introduction to formal epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · eric...
TRANSCRIPT
![Page 1: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/1.jpg)
Introduction to FormalEpistemology
Lecture 3
Eric Pacuit and Rohit Parikh
August 14, 2007
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 1
![Page 2: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/2.jpg)
Plan for the Course
X Introduction, Motivation and Basic Epistemic Logic
X Other models of Knowledge, Knowledge in Groupsand Group Knowledge
Lecture 3: Adding Dynamics, Reasoning about Knowledge inGames
Lecture 4: Logical Omniscience and Other Problems
Lecture 5: Reasoning about Knowledge in the Context of SocialSoftware
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 2
![Page 3: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/3.jpg)
Knowledge and Dynamics
Many of the examples we have discussed are dynamic — theyinvolve announcements, actions, etc.
How should we add this dynamics to our models?
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 3
![Page 4: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/4.jpg)
Knowledge and Dynamics
Many of the examples we have discussed are dynamic — theyinvolve announcements, actions, etc.
How should we add this dynamics to our models?
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 3
![Page 5: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/5.jpg)
Knowledge and Dynamics
Public Announcement
P
s
¬P
t
B
A, BA, B
P means “The talk is at 2PM”.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 4
![Page 6: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/6.jpg)
Knowledge and Dynamics
Public Announcement
P
s
¬P
t
B
A, BA, B
What happens if Ann publicly announces P?
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 4
![Page 7: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/7.jpg)
Knowledge and Dynamics
Public Announcement
P
s
¬P
t
B
A, BA, B
What happens if Ann publicly announces P?
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 4
![Page 8: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/8.jpg)
Knowledge and Dynamics
Public Announcement
P
s
A, B
What happens if Ann publicly announces P? s |= CP
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 4
![Page 9: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/9.jpg)
Knowledge and Dynamics
Public Announcement Logic
J. Plaza. Logics of Public Communications. 1989.
J. Gerbrandy. Bisimulations on Planet Kripke. 1999.
J. van Benthem. One is a lonely number. 2002.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 5
![Page 10: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/10.jpg)
Knowledge and Dynamics
Public Announcement Logic
The Public Announcement Language is generated by the followinggrammar:
p | ¬ϕ | ϕ ∧ ϕ | Kiϕ | Cϕ | [ψ]ϕ
where p ∈ At and i ∈ A.
I [ψ]ϕ is intended to mean “After publicly announcing ψ, ϕ istrue”.
I [P]KiP: “After publicly announcing P, agent i knows P”
I [¬KiP]CP: “After announcing that agent i does not know P,then P is common knowledge”
I [¬KiP]KiP: “after announcing i does not know P, then iknows P. ”
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 6
![Page 11: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/11.jpg)
Knowledge and Dynamics
Public Announcement Logic
The Public Announcement Language is generated by the followinggrammar:
p | ¬ϕ | ϕ ∧ ϕ | Kiϕ | Cϕ | [ψ]ϕ
where p ∈ At and i ∈ A.
I [ψ]ϕ is intended to mean “After publicly announcing ψ, ϕ istrue”.
I [P]KiP: “After publicly announcing P, agent i knows P”
I [¬KiP]CP: “After announcing that agent i does not know P,then P is common knowledge”
I [¬KiP]KiP: “after announcing i does not know P, then iknows P. ”
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 6
![Page 12: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/12.jpg)
Knowledge and Dynamics
Public Announcement Logic
The Public Announcement Language is generated by the followinggrammar:
p | ¬ϕ | ϕ ∧ ϕ | Kiϕ | Cϕ | [ψ]ϕ
where p ∈ At and i ∈ A.
I [ψ]ϕ is intended to mean “After publicly announcing ψ, ϕ istrue”.
I [P]KiP: “After publicly announcing P, agent i knows P”
I [¬KiP]CP: “After announcing that agent i does not know P,then P is common knowledge”
I [¬KiP]KiP: “after announcing i does not know P, then iknows P. ”
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 6
![Page 13: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/13.jpg)
Knowledge and Dynamics
Public Announcement Logic
The Public Announcement Language is generated by the followinggrammar:
p | ¬ϕ | ϕ ∧ ϕ | Kiϕ | Cϕ | [ψ]ϕ
where p ∈ At and i ∈ A.
