analyse de la complexit© et transformation de programmes
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UFR mathematiques et informatique Ecole doctorale IAEM Lorraine
Departement de formation doctorale en informatique
Analyse de la complexite et
transformation de programmes
THESE
presentee et soutenue publiquement le 17 decembre 2003
pour l’obtention du
Doctorat de l’universite Nancy 2
(specialite informatique)
par
Jean-Yves Moyen
Composition du jury
Rapporteurs : Roberto Amadio, ProfesseurNeil D. Jones, Professeur
Examinateurs : Bruno Gaujal, Directeur de recherchesClaude Kirchner, Directeur de recherchesJean-Yves Marion, Professeur (directeur)Jeanine Souquieres, Professeur (presidente)
Laboratoire Lorrain de Recherche en Informatique et ses Applications — UMR 7503
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1 U .!
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3C
7( 0 6 h1C q0
Z $U;# 2/%H 8(U f#84C
q1 =E(.8$ ;# 0C
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?
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MX X g
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021 130 5 #"( %3T 2S # . . 2. 2. 2 K ((. C ( 8
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X%F
%C
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T (C) 3 u ::= c(u1, . . . , un) T (C,F) 3 s ::= c(s1, . . . , sn) | (s1, . . . , sn) T (C,F ,X ) 3 t ::= x | c(t1, . . . , tn) | (t1, . . . , tn)! #" P 3 p ::= c(p1, . . . , pn) | x$%& D 3 d ::= (p1, · · · , pn)→ t
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!
8 -,K-, -, RS 0 6C 2 a1 → (· · · (an → a) · · · )
2. B BC !A V. f ;
(a1, . . . , an)→ a
!
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0 RS ;-,!
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t ∈ T (C,F ,X ) 7 RS
τ% 2.
t : τ%
2 t
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%ti : ai
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!
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RSe.-1 %A
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t 6I R'cN'
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t
s 8(U !e 5 D-,
t w n=l&w'Hcv m N6v7
s%
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⋃
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i<-,
t E si!
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v w Hl(rAvBm2vx;p,H*s nNr nNm =m2vXw l(mT (C,F ,X )
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σ 4C Y
X
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X T (C,F)
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T (C)
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t
x
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tσ= Y
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t !$ mJ vPOY
t 2
u<-,
u E t!
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2$ 2σ
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! 8 0mJ Nl(sum2v
t
s%& d
u
σ(v)
t!e
t→ s!
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→ K2 7
→!
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%= Z'cN'c 4-,t∗→s
t = s
t
+→s
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→ 51rjnNm NpNxys`w p Hzs n=r! = Z'cN'c D-,
t!→s
t∗→s
s
"C 2 2 !
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t, s
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t∗→s
t∗→u
% U v
!-s
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u
∗→v
!
A
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t-, V 8(UhC 2 2
s
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u
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s = v = u C `-,!
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nP #%O%7[ E(E T 2. ;-,7H RS
τ z 8X7 RS
τ!
[τ ] = t : τ | t ∈ T (C)!
T XC f. =6 & 2 !(g C 8% C 2 `(.2. 1 !
nP #%O%7[ E[V PHK 46% !0 Q N6p=sura 6 RSA.
Γ( "
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[ "
] : [τ1]× . . .× [τn] 7→ [τ ](. 6 Z !
W -,ui ∈ [τi], [
" ](u1, . . . , un) = v
8 " (u1, . . . , un)
!→v
v ∈ [τ ]
!j? %
[ "
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N,0HKYL#.70 d T ZS 2# 0 . . 2 2 * $ ( & -,. %.;. 2.. !jT RS
, , " % 2 8 0 RS /1
W T #0
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≤.2. $U(. ; ; 2.!
≤ : × → !
0 ≤ y →W T !0
S(x) ≤ 0→ 4Z.'1&0S(x) ≤ S(y)→ x ≤ y
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" (x,
J%7.)→
F (!1(x,
J%'.)
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(y,.))→
" (x ≤ y, x, y,
.)
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, x, y,.)→
F (!1(x,F [#1
(y,.))
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.)→
F (!1(y,
" (x,.))
" =
: " → " !
(J%'.
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(F (#1
(x,.))→
" (x,
(.))
9 c
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t1, . . . , tn 2 8 !T H v NKoew v M_pNxCM l(x NpNsur Z
t1, . . . , tn . F0 ; 61 .8( /.
[ "
](t1, . . . , tn)%
" (t1, . . . , tn)
!
