Distributed knowledge & beliefs

Lennart v. Luijk

Tijs Zwinkels

Jeroen Kuijpers

Jelle Prins

Overview

RecapitulationProving a system

Implicit knowledgeMessage logic (ML)

Belief Discussion

Recapitulation Commonly used symbols

├ (single flubber) used for axiom systems(K) ╞ (double flubber) used for world models(K)

Seriality

Euclidicity:

s1

s2

s3

s1

s2

s3

Proving a system

You prove a System S by proving: S├ φ M╞ φ

Soundness

Completeness

Soundness

DefinitionLet S be an axiom system for epistemic formulas, and let M be a class of Kripke models. Then S is called sound with respect to M, if S├ φ => M ╞ φ

“Everything that can be proven with

the axiom system is actually true.”

Completeness

DefinitionLet S be an axiom system for epistemic formulas, and let M be a class of Kripke models. Then S is called complete with respect to M, if M ╞ φ => S├ φ

“Everything that actually is true

can be proven with the axiom system”

(M,s)╞ Iφ (M,t)╞ φ for all t such that (s,t) є R1∩ … ∩ Rm

If φ is true in every world which can be reached by all agent from the current world w1, then φ is implicit knowledge in w1 ( (M,w1)╞ Iφ )

(A11) Kiφ Iφ (i=1,…,m)

Implicit knowledge (§2.3)

Implicit knowledge (§2.3)

Distributed Knowledge:

R4: IKK mm

m

)...(

)...(

11

1

Implicit knowledge (§2.3)

W1

W2

W3

R1,R2

R1

R2p,~q

~p, ~q

You are here

<M,w1> ╞ K1p

<M,w1> ╞ K2(pq)

<M,w1> ╞ Iq (and also Ip)

p,q

Distributed Knowledge:

(M,s)╞ Iφ (M,t)╞ φ for all t such that (s,t) є R1∩ … ∩ Rm

In normal human language: Iφ : ‘A clever man knows φ’ such as a detective

If one agent knows a b and another knows a then together they know b.

Compare this to C : ‘Any fool knows φ’

Compare this to Ki : ‘Person i knows φ’

Implicit knowledge (§2.3)

Examples

Distributed knowledge: Universe Background radiation:

• Arno Penzias and Robert Wilson have noise in their satellite dish. Thinks this is because of ‘white dielectric material’ (bird droppings)

• This radiation has been predicted years earlier by George Gamow, but didn’t have the instruments to measure the radiation.

Implicit knowledge (§2.3)

Some axioms and systems with I

Axioms: (A11) Kiφ Iφ (i=1,…,m) (R4)

Systems: KI(m)= K(m) + (A11) + KI

TI(m) = T(m) + (A11) + TI

S4I(m) = S4(m) + (A11) + S4I

S5I(m) = S5(m) + (A11) + S5I

IKK mm

m

)...(

)...(

11

1

Implicit knowledge (§2.3)

Proof: Soundness (A11):

Kip Ip:

Suppose (M,s) |= p. If t is such that (s,t) є (R1 ^ .. ^ RN), then ofcourse Rist, so (M,t) |= ip.

Mention Completeness

Message Logic

ML axiom is added to S5I(m)

(ML)“The axiom(ML) says that, if it is implicit knowledge that a state is impossible, then the stronger formula is true that some agent knows that the state is impossible.”

Counter example:

)...( 1 mKKI

W1

W2

W3

R1,R2

R1

R2q

q

~q

<M,w1>╞ I~q

but also ~(K~q)

Belief (§2.4)

(M,s)╞ Biφ (M,t)╞ φ for all t with (s,t) є Ri

© Gummbah

The escaped Knock-knock canary brought false hope to many a lonely citizen

Come in!Knock Knock!

Belief (§2.4)

(D) ¬Bi(┴)

(axiom: a knowledge base is not inconsistent) Same as :

¬ Bi(φ ^ ¬φ)

Same as S5 but no (A3), instead we have (D) KD45(m) = (R1)+(R2)+(A1)+(A2)+(D)+(A4)+(A5)

s tRi

φBiφ

Belief (§2.4)

Proof soundness of KD45

We know that the canonical model Mc(KD45(m)) posesses accessibility relations Ri

c that are serial, transitive and euclidian.

We may combine this with the observation that serial, transitive and euclidian Kripke models are models for (D), (A4) and (A5), respectively. For (A4) and (A5) we know this already. Therefore, we only have to consider the soundness of the Axiom (D).

Belief (§2.4)

Proof soundness of KD45

Suppose KD45(m) ╞ ¬Bi(┴). Then there would be an KD45(m)-model M with a state s such that (M,s)╞ Bi┴. This would mean that all Ri-successors of s would verify ┴, which is only possible if s does not have any Ri successor.

However, by seriality, we know that s does have them, so our assumption, i.e. that KD45(m) ╞ ¬Bi(┴), must be false.

Hence we have KD45(m) ╞ ¬Bi(┴).

Completeness possible to prove, not of interest here.

