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C.&0(2cX t Bo ?"6@ NNN?N#Q   C.&0(2cX t Br ?"6@ NNN?N<2|  C.&0(2cX t B`v ?"6@ NNN?NT   C.&0(2cX t H Ԕ?"6@ NNN?N a ,$D0 t B Ԕ?"6@ NNN?N"  ,$D0n ,t 3 ry 3?3ԔgggAD[X8c?"<?@@`NNN?N ,$D 0 TStates2(2ea .t 3 r?fԔL!?"<?@@`NNN?N* O Y ,$D0 hSignal = player s information = player s knowledge 5P5a /t C *Al-cloudsMr ,$D0H t 0޽h ? ̙33jj___PPT10j.@'+vbDi' = @B Dh' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,t%(D' =-o6Bwipe(up)*<3<*,tD' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*/t%(D' =-g6B fade*<3<*/tD~' =%(D' =%(DC' =4@BB5BB%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*/tD' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*/tD' =-g6B fade*<3<*/tD' =1:Bhidden*o3>+B#style.visibility<*/t%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*-t%(D' =-g6B fade*<3<*-tD[' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*(t%(D' =-g6B fade*<3<*(tD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-g6B fade*<3<* tD8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-g6B fade*<3<* tDX ' =%(D ' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<* tD' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<* tD' =-g6B fade*<3<* tD' =1:Bhidden*o3>+B#style.visibility<* t%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<* tD' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<* tD' =-g6B fade*<3<* tD' =1:Bhidden*o3>+B#style.visibility<* t%(D} ' =%(D' =%(Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*'t%(D' =-g6B fade*<3<*'tD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*'tD' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*'tD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*'t%(D ' =%(D' =%(DY' =4@BB BB%()))D' =-g6B fade*<3<*tD' =1:Bhidden*o3>+B#style.visibility<*t%(D' =%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*tD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*tD' =-g6B fade*<3<*tD' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-g6B fade*<3<*tD ' =%(D' =%(DY' =4@BB BB%()))D' =-g6B fade*<3<*tD' =1:Bhidden*o3>+B#style.visibility<*t%(D' =%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*tD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*tD' =-g6B fade*<3<*tD' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-g6B fade*<3<*tD' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*tD' =1:Bhidden*o3>+B#style.visibility<*t%(D' =%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*tD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*tD' =-g6B fade*<3<*tD' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-g6B fade*<3<*tD' =%(D5' =%(DY' =4@BB/BB%(D' =-g6B fade*<3<*tD ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*tD' =+4 8?^CB ppt_yBCBppt_y-.1B*Y3>B ppt_y<*tD' =1:Bhidden*o3>+B#style.visibility<*t%(DY' =4@BB/BB%(D' =-g6B fade*<3<*tD ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*tD' =+4 8?^CB ppt_yBCBppt_y-.1B*Y3>B ppt_y<*tD' =1:Bhidden*o3>+B#style.visibility<*t%(DY' =4@BB/BB%(D' =-g6B fade*<3<*tD ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*tD' =+4 8?