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RQU"l*G@biYfEXEK/.;/+ .X-+)2-+U%   a5558855ijl "-.-(MHHoMKK@c"!!EA  d" +2'&KKK&=&=Ka1K(Y&= js +# #!.E->.!@ !#%"!$U@HnM_hknopu+l" +1a"aqY>:ar;KE+$% qq    !Ca18qq9982a2:u | '!*m# !!mmR#GHaI,LMxxTUVZ]8abcdssi8>82?Es0,g? D2$Ʉo>k u5w_2$6nɘ؁V-j_R$e-Xhʍ$Q<22$W6`]߆U52$]P( q{Dm6$$$2$2$͗f`>.+gJi?2$cP=K?e'H@$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$2$2X1*Ft~A 0e0e    A A1 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||S" 33ff3@8H I \g4KdKd@:A 0bhpp<4!d!dl$ 0  <4BdBdl # 0Dʚ;p.ʚ;<4ddddlT# 0Xg4;d;d@:A 0Fp, pj0___PPT10 *___PPT9 z?H, 4Radcliffe Institute 2/6/06O =m%3Representation and Reasoning with Graphical Models44$oRina Dechter Information and Computer Science, UC-Irvine, and Radcliffe Institue of Advanced Study, Cambridge:pP b>4 hOutlineqIntroduction to reasoning in AI Graphical models Constraint networks Probabilistic networks Graph-based reasoning61+1+@7The Turing Test (Can Machine think? A. M. Turing, 1950),8$ (yRequires Natural language Knowledge representation Automated reasoning Machine learning (vision, robotics) for full test* q qPropositional Reasoning( If Alex goes, then Becky goes: If Chris goes, then Alex goes: Question: Is it possible that Chris goes to the party but Becky does not?\LPEP @B Sudoku  Constraint Satisfaction""(UbGraphsjInformation as Constraints7I have to finish my talk in 30 minutes 180 degrees in a triangle Memory in our computer is limited The four nucleotides that makes up a DNA only combine in a particular sequence Sentences in English must obey the rules of syntax Susan cannot be married to both John and Bill Alexander the Great died in 333 B.C.J8P'3.%Y Applications  Planning and scheduling Transportation scheduling, factory scheduling Configuration and design problems floorplans Circuit diagnosis Scene labeling Spreadsheets Temporal reasoning, Timetabling Natural language processing Puzzles: crosswords, sudoku, cryptarithmeticZ.Z"Z ZZ." s$Probabilistic Reasoning(Alex is-likely-to-go in bad weather Chris rarely-goes in bad weather Becky is indifferent but unpredictable Questions: Given bad weather, which group of individuals is most likely to show up at the party? What is the probability that Chris goes to the party but Becky does not? mP PPP. R0 Bayesian Networks: Representation (Pearl, 1988),1("  The  alarm network - 37 variables, 509 parameters (instead of 237),EA OSample DomainspWeb Pages and Link Analysis Battlespace Awareness Epidemiological Studies Citation Networks Communication Networks (Cell phone Fraud Detection) Intelligence Analysis (Terrorist Networks) Financial Transactions (Money Laundering) Computational Biology Object Recognition and Scene Analysis Natural Language Processing (e.g. Information Extraction and Semantic Parsing)qPqS; Graphical models in News: The New York Times, Dec 15, 2005 <$; Three Technology Companies Join to Finance Research in Graphical Models David Patterson, center, founding director of the Berkeley lab, talks with Prof. Michael Jordan of