Hierarchical Reinforcement Learning with the MAXQ Value Function

!"#%$&'()(+*-,/.10 2345
674
8(9;:&
<=9>('(9
?A@BDCFEHGJILKNM5OQPSRTRUPWVTOYXZ@
M5PS[NE\V]RUC^PW_`R7B\acbdBDCe[fRUPSVghXWOYPS_iXWP
j VkPSlDBD_mghRUEHRUPon_OQpDPSVkGTOQRq
bdBDVTpHE\rQrYOYGSs jutwv`xHyDyz
{h|h}h~i€k‚HƒN€„{d]…D}D†
‡\firQq‰ˆs zŠv\vD‹
JŒ ŽT]Z‘„’
“•”—–™˜š—›œš\]žš„ž'T˜Ÿ] W¡'˜› „T¢£›œš„š„ž'¤S›S¥¦”d¡Ÿ¤”„–™]ž¦›œž¦¥¦”„–§¥]›¨„žŸT–© „ª«¤ž¦¥]¬­T S¡¨©k›œž' „–© —®¯—›˜ŸT°­¤S A¡Ÿ”—±³²µ´¶
°„T¥¤S¬­šH¤W˜·–©¡Ÿ–™¤ c¤œªW¡Ÿ”—8¸Z›œ¨™¹„iª«¹„ —¥¡Ÿ–™¤ ºi“•”„8±³²µ´¶^°„T¥¤S¬­šH¤W˜·–©¡Ÿ–™¤ ”—›˜¯H¤¡Ÿ”›•š„ž'¤Š¥T°„¹„ž'›¨œ˜·T¬A›œ W¡Ÿ–§¥]˜¼»
›S˜h›½˜Ÿ¹„¯„ž'¤¹„¡Ÿ–™ „N”„–™]ž¦›œž¦¥¦”W¾S»£›œ —°›½°Šk¥¨§›œž¦›Z¡'–©¸S8˜·T¬­› W¡Ÿ–§¥]˜¼»¿›S˜`›žŸTš„ž'T˜Ÿ] W¡'›œ¡Ÿ–™¤ ¤œªŠ¡Ÿ”—8¸Z›œ¨™¹„Nª«¹„ —¥¡Ÿ–™¤ ¤ªh›9”„–™]ž¦›œž¦¥¦”„–§¥]›œ¨Šš\¤¨™–§¥¾ºN±³²µ´¶¹„ „–©À—T˜›œ —°­]Á¡Ÿ] H°„˜Ãš—žŸT¸W–™¤¹H˜N¢¤SžŸÄ¤ A”„–™]ž¦›œž¦¥¦”„–§¥]›¨„žŸT–© „ª«¤ž¦¥]¬­T S¡
¨™T›žŸ —–© „®^¯W¾¿Å–™ „®”hƽÇ9›œ]¨™¯„¨™–© —®—Æ›œ —°È›U¾W›œ ¿›œ —°¿Éµ–© W¡'¤ º3ʤS —°Š–©¡Ÿ–™¤ —˜7¹„ —°ŠTž¢”„–§¥¦”‰¡Ÿ”—=±³²µ´¶
°„T¥¤S¬­šH¤W˜·–©¡Ÿ–™¤ 7¥T›œ ­žŸTš„ž'T˜Ÿ] W¡N¡Ÿ”„µ¤šŠ¡'–©¬A›œ¨—¸U›¨©¹—ª«¹„ —¥¡Ÿ–™¤ 7›œž'½°ŠTžŸ–™¸k°`º“•”—½š—›œš\]ž°ŠÀ— —T˜Ã›c”—–©Tž'›ž·Ë
¥¦”—–™¥T›œ¨h¶Ì¨™T›œž' „–™ „®­›œ¨™®¤ž'–©¡Ÿ”„¬5ÆÂš„ž'¤Z¸T˜Ã–Í¡¦˜¥¤S W¸S]ž'®T —¥SÆW› —°J˜Ÿ”„¤Z¢½˜ÁŠš\]ž'–©¬­] W¡¦›œ¨™¨©¾d¡Ÿ”H›Z¡–Í¡½¥T›œ ¨©k›œž' ¬¹—¥¦”Ϊϛ˜·¡ŸTž¡'”—›œ "¤Sž'°Š–™ —›žŸ¾£Ð·ÑH›Z¡¦Òo¶Ó¨©k›œž' „–© —®—º5“•”„T˜Ÿ7ž'T˜Ÿ¹„¨Í¡¦˜Ô› —°ÎÁŠš\]ž'–©¬­] W¡¦˜›œž'7Á¡'] —°Šk°Õ¡Ÿ¤
˜Ÿ¹„š„š\¤žŸ¡N˜¼¡¦›Z¡Ÿ•›œ¯H˜¼¡'ž'›S¥¡Ÿ–™¤ › —° —¤ ŠË֔„–™]ž¦›œž¦¥¦”„–™¥T›œ¨œ]ÁŠT¥¹„¡Ÿ–™¤ ºi“•”„š—›œš\]ži¥¤S —¥¨™¹—°Šk˜(¢–©¡Ÿ”d›°„–™˜'¥¹—˜'˜Ÿ–©¤S ¤ªi°„T˜Ÿ–©®S J¡'ž'›S°Š]¤×D˜•–™ 5”„–©Tž'›ž'¥¦”—–™¥T›œ¨\žŸT–© Šª«¤Sž'¥]]¬­] W¡¨™T›œž' „–™ „®—º
Ø
!#"$&%'(!)'*,+.-./'01*2(3465.$78$6)'9):%'(;";"<!)'7="5.2"$9)'>(@?A213B(@"!CD+E"D()'78$9"3B
<"<!/GFH"$6):%'(;"I'01!?'(J"$K(;?L23B(;"!FE78$L<(I)'1M=NA(7!O"<(P)'Q)'7="2GFR)'M(7=(?'(I)S()'*UT
?:)'*O(P!5E):SVW'.X7!OE)B()'*2TY?Z)'*O([7=<"8\8C^]R$#%')'*_'`a"$):%'(@"#E"<J7$;<&)'7="2RP'8!H"<
3)ZN32b!c"$dO3e'`,"$(&!5E):(CRf'"$d7!)'&5.$!"$):%'(@"7!)'>/g!7=(2?'c"$c(;"2&"):"<d'`
"$d(@?A213B(@")'K5$!c"$d)'7="21):<"<A7$)'"7hViYCj'C2F"$&<"):"<(O1*2"%B`W34)'6)'7="29#)
/'0k):01*"7l`iO7="B'`"$./1!?LOm<"):"<)'"$7$<(I)'7="k\8F;"$.(O1*2"%D<(nLO(;")'*1(7!X2
/'0k*2(3oEpAq5.K)'.)Br>):p'q?Ks#(7!2[tm'0k*2(3uVir>s#t.v1tlO"<!3)'F (w'wZx \8C
y (7!)'O<Dr>):p'q?Ps#(7!2htm'01*2(3l/q?A#)c?'!z%'(!)'*g3BA(*k'`_<(nLO(@"X)'*k(7!X2AT{3):pA%
O!GO7=!")';"{z'F:"$!z$)q?'l/|?LX("$m`WO):"`}'3&O78$D(7=(;"_5'pD/'0k):01*"7_/k*)'%
)'P*2():1%C~f'3)'**z'FA)d7=!"<=TY"3B|ZX7=!"<=T{)'7="2[r>stME</g(7!U€1(J0;zJ)&€k12"<<!"E'`_"):"<(F
)d€k"<!"R'`)'7="R‚&F1)/'01):0k*2"ƒzB"<)'2"[<"<01O"2I„JV†…(‡{ˆ …@‰ŠA\^5$7$P%?'(H"$D/'0k):01*2"ƒz
"$):"D… ‡_‹ Œ5.X**0g"$<"):"<(O*"%B`W3/g!`W'3X%)'7="29Š ‹ ‚Ž><"):"<B… ‹ Fg)'6)J!5E):
`iO7="2V†…q‡Yˆ …‰Š\5.$17$‘%?'(c"$I!5E):"$):"&5.**~0g(7=(2?'(S5$()'7="2’Š ‹ ‚oD/g!`W'3B(
“<"):"<K… ‹ ”)'’(O*"dX’"):"<K… ‡#‹ CS-•’"$d/1):/g!(Fm5P5.**^)'O3h"$1):"B"$[r6st–"<‘0g
<*2?'(h^)—=˜Y™(š=›Lœ:—˜iž†šD—8›L™:Ÿ=˜{ —˜¡Aœ'˜W›c¡kŸ•™;¢=£ =¤Cl-•JO17$I/'01*2(3J!F@)'**1!5):8^):!%):"2?''b!!F)'
"$!&H)B"<!3X)'*):01<'0k%B<"):"<'CE+¥):0k<'01%"):"<&R)J<"):"<c`W3o5.$17$[)'**)'7=".*2()'¥01)'7p
"<9"$[)'3BP<"):"<[5.2"$¦/'0k):01*2"ƒz
)'1¦52"$Sb!!>!5):GC y (7!)'O¥A€k"<=T{$'2b!¦7=O;"<(
(`}'7=(3(@"#*2():%/'01*(3R7!)'>0g&7=;?'!"<(>"<h<"<A7$1)'<"7$'"<("•TY/1):"$K/1'01*2(3."$O%$>)
<")'1):9"<)'<`W'3):"MV y !"<!pZ)'d§o]R"2pA*!F (w'w'¨ \8F"$O*>0g/201*2d"<[=NA"<()'**~'`"$
(O1*2"HP"$E/k):/!."<"$A€2"<=T{$'b!h17=O;"<(¥7!)'<'C
-•>/'01):01X*<"7/1*)'X%Fk2"D)'O3B(>"$):""$B):%'(@"cpL|5."$B<"):"<B"<)'12"29"<201O"
„©)'ª"$9!5E):ª`WO17="2«C–]R$>%')'*I"<M`W!h)’/g*7=zM¬®­®¯ ‚°5.$17$«17!):"<(!FR`W'
()'7$¦<"):"<h… ‹ ª5$):")'7="2Š ‹ ‚±$O*X60gB/g!`W'3B(C]R$B'/1"3)'*~/g*7=zK¬G²3J)ZNA3J2b!("$
=NA/(7="<(h7!O3dO*):"2?'.!5):8B(7=(2?'(I0;z"$):%'(@"!Cl].$!.):3)'@z)'*%''2"$3l)(?:)'*):01*2H`W'm€%
'/"3J)'*/g*7!2(_7!*O1%"$zL)'3J7,/'%'8)'33%R)'*%''2"$3_pLq5.d)'_?:)'*O2"<!):"2&)'&/*X7=z
2"<!):"’Vit^O"<!3)'F (w'wZx \#)'>*():/'%')'3J3%J)'*%''2"$3(C.rK"#'`,"$(<d)'*%''2"$3#;?'*2?'
7=3B/1O"%)D`iO7="2pA|5.)'m"$³Zœ:£µ´1 a¶8´A·kšq˜iži™'·Cm]R$H?:)'*OH`iO7="2J¸IV†…:\(<"3):"<(^"$R=NA/g(7="<(
7!O3dO*):"2?'&!5E):¥'`m<"):"%IX¥<"):"<B…)'¥`W**2|5.%B"$&'/"3)'*_/g*7=z'C y (**3)'SV (w'¹º \#$q5(
"$):"a?Z)'*Om`WO17="2d¸ ² '`"$^'/"3)'*/*X7=z#XG"$OnLO,<*O""<."$^<:T{7!)'**( y (**3)'c(n;O1):"2­
Ø
Ø
Ø
Ø
Ø
¸ ² V†…Z\m»¼3)ZN
„IV†… ‡ ˆ …‰Š\= V†… ‡ ˆ …@‰Š\GÃM¸ ² V†… ‡ \{ÄYÅ
½¿¾{ÀYÁ
]R$D'/1"3)'*/g*7=zh7$L<(."$c)'7="2K5.$78$[)'7$2!?'("$c3J)ZNA3dO3ÆK"$D2%$;"•T{$)'¥'`~"$
(nLO):"2C
]R$^?:)'*O2"<!):"2d)'*2%''2"$13Ç<"):"~5.2"$c)'12")'*'%O(a`W'l¸«)'d)://1*(G"$ y (**3)'&(n;O1):"2
)'.)'KO/):"<`}'3dO*)BX[)'**<"):"<(…­
¸IV†…:\­È»«3)ZN
„JV†… ‡ ˆ …@‰Š\= V†… ‡ ˆ …@‰ŠA\GÃɸJV†… ‡ \{ÄYÅ
½¿¾{À{Á
-•Q)S€k2"<>;O13c0g!h'`c<"<!/1!F."$>7='(</g%/g*7=zÊVi†Cj'C2F."$6/g*7=z“"$):"[78$;(P"$>)'7="
3)ZN32b(X%&"$2%$;"•T{$)'[X'`"$1E`}'83&O*X)@\^5.**g7=;?'!%'c"<I"$D'/"3)'*g/g*X7=z'C
-•9(`W'7=(3B(;"D*2():X%Fg"$):%'(@"&2")'*X*2z[2%')';"D'`^„o)'‘va"$B):%'(@"c*2zKpA|5.c
)'[‚&Cm]R$%')'*g'`"$D):%'(@".X"**k"<*2():P)'P'/"3)'*/g*7=z'FL01O"H|5¼"$D):%'(@"3&O"E*():h"$):"
/g*7=z&0;z*XH@"<!8)'7="2m52"$B"$R(;?A23B(;"m):"$!m"$)'0LzB7=3B/1O"X%D5.2"$)%2?'(J3BA(*'`
"$(;?L23B(;"!C~]R5^%'(!)'*g)://1)'7$(E"<B(`W'7=(3B(;"*2():8%d):#3L(*2TY01)'<(P3B!"$AE)'
3BA(*UTY`W!3B!"$A!C,rKA(*UTY0k)'<(P3B!"$AE!?'(*2'/[<"):"<"X7!)'*k("3):"<(E'`a„Ë)'PÇ0;zI@"<!8)'7="%
Ì
5.2"$"$R(;?L23B(;"l)'JO1<R"$<.(<"3):"<(ViJ3BR`W'3–'`zA)'37E/1'%')'33X%;\"<&7=3/1O"<
"$R'/1"3)'*/g*7=z'C]H5/g'/1O*):m3BA(*UTY0k)'<(B3B!"$Am):/2'82"2b!(<5!!/1%JVir6;'#§¼+E"<p'(<GF
+ ~]RstRv y ):"<F y )'"<p''FH
§ A%$GF (w'w'¹ \8C
(w'w \)'Œ)'1):/"2?'[()'*UTY"X3B[zL1)'37I/'%')'33J%ŒVi
r6L(*l`}!I3B!"$AD/g!`W'3u*B<"<A7$1)'<"7h)://qNA3J):"2D'`RzA)'3X7/'%')'33J%["<9*2():
"$/g*X7=zJ2(7="*zJ5.2"$O"E€1<"H7="<O7="%)J3BL(*g'`"$D(@?A23(@"!C
]R$0g(<"•TYpLq5.3L(*2TY`}!3!"$L¦
*2():X%V ):"<pL1!F (w 'w \8C[-•<"<()'S'`H7=<"<O7="X%K)
?:)'*OD`iO7="2[)'P?Z)'*XOD2"<!):"2
F Ž*2():1%B7=<"<O17="R)'¥)'7="2ATY?:)'*O'F1'Ë`iO7="­
Ø
Ø
Ø
V†…@‰ŠA\l»
¾YÀ Á
„JV†… ‡ ˆ …@‰ŠA\= V†… ‡ ˆ …‰Š\GÓ3J)ZÁ N JV†… ‡ ‰Š ‡ \{Ä
½
]R$&?Z)'*XOJV†…‰Š\RXR"$d=NL/g(7="<(97!O3dO*):"2?'&!5E):K"$1):"5.X**0gc(7=(?'(60Lz[/g!`W'3%)'7=">Š
’<"):"<K…[)'¦"$(’`W**2q5%K"$h'/1"3)'*^/*X7=z‘"$!():`W"<!(CsO%K(`}'7=(3(@"*2():X%F~"$
`W'3&O1*)7!)''"^0gRO<("<dO/):"<
FL0g(7!)'O<.„Ç)'J ):'"lpL|5."<&"$):%'(;"!
C q5!?'!(F2`g"$
):%'(;"7!)'9;"<!)'7="#5.2"$6"$c(;?A23B(;"!Fg2"7!)'>/g!`W'3)'7="29Š¥XK<"):"<…@Fk'01!?'&"$&(O*2"%
<"):"<…(‡WF)'¥OD"$E"<JO/):"<
JV†…‰Š\E)'7!7='1%B"<"$c`}'3dO*)A­
Ø
|V†…‰Š\­È»ŽV ƒ\V†…‰Š\Gà  V†… ‡ ˆ …@‰ŠA\Ì3)ZÁ N V†… ‡ ‰Š ‡ \{Ä
½
5.$!d"$O017=/"#('"<(D"3"<!/)' EX"$B<:T{7!)'**2(‘*2():8%h):"<B/1):)'3!"<!):"D"3B
C y !"<!pZ)')'[]R"2pA*V (w'w'¨ \R/|?'D"$):"R2`a"$):%'(@".`}**q5.H)'
•=NA/1*2'8):"2P/g*7=!z c"$):"."<2(
!?'!z[)'7="26P!?'!z[<"):"<&XA€k2"<(*zB'`}"<(6)'¥2`
Ø
*3
"
"$#&% ¾ '
»)(
)'
*3
*
*$#+% ¾ '
,
,+
- .
(
Ø
"$(/m7=@?'!%'(."<"$'/"3)'*)'7="ATY?Z)'*XO`iO7="20c².52"$[/'0k):01*2"ƒz C
r6<"E(`W'7=(3B(;"^*():%&)'*2%''2"$3Jl):(2%(J"<d5^'pB5.2"$h)://qNA3J):"2J78$(3B(EO78$
)'(O)'*!"ƒ5^'pA"$):"7!)'“0gI)://1*X2("<‘)://(N3):"<h"$I?:)'*Oh`iO7="21
C .|5^!?'!qF^?'!z¦*2"<"*2
#pAq5.>"$!'!"7!)'**2zK):0gO"#"$B7=;?'!%'(7=/'/!"2(.'3
` 2 *2():%J5.$(>7=3&01(>5.2"$6?:)'*O
`iO7="2[)://1(N3):"2GC
s#zA)'376/'%')'3J3%“)'*2%''2"$13J`W'P/'01):0k*<"X7P/1*)'X%S!O 4g!P`}3 "ƒ5^5
<7!O<(6 A­’]R$
EO8<D'`,s#X3B(21)'*2"{!
z B)'["$0
EO'`,rKA(*7% ¥V y !"<!pZ)'§–].2"2pA*(F (w'w'¨ \8CR]R$7!O<
'`R3(2)'*X2"{z>"$):"d"$h)'*2%''2"$3JD(nLO2"X3BJ"$):"d/g*2zA3X)'*,"$h;O13c0g!c'`R"):"<(
ˆjRˆÈCÊr>)'@zŒ/1)'7="7!)'*R/1'01*2(36Vi)'É)'*3<"J)'**+.-/'0k*2(38\d;?'*2?'6?'!zM*):%'K"):"<></k)'7=(J`W'
5.$X7$M7=3B/k*2!"<6zA)'3X7P/'%')'33J%I`W()'201*2'CQ]R$K7!O8<K'`D3BA(*%B"$):"hŒ'!I"<
)://1*zzL)'3J7H/'%'8)'33%d3B!"$A!FL2"^l€1"l(7=():z"<&=NA/(^„JV†…q‡{ˆ …‰Š\m=NL/1*X7!2"*2z[VW"{zL/17!)'**z
)'D)h/'01):0k*2"ƒz["<)'2"63):"<8UN\8C&]R$##'`}"<(9?'!z69 8h7!O*2"!F!?'(9`}'D(;?L3B(;"#5.$!B2"
/g20k*2"<I7=<"<O7=")B/A7=(O)'*g3BA(*g'#3dO*):"<'(C
.(7=(@"(<():87$9¥/1'01):01*X<"7#/1*)'1%B$)'`}O1K<!?'!8)'*a"<(78$nLO(`}'#(7!):/1X%J`W3e"$(<
7!O(!Cf'l=N)'3B/1*2'F y O"X*2!,)'s#():(6V (w'w'¨ \,$|5É$|5É)</1):8<E0(*X2!`1!"ƒ5^'p!/1(<(;"):"2
'`,„JV†…q‡†ˆ …@‰ŠA\R7!)'¥0g=NA/1*22"<(¥"<'0")'63;) :•'#</g!(O/1HXP?Z)'*Oc2"<!):"2GC^s#()'K)'/
<_Vis#()'9§
<_XF (w'w'¹ \m$1)(?'"O2(5E)(zAl'`(7=3/%)*):%'.r6st’;"<c)<!"m'`3J)'**2!(F;"<!)'7="%cr6s
t = !C
+#'"$!d<*O"2"<¥"$"ƒ5^97!O<(cD"<6):/1/1*2zK(`W'7=(3B(;"c*2():X%¥3B!"$AD"<K)K/A7=(OAT
)'*^3BL(*,'&X3&O*X):"<'(CI- `HO78$S)K3&O1*):"<'&7!)'S0J()'*2z‘7=<"<8O7="<(F"$(S"$1<*2?'(&"$J7!O<
'`3L(*X%C‘]R$P7!O<h'`13B()'*2"ƒz97!)'“0gh/1):")'*X*2z<*2?'(’0g(7!)'O<P'`."ƒ5^‘`i)'7="<'!C¦f<"!F
0Lz6(3B/k*2qzA%P?:)'*OB`iO7="2‘)://1(N3):"2‘3!"$L1!FG"$?:)'*O`iO7="2‘7!)'S0gd!/(<(;"<(?'!z
7=3B/1)'7="*z'F1)'P?:)'*O(H*2():(P`}'R<3<"):"<(.7!)'[0g%'(!)'*X2b!(P"<%2?')'7!7!O8):"<?Z)'*XO(E`}'H'"$!(F
3J*):~<"):"<(!>
C L(7=GF'(`W'7=(3B(;",*2():%#)'*2%''82"$3~<"<p'E)"<8)'!? 4h0g!"ƒ5^!(B=NA/1*2'8):"2[Vi†Cj'C2F
Ø
Ø
Ø
"<zA%d?Z):82OE)'7="2HI?:):2OE<"):"<(H"<!#5$):"H$)://g(8\m)'h=NL/1*2"):"2‘Vi†Cj'C2F/g!`W'3%&"$
)'7="2"$):"c7!O(@"*2z6)://g():0g(<"8\8CB]R$(O*"X"$):"D"$(<J)'*%''2"$3#"<("<K2%'"$B/k):"
'`."$¥<"):"<K</1)'7=["$1):"):¥2(*2!?:)';"d"<‘'/1"3)'*DVW'J():*z@TY'/"X3)'*}\/*X7!2(!C]R$17=3c01X):"2
'`m<"<A7$)'"7c)://1(N3):"26)'¥?:)'*Oc`WO17="2K)://1(N3):"2K)'**2q5R(`}'7=(3(@"#*2():%)'*2%':T
2"$13"<6)q?'"$J=NL/g(;")'*,)'3BO;"c'`E=NA/g!2(7=)'S7=3B/1O"):"2S"$):"5O*9'"$!5.<0g
(nLO2(P"<I7=3B/1*2!"<(*2zP<*2?'D"$cr6stC
s#(</1"<R"$)'?Z)'7=(^B/'01):01*X<"7m/k*)'%)'1B(`W'7=(3B(;"l*2():8%(<():87$FL"$(<#3B!"$AT
Al):<"*X*`W()'201*z&*2|5ªXJ3)';z)://1*X7!):"2!C R(<():87$PJ7!*X)'7!)'*1/1*)'X%$)'$q5"$):"E$UT
!):78$7!)'*3B!"$AO7$h)'E$2!):87$7!)'*1")'<pJ!"ƒ5^'pLDVYEO2§Ç]a):"<'F (w'w \8F3J)'7=d)'7="DVifp'(!F
#):"!FE§
*<F (wº:Ì v '`ƒF (w '¹ \8FE)'1Œ<"):"<K):01<"<8)'7="2M3B!"$A¥V A)'7=!8'"†F (wº|x v '01*2A7pF
(w'w \c7!)'S/q?A=NA/g(@"X)'*,(O7="‘"$I7=3B/1O"):"2)'*l7="c'`E€k1%P%';A/k*)'!C[AT
<(nLO(;"*2z'F#3J)'@zQ(<():78$!V A%$F (w'w'Ì v <_F (w'w v ):(*01*%F (w'w v&s)(z)'”§ ;"<F (w'w v
#)'O<pL(7$;"!Far6(O*2()'OF y O"*2!(F):(*01*%FG§Æs#()'F (w'w vt~):&§ O<(*X*†F (w'w v O"<"<FGtm(7!O/GF
§ A%$GF (w'w \d$)q?'[=NL/g!X3B(@"<(¦5.2"$’9 4g!(;"&3B!"$Ld'`.$12!):78$7!)'*^(`W'7=(3B(;"d*():%
)'[$!):7$17!)'*/'01):01X*<"7R/1*)'X%C q5!?'!(F`WO*X*2z@T{):"<`i)'7="<'zI$!):7$17!)'*g3B!"$A[$)'Ez'!"
0g!(¥!?'(*2'/g(C
]R$J()'*m$12!):78$7!)'*,(X`}'87=(3B(@"&*2():%P3!"$LS5^O1*‘/|?LXB"$J0g(=€k"D'`H$2!):78$@z
VW`i)'<"<!I*2():8%9)'ŒO01")'<p’$):%9)'Œ(O<|\&5.$X*2h3J)'@")'X%>"$[0g(=€1"'`#(`W'7=(3B(;"
*2():1%>VW'/1"3)'*2"ƒz'F*&*2():1%J`W34"$(;?L23B(;"!Fg)'1>)'O"<3cz\8
C A/(7!2€k7!)'**2z'Fg)'9()'*
3B!"$A[`W'.$2!8):7$X7!)'*g(`W'7=(3B(;"R*2():X%B$O*P):"X<`}zh"$`W**2|5.%d(nLO2(3(@"!­
Ø
Ø
Ø
ØC
Ø
Ø DØ
Ø
Ø
Ø
Ø
Ø
Ø
Ø
O//g'"G`W'a"<(3/'8)'*@):01<"<8)'7="2C~]R$^$12!):78$@z$O*cO//g'"GO01")'<pLa"$):"=NL(7!O"<E!?'!)'*
/132"?'#)'7="21R0g!`}'D"<!3):"X%C
Ì C O//g'"#`}'d<"):"<J):01<"<8)'7="2CJ]R$3B!"$A‘$O*‘3J):p'J2"D/g20k*2&`W'c12?AO)'*aO0")'<pA
"<I2%'2(*2!?:)'@"R)'</g(7=".'`a"$&<"):"<c</1)'7='C
C $):%;|(O<'`O0")'<pA!C]R$3!"$L‘$O*93):p'h2"/201*2d"<K*2():8‘)'‘'/"3)'*m/*X7=z
`W'.)O0")'<pP)'["$(¥=T{O<D"$):"*2():([/g*X7=z`}'94g!(@"H/k):(@".")'<pL(C
8 7!2(;"R*2():X%C
#0;?A2O*2z)B$2!):87$7!)'*g3B!"$LP`W'H(`W'7=(3B(;"H*2():X%&5**k0DO(*2(
x C I
`2"R;(R'"H/q?A)B/!(1O/[q?'!AT{$!):7$17!)'*g3B!"$A!C
¹ C/"3)'*"{z'Cm]R$3!"$L[$O*[*2():8P'/"3)'*'():<TY'/"3J)'*/g*7!2(!C
¨ C
AT{$!):7$17!)'*=NL(7!O"I'`O0")'pL!C #2!):78$7!)'*")'<ph(7=3B/g2"21l7!)'[3):p'D2"E3B/g•T
201*2"<K7=3B/kO"<"$'/1"3)'*~/g*7=z'C):(*01*%‘V):(*20k*%F (w'w \$)'c$q5.‘"$1):"c3B/q?'(
/g*X7!2(E7!)'¥0gD'0")'(¥0;zPAT{$!):7$17!)'*g=NL(7!O"2['`,)B*2():(K$2!):78$@z'C
Ø
º CD*(`}'7=(3(@"l*():%C]R$R3B!"$L$O*B)q?'"$7!O<H'`3BA(*%0LzB'",(nLO2<T
X%I"$d/1'%')'33B!"<[/q?A&7=3/1*2!"<'F_7='(7="3L(*X'`^"$(@?A213B(@"BVi†Cj'C2FG„JV†…q‡Yˆ …‰Š\
)'1‘V†…(‡Yˆ …‰Š\<\8C]R$B3B!"$A9$O*>0g):01*2"<¥*2():>*X&5'pA%h5.2"$>"$B(;"2")'<p9)'
'/1/(["<(nLO2X%&"<8)'%&€1"H["$c12?AO)'*kO0")'<pA!C
C.+01*X2"{zB"<5'p5.2"$I?:)'*O#`WO17="2I):/1/(N3):"J"<(78$nLO(!Cm]R$#3B!"$LP$O1*J)q?'P"$
!7 O<'`aX3B(21)'*2"{z0LzJ'"E(nLO2X%&)d"):01O*X):!/(<(;"):"2h'`_"$?:)'*O`iO7="2I'H"$
/g*X7=z'C
w C<_():%I'`^"$B$2!):78$@z62"<(*2` C]R$B3B!"$A9$O*X6'"(nLO2c"$/'%'8)'33B!D"<[</g(7!2`Wz
"$c$2!):87$7!)'*g")'pP(7=3B/g2"2[[)'?Z)'7=c01O"H$O*XP0D):01*D"<J*2():¥"R)'O"<3BO*2z'C
x
-•M"$X/1):/g!(FE5K/((@"h)S!5 )://)'78$É"<’$2!):87$7!)'*.(`W'7=(3B(;"J*2():8%FH7!)'**2(É"$
r>+ 2 3B!"$LF"$):"P5^‘7!*)'X3 3!!"[)'**.'`&"$(<9(nLO2(3(@"h=N7=!/"P"$‘):01*2"ƒzŒ"<Œ*2():ª"$
$2!8):7$;zI2"(*2`_)'P"$D):01X*2"{zJ"<B5^'pI5.2"$h?:)'*O`WO17="2P)://qNAX3):"2h"<(7$1n;O(!C.V ‘0g(*2!?'
