
I Embedding one logic in another: coalition logic is a fragmentof ATL (tr([C ]ϕ) = 〈〈C 〉〉 © ϕ)
I Compare different models for a fixed language:
• Alternating-Time Temporal Logics: Three different semanticsfor the ATL language.
V. Goranko and W. Jamroga. Comparing Semantics of Logics for MultiagentSystems. KRA, 2004.
I Comparing different frameworks: eg. PDL vs. TemporalLogic, PDL vs. STIT, STIT vs. ATL, etc.
Eric Pacuit 167
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General Issues
General Issues
How should we compare the different logical systems?
I Embedding one logic in another: coalition logic is a fragmentof ATL (tr([C ]ϕ) = 〈〈C 〉〉 © ϕ)
I Compare different models for a fixed language:
• Alternating-Time Temporal Logics: Three different semanticsfor the ATL language.
V. Goranko and W. Jamroga. Comparing Semantics of Logics for MultiagentSystems. KRA, 2004.
I Comparing different frameworks: eg. PDL vs. TemporalLogic, PDL vs. STIT, STIT vs. ATL, etc.
Eric Pacuit 168
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General Issues
General Issues
How should we compare the different logical systems?
I Embedding one logic in another: coalition logic is a fragmentof ATL (tr([C ]ϕ) = 〈〈C 〉〉 © ϕ)
I Compare different models for a fixed language:
• Alternating-Time Temporal Logics: Three different semanticsfor the ATL language.
V. Goranko and W. Jamroga. Comparing Semantics of Logics for MultiagentSystems. KRA, 2004.
I Comparing different frameworks:
eg. PDL vs. TemporalLogic, PDL vs. STIT, STIT vs. ATL, etc.
Eric Pacuit 169
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General Issues
General Issues
How should we compare the different logical systems?
I Embedding one logic in another: coalition logic is a fragmentof ATL (tr([C ]ϕ) = 〈〈C 〉〉 © ϕ)
I Compare different models for a fixed language:
• Alternating-Time Temporal Logics: Three different semanticsfor the ATL language.
V. Goranko and W. Jamroga. Comparing Semantics of Logics for MultiagentSystems. KRA, 2004.
I Comparing different frameworks: eg. PDL vs. TemporalLogic, PDL vs. STIT, STIT vs. ATL, etc.
Eric Pacuit 170
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General Issues
General Issues
How should we merge the different logical systems?
I Combining logics is hard!
D. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev. Many DimensionalModal Logics: Theory and Applications. 2003.
Theorem ϕ↔ ϕ is provable in combinations of Epistemic Logicsand PDL with certain “cross axioms” ([a]ϕ↔ [a]ϕ) (and fullsubstitution).
R. Schmidt and D. Tishkovsky. On combinations of propositional dynamic logicand doxastic modal logics. JOLLI, 2008.
Eric Pacuit 171
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General Issues
General Issues
How should we merge the different logical systems?
I Combining logics is hard!
D. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev. Many DimensionalModal Logics: Theory and Applications. 2003.
Theorem ϕ↔ ϕ is provable in combinations of Epistemic Logicsand PDL with certain “cross axioms” ([a]ϕ↔ [a]ϕ) (and fullsubstitution).
R. Schmidt and D. Tishkovsky. On combinations of propositional dynamic logicand doxastic modal logics. JOLLI, 2008.
Eric Pacuit 172
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General Issues
General Issues
How should we merge the different logical systems?
I Combining logics is hard!
D. Gabbay, A. Kurucz, F. Wolter and M. Zakharyaschev. Many DimensionalModal Logics: Theory and Applications. 2003.
Theorem ϕ↔ ϕ is provable in combinations of Epistemic Logicsand PDL with certain “cross axioms” ([a]ϕ↔ [a]ϕ) (and fullsubstitution).
R. Schmidt and D. Tishkovsky. On combinations of propositional dynamic logicand doxastic modal logics. JOLLI, 2008.
Eric Pacuit 173
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General Issues
Merging logics of rational agency
I Reasoning about information change (knowledge andtime/actions)
I Knowledge, beliefs and certainty
I “Epistemizing” logics of action and ability: knowing how toachieve ϕ vs. knowing that you can achieve ϕ
I Entangling knowledge and preferences
I Planning/intentions (BDI)
Eric Pacuit 174
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General Issues
Merging logics of rational agency
Reasoning about information change (knowledge andtime/actions)
I Knowledge, beliefs and certainty
I “Epistemizing” logics of action and ability: knowing how toachieve ϕ vs. knowing that you can achieve ϕ
Entangling knowledge and preferences
Planning/intentions (BDI)
Conclusions
Eric Pacuit 175
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General Issues
Example
Ann would like Bob to attend her talk; however, she only wantsBob to attend if he is interested in the subject of her talk, notbecause he is just being polite.
There is a very simple procedure to solve Ann’s problem: have a(trusted) friend tell Bob the time and subject of her talk.
Is this procedure correct?
Eric Pacuit 176
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General Issues
Example
Ann would like Bob to attend her talk; however, she only wantsBob to attend if he is interested in the subject of her talk, notbecause he is just being polite.
There is a very simple procedure to solve Ann’s problem: have a(trusted) friend tell Bob the time and subject of her talk.
Is this procedure correct? Yes, if
1. Ann knows about the talk.
2. Bob knows about the talk.
3. Ann knows that Bob knows about the talk.
4. Bob does not know that Ann knows that he knows about thetalk.
5. And nothing else.
Eric Pacuit 177
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General Issues
Example
P
s
¬P
t
B
A, BA, B
P means “The talk is at 2PM”.
Eric Pacuit 178
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General Issues
Example
P
s
¬P
t
B
A, BA, B
P means “The talk is at 2PM”.
M, s |= KAP ∧ ¬KBP
Eric Pacuit 179
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General Issues
Example
P
s
¬P
t
B
A, BA, B
P means “The talk is at 2PM”.
M, s |= KAP ∧ ¬KBP
Eric Pacuit 180
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General Issues
Example
P
s
¬P
t
B
A, BA, B
Pw1 P w2
¬P w4Pw3
B
A
B
A
Eric Pacuit 181
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General Issues
Two Methodologies
ETL methodology: when describing a social situation, first writedown all possible sequences of events, then at each moment writedown the agents’ uncertainty, from that infer how the agents’knowledge changes from one moment to the next.
Alternative methodology: describe an initial situations, provide amethod for how events change a model that can be described inthe formal language, then construct the event tree as needed.
Dynamic Epistemic Logic
Eric Pacuit 182
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General Issues
Two Methodologies
ETL methodology: when describing a social situation, first writedown all possible sequences of events, then at each moment writedown the agents’ uncertainty, from that infer how the agents’knowledge changes from one moment to the next.
Alternative methodology: describe an initial situations, provide amethod for how events change a model that can be described inthe formal language, then construct the event tree as needed.
Dynamic Epistemic Logic
Eric Pacuit 183
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General Issues
Two Methodologies
ETL methodology: when describing a social situation, first writedown all possible sequences of events, then at each moment writedown the agents’ uncertainty, from that infer how the agents’knowledge changes from one moment to the next.
Alternative methodology: describe an initial situations, provide amethod for how events change a model that can be described inthe formal language, then construct the event tree as needed.
Dynamic Epistemic Logic
Eric Pacuit 184
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General Issues
Epistemic Temporal Logic
R. Parikh and R. Ramanujam. A Knowledge Based Semantics of Messages.Journal of Logic, Language and Information, 12: 453 – 467, 1985, 2003.
FHMV. Reasoning about Knowledge. MIT Press, 1995.
Eric Pacuit 185
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General Issues
The ‘Playground’
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e6
e7 e3
e2 e1 e2
e4 e2
e1 e3
e7
Eric Pacuit 186
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General Issues
The ‘Playground’
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e6
e7 e3
e2 e1 e2
e4 e2
e1 e3
e7
Eric Pacuit 187
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General Issues
The ‘Playground’
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e6
e7 e3
i
ii
j
j
e2 e1 e2
e4 e2
e1 e3
e7
Eric Pacuit 188
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General Issues
Formal Languages
I Pϕ (ϕ is true sometime in the past),
I Fϕ (ϕ is true sometime in the future),
I Yϕ (ϕ is true at the previous moment),
I Nϕ (ϕ is true at the next moment),
I Neϕ (ϕ is true after event e)
I Kiϕ (agent i knows ϕ) and
I CBϕ (the group B ⊆ A commonly knows ϕ).
Eric Pacuit 189
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General Issues
History-based Models
An ETL model is a structure 〈H, ∼ii∈A,V 〉 where 〈H, ∼ii∈A〉is an ETL frame and
V : At→ 2finite(H) is a valuation function.
Formulas are interpreted at pairs H, t:
H, t |= ϕ
Eric Pacuit 190
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General Issues
Truth in a Model
I H, t |= Pϕ iff there exists t ′ ≤ t such that H, t ′ |= ϕ
I H, t |= Fϕ iff there exists t ′ ≥ t such that H, t ′ |= ϕ
I H, t |= Nϕ iff H, t + 1 |= ϕ
I H, t |= Yϕ iff t > 1 and H, t − 1 |= ϕ
I H, t |= Kiϕ iff for each H ′ ∈ H and m ≥ 0 if Ht ∼i H ′m thenH ′,m |= ϕ
I H, t |= Cϕ iff for each H ′ ∈ H and m ≥ 0 if Ht ∼∗ H ′m thenH ′,m |= ϕ.
where ∼∗ is the reflexive transitive closure of the union of the ∼i .