Berkeley, right, and Prof. Armando Fox of Stanford. <MPPPM9Complexity of Reasoning Tasks$Constraint satisfaction Counting solutions Combinatorial optimization Belief updating Most probable explanation Decision-theoretic planning TZZFG7Tree-solving is easyTransforming into a Tree By Inference (thinking) Transform into a single, equivalent tree of sub-problems By Conditioning (guessing) Transform into many tree-like sub-problems. j:-9-Inference and Treewidth Conditioning and Cycle cutset(Search over the CutsetSearch over the Cutset (cont)(Inference vs. Conditioning By Inference (thinking)*My WorkConstraint networks: Graph-based parameters and algorithms for constraint satisfaction, tree-width and cycle-cutset, summarized in  Constraint Processing , Morgan Kaufmann, 2003 Probabilistic networks: Transferring these ideas to Probabilistic network, helping unifying the principles. 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N ,$ 0 @2 "(2  NT 1?"h UVariables: empty slots Domains = {1,2,3,4,5,6,7,8,9} Constraints: 27 all-differentE  (  ZPE 1?"1 ,$ 0 \"Constraint Propagation Inference##3 & ZD 1?"p>  @$ H  0޽h ? 3333$ $___PPT10#.|Ёm+hvD"' = @B D"' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D3' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D3' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bbox(out)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(Dv' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+  ++0+  ++0+  ++0+  +M  CC@)jd b<(  d ~ d s *,  PU   j z jh  d  jh ,$D 0N  d  jn2 d  0( "`3 CC&0  2 d  0 "`# CE&0  2 d  0 "`M CA&0  2 d  0H "` CB&0  2  d  0 "`U3 CF&0  2  d  0 "`U CG&0  N  d S mzN  d B S 0N  d  S MUUN d  S o#N d  S 41<N d S opUN d  S U2 d  0 "`qm CD&0  N d  S 3nN d S 3N d S T5 d  Z 1?"iSh  ?A graph z  jh  d  j h ,$D 0 d  Zp 1?"fh  >A tree  N  j d   j2 d  0t} "`  0 CE&0  2 d  0L "`=H CG&0  2 d  0 "` jy CB&0  2 d  0T "`j CC&0  2 d  0$ "` g CJ&0  2 d  0@ "`n CK&0  N  d S 4N !d B S   N "d S $/f2 #d  0| "` CF&0  N $d  S  4N %d S NqN &d S +2 'd  0L "`; CD&0  N (d S $"/2 )d  0` "`  : CA&0  N *d  S  ! 2 +d  0 "`RJ CL&0  2 ,d  0T "`R= CM&0  N -d S 1R2 .d  0 "`rI CH&0  N /d S 0N 0d S RMz AD {  1d  D A{ ,$D 0N AD {  2d  AD { 2 3d  0 "`%   CD&0  2 4d  0h "`cJ   CO&0  2 5d  0d B S L |^ 2 ?d  0 "`M   CL&0  T @d  c $jJ Ma 2 Ad  0{ "` 4  CC&0  2 Bd  0 "` bK  CB&0  N Cd B S 2  N Dd  S  %w 2 Ed  0 "`A n  CA&0  N Fd  S U , 2 Gd  0"` {r  CM&0  2 Hd  0,"`e {   CN&0  N Id S bY  N Jd S  ^ 2 Kd  0"` c  CG&0  2 Ld  0"`   CK&0  N Md B S  # 2 Nd  0x"`D j  CE&0  N Od S Q  N Pd  S    Qd  Zp 1?" NA graph with one cycle Qz dj h  sd   P ,$D 02 td  0"`Cf CD&0  2 ud  0T "` j CC&0  2 vd  0X""`vj# CA&0  2 wd  0p""`jx CB&0  2 xd  08""`}f CE&0   yd  Z"1?"h  FDirected graph 2 zd  0""`h CH&0  2 {d  0 ""`d CF&0  2 |d  0#""`f CG&0  2 }d  0l(""`j  CJ&0   ~d  N?"  d B N?"* d  N?"_ d B N?"d9 d B N?" d B N?"M d  N?"*M d  N?"C d B N?"M d  N?"} d  N?"dM d  N?" d  N?"# d  N?"[j[H d 0޽h ?