Upload
derick-hann
View
220
Download
0
Tags:
Embed Size (px)
Citation preview
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
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)
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.
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?
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)