Eric Pacuit 191
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General Issues
Truth in a Model
I H, t |= Pϕ iff there exists t ′ ≤ t such that H, t ′ |= ϕ
I H, t |= Fϕ iff there exists t ′ ≥ t such that H, t ′ |= ϕ
I H, t |= Nϕ iff H, t + 1 |= ϕ
I H, t |= Yϕ iff t > 1 and H, t − 1 |= ϕ
I H, t |= Kiϕ iff for each H ′ ∈ H and m ≥ 0 if Ht ∼i H ′m thenH ′,m |= ϕ
I H, t |= Cϕ iff for each H ′ ∈ H and m ≥ 0 if Ht ∼∗ H ′m thenH ′,m |= ϕ.
where ∼∗ is the reflexive transitive closure of the union of the ∼i .
Eric Pacuit 192
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General Issues
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e6
e7 e3
i
ii
j e2 e1 e2
e4 e2
e1 e3
e7
Eric Pacuit 193
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General Issues
Returning to the Example
Ann would like Bob to attend her talk; however, she only wantsBob to attend if he is interested in the subject of her talk, notbecause he is just being polite.
There is a very simple procedure to solve Ann’s problem: have a(trusted) friend tell Bob the time and subject of her talk.
Is this procedure correct?
Eric Pacuit 194
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General Issues
Returning to the Example
Ann would like Bob to attend her talk; however, she only wantsBob to attend if he is interested in the subject of her talk, notbecause he is just being polite.
There is a very simple procedure to solve Ann’s problem: have a(trusted) friend tell Bob the time and subject of her talk.
Is this procedure correct?
Eric Pacuit 195
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General Issues
Returning to the Example
Ann would like Bob to attend her talk; however, she only wantsBob to attend if he is interested in the subject of her talk, notbecause he is just being polite.
There is a very simple procedure to solve Ann’s problem: have a(trusted) friend tell Bob the time and subject of her talk.
Is this procedure correct?
Eric Pacuit 196
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t = 0
t = 1
t = 2
t = 3
m2PM m3PM
mA→C t mA→C t
mC→B
t
mC→B
t
mC→B
t
mC→B
t
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t = 0
t = 1
t = 2
t = 3
m2PM m3PM
mA→C t mA→C t
mC→B
t
mC→B
t
mC→B
t
mC→B
t
H, 3 |= ϕ
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t = 0
t = 1
t = 2
t = 3
m2PM m3PM
mA→C t mA→C t
mC→B
t
mC→B
t
mC→B
t
mC→B
t
Bob’s uncertainty: H, 3 |= ¬KBP2PM
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t = 0
t = 1
t = 2
t = 3
m2PM m3PM
mA→C t t
mC→B
t
mC→B
t t
Bob’s uncertainty + ‘Protocol information’: H, 3 |= KBP2PM
![Page 201: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/201.jpg)
t = 0
t = 1
t = 2
t = 3
m2PM m3PM
mA→C t t
mC→B
t
mC→B
t t
Bob’s uncertainty + ‘Protocol information’:H, 3 |= ¬KBKAKBP2PM
![Page 202: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/202.jpg)
t = 0
t = 1
t = 2
t = 3
m2PM m3PM
mA→C t t
mC→B
t
mC→B
t t
Bob’s uncertainty + ‘Protocol information’:H, 3 |= ¬KBKAKBP2PM
![Page 203: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/203.jpg)
t = 0
t = 1
t = 2
t = 3
m2PM m3PM
mA→C t t
mC→B
t
mC→B
t t
Bob’s uncertainty + ‘Protocol information’:H, 3 |= ¬KBKAKBP2PM
![Page 204: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/204.jpg)
t = 0
t = 1
t = 2
t = 3
m2PM m3PM
mA→C t t
mC→B
t
mC→B
t t
Bob’s uncertainty + ‘Protocol information’:H, 3 |= ¬KBKAKBP2PM
![Page 205: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/205.jpg)
General Issues
Parameters of the Logical Framework
1. Expressivity of the formal language. Does the language includea common knowledge operator? A future operator? Both?
2. Structural conditions on the underlying event structure. Dowe restrict to protocol frames (finitely branching trees)?Finitely branching forests? Or, arbitrary ETL frames?
3. Conditions on the reasoning abilities of the agents. Do theagents satisfy perfect recall? No miracles? Do they agents’know what time it is?
Eric Pacuit 205
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General Issues
Parameters of the Logical Framework
1. Expressivity of the formal language. Does the language includea common knowledge operator? A future operator? Both?
2. Structural conditions on the underlying event structure. Dowe restrict to protocol frames (finitely branching trees)?Finitely branching forests? Or, arbitrary ETL frames?
3. Conditions on the reasoning abilities of the agents. Do theagents satisfy perfect recall? No miracles? Do they agents’know what time it is?
Eric Pacuit 206
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General Issues
Parameters of the Logical Framework
1. Expressivity of the formal language. Does the language includea common knowledge operator? A future operator? Both?
2. Structural conditions on the underlying event structure. Dowe restrict to protocol frames (finitely branching trees)?Finitely branching forests? Or, arbitrary ETL frames?
3. Conditions on the reasoning abilities of the agents. Do theagents satisfy perfect recall? No miracles? Do they agents’know what time it is?
Eric Pacuit 207
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General Issues
Parameters of the Logical Framework
1. Expressivity of the formal language. Does the language includea common knowledge operator? A future operator? Both?
2. Structural conditions on the underlying event structure. Dowe restrict to protocol frames (finitely branching trees)?Finitely branching forests? Or, arbitrary ETL frames?
3. Conditions on the reasoning abilities of the agents. Do theagents satisfy perfect recall? No miracles? Do they agents’know what time it is?
Eric Pacuit 208
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General Issues
Agent Oriented Properties:
I No Miracles: For all finite histories H,H ′ ∈ H and eventse ∈ Σ such that He ∈ H and H ′e ∈ H, if H ∼i H ′ thenHe ∼i H ′e.
I Perfect Recall: For all finite histories H,H ′ ∈ H and eventse ∈ Σ such that He ∈ H and H ′e ∈ H, if He ∼i H ′e thenH ∼i H ′.
I Synchronous: For all finite histories H,H ′ ∈ H, if H ∼i H ′
then len(H) = len(H ′).
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General Issues
Perfect Recall
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e3
e7 e6
e2 e1 e2
e4 e2
e1 e3
e7
i
Eric Pacuit 210
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General Issues
Perfect Recall
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e3
e7 e6
e2 e1 e2
e4 e2
e1 e3
e7
i
i
Eric Pacuit 211
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General Issues
Perfect Recall
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e3
e7 e6
e2 e1 e2
e4 e2
e1 e3
e7
i
i
i
i
Eric Pacuit 212
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General Issues
No Miracles
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e3
e7 e6
e2 e1 e2
e4 e2
e1 e3
e7i
Eric Pacuit 213
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General Issues
No Miracles
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e3
e7 e6
e2 e1 e2
e4 e2
e1 e3
e7i
i
Eric Pacuit 214
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General Issues
No Miracles
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e1
e7 e6
e2 e1 e2
e4 e2
e1 e5
e7
i
i
i
Eric Pacuit 215
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General Issues
Ideal Agents
Assume there are two agents
TheoremThe logic of ideal agents with respect to a language with commonknowledge and future is highly undecidable (for example, byassuming perfect recall).
J. Halpern and M. Vardi.. The Complexity of Reasoning abut Knowledge andTime. J. Computer and Systems Sciences, 38, 1989.
J. van Benthem and EP. The Tree of Knowledge in Action. Proceedings of AiML,2006.
Eric Pacuit 216
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General Issues
Two Methodologies
ETL methodology: when describing a social situation, first writedown all possible sequences of events, then at each moment writedown the agents’ uncertainty, from that infer how the agents’knowledge changes from one moment to the next.
Alternative methodology: describe an initial situations, provide amethod for how events change a model that can be described inthe formal language, then construct the event tree as needed.
Dynamic Epistemic Logic
Eric Pacuit 217
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General Issues
Returning to the Example: DEL
Eric Pacuit 218
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General Issues
Returning to the Example: DEL
(M⊗ E1)⊗ E2
The initial model (Annand Bob are ignorantabout P2PM).
Private announcementto Ann about the talk.
Eric Pacuit 219
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General Issues
Returning to the Example: DEL
(M⊗ E1)⊗ E2
The initial model (Annand Bob are ignorantabout P2PM).
Private announcementto Ann about the talk.
Eric Pacuit 220
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General Issues
Abstract Description of the Event
Recall the Ann and Bob example: Charles tells Bob that the talk isat 2PM.
Eric Pacuit 221
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General Issues
Abstract Description of the Event
Recall the Ann and Bob example: Charles tells Bob that the talk isat 2PM.
