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(Anonymous, circa 1970) fif df   {-Asynchronous Computability$  3Asynchrony Complicates Life$  eFLP (  Fischer Lynch and Paterson Showed that Asynchronous computability `" Turing computability Consensus Trivial in uniprocessor Impossible with 1 asynchronous failure Reasoned directly about executions( 5  ? " #   2  ?# &r   K f Graph Theory (  Biran, Moran, Zachs 1988 Single asynchronous failure Coordination problem is a graph Problem is asynchronously computable iff graph is connectedJ   " %  "  1  % 4    ~ gMore Graph Theory(  TFischer Lynch 82, Dolev Strong 83, Merritt 85, Dwork Moses 90, & Synchronous crash failures Computation state is a graph Consensus not computable while graph is connectedn   "   " Af -f  ff f@    w hLimitations of Graph Theory(  }Asynchronous model Multiple failures? Synchronous model Problems beyond consensus? Need a more general notion of connectivity  "   " +   +  ~ iThe Topological Approach(  Borowsky Gafni STOC93 Herlihy Shavit STOC93, JACM1999 Saks Zaharoglou STOC93, SICOMP2000 Showed k-set agreement impossible Generalization of consensus [Chaudhuri 90] Open problem for several years| J          5   jThe Topological Approach(  Computations as geometric objects Use topological methods to show existence of  bad executions. Borowsky & Gafni Sperner s Lemma Herlihy & Shavit Simplicial complexes & homology Saks & Zaharoglou Brouwer s fixed-point theoremL# ?        #f ?ff ff f f  f@s   J    QkRenaming (  Process has input name (flight #) Must generate output name (altitude) Interested in comparison-based protocols: Equality: A=B? Order: A3 Planes 4 Slots$  Impossibility by reduction Assume a protocol for 3 airplanes, 4 slots Choose 0 if your name is even 1 if your name is odd Result Not all odd Not all evenM /   M  Output Complex (3 processes)$  Protocol Complex (schematic)$    Boundary = 2-Process Executions! $ ! *Protocol Complex for One Process Execution++$ +  \2-Process execution might be mapped this way & /.$ / \2-Process execution might be mapped that way & /.$ /  In general &  (  rBut a Simplicial Cannot Map a Boundary Around a Hole& QED!:9$&   * #So 3 Planes Need At Least 5 Slots! $$$ $ _The Glorious Future$  TOur work Asynchronous, wait-free, one-shot, RW memory Open problems Long-lived computations Other kinds of memory (compare&swap) Randomization Other progress conditions &  - "  g "   - g&s   , den /*  ! " # $%&'()*+,-./0123456 7!8"9#:$;%<&='>(?)@*A+B,CPa ` ` ̙33` 333MMM` ff3333f` f` f` 3>?" dZ@x?  dZ@ +@" `  n?" dZ(@   @@``PR    @ ` ` p>  >   (     ``o>ֳ>ֳ1 ?pP  C Slide Title   -  Z0coxaxa1 ?p o ;Body Text Second Level Third Level Fourth Level Fifth Level    < L  <ho8c?" PH@___PPT9"@ z*68 Z2'B  s *޽h ? X=[0y___PPT10Y+D=' = @B + Default Design 0 G?`(  -  Zdxaxa1 ?  ;Body Text Second Level Third Level Fourth Level Fifth Level    < p  01 ?  B  s *޽h ? a(80___PPT10.> 0M  p 0(   B  s *޽h ? a(80___PPT10.> P6 -K0 E=00 (     fXNxaxa1?> Nir Shavit Tel-Aviv University and Sun Microsystems Labs (Joint work with Maurice Herlihy of Brown University) Herlihy and Shavit 2004 y Z*Z Z  D   N         2  `xaxa1?0 7  T   Z    Zx>ֳ>ֳ1 ? 7The Topological Structure of Asynchronous Computability88(7 H  0޽h ? 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M *F L  `  f  61?)f  6P1?LY   `Nxaxa1?a 7  &A multitude of concurrent executions"' Z'&    s BCDExF(1?0HTlx`H0 x`0`HTH<<$$<Tl =@@ w 8   s BC1DE<FD1?  $,0,(  @* |[ 8   s BC1DE<FD1?  $,0,(  @2 Tmc 8  s BC1DE<FD1?  $,0,(  @6 (Ag 8  s BC1DE<FD1?  $,0,(  @2 c   s pB-CYDEF1?{{0,,, (((0$8$@$H P X `$h$p(x(08DLT\dpx (004<@DLPXXPH$D(<,4$0,,$$ xld\TH@80( xph`XPH@8 0(  (000@)$ d2  <1? ,    fdUxaxa1? ?  d Asynchrony"  Z (    s BCaDEF1?iV`i:i @Y   s BCDExF(1?<000$H Tl0H`xx``T<$l T <<<=@@ aw 8  s BC1DE<FD1?  $,0,(  @* [ 8  s BC1DE<FD1?  $,0,(  @2 c 8  s BC1DE<FD1?  $,0,(  @6 g 8  s BC1DE<FD1?  $,0,(  @2 1c 2  <\]?jP J  Z   Hd`8c?@ c Synchrony"  Z   d  <G/*1?@M   Tdxaxa_1?@ - lA single execution" Z &  H\c8c? do,$ 0 z"Seek help from modern mathematics! # Z# " H  0޽h ? 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H / ~ 36,000 f t  Z   q @ c BCADEF1?@@ @ PN 1  @ 1 4JN   @  xN  m @  m @ c BCDEF?DD)`'b#efsgfFbAc<c8d3e/f*h'i#k mprv{| ~  #*4;Kx(2;CN}z~oO1@ m~N  h @  hJ @ c B CDEDFL1?         #$@ h J @ c B CDEDFL1?         #$@ k : @ c BCDE<FD1?         @pk~: @ c B CDE<FD1?       @lN  q{ @  q{XN w{ @ w{f2 @ 61?w{f2 @ 61?w{f2 @ 61?w{f2 @ 61?w{f2  @ 61?w{f2 !@ 61?w{f2 "@ 61?x{ N  q w #@  q wXN  qL u $@  qL uf2 %@ 61? q tf2 &@ 61?$ q% tf2 '@ 61?+ q, tf2 (@ 61?3 r4 uf2 )@ 61?: r< uf2 *@ 61?B rD uf2 +@ 61?K rL uXN S r w ,@ S r wf2 -@ 61?S rU uf2 .@ 61?Z r\ uf2 /@ 61?b rc wf2 0@ 61?j tk wf2 1@ 61?r ts wf2 2@ 61?z t{ wf2 3@ 61? t wN  t w 4@  t wf2 5@ 61? t wf2 6@ 61? t wf2 7@ 61? t wf2 8@ 61? t wf2 9@ 61? t wf2 :@ 61? t wN  tw ;@  twf2 <@ 61? t wf2 =@ 61? t wf2 >@ 61? t wf2 ?@ 61? t wf2 @@ 61? t wf2 A@ 61?twN x{ B@ x{f2 C@ 61?x{f2 D@ 61?x{f2 E@ 61?x{f2 F@ 61?x{f2 G@ 61?x{f2 H@ 61?x{ I@ c jBC\DEF1?;;Y6T<R=Q>O?L@JAGBECAD=E8E;B0A"@?A B DEGILMOQSUWY Z[XVSRPNLK I#H'G+F4G;GAG8K=KBKGKJJMJQIUIXGwx@ ;Y\N  r x J@  r x K@ c BCDEF1? @ r v L@ c BCDEF1? @ r v M@ c BCDE,F41? @ r x2 N@ BCE$GI`TQ? `T`T`T`T`T`T`T`T`T`Ti~j~R O@ c "BICDEHFP___1?H4*    "*0 ; H%(@ )N  ~ P@  ~N  ~ Q@  ~" R@ c BC*DEF1?,, !#$%&'())u)^)+)'&$ "  #'*08CQY\@ ~ S@ c bBC#DEF???1?:: !!"q"]!,!!     ^ e o x}ux@ LN   T@  <N   U@  dN   V@   W@ c B CDE(F0???1?      @  X@ c B CDEF 1?    @  Y@ c B CDEF ___1?   @ r Z@ c BB2CDEXF`1?11 0 00+&"  )1-0@   [@ c zB2CDEtF|???1?    % 1 0,(%!   ;<@   N   \@  nN   ]@    ^@ c B CDE$F,1?     @  _@ c B C DE(F01?     @  `@ c B CDEF 1?     @  a@ c B CDEF___1?   @ 6N    b@    c@ c B?C)DEF1?''<950+"  "$%&& '(((%()'-&1&5%9#="==>>= <OP@   d@ c B=C DEF___1?%%<950*%    &(,048<KL@  N    e@    f@ c BC(DEF1?'@   g@ c BC(DEF1?'@   h@ c BC+DEF1?*@  < N   i@  " j@ c BCpDE0F81? ` _bcd eg/iFk[o@mm N  D k@  D l@ c rB<CDEpFx___1?   %14 7 8:;4$   9<@ HN  D m@  D2 n@ c BjCDE8F@1?i_U0  ,5i @cB o@ c BC@DE@FH1?yR7".9;<>*?7>J<V<Y5!$@ DN +hoy p@ +hoyN +t`y q@ +t`y* r@ c B5CDE4F<1? 4($"(.4@+t`y s@ c B(CDE,F4???1? '#  #'@+tSyN Fhol t@ Fhol: u@ c B)CDE<FD1? $(# @Fhol v@ c BCDE,F4???1?  @Fhbl w@  `^xaxa1?  YDL227 Z N 1  , x@ 1  ,NN 1  , y@ 1  ,2 z@ ENGZHI`TJQ1? `T`T`T`T`T`T`T`T`T`T  2 {@ ENGZHI`TJQ1? `T`T`T`T`T`T`T`T`T`T]  2 |@ ENGZHI`TJQ1? `T`T`T`T`T`T`T`T`T`T1  ,2 }@ ENGZHI`TJQ1? `T`T`T`T`T`T`T`T`T`T  f ~@ 61?  JN Q i s  @ Q i s CN  i f  @  i f  @ c BCDEF?DD)~'#sFA<83/*'#   #*4;Kx(2;CN~h@@ i E J @ c B C%DEDFL1?   $ $     #$@& 2  J @ c B C"DEDFL1?  ! !      #$@  : @ c BC!DE<FD1?        @  : @ c B C DE<FD1?       @   f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  XN @  p   @ @  p  f2 @ 61?@  A  f2 @ 61?H  I  f2 @ 61?O  P  f2 @ 61?W  X  f2 @ 61?^  `  f2 @ 61?f  h  f2 @ 61?o  p  XN w   @ w  f2 @ 61?w  y  f2 @ 61?~   f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  N    @   f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?& ' f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?  f2 @ 61?   @ c jBCyDEF1?;;YGTNRPQROTLUJVGXEYAZ=[8[;X0V"UTV W Z[^adfhknpsu vxtqnljfec a#_']+]4];^A^8c=cBcGcJbMbQaU`X^wx@ }3  @ c BCDEF1? @    @ c BCDEF1? @     @ c BCDE,F41? @   2 @ BCE$GI`TQ? `T`T`T`T`T`T`T`T`T`T  R @ c "BICDEHFP___1?H4*   "*0; H%(@ M* " @ c BC7DEF1?,,  #%),./023455u6^5+5320 ,'"   #'*08CQY\@  I  @ c bBC.DEF???1?::  !#$%'()+,,q-],,,+*)& $!   ^eo x }ux@  G dN H %^  @ H %^  @ c B CDE(F0???1?     @H %^  @ c B CDEF 1?    @Q %U  @ c B CDEF ___1?   @Y %^ r @ c BB2CDEXF`1?11 000+&"  )1-0@ I f  @ c zB2CDEtF|???1? %10,(%!  ;<@ M f   @ c B CDE$F,1?      @ & ,E  @ c B CDE(F01?     @!. ,?  @ c B CDEF 1?     @!2 ,5  @ c B CDEF___1?   @ > +E  @ c B?C5DEF1?''<950+"  *,/011 3343%3)3-111509.=,=)=$>>=<OP@  N  @ c B=C)DEF___1?%%<!9!5"0#*#%$$$$# # !   "$$&&'('&'(',&0%4$8"<!KL@ % N N   R  @   R  @ c BC4DEF1?3@  N  @ c BC4DEF1?3@  N  @ c BC8DEF1?7@ R " @ c BCDE0F81? ` ~ /F[@i   @ c rB<C$DEpFx___1?  !# !%147 8 :;4$   9<@0 l9 2 @ c BjC'DE8F@1?i_U0 "$%& &&,%5$i @9  B @ c BCTDE@FH1?yR7-=KMOQ*S7RJOVOYF!$@  N O   @ O  N O   @ O  * @ c B5CDE4F<1? 4($"(.4@O   @ c B(CDE,F4???1? '#  #'@O w N j   @ j  : @ c B)CDE<FD1? $(# @j   @ c BCDE,F4???1?  @j   @  `4|xaxa1?3 $s  YAA082 Z N Q ,   @ Q ,  NN Q ,  @ Q , 2 @ ENGZHI`TJQ1? `T`T`T`T`T`T`T`T`T`T 0 p 2 @ ENGZHI`TJQ1? `T`T`T`T`T`T`T`T`T`T} , 2 @ ENGZHI`TJQ1? `T`T`T`T`T`T`T`T`T`TQ , 2 @ ENGZHI`TJQ1? `T`T`T`T`T`T`T`T`T`T 0 X f @ 61? 4  < PN Yz }  @ Yz } ^JN Yz ]  @ Yz ] xN lz >  @ lz >  @ c BCDEF?DD!zp}svx2xx`sdshtmtquvw{y~{~{qjZ-|rhaV#&]9@lz > ~N    @   J @ c BC DEDFL1?       #$@  J @ c B C!DEDFL1?       #$@C O : @ c B CDE<FD1?       @\ h : @ c B CDE<FD1?         @  N    @   XN $ Q  @ $ Q f2 @B 61?P Q f2 @B 61?I J f2 @B 61?B C f2 @ 61?: ; f2 @ 61?3 4 f2 @B 61?+ , f2 @ 61?$ %  N     @    XN    @   f2 @B 61?  f2 @B 61?  f2 @B 61?  f2 @B 61?  f2 @B 61?  f2 @B 61?  f2 @B 61?  XN U   @ U  f2 @B 61?  f2 @B 61?}  f2 @B 61?v w f2 @B 61?m o f2 @B 61?e g f2 @B 61?] ^ f2 @B 61?U W N   9  @   9 f2 @ 61?8 9 f2 AB 61?0 1 f2 AB 61?( ) f2 AB 61?  ! f2 AB 61?  f2 AB 61?   N    A   f2 AB 61?  f2 AB 61?  f2 AB 61?  f2  AB 61?  f2  AB 61?  f2  AB 61?  N     A   f2  AB 61?  f2 AB 61?  f2 AB 61?  f2 AB 61?  f2 AB 61?  f2 AB 61?   A c jBCnDEF1?;;7A=G>I@KBLDNFOIPLQORTSYSWPbOpNLNORSVX[]_bdfikkmjgd~b}`z^x\uZrXoVkUfU]UWVQVX[T[NZIZFZCY?Y;W9V wx@ . \N    A    A c BCDEF1? @   A c BCDEF1? @   A c BCDE,F41? @  2 A C}IENGHI`TJ.Q? n.T}I,n.T}I,`T.T}I,`T.o p R A c "BJCDEHFP___1?(.5:?BEHI F( %(@ % N Y 8]  A Y 8] N Y 8A  A Y 8A " A c BC0DEF1?,,7. ' !$&( ) *+-%-=.N.h/~..-,*'" 7Y\@Z 8A  A c bBC*DEF???1?::79:: 9753,%  !"$ %&'&(L(k)(('&%#! ~xm d _ZYLB7ux@Y 7A LN C ]  A C ] <N C ]  A C ] dN C V   A C V  !A c B CDE(F0???1?  @C V  "A c B CDEF 1? @J M  #A c B CDEF ___1? @R V r $A c BB4CDEXF`1?  #*.1233 33" -0@C ]  %A c zB3CDEtF|???1?-,+) ( & #    "%)-/1222-;<@G ]  N  J  &A  J nN $ =  'A $ =   (A c B CDE$F,1?   @$ =  )A c B CDE(F01?    @) 8  *A c B CDEF 1?  @- 1  +A c B CDEF___1?   @7 = 6N  J  ,A  J  -A c B>C.DEF1?'' &158:<<<= <<%<';):*8+5+0,+,$--,,+ * )(&$ OP@ F  .A c B>C%DEF___1?%%  !!%!* - 024679:; < <=<<; :!8"5"($!$$##" ! KL@! F N  J  /A  J  0A c BC0DEF1?/@  H  1A c BC-DEF1?,@ G  2A c BC2DEF1?1@ J < N Yz 2  3A Yz 2 " 4A c BCDE0F81? wpzt|uwsxdyP{9$@lz   N Y 2  5A Y 2  6A c rB<CDEpFx___1?322 24679;1&    &+,.0139<@ 2 N Y   7A Y  2 8A c BjC%DE8F@1? 9ih f"d#a$]$L$=#4" @Y  B 9A c BCMDE@FH1?@[)u8EGIxJhL[KII<H9@!$@S  N j   :A j  N x   ;A x  * A j  : ?A c B(CDE<FD1? !'   @j   @A c BCDE,F4???1?   @w   AA  `Pxaxa1?_ B}  YAL991 Z N $   BA $  6N ($   CA ($  2 DA ENGHZJQ1? `T`T`T`T`T`T`T`T`T`T@( `h 2 EA ENGHZJQ1? `T`T`T`T`T`T`T`T`T`TL$  2 FA ENGHZJQ1? `T`T`T`T`T`T`T`T`T`Th$  2 GA ENGHZJQ1? `T`T`T`T`T`T`T`T`T`T(( 8P f HA 61?, 4  IA  `xaxa1? d_   oPick your own altitude.( Z  V LA ZȨxaxa8c? m,$ 0 8How many slots do we need to allow safe coordination? 08 ZZ9( 9 H @ 0޽h ?N@z@{@|@}@@@@ @ @ A DA EAFAGA X=[0___PPT10+Wg|D' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*LA%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*LAD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*LA+8+0+LA + 1  0T(  T~ T s *pP    T  BԾ "`p<$D 0   H T 0޽h ? ̙335-___PPT10 +)D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*T#I%(D' =-6B'blinds(horizontal)*<3<*T#ID ' =%(D ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*TIs%(D' =-6B'blinds(horizontal)*<3<*TIsDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*Ts%(D' =-6B'blinds(horizontal)*<3<*TsDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-6B'blinds(horizontal)*<3<*TDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-6B'blinds(horizontal)*<3<*T+ 2 P\P(  \~ \ s * I1    \ s *zd    H \ 0޽h ? ̙33y___PPT10Y+D=' = @B +  !