^CB ppt_yBCBppt_y-.1B*Y3>B ppt_y<*tD' =1:Bhidden*o3>+B#style.visibility<*t%(DY' =4@BB/BB%(D' =-g6B fade*<3<*tD ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*tD' =+4 8?^CB ppt_yBCBppt_y-.1B*Y3>B ppt_y<*tD' =1:Bhidden*o3>+B#style.visibility<*t%(DY' =4@BB/BB%(D' =-g6B fade*<3<*'tD ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*'tD' =+4 8?^CB ppt_yBCBppt_y-.1B*Y3>B ppt_y<*'tD' =1:Bhidden*o3>+B#style.visibility<*'t%(D ' =%(D' =%(Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.t%(D' =-g6B fade*<3<*.tD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*.tD' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*.t+P+0+ t0 ++0+ t0 ++0+ t0 ++0+ t0 ++0+,t0 ++0+.t0 +0 .. 8.(    H 3Ԕ?"6@ NNN?NQn9F :|   |:   < ?"6@ NNN?N  C.&0(2cX  <L ?"6@ NNN?N| C.&0(2cX  < ?"6@ NNN?N:0 C.&0(2cX  < ?"6@ NNN?N!  C.&0(2cX  < ?"6@ NNN?N#Q   C.&0(2cX  <\ ?"6@ NNN?N<2|  C.&0(2cX  < ?"6@ NNN?NT   C.&0(2cX  B Ԕ?"6@ NNN?N a   B Ԕ?"6@ NNN?N"     0e0e    B C DEXF* A@  Ԕ 8c8c     ?A)BCD|E||j*=:7D    JBmJwj@       `# "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abc! g,$D 0  <| ?"6@ NNN?N-,$  0 ?E" 2g3  < ?"6@ NNN?N8,$0 UFrom partition to knowledge 2c     0e0e    BX CYDEdF. A@  Ԕ 8c8c     ?A)BCD|E||uPyYrM rJX h (C\=@        `# "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab~ & ,$D 0L  < ?"6@ NNN?N. ,$ 0 nK(E)N 2ggg   < ?"6@ NNN?N ,$0 MHere the player knows the element of the partition that contains the states. HN 2ee9eB !@ HDo?"0@NNN?N= ,$@ 0B #@ HDo?"0@NNN?N ,$D 0H $ < ?"6@ NNN?N +,$ 0 j...and also E.@ 2 aeay % < ?"6@ NNN?Np  ,$0 The event that she knows E.d 2aeaeaB ' HDo?"0@NNN?NZ ,$D 0 ( 3 r 3?3ԔgggAD[X8c?"<?@@`NNN?N} c5,$D 0 Partitional knowledge6(2g c   N 0 N ?"6@ NNN?N,$0 ^" 2\ 2 NP ?"6@ NNN?N4T,$0 lP()" 2" * 3 r`?fԔL!?"<?@@`NNN?NCe  ,$@0   a state space P  a partition of  P()  the element of P that contains state .  K(E) when P() E.6vP,$B 8 s <hApointing9,$D 0H  0޽h ? ̙33SS___PPT10S.@'+:DGQ' = @B DQ' = @BA?%,( < +O%,( < +D ' =%(D' =%(Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*D*' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-u6Bwipe(right)*<3<*!D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-u6Bwipe(right)*<3<*#D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*D' =1:B true*a3>Bstroke.on<*D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*D' =1:B true*a3>Bstroke.on<*D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*D' =1:B true*a3>Bstroke.on<*D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D)' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-g6B fade*<3<*%D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-u6Bwipe(right)*<3<*'D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bwipe(up)*<3<*(D ' =%(DS ' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-g6B fade*<3<**D)' =4@BB BB%(D' =-g6B fade*<3<*!D' =1:Bhidden*o3>+B#style.visibility<*!