Common Knowledge in Email Exchanges Krzysztof R. Apt CWI, - - PowerPoint PPT Presentation

Common Knowledge in Email Exchanges

Krzysztof R. Apt

CWI, Amsterdam, the Netherlands, University of Amsterdam

based on joint work with

Floor Sietsma, CWI

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History: Common, Knowledge

COMMON, something that belongs to all alike, in contradistiction to proper, peculiar. KNOWLEDGE is defined, by Mr Locke, to be the perception of the connection and agreement, or disagreement and repugnancy, or our ideas. Encyclopaedia Britannica By a Society of GENTLEMEN in Scotland Edinburgh MDCCLXXI

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History: Nested Knowledge

• A. Koestler, Arrow in the Blue, p. 75, 1952.
• p. 192: “As for the diplomatic informer, he worked, of course for

the Deuxième Bureau, and the chief instructed us that we should carefully stick to the fiction that we didn’t know that he knew that we knew it.

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History: Common Knowledge (1)

Morris Friedell, On the structure of shared awareness, Working Paper, 1967. p.2: “The final section, Economics, will deal with A thinks B thinks ... reasoning in formal games.”

• A. Koestler, Arrow in the Blue, 1952.
• p. 279: “It is common knowledge that if you plant a flag on a

hitherto uncharted island or territory, you have staked a claim to that territory on behalf of the nation which the flag represents.”

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• me History: Common Knowledge (2)

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History: Email

Ray Tomlinson (BBN): 1971. Ted Meyer and Austin Henderson(BBN): BCC feature added in 1975.

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Story 1: Why BCC is Useful

FYI: Wouter Bos knows Trijntje Oosterhuis. Source: Wouter Bos e-mailt per ongeluk zijn netwerk rond. NRC Handelsblad, 7th October 2010.

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Story 2: Forwarding

I got an email from my Chinese postdoc Helen with a CC to her husband Bo.

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Story 2: Forwarding

I got an email from my Chinese postdoc Helen with a CC to her husband Bo. I forwarded it to Floor.

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Story 2: Forwarding

I got an email from my Chinese postdoc Helen with a CC to her husband Bo. I forwarded it to Floor. Floor replied to my forward to Bo and Helen with a BCC to me.

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Story 2: Forwarding

I got an email from my Chinese postdoc Helen with a CC to her husband Bo. I forwarded it to Floor. Floor replied to my forward to Bo and Helen with a BCC to me. I forwarded the last email to Helen and Bo with a BCC to Floor.

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Story 2: Forwarding

I got an email from my Chinese postdoc Helen with a CC to her husband Bo. I forwarded it to Floor. Floor replied to my forward to Bo and Helen with a BCC to me. I forwarded the last email to Helen and Bo with a BCC to Floor. Do we all have common knowledge of Floor’s reply?

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Messages

Agents: finite set Ag, Notes: further unspecified.

s(i, l, G);

message containing note l, sent by agent i to group G,

f(i, l.m, G);

forwarding by i of message m with added note l, sent to group G,

S(m) = {sender of m}, R(m) = receivers of m.

Special cases: reply: f(i, l.m, G), with G = S(m), where i ∈ R(m), reply-all: f(i, l.m, G), with G = S(m) ∪ R(m), where i ∈ R(m).

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Emails

Email: mB, with

m a message, B ⊆ Ag a set of BCC recipients.

Examples Passing a note further:

s(i, l, G)B, s(j, l, G′)B′, where j ∈ G ∪ B,

Forwarding an email:

mB, f(i, l.m, G)C, where i ∈ R(m) ∪ B,

Forwarding one’s own email:

mB, f(i, l.m, G)C, where S(m) = {i},

BCC reply-all (grrr . . .):

mB, f(i, l.m, G)C, where i ∈ B and G = S(m) ∪ R(m).