Pe1 P e2
>e3
B
BA
A
A, B
Ann knows which event took place.
Eric Pacuit 222
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General Issues
Abstract Description of the Event
Recall the Ann and Bob example: Charles tells Bob that the talk isat 2PM.
Pe1 P e2
>e3
B
BA
A
A, B
Ann knows which event took place.
Eric Pacuit 223
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General Issues
Abstract Description of the Event
Recall the Ann and Bob example: Charles tells Bob that the talk isat 2PM.
Pe1 P e2
>e3
B
BA
A
A, B
Bob thinks a different event took place.
Eric Pacuit 224
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General Issues
Abstract Description of the Event
Recall the Ann and Bob example: Charles tells Bob that the talk isat 2PM.
Pe1 P e2
>e3
B
BA
A
A, B
That is, Bob learns the time of the talk, but Ann learns nothing.
Eric Pacuit 225
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General Issues
Product Update
Eric Pacuit 226
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General Issues
Product Update
M⊗ E1
P
s
¬P
t
B
A, BA, B
E2
Pe1 P e2
>e3
B
BA
A
A, B
Eric Pacuit 227
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General Issues
Product Update
P
s
¬P
t
B
A, BA, B
Pe1 P e2
>e3
B
BA
A
A, B
P(s, e1) P (s, e2)
¬P (t, e3)P(s, e3)
Eric Pacuit 228
![Page 229: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/229.jpg)
General Issues
Product Update
P
s
¬P
t
B
A, BA, B
Pe1 P e2
>e3
B
BA
A
A, B
P(s, e1) P (s, e2)
¬P (t, e3)P(s, e3)
Eric Pacuit 229
![Page 230: Logics of Rational Agencyepacuit/talks/lograt-tutorial.pdfBasic Ingredients I What are the basic building blocks? (the nature of time (continuous or discrete/branching or linear),](https://reader034.vdocuments.net/reader034/viewer/2022042419/5f3677403c84f50569432bd7/html5/thumbnails/230.jpg)
General Issues
Product Update
P
s
¬P
t
B
A, BA, B
Pe1 P e2
>e3
B
BA
A
A, B
(s, e1) |= ¬KBKAKBP P(s, e1) P (s, e2)
¬P (t, e3)P(s, e3)
Eric Pacuit 230
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General Issues
Product Update
P
s
¬P
t
B
A, BA, B
Pe1 P e2
>e3
B
BA
A
A, B
(s, e1) |= ¬KBKAKBP P(s, e1) P (s, e2)
¬P (t, e3)P(s, e3)
B
Eric Pacuit 231
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General Issues
Product Update
P
s
¬P
t
B
A, BA, B
Pe1 P e2
>e3
B
BA
A
A, B
(s, e1) |= ¬KBKAKBP P(s, e1) P (s, e2)
¬P (t, e3)P(s, e3)
B
A
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General Issues
Product Update
P
s
¬P
t
B
A, BB
Pe1 P e2
>e3
B
BA
A
A, B
(s, e1) |= ¬KBKAKBP P(s, e1) P (s, e2)
¬P (t, e3)P(s, e3)
B
A
B
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General Issues
Product Update Details
Let M = 〈W ,R,V 〉 be a Kripke model.
An event model is a tuple A = 〈A, S ,Pre〉, where S ⊆ A× A andPre : L → ℘(A).
The update model M⊗ A = 〈W ′,R ′,V ′〉 where
I W ′ = (w , a) | w |= Pre(a)I (w , a)R ′(w ′, a′) iff wRw ′ and aSa′
I (w , a) ∈ V (p) iff w ∈ V (p)
M,w |= [A, a]ϕ iff M,w |= Pre(a) implies M⊗ A, (w , a) |= ϕ.
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General Issues
Product Update Details
Let M = 〈W ,R,V 〉 be a Kripke model.
An event model is a tuple A = 〈A, S ,Pre〉, where S ⊆ A× A andPre : L → ℘(A).
The update model M⊗ A = 〈W ′,R ′,V ′〉 where
I W ′ = (w , a) | w |= Pre(a)I (w , a)R ′(w ′, a′) iff wRw ′ and aSa′
I (w , a) ∈ V (p) iff w ∈ V (p)
M,w |= [A, a]ϕ iff M,w |= Pre(a) implies M⊗ A, (w , a) |= ϕ.
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General Issues
Product Update Details
Let M = 〈W ,R,V 〉 be a Kripke model.
An event model is a tuple A = 〈A, S ,Pre〉, where S ⊆ A× A andPre : L → ℘(A).
The update model M⊗ A = 〈W ′,R ′,V ′〉 where
I W ′ = (w , a) | w |= Pre(a)
I (w , a)R ′(w ′, a′) iff wRw ′ and aSa′
I (w , a) ∈ V (p) iff w ∈ V (p)
M,w |= [A, a]ϕ iff M,w |= Pre(a) implies M⊗ A, (w , a) |= ϕ.
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General Issues
Product Update Details
Let M = 〈W ,R,V 〉 be a Kripke model.
An event model is a tuple A = 〈A, S ,Pre〉, where S ⊆ A× A andPre : L → ℘(A).
The update model M⊗ A = 〈W ′,R ′,V ′〉 where
I W ′ = (w , a) | w |= Pre(a)I (w , a)R ′(w ′, a′) iff wRw ′ and aSa′
I (w , a) ∈ V (p) iff w ∈ V (p)
M,w |= [A, a]ϕ iff M,w |= Pre(a) implies M⊗ A, (w , a) |= ϕ.
Eric Pacuit 237
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General Issues
Product Update Details
Let M = 〈W ,R,V 〉 be a Kripke model.
An event model is a tuple A = 〈A, S ,Pre〉, where S ⊆ A× A andPre : L → ℘(A).
The update model M⊗ A = 〈W ′,R ′,V ′〉 where
I W ′ = (w , a) | w |= Pre(a)I (w , a)R ′(w ′, a′) iff wRw ′ and aSa′
I (w , a) ∈ V (p) iff w ∈ V (p)
M,w |= [A, a]ϕ iff M,w |= Pre(a) implies M⊗ A, (w , a) |= ϕ.
Eric Pacuit 238
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General Issues
Product Update Details
Let M = 〈W ,R,V 〉 be a Kripke model.
An event model is a tuple A = 〈A, S ,Pre〉, where S ⊆ A× A andPre : L → ℘(A).
The update model M⊗ A = 〈W ′,R ′,V ′〉 where
I W ′ = (w , a) | w |= Pre(a)I (w , a)R ′(w ′, a′) iff wRw ′ and aSa′
I (w , a) ∈ V (p) iff w ∈ V (p)
M,w |= [A, a]ϕ iff M,w |= Pre(a) implies M⊗ A, (w , a) |= ϕ.
Eric Pacuit 239
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General Issues
Literarture
A. Baltag and L. Moss. Logics for Epistemic Programs. 2004.
W. van der Hoek, H. van Ditmarsch and B. Kooi. Dynamic Episetmic Logic.2007.
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General Issues
Some Questions
I How do we relate the ETL-style analysis with the DEL-styleanalysis?
I In the DEL setting, what are the underlying assumptionsabout the reasoning abilities of the agents?
I Can we axiomatize interesting subclasses of ETL frames?
J. van Benthem, J. Gerbrandy, T. Hoshi, EP. Merging Frameworks for Interaction.JPL, 2009.
Skip Details
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General Issues
DEL and ETL
Observation: By repeatedly updating an epistemic model withevent models, the machinery of DEL creates ETL models.
Let M be an epistemic model, and P a DEL protocol (tree of eventmodels). The ETL model generated by M and P, forest(M,P),represents all possible evolutions of the system obtained byupdating M with sequences from P.
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General Issues
DEL and ETL
Observation: By repeatedly updating an epistemic model withevent models, the machinery of DEL creates ETL models.
Let M be an epistemic model, and P a DEL protocol (tree of eventmodels). The ETL model generated by M and P, forest(M,P),represents all possible evolutions of the system obtained byupdating M with sequences from P.