F X$. `(# B-, -, D .8( /E8 !AF 1 (U _-, g <H = _H% <-, 4 ;f.8 < ZS 2#8! 2.8. 2 % -, 2.](
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nP #%O%7[ E 8 W 46%'.7.'0J#0H1I&05TKY0/1Q
t !eQ K(. . 8 6 ;# 2
|x| = 1!
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i=1 |si|!
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i=1 |si|!
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T -,
t1, . . . , tn
% 8 j N6p=sur 2t1, . . . , tn
$C . 8 . jT (
∑ni=1 |ti|)
!
T _-, U 2. 8 Bg N9 !
nP #%O%7[ E(a 1_L 46"B0 !0J"546.'"5 !. Q N6p=sur 2S 2#8 " 2.8. 2
t1, . . . , tn 2 8 !@T!uv w*o/p M_v v7M_p=xCMl(x
,G .8( NU f$ E 1.] /. = B [ "
](t1, . . . , tn)% = 1 " "
(t1, . . . , tn) 3 -G " 2.]('
!T!uvwzo/p M_v v7M_pNxCM l(xDNpNsur 2
t1, . . . , tn V f$ 6 -,
,G7 .8$ " (t1, . . . , tn)
!
N,0HKYL#.70`8 Q ZS 2# 7 . . 2 !
: → !
(0)→ 0
(S(x)) → S(S(
(x)))
9B9
: → !
(0)→ S(0)
(S(x)) →
(
(x))
: → !
(x) → 0
" : → !
" (x) → ( (x))
T " (S(S(0)))
( f ,G 2.]( !$TG BH
" (S(S(0))) → ( (S(S(0)))) → 0
= "
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T 8;# ,G 2.]( E G 5
7- 3 (U ;# E !
6!<Q `(#z 6 ! "
(S(S(0)))% / N1sur/s6N1l(N 7
-, -G 4 2.]( K 4!
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E -,
2 t1, . . . , tn
% z 4 N6p=sur 2t1, . . . , tn
(C . V. jE(
∑ni=1 |ti|)
!
T< % C .ZS& `-, !F =(U 7"g< /% U . H ZS 2#8 2.8. 2 =
$. !ei A@ ]% ; 1. ( 2 C E(N) ≥ N + 1
!jW 0 4 g e% C ! Z * &`-, 2 8(U f#8$ 0 ` /1 2 G f G $ !-, Z' $. R0 RS`- 8% $ `-,/1 !
! #%$ '&V V XV ,V@
021 130 5 #"( %3T X 6 1.2. (( !!F !!Q ^ !@i !GQ 2 ) Q/QR=B>-+ !
. K Y 3-, 2. V f " ! ; 8 8 %&-, % H X9 2.]( H 2 -, H 2.8 B$. . 7 /! / .( 1 ! W =$U 0 8 2 % 0 4 7 $Uh /-, =$U !H? * e. $6.`- 0 8\6C ; f.8 P1 H = * _ U B0 U 8 6 P1 !i % ! GH X -, /-,X 8 K-, K 0-, X @ K ![2 ]% (. 1 2 nP #%O%7[ E Q l*+46" 3#%'#0/1 h"5[KML#0/ !
& ::=
,: i1, . . . , im;
#" : o1, . . . , on;
0 :"
0; . . . ; n :"
n; end" ::=
x |
" x | i | x
i |
i |
x ::=
::= 0| . . . |n|end
9 ;
T i1, . . . , im
tOp=m s p xfvw v rIH*mJ_v o1, . . . , on
tNpNm s p x;v wvw nNm H*s v! = I 0C = H . !
x . ; D- =%-, G 2.%,
% D =D Z.7`-, z . 8( =6 !$F - = D M_n,NKo4Hcvl(m _B f# !A 1 BC 3
Nm
Nn !T! 2 " 70 ! ,1 . 0 !j$. . /1K !HT! 2 C B f'
= * U&. $ =6 = = / `-,$.!<TG = 8 / `-,$.
0!<T 2 C E
* / K z !
nP #%O%7[ V (. !C " -, e6 $ : = * 6- S != Z'cN'c 4-, "
(i) ="
i!
T 06.Z 4 Z' 2 DC GC ( ( 7- 2.; C !F e 76. C 4 % -, B 1. = 7/ U !e+#3H . Dd/%f B 2# B X =$UZ' 8% -, -, R' 4 C 2 K 8 -, * ;-, !
nP #%O%7[ V ) [ #1 T[ !%7!0/1T! 8X
Ξ(
) Bp H Hcv s?:=rjp xfv wX$= ; 7-, .
9 ! ∈ Ξ(
)!
;!KQ ′ ∈ Ξ(
) "
( ′) ∈ " x,
x, ′′, x
′′ % ′ + 1 ∈ Ξ(
)!