I [ψ]ϕ is intended to mean “After publicly announcing ψ, ϕ istrue”.
I [P]KiP: “After publicly announcing P, agent i knows P”
I [¬KiP]CP: “After announcing that agent i does not know P,then P is common knowledge”
I [¬KiP]KiP: “after announcing i does not know P, then iknows P. ”
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 6
![Page 14: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/14.jpg)
Knowledge and Dynamics
Public Announcement Logic
The Public Announcement Language is generated by the followinggrammar:
p | ¬ϕ | ϕ ∧ ϕ | Kiϕ | Cϕ | [ψ]ϕ
where p ∈ At and i ∈ A.
I [ψ]ϕ is intended to mean “After publicly announcing ψ, ϕ istrue”.
I [P]KiP: “After publicly announcing P, agent i knows P”
I [¬KiP]CP: “After announcing that agent i does not know P,then P is common knowledge”
I [¬KiP]KiP: “after announcing i does not know P, then iknows P. ”
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 6
![Page 15: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/15.jpg)
Knowledge and Dynamics
Public Announcement Logic
Suppose M = 〈W , {Ri}i∈A,V 〉 is a multi-agent Kripke Model
M,w |= [ψ]ϕ iff M,w |= ψ implies M|ψ,w |= ϕ
where M|ψ = 〈W ′,R ′,V ′〉 with
I W ′ = W ∩ {w | M,w |= ψ}I R ′ = R ∩W ′ ×W ′
I for all p ∈ At, V ′(p) = V (p) ∩W ′
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 7
![Page 16: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/16.jpg)
Knowledge and Dynamics
Public Announcement Logic
[ψ]p ↔ (ϕ→ p)[ψ]¬ϕ ↔ (ψ → ¬[ψ]ϕ)
[ψ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ)[ψ][ϕ]χ ↔ [ψ ∧ [ψ]ϕ]χ[ψ]Kiϕ ↔ (ψ → Ki [ψ]ϕ)
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 8
![Page 17: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/17.jpg)
Knowledge and Dynamics
Public Announcement Logic
[ψ]p ↔ (ϕ→ p)[ψ]¬ϕ ↔ (ψ → ¬[ψ]ϕ)
[ψ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ)[ψ][ϕ]χ ↔ [ψ ∧ [ψ]ϕ]χ[ψ]Kiϕ ↔ (ψ → Ki [ψ]ϕ)
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 8
![Page 18: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/18.jpg)
Knowledge and Dynamics
Public Announcement Logic
[ψ]p ↔ (ϕ→ p)[ψ]¬ϕ ↔ (ψ → ¬[ψ]ϕ)
[ψ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ)[ψ][ϕ]χ ↔ [ψ ∧ [ψ]ϕ]χ[ψ]Kiϕ ↔ (ψ → Ki [ψ]ϕ)
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 8
![Page 19: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/19.jpg)
Knowledge and Dynamics
Public Announcement Logic
[ψ]p ↔ (ϕ→ p)[ψ]¬ϕ ↔ (ψ → ¬[ψ]ϕ)
[ψ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ)[ψ][ϕ]χ ↔ [ψ ∧ [ψ]ϕ]χ[ψ]Kiϕ ↔ (ψ → Ki [ψ]ϕ)
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 8
![Page 20: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/20.jpg)
Knowledge and Dynamics
Public Announcement Logic
[ψ]p ↔ (ϕ→ p)[ψ]¬ϕ ↔ (ψ → ¬[ψ]ϕ)
[ψ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ)[ψ][ϕ]χ ↔ [ψ ∧ [ψ]ϕ]χ[ψ]Kiϕ ↔ (ψ → Ki [ψ]ϕ)
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 8
![Page 21: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/21.jpg)
Knowledge and Dynamics
Public Announcement Logic
[ψ]p ↔ (ϕ→ p)[ψ]¬ϕ ↔ (ψ → ¬[ψ]ϕ)
[ψ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ)[ψ][ϕ]χ ↔ [ψ ∧ [ψ]ϕ]χ[ψ]Kiϕ ↔ (ψ → Ki [ψ]ϕ)
Theorem Every formula of Public Announcement Logic isequivalent to a formula of Epistemic Logic.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 8
![Page 22: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/22.jpg)
Knowledge and Dynamics
Public Announcement Logic
[ψ]p ↔ (ϕ→ p)[ψ]¬ϕ ↔ (ψ → ¬[ψ]ϕ)
[ψ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ)[ψ][ϕ]χ ↔ [ψ ∧ [ψ]ϕ]χ[ψ]Kiϕ ↔ (ψ → Ki [ψ]ϕ)
The situation is more complicated with common knowledge.