` `(  `~ ` s *pP   dB ` <DԔ ` # B C DE FԔ @dB ` <DԔ C  ` Bl3 ? e  )Point in high-dimensional Euclidean Space(*(*2*g  * ^2  ` 61?R H ` 0޽h ? ̙33y___PPT10Y+D=' = @B +3 !%,t(  , ,  `>ֳ>ֳ1 ?`   Simplexes"  Z $   , s BDCrDEF`8c?&qC&q @ d = ^2 , 6(1?  Z a ^2 , 61? z  ^2 , 61?J ^  , 68c?P N F .g  , )  f2  , 61? I%f2  , 61?c ,  `X vB;vB;1?g f Z      ,  `<vB;vB;1?.g f Z     TB , c $8c?J_^2 , 61?@bhF    , / ZN  )  , F w , S BCNDEF`8c? =N  @`S"  )  , S BCNDEF`8c?N @`S"  )  , c B0C>DEF8c?0> @` ) f2 , 61?  f2 , 61?VIf2 , 61? DKf2 , 61? u ^B , 6D)?@  , N6?8c?"N c Dimension"  Z    , Hh!8c?   [0" Z  , H`&8c? u  [1" Z   , H)8c? [2" Z  !, H-8c?/@ 1 [ [3" Z  ", H%8c?C  hsolid tetrahedron Z  #, H458c?s   esolid triangle Z  $, HD28c?9 v  Zline Z  %, H =8c? [point Z H , 0޽h ? X=[0y___PPT10Y+D=' = @B +24 YQ0!0@(  @ .@ s BDCrDEF`8c?&qC&q @Z ~ !  @ s BDCrDEF`8c?&qC&q @-q !  @  `XL>ֳ>ֳ1 ?w7  Complexes"  Z $   @ s BDCrDEF`8c?&qC&q @:~ = ^2 @ 61? ta ^2 @ 61?7  ^2 @ 61? d^ @ 68c?1 X ^2 @ 61? @ W $ F 2 @ /aH @  BJCDEFd @1?I @q( @  BCDEFfd @1? @j" @  B.CDEFd @8c?-E @m" @  B.CnDEF @1?-mm @'rNB @ S 1?BP @  B.CDEF|A1?- @'TB @ c $1?5:TB  @B c $1?P5f2 !@ 61?f2 "@ 61?>f2 #@ 61?\f2 $@ 61?f2 %@ 61?Xf2 &@ 61? 2^2 '@ 61?6]^2 (@ 61?Tz^2 )@ 61?)3^2 *@ 61? 1e^2 +@ 61?R@^2 -@ 61?@'^2 /@ 61?7   0@ H0[8c? m v Glue simplexes to form complexes! Z!  H @ 0޽h ? X=[0y___PPT10Y+D=' = @B +  %h(  h~ h s * ]d   , $h <h ?",$ 0 "OK to think of it as oriented path:#*"gc # ) (h <Pm ?R-X,$ 0 Also works in higher dimensions: *gc   8 w  hw  +h s BCDEF`1? @y  ,h s BCyDEF`1?xx @# ZB -h s *1?% \ ZB .h s *1?Q  ZB /h s *1? o ZB 0h s *1? 3 ZB 1h s *1?  ZB 2h s *1?v s ZB 3h s *1?o  ZB 4hB s *1? % s ZB 5hB s *1?  ZB 6h s *1?  ZB 7hB s *1? o ZB 8hB s *1? ZB 9h s *1? x l2 :h <1?H  l2 ;h <(1?S  l2 h <1? _  l2 ?h <(1?L H l2 @h <(1?  l2 Ah <(1?Z  3l2 Bh <1? w l2 Ch <1?t  l2 Dh <1?W  l2 Eh <1?]  r Th B8c?n   h C BDCDE$F.A3)? 4nDQ{AA$@`S" 3 ^B h 6D3)? P s ^B h 6D3)?# ~ _^B h@ 6D3)? Z H h 0޽h ? ̙33___PPT10r+,?1D' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$h%(D' =-6B'blinds(horizontal)*<3<*$hD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*(h%(D' =-6B'blinds(horizontal)*<3<*(h+p+0+$h ++0+(h +5  C;P %%(  ~  s *d     <t ?/ ,$ 0 nA cycle is a boundry if it goes around a  solid regionf8* gg#gc&   $ +  <ؑ ?Cnf,$ 0 Encompases no holes:*gc    v F w   w   s BCDEF`1? @y   s BCyDEF`1?xx @# TB  c $1?% \ TB   c $1?Q  TB   c $1? o TB   c $1? 3 TB   c $1?  TB   c $1?v s TB  c $1?o  TB B c $1? % s TB B c $1?  TB  c $1?  TB B c $1? o TB B c $1? TB  c $1? x f2  61?H  f2  6(1?S  f2  61? d Zf2  61? S ) f2  61? _  f2  6(1?L H f2  6(1?  f2  6(1?Z  3f2  61? w f2  61?t  f2  61?W  f2   61?]  l ! <8c?n  l " c BDCDE$F.z3)? 4nDQ{AA$@`"`  ` 3 ^B # 6D3)? P s ^B $ 6D3)?# ~ _^B %@ 6D3)? Z H  0޽h ? ̙33___PPT10r+fOD' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+p+0+ ++0+ +r  ldp $&(  ~  s *d     < ?Cnf,$ 0 |Goes around a hole:*gc    s BCDEF`1? @mP   s BCyDEF1?xx @S"  # RB  s *1?% \ RB   s *1?Q RB   s *1? o RB   s *1?3  RB   s *1?  RB   s *1? v s RB  s *1?o  RB @ s *1? % s RB @ s *1?  RB  s *1?  RB @ s *1?  o RB @ s *1? RB  s *1? x d2  <1?9 c d2  <(1?9 & d2  <1? d Zd2  <1?9 9 d2  <1? _  d2  <(1?9 3 d2  <(1?  d2  <(1?Z 3d2  <1?w d2  <1?c  d2  <1?c  d2   <1? ]  j ! B8c?n l " c BDCDE$F.u3)? 4nDQ{AA$@`"`  ` 3 ^B # 6D3)? P s ^B $ 6D3)?# ^B %@ 6D3)? J ^B &@ 6DԔ? c H  0޽h ? ̙33___PPT10+D2' = @B D' = @BA?%,( < +O%,( < +D$' =%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+8+0+ + 5 P(  ~  s *,7     s *p   H  0޽h ? ̙33___PPT10i.6f+D=' = @B +S  zr!A (  ~  s *|pP    # B ||?"6@ NNN?N,  0Each (n-1)-simplex colored by n distinct colors1*ggg gcg 1 $ $ B ||?"6@ NNN?N  _ tCorresponding to process ids(*g  v F w  %   & s BCDEF`1? @y  ' s BCyDEF`1?xx @# TB ( c $1?% \ TB ) c $1?Q  TB * c $1? o TB + c $1? 3 TB , c $1?  TB - c $1?v s TB . c $1?o  TB /B c $1? % s TB 0B c $1?  TB 1 c $1?  TB 2B c $1? o TB 3B c $1? TB 4 c $1? x f2 5 61?H  f2 6 6(1?S  f2 7 61? d Zf2 8 61? S ) f2 9 61? _  f2 : 6(1?L H f2 ; 6(1?  f2 < 6(1?Z  3f2 = 61? w f2 > 61?t  f2 ? 61?W  f2 @ 61?]  l A <8c?n  H  0޽h ? ̙33y___PPT10Y+D=' = @B +^ 6 kc ))(  B  NZD3|?"0@NNN?N ,$D 0B  NZD3|?"0@NNN?N L ,$D 0B  NZD|?"0@NNN?Nh  ,$D 0B  TD|?"0@NNN?N E ,$D 0B  TD|?"0@NNN?N k ,$@ 0B  ZZD|?"0@NNN?Nzz ,$D  0B  ZZD|?"0@NNN?Naa ,$@  0B   TD|?"0@NNN?N ,$D  0B   TD|?"0@NNN?NW,$D  0~   s *d   XB   0D8c?tt vL t-   # }F `B  0D8c?"S `B  0D8c?*[ N t-}  t-}`B  0D8c?00f2  6#8c?t}f2  68c?-}N t-}  t- `B  0D8c?00f2  68c?t}t2  6#8c?"`-}L t-} # ` < `B  0D8c?00f2  6#8c?t}f2  68c?-}L t-} # V `  `B  0D8c?00f2  68c?t}f2  6#8c?-}T    B C'DEFԔ?7n '@  S" v` ,$D 0T !  B C'DEFԔ?7n '@  S"  ` ,$D 0T "  B CDEFԔ?k5 k @  S" Pp,$D 0T #  B CDEFԔ?k5 k @  S"  ,$D 0T $  Bw CDEFԔ?&d Dw @  S" %  ,$D 0T %  Bw CDEFԔ?&d Dw @  S" F V ,$D 0^2 & 68c? >[ ^2 ' 6#8c? = [ ( B  ||?"6@ NNN?NB $,$ 0 w%Color-preserving vertex-to-vertex map"&*&a & ^ ) B ||?"6@ NNN?N/ j,$  0 z(That also carries simplexes to simplexes")*)a ) H  0޽h ? ̙33@@___PPT10@.6@ ]+hN{D+@' = @B D?' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B'blinds(horizontal)*<3<*(D:' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-6B'blinds(horizontal)*<3<*"D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-6B'blinds(horizontal)*<3<*#D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-6B'blinds(horizontal)*<3<*!D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-6B'blinds(horizontal)*<3<*%D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DR' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-6B'blinds(horizontal)*<3<*)D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DF' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DF' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DC' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DC' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DC' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+p+0+( ++0+) +hA $ 33P G\ }3(  \~ \ s *07t   nF  f\   R g\ Z03`jJ?+ L" Zc R h\ NjJ?C"?! x B i\B TDjJ?  B j\B TDjJ?  - FB k\ TDjJ?/` +B l\ TDjJ? i B m\ TDjJ?  B n\ TDjJ? +2 o\ T8jJ? 2 1 P&*g   p\ N7 jJ?C"?W 5 y R(*g  q\ N> jJ?C"?i C G  R(*g  r\ NLB jJ?C"? l R(*g  s\ BF jJ?C"? R(*g  t\ NI jJ?C"?   R(*g  u\ NH jJ?C"?+   R(*g 2 v\ TPjJ?S l  P&*g  2 w\ T+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<*\%(+8+0+\ +  .&0  (  ~  s *pP   dB  <DԔ  # B C DE FԔ @dB  <DԔ C   H ?pp  hProcess id (color)&(*2G     H ?9    kValue (input or output)$*G   2   B8c?   J  Z    HX8c? -/  W2 Z  H  0޽h ? ̙33y___PPT10Y+D=' = @B +3  ZR ;,   N(  ~  s *pP    ڬ@ T||?"6@ NNN?N`b @J8 v \v~T lz >  ܬC# v ݬ c BCDEF?DD!zp}svx2xx`sdshtmtquvw{y~{~{qjZ-|rhaV#&]9@lz > ~N    ެ   J ߬ c BC DEDFL1?       #$@  J  c B C!DEDFL1?       #$@C O :  c B CDE<FD1?       @\ h :  c B CDE<FD1?         @  N       XN $ Q   $ Q f2 B 61?P Q f2 B 61?I J f2 B 61?B C f2  61?: ; f2  61?3 4 f2 B 61?+ , f2  61?$ %  N         XN       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  XN U    U  f2 B 61?  f2 B 61?}  f2 B 61?v w f2 B 61?m o f2 B 61?e g f2 B 61?] ^ f2 B 61?U W N   9     9 f2  61?8 9 f2 B 61?0 1 f2 B 61?( ) f2 B 61?  ! f2 B 61?  f2 B 61?   N       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2  B 61?  f2  B 61?  N        f2  B 61?  f2  B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?   B s pBCnDEF1?;;7A=G>I@KBLDNFOIPLQORTSYSWPbOpNLNORSVX[]_bdfikkmjgd~b}`z^x\uZrXoVkUfU]UWVQVX[T[NZIZFZCY?Y;W9V wx@bT    C# NY  c BCDEF1? @    c BCDEF1? @    c BCDE,F41? @  2 B  C}IENGHI`TJ.Q? n.T}I,n.T}I,`T.T}I,`T.hiX B s (BJCDEHFP___1?(.5:?BEHI F( %(@iYT Y 8]  C# ]fN Y 8A   Y 8A "  c BC0DEF1?,,7. ' !$&( ) *+-%-=.N.h/~..-,*'" 7Y\@Z 8A   c bBC*DEF???1?::79:: 9753,%  !"$ %&'&(L(k)(('&%#! ~xm d _ZYLB7ux@Y 7A LN C ]   C ] <N C ]   C ] dN C V   C V    c B CDE(F0???1?  @C V  ! c B CDEF 1? @J M  " c B CDEF ___1? @R V r # c BB4CDEXF`1?  #*.1233 33" -0@C ]  $ c zB3CDEtF|???1?-,+) ( & #    "%)-/1222-;<@G ]  N  J  %  J nN $ =  & $ =   ' c B CDE$F,1?   @$ =  ( c B CDE(F01?    @) 8  ) c B CDEF 1?  @- 1  * c B CDEF___1?   @7 = 6N  J  +  J  , c B>C.DEF1?'' &158:<<<= <<%<';):*8+5+0,+,$--,,+ * )(&$ OP@ F  - c B>C%DEF___1?%%  !!%!* - 024679:; < <=<<; :!8"5"($!$$##" ! KL@! F N  J  .  J  / c BC0DEF1?/@  H  0 c BC-DEF1?,@ G  1 c BC2DEF1?1@ J N 3B s BCDE0F81? wpzt|uwsxdyP{9$@S" vH T Y 2  4C#  5 c rB<CDEpFx___1?322 24679;1&    &+,.0139<@ 2 N Y   6 Y  2 7 c BjC%DE8F@1? 9ih f"d#a$]$L$=#4" @Y  B 8 c BCMDE@FH1?@[)u8EGIxJhL[KII<H9@!$@S  N j   9 j  N x   : x  * ; c B6CDE4F<1?  5' @x   < c B'CDE,F4???1?   & @  N j   = j  : > c B(CDE<FD1? !'   @j   ? c BCDE,F4???1?   @w  ^2 J 6(1? , #  M s BDCrDEF`8c?&qC&q @C6 ^2 N 61?v , ^2 O 61? L ^2 P 61? q Y H08c?2jI ; U1 Z  Z H(8c?2 ; U2 Z  [ H8c?s  |  U3 Z @J8 V    *~T lz >  ^C# V y  _ c BCDEF?DD!zp}svx2xx`sdshtmtquvw{y~{~{qjZ-|rhaV#&]9@lz > ~N    `   J a c BC DEDFL1?       #$@  J b c B C!DEDFL1?       #$@C O : c c B CDE<FD1?       @\ h : d c B CDE<FD1?         @  N    e   XN $ Q  f $ Q f2 gB 61?P Q f2 hB 61?I J f2 iB 61?B C f2 j 61?: ; f2 k 61?3 4 f2 lB 61?+ , f2 m 61?$ %  N     n    XN    o   f2 pB 61?  f2 qB 61?  f2 rB 61?  f2 sB 61?  f2 tB 61?  f2 uB 61?  f2 vB 61?  XN U   w U  f2 xB 61?  f2 yB 61?}  f2 zB 61?v w f2 {B 61?m o f2 |B 61?e g f2 }B 61?] ^ f2 ~B 61?U W N   9     9 f2  61?8 9 f2 B 61?0 1 f2 B 61?( ) f2 B 61?  ! f2 B 61?  f2 B 61?   N       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  N       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?   B s pBCnDEF1?;;7A=G>I@KBLDNFOIPLQORTSYSWPbOpNLNORSVX[]_bdfikkmjgd~b}`z^x\uZrXoVkUfU]UWVQVX[T[NZIZFZCY?Y;W9V wx@ V t bT    C# . 9   c BCDEF1? @    c BCDEF1? @    c BCDE,F41? @  2 B  C}IENGHI`TJ.Q? n.T}I,n.T}I,`T.T}I,`T.r H s I X B s (BJCDEHFP___1?(.5:?BEHI F( %(@ I & f T Y 8]  C# * F N Y 8A   Y 8A "  c BC0DEF1?,,7. ' !$&( ) *+-%-=.N.h/~..-,*'" 7Y\@Z 8A   c bBC*DEF???1?::79:: 9753,%  !"$ %&'&(L(k)(('&%#! ~xm d _ZYLB7ux@Y 7A LN C ]   C ] <N C ]   C ] dN C V   C V   c B CDE(F0???1?  @C V   c B CDEF 1? @J M   c B CDEF ___1? @R V r  c BB4CDEXF`1?  #*.1233 33" -0@C ]   c zB3CDEtF|???1?-,+) ( & #    "%)-/1222-;<@G ]  N  J    J nN $ =   $ =    c B CDE$F,1?   @$ =   c B CDE(F01?    @) 8   c B CDEF 1?  @- 1   c B CDEF___1?   @7 = 6N  J    J   c B>C.DEF1?'' &158:<<<= <<%<';):*8+5+0,+,$--,,+ * )(&$ OP@ F   c B>C%DEF___1?%%  !!%!* - 024679:; < <=<<; :!8"5"($!$$##" ! KL@! F N  J    J   c BC0DEF1?/@  H   c BC-DEF1?,@ G   c BC2DEF1?1@ J N B s BCDE0F81? wpzt|uwsxdyP{9$@S"  V y (  T Y 2  C#  {   c rB<CDEpFx___1?322 24679;1&    &+,.0139<@ 2 N Y    Y  2  c BjC%DE8F@1? 9ih f"d#a$]$L$=#4" @Y  B  c BCMDE@FH1?@[)u8EGIxJhL[KII<H9@!$@S  N j    j  N x    x  *  c B6CDE4F<1?  5' @x    c B'CDE,F4???1?   & @  N j    j  :  c B(CDE<FD1? !'   @j    c BCDE,F4???1?   @w  J8 P  Hdh  & T||?"6@ NNN?NP  ) s BCDEF?DD!zp}svx2xx`sdshtmtquvw{y~{~{qjZ-|rhaV#&]9@<fT    *# /cJ + c BC DEDFL1?       #$@  J , c B C!DEDFL1?       #$@C O : - c B CDE<FD1?       @\ h : . c B CDE<FD1?         @  T    /# <>UXN $ Q  0 $ Q f2 1B 61?P Q f2 2B 61?I J f2 3B 61?B C f2 4 61?: ; f2 5 61?3 4 f2 6B 61?+ , f2 7 61?$ %  N     8    XN    9   f2 :B 61?  f2 ;B 61?  f2 <B 61?  f2 =B 61?  f2 >B 61?  f2 ?B 61?  