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General Issues
Example: Initial Model and Protocol
P,Qs
P,Q,Rt P,R u
Q,R vi
i
i
j
j
j
!P
!Q !R
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General Issues
Example
P, Qs
P, Q, Rt P, R u
Q, R vi
i
i
j
j
j
!P
!Q !R
(s, !P, !Q) (t, !P, !Q) (t, !P, !R) (u, !P, !R)
(s, !P)
(s) (t)
(t, !P)
(u)
(u, !P)
(v)
!P
!Q
!P
!Q !R
!P
!R
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General Issues
Example
P, Qs
P, Q, Rt P, R u
Q, R vi
i
i
j
j
j
!P
!Q !R
(s, !P, !Q) (t, !P, !Q) (t, !P, !R) (u, !P, !R)
(s, !P)
(s) (t)
(t, !P)
(u)
(u, !P)
(v)
!P
!Q
!P
!Q !R
!P
!R
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General Issues
Example
P, Qs
P, Q, Rt P, R u
Q, R vi
i
i
j
j
j
!P
!Q !R
(s, !P, !Q) (t, !P, !Q) (t, !P, !R) (u, !P, !R)
(s, !P)
(s) (t)
(t, !P)
(u)
(u, !P)
(v)
!P
!Q
!P
!Q !R
!P
!R
Eric Pacuit 247
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General Issues
Example
P, Qs
P, Q, Rt P, R u
Q, R vi
i
i
j
j
j
!P
!Q !R
(s, !P, !Q) (t, !P, !Q) (t, !P, !R) (u, !P, !R)
(s, !P)
(s) (t)
(t, !P)
(u)
(u, !P)
(v)
!P
!Q
!P
!Q !R
!P
!R
Eric Pacuit 248
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General Issues
Example
P, Qs
P, Q, Rt P, R u
Q, R vi
i
i
j
j
j
!P
!Q !R
(s, !P, !Q) (t, !P, !Q) (t, !P, !R) (u, !P, !R)
(s, !P)
(s) (t)
(t, !P)
(u)
(u, !P)
(v)
!P
!Q
!P
!Q !R
!P
!R
Eric Pacuit 249
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General Issues
Example
P, Qs
P, Q, Rt P, R u
Q, R vi
i
i
j
j
j
!P
!Q !R
(s, !P, !Q) (t, !P, !Q) (t, !P, !R) (u, !P, !R)
(s, !P)
(s) (t)
(t, !P)
(u)
(u, !P)
(v)
!P
!Q
!P
!Q !R
!P
!R
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General Issues
Example
P, Qs
P, Q, Rt P, R u
Q, R vi
i
i
j
j
j
!P
!Q !R
(s, !P, !Q) (t, !P, !Q) (t, !P, !R) (u, !P, !R)
(s, !P)
(s) (t)
(t, !P)
(u)
(u, !P)
(v)
!P
!Q
!P
!Q !R
!P
!R
Eric Pacuit 251
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General Issues
Example
P, Qs
P, Q, Rt P, R u
Q, R vi
i
i
j
j
j
!P
!Q !R
(s, !P, !Q) (t, !P, !Q) (t, !P, !R) (u, !P, !R)
(s, !P)
(s) (t)
(t, !P)
(u)
(u, !P)
(v)
!P
!Q
!P
!Q !R
!P
!R
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General Issues
Example
P, Qs
P, Q, Rt P, R u
Q, R vi
i
i
j
j
j
!P
!Q !R
(s, !P, !Q) (t, !P, !Q) (t, !P, !R) (u, !P, !R)
(s, !P)
(s) (t)
(t, !P)
(u)
(u, !P)
(v)
!P
!Q
!P
!Q !R
!P
!R
(t) |= R ∧ ¬〈!R〉>
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General Issues
Representation Result
Given a set of DEL protocols X, let F(X) be the class of ETLframes generated by protocols from X.
Theorem (Main Representation Theorem)
Let Σ be a finite set of events and suppose XuniDEL is the class of
uniform DEL protocols (with a finiteness condition). A model is inF(Xuni
DEL) iff it satisfies propositional stability, synchronicity, perfectrecall, local no miracles, and local bisimulation invariance.
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General Issues
Bisimulation Invariance + Finiteness Condition
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e1
e7 e6
e2 e1 e2
e4 e2
e1 e5
e7
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General Issues
Bisimulation Invariance + Finiteness Condition
t = 0
t = 1
t = 2
t = 3
e2 e4
e1 e5
e1 e3
e2 e1
e7 e6
e2 e1 e2
e4 e2
e1 e5
e7
Eric Pacuit 256
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General Issues
Recall that if X is a set of DEL protocols, we defineF(X) = F(M,P) | M an epistemic model and P ∈ X. Thisconstruction suggests the following natural questions:
I Which DEL protocols generate interesting ETL models?
I Which modal languages are most suitable to describe thesemodels?
I Can we axiomatize interesting classes DEL-generated ETLmodels?
J. van Benthem, J. Gerbrandy, T. Hoshi, EP. Merging Frameworks for Interaction.JPL, 2009.
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General Issues
Announcement + Protocol Information
1. A→ 〈A〉> vs. 〈A〉> → A
2. 〈A〉KiP ↔ A ∧ Ki 〈A〉P
3. 〈A〉KiP ↔ 〈A〉> ∧ Ki (A→ 〈A〉P)
4. 〈A〉KiP ↔ 〈A〉> ∧ Ki (〈A〉> → 〈A〉P)
Theorems Sound and complete axiomatizations of variousgenerated ETL models.
Conclusions
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General Issues
Announcement + Protocol Information
1. A→ 〈A〉> vs. 〈A〉> → A
2. 〈A〉KiP ↔ A ∧ Ki 〈A〉P
3. 〈A〉KiP ↔ 〈A〉> ∧ Ki (A→ 〈A〉P)
4. 〈A〉KiP ↔ 〈A〉> ∧ Ki (〈A〉> → 〈A〉P)
Theorems Sound and complete axiomatizations of variousgenerated ETL models.
Conclusions
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General Issues
Announcement + Protocol Information
1. A→ 〈A〉> vs. 〈A〉> → A
2. 〈A〉KiP ↔ A ∧ Ki 〈A〉P
3. 〈A〉KiP ↔ 〈A〉> ∧ Ki (A→ 〈A〉P)
4. 〈A〉KiP ↔ 〈A〉> ∧ Ki (〈A〉> → 〈A〉P)
Theorems Sound and complete axiomatizations of variousgenerated ETL models.
Conclusions
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General Issues
Announcement + Protocol Information
1. A→ 〈A〉> vs. 〈A〉> → A
2. 〈A〉KiP ↔ A ∧ Ki 〈A〉P
3. 〈A〉KiP ↔ 〈A〉> ∧ Ki (A→ 〈A〉P)
4. 〈A〉KiP ↔ 〈A〉> ∧ Ki (〈A〉> → 〈A〉P)
Theorems Sound and complete axiomatizations of variousgenerated ETL models.
Conclusions
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General Issues
Announcement + Protocol Information
1. A→ 〈A〉> vs. 〈A〉> → A
2. 〈A〉KiP ↔ A ∧ Ki 〈A〉P
3. 〈A〉KiP ↔ 〈A〉> ∧ Ki (A→ 〈A〉P)
4. 〈A〉KiP ↔ 〈A〉> ∧ Ki (〈A〉> → 〈A〉P)
Theorems Sound and complete axiomatizations of variousgenerated ETL models.
Conclusions
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General Issues
Merging logics of rational agency
Reasoning about information change (knowledge andtime/actions)
I Knowledge, beliefs and certainty
I “Epistemizing” logics of action and ability: knowing how toachieve ϕ vs. knowing that you can achieve ϕ
Entangling knowledge and preferences
Planning/intentions (BDI)
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General Issues
Logics of Knowledge and Preference
K (ϕ ψ): “Ann knows that ϕ is at least as good as ψ”
Kϕ Kψ: “knowing ϕ is at least as good as knowing ψ
M = 〈W ,∼,,V 〉
J. van Eijck. Yet more modal logics of preference change and belief revision.manuscript, 2009.
F. Liu. Changing for the Better: Preference Dynamics and Agent Diversity. PhDthesis, ILLC, 2008.
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General Issues
Logics of Knowledge and Preference
K (ϕ ψ): “Ann knows that ϕ is at least as good as ψ”
Kϕ Kψ: “knowing ϕ is at least as good as knowing ψ
M = 〈W ,∼,,V 〉
J. van Eijck. Yet more modal logics of preference change and belief revision.manuscript, 2009.
F. Liu. Changing for the Better: Preference Dynamics and Agent Diversity. PhDthesis, ILLC, 2008.
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General Issues
Logics of Knowledge and Preference
K (ϕ ψ): “Ann knows that ϕ is at least as good as ψ”
Kϕ Kψ: “knowing ϕ is at least as good as knowing ψ
M = 〈W ,∼,,V 〉
J. van Eijck. Yet more modal logics of preference change and belief revision.manuscript, 2009.
F. Liu. Changing for the Better: Preference Dynamics and Agent Diversity. PhDthesis, ILLC, 2008.
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General Issues
A(ψ → 〈〉ϕ) vs. K (ψ → 〈〉ϕ)
Should preferences be restricted to information sets?
M,w |= 〈 ∩ ∼〉ϕ iff there is a v with w ∼ v and w v suchthat M, v |= ϕ
K (ψ → 〈 ∩ ∼〉ϕ)
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General Issues
A(ψ → 〈〉ϕ) vs. K (ψ → 〈〉ϕ)
Should preferences be restricted to information sets?
M,w |= 〈 ∩ ∼〉ϕ iff there is a v with w ∼ v and w v suchthat M, v |= ϕ
K (ψ → 〈 ∩ ∼〉ϕ)
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General Issues
A(ψ → 〈〉ϕ) vs. K (ψ → 〈〉ϕ)
Should preferences be restricted to information sets?