>!KQ ′ ∈ Ξ(
) "
( ′) ∈ ′′, x
′′ % ′′ ∈ Ξ(
)!
"w nNl&w cm2n=l(Hzsurjv z 1 =4 $0 = !-< U 1 Z' 6 6!
nP #%O%7[ VE F ZT0/"$&%'(f1_^,&4RN,% #0 6 4w NrIHcp Os l/v Nvr=H M_nNm m2v2M H 4 F -,= = f Z' ! F -, Z' !
W Z % 8 2. D;-,8 =(U =6 ZS =U`-, !
021 1 5 !$"L T 6 . 2. 2. ;-,H /2. '
2.4 9 ! 9 !F 6 ZS 2#8 G . . 2% Z$ =X 8% H
2 `-, !
nP #%O%7[ V(V 1 1_&%O& &%'( hw l w H*s6Hzl(H*s nNr
σ C 2 !T Z$
σx ← v 5-, v
x
36I 6-σ!!? V6I %
x1 ←v1, . . . , xn ← vm
Z$ Y-, vi
=(Uxi
0 -, = !
nP #%O%7[ V[d ( %'.70J!030H&[ ! Kojsuxfv v7mvAHcnNl(m 0= !&T ` .
[]!
:: l 7
2I l6-, !
nP #%O%7[ V 8 4R J H p H NloAmn:m2p,NFNv
(
, σ,
) z%
σ 2$ 2
B 8% z . e. ⊥
2. . 8 !
nP #%O%7[ V!% PHK 46% !0 T
→ -G(. 9 ! 9 /4 * U(. BH B( =6 !
F 6 X((# 2. .8 8% +→
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∗→
K 2.$,U
9 >
" (
) ="
x . & `
, σ, →
+ 1, σx← σ(x) + 1,
" (
) =
x σ(x) = 0 8& `
, σ, → ⊥
" (
) =
x σ(x) 6= 0 (.2.8 & `
, σ, →
+ 1, σx← σ(x) − 1,
" (
) = ′ & `
, σ,→
′, σ,
" (
) = x ′ σ(x) = 0
x = 0& `
, σ, →
′, σ,
" (
) = x ′ σ(x) 6= 0
x 6= 0& `
, σ, →
+ 1, σ,
" (
) = ′ &
`
, σ, →
′, σ, (
+ 1,
)
" (
) =
& `
, σ, [ ′, ]→ ′, σ,
9 ! 9 Q&.8 `-4 !
9 C
h →
! !→
/ 2 % 2' =' Y-s
!→s′
s
∗→s′
s′ = end, σ, []!
T 6 & XC [&
]$.
[&
](v1, . . . , vn) = (σ(o1), . . . , σ(on))
& ` 0, i1 ← v1, . . . , in ← vm, []
!→end, σ, []
!
-,-, 8\!C .$. @ ; -, RS .8 ;-,*0 U8 21= 8 2 !@ 4 1-, 7 R&S (
2. E GC !
N,0HKYL#.70 % T 62 6 7 7 (U !
i . x, y
!Q
y!
0 : x end1 :
x
2 :"
y3 : 0
end
nP #%O%7[ Va W 4Z"B0 PH&4RQ
(
, σ,
)3. 0 =6 !/Q 7Hzm2p M_v J H p H
θ(
, σ,
) * h U(. !ei
C 8 (. 6 2 ! Q "
(
) 6= x ′
θ(
, σ,
) = "
(
)!
Q " (
) = x ′
σ(x) = 0
θ(
, σ,
) = x = 0!
Q " (
) = x ′
σ(x) 6= 0
θ(
, σ,
) = x 6= 0!
nP #%O%7[ V!* W 4Z"B0 #0JL#T2[T46KYKY0 BHzmp M_vvGojm2n:m2p N Nv0 $G.
s0 2 K !H .
ω = ω0ω1 . . .
-,s0 → s1 → . . .
i θ(si) = ωi
!Q
ω = ω0 . . . ωn %
s0ω→sn+1
!
021 1 8"!$#%3(nP #%O%7[ V Q W 0HKYL#1I!0 "B46.7"B #.Q & =6
v1, . . . , vm 2 !HT Hcv NoAwv M_pNx Ml(x & 2
v1, . . . , vm . H 8 $. (0, i1 ← v1, . . . , in ← vn, [])
!
nP #%O%7[ d# 1_L 46"B0 !0J"546.'"5 !. Q & =6
v1, . . . , vm !jT!uv w*o/p M_v v M_pNxCM l(xG & 2
v1, . . . , vm 0. @
D NU ; KH Z$ &45 . $. 2 2 !