%(D)' =4@BB BB%(D' =-g6B fade*<3<*#D' =1:Bhidden*o3>+B#style.visibility<*#%(D)' =4@BB BB%(D' =-g6B fade*<3<*'D' =1:Bhidden*o3>+B#style.visibility<*'%(D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-g6B fade*<3<**D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<**'%(D' =-g6B fade*<3<**'D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<**'p%(D' =-g6B fade*<3<**'pD' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*8D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*8D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0Du' =%(D' =%(D-' =4@BB8BB%()?)?D' =.I7 BBBBB[M -3.05556E-6 2.22222E-6 L 0.06997 -0.09769 *3>*B ppt_xB ppt_y=@0BBAApBB=B,H<*8D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-g6B fade*<3<*2D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<**p%(D' =-g6B fade*<3<**p+0+0+0 ++0+0 ++0+0 ++0+ 0 ++0+$0 ++0+%0 ++0+(0 ++0+00 ++0+20 ++0+*0 +@0aN0 xp@  (    << ?"6@ NNN?NLp UFrom knowledge to partition 2c    <hA ?"6@ NNN?NZH,$0 BSyntax  2ef  <E ?"6@ NNN?NA,$0 Bob knows that G.W. Bush is the president of the US and he does not know that Barbara is G.W. s wife.f 2ae\+ae aeae\aY  BQ Ԕ?"6@ NNN?N ,$D 0 The set of all sentences form a Boolean algebra (w.r.t  and ,  or ,  not ) with an operator (know)d 2a e!aeaeaeaefa$2,B  BD)?"0@NNN?N 8H ,$D 0   3 rg?fԔL!?"<?@@`NNN?N T,$D 0 zLanguage: An algebra of subsets, A with an operator K: A A. 24(2!agaefagag0   HhGH4fo?"6@ NNN?N 5 ,$D 0 9A family of sets closed under intersection and complementT: 2a ea e9=   < ?"6@ NNN?NZ  ,$ 0 _ Sentences 2 ef C   <  ?"6@ NNN?Ne,$ 0 eDeduction rules  2efD  <pk ?"6@ NNN?NVF ,$ 0 fvs. Semantics" 2effH  0޽h ?  ̙33..___PPT10-.+s;D+' = @B D+' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D\' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*  %(D' =-g6B fade*<3<*  D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*  >%(D' =-g6B fade*<3<*  >D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<* D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+0 +!0  ,(    <h ?"6@ NNN?NLp UFrom knowledge to partition 2c o  <8 ?"6@ NNN?N`O ,$ 0 The axioms of S5 knowledge:@ 2aef av  3 rX 3?3ԔgggAD[X8c?"<?@@`NNN?Nq&Y ,$D0 \0. The axioms of Boolean algebra 1. K() = 2. K(E) K(F) = K(E F) 3. K(E) E 4. K(E) = K( K(E))\s 2e"aeae3aeae3aae3ae3aaeae3a$aeae3ae3ae3aH  0޽h ? ̙33 ___PPT10.0<+LOhDI' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-g6B fade*<3<*#D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*#1%(D' =-g6B fade*<3<*#1D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*1M%(D' =-g6B fade*<3<*1MD' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*M[%(D' =-g6B fade*<3<*M[D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*[s%(D' =-g6B fade*<3<*[s+p+0+0 ++0+0 +0 aN0    , (    0|P 2British Panel Gives Rumsfeld 'Foot In Mouth' Award030 22aa  0P ADefense Secretary's Comment On 'Known Unknowns' Is Most Baffling 4B0 2@cc  0P JLONDON -- 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. 40 2 cccn  0P0   *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.  +0+c' L  0$ P` "We also know . That is to say we know there are some things we do not know. But there are also D0vcgcR  C *Arumsfeldp '  N* ?"6@ NNN?N  kmost baffling statement 2c0   N 1 ?"6@ NNN?N^ c E  tthere are known knowns 2c(   N5 ?"6@ NNN?N .Z i lthere are known unknowns 2c   B: ?"6@ NNN?N*\F \ unknown unknowns: the ones we don't know we don't know.  2cH  0޽h ? 