J. van Benthem, J. van Eijk, B. Kooi. Logics of Communication and Change.2006.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 8
![Page 23: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/23.jpg)
Knowledge and Dynamics
Public Announcement Logic
An announcement if successful if after it is publicly announced, itbecomes common knowledge. [ϕ]Cϕ.An announcement is unsuccessful if after it is publicly announced,it becomes false. [ϕ]¬ϕ.P means “it is raining.”[P ∧ ¬KP]¬(P ∧ ¬KP)Question: Which formulas are successful? unsuccessful?
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 9
![Page 24: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/24.jpg)
Knowledge and Dynamics
Public Announcement Logic
An announcement if successful if after it is publicly announced, itbecomes common knowledge. [ϕ]Cϕ.An announcement is unsuccessful if after it is publicly announced,it becomes false. [ϕ]¬ϕ.P means “it is raining.”[P ∧ ¬KP]¬(P ∧ ¬KP)Question: Which formulas are successful? unsuccessful?
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 9
![Page 25: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/25.jpg)
Knowledge and Dynamics
Public Announcement Logic
An announcement if successful if after it is publicly announced, itbecomes common knowledge. [ϕ]Cϕ.An announcement is unsuccessful if after it is publicly announced,it becomes false. [ϕ]¬ϕ.P means “it is raining.”[P ∧ ¬KP]¬(P ∧ ¬KP)Question: Which formulas are successful? unsuccessful?
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 9
![Page 26: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/26.jpg)
Knowledge and Dynamics
Public Announcement Logic
An announcement if successful if after it is publicly announced, itbecomes common knowledge. [ϕ]Cϕ.An announcement is unsuccessful if after it is publicly announced,it becomes false. [ϕ]¬ϕ.P means “it is raining.”[P ∧ ¬KP]¬(P ∧ ¬KP)Question: Which formulas are successful? unsuccessful?
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 9
![Page 27: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/27.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
What about more complicated announcements?
A. Baltag and L. Moss. Logics for Epistemic Programs. 2004.
W. van der Hoek, H. van Ditmarsch and B. Kooi. Dynamic Logic. 2007.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 10
![Page 28: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/28.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
What about more complicated announcements?
A. Baltag and L. Moss. Logics for Epistemic Programs. 2004.
W. van der Hoek, H. van Ditmarsch and B. Kooi. Dynamic Logic. 2007.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 10
![Page 29: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/29.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
Recall the Ann and Bob example: Charles tells Bob that the talk isat 2PM.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 11
![Page 30: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/30.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 11
![Page 31: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/31.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 11
![Page 32: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/32.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 11
![Page 33: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/33.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 11
![Page 34: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/34.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 12
![Page 35: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/35.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
(M⊗ E1)⊗ E2
The initial model (Annand Bob are ignorantabout P2PM).
Private announcementto Ann about the talk.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 12
![Page 36: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/36.jpg)
Knowledge and Dynamics
Dynamic Epistemic Logic
(M⊗ E1)⊗ E2
The initial model (Annand Bob are ignorantabout P2PM).
Private announcementto Ann about the talk.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 12
![Page 37: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/37.jpg)
Knowledge and Dynamics
Product Update
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 13
![Page 38: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/38.jpg)
Knowledge and Dynamics
Product Update
M⊗ E1
P
s
¬P
t
B
A, BA, B
E2
Pe1 P e2
>e3
B
BA
A
A, B
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 13
![Page 39: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/39.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 13
![Page 40: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/40.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 13
![Page 41: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/41.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 13
![Page 42: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/42.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 13
![Page 43: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/43.jpg)
Knowledge and Dynamics
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
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 13
![Page 44: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/44.jpg)
Knowledge and Dynamics
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
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 13
![Page 45: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/45.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 14
![Page 46: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/46.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 14
![Page 47: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/47.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 14
![Page 48: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/48.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 14
![Page 49: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/49.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 14
![Page 50: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/50.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 14
![Page 51: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/51.jpg)
Knowledge and Dynamics
DEL methodology: when describing a social situation, describethe initial situation, describe the event and provide a method forhow events change a model that can be described in the formallanguage, then construct the event tree as needed.