T D U . (. K =X C ; U !
i A@ 8% X%-, (( ?@j. GC / = < ((# :-, 6 R0 f P1 ! !. ( 8% 4. = $ !H+ z% D Y D 0 8 = 0((# % :S = =9 03C /`-, 0 U z% 21K-, (( !HF $.6# D ,<% U8 1-,/ ;$# K K 6g " * = K-, (U 2S, "0 -, ; (U0 P1 !8? G6I %
16 :n Nojl(H p sux s6H p=r !n,NKoAx;vOs6H _m2n,N p m2nP:=m2p N N1sur : v m_wzo/v2MAH*sutNv%G> !G? !! ) ZAH? +
_-,K-, 8 \K U 2(> 1 2. 4 U .!
F e 8% &C = 8A. _-, /. !8i A@ ]% \GC N9 = 2
x :=) (y)
-,A x
0-, X " y
2
y! D * ; 0 B. 8 XH 2
21 %
9 :
`(.X-, _. a .] 1 -," %4-, f# f#8 4. !
? 6I % !C 0H ./ # f. # 2 Z &0 " K. 8 . F (#1 " 1 !& . `-, < S fK B(. =(U ((# !
! ,V $ 5 (V (
021 130 $ LW 3#-/10J#0 F 3 T" 3 v w*H*m2nNs`w7Nn" x;v w v Mp=xCM l(x w6oAmJw vr=H;Jw1oAmJ2M2J v N Nv rIH1w nNrIH J l(sutNpNx;vr=H w M v w H 3=sum2v l/v HcnNl(Hcv n=r4M Hzs n=rM_pNxCM l(xfp xfvo/pNmBx l(r vwRHzm2n=s`w7x v w HpNl&w_w sAo$p=mBxfvw v l OVp=l(Hzm2vw
g ((# K 6I C !
T ;# 2. 8 U !jb 8 4H 2. Z `-, = .!A Y @-`- !A 8 f# 1 .]- z VE !F !!Q_ ^ !:i !GQ2 ) Q$Q6=B>,+ * .8-, E K 8 0 D D Xg !eTG .2/ 1 > !e? N \ ) ? @H?/+ 2 2S # D . . B fhH 1 X1g 0 2S # 2.8. 2 8
2# !:T
8
9 ) ZAH? +0 = f#
.8-, !
J_v wo/nNl(m7xfp4oAm2v l(tOv T ;# 0 ;X K E NS5H ( 5 ((# 7= !W ;XK7 XBg< K ZS 2#8B 2.8. 2 %. Y
4
H " $. 7 !i Z % 6C -,. !OT ! G C C 0 BC z . 1 2 -,$. -, $ 0 -, ; C .D
t% K-, .$#
1 !Ni 0C / = $. ]% G9 (U 2. -,$80 . 2 7H %$(. 8 K I B 2-1 !
F - < 6 B g< . 2. 2.]( Y( ZS 2#8 . . 2!$W z . # *%<C D. 8 N9 D-,`-,. 2. Z $U G-, !&F A / X /. .8( !,T U . ( ZS 2# 0 . .
O(n) U . B D 0g< !
? V6I % U 2. 0 V Y VVg \ e % = Z'cN'c =E /. % &5-, 2.7-, 2 7 8(U f#8 2$ =1 !
W ;XG ZS 2#8D 2.8. 2 ! Kg e%6 = - 2S&X D 7C %j 8%e 8%/ $#8%e !j h. 2 $E 4.8( .8 U 2.](!
W f K 1 K 6g< /% . 2 6 $U;# 2$ E 8%$ 7 7 $.%Z.. 2S&X A. !/T = 0 2 P1 ((.X4. B * = !Hi 2 % !C = f - U(.H % "-, 1 e f B E( 0 Y 21 K. 8 .0 . 1 %A(. . f 2 $ fA. /. !
W ;X"3 h 3g h 3 8 % ( -, 2S, * = % $ -,D 8 ' 2/ 6 X & AKD AK ( 2S&X 0 B6I 0 ;# D-, ( 0 -G # ! z ` 0( ( +DQeFbZb1 !Ob ,C =@C 1 2S&X <-, ( /. / (U `C ; .0z B ;# B -, - B6 ( 0 -4C P1 !i 2 %
4 $ B z . ;X
C> 2. 1 7 Bg !
9 =
nP #%O%7[ d ) + . 2[[T%O"3!KY0 p=x :,nNm s6H Nv 7 B Bg< /% 2S # 7 . . % B 7 $'8 !eFD Z'cN'c 0E NSEH A@ 2z% 2. 4 ((#!