"$):"Er6+ 2 5.**A5'pB5."$?:)'*OR`iO7="2I)://qNAX3):"2F;01O"m5#$1)(?'#z'!"^"<d/g!`}'83 )';zB(<():78$
["$XRn;O(<"2CÈ\
]R$,(3)'!'`L"$,/1):/g!_g'%)'b!(c)'`W**2|5.!Caf~28<"!Fq5l;"<AO7=~"$lr>+ 2Œ?Z)'*XO,`WO17="2
(7=3B/g"2PO%)B3/1*2.=N)'3B/1*)'1P`}'3J)'*2b!D2"E/A7=(O8)'*k)'[(7!*):):"2?'<(3J)'@"7!(C ‘
/|?'J"$):"&"$Ir>+ 2e(7=3/2"27!)'S!/(<(;""$I?Z)'*XO`iO7="2‘'`H)';z$2!):87$7!)'*m/*X7=z
7=X<"<(@".5.2"$P"$D%2?'(6r6+ 2Ž$12!):78$@z'C
s#(</1"<."$.):0k*2"ƒz&"<d!/((@"m"$.?:)'*OR`iO7="2'`G)'@z$!):7$17!)'*/g*7=z'FOl%')'*m"<d*2():
)K/(7!X)'*~pL1‘'`H$2!):78$7!)'*,/g*X7=z ª)>Ÿ š(´LŸ8—8žW³: !£ ‘™8¡˜ižW¤œ:£G/g*7=z'Ch]R$J=NL"c(7="2=NA/1*2'(c"$
/'/g!"(m'`G(7!O2?'(*2z'/"X3)'*/g*7!(!Ca- "l"$(hX@"<AO7=(^
) 2 *2():1%c)'*2%''82"$3 7!)'*X*2(Ir>+ 2T
)'9/q?'(D"$):"O1!D7=!")'‘7=12"2r>+ 2T 7=@?'!%'(c"<¥"$BOXn;Od(7!O82?'(*2z¥'/1"3)'*
/g*7=zh7=<"<(;"H5.2"$P"$%2?'(¥r>+ 2 $2!8):7$;z'C
]R$R=NL"^<(7="2B"):p'(O/B"$.XOE'`<"):"<):0k<"<)'7="2)'1B/|?'(l)D"$!'(3 7$1):)'7="<!2b(%"$
7="2O1!5.$X7$S)'¦):01<"<)'7="•T{"):"<Pr>+ 2$12!):78$@z‘7!)'’<"**,!/(<(;""$I?Z)'*O`WO17="2
'`l)';zK$2!8):7$X7!)'*G/g*7=z'C A"):"<):0k<"<)'7="297!)'9*2()'6"<h)h!5”pL1¥'`l7=(12")'2%3B(;"/1'01*2(3
5.$X7$P5c7!)'**"$&$2!):78$7!)'*7=(12"R)'2%13B(@"./'01*2(3¥Cm]R$E<(7="26(7=0(E"$1H/'0k*2(3o)'
/g20k*2¥<*O""<“2"!CÇ+ (O*2"IJ/1(<(;"<(Q5$7$QO%'%'(<"‘VW01O"IL(I'"I/q?'|\J"$):"P2`D"$
):01<"<8)'7="•T{<"):"<¥r6+ 2®$2!):87$;z7!)'¦!/(<(;"c"$I(7!O2?'(*2z9'/"3)'*^/g*7=z'F"$(’r6+ 2T 5.**
<"*X*7=;?'!%'d"<"$E/g*X7=zJ5.$([):/1/1*2(h"<"$&):01<"<)'7="•T{<"):"<$2!8):7$;z'C
L(7=" ¹ X7!O<(B3B6O(B7=7=!1%S$|5®O0")'<pA):K</g(7!U€k(C L3B6O0")'<pA$)q?'
():01*2)'’O(2):01*d"<!83):"%K<"):"<(!Fl)'1S7!):/"O8%¥"$X(nLO2(&<3Bh=NL"<(12c'`R"$
r>+ 2T )'*2%''2"$13[Cm]R$D=NA"<((¥)'*%''2"$3o.7!)'**2([r>+ 2T 2C
L(7=" ¨ )'((R"$dn;O("2¥>AT{$!):7$17!)'*=NA(7!O"26'`m)J*2():(Kr>+ 2–/g*7=z[)'
/|?'(R"$):"AT{$2!8):7$X7!)'*g=NL(7!O"¥7!)'K3/q?'D"$c*2():(P/g*X7=z'C
f~)'*X*2z'FH"$>/1):/g!I(*X):"<([r6+ 2 "<“'"$![$2!):78$7!)'*(`W'7=(3B(;"P*2():%S3B!"$AP)'
):"<"<(3B/""<[(@"2`WzP"$3;) :•'D(2%6XO(#¥"$d</k)'7=d'`^$2!):78$7!)'*3B!"$A!C#]R$!):d/1):<T
"7!O*X):*2zIX@"<!("%7=(7="H0g!"{5!(6r>+ 2Ž)'["$5^'ph'`,s#()'K)'1 <_V (w'w'¹ \.¥(7=3T
/g2"21)'*3!"$L1H`}'./'01):01X*<"7./1*)'1%C~]R$(<7=(7="2RO%'%'(<".)B?:):):"2P'`,r>+ 2#T 2dF
7!)'**2(ªr>+ 2T{
s <lFH5.$78$Q3/1*2(3B(;"J)'ª*X'FH(`W'7=(3B(;"•T{*2():1%S?'!82Q'`"$‘)://)'78$
)'?'A7!):"<(K0LzPs()'6)' <aC
Ø
B#m,h!g•
Ø
‘5.X**aX@"<AO7="$r>+
2Æ3B!"$AOX%I"$3B/1*d])ZNA,tl'01*2(3°$q5.‘X>f~2%O CB+ ")ZN
$1):012"m) ¹ TY0;z;T ¹ %'85'*C,]R$!.):#`}O^</g(7!)'**z@T{(2%):"<(*2A7!):"2^"$1m5'*FL3):p'(h)'
&VW(\8F y Vi*O|\8F#"BVW!(k\8FA)'%$VW(**2q5\8Cm]R$H")ZN/'01*(3Ž~!/1<A7:C-•()'7$I!/1<A'F'"$R")ZN1"):"
&)R)'3*2z;T{7$<(&<"):"<'Cm]R$!l_)./1)'<(%'!):"l'`"$l`WO*L7!):"21^Vi7$<(d)'3*z\8F:)'
"$):"#/1)'<(%'!5.X$(H"<h0g"<)'</g'"<(6"<J&'`,"$&`}O*2A7!):"2Vi)'*<h7$<(>)'13*2z\8CH]R$
")ZN3&O1<",%'"<"$H/k)'<(%'! = l*2L7!):"KVW"$ <<O7= @\8F;/k7pdO/B"$R/1)'<(%'!(F%'c"<"$R(<"X):"2
*2A7!):"2SVW"$ <(<"):"2 @\8F1)'P/1O"H|5.["$D/k)'<(%'!R"$!'CVi]ap'!!/K"$X%HO2`W'3[FL"$D")ZN
3dO<"H/17pIO/¥)'1['/P? 4‘"$D/1)'<(%'!H!?'(K2`_$|Z$H)'*2()'zI*2A7!):"<(¥):"R"$(<"):"'
&È\&]R$
!/1L(1H5.$(P"$c/1)'<(%'!H!/g2"<(¥):"."$(<"1):"2[*2A7!):"2C
]R$!&):UN[/832"2?'&)'7="2#K"$1.3J)'­Vi)@\H`WO)(?A2%):"29)'7="21."$):"3B|?'d"$&")ZN
nLO):
(*),+.-0/@2
F 1#)435-6/;8
F 7#95:;-8FA'=<?>@:;-8FGVW0\lB
) ADC EGFG3IHc)'7="2F)'9Vi7q\^J
) AK3L-NM,)PORQc)'7="C l)'78$[)'7="2
!"<!3X<"7:C“]R$!¥J)‘!5E):Œ'` `W'()'7$ª)'7="2É)'1M)'M)'"2)'*!5E):M'`Ã Ì `W'
JØ
¹
4
R
G
3
2
1
0 Y
0
B
1
2
3
4
Ø ­l]R$]a)ZNGs#3)'
O7!7=(<`WO1**2zJ(*X2?'!%"$D/k)'<(%'!(C^]R$!H)!5):8P'` Ø 2`_"$")ZNA_):"<"<(3B/"R"<J=NL(7!O"<&"$
AK3L-NM,) OQ' ADC EGFG3,HJ)'7="**2!%)'**z'C- `l)P)(?A2%):"29)'7="295O*>7!)'O<"$")ZN"<P$12"#)I5E)'**†Fg"$
)'7="26H):TY'/GFk)'P"$!cXE*2zh"$OO)'*g!5):¥'` Ø C
‘B<!!p>)I/g*X7=zh"$1):"3J)ZNA3J2b!(."$&"<'")'*~!5E):>/g!!/k<A'C#].$c'/"X3)'*_/g*X7=z¥):"<")')'
)(?'!8):%'[!5):8S/g!<"<!/¦'` Ø C Ì
C9].$!I): ¹ /g201*<"):"<(!­ Ì'¹ nLO):(!F ¹ *2A7!):"2&`}'d"$
f%O
/1)'(%'!&Vi7=O;"%"$D`}O<"):"%*L7!):"21R)'["$D")ZN}\8F1)'1 x (<"X):"2!C
]R$1G")'<p&$)'~)X3B/1*2m$!):7$17!)'*@<"<8O7="Oc5$7$c"$!E):"{5O0TY")'<pA!­*"#!"~"$/1)'(%'!
)'¥s#(*?'!R"$/1)'<(%'!(C
^)'7$6O0")'<pP;?'*2?'(.1)(?A2%):"%B"<'`~"$`WOR*2A7!):"2R)'¥"$(
/g!`W'3%) ADC EGFG3,Hd' AK3L-NM,)PORQ)'7="2C
]R$1")'<p[**O"<):"<("$!([`W'H"<(3B/g')'*):01<"<8)'7="2F1<"):"<&):01<"<)'7="2GF)'¥O01")'<pI$1):%C
]R$H"<(3/'8)'*):01<"<)'7="J,'0L?A2O Q"$.O0")'p&'`)q?L2%):"X%"<c)c"):%'!"^*2A7!):"2Xm)D"<(3B/g')'**z
=NA"<((S)'7=""$):"c7!)'S"):p'IO/‘"< "3B"<!/1D"<67=3/1*2!"<'CJ]R$J!(‘`W'c<"):"<h):01<"<)'7="2¦
/g!$):/1#*2(#'0L?LO!Cd1!#"$O01")'<p¥'`l%'!"<"X%["$/1)'<(%'!(C+«$*2d"$O01")'<pK#0g(%
=NA(7!O"<(FR"$>(<"1):"2M'`D"$6/1)'<(%'!hJ7=3B/1*2!"<(*2zM(*2!?:)'@" «2"J7!)''"h) 4(7="[)';zŒ'`D"$
):%2?:):"2S'&/1X7pAO/S(7!2!CPt!$):/1d3B'h3B/g'")';"*2z'F_5.$(S)q?L2%):"X%K"<>)¥"):%'!"*2A7!):"2
VW(2"$!H"$O7='R(<"1):"2I*L7!):"2P'`a"$/1)'<(%'!\8F*2zB"$X(@""{z'`a"$"):%'!"R*2A7!):"2
#3B/g'")';"!C#].$&`i)'7="#"$):"D63Bd7!)'("$&")ZN~7!):zA%I"$&/k)'<(%'!D)'>6'"$!D7!)'<(D2"
H'"H2(*!?Z)';"!Cmf~)'**2z'FO0")'<pI$):%B.7=2"7!)'*g0g(7!)'O<D"$)(?A2%):"26O0")'<phR$1):(P0;z
0g'"$['`"$0
"#!""$D/1)'<(%'! )'
<s#(*?'!R"$/1)'<(%'! O0")'<pA!C
‘.$1)(?'.!?'(*'/()%'):/k$7!)'*'"):"2`W'l(7=201%."$H%')'*}ZO0%')'*<"<8O7="OE'`1"$(<H")'<pA!F
5.$X7$¦5^P7!)'**"$Pr6+ 2®%'):/1$C‘f~2%O Ì $|5.)6r>+ 2®%'8):/1$’`}'"$&/'0k*2(3[C6].$%'8):/1$
7=;")'E"{5pLX^'`_A(!­,r6)ZN[A(Vi7!):"<(h0Lz"<)'%*2(8\l)' 2 L(Vi17!):"<(h0;z|?Z)'*X8\8C
r>)ZNQA(5."$ɒ7$*X(M('"<6/832"2?'¥)'7="IXM"$>3)'v#r6)ZNªL(5.2"$Q7$X*(
!/(<(@"O0")'pL!C~-•"$Xl3/1*2H/'0k*2(3[F@"$!):#`WO^O17$hO0")'<pA!­HVi)@
\ ( 9 C ,95- >V \.Vi3q?'"$
")ZNk"<d"):%'!"R*2A7!):"2 <\8FGVW0
\ K>@-.Vi3B|?'D"<&"$/1)'<(%'! = E*2A7!):"2P)'I/17pO/I"$#/1)'(%'!\8FGVi7q\
AK3L-DVi3Bq?'"<J"$D/1)'<(%'! = R(<"X):"2[)'P/1O"Eq5.P"$D/k)'<(%'!\8Fk)'Vik
\ )I)G-DVW/g!`W'3Æ"$
q?'!8)'**")'<pJ'`G/k7pA%dO/h)'1I(*X2?'!%c"$/k)'<(%'!\8C '"7="$):"E"$
( 9 C ,95-N>.")'<ph$):(I0;z
"$
*>@-)'?
AK3L-E")'<pA!C
]R$PX33B():"<[78$*(¦'`#()'7$Qr6)ZNŒA[):/2 A(!C l)'78$ 24AP!/((@")'É)'7="2
"$):"7!)'K0g&/!`}'3(["<h)'78$2!?'B2"./1):(;" = #O0")'<pC.f'=N)'3B/1*2'Fk"$
9
*>@-RLc$)'#)I78$*
( 9 C ,95- >G)+K>@-~5.$78$&!/(<(;","$R)'7="2'`)q?L%):"%D`W3Ë"$R7!O(@",<"):"<R"<c"$H/1)'<(%'! = *2A7!):"2Cd]R$B<"X7="260!"ƒ5^!(Sr6)ZN‘A()'
2±A(D#(nLO2(9>'!"<['0")'X‘)[7!*2()'
(7=3B/g"2K'`l"$&?:)'*Od`iO7="2C l)'78$‘r6)ZN>Ld7=3B/kO"<(#"$¥š™:·1˜{ ˜HžW
· @ ¡ =
· @ =·1˜~=NA/g(7="<(
7!O3dO*):"2?'!5E):P'`/g!`}'83%2"EO0")'pkCmf'=NA)'3/1*2'
F 9
4( 9 C ,95-N>V <\^("3):"<(H"$D=NA/g(7="<(
¨
MaxRoot
QPickup
QGet
QPut
MaxGet
MaxPut
QNavigateForPut(t)
QNavigateForGet(t)
Pickup
QPutdown
Putdown
MaxNavigate(t)
QNorth(t)
QEast(t)
QSouth(t)
QWest(t)
North
East
South
West
f~2%O Ì ­l+Ër6+
2Ë%'):/k$[`}'."$c])ZNAGs#3)'X
7!O3dO*):"2?'6!5E):M'`)q?L%):"%S`W3 )'@zM<"):"<9"<¦K'`D"$6`}OJ"):%'!"[*2A7!):"2 8C ^)'7$ A7=3B/1O"<("$¥š8™:·1˜{ @˜ @ ¡ =
· @ =·1˜~=NA/g(7="<(‘7!O3dO*):"2?'!5E):>'`l/g!`W'3%I2"O01")'<pkCf'
=N)'3B/1*2'F ( 9 C ,95- >G)+K>@-(V <\7=3B/kO"<(E"$=NL/g(7="<(¥7!O13&O*X):"2?'!5):8I'`~)q?L2%):"X%"<*2L7!):" œ:
· ¥˜W›A =·¦š8™:¤¡k£ !˜ižW· ˜W› K>@-&˜Yœ:— |C ¥"$c'"$!D$)'GF ( 9 C ,9L-N>G)+ AK3L-(V \R7=3/1O"<("$&=NA/g(7="<(
7!O3dO*):"2?'!5E):>'`^1)(?A2%):"%P"<¥*2A7!):"21
dœ:
· ¥˜W›A !·“š8™:¤¡k£2 (˜iž}· ˜W› AK3L-B˜YœZ— |C y '"$‘'`^"$(< 2
A( <)'<!p 9
#( 9 C ,95- >V \$|5 3dO7$¦2"5.X**~7=<"c"<6%'!"c"<6*L7!):"2 8F~)'‘"$!z95.X**,O<B"$XD"<
$(*2/["$(3e7=3B/1O"<c"$(22”?:)'*O(!C
-•P"$("H'`a"$D/k):/!qF5^D5.X**g)(zh"$):".r>)ZN¥LŠhE"$78$*P'`~r>)ZN[L
l`a"$!H) 2
AD5.$D/1):(;"R
^)'1P5.$<78$*[.Š1C
]a¥=€kd"$B<(3)';"7!D'`^"$Br6+ 2±%'):/k$‘3B'`}'83)'**2z'F*2!"DODO
"<[('"<B"$q?'!T
)'**^r6):p'|?¦(7!X2/1'01*2(3°52"$S<"):"<(JEF~)'7="&‚&F~!5E):S`iO7="SV†… ‡ ˆ …@‰ŠA\8Fm)'1"<)'2"2
/'0k):01*2"ƒzP`WO17="26„IV†…(‡Yˆ …‰Š\8C l)'78$‘r6)ZN>L
H7='(/1."<[)h/1):""2K'`l"$B<!"DQ;"<P"{5
:•;"O01<!"!­D
 ^)'1
Cc]R$<!"&
 lX"$!"#'`œš(˜ižW³: B—˜Yœ˜{ —`W'DA
FG)'9"$B<!
" #"$
<!"E'`^˜{ !Ÿ8¤žW·kœ'˜{ —=˜Yœ'˜{ —`}'C l)'7$[L
m$)'H)<!"E'`a)'7="H‚ "$):"H2"E7!)'P=NL(7!O"<=€k(h0;zh2"
7$1*>r6)ZN9L(!+
C <G!
" 9V \('"<"$B(7=(1)'@"'`A
RXK"$Br>+ 2Æ%'):/1$
C AO//g<c()'78$
'`m"$(d(7=()';"
‹ >V •\.XR`W**2q5%I)h<"):"21):z¥/g*7=z[¬ &`W'D7$L%h2"#)'7="2!
C ‘5.**
º
('"<h"$&<!"d'`H/*X7!2(D0LzS¬~CK]R$(`W'&()'78$Œ7$*XŠ ‹ ‚ F"$I/'01):01*"{zK"<8)'2"2S`WO17="2
„ V†…(‡Yˆ …‰Š\ER5^(**=€k(F)'[5c7!)'6=€kD)r>):p'q?¥(7!2P/'01*2(3 `W'.A^)'.`}*X*2q5.(­
=€k(Kq?'!D"$d<!"#'`l<"):"<(F01O")'*X*_<"):"<( É):):01<'0k%"<!3J)'*_"):"<(
52"$Pb!!J!5):GC.VY 3J)(z[7=@")'X¥'"$!#):01<'01X%"<!3)'*"):"<(.)'R5(**†CÈ\
].$<!"R'`,)'7="2.R‚ F "$7$*X[r6)ZN¥A(E'` C
].$D/'01):0k*2"ƒz"<)'"2P`WO17="2[H„ †V …(‡{ˆ …@‰ŠA\8F1`}'.Š ‹ ‚ C
].$!5E):¦`WO17="2&¸ ½ V†…Z\8F5.$78$S"$J?:)'*O`iO7="‘`}'7$1*‘r>s#t ½ Vi)'O3X%¥2"
)'1>"#(7=()';"D):=NL( 7!O"%h"$(2#€NA(6/g*X7!2(.¬_\8Cc].$.X"$d=NA/g(7="<(9!5E):9 )'7!7!O13&O*X):"<(¥5.$X*2=NA(7!O"%J)'7="2KŠh<"):"X%[<"):"<…C
-^
" ^3B/g'")';"m"<&(3/1$)'2b!"$):"l"$1lr>st !/g(1,J"$./g*7!2(m0(X%=NA(7!O"<(h0;zB)'**1'`_2"
(7=(1)'@"(C^]aJ3):p'c"$1H!/g((7=7!*():P"$&'"):"2F5c5**5R2"< Vi¬_\8C '"<c"$):"#2`_5
5!7=!%XA€k2"<E$'b!J7=O@"<(/'01*2(3J!F'"$ Vi¬_\5O*B0g.)(3UT{r>):p'q?(7!X2
/'0k*2(3[FA0g(7!)'O<D()'7$>7=3B/g2"<)'7="K7!)'¥"):p'd)?Z):8):01*2LO3&0!H'`/X32"2?'<"<!/1."<=NL(7!O"<'C
+ ›@žY =Ÿ<œ:Ÿ•š!›@ž†šœ:£1¡A™:£µžiPš D`}')Br6+ 2Ë%'):/k$[H)<!"H'`/g*7!2(E¬K
» q
¬ :‰!Å!Å!Å(‰
¬ F`W'H()'7$6r6)ZN
A'F@"$1):"17!):"<.$|5«()'7$Pr6)ZNhL.$O*XJ7$;<2"^)'7="2!Cm+Ê$12!):78$7!)'*/g*7=zBl=NL(7!O"<(
"$)'35)qzP"$):".O01O"():=NA(7!O"<(6P'1):z/'%'8)'33%B*)'%O):%'(!Cl]R$ 8) )G-/*X7=z
7$;<(d'`^2"7$*XK)'7="2"<P/g!`W'3[Fg)(z'F *>@-8CD]R$ K>@-R/g*X7=z["$(97$L<(d'`^2"78$*
)'7="2(Fm)(z'
F A C EGF 3IH@C‘]R$(S"$?
A C EGFG3,H¥)'7="2“&=NL(7!O"<(Fl17=h2"dd)6/132"?''CK+er>)ZN¦A = /g*7=zd~=NL(7!O"<(JO;"*A"$):"lr>)ZNAH(@"<!8l)"<!3X):"%<"):"<'FA):"m5.$17$/g;"!F <7=@"<* !"O
"<J2"R/1):(;"Rr6)ZNKA'C
]R$!!`}''Fa5J7!)'¦?L2!5 "$Jr>+ 2e%'):/1$¦)'c)KO0O"XB7!)'**,%'):/1$GC <a2p'JO01O"((FGr6)ZN
A(c7!)'S0g/1):)'3B!"<!82b!(CP- "$D%'):/k$F 9
#( 9 C ,95- >"):p'(dI/1):)'3B!"<!qFFa5$7$S</g(7!U€1(
5.$X7$6'`l"$`WOD*2L7!):"BV F y F "F $\"$"):%'!"D'`l"$ 9
#( 9 C ,95- >:C+ /1):)'3B!"<!82b!(9r6)ZN
Ac.(<(;")'**2z[)h$'"$1)'¥`W'<!?'!8)'*19 4!(@".r>)ZN6A(!Fkc`W'#()'7$>/g201*7=3c01X):"2
'`_?:)'*O(E'`"$/1):)'3!"<!!Cm-•h"$E7!)'< 9
4( 9 C ,95- > -^7!)'P0?A2!5([)'H)B$'"$)'1I`W'H"$D`WO
A( 9
4( 9 C ,95- > ) F 9
4( 9 C ,95-N> )~F 9
4( 9 C ,95-N> ) lFG)' 9
4( 9 C ,95- > ).CG+#S)'*2"<!8):"2?'
?A2!5”X."$):""$d?:)'*O('`m"$B/1O"R/1):)'3B!"<!8"<[r>)ZN6A
0g(7=3Bd=NA"<)h`}():"O('`l"$B<"):"<
K
 C
D5)qzPX[5.$78$P"$r>+ 2–%'):/1$[.9 4g!(@"H`W3e)'¥'):zJ/'%'8)'3eE"$):"R"$c7$X*(
'`()'7$r>)ZNLH):D´A·k™'Ÿ @ !Ÿ< :Cl]R$!z&7!)'B0gH7!)'**2()';z&'!(F@)')Dr>)ZNAH7!)'B=NA(7!O"<R()'78$
'`_2"7$1*(I3&O1*2"2/1*2H"X3B(0!`W'#2"7=3/1*2!"<(E2"O0")'<pC,]R$r6+ 2 %'):/1$h^"$!!`W'D)cpA
'`H7=3B/k*2!"<(*2z;T{</g(7!U€1(‘AT{!"<!3X<"7d/'%')'3¥C B(O*2"c'`E*():%[5**0g"<6!"<!83
)6/*X7=z9`}'()'7$Œr>)ZN’LI"$):""<(**&$q5o)'’5.$(¦"<9@?''p'[2"d7$1*(GCK]R$&5.*X*m3):p'P"$
r>+ 2o%'):/1$)¥7=3/1*2!"<(*2z;T{</g(7!U€1(‘!"<!83"7c/1'%')'3 Vi;"<!)'7="%¥5.2"$‘)¥AT{!"<!3X<"7
(;?L23B(;"8\8C
]R$LO~`i):(FO,`W'3dO*):"2d'`"$Rr>+ 2«3!"$LJ(<(;")'**2z&"$R)'3BR)',"$Rf(O)'* 2«*():%
3B!"$A'`ks#)qz)')'
#;"<hV (w'w \8C q5!?'!(F@)'d3B/g'")';"_3/q?'(3(@"q?'!,f(O1)'!* 2M*2():%.
"$.):0k*2"ƒz&"<d@"<!/!"m"$r6+ 2 %'):/1$)')!/1(<(;"):"2'`g"$?Z)'*OR`WO17="2B`W'^)$12!):78$7!)'*
/g*7=z'Cl!Er6)ZN[A
F)'[=€k¸ V†…:\l"<B0#"$=NA/(7="<(K7!O3&O1*):"2?'!5E):P`}'E`W**2|5.%
"$$!):7$17!)'*/*X7=z¬‘<"):"%Xh"):"<&…O;"*5#(;"<!3BD<"):"<c Cm].$(h5^7!)'[5.2"<"$
y (**3)'P(nLO):"2K`}')d€NL(¥/*X7=zI)'
¸
Ø
V†…:\m»Ê¸
½
V†…:\GÃ
¾{ÀYÁ
„
V†… ‡ ˆ …@‰ŠA\<¸
V†… ‡ \8‰
Ø
V \
5.$!IА»e¬ V†…:\D)'1¦„ &"$I"<)'2"/'0k):01*2"ƒz6X<"<201O"2‘O!&/g*7=z9¬~C6]R$XD%2?'(O
)I(7!O?'d(7=3/2"26'`m"$d?Z)'*O&`iO7="2><h"$):""$d?Z)'*Oc`iO7="26'`m"$L'"#L"$
?:)'*OD`iO7="2P'`"$(;"2r>st O!R$2!):78$7!)'*/g*7=zh¬~C
]R$1c(7!O2?'P=NA/(2’0(7=3(B3B'[O<!`iO*^5.$(’5^¥5.2"7$’"<‘"$[)'7="2TY?Z)'*OSVW' 2@\
!/(<(@"):"&'`1"$^?:)'*O^`WO17="2Cs=€3
V†…@‰ŠA\_"<#0gl"$=NL/g(7="<(B7!O3dO*):"?'l!5E):d`}'~r>s#t
Vi¬_\&'`/!`}'3J%6)'7="ŒŠS“<"):"<K…K)'’"$(“`}*X*2q5.X%K"$[$!):7$17!)'*/*X7=z¬ª"$!():`W"<!(C
s#=€kP"$[<(7=“"<!83 ¦"$h82%$@"•T{$1)'“h'
` ^n1CRV \&"<90g
V†…@‰ŠA\8F^5.$78$S5P5.**7!)'**"$
š™'¤.¡£2 (˜iži™:·h¶´L·š(˜iž†™:·C].$R"$&=NL/g(7="<(97!O3dO*):"2?'&!5):8K'`š8™:¤¡k£ !˜ižW· r>s#t Vi¬_\H`W**2|5.%
/g*7=zh¬Mœ•¶˜{ !ŸE=NL(7!O"X%J)'7="2¥ŠP["):"<&…
C ª2"$¥"$(<=€k12"2!FA57!)'¥!5.2"< lnkC_V \H)'
Ø
V†…@‰ŠA\m»
5.$!
¸
V†…:\
»
¸
V†…Z\Ã
½
V†…@‰¬ V†…:\<\
„ V†… ‡ ˆ …‰ \ V†… ‡ ˆ …‰ • \
À{Á J
Ø
V†…@‰Š\
` ^XH7=3B/g2"<
` ^X E/3"2?'
VÌ \
V \
„ V†… ‡ ˆ …@‰ŠA\<¸ V†… ‡ \
Vx \
ƒ¾ ÀYÁ
]R$(&7=3B/1*!"<(*2z¥=€k&"$c?:)'*O=TY`iO7="K<(3)';"7!#'`,"$r6+ 2 $2!):87$;z'Cl)'78$±Lc5.2"$
/1):(;" )'J78$*JŠd"<'(l"$.`W'3):"2
V†…@‰ŠA\,`}'()'7$h<"):"<D…#I C l)'7$hr>)ZNJL _!"O
"$ Ž?Z)'*XOD'`a"$c7$X*P7$<(K0;z[¬ ƒC
]ah7=3/1O"<c"$&?:)'*O&'`m)I$!):7$17!)'*G/g*7=zP¬SXK<"):"<…@Fk5c0g!%6):" 9
) )G-cViA \.)'
7=3B/1O"< V†…‰¬ V†…Z\<\8Cm]R$1~(nLO2(~"$):"l5^#)'<pOl7$1*BARŠ »«¬ V†…:\`}'2",?:)'*O¸ V†…:\8
C O
½
7$1*¥(7!O2?'(*zP)'pL#2"#7$*X¥Š »”¬ ½ V†…Z\H`}'2"?Z)'*O'F)'6<h6O;"*a)I*2():`mL&Š .()'7$(C
<G!"JViŠ (‰Š ‰!Å!Å!Å|‰Š \0g&"$B/k):"$9"$):"5)'D"<8)(?'!<("$O%$9"$Br>+ 2Æ%'):/1$C
|5–*2():`^ABŠ !"O8.¸ V†…:\8F"<B5.$X7$[2"E/1):(@".)'1
V†…@‰Š \)'[<J[(7!O2?'(*2z'C,]R$D?:)'*O!"O8(P0;z
½ 9
) )G-^
½ V†…@‰Š\
»
V†…:\G
Ã
V†…@‰Š \GêÅ!Å!Å:à V†…@‰Š \Gà V†…@‰Š \8Å
V¹ \
½
½ ½
f~2%O $|5.l$q5ª"$<(nLO(7='`g!5):8 ‰ ‰!Å!Å!Å;(7=(?'(J`W3–"$R/3J2"2?'R)'7="2m(7=3T
/g<(¥$!):7$17!)'**2zJX@"<B"$&O3Æ'`~"$
Ë"<!83!C
!D5.$1):"7=2"21#7!)'"$D$2!8):7$;zK!/1(<(;"D"$B?Z)'*XO`WO17="29'`)h€1NL(FG$12!):78$7!)'*
/g*7=z ‘5.X**k)(zh"$):"R)Br6+ 2”%'):/k$[E)¶8´A£W£ ƒ—˜Yœ'˜{ H%'):/1$[2`a<!/1):):"<
V†…@‰ŠA\^?:)'*O(H):"<'(
`W'R()'7$><"):"<&… ‹  C
¸
V†…Z\ »
¸
!"# $&% !˜R¬ » q¬ v » ‰!Å!Å!Å|‰' “¢ 6œ9›;ž† !Ÿ•œ:Ÿ<š=›;žiš8œ:£,¡A™'£Èž†š ; )(m· ¦™'³: =ŸKœ¶8´A£W£ƒ—˜Yœ˜{ +*-,.0/
Ÿ<œ8¡1›2^
1 œ:·B£ !˜~» ¢ ˜W›A Ÿ<™(™'˜l·k™
@ &™•¶˜W›A Ÿ•œ8¡k›354_›A =·>˜W›A =Ÿ c 'ž}—˜~³:œ:£È´1 =—a¶!™:Ÿ6 87 ¶=™:Ÿ#žW·1˜{ =Ÿ·kœ:£!*œ ·k™
; —:K
9 œ:·h¸ ;7 ¶=™'Ÿ.¡Ÿ8žW¤žW˜ižW³: <1R£2 œ•¶=*œ
K·k™
@ =—:9P—8´š!›>˜W›Lœ'˜E¸ V†…Z\žX—B˜W›A h ¡ š(˜{ 6š(´A¤d´A£2œ'˜ižW³: Ÿ <>lœ'Ÿ ™•¶m¶!™:£W£U™?,
> ž}· ¡A™:£µžišP&¬Qž}·>—˜Yœ˜{ …@3
A "B!CED ]R$[/;'`B0LzŒO7="“Œ"$6LO3&0g!B'`*!?'(*J“"$K$2!):87$;z'Cª]R$[01)'<K7!)'<>
)>$2!):78$@z¦7=@")'1%KPr>)ZN¦AI5.$<P7$X*(¦):I/132"?' 4A(c52"$S/132"?'r6)ZN
7$1*(‘Vi<!df~2%O x \8C mNL/k)'%d(nLO):"2cV Ì \8F~V \8Fk)'V x \8Fk5^&$)(?'