33  ___PPT10 ..'+D ' = @B DE ' = @BA?%,( < +O%,( < +DY' =%(D' =%(D' =A@BBBB0B%(D' =,543*3>!Bstyle.color='`B@BPB<*DY' =%(D' =%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<* DY' =%(D' =%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<* DY' =%(D' =%(D' =A@BBBB0B%(D' =,543*3>!Bstyle.color='`B@BPB<*++0+0 ++0+ 0 ++0+ 0 ++0+0 + 0 aN0    (    <L ?"6@ NNN?NLp UFrom knowledge to partition 2c ;  <0R ?"6@ NNN?N`O  The axioms of S5 knowledge:@ 2aef aB  3 r\Z 3?3ԔgggAD[X8c?"<?@@`NNN?Nq&Y  \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))\s 2e"aeae3aeae3aae3ae3aaeae3a$aeae3ae3ae3a  <Y ?"6@ NNN?N>o,$ 0 Is S5-knowledge partitional?(2g(c(g(3c(g(c($   <T{ ?"6@ NNN?N %,$0 #Partitional knowledge satisfies S5.B$(2 aefa H  0޽h ? ̙33!  ___PPT10.0<+4ϐD]' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*+p+0+0 ++0+0 +?0 aN0 !    (    3 r 3?3ԔgggAD[X8c?"<?@@`NNN?N v` K ,$D 0 o!Fairy tale: S5 implies partition2"(2 ea  3 r 3?3ԔgggAD[X8c?"<?@@`NNN?N D ,$D  0 5Hard facts: Words alone do not bring about partitions26(2 e,ar   6?"6@ NNN?N  ,$D0  s 6A  10950d ,$D 02   B ?"6@ NNN?N_y 3   <Ȕ ?"6@ NNN?N  ,$0 U. 2c0z   ,$D 0`  c 0A stork B B1! ?"6@ NNN?N *  @S5" 2g B <؝ ?"6@ NNN?N  4 BP$ 2H  0޽h ? ̙33h5`5___PPT10@5.V@+D44' = @B D3' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*Dv' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D-' =4@BB5BB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* Dv' =%(DC' =4@BB5BB%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<* D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D:' =A@BB8BB0B%()?)?D' =.I7 BBBBB[M 8.33333E-7 -3.7037E-6 L -0.22222 -0.38564 *3>*B ppt_xB ppt_y=@0BBAApBB9㽐BsE<* D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =%(Db' =4@BB BB%(D' 2=,54*3>!Bstyle.color='`B@BPB<*D' 2=,54*3>Bfillcolor=@BPB<*D' 2=1:B solid*a3>Bfill.type<*D' 2=1:B true*]3>Bfill.on<*D' 2=/u8@C*Q3> Br<*%(D' d=/u8C*Q3> Br<*%(dD' d=/u8@C*Q3> Br<*%(D' d=/u8C*Q3> Br<*%(,D' d=/u8@C*Q3> Br<*%(D ' =%(D' =4@BBBB%(D ' =+4 8?XCB ppt_xBCB ppt_xB*Y3>B ppt_x<*D' =+4 8?`CB ppt_yBCB1+ppt_h/2B*Y3>B ppt_y<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(DY' =4@BB BB%()))D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*++0+0 ++0+0 ++0+ 0 ++0+ 0 +60aN0 ** !4H*(  t 1 3 r0?fԔL!?"<?@@`NNN?Nh=R,$D0 Z  G0'  Bl ?"6@ NNN?NC ,$0 C.&0(2cX'  BT ?"6@ NNN?NCJ @ ,$0 C.&0(2cX'  B ?"6@ NNN?NC,$0 C.&0(2cX'  B ?"6@ NNN?NC ,$0 C.&0(2cX'   BX ?"6@ NNN?NCZP,$0 C.&0(2cX'   B ?"6@ NNN?NC,$ 0 C.&0(2cX'   B ?"6@ NNN?NC ,$ 0 C.&0(2cX   <\ ?"6@ NNN?N1,$0 @-3" 2g  < ?"6@ NNN?N ,$0 @-2" 2g  < ?"6@ NNN?N] ,$0 @-1" 2g  <` ?"6@ NNN?N ,$0 ?0" 2g  <  ?"6@ NNN?N ,$0 ?1" 2g  < ?"6@ NNN?NI :,$0 ?2" 2g  <  ?"6@ NNN?NZ,$0 ?3" 2g  <L ?"6@ NNN?N8,$ 0 @. . .  2c  <$  ?"6@ NNN?N@,$ 0 @. . .  2c!  < ?"6@ NNN?NC,$ 0 C.&0(2cX!  < ?"6@ NNN?NC,$0 C.&0(2cX!   < ?"6@ NNN?NC~t,$0 C.&0(2cX! ! < ?"6@ NNN?NC:0,$0 C.