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Modelling Story 2

I got an email from my Chinese postdoc Helen with a CC to her husband Bo.

e := m∅, where m := s(H, l, {B, K}),

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Modelling Story 2

I got an email from my Chinese postdoc Helen with a CC to her husband Bo.

e := m∅, where m := s(H, l, {B, K}),

I forwarded it to Floor.

e′ := m′ ∅, where m′ := f(K, m, F),

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Modelling Story 2

I got an email from my Chinese postdoc Helen with a CC to her husband Bo.

e := m∅, where m := s(H, l, {B, K}),

I forwarded it to Floor.

e′ := m′ ∅, where m′ := f(K, m, F),

Floor replied to my forward to Bo and Helen with a BCC to me.

e′′ := m′′

{K}, where m′′ := f(F, m′, {B, H}),

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Modelling Story 2

I got an email from my Chinese postdoc Helen with a CC to her husband Bo.

e := m∅, where m := s(H, l, {B, K}),

I forwarded it to Floor.

e′ := m′ ∅, where m′ := f(K, m, F),

Floor replied to my forward to Bo and Helen with a BCC to me.

e′′ := m′′

{K}, where m′′ := f(F, m′, {B, H}),

I forwarded the last email to Helen and Bo with a BCC to Floor.

f(K, m′′, {H, B}){F}.

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Assumptions

Each agent has his set of notes, can send/forward his notes, also can send/forward notes he received, also can forward messages he received.

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Some Considerations

Are all sets of emails meaningful?

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Some Considerations

Are all sets of emails meaningful? No: you can’t forward a note you did not receive.

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Some Considerations

Are all sets of emails meaningful? No: you can’t forward a note you did not receive. If one sends/forwards somebody’s else note, then one should have received it.

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Some Considerations

Are all sets of emails meaningful? No: you can’t forward a note you did not receive. If one sends/forwards somebody’s else note, then one should have received it. If one forwards a message, then one should have received it.

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States

Factual information:

FI(s(i, l, G)) := {l}, FI(f(i, l.m, G)) := FI(m) ∪ {l}.

State: s = (E, L), where

E: set of emails, L = (L1, . . ., Lk): sets of agents’ notes.

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Legal States

Legal state:

s = (E, L), such that for some partial ordering ≺ on E

for each s(i, l, G)B ∈ E, where l ∈ Li,

mC ∈ E exists such that mC ≺ s(i, l, G)B, i ∈ R(m) ∪ C and l ∈ FI(m),

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Legal States

Legal state:

s = (E, L), such that for some partial ordering ≺ on E

for each s(i, l, G)B ∈ E, where l ∈ Li,

mC ∈ E exists such that mC ≺ s(i, l, G)B, i ∈ R(m) ∪ C and l ∈ FI(m),

for each f(i, l.m, G)B ∈ E, where l ∈ Li, some m′

C ∈ E exists such that

m′

C ≺ f(i, l.m, G)B, i ∈ R(m′) ∪ C and l ∈ FI(m′),

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Legal States

Legal state:

s = (E, L), such that for some partial ordering ≺ on E

for each s(i, l, G)B ∈ E, where l ∈ Li,

mC ∈ E exists such that mC ≺ s(i, l, G)B, i ∈ R(m) ∪ C and l ∈ FI(m),

for each f(i, l.m, G)B ∈ E, where l ∈ Li, some m′

C ∈ E exists such that

m′

C ≺ f(i, l.m, G)B, i ∈ R(m′) ∪ C and l ∈ FI(m′),

for each f(i, l.m, G)B ∈ E some mC ∈ E exists such that

mC ≺ f(i, l.m, G)B and i ∈ S(m) ∪ R(m) ∪ C.