¸
V†…:\ »
V†…Z\
2F ÀHG V†…Z\Ã
¸
»
»
»
¾ƒÀYÁ
¾ƒÀYÁ
V†…@‰¬mV†…Z\<\
„JV†… ‡ ˆ …‰¬mV†…:\<\ V†… ‡ ˆ …@‰¬mV†…Z\<\_Ã
Y¾ À{Á
„JV†… ‡ ˆ …‰¬mV†…:\<\JI V†… ‡ ˆ …@‰¬mV†…Z\<\aÃM¸
w
„JV†… ‡ ˆ …@‰¬mV†…Z\<\<¸
V†… ‡ \:K
V†… ‡ \
"!$#
%!&# %!&#
f~2%O ­6]R$6r6+ 2°(7=3/2"2v 7 =‰!Å!Å!Å|‰('K('"<K"$K(n;O(17=¥'`!5):8(7=(?'(M`W3
/X32"2?')'7="2):"R"3B( ‰!Å!Å!Å|‰ =x C
Ø
Ø
V
a1
a2
an
a1
a2
an
f~2%O x ­ A3B/k*2–*2():X%­~"$01)'<7!)'c`W'.]R$!'(3
ØC
]R$#*)'"E*XX y (**X3)' = (n;O1):"2h`}'R)&€NA(h/*X7=z¬~F`}'E5.$17$¥¸
VW"$?Z)'*XO`iO7="I'`a"$
`iO**gr6stªO!H/g*X7=zI¬_\H"$OnLOD<*O"C
]aP/q?'"$1O7="2?'&7!)''FG7=!)hr6)ZN9L*)<3!5.$!6"$&"<!'FG)'6*!"#Š[»Ë¬,+1V†…:\
0gD"$c78$*[)'7="2>7$<(¥0LzP"$c7!O(@"R/g*X7=zI,
¬ +Cr6):p'&"$c1O7="2?')'O3B/"2P"$1):"#¸ V†…:\E
½
"$#?Z)'*XOR`iO7="2J`}'H):01<"<8)'7=")'7="2PŠBI<"):"<…O!^"$7!O(;"^$2!8):7$X7!)'*/g*7=zB¬~C ‘#5)';"
"</q?'"$):"¸ + V†…Z\XH"$D?:)'*OD`iO7="2P`W'.r>s#t -+1Vi¬_\8C
¸
+
V†…Z\
¸
»
»
½
¾À
V†…Z\à + V†…@‰Š\
„JV†7… qˆ …@‰¬mV†…:\<\= V†…Zˆ …@‰¬mV†…Z\<\aÃM¸
½
V†…7\{ÄÃ
¾YÀ{Á
„
+
V†… ‡ ˆ …‰Š\<¸
+
V†… ‡ \
]R$&€1"#"<!834X6"$(7=‘*d#'0")'(90Lz¥=NA/1)'X%I"$ y (**X3)'>(n;O):"6`W'&¸ V†…:\.O1%
½
"$B1O7="2?'&)'O3B/"CO//g<c"$1):"#"$&€k<"#/3"2?'c)'7="2‘78$<(90LzK¬¼Vi†Cj'C2F5.$(>7=@"<*
()'78$(m)*2():`1AE&"$Hr6+ 2Q%'):/1$k\aŠ »«¬mV†…Z\8F)'1&"$1):"m<"):"<.… 5.**@0g^"$H=NA"~<"):"<H?LX2"<(
0LzI"$):%'(;".5$([2"H=NA(7!O"<(.)'7="26Š Cl].$D<(7=¥"<!83eE'0")'(P0;zP=NA/1)'
% ^n;O):"SV x \8v
„ + V†… ‡ ˆ …‰Š\H('"<(R"$c/'01):01X*2"{zJ"$):".):01<"<8)'7=")'7="26ŠJ5.X**"<!3):"<&[<"):"<d… ‡ 5.$*`W**2|5.%
$2!8):7$X7!)'*g/*X7=zI¬~C
q5°56=NA/1)'ª"$>"<)'2"2M`W3¿…>"<M… ‡ M"<!3JI'`c"$‘O3 q?'!¥)'** T{<"<!/¼"<)'2"2
`W3 …"<K7… #`W**2|5^(90Lz63dO*2""<!/>"<8)'2"21.`W3 7… #"<K…q‡}&
C ‘0g!%K0Lz67=12"2)'*X2b(%I… !v
Ø
Ø
„ + V†… ‡ ‰…Zˆ …@‰Š\DD"$/'01):0k*2"ƒzK"$):"&5.$(‘=NA(7!O"%6Š>X<"):"<[…I)'1‘`}*X*2q5.X%¥¬~Fa"$B€1<"&<"):"<
?A2"<(P5.X**10gc…7R)'P"$)'7="26ŠJ5.X**1"<!83):"<["):"<d…q‡}C y (7!)'O<&'`a"$r6):p'|?[/'/g!"ƒz'F"$
"$J/'01):01X*2"{zK'`^"$J=T{<"<!/S"<8)'2"29"<>… "3(D"<¥/'0k):01*2"ƒz¥"$):"=NA(7!O"%¥)'7="2SŠ>
<"):"<… H5.*X*k"<!83):"<["):"<d…q‡}C
„
V†… ‡ ˆ …@‰Š\
+
»
»
AO0k<"2"O"%&"$XH@"<B"$d¸ +
¸
¾À
V†… ‡ ‰…7Zˆ …‰Š\
+
„JV†…7|ˆ …‰¬mV†…:\<\ „
+
V†… ‡ ˆ …7!‰ŠA\
`}'83&O*X)AF5^'01")'
V†…:\
+
„
¾À
»
¾À
„IV†…Zˆ …@‰¬mV†…Z\<\g V†…|ˆ …@‰¬mV†…Z\<\aÃM¸
„JV†… ˆ …@‰¬mV†…Z\<\ „
¾ƒÀ Á ¾ À
V†… ‡ ˆ … ‰Š\<¸
+
V†…8\{ÄAÃ
½
V†… ‡ \
+
q5 5c7!)'K():)'%'D"<!83R"</kO**O"H"$"<!83R;?'*2?A%…!­
¸
+
V†…:\,»
„JV†…7Zˆ …‰¬mV†…:\<\ V†…|ˆ …@‰¬mV†…Z\<\aÃM¸
¾À
q5F17=1!R"$=€k2"I'`l¸
¸
V†…7\_Ã
Ø
„
¾YÀ{Á
+
V†… ‡ ˆ …!‰Š\<¸
+
V†… ‡ \
V¨ \
V†…78\8F5.$78$P0;z ^
n;C_V \H7!)'¥0gD5R2"<"<([)'.`}*X*2q5.(­
+
V†…7\
+
½
»
¸
½
V†…78\GÃ
¾{ÀYÁ
„
+
V†… ‡ ˆ …7=‰Š\<¸
+
V†… ‡ \
‘&7!)'K)://1*2zI"$R=€2"2h01)'7p;5E):H"<J"$2%$;"•T{3B<"./1):"H'` l
nLC_V ¨ \E"<'01")'
¸
+
V†…:\
»
¾À
„IV†…|ˆ …@‰¬mV†…Z\<\= V†…7:ˆ …@‰¬mV†…Z\<\GÃɸ 1
+ V†…78\{Ä
Ø
]R$XRR"$ y (**3J)'¥(nLO):"26`}'#"$c?:)'*O&`WO17="26¸ + 5R2"<"<(K5.2"$K(</g(7="."<P) T{<"<!/9/832"2?'
)'7="26¬mV†…Z\8Cl].$/|?'(R"$cXO7="2?'D<"<!/6)'["$!!`W'/1q?'(."$"$!'(3¥C
-•K"$1./k):/!qF15&5.X**0g&7=7=!8(>5.2"$K$12!):78$7!)'*G/g*X7!2(R'`m)I/1):"7!O1*):.`W'3®"$):"5^7!)'**
Ÿ š!´AŸ8—8žW³: !£Œ™8¡˜ižW¤œ:£ÈC”]R$(<6):>$12!):78$7!)'*H/g*7!(¬–» q¬ ‰!Å!Å!Å(‰¬ ‘O7$É"$):"h()'7$«?L1O)'*
/g*7=z[¬ mX)'6'/"3)'*a/g*7=zP`W'#r>st Vi¬_\&Vi†Cj'C2Fg"$dr>st¼O17=([0;zK`}**q5.%J¬S>)'**_'`m"$
(7=(1)'@"'`,A
\8C `W'"O):"<(*2z'7F :<O1<"R0g(7!)'O<c)J/g*7=zPH(7!O82?'(*2zh'/"3)'*aL(.'"#3B()'
"$):"2"mm"$R'/"3)'*/*X7=z&`W'l"$R'2%1)'*r6st C~-•!(F;5^.5.**<!"$):"^)(7!O?'(*2z&'/1"3)'*
/g*7=zB`W'Hr6+ 2Ç%'):/1
$ uX^'"H(7=():*2zO1nLO)'h"$):"H<3#(7!O2?'(*zB'/"3)'*k/g*7!2():
0g!"<"<!R"$)'K'"$!!C
- `H)¥/g*7=z9D(7!O2?'(*2z9'/"X3)'*†Fa$q5!?'!(F~"$(S"$ y (**3)'S(n;O):"`W'c"$I'/"3)'*l/*X7=z
3dO<"H0g):"•€k([):"A
­
¸ V†…Z\,»«3J)ZN V†…@‰ŠA\8Å
½
‘&7!)'¥"$!!`W'!5R82"<D"$(7=3B/g"2P(n;O1):"2cV Ì \8F~V \8Fk)'V x \R)'
V†…@‰Š\
¸
»
»
V†…@‰Š\
»
V†…:\
¾ƒÀYÁ
†V …‰Š\GÃM¸ V†…:\
½
3)ZN ½ V†…‰Š\
À{Á „JV†…q‡Yˆ …‰ \ V†…q‡{ˆ …‰ • \
„
V†… ‡ ˆ …@‰ŠA\<¸
؊Ø
V†… ‡ \8Å
Vº \
` ^XR7=3B/g2"<
` ^X H/3J2"2?'
V
\
Vw \
10
MaxRoot
10
12
QGet
QPut
-2
TERMINATED
MaxGet
-12
-2
MaxPut
-2
QPickup
QNavigateForPut(t)
QNavigateForGet(t)
QPutdown
-1
-10
-1
Pickup
Putdown
MaxNavigate(t)
-1
0
QNorth(t)
-1
-3
-2
-2
QEast(t)
-1
North
-2
-3
East
-1
QSouth(t)
QWest(t)
-1
-1
South
West
f~2%O ¹ ­3/1O"%&"$?Z)'*XO'`)B<"):"<OX%c"$Dr>+ 2Ž$2!):78$@z'Cm].$ Ë?Z)'*O'`a()'7$/2”A
$q5¥"<h"$&*2!`W"R'`,"$A'CH+*X*G'"$!;O13c0g!R$q5”"$c?:)'*O(.0g(%!"O(¥O/6"$c%'8):/1$C
]R$$()(?LzP*X(H$|5Ê"$D/1):"$¥5.2"$["$3J)ZNA3dO3Æ?:)'*O'C
]R$H1*2zB7$)'%'#,"$):"l`}'7=3B/g2"<.)'7="21!FA¸ V†…:\~l7=3B/1O"<(J0;zd"):pLX%"$#3)ZNAX3&O3 Ê?:)'*O'C
+#^)'h=N)'3B/1*2'FL7=!^"$#2"O):"J$|5.hJf~2%O FL5.$78$55.X**('"<0;zh…!C,]R$'/"UT
3)'*k/g*7=zB`W'^"$Xl")'<ph)c(7!O2?'(*2zB'/1"3)'*$2!):78$7!)'*1/*X7=zKVW`}'E"$#%2?'(Pr>+ 2”X):%')'3I\8C
]R$!!`}''F5B7!)'‘)://1*2z ^n;O1):"2BV º \8F^V \8F_)'MV w \"<[7=3B/kO"<B2"#?Z)'*Od`iO7="&
C AO//g<&"$):"
"$/k)'<(%'!cX):&
" )'1‘5.$("<K%'¥"< y C_].$B?Z)'*XOB'`E"$D<"):"<h F0(7!)'O1<2"D5.*X*7="
O""<[3B|?'"$")ZNA~"< cF O12"."<[/k7pAO/9"$/1)'<(%'!(F º O2"#"<[3Bq?'J"$")ZNA~"< y F)'
O"."<[/1O"q5.>"$/1)'(%'!(F`W')I"<'")'*m'` O"dVi)P!5E):9'` \8&
C «$(9"$/1)'(%'!
(*?'!(FL"$D):%'(@"H%'!"R)d!5E):h'`~Ã Ì FAd"$!"?:)'*ORÃ Cmf~2%O ¹ $|5.H$|5ª"$r>+ 2
$2!8):7$;z6(7=3/(D"$?:)'*O'C]_¥7=3/1O"<B"$B?:)'*O'`H… !F 9
) )G-)'<pAD()'78$'`E2"D7$X*(F
*>@-~)'1 AK3L-,"<&(<"3):"<#"$R=NA/g(7="<(h!5):8'`_7… =C AD35-~Xl7!O(@"*2zB'")://1*X7!):01*2'F'0g(7!)'O<… H)B3B(3&0g!H'` ½ + <F2"HE)"<!3):"<(K<"):"<'C *>@-8F$|5^!?'!qF1H)://1*X7!):01*2'FA)'[2"E(<"3J):"<(
"$D7=3B/1*!"2[7=<"H'`"$")'<
p
½
V†… ‰ <\l"<B0g (Ì F0g(7!)'O<D2"E5.**7=<" O12"l"<(*2?'!
"$c7!O1<"<3B!BVW`}'#)J!"!5):8['` Ì + » (Ì \R):`}"<!7=3B/1*!"%"$ K>@-HO0")'<pC .|5^!?'!(Fg"$XR
:O<"."$&!5E):Œœ•¶=˜{ =Ÿ.7=3/1*2!"%I"$ K>@-8Fg<P2"3&O")'<p 9
K>@-H"<P(<"3):"<d"$d=NL/g(7="<(>!5E):
'`~/g!`}'83%d"$ K>@-E2"<(*2` C
Ø
Ø
Ø
Ø
Ø
Ø
Ø(Ì
JØ
Ø
Ø
Ø
9 *>@-#)'<pA2""{567$1*( A C EGF 3IHh)'1
( 9 C ,9L-N>G)+ *
>@-D"<¥(<"3):"<J"$=NA/(7="<(S!5E):C
ADC EGFG3IHSpAq5.P"$1):"P`c) A C EGFG3,H¦)'7="¼P):"<"<(3B/"<(Ê|5F2"P5.**'"¥7$)'%'‘"$‘<"):"<'FD<Œ"$
?:)'*Oh'` V†…7!‰ ADC EGFG3,H\B» Ì C 9 #A C EGFG3,H[5.*X*l!"O¦)9!5):’'` Fm0g(7!)'O<?ADC EGFG3,H>
'"c)K*2!%)'*m)'7="2¦9"$7!O(@"c"):"<'C (7='F V†…7=‰ A C EGFG3,H\#» Ì Ã » (Ì C ( 9 C ,95-N>G)+ K>@-!VW"8\[XP!?Z)'*O1):"<(¼5.2"$ h0O1ª"<MF#"$7!O(;"P/1)'<(%'!6*2L7!):"CÆ-ƒ"[pAq5P"$):"
):`W"<!h7=3B/k*2!"%S) 9
4( 9 C ,95-N>'Vic\8FR63'K)'7="2ËVW"$ A C EGF 3IH:\B5.**E0g¥(nLO2(Œ"<¦7=3B/k*2!"<
"$ *>@-8F<&
V†… !‰ ( 9 C ,95- >G),+ K>@-qVic\<\S» C- "¥"$(Ç)'<pA 9
4( 9 C ,95-N>Vic\h"<Q7=3/1O"<
"$‘=NA/g(7="<(Ê!5E):¼'`d/g!`}'83%“)
)q?L%):"<'C 9
4( 9 C ,9L-N>‘)'<pA[()'7$ '`d"[`}O/2 7$X*(
"<(<"X3):"<¥"$B7="!C ( )+ -6/>pLq5.B"$1):"J):`}"<!I%'%
'"$F"$[")ZNH5.**0g[):"J*2A7!):"2MFE<
F G V†… ‰ ( )+.-0/@\#» C 9"$'"$!d$)'F 7#95:;-pAq5D"$):"c):`W"<!c%'% l)'<"!Fa"$")ZN
5.*X*G0g&"ƒ5^6<"<!/1D)(5E)(z>`W3 "$"):%'!"!F_< G V†7… (‰ 7#95:;-<\.» Ì C 3*):#7=3B/1O"):"
F
):#/g!`}'83B(0LzB"$#'"$!E"{5 2 A(!C l)'78$h'`G"$./132"?'.r6)ZNhL(^$)')'I=NL/g(7="<(h!5E):
'` F,S/1):"7!O*X):(F,¸ V†… 8\c» Fm<
G V†7… =‰ ( )+.-0/'\&» à » CK]R$I'"$!
F
±?:)'*O(D): F F_)' #Ì Cd+E"D"$X/g;"!F 9
4( 9 C ,9L-N>'Vic\#*2?'(DO/9"<K"#1)'3BB0;z6"):pLX%["$
3)ZN3dO3–'`"$(`WO?Z)'*O(!F@5.$17$Im"$ `}3 ( ),+.-6/;Cm- "l/1)'(l"$Xm?:)'*OO/J"$"<!5$!
( 9 C ,95- >G)+K>@-(Vic\)'1I2" )'ª!"O1"$>?Z)'*XO V†… (‰ ( 9 C ,95-N> ),+ K>@-(Vic\<\K» Ì C
9
K>@-^q5Ê"):p'(R"$3)ZN3&O13 '` Ì )' (Ì F1)'1I!"OE"$):"E?:)'*O"< *>@-8C K>@-^)'1H2"
?:)'*O"< Ì "<'01")'KÃ F5.$17$[2"E!"OE"< 9
) )G-8C 9
) )G-l"<?L)'*X*2zB"):p'(."$3)ZNAX3&O3Æ'`
"$H)'1[!"O1.à )'R"$D?Z)'*OD'`,7… !C
'"<&"$1):")'.)I=TY 4g(7=".'`,7=3B/kO"%"$?Z)'*XO'F1()'78$6r>)ZNKAc7!)'K)'*X<J7=3B/1O"<c"$d)'7="2
"$):")'7$!?'("$&3)ZN3dO3e?:)'*O'Fg<h"$):"#5.$(K"7=3B(#"3Bd"<P7$L<)'7="2BVi†Cj'C2F"<P7=3/1O"<
)B%'!(zI/g*7=z\8F"$H7!)'K0g#/g!`W'3B(P0Lzh"$c)'3Bc/A7=(O"$):".XHO<(["<J7=3/1O"<?:)'*O(!C
JØ
JØ
JØ
JØ
JØ
JØ
Ø
HGB"!
Ø
JØ
Ø
JØ
JØ
Ø
!.G• ‘5.$K"<h!?'(*2'/>)h*2():1%J)'*2%''82"$3®"$):"#7!)'9X7=q?'!"$&0("(7!O2?'(*2zP'/"X3)'*G/g*7=zP`W'
"$&%2?'(Kr>+ 2 %'):/k$C]aIJ"$(F5^d3&O1<"H€1"R%)'K)J0g!"<"<!O!<")'X%'`,"$&<"<8O7="O'`
(7!O82?'(*2z'/1"3)'*$2!8):7$X7!)'*;/g*X7!2(!Caf'^)''X):zcr>staF@"$!R7!)'B0gH3)';z&'/"X3)'*A/g*7!2(!F
01O"H"$!zP)'*X*G$)(?'&"$c)'3?Z)'*XO`iO7="2Cm+#R"$`}*X*2q5.X%*2(33)B$q5.(F1$q5!?'!(F"$c)')'*'%'O
/'/g!"ƒzh;(R'"R$*P`W'.(7!O82?'(*2zI'/"3)'*$!):7$17!)'*g/g*7!2(!C
# #%$ $&% !˜¬ 6œ:·K¬ ¢ K˜)l> ™Ÿ š(´LŸ8—8žW³: !£’™8¡˜ižW¤Bœ'£~¡A™:£µžiš(ž† =—D¶=™:Ÿ *-,.0/
*'&)( 1˜W›A =·‘˜W›A !ž}ŸŸ !˜i´AŸ- ·—c· 8 I·k™'˜R¢8 c˜W› c—=œ:¤J 3
#
A "B!CED
›@žY =Ÿ<œ:Ÿ<š=›,
œ8¡¡k£µž† ’˜Y™
‘J%2?'h)'=N)'3B/1*2'C¥!D"$J3J):b!J/'0k*2(3 $|5.Sf%O ¨ )'1S2")'<L7!X):"<(
r + 2 $2!):87$;z'Cl]R$!&):"$!c'/!8):"<'!FD( )+ -6/;F81#)435-6/;F1)' 7#95:;-8C^]R$ ( )+.-0/B)' 1#)435-6/d'/g!)ZT
>
"<'D$)q?'b!!¥7=<"!F5.$X*2c"$ 7#95:;-R'/g!):"<'%2?'(D)h!5E):>'` C]R$!d):d"$!")'<pL(C- 6")'<p
‚&Fk"$&"<!31)'*G<"):"<():&"):"<(#Š[)'1+*D)'#$|5.¥6"$&3J):b!'C.f'")'<p-,hF1"$&"<!31)'*_<"):"<
/1C /6(;"<zK"< . F"$B):%'(@"D(7=(2?'(D)h!5):86'` F01O"
. FG)'>`}'")'<p Fg"$"<!3X)'*a"):"<0
(;"<z‘"<1
/F"$I):%'(;"&(7=(2?'(&)¥!5):8'` 0
C L¥"$J'/"3J)'*,/g*7=z>`}'&"$(;"2J/1'01*2(3
5O*60&`}'")'<p6‚Ž"<[)'*25)qzL"<!83):"<K"):"<ŠSVi7!)'**a"$#¬ 2 \8Fg`W3®5.$78$K")'<1
p ,e7=O*6()'78$
<"):"< . C .|5^!?'!(FA"$!):#3)'@z(nLO)'**2z'/"X3)'*/g*7!2(~`W'")'<p‚cFL<3B'`5.$78$B"<!31):"<ŠkF
<3Bd *|Fk)'6<3BcX¥)J32NL"OD'`"$"{5CX!(F`W'.=N)'3B/1*2'F1¬ 2- F5$7$K)'*25)qzL"<!3X):"<(
6<"):"<3
*qC]R$c'/1"3)'*G/g*X7!2(H`W'),o)'1
):OnLO'F1J5d7!)'>'0")'’Vi):"D*2()'<"8\R"{5h 4!(;"
(7!O82?'(*2z'/1"3)'*/g*7!2(!F,h0k)'<(“M¬ 2 )'’[01)'<(“Œ¬ 2- 5.2"$“9 4g!(;"?Z)'*O(!C 1*2z
(7!O82?'(*2zI'/"3)'*g/g*7!(E01)'<([¥¬ 2 7!)'K0'/"X3)'*`W'R"$D(;"2cr>st C JØ
JØ
Ø
Task A
Task B
a
c
b
d
Task C
Vi)@\
Root
QNorthA
QA
QB
QC
A
B
C
QSouthA
QEastA
QNorthB
QSouthB
QEastB
North
South
East
QNorthC
QSouthC
QEastC
VW0\
f~2%O ¨ ­¥r>):b!6tl'01*2(3©X**O<"<8):"%K(7!O2?'(*z'/"3)'*E/g*X7!2(d5.2"$Œ94g!(;"B!"O1!­Vi)@\d"$
3):b!'F~VW0\^"$cr>+ 2Ž%'):/1$C
] $#/1'01*2(3±"$):"E"$94g!(;"^/g*X7!2(l`W'R])'<pI+ zL2(*XI 4!(;"E/'01):0k*2"ƒzd"<8)'2"2h`iO7T
R
"2„JV†…q‡†ˆ …@‰‚D\,`W'E"$.L'"Er6st C,-ƒ`5(n;OR`W'^()'78$[O0")'<pB"$1):")'**1'`_2"^'/"3)'*1/g*7!2(
%2?'"$)'3B/'01):01*"{z"<)'12"2P`iO7="2[„IV†… ‡ ˆ …‰Š\8F1"$(["$E/'01*2(3Æ5.*X*kX)://g():(C
* &)( œ:· *-,.0/
&
$ % !˜4¢8 ˜W›A d—( !˜H™•¶œ:£}£aŸ< š!´AŸ8—8žW³: =£ ¥™8¡˜ižW¤œ:£k¡A™:£µžiš(ž† =—^¶!™:Ÿœ ž}³' =· '
›@žY =Ÿ<œ:Ÿ<š=›,
3 6 —=œGh˜W›Lœ'˜^—8´1¢!˜Yœ:—žX—cœd<"):"21):z[3)'7=>ž „ V†… ‡ ˆ …‰Š\m»«„ V†… ‡ ˆ …‰Š\¬ =‰¬ ‹ c‰Š ‹ ‚ ‰g… ‹  1 ˜W›A =· ^> ,> žW£}£G—=œGh˜W›Lœ'˜ ¿ž}—œd<"):"):z[3)'7=J$12!):78$@zB3
¶& =³' =Ÿ0J·k™ @ #žX—&œJ—˜Yœ˜iži™:·œ:Ÿ6P¤œš(Ÿ•™"E
* &)( l1 ˜W›A =·œ:£W£GŸ š(´LŸ8—8žW³: !£P™8¡˜ižW¤œ:£¡™:£Èž†š!žY —
# # %
# $ ¶ žX—&œJ—=˜Yœ'˜iži™'·kœ:Ÿ0I¤œš(Ÿ•™h›;ž† !Ÿ•œ:Ÿ<š=›I¶!™:Ÿ +
> ži˜W›PŸ —{¡ šq˜H˜Y™ œ:· ›Lœ'³: d˜W›A c—!œ:¤J &³Zœ:£µ´1 3
,
A "B!CED ]R$.`}*X*2q5.(7="*2z[`W3"$=€k2"2K'`l)P<"):"2):zK3)'7=[$2!8):7$;z'CD+E"D!?'!z6A
F"$cr6st
iV ¬_\E$)'R"$c)'3Bc/'01):01X*2"{zB"<8)'2"2P`iO7="[„ V†… ‡ ˆ …‰Š\`W'.)'**(7=(1)'@"$2!<T
:) 78$7!)'*/g*7!2(¬6)'H*%d)'H¬>(7!O82?'(*2zB'/"X3)'*1`W'H"$(7=(1)'@"HA(!C q5Ê`W'H)'@zhr6stF
Ø=x
') **g'/"X3)'*g/g*7!2(`W'.$)(?'&"$)'3Bc?Z)'*O'F<7=
^$1)(?'"$&)'3B?:)'*O'C Vi¬_\ERO1n;O'FL)'**g'/1"3)'*/g*7!(^`W'.A
+ 7=<(nLO(7='`m"$##"$):""$(2%!'`^"$dr>+ 2Æ$2!):87$;z¥$O*><!!p6"<h(1O&"$):"
!?'!z[r>)ZN[LDH)J<"):"2):zP3)'7=C^+O3%"$):"H"$cr6+ 2Ž$12!):78$@zhR)B<"):"21):zh3J)'7=
$2!8):7$;z'FA"$%')'*'`_)d*2():1%&)'*%''2"$3 $O*J0g."<d€kJ'`_"$(7!O?'(*2z'/"3)'*k/g*7!2(!C
A17=#"$!zP)'*X*$)(?'&"$)'3B?:)'*O'F"$1HR)B5(**UT{=€k(h"):%'!"!C
]ad!2?'O7$P)d*2():%c)'*2%''2"$13[F;*2!"EO7=<"<O17="^"$
R'0101•T{r6&<"<L78$)'<"7#)://(N3)ZT
"2&V y !"<!pZ)'§Ž]R2"2pA*!F (w'w'¨ \E'` lnLO):"21V w \H)'SV \8­
­È»
V 3 V \<\ V†…‰Š\Gà V \<¸ V†… ‡ \
V \
V†…@‰ŠA\
¸ V†…Z\ ­È»
V 3 V \<\<¸ V†…Z\à V \ V†… ‡ ˆ …‰ \
V \
Ø
Ø
Ø
Ø
؊Ø
]R$h(O*2"%>*2():8%9)'*2%''2"$13[Fl5.$17$’5[5.**7!)'**Hr>+ 2T FE'/g!):"<()'`}**q5.!C“+E"()'78$
7=3B/g2"<r>)ZN>A Fg78$L<B)I78$*6)'7="2‘Š¥)'6=NL(7!O"<"!C #0k<!?'d"$&(O*2"%J<"):"<J…q‡_)'
7=3B/1O"<2"E?Z)'*XOc¸ V†… ‡ \8F)'[):/1/1*2z ^n;O):"V \8C^+E".)/832"2?'r>)ZNPAF=NA(7!O"<"$)'7="F
'01<!?'"$(O*2"%d!5E):F)'K)://1*2z lnLO):"2SV \8C
«$):"l=NA/1*2'8):"2/g*7=zd3&O1<"m0g.O<(J5.$(B=NA(7!O"%cr6+
2T 6f'/3J2"2?'Hr6)ZNJL(^)'
`W'#r>)ZNKA(."$1):"$)q?'&*z[/3"2?'D7$*X(F"$1.<")'):81 *2():X%F1)'62".5.X**7=@?'!%'
"<K"$'/"3)'* Æ?:)'*O(O1!c)';z>=NA/1*2'8):"2‘/g*7=z>"$):""<(D!?'!z<"):"<=T{)'7="2’/1)'cA€k12"<(*2z
'`W"<(C q5!?'!(FG"<[*2():8¥"$[Vi*2A7!)'**2z\E'/"3J)'*_/g*7=zP`W'D)Ir>)ZN6A H"$1):"#$)'#)P7=3B/g2"<dr6)ZN
A I)'.)78$*F=N<"X%"$!'!"7!)'*_(O*2".'".)://k*2z'C «**r>+ 2T 7=@?'!%'&'.7=;?'!%'d"<
"$d'/"3)'* ?:)'*O(#`}' “- >'!#`W'#"$d<")'1): 2–*():%I(O*"."<[)://1*z'FA
E!(
"<<!#)'3B/1*2(l'`"$<"):"<"<)'12"2l„ V†…q‡{ˆ …‰ L\8FL5$!.¬>l)c(7!O2?'(*2z'/"3)'*1/g*7=z`}' ;Vi¬_\
)'92"#(7=(1)'@"!CD- `lA P#=NL(7!O"%P)'9=NL/1*'):"26/*X7=z["$):"D#'"'/1"3)'*†Fg"$(92"#5.**
/AO7=D)'3/1*2(H`W3e)B 4!(;"H/'0k):01*2"ƒzJ"<201O"C
(7='Fr>+ 2T !(E"<(3B/1*2|zI)'P=NA/1*2'):"I/g*X7=z):"H()'7$¥A#"$):"H!?'(;"O)'**2zh7=@?'!%'(
"<J)'['/1"3)'*g/g*7=zJ)'1Pz'!".<"**g):"X•€1(H"$7=12"2`W'R"$7=;?'!%'(7=d'` –*2():X%
C A%$F
)'):pLp'*)A$
F <_2"<"3)'GF)'/
Lb!!/g(<
? ): mV (w'w \R/q?A=N)'7="*2zP"$(nLO2(IpAP'`/g*7=z­
Ø
؊Ø
Ø
, % 7 Ÿ IžW·>˜W› £µžW¤dži˜ ,> žW˜W›[žW·m( ·ži˜{ ¡£U™:Ÿ<œ'˜iž†™:·?c9 ¡A™:£µžišPJž}—œ'· D¡A™'£Èž†š J—=œ'˜iž}—}¶0:ž}·
3 œš=›Kœšq˜iži™'·‘žX—& L š(´A˜{ Iž}·(l·žW˜{ !£P™•¶=˜{ =·9ž}·S !³: =Ÿ0B—=˜Yœ'˜{ ˜W›Lœ'˜ž}—D³qž}—8ži˜{ Iž}·(m·ži˜{ =£ h™•¶=˜{ =· 3
3 ·‘˜W›A &£µžW¤dži˜ ^1 ˜W›A R¡™:£Èž†š hž}— Ÿ< 8 G,> ži˜W›¥Ÿ< =—Y¡ š(˜E˜Y™h˜W›A ³Zœ:£µ´1 l¶8´A·kšq˜iži™:· ,> ži˜W›¡kŸ<™@¢œ@¢!ž}£µži˜ 3
, ›@žY =Ÿ<œ:Ÿ•š!›@ž†šœ:£ % ¡A™'£Èž†š ‘ž}—Pœ>›;ž† !Ÿ•œ:Ÿ<š=›;žiš8œ:£~¡A™:£µž†š ¥¬Ê» q¬ ‰¬ (‰!Å!Å!Å Sž}·~> ›@žiš!›’˜W›A 6 =¡k£U™'Ÿ•œ'˜iž†™:·
°
¡A™:£µž†š ¬ œ'˜R œ@š=›[·k™
@ #ž}—cœ
% ¡A™:£µž†š 3
‘¥|5Æ/q?'["$1):"`H"$[r6+ 24$2!):87$;z¦d)9<"):"2):z’3)'7=‘$2!8):7$;zS)'“2`Rr>+ 2#T (3B/1*qzAH)B$2!):78$7!)'* "<_- Œ=NA/1*2'8):"2P/g*7=z'FA"$([r>+ 2T 5.**7=@?'!%'"<B"$JViOXn;O|\m?:)'*O
`iO7="2P'`,)(7!O82?'(*2zI'/"3)'*/g*X7=z'C
¢ PœK—˜Y™qš=›LœZ—=˜ižiš[—›L™'Ÿ˜{ =—˜l¡Aœ'˜W›P¡kŸ<™@¢!£2 !¤±¶!™:Ÿ5>~›;žiš!›¦œ:£}£ @ !˜{ !Ÿ8¤žW·žX—=˜ižišc¡™:£Èž†š!žY —Pœ:Ÿ< ¢ œ+* , .0/ʛ;ž† !Ÿ•œ:Ÿ<š=›Ic¶!™:Ÿ —8´š!›6˜W›Lœ'˜ ž}—œh—=˜Yœ'˜iži™'·kœ:Ÿ0¥¤œš!Ÿ<™[›;ž† !Ÿ•œ:Ÿ<š=›I?3 % !˜
¢ œJ—! !´1 =·š8 d™•¶&š8™:·—˜Yœ'·1˜W—l¶!™:Ÿ œš=+
› *œ h·k
™ @ #—8´š!›[˜W›Aœ'˜
!"# % !˜
¡kŸ<™8¡ !ŸE#
1 œ'·[£ !˜
V \
*3
"
"$#+% ¾'
V \m»
(
œ:·
*3
"
"$#&% ¾ '
- V \ .