/@d d d d d d d d d d d d d d d  d d d d d d d d d  d d d  d d d  #d d d  d d !d  d d "d d #d $d d #d %d #d d &d d 'd (d )d d *d ,d +d -d +d .d /d d +d 0d ?d 4d 9d 5d 3d :d 5d 6d ;d 6d 4d d 3d ?d @d Bd Ad Cd Ad 3d Dd Ed Ad Fd  Hd Gd Id !6d Gd Jd "Ld ?d Md #5d Nd Od $5d Kd Pd 4vd td ~d 5wd td d 6wd xd d 7ud xd d 8vd {d d 9td {d d :td |d d ;td zd d <xd |d d =xd }d d >xd zd d ?ud }d d @vd wd d Azd }d d  3333J B ___PPT10" .Qp"+s OD ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1d %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*sd %(+} 8 LLPP HL( ( P z 9 @  9 @,$D 0  Z:" o? @ HA B red green red yellow green red green yellow yellow green yellow red&IEB  TDo?  B  TDo? @B  TDo? @B  TDo?@@2  T o?K 9 B  B ZDo?M{ P   0 py  Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (red, green, blue) Constraints: >$ q] ]   c $A (??y{ &]x (dj   BA6mapcolor S+z  * J   H,$D 0g*N  J   * JT H z #  J~  N \o?` [   0e0e    BCfDEF&3 3 o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||TVfb@`P f 8   0e0e    BCDE8FB  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||bf  BB\\tt @` zZ   *0e0e    BCDEDFL33 33 o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||8:xNxN**DFrp 4 #$@    0e0e    B8CjDEF  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||j8 @  T @   0e0e    BFCDE<FF3 3 o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||FF00XX\\F $@`H ~  N33o?> r    0e0e    BC>DE,F633 33 o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| BB>>``DB@`P<  z(   0e0e    BCpDE0F:  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||  LLphph @`HR    0e0e    BCfDE F*  o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||HfDbH@`  T EN       Z̕ o?N 6  9CN  /   /B  ZDo?[ K B  ZDo?$ no B   ZDo? * >  ! Z| o?   9A " Z o?   9B # Z4 o? | v  9D $ Z| o?  9E % ZL  o?I3 -  9F & Zh o? - ' 9G~2 ' No?n> l ~2 ( No?. Q Z  ~2 ) No?e- [ ~2 * No? : /~2 + No?\  B , ZDo?l - B - ZDo?[  B . ZDo?: y 'B / ZDo?V v n B 0B ZDo?V I v B 1 ZDo?I * S ~2 2 No? # 8 Q ~2 3 No? R  B 4 ZDo?a V | 5 T Ԕ? S 6 N \o?" < } N1?"n 7A$  6 ,iD<4___PPT9 _!Constraint Networks """ 2  Z1?" cp ,$@ 02  Z1?"g  ,$@ 02  Z1?"  ,$@ 02  Z1?",$@ 02  Z1?" {,$@ 02  Z1?"2  ,$@  02  Z1?" t7 ,$@  0B @ ZD1?"q Ic ,$@  0B  ZD1?"a ?! ,$@  0B  ZD1?"  ,$@  0B  ZD1?"P  ,$@ 0B @ ZD1?" GT,$@ 0B  ZD1?" Z,$@ 0B  ZD1?" ?M,$@ 0B  ZD1?"P  ,$@ 0B  ZD1?" O,$@ 0B @ ZD1?".,$@ 0    `#1?" \{ ,$ 0 7A    `'1?"L 3 ,$ 0 7B  T\+1?" j4 ,$ 0 7E    `/1?" ,$ 0 7G  T-1?"  ,$ 0 7D    `51?"  ,$ 0 7F    `t:1?",$ 0 7C   T>1?" ,$ 0 FConstraint graphH  0޽h ? ̙3300___PPT100+5gD.' = @B Da.' = @BA?%,( < +O%,( < +D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D'' =%(Dd'' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +,7 -K0 119:f:1(  R  0r ` | Example: map coloring Variables - countries (A,B,C,etc.) Values - colors (e.g., red, green, yellow) Constraints: D qe  $e"  c $A +??P x +d* | # #"."r`R`^|,$@ 0  TXo?"o,  Ured   T| o?", o [green    T o?"D,  Ured   T o?", D Vblue    To?"|,  Ured    To?"o ,  Ured    T Po?" o,  T&     T_o?"D ,  T&     Tho?" D,  T&    T,To?"| ,  T&    Tho?"oa   Wgreen   To?"a o  T&    THo?"Da   T&    T2o?"a D  T&    T o?"