021 1 %($& ( "($ L & '(W!- ((#5 2. 2. Y "6I8 C 8%
$. 4 .8-, ! Z' ; (DF 6 * ,<% ;XH 3 ((# D = e X 6 4H !F Z' $. % S= 2' 7 NS 56I 6C 4 ((# 51I8 U 2.!D? 8% -, S G NS DC D 8(U !,W U 8 % ;X " J+ Bg 0C 8 D 7 !
F [KYL#.705N,%'Pn0HN L#.7%'"B%'0 !"5[KYL#.705N,%'P`%'KYL#.7%'"B%'0W ;# 4 h C@A.86 2 84C DU
\ ;-, !b U f $ : !,g 46I U .!& =6 %$-, 0 7
n 8
O(n2) C
-, 66I O(n log(n))
!? j9 1 "C *%,9 fU ; 8 $ 8% :9 = 6 U 2. C@A. !
T 8 f# _-,-, % = -, U ."-, h. 2.5(. pNx :,nNm s6H Nv% ;$ KH XC !j? h &$# D /.% 8 U ; 8 $ -, N ANv4C !
nP #%O%7[ d E F [KYL#.705N,%'Pn !#0 [#"$&%'( Q
fC !&T M_n NojxfvOs6H'JBv r Hcv NoAwD
f f U . H $
-, f!
T *M_n,NKoAx;vOs6H'J1v r vw*o$p Mv f
; U 2. 4H $ 7-, f!
T -, U 2. Xg E 2S # K 2.8. 2 8 (';-, " C !
: U .H C !]b ; - U . 4C
O(n log(n))0 U8
9 * rIH*m2n NlIM Hzs n=r Xx pNx :,nNm s6H N1s l/v
1g!HF 2 <%HF !jT 7J !AJ )yFT<J ABC,+61 !eW f D = C 8% U . & * ! 7-, U .H $ ; 0 2' =' U 2.7H 4C =1 # .$. 2# C> BC !
b : 2. 7 B ,!+ 8 8 2. 4 U 2. / :-, GC % . G U 2. 4C f. G /D /.!&+ *%( % 0- U .
O(n2)!
= 7- U 2.B1C E-, O(n log(n))
!
nP #%O%7[ dV F [KYL#.705N,%'Pn%'KML .'%7"B%'0 Q
A / BC
f!eT 7Mn NojxfvPOs6H'JvPO ojxysQM s6H v1
A U 2. -
(. 2. .] 6 !$T *Mn NojxfvPOs6H'Js6NojxysQM s6H v A
U 2.7 f!
+ 5 ]% U 2. ; H $ U 2. C f. /!<T! `(.5- f# V$. 8 ; !i 2' "-, 1 8K-, . D f D. B =6 8 A>
DD
Z' % ; f ]%<-, s6NojxysQMs6Hcv N6v rIH4. Z
$ E.2. =0 ?@j. 216D
T 2. @.` !Oi A@ 8% `-, :-, Z = $
; f D6. %/ %/ / h -, R !$i 2.8-, %$
9 = 6 =6 8% 8 I 2. GH . R $
-,C 6 V(. 8 8% 8% % !!!X-, 5 U .4$ =6 2.!
i / : f ZC X =6 -G ]%N. $. /% 1 = $ % 6I85C %: V U . ; !
9 ?
$. " =9 . " h f# . ; ,/ % $ RC 2 ( A> & ( A> _ 7 ; &$ !
&0H&%'(!#46.7%OP 0HN 0/!1&%'(#!46.7%O&P (UX ((#G ! 4 2. !.]-, $. ;-,%= Z'cN'c -,
G6I GC !$D Z' A( &D$ 1 8(DKF G ((# Z' H . 6I $ %e(E %$ $ ; [D
Q z B ((# 6H . -, !C % 2' ; .8-, 0h ((#-, = <C D1L <]% D (UY ((# 0 6I 4C 0 61 ! z Y (U Z#]- A> 4-, * !
T Y-, B(. UI ; =6 ]% ;. 2e BH ((# 2 7 4C f. = 4 2.!eb 8 ;. / 7H / 1 8%f ;# 8% 8 . R
$ !
F <%U 2I88 ; % X z;surIHcv rIH*s nNr/rjpNxys6H'J ; . 2. K M_n N oAx JAH*l vsurIHcvr=Hzs n=r/rAv xuxfv6( ((#!HT ;# 4 ((# 8 D 4C !(b 2 K . R
$ C !