(
Ø
V (Ì \
( ˜a¬«¢ dœJ›@žY =Ÿ<œ:Ÿ•š!›@ž†šœ:£ % ¡A™:£µž†š [œ:·PœZ——´A¤ d˜W›Aœ'˜Eˆj¸ V†…:\!ˆGœ:·¥ˆ V†…‰Š\!ˆaœ:Ÿ< B¢™:´A·@ #¶=™:Ÿdœ:£}£ E1 1
… 1œ:·hŠ 3 4_›A =· >,ži˜W›J¡kŸ<™@¢œ@¢=žW£Èži˜ 1#œ:£ ™:Ÿži˜W›@¤ *-,.5/ I´L—8žW· ¬ œ8¡@¡k£ÈžY 9˜Y™ >,žW£}£lš8™:·³: !Ÿ I˜Y™6˜W› ´A·ž !´1 c³Zœ:£µ´1 m¶´A·kš(˜iž†™:·™•¶&œIŸ š(´LŸ8—8žW³: !£P™8¡˜ižW¤œ:£¡™:£Èž†š #¶!™:Ÿ š8™:·—8ž}—˜{ !·1˜ >,žW˜W› 3
%
Ø(¹
A "B!CED
] $&/L'`,(3B/1*2|zL#)'6):%O3B(;"#X3*):R"<h"$<;"<AO7=(¥"<P/q?'d"$d7=;?'!%'(7=B'`
R
o*():%[)'’~‚c,‚dV \JV y !"<!pZ)'§o]R2"2pA*!F (w'w'¨ v )'):pLp'*)AF '1)'Fa§ %$F (w'wZx \8C ‘
5.*X*k(3/1*2qzh"$D`W**2|5.%(O1*2"H`}3o<"<L78$)'<"7&)://qNAX3):"2P"$!'z­
#
Ø
Ø
7 (EŸ<™8¡A™:—8ži˜iži™:·!3.¶Ÿ<™:¤ !Ÿ˜W—( :œ:—&œ:· _4 —8ži˜W—8ž |£µž}—E1 9,™:·—8ž @ =Ÿd˜W›A &žW˜{ !Ÿ•œ˜iži™:·
|Vg \­È»ŽV Ø Vg \<\H 8Vg \Gà Vg\=V<V \=V\GÃ3Vg\Gà ) Vg\<\8Å
% (˜a» V \8‰!Å!Å!Åq‰ V \8‰ Vg \8‰!Å!Å!Å|‰3 |Vg \8‰ Vg \8‰!Å!Å!Å|‰ Vg\8‰ P¢8 D˜W›A d =·1˜ižWŸ #›@ž}—˜Y™:Ÿ0I™•¶D˜W›A DžW˜{ !Ÿ
œ'˜iž†™:· 3
# #%$
¶
7 œ 9 _4 › V \ —!œ'˜ižX—W¶6Pš™:·:žW˜iž†™:·— 7 9
7 ¢ 9_™:Ÿ !³: !Ÿ6œ:· ˜W›A &·k™:ž}—! d˜{ !Ÿ8¤d—3V \—=œ˜ižX—W¶6I 38V \!ˆ YÄ» 7 š 9 EžW³: !·Jœ:· ·k™:Ÿ¤ ˆ2ˆ Wˆ2ˆ™'· 1˜W›A =Ÿ . 'ž}—˜š™:·—˜Yœ:·1˜W—,‚Çœ'· ,Ž—´š=›c˜W›Lœ'˜!I - V•\!ˆ Ä#"ɂà ,¥ˆ2ˆ ˆ2ˆ - 3
7 9 _4 › =Ÿ žX—=˜W—&œJ³' šq˜Y™:Ÿ ² ^1 œD¡™Z—8ži˜iž}³' d³: š(˜Y™:Ÿ$@E1 œ:·PœJ—=šœ'£Uœ:Ÿ&% ‹  ‰ Ø \E^1 —8´š=›¥˜W›Lœ˜k¶!™:Ÿ&œ'£}£ 1
ˆ2ˆ' ² ˆ2ˆ (&")E
% ˆ2ˆ ² ˆ2ˆ (
7 98_4 › =Ÿ I 'ž}—˜W—œ[·k™:·· œ'˜ižW³: BŸ•œ'·™:¤°—( (´ !·kš8 +* ˜W›Lœ'˜š™:·³' =Ÿ =—˜Y™-|, =Ÿ<™,> ži˜W›B¡kŸ<™@¢œ;¢=žW£Èži˜ œ:·
ž}—D—8´š!›[˜W›Aœ'˜1¶!™:Ÿœ:£}£ ˆ ) Vg
\!ˆ."/;* Vˆ2ˆ ˆ2ˆ m( Ã Ø \
˜W›A !· š™'·³: !Ÿ —&˜Y™ ² ,
> ži˜W›¡kŸ•™;¢œ@¢!ž}£µžW˜ 35_4 › &·k™'˜Yœ'˜iž†™:·¦ˆ2ˆ0; ˆ2ˆ ( ; =·k™'˜{ =—dœ+^> =ž ›;˜{ I¤Bœ 'žW¤d´A¤4·k™:Ÿ¤
ˆ ‚dV
\!ˆ Å
ˆ2ˆ ‚Bˆ2ˆ (»«3)ZN
$ Vg \
]R$"<O7="OD'`"$D/;'`a'`]R$!'(3 Ì 5.*X*10gDO17="2?''F<"):"%B):"R"$*2()(?'(.'`a"$r>+ 2
"<!B)'95'pA%h"<q5E):9"$L'"!C+‘B5.**_(3B/1*2|z6)P94g!(;""X3Bd7!*L7p6):"D()'7$SLH"<[7=O;"
"$d;O3&0g!R'`mO/):"<c"<!/1R/g!`W'3B(K0;z[r>+ 2T ):"#"$):"A'CH]R$c?Z):X):01*2+
E5.*X*)'*25)qzL!`W!
"<"$c"3B7!*2A7pP'`"$c7!O(@"RA C
]a‘/|?'¥"$[01)'<¥7!)'[`}'J)';z’/3"2?'hr>)ZNŒL'Fl5¥'"<["$):" ^n;O):"ÊV \
:O<""$
<"<A7$)'"7)://(N3):"2¥)'*%''2"$3o`W'.7=3B/kO"%B"$c=NL/g(7="<(¥!5):¥`}'R/g!`W'3X%B)'7="2KŠh
<"):"<…F)'¥"$!!`W'2"R7=;?'!%'(O!H"$7=12"2E%?'(¥):0gq?''C
]a/1q?'"$(7!O?'l7!)'<'F@7=!a)';zc7=3B/g"<^r>)ZN&A g5.2"$&7$*XcA C <_!"„ V†…q‡{ˆ …‰ L\
0g&"$d(O*2"•T{<"):"</'01):01*"{z¥"<201O"[`}'/g!`W'3%I7$1*>)'7="2 [9<"):"<J…B):""3 dVi†Cj'C2F
5.$X*2I`W**2|5.%9"$[=NA/1*2'8):"2“/g*7=z¦XŒ)'**H(7=((@"A('`DA \8C y z’"$K1O7="2?'P)'•T
O3/"2Fgr6+ 2T )://k*2(¥"< P5**_7=;?'!%'"<P"$¥ViO1n;O|\E'/"3J)'*a?:)'*Od`iO7="2‘¸c ² V†…:\.5.2"$
/'0k):01*2"ƒz CmfO"$!3B''F;0g(7!)'O<#r6+ 2T l`}**q5.%
) "<_- ¦/g*7=z`W' )'I2"^(7=()';"!F
„ V†…q‡†ˆ …@‰ ;\,5.**7=;?'!%'#"<&„c
² V†…q‡{ˆ …‰ L\8F;"$#OnLOE"<)'2"2B/'01):01*"{z`iO7="2`}'^=NL(7!O"X%&78$*
6O!)[*2A7!)'**2z6'/"3)'*~/g*X7=z'C «$):"(3)'"<¥0g$q5.‘"$):""$O/):"<(nLO):"2S`}' F
lnLO):"2¦V \E5.*X*k7=;?'!%'d"<"$D'/"3J)'* ²
`iO7="2P5."$P/'01):01X*2"{z C
]aB/|?'D"$!FA5D5.**)://1*2z <G(33) C ‘c5.**(;"2`Wz"$ Kh"$*(33)d5.2"$P)B<"):"<=T{)'7="2
/1)'2.V†…@‰Š\8Cm]R$E?'(7="<' 5.**;0gH"$R7=3B/1*2!"AT{7=<"m?'(7="<' V†…@‰ŠA\a`W'l)'**…@‰Š1FL):`}"<!>aO/):"<R"<!/1!C
]R$D?'(7="<' ²5.**0gD"$D'/"3J)'*7=3B/1*2!"2T{7=<".?'(7="<' ²
V†…‰Š\8C^s#=€kD"$c3)://k% Ç"<0g
؊Ø
Ø
Ø
1
V
Ø
2
\=V†…‰Š\m»
¾{ÀYÁ
„ ² V†… ‡ ˆ …‰Š\=Vi3)ZÁ N Â
½
Ø(¨
V†… ‡ ‰Š ‡ \GÃɸ ² Á V†… ‡ \{ÄW\
½
] $XdXB)
R
O/):"<[O!B"$¥r6st )'O13%9"$):")'**H(7=()';"B?Z)'*XO[`WO17="2!FE¸ ² FE)'
½
"<)'2"2P/'0k):01*2"(!F;„c
² V†…q‡Yˆ …‰Š\8F1$)q?'&7=;?'!%'(C
]ad):/1/1*2z"$*(33)AFL5^3&O1<"m€18<"m=NA/(^"$ ÇO/):"<.`}'3dO*)&"$.`W'3 '`"$#O/):"<.O*2
P"$&*2(33)AC <_!" …0g"$c"):"<c"$):"R(O*2"E`W3o/g!`W'3%dŠh[<"):"<…C
V†…@‰ŠA\
Ø
V 3 {\ V†…@‰ŠA\_à { ¸
V 3 {\ †V …@‰ŠA\_à 2 V
­È»
­È»
Ø
V …Z\
\=V†…@‰ŠA\_à 3V†…‰Š\GÃ
)
V†…‰Š\{Ä
5.$!
)
V†…@‰ŠA\
V†…@‰ŠA\
»
»
Vi3)ZÁ N  V …@‰Š ‡ \GÃM¸
½
„ †V … ‡ ˆ …@‰ŠA\=Vi3J)ZÁ N
¾YÀ Á
½
½
V†… ‡ ˆ …@‰ŠA\=Vi3)ZÁ N  †V … ‡ ‰Š ‡ \GÃM¸ Á V†… ‡ ‰Š ‡ \{ÄW\
½
¾YÀ{Á
½
 †V … ‡ ‰ Š ‡ \GÃɸ Á V†… ‡ \{ÄW\ „ ² V†… ‡ ˆ @… ‰ŠA\=Vi3)ZÁ N  †V … ‡ ‰ Š ‡ \GÃM¸ ² Á V†… ‡ ‰Š\{ÄW\
½
½
¾{À Á
½
Á V …Z\{ÄW\ „
! 3lXR"$d94g!(7=0g!"{5!(6"$&X%*2—=œ'¤.¡£2 #¡A™:žW·1˜ …d8)(5.6)'7!7='%J"<h„ V†…q‡{ˆ …‰Š\H)'K"$
`iO**<"<01O"2„ V†… ‡ ˆ …‰Š\8Cl]R$#?:)'*O#'` ) ~
7!):/"O(E"$194!(7=0g!"ƒ5^!(¥%)'[O/):"<#O1%
"$R7!O(@",/1'01):01*X2"{z"<8)'2"21„ V†…(‡Yˆ …‰Š\~)'B7!O(;",?Z)'*OH`WO17="2'`g"$R7$X*(¸ Á V†…q‡\~)'
½
%d)'PO/):"<O%c"$'/"3J)'*k/1'01):01*X2"{z"<)'2"2„c ² V†…q‡†ˆ …@‰ŠA\^)'h"$'/"X3)'*?Z)'*O(^'`a"$
7$1*(¥¸ ² Á V†… ‡ \8C
½
‘dq5Ê?'!82`}zh"$7=12"2E'` <G(3J3) C
12"2‘Vi)@\HXH)'O3B(KP"$c7=12"2'`~"$D"$!'(35.2"$ V†…‰Š\m» <C
12"2ªVW0\Xc):"<€1(S0g(7!)'O …Pd)'3B/k*2(`W3 „ V†…(‡Yˆ …@‰ŠA\8Fl6"$h=NA/(7="<(’?:)'*OI'`R"$
9 4g!(7=XEb!!C
12"29Vi7q\^`}*X*2q5.E(7="*2z`W3±"$)'O13B/"2I"$):"H"$Jˆ V†…@‰ŠA\!ˆL)'1Sˆj¸ V†…:\!ˆ;):D0gO(C
12"2«Vik\dd"$¥7=12"2’"$):" ®B)95(2%$;"<(Œ3)ZNŒ'3 /1<(O1:T{7=@"<8)'7="2C ‘K7!)'
!?'I"$Xc0Lz¦<"):"%>5.2"$¦"$P5^(2%$;"<(Œ3)ZN¦'3 `}' *2():8%C>-ƒ"B&5(**pL|5.’"$):" 4
)65^(2%$;"<(“3)ZN¦'83u/1(O:T{7=;"<)'7="2ÊV y !"<!p:)'§4]R2"pL*X!F (w'w'¨ \8C]R$):"!F~"$!I=N<")
/g2"2?'?'(7="<' )'[)7!)'*X):
‹  ‰ \8FO78$["$):"R`W'.)'** F
$
%
Ø
ˆ2ˆ ² ˆ2ˆ
Ø
( "/%ˆ2ˆ (
² ˆ2ˆ Z‰
V
Ø
\
5.$!ÇE"$'/g!):"<'
V c\=V†…@‰ŠA\l»
¾{À{Á
„JV†… ‡ ˆ …@‰ŠA\= V†… ‡ ˆ …‰Š\GÓ3J)ZÁ N JV†… ‡ ‰Š ‡ \{ÄYÅ
½
q 5¼5#5**1$q5¼$|5¼"<!2?'#"$D7=@"<)'7="[`}'E"$ ”O/):"<#'/g!):"<' &C O^/1*X)'I"<B$|5
€1"m$q5M"<c=NL/(,"$ «'/g!):"<'m`W'l*2():X%=¼"<!3J,'`g"$É'/g!):"<'l`W'lO/):"X%¼?Z)'*XO(!C
]R$(¥5^5**g!/1*X)'7= ŽX["$c7=;"<)'7="26(n;O1):"2¥`W'–*2():%B5.2"$
BF1)'6$q5 "$):" Ž
)B5(2%$;"<(K3)ZNLT{'3e7=;"<)'7="26O!H"$c)'3BD5(2%$;" B)'["$&)'3B mC
.(7!)'**,`W3 lnLCmV Ì \#"$):
" V†…‰Š\»
V†…@‰ŠA\~ë¸ V†…Z\8Cf1O"$!3''F_"$ ±'/g!):"<'c/g!`W'3
½
2"HO/):"<(EO%d"$D'/"3)'*?:)'*O`iO7="2'`"$D7$X*PL(!F<B5^D7!)'[5.2"<"$E)'
V†…@‰Š\m»
q5 7=B"$J78$*(6'`EA .$)q?'I7=;?'!%'(F"$ #TY`iO7="29?'!82‘'`"$
V†…@‰ŠA\~ë¸ ² V†…Z\8C
½
y (**3)'P(nLO):"2K`}'r6st 7!)'¥0gD5R"<"<([)'
$
1%
V†…‰Š\m»
¾ÀÁ
„ ² V†… ‡ ˆ …‰Š\= ¸ ² V†…Z\GÌ3)ZÁ N ½
½
Øqº
V†… ‡ ‰Š ‡ \{Ä
+
' "7=B"$):"c¸ ² V†…:\R/1*)(zA#"$*2'`m"$B3J3B():"<d!5E):>`WO17="2K`W'
½
"$ Ç'/!8):"<'.7!)'¥0gD!5R82"<"<([)'
V =\ V†…‰Š\l»
q5 5!/k*)'7=
V†…‰Š\0Lz
„ ² †V … ‡ ˆ …‰Š\= ¸ ² †V …Z\_Ó3)ZÁ N ½
½
¾YÀ{Á
V†…@‰ŠA\_ÃM¸ ² V†…:\8F1)'['01")'
½
V \=V†…‰Š\m»
„ ² V†… ‡ ˆ …‰Š\=VY¸ ² V†…:\GÓ3J)ZÁ N Â
½
¾{ÀYÁ
½
Cc].$!!`W''Fg`}'DAF
V†… ‡ ‰Š ‡ \{ÄYÅ
V†… ‡ ‰Š ‡ \ÃM¸ ² Á V†… ‡ \{ÄW\8Å
½
'"<c"$):"#¸&² V†…Z\H;(.'".!/g(h6…q‡XFk<57!)'K3B|?'&2"HO"D"$D=NA/g(7="):"2K)'¥'0")'
½
V \=V†…‰Š\ »
¸ ² V†…Z\_Ã
„ ² V†… ‡ ˆ …@‰ŠA\=Vi3)ZÁ N  V†… ‡ ‰Š ‡ \ÃM¸ ² Á V†… ‡ \{ÄW\
½
½
¾YÀ{Á
½
»
¸ ² V†…Z\_üV
\=V†…‰Š\
½
+01O%h'"):"*2%$;"*2z'F5B5.**a=NA/(D"$19?'(7="<'c`W'3°)'
»±¸c²EÃ
C 3*):8*2z'F15
7!)'¥5.2"<&
V†…@‰ŠA\l»
V
†…@‰ŠA\GÃɸ ² V†…:\EP?'(7="<'#`}'3)' »
M
Ã
¸
C
²
½
q5–57!)'‘O0k<"2"O"<"$(<"ƒ5^¥`W'3dO*)';"<P"$3)ZN>'3 /1<(O:T{7=@"<)'7="‘`}'83&O*X)J`W'
cF ^n;C_V
\H"<'0")'
Ø
ˆ2ˆj¸ ² Ã
V
²
Ãɸ ² \!ˆ2ˆ
( ")%ˆ2ˆj¸ ² Ã
(
VY¸ ² à ² \!ˆ2ˆ |Å
]R$&¸&²#"<!3.7!)'7=(*†F1)'¥5^c$1)(?'
'
ˆ2ˆ
²
ˆ2ˆ
( ")%ˆ2ˆ
²
(
ˆ2ˆ ZÅ
f~)'*X*2z'F2"m,()'zd?'!82`}zPVW|\8F;"$.3B<"m3/'")'@"m7=12"2C y zd)'O3B/"2GF'"$"<_- S/*X7!2(~
"$E7$X*cA(7=@?'!%'H5.2"$&/'01):01X*2"{z "<'/1"3)'*;/*X7!2(G`W'~"$E7$*X(C_].$!!`W'^„ V†… ‡ ˆ …‰Š\
7=;?'!%'(#"<J„c
² V†…(‡{ˆ …@‰ŠA\E`}'.)'*X*G…(‡W‰…‰k)'¥Šh5.2"$P/'01):01*"{z )'16¸ V†…:\E7=;?'!%'(5.2"$P/'0k):01*2"ƒz
½
"<¸&² V†…:\D`W'B)'**^78$*¦)'7="2BŠkC>].$!!`W''Fˆ ) ˆ_7=@?'!%'(B"<9b!!95.2"$¦/'01):01X*2"{z C ‘¥7!)'
½
"<2?A)'**zJ7=<"<O17="R)<(nLO(7= ;a»Žˆ ) 8ˆL"$1):"R0gO"$H7=;?'!%'(7='Fg<
Ø
Ø
*
ˆ ) V†…‰Š\!ˆ
Ø
" * "/*;Vˆ2ˆ
Ø
( Ø \8Å
V†…@‰ŠA\!ˆ2ˆ ^Ã
;F <57!)'I7=7!*OH"$):" 7=;?'!%'("< ² 5.2"$
/'0k):01*2"ƒz C y z[O17="2FL5^c7!)'67=7!*O#"$1):"R"$R$*E`W'.)'**A(RP"$r6+ 2Ž17!*O%
"$R;'"lL'F;<"$R?:)'*O.`WO17="2!/((@"<(J0;z"$.r>+ 2«%'):/1$J7=@?'!%'("<&"$.OXn;O?:)'*O
`iO7="2P'`,)(7!O82?'(*2zI'/"3)'*/g*X7=z'C ‘P'"<h"$):"d` ¿Xc'"d)6<"):"):z3)'7=9$2!8):7$;z'F~"$(¦)63X*):c):%O3(@"&5^O*XS$|5
"$):"mr>+ 2#T 7=@?'!%'(m"<—=™'¤ ,(7!O2?'(*2zD'/"3J)'*$2!):78$7!)'*;/g*7=z`W' C q5!?'!(F;"$R)'7="O)'*
/L'`5O*I09 8h7!O*2"!F@0g(7!)'O55O*1 =j"lpAq5¼=N)'7="*2z5$7$I/g*7=zB5O*I0g <7$<( 0Lz"$
"<_- ª2"<!):"26):".()'78$6A
C y (7!)'O<<3B&(7!O2?'(*2zh'/"3)'*/g*X7!2(H7!)'¥0gc):0k2"<):*zJ5'<
"$)'9'"$!8!F_2"D3B/g'")';"`}'D"$(2%!'` "<[(Od"$):"D!?'!z9'/"3)'*/g*7=zK`}')[78$*
A IE/1):"H'`,)'¥)'7!7=!/"):0k*2zP%';A[/g*7=zI`}'R"$/1):(;"RA C
+#*2%''2"$13 r>+ 2#T 7!)'h0g.=NA"<((h"<B)'7!7=(*2!):"<*():%&"$$2%$!A(^'`G"$#%'):/1$I0;z
)d"<(7$1n;O#"$1):"5D7!)'** <)'**g<"):"<(RO/):"X7% AC ª$(P)'[)'7="2[ŠJE7$<(P`W'Rr>)ZNhA
mP<"):"<
…@F"$c=NA(7!O"2K'`,ŠP5.**g3B|?'&"$c(@?A23(@"R"$O%$K)J<(n;O(7=c'`,<"):"<(D7… =‰!Å!Å!Åq‰… ‰… »Ç… ‡ C
- `mŠ[5)'1!(K"$&0g(<"#):01<"<)'7="D)'7="2>"<P7$L<B9… !F"$(92"$O*K)'*X<I0g&"$&0g(<"#)'7="2>"<
7$;<Vi):"BA •\’<"):"<(… "$O%$’… C (7='F lnLO):"2«V \d7!)'’0h)://1*(’)'**l'`H"$(<
<"):"<(5.2"$¥…»«… m)'1K…(‡»¼… C
‘$1)(?'.?'!8U€1()'**'`"$.7=2"~'` <G(33)
Ø
Ø
Ø
•- >OD3B/1*2(3(@"):"2GF)'()'78$‘7=3B/g2"<B)'7="2‘=NL(7!O"<(F_2"7=<"<O17="#)P*Xp'(6*X<".'`^"$
<(nLO(7='`~/832"2?'<"):"<("$):"R5!?A2"<([)'1["$c)'L7!):"<(K!5E):H"$):".5^!c(7=(2?'(C]R$
*"mm!"O(B5.$(B"$7=3/2"<.)'7="2I"<!31):"<(!C,]R$R/1):(@"r6)ZNJA.7!)'I"$(J/A7=(m()'78$
V†… (‰V†… Zˆ … •‰Š ƒ\<\R/1)'2#["$*<")'K)://k*2z lnLO):"2¦V \8C]R$c/1):(;"r>)ZN6A&)://g(1H"$
<"):"<=TY!5E):*X<""$):"m2"(7=(2?'(,`W3Ë2",78$*(&)'d/1)'<(~"$(3Ë"<2"/1):(;"~5.$(2"~"<!3X):"<(!C
+#**g=NL/g!X3B(@"HXP"$H/1):/g!H(3B/k*2qzP)'**<"):"<(#O/):"%C
):(*01*%¥V (w'w \.;"<L1O7=(R)B(*X):"<(F01O"3B'/gq5!`iO*†F3B!"$L[`W')'7!7=(*!):"%J$12!):78$7!)'*
(`}'7=(3(@"c*():%P"$):"&$7!)'**0
<)'**,%')'*&O/):"%C –].$3B!"$ASDO2"):01*2B`W'c)¥r>+ 2
$2!8):7$;z7=@")'X%¥1*2z)¥L'"d")'<pS)'Sh*!?'(*l'`R7=3B/g"<J")'<pA!C6]a6O1!<")'1)'**^%')'*
O/):"%FEO//g<K"$1):"I`W'P()'7$«/X32"2?'¥)'7="2GF."$!9):9<!?'!)'*7=3B/g2"<>")'<pAh"$):"[7=O*
$)q?'c;?''p'(P"$):"H/3J2"2?')'7="Cm-•[)'**%')'*RO/):"%F;5.$(!?'!R)/3J2"2?')'7="[E=NL(7!O"<(F
lnLO):"2«V \&d)://1*2(¦!?'!z¦7=3B/g2"<h")'<p"$1):"d7=O*X¦$)q?'[@?''p'(’"$):"/832"2?')'7="C
AO"<"<GFRtm(7!O/_FR)' AX%$ÇV (w'
w \I/q?'>"$):"I()'78$ª'`D"$>7=3/2"<K")'<pAh5.**H7=;?'!%'‘"<’"$
'/"3J)'* Ž?:)'*O(HO1!R)'**g%')'*O/):"%C
+#**A%')'*mO/g1):"%#5^O*X5^'p"$Rr>+ 2¼$2!8):7$;zc`W'l7=3B/g2"<H")'pLm)'*X*A'`k5.$<R7$X*(
):h/3J2"2?'J)'7="21!/
C q5!?'!(F^)'d5h$1)(?'[<!(Fm):"B$2%$!&*2!?'(*d¦"$P$2!):78$@z'F,A !(
"<['0")'9)'3/1*2('`l(O*2"<"):"<(8)(5.‘)'7!7='1%P"<¥„c
² V†…(‡Yˆ …@‰ŠA\`W'7=3B/g2"<")'<pAŠkC&+#**%')'*
O/):"%d7!)''"R/|?LD"$(<c)'3/1*2(!F<J"R7!)''"R0g)://1*X2(P):"R"$(<c$12%$!R*2!?'(*(C
Ø
Ø
Ø
Ø
1'Hg
8
O!?'(*2'/13(@"E<c`i):H$)')'O3B(J"$):"!?'!zJr>)ZNIA$)')'7!7=(E"<d"$.`iO**1<"):"<'`G"$O1!<T
*2zA%r>st
C <G!"Ol|5Q7=!^X@"<AO7!X%D)'J):01<"<8)'7="2J`iO7=" V†…‰Š\~`W'l()'78$Pr6)ZNJA
<V†…‰Š\.3):/1R"$c`iO**G<"):"<B…d;"<I)J!"'`~`W():"O(#"$):"):0k<"<)'7="`W'3):"2
):0gO","$R<"):"<"$):"l~(<(@"X)'*`}'l!/(<(;"%"$ V†…@‰ŠA\~?Z)'*O(!