|a   T&    T)o?"oa  Vblue   TH6o?"oa  Wgreen    T`ʈo?"Da  ogre    To?"Da  oblu   Txo?"|a  Ured   To?"o Vblue   To?"o [green   To?"D Yred   To?"D [green    Tso?"| Yred   Tto?"o \    T耈o?"o `     To?"D `   ! To?"D `   " Tro?"| `   # TPo?"o ZE&   $ T:o?"o WD  % TBo?"D WC  & TLo?"D WB  ' TUo?"| WA B ( Ho ?"|B ) B1 ?"|B * B1 ?"|B + B1 ?"|B , B1 ?"|a a B - B1 ?"|  B . B1 ?"|, , B / Ho ?"|B 0 Ho ?"||B 1 B1 ?"B 2 B1 ?"DDB 3 B1 ?"B 4 B1 ?"ooB 5 Ho ?"v 6 N?40 f 7 No?#" `0 $  c 6| UD<4___PPT9 k+Constraint Satisfaction Tasks $," ( e T1?"` h,$D 09 f TD1?"i X  qAre the constraints consistent? Find a solution, find all solutions Count all solutions Find a good solutionrrH  0޽h ? ̙33 ___PPT10+BD' = @B D' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(+   06\(  x  c $` PU     c $ P <$ 0  H  0޽h ? 3333]U___PPT105..`\+[W_D' = @B D' = @BA?%,( < +O%,( < +D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*'A%(D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*Ac%(D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*c%(D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D)' =%(D' =ABBB@B0B%())))?D' =1:Bvisible*o3>+B#style.visibility<*8%(+8+0+ +  s <(  ~  s *3 PU   ~  s *  P   H  0޽h ? f3f___PPT10i.X03c+D=' = @B +?   j1b1/=t(P0(  (x ( c $Q PU    ( N,?"c Y!Party example: the weather effect""x ( c $< PcE   : ( Z[1?"Z ,$ 0 nNP(W,A,C,B) = P(B|W) P(C|W) P(A|W) P(W) P(A,C,B|W=bad) = 0.9 0.1 0.5O     $$(((( ( ZZ1?"   @$  l  \ ?(  \,$D 0T  \ 3(#  \ ( ZH^1?" \ I P(A|W=bad)=.9  2 (  fp1?" } 9W2 *(  `ho1?" 9A +( T1?"}aa@  Z >( Z (  `D1?"Z I P(C|W=bad)=.1  2 ,(  f1?" } 9W2 -(  f41?" 9C .( Z1?"}QQT  :\ 5(#  \ ( ZȂ1?"A\ I P(B|W=bad)=.5  2 /(  `@1?" :} 9W2 0(  `1?": 9B 1( T1?"} l   @( RZ,$D 02 (  fh1?"6Z &  9W "(  `(1?" z@  @P(W)  #(  `81?"n. BP(A|W)  $(  `D1?"Py BP(C|W)  %(  `E1?"  BP(B|W) 2 7(  fK1?"I~ 9B2 8(  f4O1?"[ 9C2 9(  f$E1?"" s 9A :(B Z1?" YI ;( Z1?" d[ <( Z1?" E r (% Y  s( #"&rrr% (Y ,$D 0 o( T4^1?" Y  N.9  @` m( Tg1?"   M1  @` k( T1?"(   Obad  @` B( Ta1?") Y  N.1  @` C( T1?")   M0  @` D( T8p1?"()   Obad  @` E( Ttx1?"} Y)  O.99  @` F( T81?"} )  M1  @` G( T %1?"(} )  Pgood  @` H( T-1?" Y}  O.01  @` I( T`61?" }  M0  @` J( T?1?"( }  Pgood  @` K( T1?"% Y  jP(A|W)2 '  @` L( Td1?"%   QA  @` M( T1?"(%   QW  @`B N( Ho ?"(% Y% B O( Ho ?"( Y B P( H1 ?"(} Y} B Q( H1 ?"() Y) B R( Ho ?"( Y B S( Ho ?"(% ( B T( H1 ?"%  B U( Ho ?"%  B V( Ho ?"Y% Y B l( H1 ?"( Y r W(@  `.JG>HLI 1?" B,$D  0H ( 0޽h ?o(*(+(,(-(.(/(0(1((7(:((8(;((9(<( 3333F > ___PPT10 ++hLD ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*?(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*@(%(D' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*W(%(D' =-s6Bwipe(down)*<3<*W(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*s(%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(+8+0+( +* v-K0 NF0!&\ (  \ x \ c $g PP    [ \ TA ?"H B ,$ 0 7P(S, C, B, X, D) = P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B)B8$%2 \ # lXF ??Pp = lung Cancer  2 \ # l`J ??  9Smoking2 \ # lM ?? P0  7X-ray2 \ # lH ??PP ] < Bronchitis  2 \ # l\P ?? @  :Dyspnoea  F     \      \ B Z o?"RP  \  Z o?"  [  \  Z o?"p@   \ B Z o?"@  \  Z o?"F  \ B Z o?" ] 0   \ # @g$ ,$D 0 \   `Z?"0   BP(D|C,B)   \   `^?" `4 @P(B|S) \   ` b?" = T >P(S) \   `Te?"   BP(X|C,S)   \   `h?"`4 @P(C|S) \  c $A "??@@e8 "$@ 0Nz P   \    ,$D 0nN P   \  P  &B \  Z E^FG!Q.RU*VW))?<<P  iA}C? }C`Ts}CR}C`TiA`TR @`0*!`T0*}C`T!ZZ$ `T GSBPubRRectCalloutP   \   `m?" <CPD:NT ` $   \ #  L !\   `$q ??` $  X C B P(D|C,B) 0 0 0.1 0.9 0 1 0.7 0.3 1 0 0.8 0.2 1 1 0.9 0.1 *XY @`N   "\   B #\  Z DjJ?"bcB $\  Z DjJ?" r %\ N ?"p ,$D 0 &\  `}?" (,$  0 DBelief Updating: P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ?zE  H \ 0޽h ?o \  \  \  \ \ \  333391___PPT10+PD' = @B D@' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\ %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\ %(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\ %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%\ %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\ %(D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&\ %(D' =-m6Bbox(in)*<3<*&\ +p+0+\  ++0+&\  +_  z-K0 nf@ WX (   x  c $\ PT  L   # S2   N?" :PCWP 2   N蝶?:K 8CO 2   NX?y  T :HRBP 2   N?p %  ;HREKG 2   N? ( ;HRSAT 2   Nܧ? g  ? ERRCAUTER   2   N?g x  8HR 2   N?  =HISTORY 2   N? ? l  >CATECHOL   2   Nd? >:  :SAO2 2   N4?P .  <EXPCO2 2   NP?0 `   <ARTCO2 2   N4Ŷ?- (  ;VENTALV2   N佶? z  JVENTLUNG& 2   N෶? ( >VENITUBE   2   Nu?i @ DISCONNECT   2   N?n ZD ? MINVOLSET   2   NԶ?  >VENTMACH   2   NDض?  j @ KINKEDTUBE   2   Nܶ? r @ INTUBATION   2   N?Pr M PULMEMBOLUS&  2   Nh? @ 9PAP 2   Nh?p GSHUNT&2   N0?Z   A ANAPHYLAXIS   2   N?`" <MINOVL 2   N?W   ;PVSAT 2   N?"  :FIO2 2   NH?  r ;PRESS 2 !  NX? @  B INSUFFANESTH   2 "  N?` @  ETPR&2 #  N ? l  ? LVFAILURE   2 $  N$ ? K  C ERRBLOWOUTPUT 2 %  N`?   A STROEVOLUME   2 &  Nt?& a  @ LVEDVOLUME    '  3 0e0e     ?B0CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'p?A)BCD|E||00H@ s (  3 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'p?A)BCD|E||`H@ _i )   0e0e    BCDEF V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||`@ ZE *  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||HH@ : +  # 0e0e     ?B`CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||0H`@  ,  # 0e0e     ?BCgDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||/Rg@   -  # 0e0e     ?BCLDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||)7L@ T tD  .  # 0e0e     ?BC|DEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||*b|@  p  /  # 0e0e     ?B CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||` @ + ]0  0  # 0e0e     ?B`CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||``0@ xt 1 B # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||H@ @ o T  2  # 0e0e     ?BCDEF   V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||<Px@\@  r` @ 3  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||`@ 0 @ 4  # 0e0e     ?BCDEF   V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||U"o(@  q k  5  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||H@   `  6  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||xH@  ` `  7  # 0e0e     ?B CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||  `@    8  # 0e0e     ?BjCDEF   V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||T Wj@  !  