f2 @B 61?  XN U   A U  f2 BB 61?  f2 CB 61?}  f2 DB 61?v w f2 EB 61?m o f2 FB 61?e g f2 GB 61?] ^ f2 HB 61?U W N   9  I   9 f2 J 61?8 9 f2 KB 61?0 1 f2 LB 61?( ) f2 MB 61?  ! f2 NB 61?  f2 OB 61?   N    P   f2 QB 61?  f2 RB 61?  f2 SB 61?  f2 TB 61?  f2 UB 61?  f2 VB 61?  N    W   f2 XB 61?  f2 YB 61?  f2 ZB 61?  f2 [B 61?  f2 \B 61?  f2 ]B 61?   ^ s pBCnDEF1?;;7A=G>I@KBLDNFOIPLQORTSYSWPbOpNLNORSVX[]_bdfikkmjgd~b}`z^x\uZrXoVkUfU]UWVQVX[T[NZIZFZCY?Y;W9V wx@_ ` s BCDEF1?  @>G a s BCDEF1?  @>G  b s BCDE,F41? @>I2 c  C}IENGHI`TJ.Q? n.T}I,n.T}I,`T.T}I,`T.BXCYX d s (BJCDEHFP___1?(.5:?BEHI F( %(@Yv T Y 8A  f#  V" g c BC0DEF1?,,7. ' !$&( ) *+-%-=.N.h/~..-,*'" 7Y\@Z 8A  h c bBC*DEF???1?::79:: 9753,%  !"$ %&'&(L(k)(('&%#! ~xm d _ZYLB7ux@Y 7A RT C ]  i#  P<N C ]  j C ] dN C V  k C V  l c B CDE(F0???1?  @C V  m c B CDEF 1? @J M  n c B CDEF ___1? @R V r o c BB4CDEXF`1?  #*.1233 33" -0@C ]  p c zB3CDEtF|???1?-,+) ( & #    "%)-/1222-;<@G ]  T  J  q#  a]nN $ =  r $ =   s c B CDE$F,1?   @$ =  t c B CDE(F01?    @) 8  u c B CDEF 1?  @- 1  v c B CDEF___1?   @7 = 6N  J  w  J  x c B>C.DEF1?'' &158:<<<= <<%<';):*8+5+0,+,$--,,+ * )(&$ OP@ F  y c B>C%DEF___1?%%  !!%!* - 024679:; < <=<<; :!8"5"($!$$##" ! KL@! F N  J  z  J  { c BC0DEF1?/@  H  | c BC-DEF1?,@ G  } c BC2DEF1?1@ J N  s BCDE0F81? wpzt|uwsxdyP{9$@S" <f(8 T Y 2  #    c rB<CDEpFx___1?322 24679;1&    &+,.0139<@ 2 N Y    Y  2  c BjC%DE8F@1? 9ih f"d#a$]$L$=#4" @Y  B  c BCMDE@FH1?@[)u8EGIxJhL[KII<H9@!$@S  N j    j  N x    x  *  c B6CDE4F<1?  5' @x    c B'CDE,F4???1?   & @  N j    j  :  c B(CDE<FD1? !'   @j    c BCDE,F4???1?   @w    H3GHg1? PSt F Z   Bp51?C"?} U2 Z   H:GFH|1? C"?gG F Z   H<8c?  U1 Z   HAGWHX1? C"?  F Z   HC8c? ] <  U3 Z H  0޽h ?o`  c ̙33y___PPT10Y+D=' = @B +:  :: D 9(  ~  s *Qdt    ٶ s BDCrDEF`8c?&qC&q @6   @ T||?"6@ NNN?N- b NJL v # B  ~T lz >  C# v  c BCDEF?DD!zp}svx2xx`sdshtmtquvw{y~{~{qjZ-|rhaV#&]9@lz > ~N       J  c BC DEDFL1?       #$@  J  c B C!DEDFL1?       #$@C O :  c B CDE<FD1?       @\ h :  c B CDE<FD1?         @  N       XN $ Q   $ Q f2 B 61?P Q f2 B 61?I J f2 B 61?B C f2  61?: ; f2  61?3 4 f2 B 61?+ , f2  61?$ %  N         XN       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  XN U    U  f2 B 61?  f2 B 61?}  f2 B 61?v w f2 B 61?m o f2 B 61?e g f2 B 61?] ^ f2 µB 61?U W N   9  õ   9 f2 ĵ 61?8 9 f2 ŵB 61?0 1 f2 ƵB 61?( ) f2 ǵB 61?  ! f2 ȵB 61?  f2 ɵB 61?   N    ʵ   f2 ˵B 61?  f2 ̵B 61?  f2 ͵B 61?  f2 εB 61?  f2 ϵB 61?  f2 еB 61?  N    ѵ   f2 ҵB 61?  f2 ӵB 61?  f2 ԵB 61?  f2 յB 61?  f2 ֵB 61?  f2 ׵B 61?   صB s pBCnDEF1?;;7A=G>I@KBLDNFOIPLQORTSYSWPbOpNLNORSVX[]_bdfikkmjgd~b}`z^x\uZrXoVkUfU]UWVQVX[T[NZIZFZCY?Y;W9V wx@bT    ٵC# NY ڵ c BCDEF1? @   ۵ c BCDEF1? @   ܵ c BCDE,F41? @  2 ݵB C}IENGHI`TJ.Q? n.T}I,n.T}I,`T.T}I,`T.hiX ޵B s (BJCDEHFP___1?(.5:?BEHI F( %(@iYT Y 8]  ߵC# ]fN Y 8A   Y 8A "  c BC0DEF1?,,7. ' !$&( ) *+-%-=.N.h/~..-,*'" 7Y\@Z 8A   c bBC*DEF???1?::79:: 9753,%  !"$ %&'&(L(k)(('&%#! ~xm d _ZYLB7ux@Y 7A LN C ]   C ] <N C ]   C ] dN C V   C V   c B CDE(F0???1?  @C V   c B CDEF 1? @J M   c B CDEF ___1? @R V r  c BB4CDEXF`1?  #*.1233 33" -0@C ]   c zB3CDEtF|???1?-,+) ( & #    "%)-/1222-;<@G ]  N  J    J nN $ =   $ =    c B CDE$F,1?   @$ =   c B CDE(F01?    @) 8   c B CDEF 1?  @- 1   c B CDEF___1?   @7 = 6N  J    J   c B>C.DEF1?'' &158:<<<= <<%<';):*8+5+0,+,$--,,+ * )(&$ OP@ F   c B>C%DEF___1?%%  !!%!* - 024679:; < <=<<; :!8"5"($!$$##" ! KL@! F N  J    J   c BC0DEF1?/@  H   c BC-DEF1?,@ G   c BC2DEF1?1@ J N B s BCDE0F81? wpzt|uwsxdyP{9$@S" vH T Y 2  C#   c rB<CDEpFx___1?322 24679;1&    &+,.0139<@ 2 N Y    Y  2  c BjC%DE8F@1? 9ih f"d#a$]$L$=#4" @Y  B  c BCMDE@FH1?@[)u8EGIxJhL[KII<H9@!$@S  N j    j  N x    x  *  c B6CDE4F<1?  5' @x    c B'CDE,F4???1?   & @  N j    j  :  c B(CDE<FD1? !'   @j    c BCDE,F4???1?   @w  d2  <(1?[ ,   s BDCrDEF`8c?&qC&q @ 6 B d2  <1? , d d2  <1? L C d2   <1?  /    Nt8c? =  U1 Z    Ny8c?  U2 Z    Nd}8c?   U3 Z NJL V   #  Y~T lz >  C# V y   c BCDEF?DD!zp}svx2xx`sdshtmtquvw{y~{~{qjZ-|rhaV#&]9@lz > ~N       J  c BC DEDFL1?       #$@  J  c B C!DEDFL1?       #$@C O :  c B CDE<FD1?       @\ h :  c B CDE<FD1?         @  N       XN $ Q   $ Q f2 B 61?P Q f2 B 61?I J f2 B 61?B C f2  61?: ; f2  61?3 4 f2 B 61?+ , f2  61?$ %  N         XN       f2  B 61?  f2 !B 61?  f2 "B 61?  f2 #B 61?  f2 $B 61?  f2 %B 61?  f2 &B 61?  XN U   ' U  f2 (B 61?  f2 )B 61?}  f2 *B 61?v w f2 +B 61?m o f2 ,B 61?e g f2 -B 61?] ^ f2 .B 61?U W N   9  /   9 f2 0 61?8 9 f2 1B 61?0 1 f2 2B 61?( ) f2 3B 61?  ! f2 4B 61?  f2 5B 61?   N    6   f2 7B 61?  f2 8B 61?  f2 9B 61?  f2 :B 61?  f2 ;B 61?  f2 <B 61?  N    =   f2 >B 61?  f2 ?B 61?  f2 @B 61?  f2 AB 61?  f2 BB 61?  f2 CB 61?   DB s pBCnDEF1?;;7A=G>I@KBLDNFOIPLQORTSYSWPbOpNLNORSVX[]_bdfikkmjgd~b}`z^x\uZrXoVkUfU]UWVQVX[T[NZIZFZCY?Y;W9V wx@ V t bT    EC# . 9  F c BCDEF1? @   G c BCDEF1? @   H c BCDE,F41? @  2 IB C}IENGHI`TJ.Q? n.T}I,n.T}I,`T.T}I,`T.r H s I X JB s (BJCDEHFP___1?(.5:?BEHI F( %(@ I & f T Y 8]  KC# * F N Y 8A  L Y 8A " M c BC0DEF1?,,7. ' !$&( ) *+-%-=.N.h/~..-,*'" 7Y\@Z 8A  N c bBC*DEF???1?::79:: 9753,%  !"$ %&'&(L(k)(('&%#! ~xm d _ZYLB7ux@Y 7A LN C ]  O C ] <N C ]  P C ] dN C V  Q C V  R c B CDE(F0???1?  @C V  S c B CDEF 1? @J M  T c B CDEF ___1? @R V r U c BB4CDEXF`1?  #*.1233 33" -0@C ]  V c zB3CDEtF|???1?-,+) ( & #    "%)-/1222-;<@G ]  N  J  W  J nN $ =  X $ =   Y c B CDE$F,1?   @$ =  Z c B CDE(F01?    @) 8  [ c B CDEF 1?  @- 1  \ c B CDEF___1?   @7 = 6N  J  ]  J  ^ c B>C.DEF1?'' &158:<<<= <<%<';):*8+5+0,+,$--,,+ * )(&$ OP@ F  _ c B>C%DEF___1?%%  !!%!* - 024679:; < <=<<; :!8"5"($!$$##" ! KL@! F N  J  `  J  a c BC0DEF1?/@  H  b c BC-DEF1?,@ G  c c BC2DEF1?1@ J N dB s BCDE0F81? wpzt|uwsxdyP{9$@S"  V y (  T Y 2  eC#  {  f c rB<CDEpFx___1?322 24679;1&    &+,.0139<@ 2 N Y   g Y  2 h c BjC%DE8F@1? 