C «2"$B"$1,):01<"<)'7="B`WO17="2F
"$ËL`W'R)'7="2¥ŠI5.**k<"<'"$):0k<"<)'7="<( Ë`WO17="2 V V†…@‰ŠA\8‰Š\5."$P?Z)'*O#`W'R()'78$
"7="H):01<"<)'7="<"):"< V†…@‰ŠA\8C
L"):"<h):01<"<)'7="¦$)'c"$!J)'?Z)';"):%'(!CKf~2"!F_"$h)'3BO;"'`R3B(3B'z‘!(("<6"<'J"$
?:)'*O(_7!)'d0glO01<")';")'**2z#(O7=(GFZ0g(7!)'O^2(*2!?Z)';"_<"):"<E`W'3):"ca'"_!/1(<(;"<(C A(7=F
"$)'3BO;"^'`=NA/1*2'8):"2I!((J"<B*2():8"$
?Z)'*O(^7!)'h0gR(O17=(FL7=."$#LO3&0!^'`G?:)'*O(
"$):"E!(h"<0#*2():(h^(O7=(GC~f~1)'**2z'F@"$#*2():8(IO0")'<p7!)'h0g(O<(h3B'#()'*2z'FL0g(7!)'O<
2"R!/g(1^6)B3)'**!R<!"R'`,<"):"<c?:):):0k*2(!C
!E5.$):"E7="27!)'P"$D):01<"<)'7="<([r>+ 2”$12!):78$@zJ7='(7="*2zJ!/(<(;"E"$?Z)'*O`iO7T
"26'`l)h/g*7=z S].$cp'!z¥"<[)'5^!X%J"$XnLO(<"2>"<h'"7=&"$1):"#"$0
V†…‰Š\R?Z)'*O(#):&*z
!?:)'*O):"<([h<"):"<(R…"$):"E(O*2"`}3±=NA(7!O"%):01<"<8)'7="R)'7="2PŠ1Cm]R$!!`}''FL"$):0k<"<)'7="2P`iO7T
"2 •V†…@‰ŠA\1*2z6!(1"<¥(Od"$1):"2`m"{5K 4!(;"D<"):"<(&… )'… 7!)'‘(O*2"`W3 =NL(7!O"%
)'7="2‘Š¥)'1>`m"$!zK$)q?'9 4g!(;" V†…‰Š\.?Z)'*XO(!Fg"$(>"$B):01<"<)'7="6`iO7="63dO<"D"%O$
0g!"{5!(>"$(3[C#-ƒ`l7… )'9… $)(?'d"$)'3B5
V†…@‰Š\R?:)'*O((F1"$(>"$d):01<"<)'7="2>`iO7="267!)'93):/
"$(3o"<J"$c)'3B&):01<"<)'7="."):"<'C
+ŽX3*):R):%O13B(@"#)://1*(H"<J/832"2?')'7="!CH]R$d):01<"<)'7="2K`WO17="2¥`W'./132"?')'7="2
$1)'"$`W'3
V†…:\#- `H)[/3J2"2?'B)'7="2 (7=(2?'(d9 4g!(;"D=NL/g(7="<(!5):15.$(9=NA(7!O"<(’
"7="H<"):"<(7… .)'¥… F"$( V†…:\E3dO<"R"%O$J0g!"{5!(¥"$<c<"):"<(!C
]aP`}'3J)'*2b!c"$1!Fg=€kd
 q… ) V†…‰Š\H» |…q‡,ˆL„ V†…q‡†ˆ …@‰ŠA\ "<P0gc"$!"'`<"):"<("$):"D7!)'
(O1*2"E`}3)://1*2zA%):01<"<)'7="#)'7="2¥ŠP[<"):"<…):"#L
O!R$12!):78$7!)'*g/g*7=zJ¬~C
E)'K78$*¥)'7="6ŠkC
¶˜W›A ~¶!™:£W£U™ >,žW· ˜)>l™š™:·:žW˜iž†™:·—›L™:£ "1m˜W›A =·6˜W›A * , .0/ Ÿ<œ8¡1›+>,ži˜W›Iœ@¢=—˜iŸ<œš(˜iž†™:·d¶8´A·kšq˜iži™:·—
†V …@‰ŠA\Kœ:· V†…Z\¥šœ:·«Ÿ ƒ¡kŸ —( =·1˜d˜W›A K³Zœ:£µ´1 D¶´A·kš(˜iž†™:·¼™•¶Kœ:· [¡A™:£µžišPK¬ >~›L™:—! 6³Zœ'£È´1 ¶8´A·kšq˜iži™:·¼šœ'·¼¢8 Ÿ ƒ¡kŸ —( =·1˜{ K¢@J˜W› * , .0/ Ÿ<œ8¡1› >,ži˜W›¥·k™Pœ@¢=—˜iŸ<œš(˜iž†™:·¶´A·kš(˜iž†™:·—
!"#
Ø(w
7 œ 9 _™:ŸHœ:£}£Lš™:¤¡A™Z—žW˜{ *œ #·k™
@ =— 1Gœš(˜iž†™:·—~Š1—˜Yœ˜{ —… ‹  1œ'· :ž}—˜ižW·kš(˜k—˜Yœ˜{ —…7!‰… - ‹
>~› =· !³: =Ÿ V†…7!‰ŠA\ » V†… - ‰Š\ži˜mž}—c˜W›A š8œZ—! d˜W›Lœ˜ V†…7!‰ŠA\ » V†… - ‰Š\
* œ
·™
@ —œ:· :ž}—˜ižW·kš(˜—=˜Yœ'˜{ —R…7œ'·… - ‰B>~› =· !³: =ŸR¸ V†…78\ »Ê¸
7 ¢ 9_ ™:Ÿœ:£W£¡kŸžW¤dži˜ižW³: 
š8œZ—! d˜W›Aœ'˜ V†…\ » V†… \<3
A "B!CED y z]R$!'(3 Ø F"$!J=N<"&?Z)'*O(`W'&"$e)'1’¸®`iO7="2cO7$¦"$):"B¸
|… ) V†…@‰Š\
V†… \žW˜~ž}—#˜W› -
!7 )'¦0J!/T
(<(;"<(«0LzÉ"$‘r>+ 2 $12!):78$@z'C O//g<>"{5M<"):"<(K…7P)'¼… $1)(?'94g!(;"P?Z)'*O([¸ V†…7\
)'S¸ V†… \8CB]R$(F"$!3&O"D=NA"3BA .O78$9"$):"D(2"$! .)P/3J2"2?'r6)ZN‘A)'
¸ V†…7\ »”¸ V†… \H' E.)I7=3B/g2"<dr6)ZNKAc)'1 V†…!‰Š\ »
V†… ‰Š\E`W'3Bd78$*K)'7="26ŠkC
]R$H7=12"2,d"$H"$!'(3–(OE"$):"m"$X,7!)'<'F@"$R):0k<"<)'7="2B`iO7="`}'lL
^Vi)'1)'7T
"2PŠ\,5.X**1X<"%OX$0g!"{5!(P"$(<#"{5<"):"<(!C(7='F"$!#=N<"?Z)'*O(l`W'E"$#"):"<=T{):01<"<)'7="<(
2o$12!):[email protected]"$
= )' ¸ = cO7$"$):"D"$B?:)'*OB`iO7="29'`E¬Œ7!)'‘0gB!/((@"<(‘0Lz>"$r6+
"$%?'(¥):01<"<)'7="[`WO17="2!C Ø
]R$EX3B/g'")'7=E'`k"$H(O*2",<"):"<(^
 q… ) V†…‰Š\5E)'~€1"~'"<(B0;zs#()')' <a[V (w'w'¹ \8F;5$!
"$!z9):B7!)'**2(9"$/g!2/k$!)'*<"):"<(D'`EO0")'<p>Š1C ‘B5.*X*a<!B"$1):"r6+ 2T 3):p'(cn;O"<"<%
)'O13B/"2):0gO"."$&(*):"2?'d?:)'*O(R'`l"$(<d(O*2"<"):"<(#`}'#()'7$‘):01"<)'7="")'pkF)'6"$):"."$(<
)'O13B/"2H*X32""$):01*X2"{zI'`"$r6+ 2–3!"$LP"<€kP"$'/1"3)'*g/g*7=z'C
]R$!'(3
;(^'"m%O):)';"<!#"$):"l)'*2%''"$3–r>+ 2T 5.*X*L7=;?'!%'#"<c"$.(7!O?'(*2z&'/1"3)'*
/g*7=zO1!l"$%2?'(P<"):"<):0k<"<)'7="2(C .|5^!?'!(F10;z=NA"<(X%d)'P):%O13B(@"E`W3 y !"<!p:)'R)'
]R"2pA*!F'5#7!)'P/1q?'#"$#`}**q5.%&*2(33J)AF;5.$17$I<"):"<(E"$):"E`}'E7=!")'Pr>stl!F2`"$):01<"<8)'7="<(
Ç`iO7="27!)'P!/(<(@""$'/"3J)'*1?:)'*O#`WO17="2 ²|FA"$( Ë*2():X%5.*X*17=;?'!%'"< ²R5.2"$
/'0k):01*2"ƒz CH]R$XE(O*2"!/g(E¥"$c`i)'7="."$):""$c):01<"<8)'7="<(
–`iO7="2¥XR<"<'(>)'.)J"):01*2'C
y )'2>V (w'w'¹ \^/(<(;"^)d7=O;"<!=N)'3B/1*2`}'"$7!)'<5$!R"$):01<"<8)'7="<( ?:)'*O(^):!/(<(;"<(
)'0Lz6)[*():`iO7="2G&
C X#7=O1@"<!=N)'3B/1*B(*2(#9"$`i)'7="D"$):"D"$B*():#`iO7="29;"<AO7=(
)'¥O1(2):0k*2#7=O/1*%d0g!"{5!(¥"$
Ë?:)'*O(H'`"{5I<"):"<((C
Ø
Ø
# #%$ % !˜ ¢8 #œ&—˜Y™(š!›LœZ—˜iž†š#—8›L™:Ÿ=˜{ —˜k¡œ'˜W› *'&)(M—8´š!›B˜W›Aœ'˜,œ:£}£@ !˜{ !Ÿ8¤žW·žX—=˜ižiš¡A™:£µžiš(ž† =—#œ:Ÿ l¡kŸ<™8¡ =Ÿ3
(
% ˜ c ²K¢ I˜W›A h™8¡˜ižW¤œ:£ ¶´A·kš(˜iž†™:·h¶=™:Ÿ 1#œ:·K£ !˜ V†…@‰ŠA\I¢ hœ:·Œœ@¢—=˜iŸ•œ@š(˜iž†™:·[¶8´A·kšq˜iži™'·S—´š!›9˜W›Aœ'˜_¶!™:Ÿ
œ:· ¡Aœ'ž}Ÿ™•¶—˜Yœ'˜{ =—E…7cœ:·… œ:·œšq˜iži™'·JŠ1~ž ¶3 ² V†…7!‰ŠA\ ») ² V†… ‰ŠA\#˜W›A !· V†…7(‰Š\ » V†… ‰ŠA\<3 1´|¡¡™Z—! ¬mV†…:\Ež}—Hœ—˜Yœ'˜iž†™:·kœ'6Ÿ &—˜Y™(š!›LœZ—=˜ižiš ¡k£2™:Ÿ<œ'˜iži™'·¡A™:£µžišPD—8´š!›˜W›Lœ˜a˜W›A !Ÿ< .ž}—.œD·k™:· q =Ÿ<™.¡kŸ<™@¢œ;¢=žW£Èži˜ d™•¶ ; š!´˜iž}·
=³' =0Ÿ œš(˜iž†™:·KžW·> !³: =0Ÿ Ÿ< œš=›Aœ@¢=£ —˜Yœ'˜{ 354_›A =· Æ£ œ:Ÿ·žW· >,ži˜W›Pœ;¢—˜iŸ<œšq˜iži™:·¶´A·kš(˜iž†™:·
>,ž}£W£gš8™:·³' =Ÿ c˜Y™
5
² >,žW˜W›d¡kŸ<™@¢œ@¢=žW£Èži˜ 7 œZ—8—8´A¤ž}· ˜W›A &£2 œ:Ÿ·ž}· Ÿ•œ'˜{ =——=œ˜ižX—W6¶ (· 3 7 9
9 3
#
A "B!CED ,
1!_"$%2?'(d):01<"<)'7="c`iO7="2 lF:"$E<"):"<E</1)'7=E7!)'&0gm/k):"2"2(&;"<(nLO2?:)'*2(7=
7!*)'( Z!‰!Å!Å!Åq‰8 O7$B"$1):"l`W'l)'@z/1)'m'`"):"<(…7!‰… XB7!*)'^ )'!?'!z)'7="2JŠkF! ² V†…!‰Š\m»
²:V†… ‰ŠA\8C><G!" V‰ŠA\!/(<(;"H"$JVi7=33B\Ë?:)'*O`W'.)'**<"):"<(#… ‹  C1!E"$<")'1):
#T{*():%BO/):"<O1*2
Ø
JV‰ŠA\H­È»ŽV 3 ƒ\ V‰ŠA\aà  V†… |ˆ … <‰Š\GÓ3J)ZÁ N JV '‰Š ‡ \{ÄY‰
½
5.$! l"$.(nLO2?:)'*2(7=#7!*)'l"$):"7=;")'E… a)'P .Xl"$(n;O12?Z)'*(7=.7!*)'^"$):"E7=@")'X… (F
"$#<"):"<"$):"^(O*2"<(J`}3 `}*X*2q5.X%"$=NL/1*'):"2I/*X7=zdXI… Cm].$*2(3J3)7!*)'3Jm"$):""$lO*2
5.*X*7=;?'!%'&"<"$ Ž!/(<(;"):"2['`~"$c'/"3)'*g/g*7=z ²|C
" Æ*2():8%PO/):"<()://1*(K"< ÆO1!
y !"<!p:)'c)'1>].2"2pA*dV (w'w'¨ F(7="2 ¨ C º \$|5Ž"$):
)¥<"):"):z9<"<L78$)'<"7=NL/1*'):"2‘/g*7=z>¬“5.**~7=;?'!%'J"<K"$JOXn;O<*O"29"<¥"$B`W**2|5.%
(nLO):"2!­
JV ‰Š\m»
V†…Aˆ Š1‰ \
„JV†… ‡ ˆ …@‰ŠA\= V†… ‡ ˆ …‰Š\GÓ3J)ZÁ N JV '‰Š ‡ \{Ä
V =x \
¾À
¾ À{Á
½
Ø
Ì
Ø
5.$! #^"$.(nLO2?:)'*2(7=#7!*)'"$):"E7=@")'XH… ‡ )'I5.$! V†…Aˆ Š1‰ •\lXl"$<"<()'z;T{<"):"</'0k):01*2"ƒz
"$):"D"$J=NL/1*'):"29/g*7=z¥5**a0B‘"):"<I…%?'(9"$):"c2"DD9"$B(n;O?Z)'*2(17=7!*X)'D'`E<"):"<(d )'¥"RH7$L%B"<=NL(7!O"<&)'7="26Š1C
(7=&)'*X*G"$):"(3)'R"<J0gc$|5.KXH"$):" c²D):"<€1(
^n;CV =x \8CR]_h<!c"$X!F1'"<&"$):
" c²D
"$c*O"2P"<
² V†…@‰ŠA\l»
„JV†… ‡ ˆ …@‰ŠA\= V†… ‡ ˆ …‰Š\GÓ3J)ZÁ N ² V†… ‡ ‰Š ‡ \{Ä
V (¹ \
¾YÀ{Á
½
Ø
Ø
y (7!)'O< ² 7!)'¥!/1(<(;" ² F5c7!)'¥=TY=NA/(H"$HP"<!83R'` ² Clf'#)'@z[<"):"<… ‹  F
² V‰ŠA\l»
¾{À{Á
„JV†… ‡ ˆ …@‰Š\= V†… ‡ ˆ …@‰ŠA\GÒ3J)ZÁ N ² V '‰Š ‡ \{ÄY‰
½
Ø
V (¨ \
5.$!B…q‡ ‹  C q5 5^pAq5–"$1):")'**~<"):"<(c… ‰… ‹ 
^$1)(?'"$B)'3 ²&?Z)'*XO(!F<[57!)'9"):p'
)';zJ):012"<8):z5^(2%$;"<(PO3 '`"$<#?Z)'*XO(!C~- I/1):"X7!O*):(F;5^7!)'h"):p'#"$.5(2%$@"<(PO13 )'7!7='1%
"< V†…Aˆ Šk‰ \8­
² V ‰Š\m»
V†…Aˆ Š1‰ \
„JV†… ‡ ˆ …@‰ŠA\= V†… ‡ ˆ …‰Š\GÓ3J)ZÁ N ² V ‰Š ‡ \{Ä
V qº \
¾À
¾ À{Á
½
Ø
]R$XH:<O1<" l
nLC_V =x \8C
Ø
y !"<!p:)'J)'1Œ]R2"pL*X&</g(7!O*X):"<‘Vi(7="2 ¨ C º C ¨ \B"$):"2"BB'"B/g01*2h"<‘(3Bq?'6`}3 "$
*2(33J)6"$[(n;O12(3B(;""$):""$[=NA/1*2'):"“/*X7=z0g[<"):"2):z'CŒ]R$1BO`W'"O):"<'F^0g(7!)'O<
2`H"$[*2(3J3)67=O1*S0gh=NL"<(1(“"<‘)://1*2z‘"< "<_- Ë/g*7!(!F~"$(“2"d7=O1*S0gP7=3c0k(’5.2"$¦"$
7=<"<8O7="2d&].$!'(3 Ì "</|?'^"$1):"r>+ 2#T 5.**7=;?'!%'E5.2"$&/'0k):01*2"ƒz "<#"$^(7!O2?'(*z
'/"3J)'*;$!):7$17!)'*@/g*X7=z7=1<"<(;"~5.2"$&"$r>+ 2ª$2!):78$@zhVi2`"$E):01<"<)'7="2`iO7="_):"X<`}z
"$J7=12"2'`E].$!'(3
\8Ch-0(*X2!?'B"$):"c"$1*2(33J)[7!)'0gJ<:TY=NA"<((F01O"D"$1D(3)'X)
O0 :•(7="H`W'R`iO"OD(():7$C
- "HE@"<!(<"%d"<B'"<D"$):"Hh"$5^'p'` AX%$>V (w'w'Ì \R)' ):(*201*X%IV (w'w \5$!D(7=3B/g:T
2"R3J*):H"<Pr6+ 2Ž5!&€1<"@"<LO17=(F()'7$>'`,"$d):01<"<)'7=")'7="2#$)'R1*2zPc"<!3J)'*
<"):"<'CB<(nLO(;"*2z'FG]R$!'(3
)'O"<3J):"7!)'**2zK):"•€1(F)'9"$5
?:)'*O():""$B):01<"<8)'7="*2!?'(*
5!c):01*2D"<I<"<'*zJ?:)'*OD`W'R()'78$6):01"<)'7=".)'7="2C
<_!"¥O[7=!h5$):"¥<"):"<):01<"<8)'7="2¥):):/1/'/X):"<6`W'["$])ZNA")'<pCof'¥"$ ( ),+.-6/;F
1#)435-6/;F 7495:;-F)' < >@:;-#/3J2"2?'c)'7="21!F"$B):01<"<)'7="9`WO17="297!)'‘3):/‘)'**~<"):"<(;"<[)h%*2
<"):"<'Fa0(7!)'O1<"$B!5E):‘#"$7=<")';"D?:)'*O Cf' ( ),+.-6/;F 1#)43L-0/@F 7495:;-FG)' <?>@:;-FG"$
):01<"<8)'7="2`iO7="2zA2(*1 :<O"m"{5c`W():"O(!­~"$7!O(;"m*2A7!):"2'`g"$R")ZN1)'1B"$.7!O(@","):%'!"
*2A7!):"2[VW5.$78$c5E)'~/1)'(&)',)/1):)'3B!"<!=\8Cm]R$a):01<"<)'7="2d`iO7="2d):"<€1("$^"$!'(3Ë0g(7!)'O<
"$
Ë?Z)'*O!/g(E*2zI["$(O*2"%*L7!):"2P'`~"$c")ZNAG)'1["$D"):%'!"#*2L7!):"C
f' ( 9 C ,95- >G),+ K>@-F"$&):01"<)'7="2>`WO17="2K7!)'>!"O:O<""$d7!O(;"*2A7!):"2>'`,"$d/1)'•T
<(%'!)'K"$&("):"26*2L7!):"CR]R$.H0g(7!)'O<d):`}"<! ( 9 C ,95- >G)+K>@-E€k$(H=NA(7!O"%F"$
")ZNGR)'*25E)(zA.*L7!):"<(¥):"."$c7!O(@"R*2A7!):"2['`~"$D/k)'<(%'!(Fk<B"$R):"•€1(H"$D"$!'(3[C
f' ( 9 C ,95- >G)+ AD35-8F:"$l):0k<"<)'7="2c`WO17="2D7!)'c3):/&)'**<"):"<(;"<.)R%*2~?Z)'*Om0g(7!)'O<m):`W"<!
7=3B/1*!"%c"$ 9
4( 9 C ,9L-N>:FL"$!m*z)c%*2)'7="2I(3)'X%"<d0g./g!`W'3B(>NV AD35- MI) ORQ'\8F<
"$d«?Z)'*OXR7=<")';"!C
]R$):01<"<)'7="2`WO17="29`}' 9 #A C EGFG3,HJ3dO<"X<"%OX$K0g!"ƒ5^!(‘"$J<O7=*L7!):"2S)'‘)'**
AT{O7=*L7!):"21VW0g(7!)'O<)/g()'*2"ƒz'` Xm%2?'(I`}'EAT{O7=*L7!):"218\8C,+)'*'%'O*2z'FL"$
):01<"<8)'7="2P`WO17="2P`W' 9 #AD35- MI) OQc3&O1<"H<"%O$J0!"ƒ5^!(¥"$D<"):"<ch5.$X7$h"$D/1)'(%'!R
P"$c")ZNAG)'1["$D")ZN_H):"R"$c(<"1):"2[*2A7!):"2¥)'¥)'*X*g'"$!<"):"<((C
«2"$d"$R):01<"<8)'7="2`WO17="2a%2?'(B):0gq?''F@].$!'(3
~):"•€k(&`W',"$E'/"3J)'*;/g*X7=z'
C (7=
"$r6+ 2±$2!8):7$;zK7!)'>!/((@"#"$d'/"3)'*a?Z)'*XO&`iO7="K)'9r>+ 2T 5.X**_7=;?'!%'B"<P"$
Ø
Ø
JØ
JØ
DØ
Ì
Ø
2
Optimal
1
Reward per Step
0
-1
Hierarchical
-2
-3
Flat
-4
-5
0
500
1000
Trial
1500
2000
f~2%O º ­3B/k):<P'`k):".)'[$12!):78$7!)'* 2Ë*2():8%["$&3B/1*#])ZNAG")'<pkC
'/"3J)'*a/g*X7=z'C#- "D;"<!(<"%h"<['"<"$):"D"$![AK"$$12!):78$@z¥"$1):"7!)' <<! P"$
7=3B/1*!"<#<"):"<'`"$#(;?L23B(;"!FL)'Jz'!"E"$#$12!):78$@z7!)'h<"**k*2():J"$'/1"3)'*?:)'*O`WO17="2C
]a‘?'!2`Wz"$=NA/g!3B(;")'**2z'F,5[3B/1*(3B(@"<(“r>+ 2#T `}'B"$P])ZNH3)'“)'Œ7=3B/k):(
"$[r>+ 2 $12!):78$@z«VW5.2"$’"$[):01<"<)'7="B*<"<(“):0gq?'|\&"<
):" °*2():%
C ‘¥(3B/1*2|z'(“"$
y *2"<b(3)'“=NA/1*2'8):"2’/*X7=z5.2"$“)'Œ12")'*m"<(3/!8):"OP'` ¹ C l)'78$Mr6)ZNŒA c3)';")')
<!/1):8):"<d"<(3/!8):"O&)'16(7=()'<(D2".0LzK)J€NA(K`i)'7="<'):`}"<!()'78$‘O7!7=(<`iO*a=NL(7!O"2>'`mO0")'<p
C>f'k):" ®*2():8%F"$h"<(3B/g!):"OI5)'3dO*2"2/1*X2(90;z C w ):`W"<!()'7$ŒO7!7=(<`WO1*m"<)'*Yv,5.2"$
r>+ 2dFg"$"<(3B/g!):"O&5E)'D3dO*2"2/1*X2([0Lz C ¨?º C&]R$(<&?:)'*O(#5!B7$(9"<h'/1"32b!d"$d):"<
'`~"$*2():1%5.$*!")'X%B7=;?'!%'(7=d"<"$'/"X3)'*g/g*7=z'C
f~2%O º $|5.."$&/g!`W'3)'7=&'`m"$(<"ƒ5^[3B!"$A!
C ‘B<!"$):""$d$!):7$17!)'*_r>+ 2#T )'*2%''"$3u7=;?'!%'(B3dO7$¦`W)'"<!&"$)' k):6" ®*2():%C y '"$“3B!"$A&7=@?'!%'P"<>"$J'/1"3)'*
/g*7=z'C
-•&3B^/1'01*2(3(Fq5^5E)'@""<@"<LO17=^<"):"<E):01<"<)'7=""$):"a?A2*):"<]R$!'(3 C,f'~=NA)'3/1*2'F
7=X!H)B3BL1U€k7!):"2I"<B"$D")ZNg/'01*2(3±"$):"H5D5.*X*17!)'**g"$DfO(*]a)ZNg/'01*(3 Vi!f%O \8C
-•"$c3)'Fa()'7$ ( )+.-0/@F 1#)435-6/;F 7#95:;-' <?>@:;-D)'7="2¦7=1O3B(cIO12"'`H`WO(*YCI]R$J")ZNA7!)'
'0")'S3B'B`iO(*~0;z>%'%["<6*2L7!):"f¼)'9=NA(7!O"%¥"$B/3"2?'d)'7=" #C 3IH@FG5$7$>€**#"$
`iO(*")'pPO/¥"< (Ì O2"E'`~`iO(*†C,- `a"$")ZNAa):"<"<(3B/"."<I3):p'd)3Bq?'&5.$(P"$c`WO(*_*2!?'(*GHb!!Fk)
/g()'*2"ƒz¥'` Ì %2?'(FG)'16"$d!/k<Ac"<!3J):"<(!C- `m"$")ZN~):"<"<(3B/"D"< #C3,H5.$(92"'"
):"H*2A7!):"2Pf^FA"$(P)d/g()'*2"ƒz'` ^%2?'(Cl].$<"):"</1)'7=`}'E"$^/'0k*2(3 X3dO78$h*X):%'!E"$)'
`W'd"$P3B/1*2I")ZNA/'01*2(3[F~0g(7!)'O<P"$![): =x /g20k*2`iO(*E*2!?'(*[V "$O%$ (Ì \8F<9"$!