Alternatively: 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.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 15
![Page 52: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/52.jpg)
Knowledge and Dynamics
DEL methodology: when describing a social situation, describethe initial situation, describe the event and provide a method forhow events change a model that can be described in the formallanguage, then construct the event tree as needed.
Alternatively: 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.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 15
![Page 53: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/53.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 16
![Page 54: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/54.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 17
![Page 55: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/55.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 17
![Page 56: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/56.jpg)
Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 17
![Page 57: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/57.jpg)
Knowledge and Dynamics
The ‘Playground’: Notation
I Let Σ be any set. The elements of Σ are called events.
I Given any set X , X ∗ is the set of finite strings over X and Xω
the set of infinite strings over X . Elements of Σ∗ ∪ Σω will becalled histories.
I Given H ∈ Σ∗ ∪ Σω, len(H) is the length of H.
I Given H,H ′ ∈ Σ∗ ∪Σω, we write H � H ′ if H is a finite prefixof H ′.
I FinPre(H) = {H | ∃H ′ ∈ H such that H � H ′} is the set offinite prefixes of the elements of H.
I ε is the empty string and FinPre−ε(H) = FinPre(H)− {ε}.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 18
![Page 58: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/58.jpg)
Knowledge and Dynamics
The ‘Playground’: Notation
I Let Σ be any set. The elements of Σ are called events.
I Given any set X , X ∗ is the set of finite strings over X and Xω
the set of infinite strings over X . Elements of Σ∗ ∪ Σω will becalled histories.
I Given H ∈ Σ∗ ∪ Σω, len(H) is the length of H.
I Given H,H ′ ∈ Σ∗ ∪Σω, we write H � H ′ if H is a finite prefixof H ′.
I FinPre(H) = {H | ∃H ′ ∈ H such that H � H ′} is the set offinite prefixes of the elements of H.
I ε is the empty string and FinPre−ε(H) = FinPre(H)− {ε}.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 18
![Page 59: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/59.jpg)
Knowledge and Dynamics
The ‘Playground’: Notation
I Let Σ be any set. The elements of Σ are called events.
I Given any set X , X ∗ is the set of finite strings over X and Xω
the set of infinite strings over X . Elements of Σ∗ ∪ Σω will becalled histories.
I Given H ∈ Σ∗ ∪ Σω, len(H) is the length of H.
I Given H,H ′ ∈ Σ∗ ∪Σω, we write H � H ′ if H is a finite prefixof H ′.
I FinPre(H) = {H | ∃H ′ ∈ H such that H � H ′} is the set offinite prefixes of the elements of H.
I ε is the empty string and FinPre−ε(H) = FinPre(H)− {ε}.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 18
![Page 60: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/60.jpg)
Knowledge and Dynamics
The ‘Playground’: Notation
I Let Σ be any set. The elements of Σ are called events.
I Given any set X , X ∗ is the set of finite strings over X and Xω
the set of infinite strings over X . Elements of Σ∗ ∪ Σω will becalled histories.
I Given H ∈ Σ∗ ∪ Σω, len(H) is the length of H.
I Given H,H ′ ∈ Σ∗ ∪Σω, we write H � H ′ if H is a finite prefixof H ′.
I FinPre(H) = {H | ∃H ′ ∈ H such that H � H ′} is the set offinite prefixes of the elements of H.
I ε is the empty string and FinPre−ε(H) = FinPre(H)− {ε}.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 18
![Page 61: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/61.jpg)
Knowledge and Dynamics
The ‘Playground’: Notation
I Let Σ be any set. The elements of Σ are called events.
I Given any set X , X ∗ is the set of finite strings over X and Xω
the set of infinite strings over X . Elements of Σ∗ ∪ Σω will becalled histories.
I Given H ∈ Σ∗ ∪ Σω, len(H) is the length of H.