&0(2cX " < ?"6@ NNN?N,$0 ?4" 2g # <X# ?"6@ NNN?N,$0 ?5" 2g $ <' ?"6@ NNN?N-,$0 @-5" 2g % <l+ ?"6@ NNN?N|,$0 @-4" 2g + </ ?"6@ NNN?N~ % ,$"0 pA0  the algebra generated by all arithmetic sequences. n9 2goc$5ac, . <9 ?"6@ NNN?N6 V ,$ 0 &dAn arithmetic sequence: {a + zd | z Z} for d 0.3 2aeaeeaaeaeaeeaeaaea / <I ?"6@ NNN?Ni ,$0 6 = the set of integers, Z. 2 0 <Q ?"6@ NNN?N. D Example 1  2 c$] 2 <V ?"6@ NNN?Ng * ,$0 e.g. -2 + 3zT 2aeae  3 <\ ?"6@ NNN?N* { q,$#0 ,8K: A0 A0 is the identity. 2e̙agoagoa 4 Bf 3Ԕ?"6@ NNN?NO,$D$0 0K is generated by the partition into singletons.F1(2e̙a/e0H  0޽h ? ̙33cb___PPT10b./Ї+2;.D[' = @B DZ' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-g6B fade*<3<*/Dp>' =%(D' =%(D' =A@BB*BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-g6B fade*<3<*1D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*1D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*1D ' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-g6B fade*<3<*!D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-g6B fade*<3<*"D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-g6B fade*<3<*#D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-g6B fade*<3<*%D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-s6Bwipe(left)*<3<*.D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-g6B fade*<3<*2Dl' =%(D' =%(D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*D' =A@BBBB0B%(D' =,54*3>!Bstyle.color='`B@BPB<*!D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-g6B fade*<3<*+D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-g6B fade*<3<*3DP' =%(D' =%(D' =A@BB*BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*4++0+10 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+ 0 ++0+!0 ++0+!0 ++0+"0 ++0+#0 ++0+$0 ++0+%0 ++0++0 ++0+.0 ++0+/0 ++0+20 ++0+30 ++0+40 +0 ;{; 08:(  ] 1 C xL 3?f3ԔgggAD[X8c?"<?@@`NNN?N t ,$D0 =  2aOl = "i=x,$D0H  3 r?fԔL!?"<?@@`NNN?N= Z  G0  <T ?"6@ NNN?N  8 C.&0(2cX  <@ ?"6@ NNN?NJ @ 8 C.&0(2cX  < ?"6@ NNN?N8 C.&0(2cX  < ?"6@ NNN?N  8 C.&0(2cX  <$ ?"6@ NNN?NZP8 C.&0(2cX  < ?"6@ NNN?N8 C.&0(2cX   < ?"6@ NNN?N  8 C.&0(2cX   < ?"6@ NNN?Nt1n @-3" 2g   < ?"6@ NNN?Nt n @-2" 2g   < ?"6@ NNN?N] t n @-1" 2g   < ?"6@ NNN?N t n ?0" 2g  < ?"6@ NNN?N t n ?1" 2g  < ?"6@ NNN?NI t:n ?2" 2g  < ?"6@ NNN?NtZn ?3" 2g  < ?"6@ NNN?N_ @. . .  2c  < ?"6@ NNN?Ng @. . .  2c  <\ ?"6@ NNN?N8 C.&0(2cX  <$  ?"6@ NNN?N8 C.&0(2cX  <4 ?"6@ NNN?N~t8 C.&0(2cX  < ?"6@ NNN?N:08 C.&0(2cX  <| ?"6@ NNN?Ntn ?4" 2g  <  ?"6@ NNN?Ntn ?5" 2g  < ?"6@ NNN?N-tn @-5" 2g  <h! ?"6@ NNN?N|tn @-4" 2g  < % ?"6@ NNN?Ngb,$0 ^A  the algebra generated by A0 and the set P. 0 2gc$ago aeaX   B- fԔ?"6@ NNN?NT Q,$D0 tK(E) = F6 (2g 3c  # Z= vk @?"6@ NNN?N )V,$D 0G $ N1 ?"6@ NNN?N ,$ 0 WP  2e % <7 ?"6@ NNN?N. D Example 1  2 c$ & <x: ?"6@ NNN?N< ,$0 LEach E A can be uniquely decomposed:~' 2ag &P ' TZlB ?"6@ NNN?NR O ,$  0 Z" 2P ) TZG ?"6@ NNN?NR O ,$ 0 Z" 2 * <|K ?"6@ NNN?N7 W ,$ 0 $E = X F Yl 2 aG + N R ?"6@ NNN?Nw v ,$ 0 WP  2eY , NV ?"6@ NNN?Nw  ,$0 i P0 2aeP - TZ[ ?"6@ NNN?NO  ; L ,$  0 Z" 2b / N_ ?"6@ NNN?Nh - ,$ 0 rA0: 2go 0 <e ?"6@ NNN?N   U  2a0 3 BlGHn?"6@ NNN?N0~+ ,$D0 L1. K() = 2. K(E) K(E ) = K(E E ) 3. K(E) E 4. K(E) = K( K(E))tR 2gcg3cgcg3ccg3cg3ccgcg3ccgcg3cg3cg3cQ 4 BhGHo?"6@ NNN?N W ,$D0 $DK is the identity on A0 K (P) = # 2g3cgocg3c9 5 Bn ?"6@`NNN?N4 ; 3,$D0 U2 2c$P 6 <ԖԔ?"6@ NNN?Nm,$@0 rK 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.j0U2 2e3a e eeae3aeaeaea eae3e6D S B 7 BDԔ?"0@NNN?N ,$D 0B 8 BDԔ?"0@NNN?N9 ,$D 0H  0޽h ?