): º F /201*2<"):"<(!CJ- 9"$"):"%¥<"):"<'Fa"$J)'3BO1@"D'``WO(*,D7$(O2`W'3*z[)'3*z
0g!"{5!( ¹ )' (Ì O12"!Cf?'cO2"R/q?A(R(O%$[`iO(*`W'."$c")ZNA_"<J()'78$6*2A7!):"26f“`}3®)'@z
7=(**†C
]aD(2%d):01<"<)'7="~`}'~"$Xa!5“")'<pF5E`W)'7=H)7$X7='­$O*5E/|?L^"$E7!O(@"~)'3BO;"
Ø
Ø
JØ
Ø
Ì'Ì
JØ
Ø
MaxRoot
4
R
G
3
QPickup
QGet
QRefuel
QPut
MaxGet
MaxRefuel
MaxPut
QNavigateForGet(t)
QNavigateForRefuel(t)
QFillup
QNavigateForPut(t)
QPutdown
2
F
1
0 Y
Pickup
Fillup
Putdown
0
B
1
2
3
4
MaxNavigate(t)
QNorth(t)
QEast(t)
QSouth(t)
QWest(t)
North
East
South
West
Vi)@\
VW0\
f~2%O ­,]R$.fO(*])ZN/'0k*2(3[­Vi)@\,"$R5'*5."$!5É*2A7!):"2If^FVW0\"$.7='(</g%r>+
):%'8)'3
2
'`l`iO(*a"<["$d/3"2?'&A(%( )+.-0/@FK1 )43L-0/@F 7#9L:;-FG)' <?>@:;-)'>"$(2+#T{Ld/1):(;"
É]_[):"X<`}z
]R$!'(3
"$!((Fl0(7!)'O1<P'"$!5.h"$(<¥A(7!)''"B!/(<(@""$[=NA/g(7="<(“!5E):
VW'Cj%C2F_¸ V†…:\<\H5.$(¥"$d)'3BO;".'`,`iO(*_Rb!!9ViYCj'C2F<I"$&=NL"#)'7="2K5**g%'!")J!5E):K'` Ì \
'D"$=NA/(7="<(‘7="•TY"<:TY%'K5.$(>"$B)'3BO1@"'``WO(*#%'():"<!"$1)'9b!!C >"$'"$!$1)'Fg"$
7$7=B):0O"5.$17$9)'7="2>"<[/g!`W'3®1O% 9 #( 9 C ,95-N>&()'**z¥!/g((;".'`l"$B)'3BO;"#'`
`iO(*†
C ‘5O*‘*2p'"<K)q?'"$ 8I7!2(17=zK'`E*2():%h<!/k):):"<(*2z‘$q5 "<K)(?A2%):"<`W3 $Æ"< y
5.$("$!): ¹ O12"~'``iO(*(3)'X%F ¨ O2"~'``iO(*(3)'%F º O12"~'``WO(*(3)'1%F@)'I<
C^1<(n;O(@"*2z'FA57$D"<B%2?'"$(<A(*2z"$`W'3):"2I"$!zJ5!D%2?'([XI"$3B/1*2
])ZNA")'pk­m"$c7!O(;"R*2A7!):"2['`~"$D")ZN_)'P"$"):%'!"*L7!):"2¦V F y 8
F "d4F $F'#f,\8C
]R$1^(7!2P7=():"<(H5$):"E5^D7!)'**"$ ž† !Ÿ•œ'Ÿ•š!›@žiš8œ:£ Ÿ :ži˜ ,—8—8ž ·¤J =·k
˜ (EŸ<™@¢=£ =¤C «$(P"$")ZN
=N$)'O<"a`WO(*;)'c(7=(2?'("$ Ì /(1)'*2"{z'F:"$(<EA(7!)' =j"a!/(<(;"_"$
•()'< .`W'~"$G/g()'*"{z'F
0g(7!)'O<"$!z>$)(?'Jh5E)(z>'`^!"<(7="%["$):"D"$X#"):"<B5)'19 4!(@"`W3 )[<"):"<B5.2"$ (Ì O"'`
`iO(*g(3)'1%C
+#**_X'"*2<"!F$|5^!?'!qF0(7!)'O1<c"$d()'*/'0k*2(3."$):"DA($12%$!X¥"$d"<!¥VW'Cj%C2F K>@-F
AK3L-F_)' > 3 > \#$O*9'"$)q?'J;?''p'( 9
4( 9 C ,9L-N>9"$D2"O):"2>"<¥0g!%95."$Cd].$<
A(3dO<"*2():8‘"<K;?''p' 9
4( 9 C ,9L-N>*2z9‘<"):"<(&5.$!2"$)'D(O%$‘`WO(*,"<K7=3B/1*!"<"$
)q?L2%):"S")'<pC (7='F~"$I<*XO"2‘"$):"&5^h$)(?'P)''/"<(’D"< @ š™:¤¡A™Z—( ¥˜W› hŸ< >lœ:Ÿ ¶8´A·kšq˜iži™'·C
f'#()'7$6!5E):["$):".7!)'K0g%2?'(F"$c(2%!H'`"$cr6+ 2 $2!):78$@zP3dO<".(7!XD5.$78$¥A(
$O1*J(7=(2?'D"$D7=(2"VW'H01*X)'3B|\l`}'H"$!5E):C,-•h"$")'<pF"$7=("E)'2%(P)'R$|5.P
]):01*2 C
]R$1(nLO2()678$)'%'h"<6"$h(7=3/2"2(nLO):"21!0
C <G!"d V†…q‡{ˆ …‰Š\D0g"$I/g'"2'`R"$
Ø
Ì
Ø
Ø ­.!5):K`WO17="2[(7=3B/g"2P`}'."$fO(*a]a)ZN")'<pC
]):01*2
R!5E):¥'ta(1)'*2"{z
<"R'`~/832"2?')'7="2
<"R'`,*X*2!%)'* A C EGF 3IH
<"R'`,*X*2!%)'* AD35- MI) ORQ
<"R'`,*X*2!%)'* #C 3IH
R!5E):¥`W'.(*X2?'!z
,N$)'O<"X%`WO(*
+#3BO;"
JØ
JØ JØ JØ Ã Ì Ì
R(/1201*2#A(
/X32"2?'r6)ZN¥A(
ADC EGFG3,H
AK3L-NM,) OQ
#C 3,H
AK3L-NM,) OQ
*>@8- F AD35-8F'
> 3 >
!5E):["$):"H)'2%(["<JA Cl].$(P5^c5R2"<D"$D`W**2|5.%­
¸
»
V†…‰Š\
»
3 )ZÀ{NÁ „I½ V†…(
‡Yˆ V†……@‰ ‰•ŠA\ \
V†…:\
„
¾ƒÀ Á
V†…q‡†ˆ …@‰ \
V†… ‡ ˆ …@‰ŠA\= 
V†… ‡ ˆ …@‰ŠA\_ÃM¸
2` ^H7=3/2"<
2 ` ^ E/132"?'
V†… ‡ \{ÄYÅ
Ø
V \
Ø
V (w \
•- K3B".3)'1!F50g(*2!?'c2".5.**0g()'<z[`W'."$&(2%!'`~"$c$2!8):7$;zP"<I(7=3B/g<&"$
!5E):¥`iO7="CE-ƒ".$1)'.0g!(K"<)'2%$;"<`}'5E):6K)'**G'`,"$c/'01*2(3H5&$1)(?'<"O2(GC q5!?'!(F)'
;"<!(<"%h/1'01*2(3 `W'`iO"O(<():87$"<6!?'(*2'/S)')'*%''2"$3 "$):"&7!)'<*2?'B"$$12!):78$7!)'*
7=(")'2%3(@"B/'01*2(3©)'O"<3O*2z'C [/201*X2"{z95O*Œ0gP"<S$)q?'6A(B*2|5^!h“"$
$2!8):7$;zB/1)'"$O=NA/1*)'(B/g'"2I'`G"$#!5):8IO/h"$#"<!'C L/g(7!U€7!)'**2z'F"$*2():1%c8O*2R`W'
ŽL(E5O*P0g(7=3B
Ø
V†…@‰Š\­È»ŽV 3 \
V†…@‰Š\Gà  V†… ‡ ˆ …@‰ŠA\ ¸
V†…:\GÓ3)ZÁ N †V … ‡ ‰Š ‡ \{ÄYÅ
½
½
-•¥"$.O*2'F"$&nLO)'@""{z¥V†… ‡ ˆ …@‰ŠA\ ¸ V†…Z\RH"$d)'3BO;".'`,"$&!5E):¥"$):"#5E)''"=NL/k*)'(
½
0Lzd"$.7$X*BARŠhVi)'1J2"m(7=()';"8\8Cm]R$3@à O/g!7=82/"I¸ ½ 17!):"<(~"$):"E¸ ½ $1)'m)'*2()'z
0g!(6O/g1):"<(¥0Lz¥)://1*2zA%"$1.)'3&O1*2c):"#"$d*q5!*!?'(*.'`l"$&$!):7$;z'C `m7=O<d"$*2q5!
*2!?'(*'`,"$d$2!):78$@z[5*3
* •"<z!J"<h=NA/1*)'["$/
<O=NA/1*)'X):01*2 !5E):0Lz¥7$)'%X%J"$(
)'
¸e?Z)'*O(!C q5!?'!(F"$$'/gB"$):"D"$):"<"<(3B/"5**7!)'7=(*,()'7$S'"$!O"D|?'!c"3["$):"
"$B*2|5^!*2!?'(*X5.**a<"***2():>"$B7='(7=
" ±)'1‘¸e?:)'*O()'>"$BO=NA/1*)'1):01*2!5E):#5.X**_0g
/1)'(¥O/["$"<!cO;"*"$!zP()'7$6)AD"$):".7!)'¥O7!7=(<`iO**2zI!/((@"H"$(3¥C
«2"$h"$(7=3B/g<(h!5E):FA5^)://1*X2(Jr>+ 2#T "<d"$#fO(*])ZN3)'C ‘(3B/1*2|z'(P"$
`W**2q5%<"):"<):01<"<)'7="!C,]R$ Ê`iO7="2B'` A C EGF 3IHD2%'(l"$R/1)'(%'!(<"):"2J)'`iO(*†F
01O"~2"3&O"apAq5Œ"$<O7=*L7!):"2)'d")ZNA*2L7!):"dXc'!a"</(17="a"$^ 4g(7=",'`1*X*2!%)'2
* ADC EGFG3,H
)'7="2(C 3*):8*2z'F AD35- MI) ORQ2%'("$E/1)'(%'!m<O7=H*2A7!):"2)'"$E`iO(*†F')' C 3,HH2%'(
"$h<O7=J)'1S(<"X):"2*2A7!):"2&)'"$J`iO(*†C ( 9 C ,95-N>G)+ K>@-7!)'¦!/(<(@"d2"
e`WO17="2
0Lz6)[%*2&?:)'*O'F0g(7!)'O):`}"<!c)PO17!7=(<`iO* ( 9 C ,95-N>:F*2z6) ADC EGFG3,HI(3)'1"<¥7=3B/1*2!"<B"$
K>@)'7="2C^]R$D)'3BE"<8OD`}' ( 9 C ,95-N> ),+ AK3L-H)' ( 9 C ,95- >G),+ > 3 > C y (7!)'O<c'`a"$$12!):78$7!)'*
7=(")'2%3B(;"!F K>@-)'1 >N35> 1!("<K<!B"$B(;"2<"):"<'F01O" AD35-#7!)'%')'**~'`"$
<"):"<`W'3):"2GF0(7!)'O1<7=d2"RO7!7=!((F1"$&")'<p[.7=3B/k*2!"<(CE+**G'`"$(<&):01"<)'7="23B()'
"$):"B"<()'S'`#)6<!"'`.!?'( º F TY(*2(3B(;" ®`WO17="2PV x@w F ?:)'*O(8\`}'k):" 4*2():1%F~"$
r>+ 2–$2!):78$@zh(nLO2(E1*2z F Ì'¹ ?Z)'*XO(H"<!/1(<(;"R"$
Ž)'16¸Ž`iO7="!C
f~2%O w 7=3B/1):(G"$m1*,/g!`W'3)'7=,'` k):"a)'c$!):7$17!)'* M*():%Caf' k):" Œ*2():8%F
5(3B/1*2|z'( y *2"<b(3)'1=NA/1*2'):"25.2"$‘)'SX2")'*"<(3B/g!):"OB'` ¹ CP]R$X5)'&(7=()'<(S0;z‘)
`i)'7="<'.'` C w'wº ):`W"<!R()'7$>O7!7=(<`WO1*k"<8)'*†C ‘=NA/g!3B(;"<(P5.2"$[3)';z[9 4g!(;"E7=L*%7$(1O*2(!F
Ø
Ø
ÌZx
2
Mean Optimal
Performance
1
0
Reward per Step
-1
-2
Hierarchical
-3
-4
Flat
-5
-6
-7
-8
0
5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
Trial
f~2%O w ­
*H/!`}'3J)'7=R'` k):")'h$2!):78$7!)'* 2 *2():1%DI"$#f1O(*k])ZN1")'<pC l)'78$P7!O?'
3BL'"$(KO%) Ì TY"<8)'*~3Bq?A%J)q?'!):%''C].$&$'82b!@")'*a*$q5.."$&)q?'!):%'B/g!`W'3)'7=
'`~"$D'/"3J)'*g/*X7=z'C
01O"55!O):01*d"<[%'!" k):" 2±*2():1%h"<K7=@?'!%'J"<¥"$B'/1"3)'*~/g*7=z¥52"$ ¹ F "<8)'*!C
]R$X,5)'"$`W)'"<(<"7=L*%c7$(O*2R"$1):"^5E)'E):01*2R"<):"<")'9Vi):"*2()'"^082 1z\a"$'/1"3)'*=NA/g(7="<(
!5E):Cmf'$2!):78$7!)'* Ë*2():8%FL5^(3B/k*2qz'(¥)<!/k):):"<D"<(3B/g!):"OD`W'E()'7$6r6)ZN[A'C,]R$
<"):"X%I"<(3/!8):"Oc`W'D)'**L(.5E)' ¹ =N7=!/" 9
)I)G-F5.$78$KO<( C l)'78$9L&(7=()'(
2"l"<(3B/g!):"O5.$(J"^O7!7=(<`WO1**2zd()'7$(P)c%')'*k"<!83)'*"):"<'C 9
)I)G-~5E)'E7=;*2(I0Lz)`i)'7="<'
'` C w'w '¨ F"$c<(7=1K*2!?'(*ar6)ZNKA():" C w'wº Fg)' 9
4( 9 C ,95- >D):" C w'w'¹ CR- K)'**G7!)'<(!Fg)J*():%
):"<'` »
5E)'R(3B/k*2qz'(Fk7=D)'**G)'7="R)'[!5):1R):c!"<!83"7:C
]R$(<^7=L*%H8):"<(a5!7$<(d<"$):"_"$*2q5!~r6)ZNcA(a"$l%'):/1$d7!)'0g(7=3B()'<):01*z
7=3B/g!"<(;"R):"H"$(2EO0")'<pAE0g!`}'"$DA(E$2%$!EXJ"$%'):/k$P"<z"<*2():GC- `a7!):^'"H"):p'(F
)[r>)ZN9A .3)qz>7=17!*O"$):"c)[O0")'<p>ŠK?'!z>=NA/(12?'¥VW0g(7!)'O<B"$BO01")'<p>$1)'D'"Dz'!"
*2():(¥)%';A¥/g*X7=z\8F)'["$!!`}''F12"R<!"."$
”?:)'*OD`W'ŠI?'!z[*2q5cC ª$(["$H;"<!)'7=".5.2"$
y *2"<b(3)'S=NL/k*2'):"2F_"$B(O*""$):"c"$O0")'<p93)qz9!?'!c0gB"<2():%)'XC (7='Fa5*z
/g!`W'3B(S)'“O/):"<I`}')2eAh2`H"$):"LI7=3B/1*!"<(’"&O0")'<p‘5.2"$¦)'“)(?'!):%'¥):01*O"<
y (**3)'P!'R/g!.<"<!/¥'`,*2(E"$)' C Ì C#Vi].$E/1):)'3!"<!.5E)'.'"H"O(¥):".)'**†CÈ\
Ø
Ø
Æ
g'm
m &8
88 ”8
-•<"<()''`H17='/g'):"%P`iO(*m;"<["$J")ZN,/'01*2(3[F_7=1!c)¥?Z):X):"2‘'`E"$'%)'*,")ZN,")'<p
O78$["$):"#2`_"$c")ZNA_OH;"<c'`"$5)'*X*E'X(E'`~"$c3):b!'Fk"$!/1LD"<!83):"<(H52"$¥)
!5E):['` ¹ C ª"$["$R78$)'%''Fk"$'/"3J)'*/g*7=zIXRJ*2%'!.)B(7!O82?'(*2zh'/"3)'*$12!):78$7!)'*
/g*7=z'F0g(7!)'O<d5.$(6"$&")ZN~/!`}'3J%J) 9
4( 9 C ,9L-N>V \H)'92"#3B'd"$)'6€1?'d"<!/1)(5E)(z
`W3Æ"$D"):%'!
" F"$D*2A7!)'**2z'/1"3)'*g/g*7=zJX^"<B8OP@"<BD'`_"$D5E)'**!C,]R$1E)'7="2["<!3X):"<(
Ì'¹
"$ 9
4( 9 C ,9L-N>&5.2"$>)I!5):>'` ¹ F5.$!()')q?L%):"%[)'**a"$5)qz6"<P"$"):%'!"R5O*9$)q?'
"<!3X):"<([5.2"$[)J3&O78$¥3B'&!%):"2?'!5E):C
]R$h/1'01*2(3 "$):"J"$ 9 #( 9 C ,95-N>hO0")'<p’$)'((’"<!3)'*E<"):"<(>Vi†Cj'C2F()'7$X%‘"$
"):%'!"8\^)'IO(("<!31)'*<"):"<(DVi†Cj'C2FA$2"<"%)c5)'*X*}\8C,]R$.(2%!m'`"$#$2!):87$;zd5O*I*2p'
9
4( 9 C ,9L-N>"<J*2():8P"$/g*7=z"$1):"R"<!31):"<(.[(2(P"<!3X)'*g<"):"<(!C
s)(z)' )' ;"<ŽV (w'w \h)'(<(É"$1h/'01*2(3 ªf(O)'+
* 2 *2():X%S0LzÉ</g(7!2`WzLX%S)’<!"
'`a(2(J"<!3)'*<"):"<(E`W'E()'7$¥O0")'<p Cm]R$!z"$(h/|?L(I)&/g()'*2"{zJ'` /X@"^"<
")'<p E2`~2"R(;"<!(6)'6O(2(P"<!3X)'*G<"):"<B… ‹ CH]R$R"<(7$Xn;O&$)'R"$c/'/g!R 4g(7=" ªq5
"$c*L7!)'**z'/"3)'*G/*X7=zJH/k):"H'`,)%';A¥(7!O2?'(*2zI'/"3)'*g/g*X7=z'C
]R$!J)¥/'01*2(3 5.2"$9"$c<"<):"<!%'zS9"$Ir>+ 2$2!):87$;z'F_0g(7!)'O<'`E"$I?Z)'*XOB`WO17="2
(7=3B/g"2Ch- ¦)KO01")'<p95.$!"$IO(2(‘"<!83)'*m"):"<(d7!)'1'"c0gJ7=3B/k*2!"<(*2z‘)(?'(F
"$l=NL"<8)c/g()'*2"ƒzB5.** <7=;")'31):"< "$.?:)'*O`WO17="2h¸ !"O(J0;z 9
4( 9 C ,95- >:FA)'
"$H7=O1*¥7$1)'%'"$'/1"3)'*g/g*7!2(E[L(.$2%$!RP"$c$2!):87$;z'C
+Ç0g!"<"<!R3B!"$A["<=€k'FA`W'H()'7$Kr>)ZN[L#r>st F)/1):8)'**2(*gr6):p'q?[(7!I/1'01*2(3
52"$“"$K)'3K<"):"<(!FR)'7="21!FH)'Œ"<)'2"2“/'0k):01*2"(&)' 01O"B52"$Œ)S<(7=1Œ!5E):
`iO7="2  "$):"lmb!!=N7=!/"l`W'lO(2(d"<!3X)'*<"):"<(!FL5.$!R",/q?A(l)*):%'H/g()'*2"ƒz'CRV ‘
O<(I)c/g()'*"{z'` /g@"\8C~+E"EA FL"$*2():X%)'*2%''2"$13–5.*X*<!!p)d*2L7!)'*X*2z'/"3)'*1/*X7=z
¬ `W' C q5!?'!(F):"."$&)'3B&"3B'F1"R5.***():¥"$/
<O7=;")'3):"<( J?Z)'*XOd¸ '`,"$1H/g*7=z'F
)'["$1HO7=;")'3):"<(¥?Z)'*OD5.**0gD/1)'<([O/¥"$$2!8):7$;zI"<J"$c)'7=(<"<'#A(!C
]R$1G%2?'(~O)'*2%''"$3 r>+ 2#T 2dC'+E"()'78$B7=3B/g2"<r6)ZN&L'F:53J)'@")'Xc"ƒ5^"):01*2( V†…‰Š\
)' V†…@‰Š\8C>]R$I)'*2%''"$3u78$;(B)'¦)'7="2“Š‘"<>/g!`W'3[F'01<!?'(d"$h(O*"%¥<"):"<¥…q‡l)'
/g!`W'3E"$D`W**2|5.%­
Ø
Ø
JØ
Š À ­È»
†V …@‰ŠA\
­È»
È­ »
V†…@‰ŠA\
3)ZÁ N Â
½
V 3
V 3
Ø
Ø
<
V†… ‡ ‰Š ‡ \GÃɸ Á V†… ‡ \{Ä
V Ì \
\ †V …‰Š\Gà   †V … ‡ G
\ ӏ V†… ‡ ˆ …‰Š\Gà †V … ‡ ‰Š À \GÃM¸ ½
V†… ‡ \{Ä
\ V†…‰Š\Gà   †V … ‡ ˆ …‰Š\Gà †V … ‡ ‰Š À \GM
à ¸ ½
V†… ‡ \{Ä
VÌ \
V Ì'Ì \
V Ì \
½
DØ
!&Š À H"$&)'7="2K"$):".5O*[0gD"$&%'!(z[)'7="2>)'7!7='%"<J"$&7!O(;" Ž)'1>¸ ?Z)'*XO(!C
' ±):BO/):"<(9OX%IŠ À Ccf' BF
"$##(<(;")'**2z¥"$r>+ 2 T )'*2%''2"$134):/1/1*2(
y '"$ Æ)
"<"$XHL'C
]R$7='(</gX%d(7=3B/g2"2P(nLO):"2.):
V†…‰Š\
»
V†…‰Š\
»
†V …‰Š\
V†…‰Š\
»
¸ †V …:\
»
»
¸ ½ †V …:\à V†…‰Š\
¸ ½ V†…:\à V†…‰Š\
„ V†… ‡ ˆ …@‰ŠA\=  †V … ‡ ˆ …@‰ŠA\_ӏ V†… ‡ ˆ …@‰Š\GÓ3)ZÁ N †V … ‡ ‰Š ‡ \{Ä
¾ƒÀYÁ
½
„ V†… ‡ ˆ …@‰ŠA\=  †V … ‡ ˆ …@‰ŠA\_Ó3)ZÁ N V†… ‡ ‰Š ‡ \{Ä
¾ƒÀYÁ
½
V†…@‰Š\
5$!ŠB»«):%3)ZN + †V …@‰)g\
)'1 lR7=3/2"<
ÀYÁ „JV†… ‡ ˆ …@‰ \  †V … ‡ ˆ …‰ •\ ` ^XE/3"2?'
V ÌZx \
V Ì'¹ \
V Ì'¨ \
V ̺ \
V Ì
\
] $/g*7=z6):"Lq5 =€k(>0;z>¬ V†…Z\#»Ž):%3)ZN †V …@‰ŠA\#)' BFF_)'1S¸o):7=3B/kO"<(
R
½
5.2"$P(</g(7="H"<J"$E/g*X7=z'C
+E"R"$c*2():`~L(!F"$c*2():8%)'*2%''2"$13eO78$)'%'(­
Ø
¸ ½ †V …:\­È»ŽV 3 <\ ¸ ½ V†…:\à  ½ V†… ‡ ˆ …@‰ŠA\8Å
Ì'¨
V Ì'w \
y zK;"<AO7!%B"$dzL;"$!"7&/g()'*2"{z d Fg"$d(2%!#'`,"$r6+ 2±$2!8):7$;z¥7!)'>(Oc"$):"
"$R(2(&/*X7=zc~)(7!O82?'(*2z'/1"3)'*A/g*7=zJVW5."$(</g(7="~"<D"$~)'(/g()'*2"ƒz\8C,+#):%O3B(;"
():*z&(;"7!)'*L"<"$H/1;'`k'`].$!'(3 Ì 7!)'B/1q?'H"$):"^2`1"$E(7!O2?'(*2z&'/"3)'*A/g*X7=zc,OXn;O'F
"$(Pr>+ 2T 2Ç5**17=;?'!%'D"<B2"!Cm]R$*z7$)'%'D"<"$/;'`G"<B$|5«"$):
" V†…‰Š\^7=@?'!%'(
"<"$
Ë?:)'*O(H'`"$'/"X3)'*g/g*7=zJ`W'R"$3L2€1([r6st C
'"7="$):"` » <["$):"D"$!):B¥O(2(6"<!3)'*~<"):"<((F_"$(  XD)'*25)qzLb!!F
)' » C&].$&=NA/g!3B(;"$|5.‘9f~2%O( º )' w )'7="O)'**2z6(3B/1*2|z'(‘r6+ 2T 2dF0kO"D7=
"$!c5^!JO(2(h"<!31)'*"):"<(!F1"$XR3)'19 4!(7=D5.$):"<L!?'!(C
o#e m'HKB lm&g#
‘E$)(?'E$|5."$):"a"$^r>+
2M$!):7$;z7!)'d*2():)'&'/"3)'*@/g*7=z2`;"$1):"a/g*7=z#7!)'c0gm=NA/(<(
') B)>(7!O2?'(*2z‘'/"3J)'*$12!):78$7!)'*^/*X7=z'C.|5^!?'!(F"$![):["O):"2d’5.$78$S"$h'/1"3)'*
/g*7=z>D)'*3B<" ª01O"'"nLO2"<0ª$12!):78$7!)'*†Cf'd=NA)'3B/k*2'F_7=X!Dz'!"&)''"$!c3BAU€k7!):"29'`
"$d]a)ZNG")'<pVW"$1
€k7pA*2d]a)ZNG/'01*2(3 @\EX[5.$78$¥)'.;6)'."$c")ZNAa/17pLRO/K"$/k)'<(%'!)'
3B|?'(DBnLO):'F"$/1)'(%'!7!)'9)'3J*2z¥7$)'%'B"$B(<"):"652"$K/1'01):01*X2"{z C Cd]R$
7$1)'%'&7=3B(#):`W"<!"$&$2!):78$7!)'*/g*X7=zI$)'7=332"<"<(K"<J=NA(7!O"% ( 9 C ,95-N> ),+ AK3L-(V <\H`}'"$
'2%X)'*1("):"2C,+#E)c(O*2"!FL"$ 9
#( 9 C ,95- >O0")'pB5.**"):p'D"$")ZN"<d"$*h(<"X):"2C
]R$(¦7=@"<*^5**,!"OS"< 9
4AK3L-Fa5$7$S5.**l@?''p' ( 9 C ,95- >G),+ AD35-"<93Bq?'P"$h")ZNA"<>"$
!5 (<"):"2GC
‘pA|5–`}3°]R$!'(3 Ì "$):"`^r>+ 2T )://1*2(>"<["$X/'01*2(3¥Fg"#5**_€k>"$OXn;O
(7!O82?'(*2zP'/"3J)'*_$2!8):7$X7!)'*G/g*7=z'Fk01O"."$1R5.X**G'"#0gc"$&'/"3)'*a/g*7=z'C ):(*20k*% = cV (w'w \
5'pP¥"$&
#2!):87$7!)'*as<")'7="< "#)'*EV s "\R3B!"$AK$|5^(K"$):"H¡™:£}£µžW· L š(´A˜iž†™:·h7=O*¥0g
O<(¥"<=NL(7!O"<c)$2!8):7$X7!)'*g/*X7=zIT{$2!):78$7!)'**z'C
-•/g**%.=NL(7!O"F5H?L2!5Œ"$Rr6+ 2«$2!8):7$;z&X3B/1*2z)',)!/1(<(;"):"2B'`1"$
ª`WO17="2C
+E"B()'7$É<"<!/GFm5¥7=3B/1O"<P"$P/832"2?'h)'7="2“5.2"$’"$[$%$(<" 4?Z)'*XO'CS]R$X&7!)'“0gPP0;z
<"):"X%6):" 9
) )G-)'1’7=3/1O"%K"$I/k):"$ªVW`W3 L'"&"<9*2():`8\5.2"$¦"$I$%$(<
" ®?:)'*O'/
C ‘
"$(‘=NA(7!O"<"$1):"/3J2"2?'&)'7="S)'9!/g():"D"$B/1L7=():%)'ÉVi<"):"%¥):" 9
) )G-<\."<K78$L<
"$c=NL"R)'7="2GC
]R$1E/g!32"EOE"<`W**2|5Ê"$D%'!(zh/g*X7=z5.2"$P(/(7="H"<J"$c*2():([r6+ 2Ž(7=3/2"2C
y (7!)'O<D"$X7=1<"2"O"<(E"<!/['`_/g*7=z3/q?'(3(@"V y !"<!p:)'!F (w'w'¹ \8Fk2"H%O):)';"<!(["<%2?'
)D0g!"<"<!l/g*X7=zc"$)'B"$.$2!):87$7!)'*A/g*X7=z&2`"$.$!):7$17!)'*A/g*7=z&Xm'"m'/"X3)'*†C~]R$,`}'3J)'**2z
/|?'(R"$c`}**q5.%­
Ø
Ø
!"# _™:Ÿœ:£W£a—=˜Yœ'˜{ —c… 1R˜W›A B³:œ:£µ´ J™•¶B˜W›A ¡A™:£µžišP6š™:¤¡k´˜{ 9¢ &¡A™:£W£ÈžW· L š!´˜iž†™:·’™<¶˜W› *-,.0/
›@žY =Ÿ<œ:Ÿ<š=›,Pž}—+±˜W›A &³:œ:£µ´ d™•¶&˜W›A R¡A™:£µžišPPš8™:¤¡k´˜{ K¢ B›@žY =Ÿ<œ:Ÿ<š=›@ž†šœ:£l ; š(´˜iži™:· 3
(7='F,/g**%P=NA(7!O"2¦'`R)Kr>+
2%'):/k$S7!)'¦/AO7=J)6AT{$2!8):7$X7!)'*,/g*7=z>"$):"dX0g!"<"<!
"$)'["$c$2!):87$7!)'*g/g*X7=z!/((@"<(K0;zh"$D%'):/1$GC
]a‘?'!2`Wz"$d=NA/g!3B(;")'**2z'F,5[3B/1*2(3(@"<(’"$[f~7pA*2[])ZNAH3J)'Cf~2%O 7=3B/k):(
"$D/g!`W'3)'7=D'
` k):" 2Ž*2():%)'1[$2!):78$7!)'* 2Ë*():%["$1H3BAU€1(h")'<pCl]R$D'/1"3)'*
/g*7=zh7!)'6)'78$2!?'&)'6)(?'!):%'!5E):[/g!."<!/6'` C qº:Ì v01O"R"$c(7!O?'(*2zh$12!):78$7!)'*g/g*7=zP7!)'
*2z¦)'7$2!?' C Ì C !):7$17!)'
* 2*():%>):01*2h"<‘):"<")'“"$*2!?'(*):/1X*2z'C9f*X):" 24*():%
)://)'7$("$d'/"3dO3[Fg01O"L(D'"#()'7$‘2"#5.2"$1 F "<8)'*!C ‘B"O(6()'7$)'*%''2"$3®"<
'/"3J2b!2"l/g!`}'83)'7='C ‘(3B/1*2|z'(I)d*2():X%D):"<#'` C '¹ )'J(7!)qz'(P"$2")'*A"<(3/!8):"O
'` ¹ C 0;zh)`W)'7="<''`lC x@¨ VW`W' k):
" 2c\l)'19C Ì
VW`}'$2!):78$7!)'* 2\l5.$(!?'!.)d%')'*G"<!3X)'*g<"):"<
5E)'R()'78$(C
]aK!?:)'*O):"<h/g**%P=NA(7!O"2F~5€18<""<)'("$hr6+ 2®$2!):78$@z9`W' "<)'*Xc)'S"$(
7=;";O1%‘"$¥"<8)'%‘52"$“/g**%>=NA(7!O"2C«f~2%O
$|5.B"$):"I"$!KJ)'É")'*E*2'`
ØØ
Ø
D؊Ø
Ø
Ø
؊Ø
̺
Ø
1.4
Optimal Non-Hierarchical
1.2
Reward per Step
1
Optimal Hierarchical
Hierarchical
0.8
Flat
0.6
0.4
0.2
0
0
Ø
1000
2000
3000
4000
5000
Trial
6000
7000
8000
9000 10000
f~2%O ­D*&/g!`}'83)'7=B'` k):")'‘$2!8):7$X7!)'* 2o*2():%I‘"$fX7pA*2]a)ZN~")'<pCl)'78$
7!O?'ID"$J)q?'!):%'P'` O1!v_"$J!"O`}3u()'7$¦O95!I3B;'"$(‘0;z‘) Ì TY"<)'*l3q?A%
)(?'!8):%''C
Ø
/g!`W'3)'7=¥5$(M56<52"7$Q"<S/g**X%9=NL(7!O"C y O"J"$(M/g!`W'3)'7=6(7=q?'!h):/k*2z“)'
()'78$(c"$J*2!?'(*m'`E"$J'/"3)'*mAT{$2!):87$7!)'*~/g*X7=z'C- "$3J)'FG/g**X%P=NL(7!O"'`E"$
(7!O82?'(*2zI'/"3)'*$!):7$17!)'*g/g*7=zJ/1L1O7=(E"$%*2'0k)'**2z'/"X3)'*/g*7=z'C
]R$.2"X)'**2,'`/g!`W'3)'7=.5.$(B5<52"7$J"<c/g**%=NA(7!O"2h7!)'J0gH=NA/1*)'()'l`W**2|5.!C
sO%K$2!):87$7!)'*l"<)'X%F_"$P3B'h):01<"<)'7=" 2L(cS"$I%'8):/1$S$)q?'P*2z‘*2():(S"$(25
?:)'*O(a5^(**A&<"):"<(,5.$!l"$!zd):`}(nLO(;"*2zD=NA(7!O"<(C,f',=N)'3B/1*2'F7=!"$ 9
#AD35-GO0")'<pC
sO%#"<8)'%F"m"<(,"<c0RX@?''p'(J*2z"$R`WOl<"):"<( 4F "dF y F@)'
$Ê3J3B():"<(*2zd):`}"<!^"$
/1)'(%'!#$1)'0g!(K/17p'(9O/_
C .(17='Fg"$)'#*2():(6%'LL
?Z)'*O(.`W'"$(<d"):"<(!C y O"K'"$!