M,w |= 〈 ∩ ∼〉ϕ iff there is a v with w ∼ v and w v suchthat M, v |= ϕ
K (ψ → 〈 ∩ ∼〉ϕ)
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General Issues
Defining Beliefs from Preferences
I Starting with the work of Savage (based on Ramsey and deFinetti), there is a tradition in game theory and decisiontheory to define beliefs and utilities in terms of the agent’spreferences
I Typically the results come in the form of a representationtheorem:
If the agents preferences satisfy such-and-suchproperties, then there is a set of conditionalprobability functions and a (state independent)utility function such that the agent can be assumedto act as an expected utility maximizer.
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General Issues
Defining Beliefs from Preferences
I Starting with the work of Savage (based on Ramsey and deFinetti), there is a tradition in game theory and decisiontheory to define beliefs and utilities in terms of the agent’spreferences
I Typically the results come in the form of a representationtheorem:
If the agents preferences satisfy such-and-suchproperties, then there is a set of conditionalprobability functions and a (state independent)utility function such that the agent can be assumedto act as an expected utility maximizer.
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General Issues
Thus logical properties of beliefs can be derived from properties ofpreferences.
S. Morris. The Logic of Belief and Belief Change: A Decision Theoretic Ap-proach. Journal of Economic Theory (1996).
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General Issues
The Framework
Let Ω be a set of states.
An act is a function x : Ω→ R. Let <Ω be the set of all acts.
xw for w ∈ Ω means that if the true state is w , then the agentreceives prize x .
We write x w y the agent prefers x over y provided the truestate is w
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General Issues
Belief Operators
A belief operator is a function B : 2Ω → 2Ω
For E ⊆ Ω, w ∈ B(E ) means the agent believes E at state w
B is normal if
I B(Ω) = Ω
I B(E ∩ F ) = B(E ) ∩ B(F )
Possibility function: P : Ω→ 2Ω: set of states the agent considerspossible at w
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General Issues
Defining Beliefs from Preferences
For E ⊆ Ω and two acts x and y , let (xE , y−E ) denote the new actthat is x on E and y on −E .
B reflects ww∈Ω provided for each E ⊆ Ω
B(E ) = w | (xE , y−E ) ∼w (xE , z−E ) for all x , y , z ∈ <Ω
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General Issues
Defining Beliefs from Preferences
For E ⊆ Ω and two acts x and y , let (xE , y−E ) denote the new actthat is x on E and y on −E .
B reflects ww∈Ω provided for each E ⊆ Ω
B(E ) = w | (xE , y−E ) ∼w (xE , z−E ) for all x , y , z ∈ <Ω
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General Issues
Theorem If the preference relations are complete and transitive,then the derived belief operator is normal.
S. Morris. The Logic of Belief and Belief Change: A Decision Theoretic Ap-proach. Journal of Economic Theory.
Conclusions
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General Issues
Merging logics of rational agency
Reasoning about information change (knowledge andtime/actions)
I Knowledge, beliefs and certainty
I “Epistemizing” logics of action and ability: knowing how toachieve ϕ vs. knowing that you can achieve ϕ
Entangling knowledge and preferences
Planning/intentions (BDI)
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General Issues
Some Literature
Stemming from Bratman’s planning theory of intention a numberof BDI logics:
I Cohen and Levesque; Rao and Georgeff; Meyer, van der Hoek(KARO); and many others.
Some common features
I Underlying temporal model
I Belief, Desire, Intention, Plans, Actions are defined withcorresponding operators in a language
J.-J. Meyer and F. Veltman. Intelligent Agents and Common Sense Reasoning.Handbook of Modal Logic, 2007.
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General Issues
Some Literature
Stemming from Bratman’s planning theory of intention a numberof BDI logics:
I Cohen and Levesque; Rao and Georgeff; Meyer, van der Hoek(KARO); and many others.
Some common features
I Underlying temporal model
I Belief, Desire, Intention, Plans, Actions are defined withcorresponding operators in a language
J.-J. Meyer and F. Veltman. Intelligent Agents and Common Sense Reasoning.Handbook of Modal Logic, 2007.
Eric Pacuit 280
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General Issues
Bratman’s Planning Theory of Intention
M. Bratman. Intentions, Plans and Practical Reason. Harvard University Press(1987).
A plan is a conduct-controlling mental attitude
An intention is a component of a future-directed plan.
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General Issues
Bratman’s Planning Theory of Intention
M. Bratman. Intentions, Plans and Practical Reason. Harvard University Press(1987).
A plan is a conduct-controlling mental attitude
An intention is a component of a future-directed plan.
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General Issues
Bratman’s Planning Theory of Intention
M. Bratman. Intentions, Plans and Practical Reason. Harvard University Press(1987).
A plan is a conduct-controlling mental attitude
An intention is a component of a future-directed plan.
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General Issues
Bratman’s Planning Theory of Intention
An agent commits to a (partial) plan that is
1. means-end coherent,
2. consistent with the agent’s current beliefs and
3. stable (i.e., plans normally resist reconsideration) “an agent’s
habits and dispositions concerning the reconsideration or
nonreconsideration of a prior intention or plan determine the
stability of that intention or plan”. Furthermore, “The stability of
[the agent’s] plans will generally not be an isolated feature of those
plans but will be linked to other features of [the agent’s]
psychology”
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General Issues
Bratman’s Planning Theory of Intention
An agent commits to a (partial) plan that is
1. means-end coherent,
2. consistent with the agent’s current beliefs and
3. stable (i.e., plans normally resist reconsideration) “an agent’s
habits and dispositions concerning the reconsideration or
nonreconsideration of a prior intention or plan determine the
stability of that intention or plan”. Furthermore, “The stability of
[the agent’s] plans will generally not be an isolated feature of those
plans but will be linked to other features of [the agent’s]
psychology”
Eric Pacuit 285
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General Issues
Bratman’s Planning Theory of Intention
An agent commits to a (partial) plan that is
1. means-end coherent,
2. consistent with the agent’s current beliefs and
3. stable (i.e., plans normally resist reconsideration) “an agent’s
habits and dispositions concerning the reconsideration or
nonreconsideration of a prior intention or plan determine the
stability of that intention or plan”. Furthermore, “The stability of
[the agent’s] plans will generally not be an isolated feature of those
plans but will be linked to other features of [the agent’s]
psychology”
Eric Pacuit 286
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General Issues
Bratman’s Planning Theory of Intention
An agent commits to a (partial) plan that is
1. means-end coherent,
2. consistent with the agent’s current beliefs and
3. stable (i.e., plans normally resist reconsideration)
“an agent’s
habits and dispositions concerning the reconsideration or
nonreconsideration of a prior intention or plan determine the
stability of that intention or plan”. Furthermore, “The stability of
[the agent’s] plans will generally not be an isolated feature of those
plans but will be linked to other features of [the agent’s]
psychology”
Eric Pacuit 287
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General Issues
Bratman’s Planning Theory of Intention
An agent commits to a (partial) plan that is
1. means-end coherent,
2. consistent with the agent’s current beliefs and
3. stable (i.e., plans normally resist reconsideration) “an agent’s
habits and dispositions concerning the reconsideration or
nonreconsideration of a prior intention or plan determine the
stability of that intention or plan”. Furthermore, “The stability of
[the agent’s] plans will generally not be an isolated feature of those
plans but will be linked to other features of [the agent’s]
psychology”
Eric Pacuit 288
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General Issues
Bratman’s Planning Theory of Intention
Central to Bratman’s theory is the idea that these partial plansdirect the agent’s deliberation by “constrain[ing] what options areconsidered relevant”:
“plans narrow the scope of the deliberation to a limitedset of options. And they help to answer a question thattends to remain unanswered in traditional decision theory,namely: where do decision problems come from?”
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General Issues
A Methodological Issue
What are we formalizing? How will the logical framework be used?
Two Extremes:
1. Formalizing a (philosophical) theory of rational agency:philosophers as intuition pumps generating ”problems” for thelogical frameworks.
2. Reasoning about multiagent systems. Three main applicationsof BDI logics: 1. a specification language for a MAS, 2. aprogramming language, and 3. verification language.
W. van der Hoek and M. Wooldridge. Towards a logic of rational agency. LogicJournal of the IGPL 11 (2), 2003.
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General Issues
A Methodological Issue
What are we formalizing? How will the logical framework be used?
Two Extremes:
1. Formalizing a (philosophical) theory of rational agency:
philosophers as intuition pumps generating ”problems” for thelogical frameworks.
2. Reasoning about multiagent systems. Three main applicationsof BDI logics: 1. a specification language for a MAS, 2. aprogramming language, and 3. verification language.
W. van der Hoek and M. Wooldridge. Towards a logic of rational agency. LogicJournal of the IGPL 11 (2), 2003.
Eric Pacuit 291
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General Issues
A Methodological Issue
What are we formalizing? How will the logical framework be used?
Two Extremes:
1. Formalizing a (philosophical) theory of rational agency:philosophers as intuition pumps generating ”problems” for thelogical frameworks.
2. Reasoning about multiagent systems. Three main applicationsof BDI logics: 1. a specification language for a MAS, 2. aprogramming language, and 3. verification language.
W. van der Hoek and M. Wooldridge. Towards a logic of rational agency. LogicJournal of the IGPL 11 (2), 2003.
Eric Pacuit 292
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General Issues
A Methodological Issue
What are we formalizing? How will the logical framework be used?
