where do informational partitions come from? tho saugh i... alle the mervelous signals of the goddys...
Post on 01-Apr-2015
212 Views
Preview:
TRANSCRIPT
Where do
informationalpartitions come from?
Tho saugh I ... alle the mervelous signalsOf the goddys celestials Chaucer, The House of Fame (c1384)
.
....
..
StatesStates
Signal = player’s information = player’s knowledge
Signal = player’s information = player’s knowledge
.
....
..
E
From partition to knowledge
K(E)
Here the player knows the element of the partition that contains the states.
...and also E.
The event that she knows E.
Partitional knowledge
Partitional knowledge
ω
(ω)
Ω – a state space – a partition of Ω(ω) – the element of that contains state ω. ω K(E) when (ω) E.
Ω – a state space – a partition of Ω(ω) – the element of that contains state ω. ω K(E) when (ω) E.
From knowledge to partitionSyntax
Bob knows that G.W. Bush is the president of the US and he does not know that Barbara is G.W.’s wife.
The set of all sentences form a Boolean algebra (w.r.t “and”, “or”, “not”) with an operator (know)
Language:
An algebra of subsets, Awith an operator K: A A.
Language:
An algebra of subsets, Awith an operator K: A A.
A family of sets closed under intersection and complement
Sentences
Deduction rules
vs. Semantics
From knowledge to partition
The axioms of S5 knowledge:
0. The axioms of Boolean algebra
1. K() =
2. K(E) K(F) = K(E F)
3. K(E) E
4. ¬ K(E) = K(¬ K(E))
0. The axioms of Boolean algebra
1. K() =
2. K(E) K(F) = K(E F)
3. K(E) E
4. ¬ K(E) = K(¬ K(E))
British Panel Gives Rumsfeld 'Foot In Mouth' Award
Defense Secretary's Comment On 'Known Unknowns' Is Most Baffling
LONDON -- He may not know it -- or he may know that he knows it -- but Secretary of Defense Donald Rumsfeld has won this year's "Foot in Mouth" award for the by a public figure.
Britain's Plain English Campaign cited Rumsfeld's comment on Iraq, when he said the following during a Pentagon briefing:
"Reports that say that something hasn't happened are always interesting to me, because as we know, , there are things we know we know.
"We also know . That is to say we know there are some things we do not know. But there are also
most baffling statement
there are known knowns
there are known unknowns unknown unknowns: the oneswe don't know we don't know.”
From knowledge to partition
The axioms of S5 knowledge:
0. The axioms of Boolean algebra
1. K() =
2. K(E) K(F) = K(E F)
3. K(E) E
4. ¬ K(E) = K(¬ K(E))
0. The axioms of Boolean algebra
1. K() =
2. K(E) K(F) = K(E F)
3. K(E) E
4. ¬ K(E) = K(¬ K(E))
Is S5-knowledge partitional?Partitional knowledge satisfies S5.
Fairy tale:
S5 implies partition
Fairy tale:
S5 implies partition
Hard facts:
Words alone do not bring about partitions
Hard facts:
Words alone do not bring about partitions
.S5
... . .. .-3 -2 -1 0 1 2 3
. . . . . . . ...4 5-5 -4
A0 – the algebra generated by all arithmetic sequences.
An arithmetic sequence: {a + zd | z Z} for d 0.
Ω = the set of integers, Z.
Example 1
e.g. -2 + 3z
K: A0 A0 is the identity.
K is generated by the partition into singletons.
... . .. .-3 -2 -1 0 1 2 3
. . . . . . . ...4 5-5 -4
A – the algebra generated by A0 and the set P.
K(E) = F
P
Example 1
Each E A can be uniquely decomposed:
E = X F Y
P ¬ P
A0
1. K() =
2. K(E) K(E’) = K(E E’)
3. K(E) E
4. ¬ K(E) = K(¬ K(E))
K is the identity on A0
K (P) =
2K satisfies S5.
The candidate for a partition: singletons.
If K is partitional then at state 3 the player knows P (because it contains {3}).
Therefore K is not partitional.
.
....
..
From partition to knowledges common
1 -
2 -
c - coarser than 1 and 2
finest than any such partition
Kc - narower than K1 and K2
For each E, Kc(E) K1(E), K2(E)
broader than any such knowledge
the common knowledge partition.
Agreeing to disagree, Aumann (1976)
From partition to knowledges common
1 -
2 -
c - coarser than 1 and 2
finest than any such partition
Kc - narower than K1 and K2
For each E, Kc(E) K1(E), K2(E)
broader than any such knowledge
the common knowledge partition.
Agreeing to disagree, Aumann (1976)
Is it possible to prove syntactically the existence of Kc for S5-knowledge K1 and K2 ?
Is it possible to prove syntactically the existence of Kc for S5-knowledge K1 and K2 ?
NO!
Is there a set S of axioms of knowledge such that
K satisfies S K is partitional
For S = S5:
For S = {K(E) = E}: NO!
top related