I Given H,H ′ ∈ Σ∗ ∪Σω, we write H � H ′ if H is a finite prefixof H ′.
I FinPre(H) = {H | ∃H ′ ∈ H such that H � H ′} is the set offinite prefixes of the elements of H.
I ε is the empty string and FinPre−ε(H) = FinPre(H)− {ε}.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 18
![Page 62: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/62.jpg)
Knowledge and Dynamics
The ‘Playground’: Notation
I Let Σ be any set. The elements of Σ are called events.
I Given any set X , X ∗ is the set of finite strings over X and Xω
the set of infinite strings over X . Elements of Σ∗ ∪ Σω will becalled histories.
I Given H ∈ Σ∗ ∪ Σω, len(H) is the length of H.
I Given H,H ′ ∈ Σ∗ ∪Σω, we write H � H ′ if H is a finite prefixof H ′.
I FinPre(H) = {H | ∃H ′ ∈ H such that H � H ′} is the set offinite prefixes of the elements of H.
I ε is the empty string and FinPre−ε(H) = FinPre(H)− {ε}.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 18
![Page 63: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/63.jpg)
Knowledge and Dynamics
The ‘Playground’: Notation
I Let Σ be any set. The elements of Σ are called events.
I Given any set X , X ∗ is the set of finite strings over X and Xω
the set of infinite strings over X . Elements of Σ∗ ∪ Σω will becalled histories.
I Given H ∈ Σ∗ ∪ Σω, len(H) is the length of H.
I Given H,H ′ ∈ Σ∗ ∪Σω, we write H � H ′ if H is a finite prefixof H ′.
I FinPre(H) = {H | ∃H ′ ∈ H such that H � H ′} is the set offinite prefixes of the elements of H.
I ε is the empty string and FinPre−ε(H) = FinPre(H)− {ε}.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 18
![Page 64: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/64.jpg)
Knowledge and Dynamics
History-based Frames
DefinitionLet Σ be any set of events. A set H ⊆ Σ∗ ∪Σω is called a protocolprovided FinPre−ε(H) ⊆ H. A rooted protocol is any setH ⊆ Σ∗ ∪ Σω where FinPre(H) ⊆ H.
DefinitionAn ETL frame is a tuple 〈Σ,H, {∼i}i∈A〉 where Σ is a (finite orinfinite) set of events, H is a protocol, and for each i ∈ A, ∼i is anequivalence relation on the set of finite strings in H.
Some assumptions:
1. If Σ is assumed to be finite, then we say that F is finitelybranching.
2. If H is a rooted protocol, F is a tree frame,.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 19
![Page 65: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/65.jpg)
Knowledge and Dynamics
History-based Frames
DefinitionLet Σ be any set of events. A set H ⊆ Σ∗ ∪Σω is called a protocolprovided FinPre−ε(H) ⊆ H. A rooted protocol is any setH ⊆ Σ∗ ∪ Σω where FinPre(H) ⊆ H.
DefinitionAn ETL frame is a tuple 〈Σ,H, {∼i}i∈A〉 where Σ is a (finite orinfinite) set of events, H is a protocol, and for each i ∈ A, ∼i is anequivalence relation on the set of finite strings in H.
Some assumptions:
1. If Σ is assumed to be finite, then we say that F is finitelybranching.
2. If H is a rooted protocol, F is a tree frame,.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 19
![Page 66: Introduction to Formal Epistemologyai.stanford.edu/~epacuit/classes/esslli/formep-lec3.pdf · Eric Pacuit and Rohit Parikh August 14, 2007 Eric Pacuit and Rohit Parikh: Introduction](https://reader033.vdocuments.us/reader033/viewer/2022042309/5ed6b19424f2a26584579391/html5/thumbnails/66.jpg)
Knowledge and Dynamics
History-based Frames
DefinitionLet Σ be any set of events. A set H ⊆ Σ∗ ∪Σω is called a protocolprovided FinPre−ε(H) ⊆ H. A rooted protocol is any setH ⊆ Σ∗ ∪ Σω where FinPre(H) ⊆ H.
DefinitionAn ETL frame is a tuple 〈Σ,H, {∼i}i∈A〉 where Σ is a (finite orinfinite) set of events, H is a protocol, and for each i ∈ A, ∼i is anequivalence relation on the set of finite strings in H.