? " X &$# 2. . z% z . X .!W ; H% ((# 1 2S NU `-, 2f.8 0 2. ; % _ 2 % !!!P1 2. 6 =6 0 U % (U U . G $. G fSLK /1 !( :-, !. ; C (E((# % = H.f $. $ -, = ;6. _ !
> 1 ]%$T !eF 2. )yF AB@/+<-, 4((# 7 2. f 2 K C
min0 -, ! f$ G 8(U 21j 2 H . * $
-, 4C O(min(x, y))
!T! .1 N9 . 1 z 7-,e 1 0 0
C ((#0 !(FD $.8 `-, #<9 A . 2 =V C K ((#B *%$ = (C 2 `- f#8 & !HF B- . ( & $C `-,% = G / DH $ =6 2. C !
W 4 C % C / 4 7H ((# 4-, 8% -,`-, % f' 7 !FD BE -, $ f / D ;# .%H ;X 4 ((# B !@Q ( /# 6) !M0 % 8 f' 2+ DQ
0 +DQ 1 ) M0 A5A-+-,*% % f f B-,
/ 2 = H = !
9 @
f
F $. ; K U 2.% 2' =' X 0 1C 0-, I ;. 7 4 8 $.%<-, 7C =6# 4 5C !@F 2./(. / (E ((#7 *% B-, ((# . 2. = 2. .] !?K. G U . ZG !C ; .$. <-, 2K f#8 1(. 2K 8X!/i A@ 8%e f#8BX(.5 8X 8 N9 I = $.7 C 2. `-!
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" -$ &V $ '&V $ * 5
&(*I! !$ / 0 -,<-, 4((# B21 ('c(. K /
C 0 fUV 4 ;# ;. ; 8 f .!F Bg 6 4C Y E-, 0 8X
H . 0 1% ZS 2#8 . . %$6I ,/ 8 8%$ E = ;. 0-, = < 2 ; 8 !
nP #%O%7[ d(d + . 2[[T%O"3!KY0/1 !( !PH&05TKY%7!%71_&0/1 Np M &surjv v (l(m sur:6rAn=r 3 J Hcvm N1sur/s`w Hcv 5- 2
(Σ, Q, q0, qf , F )
Σ, Q, q0, qf f
=$UE K$. 2 F
0C Q× Σ × Σ → Qn ×∆ ×∆ × Σ
0∆ = ←, ↓,→
1 ! D - *% A. B * V 8(U NS. 7C V 0 7 f#;. %&.8-, ?@j. fU ; P1 !
w Ow H Nv vmJ2J2M m s6Hzl(m2vBrAn=r 3 J Hcvm N1sur/s`w Hcv ZS 2#87 . . (' ,/ ! Np M &surjv Mn No H v l(m_wrAnNr 3 J H v m N1sur/s`w H v : :-,( H
= ('c(. 1 ou
2- * B 8(U5 4 ;# ;. !
T U . 5 / (' $. 2 . V U . NU ; 7 U&. ; * $ !
T 1C f. $ $' (.2 B -, . =8N6p Os6N1l(N .2 z $ (' $. 2 !
T fUaH 2 C@j. 2. Z ; $. 9 Z6 C Y ;# X(.Y 8X 2. ;-, !(+ z% C@A. .2f' / ; -, 8
0
1!H? ; 4 <
; .2 .2.4. / ;. 6 ? !AQ 8) Q @B@/+ V> !e? F !,W 5) ?DW c5;,+ !&T (. 8 0 0 K)!$M0 i !$M ) M7M A5:/+ !&i
9 A
3C $. ;-, B f#8 D1(. 0 $U 2. Z F0 W T !0 46.'1&0(1 ; _ /. W T #0
> 46.710 !T! _ 2. Z K NU ;X$ <B(. 6I <I : 2. -, / ('c(. !i `-, f< _ =
D6I U 2.! `(. #$U `(.K-, /E U `-!
N,0HKYL#.70 a 9 !KQ& -,* V 7H 4 4 7g< =0 * . R0 5-1
Z (U ;# 2/ 4 8 : * 7. 2 8 $ !< C 8 0 V$. 2 4-, j . Z 8 2$ 8 !(i @ 8% 2(> KH . 4X 7- e-2 7 $U;# $ /K 1H 1( X. Z 2$ !:Q ]% ; .2 46 !N !AQ %S 5 - BY .2 W T !0 !T Z. KX D-,/-,
2. 2. 2 !$T .
q0, q1
q2. 2 1 (U ;# $Y f /E- f. Z !/W /%q0
B 7 f# h(. Xq1
q2
-,<. 0
Y1
Z 7 8(U f#8 $ !(i 2 % 7z .