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Language to Discuss Emails

ϕ ::= m | i ◭ m | ¬ϕ | ϕ ∧ ϕ

Abbreviation:

mB ::= m ∧

• i∈S(m)∪R(m)∪B

i ◭ m ∧

• i∈S(m)∪R(m)∪B

¬i ◭ m

Semantics Take s = (E, L).

s | = m

iff

∃B : mB ∈ E s | = i ◭ m

iff

∃B : mB ∈ E and i ∈ S(m) ∪ R(m) ∪ B s | = ¬ϕ

iff

s | = ϕ s | = ϕ ∧ ψ

iff

s | = ϕ and s | = ψ

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Epistemic Language

Add CAϕ to the language discussing emails. Intuition: CAϕ iff group A commonly knows ϕ. Semantics Take s = (E, L).

s | = CAϕ

iff

s′ | = ϕ for any legal s′ s.t. s ∼A s′. ∼G is the reflexive, transitive closure of

i∈A ∼i.

How to define ∼i?

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∼i Relation on Emails

mB ∼i m′

B′

iff

i ∈ S(m), B = B′ and m = m′, or i ∈ R(m) \ S(m) and m = m′, or i ∈ B ∩ B′ and m = m′.

Example If i ∈ B ∩ B′, then i cannot distinguish mB from m′

B′.

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∼i Relation on Legal States

(E, L) ∼i (E′, L′)

iff

Li = L′

i,

∀mB ∈ E (i ∈ S(m) ∪ R(m) ∪ B → ∃mB′ ∈ E′ mB ∼i mB′), ∀mB′ ∈ E′ (i ∈ S(m) ∪ R(m) ∪ B′ → ∃mB ∈ E mB ∼i mB).

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Common Knowledge of a Message

Theorem

s | = CAm iff

there is an email shared by the group A that proves the existence of m. Formally:

EA := {mB ∈ E | A ⊆ S(m)∪R(m) or ∃j ∈ B : (A ⊆ S(m)∪{j})}. EA is the set of emails that group A shared. s | = CAm iff ∃m′

B ∈ EA : m′ → m is valid.

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Common Knowledge of an Email

Theorem Assume |A| ≥ 3.

s | = CAmB iff

there is an email shared by the group A that proves the existence of m,

s | = CAm iff ∃m′

B ∈ EA : m′ → m is valid.

for every agent j ∈ B there is an email shared by A that proves that j forwarded m,

∀j ∈ B ∃m′

B ∈ EA : m′ → j ◭ m is valid.

mB involves all agents,

Ag = S(m) ∪ R(m) ∪ B.

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Conclusions Concerning Story 2

We have

s | = C{B,H,K,F}f(F, m′, {B, H}){K},

where

m′ := f(K, m, F), m := s(H, l, {B, K}).

I forwarded Floor’s reply to Helen and Bo with a BCC to Floor:

f(K, m′′, {H, B}){F},

where m′′ := f(F, m′, {B, H}), I should have forwarded Floor’s reply to Helen and Bo and Floor:

f(K, m′′, {H, B, F})∅.

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Can one Simulate BCC?

Yes and No.

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Can one Simulate BCC?

Yes and No. Yes:

mB ≡ m, f(i, m, j1), . . ., f(i, m, jk),

where S(m) = {i} and B = {j1, . . ., jk}.

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Can one Simulate BCC?

Yes and No. Yes:

mB ≡ m, f(i, m, j1), . . ., f(i, m, jk),

where S(m) = {i} and B = {j1, . . ., jk}. No: the mailboxes of the BCC recipients differ. So for j ∈ B

mB, f(j, m, G) is legal, while m, f(i, m, j1), . . ., f(i, m, jk), f(j, m, G) not.

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Can one Simulate BCC? (ctd)

Take state s with

Es = {s(1, l, 2){3}}.

Then s |

= K3¬K2K3s(1, l, 2).

Take the simulation state t. So

Et = {s(1, l, 2), f(1, s(1, l, 2), 3)}.

Then t |

= K3¬K2K3s(1, l, 2).

This observation can be made more general.

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Current Research

Decidability of semantics. Sound and complete axiomatization. Operational semantics.

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THANK YOU

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Dzi˛ ekuj˛ e za uwag˛ e

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