9ih f"d#a$]$L$=#4" @Y  B i c BCMDE@FH1?@[)u8EGIxJhL[KII<H9@!$@S  N j   j j  N x   k x  * l c B6CDE4F<1?  5' @x   m c B'CDE,F4???1?   & @  N j   n j  : o c B(CDE<FD1? !'   @j   p c BCDE,F4???1?   @w  JL P  q#  dh  r N||?"6@ NNN?NP  s s BCDEF?DD!zp}svx2xx`sdshtmtquvw{y~{~{qjZ-|rhaV#&]9@<fT    t# /cJ u c BC DEDFL1?       #$@  J v c B C!DEDFL1?       #$@C O : w c B CDE<FD1?       @\ h : x c B CDE<FD1?         @  T    y# <>UXN $ Q  z $ Q f2 {B 61?P Q f2 |B 61?I J f2 }B 61?B C f2 ~ 61?: ; f2  61?3 4 f2 B 61?+ , f2  61?$ %  N         XN       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  XN U    U  f2 B 61?  f2 B 61?}  f2 B 61?v w f2 B 61?m o f2 B 61?e g f2 B 61?] ^ f2 B 61?U W N   9     9 f2  61?8 9 f2 B 61?0 1 f2 B 61?( ) f2 B 61?  ! f2 B 61?  f2 B 61?   N       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  N       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?    s pBCnDEF1?;;7A=G>I@KBLDNFOIPLQORTSYSWPbOpNLNORSVX[]_bdfikkmjgd~b}`z^x\uZrXoVkUfU]UWVQVX[T[NZIZFZCY?Y;W9V wx@_  s BCDEF1? @>G  s BCDEF1? @>G   s BCDE,F41? @>I2  C}IENGHI`TJ.Q? n.T}I,n.T}I,`T.T}I,`T.BXCYX  s (BJCDEHFP___1?(.5:?BEHI F( %(@Yv T Y 8A  # V"  c BC0DEF1?,,7. ' !$&( ) *+-%-=.N.h/~..-,*'" 7Y\@Z 8A   c bBC*DEF???1?::79:: 9753,%  !"$ %&'&(L(k)(('&%#! ~xm d _ZYLB7ux@Y 7A RT C ]  # P<N C ]   C ] dN C V   C V   c B CDE(F0???1?  @C V   c B CDEF 1? @J M   c B CDEF ___1? @R V r  c BB4CDEXF`1?  #*.1233 33" -0@C ]   c zB3CDEtF|???1?-,+) ( & #    "%)-/1222-;<@G ]  T  J  # a]nN $ =   $ =    c B CDE$F,1?   @$ =   c B CDE(F01?    @) 8   c B CDEF 1?  @- 1   c B CDEF___1?   @7 = 6N  J    J   c B>C.DEF1?'' &158:<<<= <<%<';):*8+5+0,+,$--,,+ * )(&$ OP@ F   c B>C%DEF___1?%%  !!%!* - 024679:; < <=<<; :!8"5"($!$$##" ! KL@! F N  J  ¶  J  ö c BC0DEF1?/@  H  Ķ c BC-DEF1?,@ G  Ŷ c BC2DEF1?1@ J N ƶ s BCDE0F81? wpzt|uwsxdyP{9$@S" <f(8 T Y 2  Ƕ#  ȶ c rB<CDEpFx___1?322 24679;1&    &+,.0139<@ 2 N Y   ɶ Y  2 ʶ c BjC%DE8F@1? 9ih f"d#a$]$L$=#4" @Y  B ˶ c BCMDE@FH1?@[)u8EGIxJhL[KII<H9@!$@S  N j   ̶ j  N x   Ͷ x  * ζ c B6CDE4F<1?  5' @x   ϶ c B'CDE,F4???1?   & @  N j   ж j  : Ѷ c B(CDE<FD1? !'   @j   Ҷ c BCDE,F4???1?   @w   Ӷ N@GH}g1? -St  F Z  Զ Hػ1?C"?Xa  U2 Z  ն NGFH{1? C"?g  F Z  ֶ N$8c? U1 Z  ׶ NGWHW1? C"? Z F Z  ض N8c?0 % 9 U3 Z  ڶ N8c?+@  4 U4 Z d2 ۶ <1? : NJL V  ޶#  V~T lz >  ߶C# V y   c BCDEF?DD!zp}svx2xx`sdshtmtquvw{y~{~{qjZ-|rhaV#&]9@lz > ~N       J  c BC DEDFL1?       #$@  J  c B C!DEDFL1?       #$@C O :  c B CDE<FD1?       @\ h :  c B CDE<FD1?         @  N       XN $ Q   $ Q f2 B 61?P Q f2 B 61?I J f2 B 61?B C f2  61?: ; f2  61?3 4 f2 B 61?+ , f2  61?$ %  N         XN       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  XN U    U  f2 B 61?  f2 B 61?}  f2 B 61?v w f2 B 61?m o f2 B 61?e g f2 B 61?] ^ f2 B 61?U W N   9     9 f2  61?8 9 f2 B 61?0 1 f2 B 61?( ) f2 B 61?  ! f2 B 61?  f2 B 61?   N       f2 B 61?  f2  B 61?  f2  B 61?  f2  B 61?  f2  B 61?  f2  B 61?  N       f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?  f2 B 61?   B s pBCnDEF1?;;7A=G>I@KBLDNFOIPLQORTSYSWPbOpNLNORSVX[]_bdfikkmjgd~b}`z^x\uZrXoVkUfU]UWVQVX[T[NZIZFZCY?Y;W9V wx@ V t bT    C# . 9   c BCDEF1? @    c BCDEF1? @    c BCDE,F41? @  2 B C}IENGHI`TJ.Q? n.T}I,n.T}I,`T.T}I,`T.r H s I X B s (BJCDEHFP___1?(.5:?BEHI F( %(@ I & f T Y 8]  C# * F N Y 8A   Y 8A "  c BC0DEF1?,,7. ' !$&( ) *+-%-=.N.h/~..-,*'" 7Y\@Z 8A   c bBC*DEF???1?::79:: 9753,%  !"$ %&'&(L(k)(('&%#! ~xm d _ZYLB7ux@Y 7A LN C ]    C ] <N C ]  ! C ] dN C V  " C V  # c B CDE(F0???1?  @C V  $ c B CDEF 1? @J M  % c B CDEF ___1? @R V r & c BB4CDEXF`1?  #*.1233 33" -0@C ]  ' c zB3CDEtF|???1?-,+) ( & #    "%)-/1222-;<@G ]  N  J  (  J nN $ =  ) $ =   * c B CDE$F,1?   @$ =  + c B CDE(F01?    @) 8  , c B CDEF 1?  @- 1  - c B CDEF___1?   @7 = 6N  J  .  J  / c B>C.DEF1?'' &158:<<<= <<%<';):*8+5+0,+,$--,,+ * )(&$ OP@ F  0 c B>C%DEF___1?%%  !!%!* - 024679:; < <=<<; :!8"5"($!$$##" ! KL@! F N  J  1  J  2 c BC0DEF1?/@  H  3 c BC-DEF1?,@ G  4 c BC2DEF1?1@ J N 5B s BCDE0F81? wpzt|uwsxdyP{9$@S"  V y (  T Y 2  6C#  {  7 c rB<CDEpFx___1?322 24679;1&    &+,.0139<@ 2 N Y   8 Y  2 9 c BjC%DE8F@1? 9ih f"d#a$]$L$=#4" @Y  B : c BCMDE@FH1?@[)u8EGIxJhL[KII<H9@!$@S  N j   ; j  N x   < x  * = c B6CDE4F<1?  5' @x   > c B'CDE,F4???1?   & @  N j   ? j  : @ c B(CDE<FD1? !'   @j   A c BCDE,F4???1?   @w   B NGWHW1? C"? W F Z  C N8c?,2 5 U4 Z H  0޽h ?ݵIӶ ն ׶  B ̙33y___PPT10Y+D=' = @B + 7  %T(      `xaxa1?@ $input complex (all possible flights)8% Zcg % $    `xaxa1?@a  ,output complex (all consistent slot choices)8- Zcg - ~ 1 s *d    2  `xaxa1?z z  dTask spec relation" Zc  ^2 4 61?  ^ 5 68c? + A ^ 7 68c? [ K 8 1  Sf   68c?9 1 TB 3 c $8c? -' f 8 68c? c f2 9 61?v  f2 : 61?  TB ; c $8c?  f D 68c?91 TB E c $8c? f2 F 61?ETB G c $8c?   J  `vB;vB;1?  nI  }AA082" Z      K  ` vB;vB;1? Q c  }AL991" Z     ZB L s *8c?:$ ZB M s *8c?  ZB N s *8c? ZB O s *8c?  d 6 <8c? 0j d2 < <1?I 7 d2 = <1?; Y RB >@ s *8c? _5 d2 ? <1? UD d2 @ <1? 48   A  f0vB;vB;1?  u2 Z       B  fX;vB;vB;1?  u1 Z     RB C s *8c? W-   P  fEvB;vB;1?   v 1 Z       Q  fUvB;vB;1?   u2 Z      R  BCbDEF)?b8\ 5@  S"  z  z H  0޽h ? X=[0y___PPT10Y+D=' = @B + 8  :: Z^9(     `0axaxa1?>:Y  ] input complex  ZC     `exaxa1?' 7 ^output complex  ZC  RB  s *)? . ^2  61?U   `ivB;vB;1? j  oP,6 Z      ^2  61?U_^2  61?l ' jR  @ BG/*1?R jR  @ BG/*1?2R dR   <G/*1?>  dR   <G/*1?> 2 dR   <G/*1?#jR @ BG/*1? k ^2  61?^2  61?(^2  61?- } ^2  61?h^2  61?Q  ^2  61? <o ^2  61?  ^2  61? a ^2  61?S^2  61? h ^2  61?  RB @ s *)?URB @ s *)?}, 9   `L{vB;vB;1? Rf ,$ 0 u1 Z     R @ BG/*`1?W ,$@ 0R @ BG/*`1?4W ,$@ 0R  <G/*`1?C  ,$D 0R   <G/*`1?C 4 ,$@  0R ! <G/*`1?(,$@ 0R "@ BG/*`1? p ,$@ 02 # 61?,$@ 02 $ 61?