9  # 0e0e     ?BCDEF   V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||h pL@     :  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||<x@ p @  ;  # 0e0e     ?BWCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||'W@ t  R <  # "0e0e     ?BxCwDELF$   V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||f<fi0%do-<\K5i 0xw@       x~ =  # 0e0e     ?BPCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||PPH@ ` @  >  # 0e0e     ?B CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||,! @ 9   ?  # 0e0e     ?BlCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||Vrl@  , @  # 0e0e     ?B0CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||00H@  / A  # 0e0e     ?BPCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||HP@ ` .  B B # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||`@   C  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||H@ l   D  # 0e0e     ?B`CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||0H`@ _  E  # 0e0e     ?BmCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||h$m@  K   F  # 0e0e     ?B CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||` @ H )7 G  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||H@  . H  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||x`@ T : I  # 0e0e     ?B8C(DEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||81A(@ p  J  # 0e0e     ?B CDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||  `@ P[`" K  # 0e0e     ?B+CDE(F   V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| +n"R}W @   Nn y p L  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||H@  2 M  N,?| ql  A HYPOVOLEMIA   2 N  Nܚ?p" ECVP& O  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||H@ Zr  2 P  N?lN} 8BP  Q  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||)@    R  # 0e0e     ?BCDEF  V 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||k@    S   B`CDEFV[!z`@   , xB T  HDVl U   BC+DEFV+@ 5`xB V  HDV  xB W  HDV  X 6# U T"Monitoring Intensive-Care Patients## H  0޽h ? a(y___PPT10Y+D=' = @B +}  y p$(  r  S  PU   r  S 4 P   H  0޽h ? ̙33___PPT10i.kY8+D=' = @B +i  {-K0 h`p (  p x p c $`> Pw   x p c $I P   h p 3 @,jpeg&filename=berk583j*P p C (A&berk583s H p 0޽h ? 3333___PPT10i.+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*  %(++0+  ++(+  0+s+0+   +! -K0   #-- (   ~  s * PU     08?7 NBelief updating (sum-prod) 2  0 N BMPE (max-prod) 2  0Г~ zFCSP  consistency (projection-join)$ 2$  0L F C#CSP (sum-prod) 2 F  0    u 2   6d"` BP(X)"0# 2    6"`@(`  DP(Y|X)"0# 2    6`""` @ `  VP(Z|X)40# + T   B c $@T    c $ @2    6,&"` 0  DP(T|Y)"0# 2   6#"`@ `0  DP(R|Y)"0# T  B c $`6 2 T   c $6  2   6,"`N n 0  VP(L|Z)40# + 2   61"`  0  DP(M|Z)"0# T  B c $6  T   c $| 6 N     s *A< #??"`V  ux #d   s *A= #??"`T % x #df   s *A> #??"`C  x #df   s *A? #??"`x  x #df   s *A@ #??"`C k x #d   s *AA #??"`v k x #d$   6A3 #? ?"6?