Two Extremes:
1. Formalizing a (philosophical) theory of rational agency:philosophers as intuition pumps generating ”problems” for thelogical frameworks.
2. Reasoning about multiagent systems.
Three main applicationsof BDI logics: 1. a specification language for a MAS, 2. aprogramming language, and 3. verification language.
W. van der Hoek and M. Wooldridge. Towards a logic of rational agency. LogicJournal of the IGPL 11 (2), 2003.
Eric Pacuit 293
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General Issues
A Methodological Issue
What are we formalizing? How will the logical framework be used?
Two Extremes:
1. Formalizing a (philosophical) theory of rational agency:philosophers as intuition pumps generating ”problems” for thelogical frameworks.
2. Reasoning about multiagent systems. Three main applicationsof BDI logics: 1. a specification language for a MAS, 2. aprogramming language, and 3. verification language.
W. van der Hoek and M. Wooldridge. Towards a logic of rational agency. LogicJournal of the IGPL 11 (2), 2003.
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General Issues
C & L Logic of Intention
1. Intentions normally pose problems for the agent; the agentneeds to determine a way to achieve them.
2. Intentions provide a “screen of admissibility” for adoptingother intentions.
3. Agents “track” the success of their attempts to achieve theirintentions.
4. If an agent intends to achieve p, then
4.1 The agent believes p is possible4.2 The agent does not believe he will not bring abut p4.3 Under certain conditions, the agent believes he will bring about
p4.4 Agents need not intend all the expected side-effects of their
intentions.
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General Issues
C & L Logic of Intention
(PGOALip) := (GOALi (LATERp)) ∧(BELi¬p)∧[BEFORE((BELip) ∨ (BELi¬p))¬(GOALi (LATERp))]
(INTENDia) := (PGOALi [DONEi (BELi (HAPPENSa))?; a])
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General Issues
Methodological Issues
A third alternative:
3. Start from an explicit description of what is being modeled.
Database/Planner Picture: Planner using a database to maintainits current set of beliefs.
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General Issues
Methodological Issues
A third alternative:
3. Start from an explicit description of what is being modeled.
Database/Planner Picture: Planner using a database to maintainits current set of beliefs.
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General Issues
Planning vs. Database Management
1. How does an agent generate new intentions?
2. Given that the agent’s intentions specify a partial plan, howand when is the plan “filled out”?
3. How does an agent choose a particular action (that is underits control) given its current intentions?
4. How should an agent maintain its current state of beliefs andintentions in the presence of new information or newintentions?
5. When should an agent reconsider its intentions?
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General Issues
Thomas Icard, EP and Yoav Shoham. Intention and Belief Revision. in prepara-tion.
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General Issues
Our Framework
I What type of information does a planner provide? How do werepresent a plan?
I Sources of beliefs
I Sources of dynamics: What can cause an agent’s database tochange?
I Changing/amending plans vs. revising/updating beliefs
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General Issues
Elements of a Logic of Intention Revision
I Beliefs in a dynamic environment: certainty (irrevocableknowledge, hard information), belief (revisable, softinformation), safe belief
I Three views of actions: PDL (state changing), Temporal (layout time and actions are sequences of time points), STIT(choices, or actions, constrain the future).
I Two types of beliefs: those about the state of the world andthose about the future which are governed by the agent’splans
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General Issues
Elements of a Logic of Intention Revision
I Beliefs in a dynamic environment: certainty (irrevocableknowledge, hard information), belief (revisable, softinformation), safe belief
I Three views of actions: PDL (state changing), Temporal (layout time and actions are sequences of time points), STIT(choices, or actions, constrain the future).
I Two types of beliefs: those about the state of the world andthose about the future which are governed by the agent’splans
Eric Pacuit 303
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General Issues
Elements of a Logic of Intention Revision
I Beliefs in a dynamic environment: certainty (irrevocableknowledge, hard information), belief (revisable, softinformation), safe belief
I Three views of actions: PDL (state changing), Temporal (layout time and actions are sequences of time points), STIT(choices, or actions, constrain the future).
I Two types of beliefs: those about the state of the world andthose about the future which are governed by the agent’splans
Eric Pacuit 304
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General Issues
Elements of a Logic of Intention Revision
I Beliefs in a dynamic environment: certainty (irrevocableknowledge, hard information), belief (revisable, softinformation), safe belief
I Three views of actions: PDL (state changing), Temporal (layout time and actions are sequences of time points), STIT(choices, or actions, constrain the future).
I Two types of beliefs: those about the state of the world andthose about the future which are governed by the agent’splans
Eric Pacuit 305
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General Issues
Intention Revision
Many of the frameworks do discuss some form of intention revision.
W. van der Hoek, W. Jamroga and M. Wooldridge. Towards a Theory of Inten-tion Revision. Synthese, 2007.
I Beliefs are sets of Linear Temporal Logic formulas (eg., ©ϕ)
I Desires are (possibly inconsistent) sets of Linear TemporalLogic formulas
I Practical reasoning rules: α← α1, α2, . . . , αn
I Intentions are derived from the agents current active plans(trees of practical reasoning rules)
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General Issues
Intention Revision
Many of the frameworks do discuss some form of intention revision.
W. van der Hoek, W. Jamroga and M. Wooldridge. Towards a Theory of Inten-tion Revision. Synthese, 2007.
I Beliefs are sets of Linear Temporal Logic formulas (eg., ©ϕ)
I Desires are (possibly inconsistent) sets of Linear TemporalLogic formulas
I Practical reasoning rules: α← α1, α2, . . . , αn
I Intentions are derived from the agents current active plans(trees of practical reasoning rules)
Eric Pacuit 307
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General Issues
Intention Revision
Many of the frameworks do discuss some form of intention revision.
W. van der Hoek, W. Jamroga and M. Wooldridge. Towards a Theory of Inten-tion Revision. Synthese, 2007.
I Beliefs are sets of Linear Temporal Logic formulas (eg., ©ϕ)
I Desires are (possibly inconsistent) sets of Linear TemporalLogic formulas
I Practical reasoning rules: α← α1, α2, . . . , αn
I Intentions are derived from the agents current active plans(trees of practical reasoning rules)
Eric Pacuit 308
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General Issues
Intention Revision
Many of the frameworks do discuss some form of intention revision.
W. van der Hoek, W. Jamroga and M. Wooldridge. Towards a Theory of Inten-tion Revision. Synthese, 2007.
I Beliefs are sets of Linear Temporal Logic formulas (eg., ©ϕ)
I Desires are (possibly inconsistent) sets of Linear TemporalLogic formulas
I Practical reasoning rules: α← α1, α2, . . . , αn
I Intentions are derived from the agents current active plans(trees of practical reasoning rules)
Eric Pacuit 309
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General Issues
Intention Revision
Many of the frameworks do discuss some form of intention revision.
W. van der Hoek, W. Jamroga and M. Wooldridge. Towards a Theory of Inten-tion Revision. Synthese, 2007.
I Beliefs are sets of Linear Temporal Logic formulas (eg., ©ϕ)
I Desires are (possibly inconsistent) sets of Linear TemporalLogic formulas
I Practical reasoning rules: α← α1, α2, . . . , αn
I Intentions are derived from the agents current active plans(trees of practical reasoning rules)
Eric Pacuit 310
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General Issues
Intention Revision
Many of the frameworks do discuss some form of intention revision.
W. van der Hoek, W. Jamroga and M. Wooldridge. Towards a Theory of Inten-tion Revision. Synthese, 2007.
I Beliefs are sets of Linear Temporal Logic formulas (eg., ©ϕ)
I Desires are (possibly inconsistent) sets of Linear TemporalLogic formulas
I Practical reasoning rules: α← α1, α2, . . . , αn
I Intentions are derived from the agents current active plans(trees of practical reasoning rules)
Eric Pacuit 311
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General Issues
Intention Revision
Many of the frameworks do discuss some form of intention revision.
W. van der Hoek, W. Jamroga and M. Wooldridge. Towards a Theory of Inten-tion Revision. Synthese, 2007.
I Two types of beliefs: strong beliefs vs. weak beliefs (beliefsthat take into account the agent’s intentions)
I A dynamic update operator is defined ([Ω]ϕ)
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General Issues
Our Framework
1. At a fixed moment, a choice situation describes the currentstate-of-affairs (i.e., facts about the state-of-the-world), thetree of options that are available to the agent (i.e., thedecision tree) and how actions change state of the world (i.e.,the effect that performing an action will have on thestate-of-the-world).
2. At a fixed moment, a model describes the agent’s (current)beliefs (about the current state-of-the-world and what willbecome true in the future including options that will becomeavailable) and the agent’s (current) instructions from thePlanner (about future choices).
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General Issues
Our Framework
1. At a fixed moment, a choice situation describes the currentstate-of-affairs (i.e., facts about the state-of-the-world), thetree of options that are available to the agent (i.e., thedecision tree) and how actions change state of the world (i.e.,the effect that performing an action will have on thestate-of-the-world).
2. At a fixed moment, a model describes the agent’s (current)beliefs (about the current state-of-the-world and what willbecome true in the future including options that will becomeavailable) and the agent’s (current) instructions from thePlanner (about future choices).