<"):"<(!F~2"c$1)''"c*2():8(‘?'!z9%'LL o?Z)'*XO(!C6Vi]R$=NL"<(12‘'`Hr>+ 2T 2e"<¥/g!`W'3°)'**m"):"<(
O/):"%&(1O7=(H"$R/'01*2(3Æ<3!5.$):"!F01O"R;(.'"R(*3J):"<D2"!CÈ\
!/g**X%F 9
4AK3L-#5.**~|5 0gB=NA/(7="<(S"<67=3B/kO"<%'LL Æ?:)'*O(S)'**,<"):"<(!C (7='F
5.$(J5^#0g!%B/g**X%D=NL(7!O"2FL5^#!(J"<7=;";O."<)'1%c<&"$):"E):01<"<)'7="EA(l7!)'h*2():I"$
7='(7=6
" ª?:)'*O(mB)'*X*<"):"<(!CmsOX%/*X*%#=NL(7!O"2F;`k)7=3B/g"<H")'<pB):"lA
a,X@"<!8O/"<(F
"$(Œ"$
?:)'*O():"I"$):"JA¥)'É):"I)'**HA():0g|?'62"“"$K$2!):87$;z“7!)'1'"0g¥O/):"<(C
(7='Fl"<)'X%>O%K/g**%¥=NL(7!O"2“5.**^*2z¦7=@?'!%'K"<>"$h'/1"3)'*E$2!):87$7!)'*/g*7=z`
()'7$>7=3B/g2"<L ^<"**"<!3X):"<(R5."$[<3BcTYb!!B/'01):0k*2"ƒz'C
L"<87="*2z¥</g():pA%F7=@"X;O(9"<8)'%hOX%/g**%=NL(7!O"2‘#'"D(7=():z'F_0(7!)'O1<B]R$!:T
(3 Ì $q5"$):"G"$mr>+ 2“%'):/1$5^O*X!?'(@"O)'*X*2z#7=;?'!%'"<R"$,?:)'*O,`iO7="2'`A"$,(7!O2?'(*z
'/"3J)'*g/*X7=zI2`a5^c(3B/1*2|z'(K$2!8):7$X7!)'*k"<8)'%B*2%B(O%$
C q5!?'!(FP/)'7="7='F15^5E)';"R"<
<5."7$["</g**X%c=NA(7!O"2K0g!`}'D"$?Z)'*XO`WO17="2[$)'.7=;?'!%'(C
+Ë/g'"<(;")'*G/'01*2(3e52"$[/g**%X¥?'!z¥*):%'&r>+ 2 $2!):87$2(R"$1):"2".3J)(z¥0g(7=3Bd7=3T
/1O"):")'**2z[=NA/g(2?''Cc+ 3X*):/1'01*2(34):<(VW`W')P3X*):()'<k\>"$ .
+ –):87$2"<(7="O
Ì
1.4
Optimal
1.2
1
0.8
Polling Execution
0.6
0.4
0.2
0
0
1000
2000
3000
4000
؊Ø
5000
Trial
6000
7000
f~2%O
­ *./g!`W'3)'7=D["$fX7pA*2]a)ZN")'<pCl]R$€k<"
7!)'**2z'Cm].$D(3)'1%"<)'*H):"<)'(P5$*2#/g**%C
8000
Ø
9000 10000
"<X)'*H):D"<)'([$2!):78$UT
Ø
V <_)'F !5(**†F§ .<(;01*2L3[F (w º \8F_5.$78$K6/17!2/1*2D3dO<"7$(7p6)'**a'`l2"/AO7="2KO*2():"
()'7$6"3B&<"<!/K"<J!&2`"$!z[7!)'[€1'C^f'"O1):"<(*2z'F"$c):")ZT{82?'( ^] ª)'*2%''2"$3 Vif'%'z'F (w
'Ì \
m):0k*2H"<&3/1*2(3B(;"~"$m78$(7pB=NA"<(3B(*2z 8I7!(@"*2z'FL<"$):"l*2zd/1):"m'`g"$.5^'pA%3(3B'zd"$):"
$)q?'78$)'%'(9!(>"<h0d7$(7p'(C+#63/'")'@"%')'*a`W'#`WO"Oc(():7$‘."<[!?'(*2'/>0'"<"<3BT{O/
3B!"$AH`W'.17=(3B(;")'*g!?Z)'*O1):"2['`~"$r>+ 2 $2!):87$;zh<B"$1):"R/g**%HX3*):*zB 8I7!2(;"!C
g
B Q'&
g
Ø
r>)'@zP(<():87$!.$)(?'&0!(P=NA/1*2'8%3B!"$AH`W'.$2!):87$7!)'*(X`}'87=(3B(@"R*():%B)'P(*):"<(
O(!C¼-•É"$XJ<(7="2GFR5^6=NL/k*2'63B6'`D"$>7=(7="2I0!"ƒ5^!(ª"$9r>+ 2 $2!8):7$;zM)'
/!?A2OE5'pC
]R$Rr6+ 2Ê$12!):78$@zm)12(7="m(7=(1)'@"l'`"$H5^'p0Lz A%$¥V (w'w'Ì \8F ):(*20k*%BV (w'w \8F)'
s)(z)'¦)' ;"<MV (w'w \8C A%$>$|5^(‘$q5Ž"<['0")'9)P"ƒ5^:T{*2!?'(*m?Z)'*Od`iO7="6(7=3/2"2
`W'c)k):%6")'<p K‘5.$X7$‘)'):%'(;"d3dO<"1)(?A2%):"<J"<K/(7!2€1( k):%K*2A7!):"2c‘)¥</g(7!U€1(9'8!(C
):(*20k*% = +
#2!):87$7!)'*^s<")'7=I"<?
"#)'*3B!"$AS)'*X<K%)q?'P)6"{5:T{*2!?'(*?:)'*O`iO7="2¦(7=3B/g:T
2"C[- ¦)'2"F$J/1!!("$I()K'`H=NL(7!O"%>"$I$!):7$;z‘0Lz‘/*X*%¥1<"<()''`H0LzS)
/A7=(O7!)'**2T{)'ATY!"O8&X7!2/1*XVi)'*2"$O%$"$H<"17="2c0g!"ƒ5^!("$(H"{5D=NL(7!O"3B!"$A
%'L(.01)'7p¥):"#*()'<"."< ):"<pL1 =V (w 'w \<!"):"2k\8C- "H()'z¥"<I7=():"<r>+ 2 $2!):87$2(R`}'#"$
")'<pA.<"O12(h0;0
z AX%$P)' ):(*20k*%C
]R$f(O)'
* 2”3!"$L¥'`s)(z)'>)' #;"<V (w'w \H5E)'R"$€1<"H3B!"$A["<=NA/1*2'#(7!O2?'(*z
'/"3J)'*~/*X7!2(#)'9"<K<"OzK<"):"<J):01"<)'7="2CB]R$!):B"{5K3;) :<'c9 4g!(7=(0g!"{5!(f(O1)'*
2Æ)'‘r>+
2dC_f~2<"!Faf(O)'>
* 2o;(c'"D/AO7=B)P?:)'*OB`iO7="2‘(7=3/2"2C]R$3):p'(&2"
3B/g201*2H"</g!`W'3 /g**X%=NA(7!O"2P'`G"$$12!):78$@z'>
C L(7=1FA"$0($1)(?A2'^'`<"):"<):01<"<8)'7="2
Ø
Ø
Ø
Ø
Ì'w
Ø
‘f(O1)'*2Æ"$B'/1/2"<'`Er>+ 2dCG-•‘r6+ 2dF_A(D():c"$BL'"D'`^"$I$2!):78$@z6"{zL/17!)'**z
!(P"<B$)q?'D"$D3B<"H!")'*2(h?A2!5«'`_"$D<"):"<0ª"$!zI'`W"<([7!)''"H0g(=€k"l`W3o<"):"<):01<"<8)'7="2
5.2"$O"B*2%>'/"3J)'*2"{z'C’-•Œ7=@"<)'"!FA(B():J"$¥*2()(?'(J'`."$["<!K7!)'“"{zL/17!)'**zS0gP%2?'(
$2%$1*2zI):01"<)'7="<(K?A2!5..'`~"$c<"):"<'CR]R$1R3B/1q?'(."$c):01*X2"{zh"<J"<)'1<`}!O7$6O0")'pL.`}3
/'0k*2(3Æ"<J)''"$!qC
-•6)I<!2('`,/1):/g!!
F AO"<"<>)'6$1R7=**2():%O(V O"<"<F (w'w'¹ vGtm(7!O/9§ O"<"<F (w'w v O"<"<
!")'*†C2F (w'w \$)(?'"O2(9)PpA>'`O0")'<p6"$):"D"$!z97!)'**,)' •'/"2C Ë+#9'/"2‘#=N)'7="*2z9"$
)'3BP)'d)6O0")'<pS"$Ir>+ 24$12!):78$@z'F,)'*2"$O%$¦S"$(2&5^'pF,"$!z$)q?'P/3):8*2z6"O2(
'/"21B"$):"=NA(7!O"<¥€1NL(FEAT{*():(Œ/g*7!(!C AO"<"< V (w'w'¹ \J$|5^(Q$q54)‘"<)'2"2Œ3BA(*
`W')'9'/1"297=O*>0gd17='/g'):"<(9@"<h?:)'*OB2"<!):"296"$/1):(;"")'<p6X>)h5E)(z6"$):"D/g!32"
3UN%'`,'/"R)'K/3"2?'D)'7="2!CE]R$(O*2"%d?:)'*O&2"<!):"2¥"**€k1H"$'/"X3)'*?:)'*O
`iO7="2ª`W'["$‘'2%1)'*r>staC+'"$!K)'?:)'@"):%''`&"$(<3BA(*UTY0k)'<(¼3B!"$APP"$):"["$!z
7=;?'!%'J>3)';zK`W!5^!2"<!8):"2#"$)'‘3BA(*UTY`W!3B!"$A#01)'<(> 2±*2():%
C q5!?'!(Fa"$!
):.!?'!)'*1X)'?:)'@"):%'(,"<&3BL(*UTY01)'<(3!"$L1!­Vi)@\mB)D(`}'7=(3(@"m*2():8%<!"<"%FL2",5O*
0g&(7=():z¥"<P*2():6"$&3L(*X!FVW0\R"$3BA(*R(n;Oc)h*):%'&)'3O@"#'`l3B(3'z["<[<"<''F)'
Vi7q\()'78$6<"<!/K'`?:)'*O"<!):"2[5."$P"$(<c3BA(*H7!)'¥0nLO2"<D=NA/g(2?''C
#)'O<pL(7$;"!F'rK(O1*2()'OF y O"*2!(F ):(*201*X%H)'s#()'IV (w'w \a)'*<H=NA/1*2'l)R3BA(*UTY01)'(D)://)'78$C
-•c"$(_5'pkF:"$!z&O<l"$3BA(*_'`"$EO0")'<pAEVW`}**q5.%H€NA(c/g*7!(G¬_\"<7=<"<8O7=")#3BA(*'`
„ V†…(‡Yˆ …‰Š\l)'J V†…q‡{ˆ …‰Š\RVW"$.=NA/(7="<(I!5E):'`/g!`W'3X%O0")'pŠI<"):"<D…)'1"<!3X):"%
’… ‡ \`W'c"$h/1):(;"c")'p CK]R$(S"$!z7!)'’<*2?'h"$D/1):(@"d")'<p‘0Lz‘?Z)'*OJ2"<!):"2’7=X!%
*2zI"$<"):"<(.… ‹  |… ) ½ V†…:
\ Š>Vi†Cj'C2F"$D<"):"<(."$):"H(O*"`W3Æ=NA(7!O"%B"$(<3)'7=)'7="218\8C
]R$XH;('"R%2?'d)7=3B/1*!"<c/g*7=zIq?'!"$(;"2c"):"<&</1)'7='Fk01O"R"$c7=3B/1O"<(K?Z)'*XOD`WO17="2
`W'^")'<p ~7!)'I07=3c01X(5."$"$?Z)'*OR`WO17="2,'`"$O01")'<pL^Š"<d7=3/1O"<)c=T{<"<!/P%'!(z
/g*7=z'C~- `G"$5!D)://1*2(J"<)'P):012"<):8*2zB!!/¥$2!8):7$;z'F2"^5O*h0g#?'!zIX3*):^"<d"$r>+ 2
)://)'7$¥52"$P/g**%C
]R$Bp'!z9 4g!(7=B0g!"{5!("$(D5^'p‘)'Sr6+ 2o"$):"cr>+ 2o(3B/1*2|zLc* 2o*():%
):"I)'*X*.*2!?'(*(F5$!()'I"$(2J)://1)'7$Q(nLO2(B hK*2():8%‘'`3BA(*J`}'I"$6O0")'<pA!C¼f~2<"
3BA(*c'`H"$I*2():`RO0")'<pAc3dO<"&0I*2():(’)'S*2?'(S?A)[?:)'*Oh2"<!):"2GC¥]R$("$J"<)'2"2
/'0k):01*2"(m„ V†…q‡Yˆ …‰Š\^)'J!5):1 V†…(‡Yˆ …@‰ŠA\l`W'"$/1):(@"E")'<pAE7!)'P0#7=<"<O17="<([)'I*2?'(
?A)?Z)'*XO2"<!):"2GCm]R$E/L7=(.E!/g():"<(KO;"*"$;'"RA.<*2?'(C
+#@"<!(<"%2(7="`W'`iO"O^5'pca"<D)://1*2zD1*3BL(*UTY01)'<(d3B!"$AO7$)',+ ~]Rst
V y ):"<h!"#)'*†C2F (w'w'¹ \.'#tl2'2"b!(0
L5^!!/k%6Vir6;'§ +E"<p'(<F (w'w \K)I3*X):H5)qz'C.]R$&01)'7
()K5^O1*0g"<9)://1*2z‘<"<A7$1)'<"7h)://qNA3J):"2S3B!"$Lc"<>7=1<"<O7="3BA(*'`R„ V†…(‡Yˆ …@‰ŠA\c)'
 V†… ‡ ˆ …‰Š\`W'.)'**A(E5.$X*2`W**2|5.%B) "<_- É=NL/k*2'):"2P/g*7=zI):"R()'7$6*2!?'(*†Cl+`}"<!.()'7$K)'7="F
"$c3L(*XE5^O*X[0gO/):"<(¥)'["$(¥<3Bd;O3&0g!H'`~?Z)'*O2"<!):"26O/):"<(H5O*P/g!`W'3B(C
AO78$9)'>):/1/)'7$9$O*K)'*X<h7=;?'!%'"<[)I(7!O2?'(*2z['/"3J)'*_/g*7=z[`W'"$$2!):78$@z'CD+##5.2"$
'"$!3BL(*UTY01)'<(3!"$L1!F|2"a$O*(n;Om3)';z#`W!5!a)'7="2_"<.0g,=NA(7!O"<(&XD'!G"<#7=@?'!%'
5.$([7=3/1):(¥52"0
$ 2Ë*2():1%C
AO"<"<F1tm(7!O/6)'1
A%$V (w'w \.(7=(;"*2z[<"O(¥) 2Ž*():%)'*%''2"$3o`W'*():%B)J/*X7=z
q?'!^)%?'(d!"'`k):01<"<)'7="m)'d/832"2?'^)'7="2!C~]R$!zc)'*X<D<"O2(c"$E/'01*(3 '`1*():%/g*7!2(
`W'E7=3/2"<.")'<pAE)'**k'`G5.$<O01")'<pL):#/3J2"2?'R)'7="2(Cl]R$!zB/q?'(h"$7=@?'!%'(17=D'`_)'**
%')'*O/):"%P"$H7!)''C^f~)'**z'FL"$!zh/1q?'(6)?Z):X)'@"R'`~]R$!'(3 x `W'R/g**%d=NA(7!O"2C
+#'"$!D;"<!(<"%J)://)'78$>"<h$!):7$17!)'*G(X`}'87=(3B(@"*2():1%"$
+#r V #2!):78$@z¥'`
+01<"<)'7="r6)'78$(8\m3B!"$A'`t~):^)'1 .O1<(**_V (w'w \8Cl- J"$m3B!"$LFL)c$!):7$;z?'!zB3X*):,"<
"$dr6+ 2±$2!):87$;zPX7="<O7="<(
C l)'7$9A
H.%2?'(>)J/1):")'*G/g*7=zP"$):"3)qz¥pLq5Ë$q5”"<
7$;<)'7="2aD<3Bl"):"<(_01O"G'"_'"$!8!Ct~):a)'+
.O1<(**';"<AO7=,).3!"$L'`;"<)'1<`}'83%
"$&$2!):87$;z9X@"<K) k):"&r>st”5.$!I"$J<"):"<(7=X<"'`H"$h7=•TY/LO17="'`H"$I"):"<(&S"$
'2%X)'*Gr>s#tª)'K"$/
•/'%')'37=O;"<!6I'`~"$&L(.["$d$2!):78$@zP'`~/1):"X)'*/g*7!2(!C m?'!z
Ø
Ø
Ø
Ø
Ø
Ø
Ø
Ø
Ø
2
Optimal
1
Reward per Step
0
MAXQ
HAMQ
-1
-2
-3
Flat
-4
-5
0
500
1000
Trial
Ø
f~2%O (Ì ­H3B/1):X<P'` #+#r 2–)'¥r>+
1500
2000
2T 2–*2():X%["$3/1*2D]a)ZN")'<pC
/g*7=z"$1m"<)'1<`}'83B(Jr>stŒl)&$2!):78$7!)'*/g*X7=z7=1<"<(;"l5."$"$%2?'(I$12!):78$@zB)'I"$
%2?'(>/1):"X)'*_/g*7!(!C]R$!z["$(9)://1*2/
z 2 *2():1%J6"$"<)'`}'3(6r>staF5$7$K5.**a7=@?'!%'
"<‘"$P'/"3J)'*$12!):78$7!)'*/g*7=z7=X<"<(@"B52"$’"$P%2?'(Œ$2!):87$;z'C `}'"O):"<(*2z'F^5.$(’"$
$2!8):7$;z>X k):"<"<((F3)'@z>'`H2"D)'?:)'@"):%'(d):J*<"!Cf'c=N)'3B/1*'F_`W'3):"2‘):0gO"D"$€k)'*
!5E):“3&O1<"B<"**l"<)(?'!¥"$h(@"2P"):"<¥</1)'7=P`}3 "$h%')'*E0k)'7p’"<>"$PX2")'*^<"):"<'Fl/1)'%
"$O%$)'**<"):"<()'*2%"$H5)qz'Cm- 1!(F0g(7!)'O<H"$.O0")'1
p •/'%')'3 7=O;"<!6c$)(?'0!(J)'(
"<¥"$I<"):"<'Fa"$!5E):`W'3):">3J)(z‘!("<K"<)(?'(*l!?'(`iO"$!D"$)'¦2"D5O*‘O! k):
" 2
*2():1%C
‘3/1*2(3B(;"<( #+#r 2”*2():8%&`}'E"$D3/1*2.])ZNA")'<ph)'P7=3B/1):(["^"<Br>+ 2T 2”*():AT
%Fk)'.$|5.¥Kf%O (Ì C ‘d1['"R/q?A)';z[/1):")'*/g*X7=zI`W'3):"P`}' +#r 2C (7='F
"$&*2zh5)qzK[5.$17$["$&$2!8):7$;z[7=<"<)'1R"$/g*X7!2(R7=X!([0Lz +#r 2 .0;zP(<"<87="%
"$<!"'`^)q?Z)'*X):01*2c/132"?'c)'7="2!Cf'D=NA)'3B/k*2'FO% 9
4( 9 C ,95-N>:Fk"$ ADC EGFG3IH)' AD35- MI) ORQ
)'7="2d):P'")(?:)'*):01*2'C ‘[7!)'’<!h"$I 4g(7="'`H"$(<h(<"<X7="2cX"$I`W)'7="d"$):"d"$I/g!`W'<T
3)'7=&'` +#
r 2–*2():8%12")'**2zI3):"7$(#"$&/g!`}'83)'7='`mr>+ 2#T 2d$
C q5!?'!(F #+
r 2Ž"):p'(
3dO7$¥*%'!H"<7=@?'!%'c"<)'['/"X3)'*g/g*7=zB"$)'¥r>+ 2T 2dF0g(7!)'O<D`}'3J):"2P):0O""$€k)'*
!5E):S3dO<"&<"*
* •"<)q?'!< 9)'**,'`H"$J;"<!3B():"<I<"):"<(‘"$/1'01*2(3[Fa5.$!()'&r>+ 2#T 2dF
`W'3):"):0gO"l"$.€k)'*1!5):8J3dO<"l1*2z/'/1):%):"<#`}3 9
4AK3L-NM,)PORQDO/I"< AK3L-m)'1J"$(J"<
*>@-8C- 1!(F'"$E€1%OH$q5~"$):" +#
r 2ª(nLO2(*2%$;"*2zc*%'!,"$)'
k):>
" 2ª*2():8%#"<7=@?'!%'
"<>"$I'/1"3)'*^/*X7=z'Fa0g(7!)'O #+
r 2®;"<L1O7=(c)>3)'**^;O13c0g!c'`)'2"21)'*l<"):"<P"<)'2"2
7='(/1%¥"<9O01O"7!)'*X*&)'1S!"O!C¦Vi+#**l"$!P3B!"$Ac5!h"O(¦"<6'/1"32b!I"$(2
/g!`W'3)'7='CÈ\
«$(B"$
+#r $2!):78$@z ):"<"<((F@"$.(;"2"{zd'`g"$R12?LXO)'*@O0")'<pAmm(<"<qz'(GCm]R$
3):p'(2"m3B/g201*2^"<c$):O0")'<pA,0g!"{5!(J19 4!(@",/1):(@"m")'pL^'m"<c"<8)'<`W!l*2():(JO0")'<pA
`W3&/'01*2(3e"<P)''"$!qC.- "#)'*h3):p'(2"#9 8I7!O1*2"R"<h=NL/1*2".)';z¥pLX¥'`l"):"<):0k<"<)'7="2>
Ø
DØ
S1
S4
x x
x
x
x
x
x x
S2
S3
Ø
f 2%O
~
­~+d=N)'3B/1*2E3):b!E")'<pc"<X**O<"<8):"<m"$H3B!"$Ad'`1s#()'B)'<_C l
)'78$;3Ë7='(</g
"<J)!"R'`"):"<( F)'["$/g!2/1$!8)'*k<"):"<(#):3):p'(¥0LzJN= !C
"$?:)'*O`iO7="2P!/(<(;"):"2C
s#()')'
<ahV (w'w'¹ \m(7=0)'B;"<!(<"% <)'*2"<!1):"%'/"3b():"2 #)://)'78$"<$12!):78$7!)'*
(`}'7=(3(@"l*():%Ca- "$(2l)://1)'7$F;"$.<"):"<#</1)'7=#‘'`g"$R'2%X)'*r6st’,/1):"2"2(B;"<
)¥X :•X@"<!"'`EO01/1)'7=(&
 !‰8 ‰!Å!Å!Å(‰8 Fa)'c$q5‘f~2%O
Ch]R$!zK"$(=€kB"$d¡ =Ÿž ¡k›A =0Ÿ '`R<!"
 F
„ 8VY
 {\"<>0g"$I!"&'`R<"):"<(d"$):"7!)'¦0J()'7$(’0Lz‘I)'7="S`W3©
 ƒC <_!"d„ ('"<c"$cOP'`a"$c/!82/1$!(l'`,)'**g'`~"$d Cl].$(2R)'*2%''"$3Æ5^'pL.0;zP)'*2"<!):"X%B0g!"{5!(
"{5/1$)'<((C~].$R/A7=(lX2")'*b!(0;z3):pA%&%O(<(^`W'^"$#?:)'*O(m'`_"$<"):"<(HX
„ Cm].$(<
?:)'*O(7!)'9"$(>0g&?A2!5(>)'=€%h):01<'01X%J"<!3J)'*a!5):8#`}'()'7$!%2 <P"$):"()'78$
!%2¥0(7=3(.2"E|5.Kr>):p'q?¥(7!2P/'01*2(3 C
]R$(Fa"$B€1"/1$)'<'Fa()'7$’'`E"$(<hr6stl&c*2?'("<6/AO7=")'*/*X7!2(D¬ `W'c()'78$
!%2GCR- 6"$d<(7=>/1$)'<'F"$(<d2"X)'*/g*7!2(R):d$(*X[€NA(6)'1K7=X!(K)'):01"<)'7="#3J)'7=
'/g!):"<'D`W'3B|?L%¥)'3B%["$<"):"<(&‘
„ C].$=€k(D)'‘):0k<"<)'7="cr>staF_5.$<"<)'2"2
/'0k):01*2"(m):D!"<!3X(>Vi)'HXhr>+ 2\l0;zI"$V}€1NL(k\l/g*7!(m'`_"$78$*I")'<pA!Cm]R$X^):01"<)'7="
r>s#tË7!)'S0*2?'("<>!"<!3!5–?:)'*O(`W'()'78$S'`H"$I<"):"<(&X
„ Ch]R$B€k<"D/1$)'J7!)'
"$([0g!/g():"<(["<J*():[!5Ê/g*7!(¬ - F)'K<B¥O;"*"$/L7=(H7=;?'!%'(!CE]R$X^/1L7=(R7!)'
0gD/|?'(["<I7=;?'!%'c"<J"$'/"3J)'*g/*X7=zI)'P?:)'*OD`iO7="P`}'"$D'2%1)'*r6stC
]R$J)'?:)'@"):%'h'`E"$Js#()'¦)'1
<_X)://)'78$S"$):"c()'78$¦r>stÇ"$1):"&3dO<"0g<*2?'(¦7!)'S0g
p'!/"H3)'*X*k<"$):"H2"H7!)'P0g#`}()'01*2z<*2?'(PO%c"):0k*2=TY01)'<(P!/1(<(;"):"2'`_"$?:)'*O#`WO17="2C
+#'"$!a)'?Z)';"):%'Xg"$):"a`}'3J):"2D):0gO"€k)'*:!5):1G7!)'0gm/'/1):%):"<():/k*2z.):"a"$l):01"<)'7="
*2!?'(*†F0g(7!)'Oc2"R7!)'