@`NNN?Nix #d   s *AB #??"`0bx #df   s *A8 #??"` Sk x #d   s *AC #??"` N x #df   s *A9 #??"` /k x #d    s *AD #??"` 1 x #dfB !  ZDjJ?"'B " B ZDjJ?" B # B ZDjJ?"  B $ B ZDjJ?"@  B %  ZDjJ?"  B &  ZDjJ?" l B ' ZDjJ?"l  B ( ZDjJ?" l d  B ) ZDjJ?" l  B * ZDjJ?"Lh  B + ZDjJ?" LB , ZDjJ?"L -  `@ ?? g-Trees are processed in linear time and memory..H  0޽h ?o                       ̙33___PPT10i.?+D=' = @B +  -K0 p.8$(  r  S M PU   r  S N P   H  0޽h ? 3333___PPT10i.%`)+D=' = @B +r  b?Z?.88;(     0e0e    BC,DEF&  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||Q,?8-Q@`"<  Q ,$D 0   0e0e    BCJDEF&  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||`Jl$#`J@`"<   ,$D  0   0e0e    BCADEF&f  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||,Abl@`"< x ,$D  0   0e0e    B&C8DEF&  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||0c8& 0@`"< T,$D 0   0e0e    B1CDE$F.  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| ]1b(!\3*@`"< ` <U ,$D 0r   0e0e    BCDEF  8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||ll @`"< E%,$D 0x  c $` PU   2   6|bS"`? F  CE&0  2   6eS"`?Y p   CK&0  2   64iS"`?4 CF&0  2   6lS"`? 7 <  CL&0  f   S C"? sp f @ S C"?q f  S C"?r f  S C"? P 2  6hqS"`? sZ  CH&0  f  S C"?  2  6S"`?>U  CC&0  2  6S"`?kk CB&0  f @ S C"?f  S C"?bRb2  6S"`? CA&0  f  S C"?M2  6 S"`?R  CM&0  2  6 S"`? CG&0  2  6TS"`? e  CJ&0  f  S C"?Z  2  6,S"`? b CD&0  f  S C"?3Mf  S C"?Mf   S C"?{ f ! S C"?_0 f "@ S C"? J f # S C"?kf $@ S C"?  f %@ S C"?30 f & S C"?mf '@ S C"?Rb (@ N3Ԕ?"Z  ,$D 0 2 ) <zS"`?; {,$D 0 EABC&0   2 * <0S"`?sc;,$D 0 FBDEF&0   + N1?"E,$D 0 2 , <0S"`?;%,$D 0 EDGF&0   -@ N1?"$,$D  0 2 . <$fS"`?: ,$D  0 EEFH&0   / N1?";,$D  0 2 0 <4S"`? >M ,$D 0 EFHK&0   1 N1?"  ,$D 0 2 2 <3S"`?  ,$D 0 DHJ&0   3@ N1?"0 ^  ,$D 0 2 4 < S"`? ( ,$D 0 EKLM&0   5 N1?"0  ,$D 0 6 Z1?"` -,$ 0 <treewidth = 4 - 1 = 3 treewidth = (maximum cluster size) - 1@=  '$  R 7  0e0e    BCDEF3 ) 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||W4X Hf@  "< ,[ p1 ,$D  0 8 Z|1?" (,$ 0 AInference algorithm: Time: exp(tree-width) Space: exp(tree-width)dB,H  0޽h ?@               ! " # $  %&'()*+,*-*./.01023045 3333 33___PPT102.+u%D0' = @B Dy0' = @BA?%,( < +O%,( < +D%' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-o6Bwipe(up)*<3<*7D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(++0+) ++0+* ++0+, ++0+. ++0+0 ++0+2 ++0+4 ++0+6 ++0+8 +  0.,(  ~  s *H` PU    F     97 k2  6bS"`? CC&0  2  6ԓS"`?G  CP&0  2  6S"`?5 CJ&0  2  6S"`? CA&0  2  6DS"`?A CL&0  n   S C"?cGn  B S C"?uun   S C"?DVn   S C"? n   S C"?(72  6S"`? 0 CB&0  n  S C"?v 2  6,S"`?m CE&0  2  6S"`?:= CD&0  n B S C"?$n  S C"?92  6{S"`?T CF&0  n  S C"?;2  6\zS"`?jX CM&0  2  6S"`?R? CO&0  n  S C"??Rn  S C"?jD2  6dS"`?  CH&0  2  6NS"`? CK&0  n  S C"?&72  6"S"`?0B CG&0  n  S C"?)Nn  S C"?4n   S C"?9n ! S C"?!$Tn " S C"?Tn # S C"?j2 $ 6h'S"`?j8  CN&0  n % S C"?Nn &B S C"?k%n 'B S C"?Nz  z%  ( j hk,$D 02 ) 6+S"`?<2 CC&0  2 * 6/S"`?f| CP&0  2 + 6`.S"`?Yj CJ&0  2 , 6 6S"`?x%  CL&0  n - S C"?Bn .B S C"?,n / S C"?