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General Issues
Our Framework
3. Dynamic operators representing each of the situations thatmay cause a change in beliefs and/or plans: learning a truefact, doing an action and receiving instructions from thePlanner. These operators will describe how to relate modelsat different moments.
Skip Details
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General Issues
Choice Situations
Mw = (W , Raa∈Act,V ,w)
t = 0
t = 1
t = 2
t = 3
w
a b
c d
c ′
a′ b′
d ′ e
...
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General Issues
Choice Situations: L1
ϕ := p | ϕ ∧ ϕ | ¬ϕ | 〈a〉ϕ
I Mw p iff w ∈ V (p)
I Mw ϕ ∧ ψ iff Mw ϕ and Mw ψI Mw ¬ϕ iff Mw 2 ϕI Mw 〈a〉ϕ iff ∃x wRax and Mx ϕ.
Notation: If α = a1a2a3 · · · an, 〈α〉ϕ := 〈a1〉 · · · 〈an〉ϕ
Nϕ :=∧
a∈Act[a]ϕ [t]ϕ :=
t times︷ ︸︸ ︷N...N ϕ
Pϕ :=∨
a∈Act〈a〉ϕ 〈t〉ϕ :=
t times︷ ︸︸ ︷P...P ϕ
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General Issues
Choice Situations: L1
ϕ := p | ϕ ∧ ϕ | ¬ϕ | 〈a〉ϕ
I Mw p iff w ∈ V (p)
I Mw ϕ ∧ ψ iff Mw ϕ and Mw ψI Mw ¬ϕ iff Mw 2 ϕI Mw 〈a〉ϕ iff ∃x wRax and Mx ϕ.
Notation: If α = a1a2a3 · · · an, 〈α〉ϕ := 〈a1〉 · · · 〈an〉ϕ
Nϕ :=∧
a∈Act[a]ϕ [t]ϕ :=
t times︷ ︸︸ ︷N...N ϕ
Pϕ :=∨
a∈Act〈a〉ϕ 〈t〉ϕ :=
t times︷ ︸︸ ︷P...P ϕ
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General Issues
Choice Situations: L1
ϕ := p | ϕ ∧ ϕ | ¬ϕ | 〈a〉ϕ
I Mw p iff w ∈ V (p)
I Mw ϕ ∧ ψ iff Mw ϕ and Mw ψI Mw ¬ϕ iff Mw 2 ϕI Mw 〈a〉ϕ iff ∃x wRax and Mx ϕ.
Notation: If α = a1a2a3 · · · an, 〈α〉ϕ := 〈a1〉 · · · 〈an〉ϕ
Nϕ :=∧
a∈Act[a]ϕ [t]ϕ :=
t times︷ ︸︸ ︷N...N ϕ
Pϕ :=∨
a∈Act〈a〉ϕ 〈t〉ϕ :=
t times︷ ︸︸ ︷P...P ϕ
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General Issues
Adding Beliefs
Standard picture where worlds are choice situations
Mw Nv : Choice situation Nv is at least as plausible as Mw .
1. Beliefs are about available options, current and future state ofaffairs: Bp ∧ B〈a〉〈b〉q
2. Immediate options are known.
3. In the static model, restrict the language to only talk aboutcurrent beliefs: 〈a〉Bϕ is not well-formed
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General Issues
Adding Beliefs
Standard picture where worlds are choice situations
Mw Nv : Choice situation Nv is at least as plausible as Mw .
1. Beliefs are about available options, current and future state ofaffairs: Bp ∧ B〈a〉〈b〉q
2. Immediate options are known.
3. In the static model, restrict the language to only talk aboutcurrent beliefs: 〈a〉Bϕ is not well-formed
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General Issues
Adding Beliefs
Standard picture where worlds are choice situations
Mw Nv : Choice situation Nv is at least as plausible as Mw .
1. Beliefs are about available options, current and future state ofaffairs: Bp ∧ B〈a〉〈b〉q
2. Immediate options are known.
3. In the static model, restrict the language to only talk aboutcurrent beliefs: 〈a〉Bϕ is not well-formed
Eric Pacuit 322
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General Issues
Belief Structures
B = (S ,,Mw )
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General Issues
Belief Structures
B = (S ,,Mw )
t = 0
t = 1
t = 2
t = 3
w
a b
c d
c ′
a′ b′
d ′ e
...
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General Issues
Belief Structures
B = (S ,,Mw )
Equally plausible
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General Issues
Belief Structures
B = (S ,,Mw )
More plausible
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General Issues
Belief Structures
B = (S ,,Mw )
Beliefs
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General Issues
Belief Structures
Language (L2): ϕ := χ | ϕ ∧ ϕ | ¬ϕ | B(ϕ), χ ∈ L1
Structures B = (S ,,Mw ) is a belief structure if:
(i) S a set of choice situations
(ii) is a plausibility ordering (reflexive, transitive, well-founded)
(iii) Mw ∈ S .
(iv) If wRax for some x in M, then for all Nv ∈ S s.t. Mw Nv ,there is some x ′ for which vRax ′ in N .
(v) If Mw Nv and vRax for some x in N , there is somex ′ ∈W such that wRax ′ in M.
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General Issues
Belief Structures
Language (L2): ϕ := χ | ϕ ∧ ϕ | ¬ϕ | B(ϕ), χ ∈ L1
Structures B = (S ,,Mw ) is a belief structure if:
(i) S a set of choice situations
(ii) is a plausibility ordering (reflexive, transitive, well-founded)
(iii) Mw ∈ S .
(iv) If wRax for some x in M, then for all Nv ∈ S s.t. Mw Nv ,there is some x ′ for which vRax ′ in N .
(v) If Mw Nv and vRax for some x in N , there is somex ′ ∈W such that wRax ′ in M.
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General Issues
Belief Structures
Language (L2): ϕ := χ | ϕ ∧ ϕ | ¬ϕ | B(ϕ), χ ∈ L1
Structures B = (S ,,Mw ) is a belief structure if:
(i) S a set of choice situations
(ii) is a plausibility ordering (reflexive, transitive, well-founded)
(iii) Mw ∈ S .
(iv) If wRax for some x in M, then for all Nv ∈ S s.t. Mw Nv ,there is some x ′ for which vRax ′ in N .
(v) If Mw Nv and vRax for some x in N , there is somex ′ ∈W such that wRax ′ in M.
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General Issues
Belief Structures
B χ, iff Mw χ.
B ϕ ∧ ψ, iff B ϕ, and B ψ.
B ¬ϕ, iff B 1 ϕ.
B B(ϕ), iff for all Nv ∈ Min(S), B,Nv ϕ.
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General Issues
Completeness
1. Standard proof works for the class of choice situations
2. The class of belief structures is also easily axiomatized (ϕmeans ϕ is true an all worlds at least as plausible as thecurrent world):
• KD45 for B• 〈a〉> → (〈a〉>)• ♦(〈a〉>)→ 〈a〉>
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General Issues
InstructionsAt each moment there are instructions from the Planner: Weassume that at each moment, there are some instructions aboutfuture choices that the agent has agreed to follow (if he can).
1. A complete plan, for each moment the specific action a ∈ Actthe agent will perform.
2. The instructions may be partial: finite list of pairs (a, t) wherea ∈ Act and t ∈ N.
3. The instructions may be conditional: do a at time t providedϕ is true.
4. Rather than instructing the agent to follow a specific (partial,conditional) plan, the Planner simply restricts the choices thatare available to the agent in the future.
5. The Planner may provide a more complicated structure(subplan structure, goals, etc.)
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General Issues
InstructionsAt each moment there are instructions from the Planner: Weassume that at each moment, there are some instructions aboutfuture choices that the agent has agreed to follow (if he can).
1. A complete plan, for each moment the specific action a ∈ Actthe agent will perform.
2. The instructions may be partial: finite list of pairs (a, t) wherea ∈ Act and t ∈ N.
3. The instructions may be conditional: do a at time t providedϕ is true.
4. Rather than instructing the agent to follow a specific (partial,conditional) plan, the Planner simply restricts the choices thatare available to the agent in the future.
5. The Planner may provide a more complicated structure(subplan structure, goals, etc.)
Eric Pacuit 334
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General Issues
InstructionsAt each moment there are instructions from the Planner: Weassume that at each moment, there are some instructions aboutfuture choices that the agent has agreed to follow (if he can).
1. A complete plan, for each moment the specific action a ∈ Actthe agent will perform.
2. The instructions may be partial: finite list of pairs (a, t) wherea ∈ Act and t ∈ N.
3. The instructions may be conditional: do a at time t providedϕ is true.
4. Rather than instructing the agent to follow a specific (partial,conditional) plan, the Planner simply restricts the choices thatare available to the agent in the future.
5. The Planner may provide a more complicated structure(subplan structure, goals, etc.)
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General Issues
InstructionsAt each moment there are instructions from the Planner: Weassume that at each moment, there are some instructions aboutfuture choices that the agent has agreed to follow (if he can).