/ 34 ̙33``___PPT10`./Ї+ߑxD\' = @B D\' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-o6Bdissolve*<3<*5D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-g6B fade*<3<*"D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bwipe(left)*<3<*#D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*$D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-g6B fade*<3<*&D' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-g6B fade*<3<*1D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*1D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*1D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*-D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*-D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-g6B fade*<3<*/DP' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*'D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*'D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(DP' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*)D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*)D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(DP' =%(D' =%(D' =A@BB*BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*4D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*4D' =-g6B fade*<3<*4D' =%(D' =%(DV' =A@BB5BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*3D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*3D' =-g6B fade*<3<*3D' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*3D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*3D' =-g6B fade*<3<*3D' =1:Bhidden*o3>+B#style.visibility<*3%(D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-g6B fade*<3<*6D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-g6B fade*<3<*6D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6<%(D' =-g6B fade*<3<*6<D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-s6Bwipe(down)*<3<*7D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*7D' =1:Bhidden*o3>+B#style.visibility<*7%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6<%(D' =-g6B fade*<3<*6<D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-o6Bwipe(up)*<3<*8D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*8D' =1:Bhidden*o3>+B#style.visibility<*8%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-g6B fade*<3<*6++0+10 ++0+0 ++0+ 0 ++0+$0 ++0+&0 ++0+'0 ++0+)0 ++0+*0 ++0++0 ++0+,0 ++0+-0 ++0+/0 ++0+30 ++0+30 ++0+50 ++0+60 +0 W>O>0 (N@=(  l :|'  4zgb ,$D0  BT ?"6@ NNN?Noe2  C.&0(2cX  B ?"6@ NNN?Nv|l C.&0(2cX  B< ?"6@ NNN?N:0 C.&0(2cX  B ?"6@ NNN?N/% 4  C.&0(2cX  B ?"6@ NNN?N1= '  C.&0(2cX   B ?"6@ NNN?N<26  C.&0(2cX   BX ?"6@ NNN?Nj6 `  C.