&2 0 6hVS"`?B7 CB&0  n 1 S C"?2 2 6XS"`?s  t CE&0  2 3 6mS"`? CD&0  n 4B S C"?~.n 5 S C"?[2 6 6gS"`? N  CF&0  n 7 S C"?5  2 8 6\jS"`? CM&0  2 9 6sS"`?F CO&0  n : S C"?2 ; 6US"`?zP' CH&0  2 < 6FS"`?x,%  CK&0  n = S C"?2 > 6BS"`?7 CG&0  n ? S C"?rn @ S C"?Qrn A S C"?~<n B S C"?2 C 6ES"`?? CN&0  n D S C"?xn EB S C"?c-n FB S C"?Uz p   G M ,$D  0B H ZD)?"s   I Z1?"p @  9Az ' , % J  j h5,$D  02 K 6DS"`?    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B$ n _ S C"?/0 b n ` S C"?  x 2 a 6D$S"`?xD %  CN&0  n b S C"?\ *n cB S C"? y1 n dB S C"? \  z ,V  e  ,$D  0B f  `ZD)?",,V  g Z01?"A  9Bz M1 K  h  97 5,$D  02 i 64S"`?{ (  CP&0  2 j 68S"`?i   CJ&0  2 k 6d;S"`?/u CL&0  n lB S C"?n m S C"?\ H2 n 6? +;@!3)A"8CB#<+D$9*E%+)F&N[O'NLP(RQS)QKT*UQV+XWY,M\]-MZ^.RK_/Wa`0[Mb1XLc2MKd3kwl4kim5onp6qnr7tsu8jxy9jvz:s|{;wj}<ti~=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ 3333!z!___PPT10Z!.U+D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%())))?D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =-s6Bwipe(left)*<3<*GD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =%(Df' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =-o6Bwipe(up)*<3<*eD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*J%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(DW' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+ +W  -F%F.pp=(  ~  s * PU   z / H   y/ ,$D 0   `d??"`H  FA=yellow" 0   2  Z?? / @ B T??9 z ( 9PH   ( P ,$D 0   `33??"`vPH  EA=green"0     T??( 9ct    `t ??4 u,$ 0 V Inference may require too much memory Condition (guessing) on some of the variables>W1 )]z J pp    p,$D 02   <L"` B   ?C"0# 2   <x"`*   ?B"0# 2  <("`0 .  ?K"0# Z  s * u Z  s *u E 2  <,"` i ?G"0# Z  s *% 0 2  <,"` pp  ?L"0# 2  <B"`0R   ?D"0# Z  s * { Z  s *` E 2  <E"`4J   ?F"0# Z  s *E1  2  <I"` |  ?H"0# Z  s *e % 2  <xV"`J   ?M"0# 2  <Y"` p ?J"0# Z B s *1  2  <]"`^J   ?E"0# Z  s * { Z   s *e 0 Z !B s * {u Z " s *  Z # s *E  Z $ s *u ` Z %B s *1 ` Z & s *u 1  'B T??   ( T??  ]z  J (p )  (,$D 02 * <d"` B X  ?C"0# 2 + <h"` * R  ?B"0# 2 , <\k"`P0   ?K"0# Z - s *R Xu Z . s *Ru  2 / <,p"` Hi ?G"0# Z 0 s *% 0 2 1 <g"` (p  ?L"0# 2 2 <"`R ~  ?D"0# Z 3 s *X 3 Z 4 s *l`  2 5 <"` J   ?F"0# Z 6 s *1  2 7 <"` S|  ?H"0# Z 8 s * % 2 9 <"`J h  ?M"0# 2 : <""` 8p ?J"0# Z ;B s *1  2 < <!"`J   ?E"0# Z = s * 3 Z > s * 0 Z ?B s *R 3u Z @ s *X  Z A s *  Z B s *Ru l` Z CB s *l1 ` Z D s *Ru 1  EB T??= f  F T??~  z   G  ,$D 0rT < H#  2 I <,"`b. ?A"0# 2 J <`/"`lx ?C"0# 2 K <2"`f` ?B"0# 2 L <5"`df ?K"0# Z MB s *fxZ N s *yZ O s *f-lZ P s *fr2 Q <X:"`r\  ?G"0# Z R s *2 S <"`z< ?L"0# 2 T <"` ?D"0# Z U s *lG-Z V s *r2 W <"` ?F"0# Z X s *r2 Y < "`g ?H"0# Z Z s *12 [ <"`| ?M"0# 2 \ <@"`zL ?J"0# Z ]B s *2 ^ <X"`* ?E"0# Z _ s *jGZ ` s *1yZ a s *1Z b s *yZ c s *GxDZ d s *l-byZ eB s *fGZ f s *lj-Z g s *rZ h s *fZ iB s *xZ j s *xrZ k s *xZ lB s *Z m s *f nB T??QPz o T??5= p  `H ??P` PGraph Coloring problem0 H  0޽h ? IKMILNKJOKQPLSRJTUQWVQYX S[Z Y\] T^_ I[` L[aISbITcIJdTKeJ^f\QgWKhIWiQIjYIkYWlYKmLYnT[o     ! "#$%&' ( !) 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