q3- 2I 5$. $ 0
-,6 !W! * . qf
/ * . G 5 X V(.; G Y ! Σ = 0,1,t Q = q0, q1, q2, q3, qf F
K /. =0x, y
$. f 7-, %6%(. 0
11
| T 1
T 2
| 1
2
|
q0 t y q3 ← ← yq0
%y q1, q2 ↓ ↓ y
q1 x y q0 → → 0q2 x y q0 → → 1q3 t y qf → → yq3
%y q3 ← ← y
;!T 2S # D 2.8. 2 2 2 0(.; ` .DH U 1 ;.!,b / .C 2 ;. 2. 0 . . ! 21 ! ' ] JZ ]% U X =@-, ! e z U % 4 W T #0 K XH 2. f'% 46.'1&0 D-, /-,!,T ZS 2# * U X -C X-< = % ; 6 f# (. K ;.!AQ z U = % z h .2 ( W T #0 !HQ<% 2. Z 46.710 !T RS , , " (α, β), " !0T 2 8
C = 0 : ,S : → , W #0 : , 46.'1&0 : , 4R : → , J[ : → , : × → ,⊥ : , ( 46%' : α× β → " (α, β)
C (.
:
"(
, " )→ " ( , " )
(( 46%'
(W #0
,.))→
( 46%O( 46.710 , . )
( 46.'1&0 , . )→ ( 46%O
(W T #0
,.)
: × " ( , " )→ " ( , " )
(W T #0
,( 4Z%O
(y,.))→
( 4Z%O(W T !0
,.)
(x, y)→ y
;$c
: × →
0 = 0→W T !0
0 = S(y)→ 4Z.'1&0S(x) = 0→ 4Z.'1&0
S(x) = S(y)→ x = y
" : × ×
" ( W #0 , ϕ, ϕ′)→ ϕ" ( 46.710 , ϕ, ϕ′)→ ϕ′
" : × " →
" (x,
J%'.)→ ⊥
" (x,F (!1
(( 4Z%O
(y, ϕ),.))→
" (x = y, ϕ,"
(x,.))
: " ( , " )× → " ( , " )
(( 46%'
(x,.), ϕ)→
(x,
(ϕ,.))
: × × " → " ( , " )
(⊥, x,
.)→
( 46%O(W T !0
,F [#1
(( 46%'
(x,W T #0
),.))
(⊥, x,
.)→
( 46%O( 46.'1&0 , F (!1 ( ( 4Z%O (x, 46.'1&0 ), . ))
(ϕ, x,
.)→
( 46%O(ϕ,.)
: × " → " ( , " )
(W T #0
,.)→
( 46%O(W T #0
,.)
( 46.'1&0 , . )→ ( 46%O
( 46.710 , . )
(JZ
(ϕ),.)→
(
(ϕ,.))
(]
(ϕ, ϕ′),.)→
(
(ϕ,.), ϕ′)
( 4R (x),
.)→
(
" (x,.), x,
.)
T 3(.2 2 0(3C 0-, # K 2.8. 2 K
!/i 80I `-/.=$U56I .8U X .2 .0 4KC P1 ?@j. !i W T !0
> 4Z.'1&0 ϕ > ⊥
U ϕ% 2S # ('
(. 7- 0 5 = e z U = f !>!T E 8 Z * $ a V (U !i C 8% -, U(.
7 .2 (C . Y. < 6 . 0 Y 8 1 !AF 1 - ; /.2 8%4-, 6 @ 8 & 0 *% " ('c(. 4C * $ Y 7 8(U X !i .
x, y!
Q y!
0 : 1 ou end1 :
x
2 :"
y3 : 0 ou end
end
; 9
*L I nP #%O%7[ d 8 W 0HKYL#1 L (.O^,[KY%746. VC U 2. 8 1 $. fSLK % = Z'cN'c
O(na)!
YC V , " $ (' $. 2 h U . $. ;S IK !
W $ % $ = I 7 7 U .5C 5 U 2.vPO ojxysQM s6H v!
8 G-, 7(. : U .K(. / U 2.D !C 8% U .D $ %( z%,( " ((# !$ j$. U 2. H((# 4 fU" / !
nP #%O%7[ d% 1_L 46"B0 .'!2[4RT%'3#KY% #0 C 3 6879" U . 0 2h a ag 1
O(log(n))!
KC ,6 79$" ! /(' $. 2 0 U .
O(log(n))!
F 0 8(U K U . 8 `-, U 4-, ((# 6 D Bg !(i A@ ]%6(. Y = ((# ; 4 U .4; 4
n!
nP #%O%7[ d a 1_L 46"B03L [.'^,(KY%746. 6C X ! U . B $. 7 ;SLK % 2' ='
O(na)!