Belief (§2.4)

W1

W2

W3

R1

R2

p

~p

p

R2

R1

<M,w1>╞ B1p

<M,w1>╞ B2¬p

Discussion

Logical Omniscience (§2.5)

L01-L010 given, give criticism on L05-L09

Knowledge & Belief (§2.13) “logics gone bad” Combining knowledge & beliefs (axiom system KL)

Both sound systems

Both systems have axioms that are good, but not watertight

Combination of the two shows the flaws in the axioms

Result: Wrong example: 2.13.6 T4: Kip ↔ BiKip Is this a valid theorem in KL?

Proof: Ki φ BiKi φ

1) KL(i) ├ Kiφ KiKiφ (A4)

2) KL(i) ├ Kiφ Biφ (KL14)

3) KL(i) ├ (Kiφ Biφ) (KiKiφ BiKiφ) (A1)

4) KL(i) ├ KiKiφ BiKiφ (MP 2,3)

5) KL(i) ├ (Kiφ KiKiφ) (Kiφ BiKiφ) (HS

short)

6) KL(i) ├ Kiφ BiKiφ (MP 4,5)

Short proof: BiKi φ Ki φ

1) KL(i) ├ BiKiφ ¬Bi¬Kiφ (D “¬Bi(┴)” in its form ¬(Biφ ^ Bi¬φ) and prop.

logic)

2) KL(i) ├ ¬Bi¬Kiφ ¬Ki¬Kiφ (KL14 “Kiφ Biφ” and prop. logic: contraposition)

3) KL(i) ├ ¬Ki¬Kiφ Kiφ (A5/KL3 “¬Kiφ Ki¬Kiφ” and prop. logic:

contraposition)

4) KL(i) ├ BiKiφ Kiφ (from 1,2,3 by prop. logic: hypothetical syllogism)

Problems with K&B Example:

Homeopathic dilution

Two persons live in axiom system KL (Hippie Tijs and scientist Lennart)

Both take the same homeopathic medicine to releave them from extreme fatigue due to too much work at their university

Tijs believes he knows it works (BtKtw) Lennart believes he knows it doesn’t work (BLKL¬w)

One dies and one survives. Who will survive?

Problems with K&B

Another Example:

Two persons live in axiom system KD45(m) (Hippie Tijs and scientist Lennart)

Both take the same homeopathic medicine to releave them from extreme fatigue due to too much work at their university

Tijs believes he knows it works (BtKtw) Lennart believes he knows it doesn’t work (BLKL¬w)

What happens now?

FIN

Full Proof of: BiKip Kip

To prove: BiKip Kip

1) KL(i) ├ ¬Bi(┴) (D)

2) KL(i) ├ ¬Bi(┴) ¬(Biφ ^ Bi¬φ) (A1)

3) KL(i) ├ ¬(Biφ ^ Bi¬φ) (mp 1,2)

4) KL(i) ├ ¬(Biφ ^ Bi¬φ) ¬(BiKiφ ^ Bi¬Kiφ) (A1)

5) KL(i) ├ ¬(BiKiφ ^ Bi¬Kiφ) (MP 3,4)

6) KL(i) ├ ¬(BiKiφ ^ Bi¬Kiφ) ¬BiKiφ V ¬Bi¬Kiφ (A1)

7) KL(i) ├ ¬BiKiφ V ¬Bi¬Kiφ (MP 5,6)

8) KL(i) ├ (¬BiKiφ V ¬Bi¬Kiφ) (BiKiφ ¬Bi¬Kiφ) (A1)

9) KL(i) ├ BiKiφ ¬Bi¬Kiφ (MP 7,8)

10) KL(i) ├ (Kiφ Biφ) (KL 14)

11) KL(i) ├ (Kiφ Biφ) (Ki¬Kiφ Bi¬Kiφ) (A1)

To prove: BiKip Kip

9) KL(i) ├ BiKiφ ¬Bi¬Kiφ

10) KL(i) ├ (Kiφ Biφ)

11) KL(i) ├ (Kiφ Biφ) (Ki¬Kiφ Bi ¬Kiφ)

12) KL(i) ├ (Ki¬Kiφ Bi¬Kiφ) (MP 10,11)

13) KL(i) ├ ¬Bi¬Kiφ ¬Ki¬Kiφ (Contraposition of 12)

14) KL(i) ├ ¬Kiφ Ki¬Kiφ (A5/ KL3)

15) KL(i) ├ ¬Ki¬Kiφ Kiφ (Contraposition of 14)

| 16) KL(i) ├ BiKiφ (Assumption)

| 17) KL(i) ├ ¬Bi¬Kiφ (MP 16, 9)

| 18) KL(i) ├ ¬Ki¬Kiφ (MP 17, 13)

| 19) KL(i) ├ Kiφ (MP 18, 15)

To prove: BiKip Kip

| 16) KL(i) ├ BiKiφ (Assumption)

| 17) KL(i) ├ ¬Bi¬Kiφ (MP 16, 9)

| 18) KL(i) ├ ¬Ki¬Kiφ (MP 17, 13)

| 19) KL(i) ├ Kiφ (MP 18, 15)

20) KL(i) ├ BiKiφ Kiφ ( intro 16-

19)