*,$@ 09 %  `vB;vB;1?^_ ,$ 0 u1 Z     9 &  `(vB;vB;1?u ?" ,$ 0 u1 Z     < '  `@vB;vB;1?l ,$ 0 x 1 Z     < (  `vB;vB;1? 8 ,$  0 x 1 Z     9 )  `dvB;vB;1?1  ,$  0 u2 Z     9 *  `vB;vB;1? B ,$ 0 u3 Z     9 +  `vB;vB;1? ,$ 0 u2 Z     9 ,  `vB;vB;1?,$ 0 u3 Z     < -  `hvB;vB;1?,$ 0 x 2 Z     < .  `vB;vB;1?  ,$ 0 x 3 Z     2 / 61?2  ,$@ 02 0 61?k,$@ 02 1 B1?V  ,$D 0 P&*G, 2 2 61?$ ?t ,$@ 02 3 61?  ,$@ 02 4 61? c ,$D  02 5 61?U,$@ 02 6 61? k ,$@ 02 7 61?  ,$@  0d F <8c?y+8 Q8 K  ^P c ~R L NWFuG%jJ?- h xR 9 H!8G/*1?# xR : H!8G/*1?% m xR ; H!8G/*1?b l2 < <1?] 1 rR = BG/*1? l2 > <1? 0+ l2 ? <1?a  rR @ BG/*`1? ~R B NuG/*jJ?z~R C NuG/*jJ?R ~R D NuG/*jJ?Y ~R E NuG/*jJ?7  l2 G <1?d & r2 H B)?l2 I <1?EA2 J H<1?= L" Z# ~R K NqG*jJ?9 e l2 M <1?o  l2 N <1?  l2 O <1?  l2 P <1? `B Q 08c? 8 `B R 08c?Qu`B S 08c? 0 `B T 08c? K  U  fvB;vB;1?j uAL991" Z       V  f0vB;vB;1?bF %  uAA082" Z       W  f<vB;vB;1?\  uDL227" Z      2 X B`1? \,$@  0 P&*G, 9 Y  `vB;vB;1?I,$! 0 u1 Z      Z  BCbDEF)?b8\ 5@  S"  w ~ [ s *#    H  0޽h ? X=[0MRER___PPT10%R+DN' = @B D|N' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-6B'blinds(horizontal)*<3<*5DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-6B'blinds(horizontal)*<3<*-DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-6B'blinds(horizontal)*<3<*'DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-6B'blinds(horizontal)*<3<*2DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-6B'blinds(horizontal)*<3<*.DM ' =%(D ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B'blinds(horizontal)*<3<*(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-6B'blinds(horizontal)*<3<*)DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-6B'blinds(horizontal)*<3<*7DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-6B'blinds(horizontal)*<3<*4D^0' =%(D ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-6B'blinds(horizontal)*<3<*"DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-6B'blinds(horizontal)*<3<*3DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-6B'blinds(horizontal)*<3<**DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-6B'blinds(horizontal)*<3<*1D ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-6B'blinds(horizontal)*<3<*/DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-6B'blinds(horizontal)*<3<*+DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-6B'blinds(horizontal)*<3<*6DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-6B'blinds(horizontal)*<3<*&DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DQ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-6B'blinds(horizontal)*<3<*0DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-6B'blinds(horizontal)*<3<*#DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-6B'blinds(horizontal)*<3<*+DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-6B'blinds(horizontal)*<3<*%DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-6B'blinds(horizontal)*<3<*,D ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-6B'blinds(horizontal)*<3<*!DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'blinds(horizontal)*<3<*XDT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =-6B'blinds(horizontal)*<3<*Y+H+0+ ++0+% ++0+& ++0+' ++0+( ++0+) ++0+* ++0++ ++0++ ++0+, ++0+- ++0+. ++0+1 ++0+X ++0+Y +V 9  t(     `dxaxa1? [ input complex Z     `hxaxa1? \output complex Z   F wx    rR  B!8G/*1?E=rR  B!8G/*1?d rR  B!8G/*1?>Ff2  61?BYlR   <G/*1?f/sf2   61?Vf2   61?DOlR   <G/*`1?h3w    `wvB;vB;1?Pj  mR Z      xR  HuG/*jJ?L.xR  HuG/*jJ?F3xR  HuG/*jJ?LuxR  HuG/*jJ?X) f  68c?wf2  61?FSl2  <)?,7f2  61?&f2  B|1?$-d L" Z# xR  HqG*jJ?NxxR  HWFuG%jJ?X+ f2  61? _ f2  61?Cf2  61?Cf2  61?ARZB  s *8c?#ZB  s *8c?P3ZB  s *8c?K  ZB   s *8c? 2  ! s BCDE F1?@  RB " s *)?    x s d0e0e #" 0epP   j L .j  z#  Z 5 ~R { N`1?pR ~R | N`1? R ~R } N`1? y ~R ~ N`1?Z p ~R  N`1? yH ~R  N`1? m ~R  N`1?Z ~R  N`1? ~R  N`1? E ~R  N`1?  ~R  N`1?:v ~R  Nf1?m ~R  N`1?C  ~R  N`1?  ~R  N`1?e ~R  N`1?  ~R  N`1? n ~R  N`1?.  ~R  N`1?j ~R  N`1?Rp ~R  N`1?m? ~R  N`1?Np ~R  N`1? + ~R  N`1?p; ~R  N`1?v   N ?W [+  U2$*G    N, ?W _+  U3$*G    N ?P C  $  U4$*G    T8` ?W +  U2$*G    N ?W +  U3$*G    N ?P = $  U1$*G    N< ?P $  U4$*G    N ?P $  U1$*G    N ?P $  U1$*G    N ?W _!+  U2$*G    N< ?W a+  U3$*G    N ?W +  U2$*G    N< ?W  +  U3$*G    N ?W [ +  U3$*G  B  # lDf)?8 ^ 8 ,$D 0B  # lDf)?% % ,$@ 0  s p0e0e    BCDE(F @  ) 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 4xX 75Y=T!JG @   S"   D ,$@ 0  s p0e0e    BCDE(F @  ) 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| 4xX 75Y=T!JG @   S"  | ,$D 0  N ?P 3$  U4$*G    N( ?P -$  U1$*G    N ?+   U4$*G    Nt ?+ Z  U1$*G    N ?+ T  U4$*G    N ?+   U1$*G    NP ?+   U1$*G    N ?+   U4$*G    N ?+ J  U1$*G  2  B1? 1  P&*G, 2  B1? 6d  P&*G, 2  B1?   P&*G, 2  B,1? :  P&*G, 2  B1?   P&*G, 2  B 1? Iw  P&*G, 2  B|1?   P&*G, 2  B1? k   P&*G, 2  B\1?   P&*G, 2  B1? '  P&*G, 2  B 1? t  P&*G, 2  B 1? 6d  P&*G, 2  B$ 1?   P&*G, 2  B 1?   P&*G, 2  B 1?+ 0 ^ X  P&*G, 2  BP 1?% c R  P&*G, 2  B 1?" O  P&*G, 2  B, 1?" 9gO  P&*G, 2  B! 1?" )O  P&*G, 2  Bd% 1?" vO  P&*G, 2  B ( 1?" O  P&*G, 2  B& 1?    P&*G, 2  B. 1? # Q   P&*G, Q@l  z 5#,$D 0  68c?C"?vG?N  n  $  C@ % s BCDE<FD1?&=Lc.r=Lcr @l f2 & 61?  `2 ' 01?  f2 ( 61?. ]U 2 )  ` @1?7" 2 *  `L @1?tN   + s BCDE,F41? T[.?|I.@B , s B6C*DE$F,1? &&Lr 5)@OM   - s BCDEF$1?r&TD=b@ [  . s B>CDE$F,1? ==&=c@h   / s B)CDE$F,1? ((j5=j@b@\ NB 0B S 1? 1 s B'C_DE$F,1? D&r&&&0^@b@.  2 s BCUDE F(1?5&[5=ELT@a   3 s BqCDEF1?!!?|CiBYh5pppaYB Zt4W[Z&p-=1CD@@BM `2 4 01?o"I`2 5 01?F w `2 6 01?d 4 `2 7 01?y]  `2 8 01?  `2 9 01?8 j `2 : 01?.$ SU `2 ; 01?J+ \ `2 < 01?s ! f2 = 61?k (* Z2 > s *1?  ? f2 ? 61?L ~ NB @B S 1?4  NB A S 1?