Some assumptions:
1. If Σ is assumed to be finite, then we say that F is finitelybranching.
2. If H is a rooted protocol, F is a tree frame,.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 19
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Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 20
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Knowledge and Dynamics
History-based Models
An ETL model is a structure 〈H, {∼i}i∈A,V 〉 where 〈H, {∼i}i∈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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 21
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Knowledge and Dynamics
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 then
H ′,m |= ϕ
I H, t |= Cϕ iff for each H ′ ∈ H and m ≥ 0 if Ht ∼∗ H ′m then
H ′,m |= ϕ.
where ∼∗ is the reflexive transitive closure of the union of the ∼i .
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 22
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Knowledge and Dynamics
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 then
H ′,m |= ϕ
I H, t |= Cϕ iff for each H ′ ∈ H and m ≥ 0 if Ht ∼∗ H ′m then
H ′,m |= ϕ.
where ∼∗ is the reflexive transitive closure of the union of the ∼i .
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 22
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Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 23
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Knowledge and Dynamics
Returning to the Example
The events are
1. Ann receives a message that the talk is at 2PM (m2PM)
2. Ann receives a message that the talk is at 3PM (m3PM)
3. Ann tells Charles the talk is at 2PM (mA→C )
4. Charles tells Bob the talk is at 2PM (mC→B)
5. Nothing happens (t)
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 24
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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
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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 |= ¬KBKAKBP2PM
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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 |= ¬KBKAKBP2PM
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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 |= ¬KBKAKBP2PM
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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 |= ¬KBKAKBP2PM
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Knowledge and Dynamics
Learning from the Protocol
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 26
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Knowledge and Dynamics
Learning from the Protocol
t = 0
t = 1
t = 2
t = 3
r t
wg t wg t
W W W ¬WR R ¬R ¬R
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 26
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Knowledge and Dynamics
Learning from the Protocol
t = 0
t = 1
t = 2
t = 3
r t
wg t wg t
W W W ¬WR R ¬R ¬R¬KW ∧ ¬KR
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 26
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Knowledge and Dynamics
Learning from the Protocol
t = 0
t = 1
t = 2
t = 3
!W !W !W
r t
wg t wg t
W W WR R ¬R
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 26
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Knowledge and Dynamics
Learning from the Protocol
t = 0
t = 1
t = 2
t = 3
!W !W !W
r t
wg t wg t
W W WR R ¬RKW ∧ ¬KR
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 26
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Knowledge and Dynamics
Learning from the Protocol
t = 0
t = 1
t = 2
t = 3
!W !W !W
r t
wg t wg t
W W WR R ¬R
r t
t t
W ¬WR ¬R
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 26
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Knowledge and Dynamics
Learning from the Protocol
t = 0
t = 1
t = 2
t = 3
!W !W !W
r t
wg t wg t
W W WR R ¬R
r t
t t
W ¬WR ¬R¬KW ∧ ¬KR¬KW ∧ ¬KR
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 26
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Knowledge and Dynamics
Learning from the Protocol
t = 0
t = 1
t = 2
t = 3
!W !W !W
r t
wg t wg t
W W WR R ¬R
!W
r t
t t
R
¬R
W
¬W
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 26
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Knowledge and Dynamics
Learning from the Protocol
t = 0
t = 1
t = 2
t = 3
!W !W !W
r t
wg t wg t
W W WR R ¬R
!W
r t
t t
W
¬R
R
¬W
KW ∧ KRKW ∧ ¬KR
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 26
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Knowledge and Dynamics
Parameters of The Model
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? Synchronization?
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 27
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Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 28
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Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 28
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Knowledge and Dynamics
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 and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 28
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Knowledge and Dynamics
Some Literature
J. Halpern and M. Vardi. The Complexity of Reasoning about Knowledge andTime I. Journal of Computer and System Science (1989).
J. van Benthem and EP. The Tree of Knowledge in Action: Towards a CommonPerspective. Proceedings of Advances in Modal Logic (2006).
J. van Benthem, J. Gerbrandy and EP. Merging Frameworks for Interaction:DEL and ELT. TARK 2007.
Eric Pacuit and Rohit Parikh: Introduction to Formal Epistemology, Lecture 3 29