&0(2cX  < ?"6@ NNN?N8 UFrom partition to knowledge 2c    ~0e0e    B}CDE4F @  o 8c8c     ?A)BCD|E||  '92T&DbT1f$}& @    # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abV r,$D0  < ?"6@ NNN?N  H,$0 ;s 2c    <@ ?"6@ NNN?Nn ,$0 @common 2c  "  X0e0e    BCDEF A@  o 8c8c     ?A)BCD|E||b {%>@  # "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab x ,$D 0ll  r  7 ,$D0 $ H Ԕ?"6@ NNN?N  . H Ԕ?"6@ NNN?N 0 H Ԕ?"6@ NNN?N  1 H Ԕ?"6@ NNN?N7D r `l 'Au  6SR ,$D 0 ' H Ԕ?"6@ NNN?Nk   ( H Ԕ?"6@ NNN?N'E  2 B Ԕ?"6@ NNN?NIA  3 B Ԕ?"6@ NNN?N#gu  8  j0e0e    B-C#DEF& A@ f Ԕ 8c8c     ?A)BCD|E||#-#-@`# "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab/ ,$@0 9 N fԔ?"6@ NNN?Ns,$D0f < < ?"6@ NNN?NQ ,$ 0  P1 -b 2d = <  ?"6@ NNN?N Q k @,$ 0  P2 -` 2P > < ?"6@ NNN?NQ ,$0 r Pc -L 2ff ? H Ԕ?"6@ NNN?N } ,$@ 0 A H Ԕ?"6@ NNN?Nn ,$D0 B < ?"6@ NNN?N ,$ 0 .coarser than P1 and P2  2 aa4 D <$ ?"6@ NNN?N q,$ 0 Vfinest than any such partition 2a^ G <) ?"6@ NNN?NS ^ 4 s ,$0 Kc -F 2efmfa H <. ?"6@ NNN?NJ j ,$0 narower than K1 and K2  2eoaa\ I B7 oԔ?"6@ NNN?N KC,$D0 x@For each E, Kc(E) K1(E), K2(E)! 2 aefmfaefaa I J <D ?"6@ NNN?N 8,$0 kbroader than any such knowledge0 2eoaB K BD)?"0@NNN?N  ,$D 0` M <HJ ?"6@ NNN?N] b ,$ 0  the common knowledge partition.F! 2y N <O ?"6@ NNN?N4 G. ,$0 #Agreeing to disagree, Aumann (1976)0$ 2cc$H  0޽h ? ̙33MM___PPT10L.@'+DI' h= @B DI' = @BA?%,( < +O%,( < +D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-u6Bwipe(right)*<3<*"D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(Dc' =4@BB/BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-g6B fade*<3<*7D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*7D' =1:Bhidden*o3>+B#style.visibility<*7%(D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-g6B fade*<3<*6D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-g6B fade*<3<*7D ' =%(D6 ' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-g6B fade*<3<*<D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-g6B fade*<3<*=D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =-g6B fade*<3<*?D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =-g6B fade*<3<*AD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =-g6B fade*<3<*>D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-g6B fade*<3<*9D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =-s6Bwipe(left)*<3<*BD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-s6Bwipe(left)*<3<*DD$' =%(D' =%(Dt' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*M%(D' =-s6Bwipe(left)*<3<*MD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =-g6B fade*<3<*ND\' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*K%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*KD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*KD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =-g6B fade*<3<*GDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =-g6B fade*<3<*ID' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =-g6B fade*<3<*J++0+0 ++0+ 0 ++0+<0 ++0+=0 ++0+>0 ++0+B0 ++0+D0 ++0+G0 ++0+H0 ++0+I0 ++0+J0 ++0+M0 ++0+N0 +q.0 _'W' *&(     <  ?"6@ NNN?N8 UFrom partition to knowledge 2c     ~0e0e    B}CDE4F @  o 8c8c     ?A)BCD|E||  '92T&DbT1f$}& @    # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abV r   < ?"6@ NNN?N  H ;s 2c    <̅ ?"6@ NNN?Nn  @common 2c z   ^0e0e    BCDEF @  o 8c8c     ?A)BCD|E||b {%>@  # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab x 2  < ?"6@ NNN?NQ   P1 -b 20  <H ?"6@ NNN?N Q k @  P2 -` 2  < ?"6@ NNN?NQ  r Pc -L 2ff  H Ԕ?"6@ NNN?N }   H Ԕ?"6@ NNN?Nn |   <T ?"6@ NNN?N  .coarser than P1 and P2  2 aa ! < ?"6@ NNN?N q Vfinest than any such partition 2a* " <į ?"6@ NNN?NS ^ 4 s  Kc -F 2efmfa # <, ?"6@ NNN?NJ j  narower than K1 and K2  2eoaa( $ B0 oԔ?"6@ NNN?N KC x@For each E, Kc(E) K1(E), K2(E)! 2 aefmfaefaa  % <H ?"6@ NNN?N 8 kbroader than any such knowledge0 2eoaB & BD)?"0@NNN?N  , ' <l ?"6@ NNN?N] b   the common knowledge partition.F! 2E ( < ?"6@ NNN?N4 G.  #Agreeing to disagree, Aumann (1976)0$ 2cc$ ) 3 r 3?3ԔgggAD[X8c?"<?@@`NNN?N,,$D0 VIs it possible to prove syntactically the existence of Kc for S5-knowledge K1 and K2 ?W 2a eaefmfefaefaa7 * < ?"6@ NNN?N $  [NO!" 2g(H  0޽h ? ̙33___PPT10.@'+R$D.' = @B D' = @BA?%,( < +O%,( < +D ' =%(D' =%(Dp' =A@BB/BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-g6B fade*<3<*)D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*)D' =+4 8?bCB#ppt_y-.1BCB#ppt_yB*Y3>B ppt_y<*)+8+0+)0 +l0    6 (  Y  BXg ?"6@`NNN?N8X 3Is there a set S of axioms of knowledge such that @4 2aef$a31  <g ?"6@ NNN?N$[ D  K satisfies SD 2e3 aef R  6?"6@ NNN?Nn <  Nhg ?"6@ NNN?N$! tD K is partitional0 2e3a ol 3   3 ,$D0  BDgG)VH1f?"6@ NNN?N3  For S = S5: For S = {K(E) = E}: 2aef aefae3 a"   6?"6@ NNN?NU / "   <?"6@ NNN?Ny 3 *    <D ?"6@ NNN?NH%    [NO!" 2g(H  0޽h ? ̙33V N ___PPT10. .ھ+(D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =%(D' =%(DC' =4@BB5BB%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<* D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<* D' =-g6B fade*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(+=0 p M(  X  C P<     S  PH .*   OAH  0re"f ? ̙3380___PPT10.Brh ;u~Gqw0Zm e@hR@-$$iXVj1Oh+'0 `h|    Samet Dov Dov Samet245Microsoft PowerPoint@Ѝ@t~n!@0"7mG g  r%  y---$x x ----$ xx ----$xx----$x&x&----$&&x+x+&----$++x0x0+----$00x5x50----$55x8x85----$88x=x=8----$==xAxA=----$AAxDxDA----$DDxIxID----$IIxLxLI----$LLxPxPL----$PPxSxSP----$SSxXxXS----$XXx[x[X----$[[x`x`[----$``xexe`----$eexjxje----$jjxoxoj----$ooxtxto----$ttx{x{t----${{xx{----$xx----$xx---'--$aa---- $aa----$^^--f-- $^^--'@Times New Roman-. 2 07Where do ."System 9-@Times New Roman-. 2 ?. informational .-@Times New Roman-. %2 >partitions come from  .-@Times New Roman-.  2 >|?.-՜.+,0    )On-screen ShowSamet>= d DavidArialTimes New RomanSymbolITC Zapf ChanceryArial Unicode MSDefault DesignSlide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9 Slide 10 Slide 11 Slide 12 Slide 13  Fonts UsedDesign Template Slide Titles !_j Dov SametDov Samet  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghiklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesTCurrent UserSummaryInformation(PowerPoint Document(jjDocumentSummaryInformation8