1. A complete plan, for each moment the specific action a ∈ Actthe agent will perform.
2. The instructions may be partial: finite list of pairs (a, t) wherea ∈ Act and t ∈ N.
3. The instructions may be conditional: do a at time t providedϕ is true.
4. Rather than instructing the agent to follow a specific (partial,conditional) plan, the Planner simply restricts the choices thatare available to the agent in the future.
5. The Planner may provide a more complicated structure(subplan structure, goals, etc.)
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General Issues
InstructionsAt each moment there are instructions from the Planner: Weassume that at each moment, there are some instructions aboutfuture choices that the agent has agreed to follow (if he can).
1. A complete plan, for each moment the specific action a ∈ Actthe agent will perform.
2. The instructions may be partial: finite list of pairs (a, t) wherea ∈ Act and t ∈ N.
3. The instructions may be conditional: do a at time t providedϕ is true.
4. Rather than instructing the agent to follow a specific (partial,conditional) plan, the Planner simply restricts the choices thatare available to the agent in the future.
5. The Planner may provide a more complicated structure(subplan structure, goals, etc.)
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General Issues
InstructionsAt each moment there are instructions from the Planner: Weassume that at each moment, there are some instructions aboutfuture choices that the agent has agreed to follow (if he can).
1. A complete plan, for each moment the specific action a ∈ Actthe agent will perform.
2. The instructions may be partial: finite list of pairs (a, t) wherea ∈ Act and t ∈ N.
3. The instructions may be conditional: do a at time t providedϕ is true.
4. Rather than instructing the agent to follow a specific (partial,conditional) plan, the Planner simply restricts the choices thatare available to the agent in the future.
5. The Planner may provide a more complicated structure(subplan structure, goals, etc.)
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General Issues
Dynamics
There are three sources of dynamics:
1. Nature can reveal (true) facts about the current choicesituation (eg., facts that are true, choices that areavailable/not available in the future).
2. The agent can decide to perform an action (which in turnforces Nature to reveal certain information such as whichactions become available).
3. The Planner can amend the agent’s current set of instructions.
We assume that only doing an action moves time forward.However, all three types of events may change the agent’s beliefsand current instructions.
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General Issues
Dynamics
There are three sources of dynamics:
1. Nature can reveal (true) facts about the current choicesituation (eg., facts that are true, choices that areavailable/not available in the future).
2. The agent can decide to perform an action (which in turnforces Nature to reveal certain information such as whichactions become available).
3. The Planner can amend the agent’s current set of instructions.
We assume that only doing an action moves time forward.However, all three types of events may change the agent’s beliefsand current instructions.
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General Issues
Dynamics
There are three sources of dynamics:
1. Nature can reveal (true) facts about the current choicesituation (eg., facts that are true, choices that areavailable/not available in the future).
2. The agent can decide to perform an action (which in turnforces Nature to reveal certain information such as whichactions become available).
3. The Planner can amend the agent’s current set of instructions.
We assume that only doing an action moves time forward.However, all three types of events may change the agent’s beliefsand current instructions.
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General Issues
Dynamics
There are three sources of dynamics:
1. Nature can reveal (true) facts about the current choicesituation (eg., facts that are true, choices that areavailable/not available in the future).
2. The agent can decide to perform an action (which in turnforces Nature to reveal certain information such as whichactions become available).
3. The Planner can amend the agent’s current set of instructions.
We assume that only doing an action moves time forward.However, all three types of events may change the agent’s beliefsand current instructions.
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General Issues
0 i i + 1 i + 2 i + 3 i + 4
ab
c
d
e f
a b e f
a e c f
a
I = (b, i + 1), (d , i + 2)
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General Issues
0 i i + 1 i + 2 i + 3 i + 4
ab
c
d
e f
a b e f
a e d f
a
I = (b, i + 1), (d , i + 2)
Add (f , i + 3)
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General Issues
Selection Function
Say a set of beliefs B and a set of instructions I is coherent if theagent doesn’t believe the instructions are impossible.A selection function γ maps a set of beliefs B and instructions toa set of instructions: γ(B, I ) = I ′
1. γ(B, I ) ⊆ I .
2. γ(B, I ) is coherent with B.
3. additional principles.....
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General Issues
Selection Function
Say a set of beliefs B and a set of instructions I is coherent if theagent doesn’t believe the instructions are impossible.A selection function γ maps a set of beliefs B and instructions toa set of instructions: γ(B, I ) = I ′
1. γ(B, I ) ⊆ I .
2. γ(B, I ) is coherent with B.
3. additional principles.....
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General Issues
Where we are going
AGM-style principles and representation theorem; Modal-stylecompleteness (with dynamic operators get considerably moretechnical: reduction axioms are not available).
Moving to complex plans (with choice, concatenation and test):
1. The notion of Belief-Plan consistency must be updated
2. Define intentions semantically: the agent “intends a, t just incase it is a necessary component of the current plan”.
3. Many agents
4. .....
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General Issues
Where we are going
AGM-style principles and representation theorem; Modal-stylecompleteness (with dynamic operators get considerably moretechnical: reduction axioms are not available).
Moving to complex plans (with choice, concatenation and test):
1. The notion of Belief-Plan consistency must be updated
2. Define intentions semantically: the agent “intends a, t just incase it is a necessary component of the current plan”.
3. Many agents
4. .....
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Conclusions
ConclusionsWe are interested in reasoning about rational agents interacting insocial situations.
What do the logical frameworks contribute to the discussion onrational agency?
I Normative vs. Descriptive
I refine and test our intuitions: provide many answers to thequestion what is a rational agent?
I (epistemic) foundations of game theoryLogic and Game Theory, not Logic in place of Game Theory.
I Social Software: Verify properties of social procedures
• Refine existing social procedures or suggest new ones
R. Parikh. Social Software. Synthese 132 (2002).
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Conclusions
ConclusionsWe are interested in reasoning about rational agents interacting insocial situations.
What do the logical frameworks contribute to the discussion onrational agency?
I Normative vs. Descriptive
I refine and test our intuitions: provide many answers to thequestion what is a rational agent?
I (epistemic) foundations of game theoryLogic and Game Theory, not Logic in place of Game Theory.
I Social Software: Verify properties of social procedures
• Refine existing social procedures or suggest new ones
R. Parikh. Social Software. Synthese 132 (2002).
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Conclusions
ConclusionsWe are interested in reasoning about rational agents interacting insocial situations.
What do the logical frameworks contribute to the discussion onrational agency?
I Normative vs. Descriptive
I refine and test our intuitions: provide many answers to thequestion what is a rational agent?
I (epistemic) foundations of game theoryLogic and Game Theory, not Logic in place of Game Theory.
I Social Software: Verify properties of social procedures
• Refine existing social procedures or suggest new ones
R. Parikh. Social Software. Synthese 132 (2002).
Eric Pacuit 351
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Conclusions
ConclusionsWe are interested in reasoning about rational agents interacting insocial situations.
What do the logical frameworks contribute to the discussion onrational agency?
I Normative vs. Descriptive
I refine and test our intuitions: provide many answers to thequestion what is a rational agent?
I (epistemic) foundations of game theoryLogic and Game Theory, not Logic in place of Game Theory.
I Social Software: Verify properties of social procedures
• Refine existing social procedures or suggest new ones
R. Parikh. Social Software. Synthese 132 (2002).
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Conclusions
ConclusionsWe are interested in reasoning about rational agents interacting insocial situations.
What do the logical frameworks contribute to the discussion onrational agency?
I Normative vs. Descriptive
I refine and test our intuitions: provide many answers to thequestion what is a rational agent?
I (epistemic) foundations of game theoryLogic and Game Theory, not Logic in place of Game Theory.
I Social Software: Verify properties of social procedures
• Refine existing social procedures or suggest new ones
R. Parikh. Social Software. Synthese 132 (2002).
Eric Pacuit 353
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Conclusions
ConclusionsWe are interested in reasoning about rational agents interacting insocial situations.
What do the logical frameworks contribute to the discussion onrational agency?
I Normative vs. Descriptive
I refine and test our intuitions: provide many answers to thequestion what is a rational agent?
I (epistemic) foundations of game theoryLogic and Game Theory, not Logic in place of Game Theory.
I Social Software: Verify properties of social procedures
• Refine existing social procedures or suggest new ones
R. Parikh. Social Software. Synthese 132 (2002).
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Conclusions
Conclusions
I Many types of informational attitudes: “hard” knowledge,belief, belief about the future state of affairs, “intention”based beliefs, revisable beliefs, safe beliefs.
I Where does the “protocol” come from? What do the agentsknow about the protocol?
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Conclusions
Logics of Rational Agency
I What’s going on in the area:www.loriweb.org
I Special Issue of Synthese: Knowledge, Rationality andInteraction. Logic and Intelligent Interaction, Volume 169,Number 2 / July, 2009(eds. T. Agotnes, J. van Benthem and EP)
I New subarea of Stanford Encyclopedia of Philosophy on logicand rational agency(eds. J. van Benthem, EP, and O. Roy)
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Conclusions
Calls for....
I Papers: LOFT 2010. University of Toulouse, July 21 - 23.Deadline: March 15, 2010.
I Ph.D. position: TiLPS, Tilburg University, “A formalanalysis of social procedures”. Deadline: October 15 (tostart in February).
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Conclusions
Thank You!
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