<<pA2/¥|?'! "$€NL(¥/*X7!2(EP"$cO0")'<pA!C
- ";"<!(<"X%P"<K'"<"$):"&‘r6+ 2dFG"$<zL;"$!"7!5):`WO17="2  /k*)(z9)[*23*):
"<P"$2")'*G%O(<(#`W'#"$&?:)'*O('`m"$<"):"<(6
„ CD]R$(<&<zA;"$!"7&!5):8€N¥"$hŸ< !£Uœ'˜ižW³: ³Zœ'£È´1 —'`E"$"<!3J)'*,<"):"<( '`()'7$’O0")'p Ch]R$r>+ 2#T 2e)'*2%''82"$3°"$(S*2():)¥/*X7=z
`W'()'78$¥O01")'<p F)'h"$/1):(@"")'<pAE"$([O<#"$</g*7!2()'E):01<"<)'7="H'/g!):"<'8E"<*():h"$(2
/g*7!2((C-•P 4g(7="!Fkr>+ 2T 2–7!)'¥0g?L!5^(¥)')'[*X'FA(`W'7=(3B(;"R*2():X%&?'!82['`~D`iO**
2"<!):"['`a"$cs#()'K)' <aP/A7=(O'C
«$!I"$J"ƒ5^93B!"$Ac19 4!qF$|5^!?'!qF~X"$):"s#()'¦)' <_"$(¦O/g1):"<"$J?:)'*O(c'`H"$
/g!2/1$!)'*~<"):"<(!F,5.$!()'cr>+ 2T 2L(c'"d7$)'%'I"$I<zL;"$!"7J!5E):`iO7="21!CI]R$Js#()'
Ø
Ø
'Ì
)' <a9)://)'78$S7=O*X90X3B/1*2(3B(;"<(95.2"$1>"$r6+ 2Æ`W)'3B!5'p‘3/1*2z[0Lz>/g!3J2"<"%h"$
<zA@"$!"7d!5):8."<[0gO/):"<(60;z¥"$d/k):(@"")'<pA!C !"$#)://)'7$F5B7$L<¥™:· #'`^"$
"<!3X)'*,<"):"<(&… ‹ "<[0B"$[Ÿ< †¶! !Ÿ< !·kš8 h˜{ !Ÿ8¤žW·kœ:£,—=˜Yœ'˜{ D`W'cO0")'<p C <G!"d¸ V†… \#0g"$B?:)'*O
'`."$1d!`}!(7=¥<"):"<¥):`W"<!"$ TY"$M2"<!):"“'`."$[s#()'M)' <aS/1L7=(1O'CS]R$(’=€kP"$
<zA@"$!"7.!5):P'`_"$!`}!(7=D<"):"<… "<0#b!!­  V†… \m» Cm+P=€."$D<zA;"$!"7!5):8
'`m)'**a'`,"$c'"$!#"<!83)'*a<"):"<(D… ‹ "<h0g"$9 4g!(7=0g!"ƒ5^!(>"$d?Z)'*XO'`l<"):"<B…)'6"$
?:)'*OD'`"$!`W!(7=c"):"<'­  V†…:\m»Ê¸ V†…:\ ¸ V†… \8C
-•@"<!(<"%*2z'FL"$#r6+ 2 ?Z)'*O.`iO7="2I(7=3B/g2"2J/|?L(l)5E)(zI'`G</g!(%&O/J"$#s#()'
)' <_I/A7=(O'C-•I"$#s#()'[)' <_XJ/L7=(O'F;O//g<."$):"Eh2"<!):"2 FA"$#?Z)'*O(^'`a)'**1'`
"$/g!/1$!)'*g<"):"<(#`W'.O0")'p Fk
„ VY
 †\8Fk5^!&7=()'<([0LzP)J7=<")';"#)'3BO1@
" @CH]R$(¥X["$
s#()'K)'/
<_[):/1/)'7$GF"$'/"X3)'*/g*7=zI`}'.O0")'<p ^3dO<"H0gD(7=3/1O"<(C y O".P"$cr>+ 2
)://)'7$Fk"$Ÿ =£2œ'˜iž}³' .?Z)'*O(H'`~"$c<"):"<([
„ 8VY
 {\5O*¥'"$)(?'d7$)'%'(GCRE<(nLO(@"*z'F"$
<zA@"$!"7d!5):86`iO7="  5O*9'"D$)q?'7$)'%'(FG)'>"$!5O*>0gd[!(>"<[(7=3/1O"<
"$h?Z)'*XOI`iO7="¦)'’/g*7=z‘`W'BO0")'p C A2"O):"c'`."$&pL1¦7!)'’()'*zS):<P2`H"$h?:)'*O
`iO7="2[`W'.c'`~"$c'"$!O01")'<pL7$)'%'(!C m?'(62`~"$(*):"?'c?:)'*O(H'`,"$c/!82/1$!)'*"):"<(
7$1)'%'[<3B!5.$):"!F,"$J/!?A2Oc?Z)'*XOJ`iO7="2«Vi)'¦/g*7=z\#5.**l$)(?'P?'!zS3)'** y (**X3)'!'!F
)'¥7=1<(n;O(@"*2z'F1?'!zP`}!5 y (**X3)'P01)'7pAO/1R5.**0gD!((¥"<IO/):"<"$?:)'*OD`iO7="2C
Lh5d7!)'><!"$):"#"$&r>+ 2 ?:)'*O&`WO17="2K(7=3B/g2"26)'K"$dr>+ 2#T 2 *():%I)'*2%':T
2"$13Æ/q?AD)3' 8I7!2(;"!F*X#/A7=(OD`W'.3B/k*2(3B(;"%d"$s#()'6)' <a[)://)'7$C
-•K"$&r>+ 2 `W)'3B!5'pkF"$!d#I!(6"<P<"%O$h0g!"{5!(>"$c"ƒ5^P/1$)'(R'`m"$ds#()'
)' <_XJ3B!"$A­m)'***2!?'(*^'`_"$#$!):7$;zJ):D)'*25E)(zAE/g!`W'3
% 2 O/g1):"<(!C .(7='FA"$D<zA;"$!"7
!5E):[`iO7="2[7!)'K0g7=3/1O"<(¥1*)'7!7='%"<
 ½ †V …:\,» ¸ V†…Z\ ¸ V†… \
V \
½
5.$! XH"$c/k):(@"")'<p['`mO01")'<p[Š1F… H"$c!`W!(7=&"<!3)'*a<"):"<d`}'#O0")'<p[ŠkFk)' ¸ ½
"$B?:)'*O`iO7="2>`W'D"$B/1):(;"")'<p
Vi7=3B/1O"<(9`W3 @\8C]R$!XD)I/g'"<(;")'*~/'0k*2(345.2"$
"$)://1)'7$Œ$|5^!?'!q­¥2`.)9O0")'p¦$)'3dO*2"2/1*J/1):(;"")'<pA!Fl"$(“"$<¥3&O1*2"2/1*2J/1):(;"B5.**
€1%$;"6h5.2"$¥()'78$9'"$!"<[=€k&"$dzL;"$!"7&!5E):K`iO7="C]R$&<zA@"$!"7c!5):17=3B/kO"<(
0Lz ^n1CaV \R7=O* kO17="O):"<5.*X*2z)'[/1!?'(@"r6+ 2T 2Ë`W3e7=;?'!%%C
- ""$!!`W'd(7=():zK"<h=€"$<zA@"$!"X7!5E):6"<J0g&<3d7=3B/13<c?Z)'*Oc"$):"#5.**
7=;?'!%''CŒ]R$d7!)'“0gP)'7!7=3/1*$(¦0Lz¦3J)'@")'X%>)>7!O(;"d?:)'*OP`W'  ½ V†…Z\8F)'“O/):"%>2"
7=(3B(@")'*X*2zJ5.$(!?'!#¸V†…Z\VW'D¸V†… \<\EXH7$)'%'(G­
½
 ½ V†…:\^­È»–V 3 Y\  ½ à  ¸ V†…:\ ¸ V†… \{ÄY‰
V \
½
5.$! Gm)&*2():%#8):"<./1):)'3!"<!l"$):"3&O"l$Xpc"<b!!&)'7!7='X%"<c"$.OO)'*17=2"21RVi)'
lnLO):"29V (Ì \<\8
C «2"$J"$1l78$)'%''F3dO*2"2/1*H/1):(;"^")'<pA5.*
* •/1O** "$.?:)'*O#'`  ½ "<|5):8P"$(2
7=3B/1O"<(S?:)'*O'F~)'’2"5.*X*,!?'(@"O1)'**2z7=;?'!%'P¦<3I7=3B/3<?:)'*O'C¥].$c$O*‘(1O
"$):"r6+ 2T 2 7=@?'!%'(E)'^5^(*X*†C <_!"^Ol7!)'**"$Xm3BAU€1(B)'*2%''82"$3–r>+ 2#T{s <lC;-•"$.7!)'<.5$!
)'**~/1):(@"c'`")'p9Š>):%'!I"$B(*X):"2?'?:)'*O(D'`E"$B"<!3J)'*,<"):"<(d ½ Far>+ 2T{
s <ª$O*
7=;?'!%'c"<h)B$2!):78$7!)'**z'/"3)'*G/*X7=z'CRVif'3)'*_?'!U€k7!):"2¥'`"$1HH*2!`W"H`}'`WO"O(<():78$CÈ\
s#(</1"<&"$&`W)'7="#"$):"#5^7!)'97$)'%'dr>+ 2#T 2–"<P3B/k*2(3B(;"R"$s()'9)'/
<_63B!"$AF"$!
)'‘3B/g'")';"#()'<'""<¥3):p'B"$17$)'%''C y z>$*%J"$<zA@"$!"X7&!5):1#€NL(GFgr>+ 2
3):p'(J"$KO01")'<pLB7=;"<=NA"•TY`}!'F)'Œ"$!!`W'KX!/g((;"&'`#"$¥/k):(@"B")'pL!Cª]R$3):p'(I2"
()'2!"<9$):h"$PO0")'pLd)'3B%93)';zS9 4g!(;"c/1):(;"d")'<pL(C6- "d)'*X<>3J):p'(B2"c()'<zS"<>=T{O<
"$B*2():(6O0")'<pA#XK`iO"Od(`}'7=(3(@"*2():X%J/'01*2(3!Cf~)'**z'F2"O/1/'".%'():"<!&<"):"<
):01<"<8)'7="2F0g(7!)'O<H"$RO0")'<pL(,'"m!(B"</
<O!")' "$.7=;"<=NL"lX5.$78$d"m~=NL(7!O"<(C
Ø
DØ
Ø
All policies
All Hierarchical Policies
Optimal
Hierarchical
Policy
Recursively Optimal Policy
Ø
f~2%O =x ­l+Ê7=3B/k):<I'`G"$#/g*7!2(^*2():(I0;zJ"$#r>+
Optimal Policy
2dF +#rSF)'hs#()'[)' <_I3B!"$A!C
+#'"$!¥5E)(zª'`d"$pA%’):0O"["$PO9XI"<Q7=![$|5u`W'3):" kq5.[):O1«"$
$ 2!8):7$;z'C.-•6r>+ 2dFg?Z)'*O&`W'3):"K!?'! 1|5. <X@"<7h)IO0")'<p[`W3®"$&/k):(@"#")'<pA!C]R$
(O(D"$J7=;"<=NA"•TY`}!h/'/g!"ƒz6'`E"$IO0")'<pA!C y O"&‘"$Js#()'¦)' <_X3B!"$AÉVi)'X‘"$
#+
r 2Ž3B!"$L¥)'.5^(*X*}\8F1`W'3):"[):0O"R"$c<"):"<("$):".):?LX2"<(P`W**2q5%B"$c7=3/1*2!"2['`
)O0")'p 1q5.01)'7pI;"<"$O0")'<pJ"$O%$I"$?Z)'*XO(l'`a"$#/!82/1$!)'*1<"):"<(!Cl-•I'!E"<B*2():
)'SVi$!):7$17!)'**2z\l'/"3)'*g/g*7=z'F"$1H`W'3):"2 kq5 3dO<"HA7!7!O(C
1|5.%¥X@"<>2"`W3
`W'"O):"<(*2z'F~S'!c`W'c"$IO01")'<p9"< <O1!<")' ["$IX`}'83):"2
"$#7=@"<=NA"!F"$O0")'<pB3dO<"#ViX%'(!)'*W\,$):."$#)'3B"):"<#</k)'7=#)'^"$7=;"<=NA"O)'*")'<pL(Cm]R$
c5.$@z9"$ +#r¿)'¦s()' <_’3B!"$Ac7!)'1'"&O//g'"<"):"<P):01<"<)'7="2¦`W'dO0")'<pAc)'¦<"**
%O):)';"<!¥'/"X3)'*E/g*7!2(!C“f'=N)'3B/1*'F7=!)‘3):b!¥5'*“52"$’5.[L'5E)(zA!C AO//g<
"$JO0")'<p •5)'*p@TY"$O%$AT{L'5E)(!z KD=NA(7!O"<(’‘"ƒ5^>9 4g!(@"D;3c$)(?A%¥=N)'7="*2z9"$h)'3B
$):/gh)'¦2b!'/
C q5!?'!(Fm0g(7!)'O<h'`R):)'%'(3B(@"'`H"$POOX%P;3J!F~"$I'/1"3)'*^/*X7=z
(nLO2("$):"6;34+&F"$&'/"X3)'*_/g*X7=zh`W' •5)'*p@TY"$O%$AT{L'5E)(!z Ip'!!/k"<I"$*2!`W"(%'d'`
"$L'5)qz'FA5.$*2L3 y FA"$'/"3)'*k/g*7=zp'!!/1"<d"$82%$@"^(%''`G"$#;'5)qz'Cl]R$(I"$
<"):"<h`}'1•5)'*p@TY"$O%$AT{L'5E)(!z K3dO<"&X7!*O"$I(;"2"{z9'`H"$J7!O(;"cL3[CP+o7=@"<=NA"•TY`}!
?'!['` •5E)'*2p;TY"$O%$T{;'5)qz J5^O*X['".!(SVi)'1¥7!)''".O1<|\^"$RX`}'83):"2C
#`^7=O<'FG2`,"$1<"17="2K?'!z>3B/g'")';"!F"$(6"$r>+ 2±$2!):78$@z67!)'903BAU€1(K"<
$)q?'J"{599 4g!(@" •5E)'*2p;TY"$O%$T{;'5)qz >O0")'pL!Fa`W'cp'!!/1X%¥"<K"$J*2!`W"c)'S)''"$!c`W'
p'!!/1%J"<J"$c2%$;"!CR]R$c/1):(;".")'<pA.5**g"$(>$)(?'d"$&'//g'"O2"ƒzI"<P7$;<&5.$78$KO0")'<pP"<
O<I95.$78$7=;"<=NL"!F~5.$X*2"$JO01")'<pL&7!)'(3J)'7=;"<=NL"&`}!'CP]R$XD)://)'78$'0L?L2O1*2z>/1O"
)'9)'(>01O8(K>"$B(2%!'`m"$B$2!8):7$;z¥"<¥)';"7!2/1):"<5.$):"#"$B3B/g'")';"O0")'<pA#5.**
0g'Cm+Ê"<'/1X7`W'E`WO"O#(<():78$¥"<€kI5)qzLR'`a)'O"<3):"X7!)'**2zI7=;?'!"%BDO0")'<ph;"<!?'!)'*
3B'P</g(7!)'*b!(SO01")'<pL&5.$(S"$&5^O1*0gPO<!`iO*†C¥tm(O3):01*z93!"$L1c(*X):"<(¦"<>"$Ps#()'
)' <a[)://)'7$¥)'Kr6+ 2T{
s <’5.**0gD):01*2D"<J)'7!7=3B/k*$P"$X!C
‘>7!)'É<!'FE"$!!`W''FE"$):""$!KB)‘`iO1)'3B(@")'*R"<)'!? 4¼0g!"{5!(M'/"3J)'*2"{z“)'M7=;"<=NA"
!/(1(7='CÊ- `56%2?'‘O/ª7=@"<=NA"¥!/(1(7='FE5^‘)'*¦%2?'9O/«$):X%FH=T{O<'F)'«<"):"<
):01<"<8)'7="2F1)'K7=<(nLO(@"*zh*2():8%5.*X*1"):p'd3&O78$K*2%'!(C
f~2%O =x O133):2b!(R"$c(*X):"2$/h0g!"ƒ5^!(6"$c/g*X7!2(H*2():(¥0;zP"$dr6+ 2dF #+#rSFg)'
s#()'k <_X‘3B!"$A!C]R$s#()'k <_X9/A7=(Od5.**G€k6"$'/"3)'*/g*7=z'Fg"$ #+r 3B!"$A95.**
*2():P"$D0("H/*X7=z"$):"R7!)'[0gD!/1(<(;"<([0;zI"$$2!8):7$;z'F1)'[r>+ 2Ž5.***2():[)(7!O2?'(*z
Ø
Zx
'/"3J)'*H/g*7=z'F5.$78$M3)(zŒ0g[!?'(Q5^'8<K"$)'É"$¥'/"X3)'*R$2!8):7$X7!)'*H/g*7=z'C «2"$Q7!):'FH"$
(%!~'`"$H$2!):87$;zc7!)'(O^"$):",)'**L"$!E3!"$L1_€c%'LAd):/1/(N3):""<#"$'/1"3)'*
/g*7=z'C ‘D!(J"<d%)'I3B'#=NL/g!2(17=R5.2"$J"$
#+ru)'Ir>+ 2 3!"$L1m"<7=|?'!E$q5«$):
2"RE`W'.(2%!H"<I7=<"<O7="H%'LAK$2!8):7$(!C
«
8
"!
H <G!"HO!?A"^"$DX7="<!)&`}'R$!):7$17!)'*1(`W'7=(3B(;"E*():%d3B!"$A"$):"H5!/(<(;"<(
6"$B;"<L1O7="2C]R$/1):/g!$)'$q56"$1):"#"$r>+ ±
2 3B!"$A>O//g'""<(3B/g')'*~)'9<"):"<
Ø
):01<"<8)'7="2I)'1"$#$):%)'(O1<.'`G*2():8(JO0")'<pAVi7=2"<!8) F Ì FL)' \8Cl]R$./1):/g!l$)')'*<
$|5.9=NA/!83B(;")'**2z["$):"r>+ 2o*2():3B' 8I7!(@"*2z6"$)'‘<")'1): 2±*2():%9Vi7="<!2 x \8C
‘h/(<(;"<(¦)'S*B(`W'7=(3B(;"&*2():8%¥)'*2%''82"$3ur>+ 2T )'1/|?'(S"$1):"d2"&7=@?'!%'(
5.2"$’/'01):01*"{z
"<)9(7!O82?'(*2z'/1"3)'*/g*7=z`W'B"$[$2!):87$;zªVi7=2"<!8) ¹ )' º \8C ‘6)'*<
$|5^(J"$):"lr>+ 2Ê7!)'JO//g'"~"$.AT{$12!):78$7!)'*GVW/g**%;\=NA(7!O"2J'`"$.$2!):87$;zPVi7="<!2
¨ \8C .|5^!?'!qFa"$B/k):/!$1)'D'")'(<(9"$O(#'`^?:)'*O`iO7="29)://(N3):"29'c*():%
"$c$12!):78$@z‘Vi7=2"<!X) )' w \8C
]R$1D/1):/g!c$)'&3)'"ƒ5^>p'!z7=;"<201O"!CIf~28<"!Fa"$Jr6+ 2e?Z)'*O`WO17="2(7=3/2"2
$|5.d$q5±"$h?Z)'*OJ`iO7="2c'`R?L1O)'*~O0")'<pAc7!)'’0gI7=3&01("<>zL2(*XS)¥?:)'*OI`WO17="2
`W'#"$d(;"2d")'<p
C L(7=Fg"$r>+ 2T 2±*():%h)'*2%''2"$13®)'92")'L7!):"<(9/1;'`i'`l7=@?'!T
%'(7=c$q5 $q5 1*#(`W'7=(3B(;"R*2():1%B7!)'¥0gD7=O7="<(K3dO*2")'!O*2zJ):".)'**_*2!?'(*E'`~"$
$2!8):7$;zI5$*2<"**g(1O%d7=;?'!%'(7=d"<J)B(7!O2?'(*2zI'/"3)'*g/g*X7=z'C
]R$H/1):/g!m$)',)'*<X7!O<(<3R'`k"$R(2%"<)'!? 4gmXB$2!):78$7!)'*L(`}'7=(3(@"m*2():8%C
-•/1):"X7!O*):(Fa5^h$)(?'P<!(¦"$):"&2"&c(7=():z‘"<6)'7=U€k7=I"$I7=;"<=NL"•TY`W!P/'/g!"ƒz9'`RO0")'<pA
"<‘%)'“"$[):01X*2"{z‘"<*():“)9/*X7=z‘"$):"d'/"3J)'*^52"$’(</g(7=""<‘"$P%2?'(Œ$2!):87$;z'C¦+#*2%
5.2"$I"$7=@"<=NA"•TY`W!D/'/g!"ƒz'FL"ƒz;/1X7!)'**2z)'*<*2<(3dO7$h'`_"$0g(=€1"^'`_<"):"<D):01<"<8)'7="2[)'
(O):01*2"ƒz'C
-•3B/g'")';"J2(7="B`W'J`iO"O[(():7$ª7!*O1P?'!2`WzL%‘"$):""$6(O*2"J“"$J/1):/g!J7!)'
0g[=NL"<(1(M"<¦"$KA€2"<=T{$'b!’17=O;"<(M7!)'<'FH"<(<"X%"$K):0k*2"ƒz¦'`Dr>+ 2 "<5'p“5.2"$
?:)'*O`WO17="2P)://qNAX3):"2[3B!"$A!F!?'(*'/1%d*#3BA(*UTY0k)'<(P)'*2%''2"$3JE`}'Rr>+ 2F1)'
<"OzL%d"$r6+ 2T{
s <“)'*2%''2"$3¥C
Ø
! # P].$B)'O"$'"$)'pL l7E$|5.9`W'c3)';z>$(*/`WO1*aX7!O#'`^"$X5^'p
)'m)'*2(@"X) y )(z'!qF ª*X*)'3<a)'%'`}'8F)'1 ‘(*2!zt^78$'"c`W'&$(*/`WO1*,7=33B(;"&¦)'S():8*2!
):`W"!C[]R$O//g'"D'` %')';" =x T w'¹ T T ¹'¹º )''` Af¼%')';"c- R-ƒT w'¨'Ì'¨'¹
Zx %'):"<!`iO**z
)'7pA|5.*2(%'(C
`Ø
Ø
Pl'l Ø
y )'2F!<mC#CGV (w'w'¹ \8C R(XO)'*)'*2%''2"$3J!­ .(`W'7=(3B(;"H*2():%5.2"$I`WO17="2h)://qNA3J):"2C_-•
tl2("!F|+&C2FZ§ .O1<(**†FCLV^!CÈ\8F2*œ@š=›@žW· % œ:Ÿ·žW·
(EŸ<™(š 'ž}· —.™•¶˜W› 4 >^ !£ ¶=˜W› ·1˜{ !Ÿ8·œ'˜iži™'·kœ:£
,™'·(¶! !Ÿ< !·kš8
A)'[f)'7!7=FE+&CrK'%)' )'O`i3)'C
Ø
DØ (Ø
y ):"<F.+&C "C2F y 8)'"<p''F C C2F.§ A%$FCltCDV (w'w'¹ \8C <_():%S"<’)'7="PO%‘()'*UTY"3B>zA)'37
/1'%')'33X%C ,#Ÿ=˜iž (^š(žiœ'£ ·1˜{ =£W£Èž !·kš8 !F F C
Ø
C_V (w'¹º \8C & :·kœ:¤ž†š)(HŸ•™ Ÿ<œ:¤¤džW· CEtl7=!"<
y (**3)'GF C R
Ø
2?'!82"{zPtm(!C
œ'·6™¡˜iž}¤œ:£š™:·1˜iŸ<™:£µCd+E"$()0A7!2(;"U€k7:F y (*3B;"!F
y !"<!p:)'!F_sdCGtCmV (w'w'¹ \8C3& :·kœ:¤ž†š¡kŸ•™ Ÿ<œ:¤¤džW·
r>+&C
'¹
y !"<!p:)'!Fs&CHtC2FR§ ]R2"pL*X!F
y (*3@"!F1r>+&C
Ø
C
=´AŸ•™?& :·kœ:¤ž†š+(EŸ<™ Ÿ•œ:¤¤žW· C±+E"$()A7!2(;"U€k7:F
CV (w'w'¨ \8C
Ø
y O"*!(F_#C2F_§os#():(FClV (w'w'¨ \8CJ+//qNAX3):"%h?Z)'*XO"<!(‘<"<8O7="O(9zA)'37d/'%')'3T
3J%C±- (HŸ•™(š 3 ˜W› ·1˜{ !Ÿ8·kœ˜iži™:·œ:£ ,™:·(¶( =Ÿ =·š8 ’™:· *œš!›@ž}·g % œ:Ÿ·žW· FD//_C ¹Zx L¨'Ì C#rK'%)'
c)'O`W3J)'C
Ø DØ
EO2'F BC2Fm§4]a):"<'F+cCRV (w'w \8C TY/1*)'­I]R$h'/(’/1*X)'%6):7$"<(7="O'C , Ÿ˜iž (^š!ž†œ:£ ·1˜{ =£W£µž =·kš =F
V \8F x@w '¨ C
Ø
Ø
s)(z)'FtC2F;§ ;"<F#"dCkV (w'w \8C_f(O1)'*(X`}'87=(3B(@"^*2():%C1-•+, '³Zœ:·kš —.žW· =´AŸ•œ'£ ·(¶!™:Ÿ¤Bœ˜iži™:·
(HŸ•™qš8 —8—8žW· 2Z—=˜{ =¤d—E1 'F //GC ̺ L̺?
Cr6'%)' )'O`W3)'1F A)'¥f)'17!7=FE+&C
\Ø
Ø
s#()'FD]DC2FR§ <_GF C T &CDV (w'w'¹ \8Cos#(7=3B/g2"«"<(7$Xn;O(P`W'[/1*)'1%SXª"<L78$)'<"7‘3J)'!C
]a(7$GC@!/GCA;T w'¹ T FAs!/k):"3B(;"l'`_3B/1O"<! A7!2(7='F y q5. 12?'!2"ƒz'F;tlq?A(7='F .$A
-•*)'1C
Ø
Ø
f~2p'(!F c
C .C2F ):"!FtaC .C2F1§
*X<F C CGV (wº:Ì \8C<G():1%)'P=NL(7!O"%%'(!)'*b!(P'0g'"/k*)'!C
,Ÿ˜iž (^š!ž†œ:£ ·k˜{ =£W£Èž !·k8š !F F 'Ì ¹ LÌ
C
DØ
Ø
f'%'z'F.#C <lC.V (w
'Ì \8C R!"<'­6+`i)'<"J)'*2%''82"$3 W` 'B"$¥3)';z’/k):"<"<!kZ3)';z¦'0 :•(7="/k):"<"<!M3):"78$
/1'01*2(3[CJ,#Ÿ=˜iž (^š!ž†œ:£ ·1˜{ !£}£µž =·š8 =F V \8F qº º C
Ø Ø
Ø
2C F y O"X*2!(FLC2F ):(*20k*%F <lC2FL§ s()'GFL]DCV (w'w
\8C#2!):78$7!)'*<*XO"2
'`ar6):p'|?h(7!2I/A7=(<(O%&3)'7=:T{)'7="!C~]_(78$CA!/GC2F y q5. 2?'!"{z'Fs#!/1):"3B(;"
'`l3/1O"<!A7!2(17='F1tm|?LX(7='F R-C
#)'O<pL(7$;"!FrSC2FLrK(O*()'OF
Ø
)'):pLp'*)AFR]DC2F ')'FErSCl-8C2FR§ %$F gCltC#V (w'wZx \8C Œ"$67=@?'!%'(17=>'`D<"<A7$1)'<"7¥2"<!8):"2?'
zL)'3J7#/'%')'33J%)'*2%''"$3!C =´AŸ•œ'£ ,™:¤¡k´˜Yœ'˜iž†™:·F V ¨ \8F '¹ (Ì C
Ø
):(*20k*%F
ØŠØ (Ø `Ø
<mCataClV (w'
w \8C #2!):78$7!)'*~(`}'7=(3(@"c*():%­tm(*31):z[(O*"!C- (HŸ•™qš8 :žW· —
<™ ¶#˜W›A +4k !·1˜W› ·1˜{ !Ÿ8·œ'˜iži™'·kœ:£ ,™:·(¶( =Ÿ =·š8 &™:·*œ@š=›@žW· % œ:Ÿ·žW· FA//GC (¨º qº? A)'Jf)'7!7=F1E+&C
r6'%)' )'O`i3)'C
Ø (Ø
'0k*2L7pkFA#C@+&CkV Ø(w'w \8C <G():8%):01"<)'7="2J$12!):78$2(~`W'l/'0k*2(3Ž<*2?A%Ck- (HŸ•™qš8 :žW· —™•¶˜W› ž ›;˜W› œ'˜iž†™:·kœ:£ ,™'·(¶! !Ÿ< !·kš8 &™:·,#Ÿ˜iž (^š(žiœ:£ ·1˜{ !£}£µž =·kš =FA//GC w'ÌLw'Ì
y <"<Fr>+cC+#++-ltm(!C
D'` F C . CmV Ø(w
'¹ \8CJr6)'7=:TY'/!8):"<'!­+ 5^():p93B!"$A9`}'*():%C ,#Ÿ˜iž (^š(žiœ:£ ·k˜{ =£W£Èž !·kš8 !F .
V Ø \8F
'¹Aº'º C
<_)'F
C R
C2F
Ø
Ø
!5^(**YFZ+&C2F§R(@01*;3[FZtC?C;V (w
º \8C .
+c­q+#c):78$2"<(7="O^`}'a%'(!)'*;@"<(*X*2%'(7='C
·k˜{ =£W£Èž !·kš8 !F V \8F L¨Zx C
,Ÿ˜iž (^š!ž†œ:£
Ø Ø
¶!™:ŸŸ<™@¢™˜W—&´L—8žW· · =´AŸ<œ:£_·g !)˜ >l™:Ÿ q—C^tl$GC s&C"$(!FkE):8!%2
?'!2"ƒz'Fs!/k):"3B(;".'`m3B/1O"<!A7!2(7='Ft^2"<"<01O%$Ft+&C
<_XF <mC T CV (w'w \8CR !ž}·q¶=™:Ÿ<š8 !¤J =·1˜E£ œ:Ÿ·žW·
r6(**2
Ø
r6;''F~+&C ÇC2F~§o+E"<p'(<Fl#C "dC^V (w'w \8C[tm'2"2b!(9<5!!/1%­R(`W'7=(3B(;"c*2():8%P5.2"$S*2(
1):"))'¥*2(H"3B'C *Sœš=›@žW· % œ:Ÿ·žW· F F 7 C
Ø
Ø (Ø
t~):(FC2F§ .O(**†F$C~V (w'w
\8C+R(`}'7=(3(@"*2():X%5.2"$6$2!):87$2('`m3)'78$(!C- žW·
!´AŸ•œ:£ ·(¶=™'Ÿ8¤œ'˜iž†™:·1(HŸ•™(š —8—8žW· 2Z—˜{ !¤d—8F ,*†C E)'3&0%''Fr>+&C1rK-•]Êtl(!C
Ø
Ø
؊Ø
, :³Zœ:·š8 —
tm(7!O/GF;sdC2FA§ AO"<"<GF CgV (w'w
\8Car>O*2"UTY"X3BR3BA(*,`W'l"<(3B/g')'*X*2zd):0k<"<)'7="/1*)'X%C-•, : ³Zœ:·š8 —
žW·
!´AŸ•œ:£ ·(¶=™'Ÿ8¤œ'˜iž†™:·1(HŸ•™(š —8—8žW· 2Z—˜{ !¤d—8F ,*†C
E)'3&0%''Fr>+&C1rK-•]Êtl(!C
'¨
Ø
tlO"<!3)'Fr¦C <mC_V (w'wZx \8C 
* œ'Ÿ'™:³ & š!ž}—8ž†™:·'(EŸ<™(š ——( —C
Ø
C «*2!z[§
L1!F
!5 $m'pkC
, (
A)'7=!'"†F R
CAs&CV (wº|x \8C,tl*)'1%cXI)d$2!):87$;z'`_):01<"<8)'7="2P</1)'7=(!C #Ÿ˜iž ^š(žiœ:£ 1· ˜{ !£}£µž =·kš =F V Ì \8F
؊Ø(¹(Ø '¹ C
Ø
A%$F C2F
)'):pLp'*)AFL]C2F <a2"<"3)'FLrSC <mC2FL§ Lb!/g(<?:):†FL#CgV (w'w \8C;?'!%'(7=#(O*"m`W'^%*2=T{<"<!/
TY/*X7=z(`}'7=(3(@"•T{*2():8%)'*2%''2"$13!C~]_(78$C!/GC2F 2?'!2"{zI'`~*2'8)'Fs#!/1):"3B(;"
'`l3/1O"<!A7!2(17='F y O*!(F1 C
Ø
A%$F C^taC#V (w'w'Ì \8CÊ]_)'1<`}!J'`*2():%>0LzŒ7=3B/g%*O"2'`D(*(3B(@")'*<(nLO(@"X)'*H")'<pA!C
Sœš=›;ž}· œ:Ÿ·žW· F F 'Ì 'w C
*
%
Ø
3 ˜W›
AO"<"<GFcC CmV (w'w'¹ \8Cd]Rs±3BA(*!­3BL(*%I"$5^'*X6):")P3UNA"Od'`l"X3Bd7!)'*2(!C-• E
( Ÿ<™(š
·k˜{ =Ÿ·kœ'˜iž†™:·kœ:£ ,™:·(¶( =Ÿ =·kš B™:· œ@š=›@žW· œ:Ÿ8·1ž}· F1//_C ¹ L¹'w Cgr6'%)' )'O`i3)'C
*
AO"<"<GF C C2FGtm(7!O/GFGsdC2Fa§
Ø
DØ
%
A%$GF gC,V (w'w
\8C y !"{5!(Sr>stlD)'A(3UT{r>s#t^!­<G():X%Fg/1*)'AT
1%F)'h!/((@"%dpA|5.*2(%'D):"R3dO*2"2/1*."<(3B/g')'*7!)'*(!C,]_(78$C!/GC2F 2?'!2"{zI'`ar>)'•T
)'7$LO<!"<"!Fs!/k):"3B(;".'`m3B/1O"<!)'[-•`}'83):"20
A7!2(17=(!F+3J$!<"!F1r>+&C
Ø
C;C cCV (w 'w \8C % œ'Ÿ8·žW· ¶8Ÿ<™:¤ &d !£Uœ R >lœ:Ÿ : —8CGtl$GC s&C@"$(!F % = ^E**2!%''F #
NL`W'C
Vi]a0gD!/8@"<(P0LzPrK-•] l
t (!CÈ\8C
):"<pA!F#C
º