[ [ `2 B 01?R y `2 C 01? DF `2 D 01?S E s BCDE(F01? &E&cLzk@b  F s B.CDE$F,1? &'E-l----@l@ 2 G CUENGgHZJOQ1? `TU]TU`TU]TUO]TUO}   H s BC/DEF1?  &. @t  `2 I 01?  `2 J 01? N  K s BC DE(F01? rS&=D&c @z l2 L <G1?{ xf 2 M BUENGU6 HggI)Q1? ɬUUSɬUUS)`TUS)`TB  2 N  BTENGHa,ZIAJQ1? T`TT`TT`TT`TAT`TAR l 2 O E$GH@ I`TQ1? `TS`TS`T`T`TS`T`TL w 2 P ENGHgQ1? `T`TS`T`TS`T`TS`T ; TB Q c $1?? [ 2 R  BTENGH9ZIUJQ1? T`TT`TT`TT`TUT`TUN& = 2 S E$GH@ I`TQ1? `TS`TS`T`T`TS`T`T  ,G 2 T BTENG IQ1? 'TT`T'TT`T`TT`T`T+ s  U s B~CPDE$F,1? OW:}BEwO@ll2 V <G1?I  2 W  BTCTENGgHS=ZIZJ:Q1? ګTTTTګTTTTZ:TTZ:tv ` 2 X  BpUENGH}ZIJQ1? pU`TpU^TpU`TpU^TpU^TW  l2 Y <G1?E\ `2 Z 01? B  [ s BCDE(F01? .L=j[y@  i NB \ S 1? /=  ] s BkCkDEF 1?jj55bj @~   ^ s BkCkDE F(1?jS=&&5L[j@~  `2 _ 01?. `2 ` 01?P  l2 a <G1?P n 2 b CTENG@gHZJdQ1? `TT`TT`TT`TTd`TTd N  c  `|J //8c?g_ y1" Z  d  `xT //1?   y1" Z  e  `^ //1?v   y4" Z  f  `_ //1? 2J  y4" Z  g  `p //1?   y4" Z `2 h 01?l  `2 i 01?< i `2 j 01?k  `2 k 01?jV } `2 l 01?U |`2 m 01? A  n  `t //8c? n  y1" Z  o  `j //1?Go ,  y3" Z  p  ` //1?@_ y3" Z  q  ` //1? o p,  y3" Z  r  ` //1?v z  y2" Z  s  ` //1?W y2" Z  t  ` //1? j&  y2" Z  u  ` //1?F   y2" Z  v  ` //8c? =@  y1" Z  w  `l //1?& ?  y4" Z  #  BCbDEF)?b8\ 5@  S"  m  H  0޽h ?GLMNOPRS T V W X Yab X=[0  ___PPT10 +D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+  0 P(  ~  s * ,d     s * p    H  0޽h ? ̙33y___PPT10Y+D=' = @B +): zr9 (     `(Wxaxa1?n ~$Asynchronous Computability Theorem "% Z%$$ J  T"xaxa1? ,$ 0 A task has a wait-free solution if and only if one can chromatically subdivide its input complex so that there exists a chromatic simplicial map to its output complex that refines the task specification map. p Z  #!#*#A.   D Ml   ۼ ,$@ 0d3T  +  Z#   : [ c BCDE<FD1?&=Lc.r=Lcr @ n f2 \ 61?  @ `2 ] 01? 6 . \ f2 ^ 61? f ; 2 _  ` @1? _ # 2 `  `L @1?   a c BCDE,F41? T[.?|I.@ " Q   b c B7C*DE$F,1? '&Lr 6)@/ f  c c BCDEF$1?r&TD=c@   d c B>CDE$F,1? ==&=c@      e c B)CDE$F,1? ((j5=j@ O  f c BCDE F1?@] T ^ U NB gB S 1?] [ l [   h c B'C_DE$F,1? D&r&&&0^@  i c BCTDE F(1?5&[5<DKS@ .  j c BqCDEF1?!!?|DiBYh4pqpaYB Zt4W[Z&q-=rxCD@# v   k  hB/CDE\Fdd @1?/Z'.}&(0ZGnO//}vV>//0@` g ; `2 l 01?J _ p } `2 m 01? ~ `2 n 01?  +' `2 o 01?  " `2 p 01?  `2 q 01? = `2 r 01?  `2 s 01?  8 `2 t 01? } `2 u 01? n 3 `2 v 01?, h `2 w 01?} X x  (BCDE<FD[d @1?H5xrP  (r E @- f2 y 61? Z Z2 z s *1?* f2 { 61?p C i NB |B S 1?l 0 t NB } S 1? N N `2 ~ 01?p G e `2  01?  `2  01? 8   c BCDE(F01? &D&cLyk@- n    c B.CDE$F,1? ''F-l----@& T 2  ENGHZJQ1? `T`T`T`T`T`T`T`T`T`T i   c BC/DEF1?  &. @l a `2  01? Z x `2  01? > > d `2  01?9 ^   c BC DE(F01? rT&=D&c @ [ f f 2  BTENG IiQ1?  TT`T TT`Ti`TT`Ti`T# `2  01?u  l2  <G1? ; 2   BTENGHa,ZIAJQ1? T`TT`TT`TT`TAT`TA G 1 2  E$GH@ I`TQ1? `TS`TS`T`T`TS`T`T  2  BTENG^ IUQ1? TT`TTT`TU`TT`TU`TP P TB  c $1?S j 2   BTENGH9ZIUJQ1? T`TT`TT`TT`TUT`TU/ / 2  E$GH@ I`TQ1? `TS`TS`T`T`TS`T`T  2  BTENG\ H-gIQ1? 'TTS'TTS`TTS`T !%    c B}CPDE$F,1? OW9|AEwO@h 2  BWUENG`X IQ1? WUWU`TWUWU`T`TWU`T`T "C 2  ENGHZJQ1? `T`T`T`T`T`T`T`T`T`T '  2  ENGHZJQ1? `T`T`T`T`T`T`T`T`T`T : % 2  ENGHgQ1? `T`TS`T`TS`T`TS`Th { P `2  01?F  d ;   c BCDE(F01? .L=k[z@X !   NB  S 1?Q  w 8   c BlCkDEF 1?kk65bj @ ! R   c BlCkDE F(1?kT=&&5L[j@ ! R f2  61?B  ` , `2  01?  . 2  BKTENGHteQ1? `TKTP`TKTP`TKTP`T  ! 2   BTCTENG(gH )ZI=JdQ1? ݫTTTTݫTTTT=dTT=dh !    BJCDEFd @1?I @*T    BCDEFfd @1? @T:    B.CDEFd @8c?-E @*+ TB  c $1?z TB  c $1?5 5   s B.CmDEF1?-ll @*a+ ZB  s *1?w8 ZB B s *1? 2   s BClDEF1?Kk @S" &`ZB  s *1?d&ZB B s *1?,&ZB  s *1?bZB  s *1?,_hZB  s *1?% ZB  s *1?f_# ZB  s *1?!W ZB  s *1? I l2  <1?Wtl2  <1?Dl2  <1? 9 l2  <1?7 ] l2  <1?' A l2  <1?% A T   # V  l2  <p? r f2  6p? ZB  s *8c?w  4   f'"UU1? VD ! Simplicial map$ Z'"   ##55FFT  !  ż# w < 2 Ƽ  BaENGHbI JQ1? @a`TamS@a`TamS amS r ! TB Ǽ c $1? r l2 ȼ <1? ,l2 ɼ <1?<l2 ʼ <1?QOxqZB ˼ s *1? l2 ̼ <1?l2 ͼ <1?7])ZB μB s *1?K"ZB ϼ s *1?plZB м s *1?hhl2 Ѽ <1? 3l2 Ҽ <1?"l2 Ӽ <1?l2 Լ <1?`9ZB ռ s *1?( ZB ּ s *1?*ZB ׼ s *1?. M ZB ؼ s *1?. = l2 ټ <1?%} A l2 ڼ <1?! =' 2C8   9 T  0  ¼# WKj2 ü  BYaC"XENG`reHbI JQ1? bYa"XDa!WbYa"XDa!W Da!W  0 TB ļ c $1?  T   ޼#  ) l2 ߼ <fp?A f2  6p? ZB  s *8c?  &   f4"UU1?   y task spec"  Z f  ##55FF   f(;"UU1?9 ~Input complex" Zf  ##55FF   fD"UU1?t } output complex" Zf  ##55FF   BJCDEFd @1?I @h   BCDEFfd @1? @bo    B.CDEFd @8c?-E @fa TB  c $1?VHdTTB  c $1?k bk h  s B.CmDEF1?-ll @.a jZB  s *1?ZB B s *1?G l2  <1?S#q@l2  <1?Y|l2  <1?V _n |l2  <1?}l2  <1?\ u l2  <1?@  s BCDE<FD1?&=Lc.r=Lcr @=l2  <1? +f2  61?P Eq gl2  <1?M?u2   f @1? 2   fL @1?+    s BCDE,F41? T[.?|I.@ y3  s B7C*DE$F,1? '&Lr 6)@4e  s BCDEF$1?r&TD=c@ y  s B>CDE$F,1? ==&=c@/7  s B)CDE$F,1? ((j5=j@}TB B c $1?]j  s B'C_DE$F,1? D&r&&&0^@u  s BCTDE F(1?5&[5<DKS@8   s BqCDEF1?!!?|DiBYh4pqpaYB Zt4W[Z&q-=rxCD@g Pn   nB/CDE\Fdd @1?/Z'.}&(0ZGnO//}vV>//0@ If2  61?Lmf2  61?ef2  61?9f2  61?f2  61?uf2  61?"f2   61?. O f2   61?%f2   61? af2   61?S8uf2   61?1gf2  61?l2  <1?;`2  01?Nl2  <1?msTB B c $1?jbTB  c $1?||f2  61?mvf2  61?yf2  61?AZ  s BCDE(F01? &D&cLyk@Q  s B.CDE$F,1? ''F-l----@Ks2  ENGHZJQ1? `T`T`T`T`T`T`T`T`T`T  s BC/DEF1?  &. @jf2  61?f2  61?] M~ mf2  61?[|  s BC DE(F01? rT&=D&c @e2  BTENG IiQ1?  TT`T TT`Ti`TT`Ti`T f2   61?5Or2 ! BG1?OI2 "  BTENGHa,ZIAJQ1? T`TT`TT`TT`TAT`TA&I2 # E$GH@ I`TQ1? `TS`TS`T`T`TS`T`T9  2 $ BTENG^ IUQ1? TT`TTT`TU`TT`TU`T CpZB % s *1?r2 &  BTENGH9ZIUJQ1? T`TT`TT`TT`TUT`TU4d2 ' E$GH@ I`TQ1? `TS`TS`T`T`TS`T`T@2 (  BTENG\ H-gIQ1? 'TTS'TTS`TTS`T?7 ) s B}CPDE$F,1? OW9|AEwO@ 2 * BWUENG`X IQ1? WUWU`TWUWU`T`TWU`T`TQ2 + ENGHZJQ1? `T`T`T`T`T`T`T`T`T`T9{2 , ENGHZJQ1? `T`T`T`T`T`T`T`T`T`TI2 - ENGHgQ1? `T`TS`T`TS`T`TS`Tg~f2 . 61? L m / s BCDE(F01? .L=k[z@ V70TB 0 c $1? 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