"C , " = 6 $ (' $. 2 5 U .
O(na)!
nP #%O%7[ d* N L [#0/%70H.7.'0/1 C - k
U . $. B 1H U 8
k% 2' ='
O(exp(exp(. . . (exp(n)) . . .)))
k U ;X ;-,$.!
:C $, - k = / ('c(. ! U .
$.H U = k!
BC 0 - k$" ! U . 0 $. D DH U
8k!
T ('c(. U 2. U K$. . =$U (. e !
4 3#%'PHT4RT" 3!%70 1L 46"B0 W 0/KML 1 6 79 !
⊆, 6879 $" !
⊆
⊆,
⊆ !
⊆, "!
W 3#P/ZT-/KY0`!0 ,4 .,%O&""3 4. a "!
=, "
!%'0J#03.'4 3!%7PH46" 3#%'0 1L#46"50 W 0/KML#1
(N) !
⊆ - 1
⊆, - 1
⊆ - 1
"!⊆ . . . ⊆ - k
⊆, - k
⊆ - k
"!. . .
6879"!6=#$" !
6= - 1$" !
6= . . . 6= - k"!
. . .
FJANnNr/w2H*m2p Hzs n=r g $ $. 2 X. @ $ (' $. 2 0 / 4 fU ;/1 % -, U 2.a(. , U 2. $'(. / !
;B;
$ 8 ;0H X-, 0H U(. 0z 2# P1 !F B-, * ; 5 8 e !
?D E
$.% U 3 NU ;X$kE D C@A. ;!HQ @ '
$ V z " $. E
V $ ; 8 -,kE % BC .8
2. =Y 2C K h $. V h !!<F -,
* ; 3 0 7 (. B U Z U .!
T! . ."!=, " U 2. (. 0(
1 !/ !eQ ) Q Z?Tc-+ !W ;XG $ ('c(. $ (.2 %$C 8%, -, fU
;% E6. U 2 8 ; 2. !jF 2' /Y `- =6 E 1( Z_ _
!(T! $. _ z $'
C Y =h _ _ ;S B 1 . !ei @ 8%A1I8 X 1 C@A. ; U *% * $ V$. 2 VC -, C !eb " B 3 _0- * (C *% @-< 8 H (C 8 ;# !$? X6I % * $ (' $. 2 3 X U .2 z $ X$. 2 D 31.8 f e" 2. Z $.R9Z &!
T U !H 2 7 8 ;7 6C 0 ;# /1 !HW U 8 % C 5-, /" 2.0
n K 4
2n - 1 !
! " !=b !C 2 Z ((#KH = C = $. .!
D z 8 % = ; !Q * ; 6 79 !
⊆, 6879" f#8
L = NL? Q z ; ⊆, 2 2# .f#8 ;#
P = NP ? F
;# 9 7K #; z (C `-,$. ;-,!(i A@ `(# ./. -, f#8 f/ ;S IK RC
-, / "
5I 2. f !+ "-, ;# , 8% * % ;$ -, U *% Y-, 8 EI # 0
2100 A. 110GHz
010
_H A. / P1 f H E _V $. A@ 221 !W 3#P/ZT-/KY0 V x vOM_v o Hzs n=r-v6879" vAH , 6879 $" !:xfv w M xfpNww vwv M_n,NKoAxfvPOs6H;JojmJ w v rIH'J_v w4w nNr=HxfvwRN Nv wo/nNl(mxfvw N6p M(surAvw/v$l(m sur : v H0x;v ww Ow H Nv w v"mPJPJ2M m s6Hzl(mv vw H 3Nsum2vVw sDn=r r pNl(HcnNm s`w v l/v"x;v wVp=x :n=m s6H (N6vw l(r-vw v l O Nn" x;v v M_p=xCMl(xywnNr n AH*s v rIHxfv wRN ANvwRM xfpOw_w vwvFMn NojxfvPOs6H'Jwcv r7Hcvm Nv v nNr4M H*s nNr$wFJANnNr/w2H*m2p Hzs n=r F (U5((# f X ;SLK !
Kv NpNm l/v6879 $" ! = 2.Z. 2S # B 2.8. 2 2. C (U -, 2 8 # 0) Zc5c/+61 !
,V@ B V _BY ,V@ B , T C ; . . . (( g!AQ& 8
1923) Q&/;$>/+ !/T ; (
H B(. 1. .4C J !$W. 0 7 1966
) W!. =5=-+ !nP #%O%7[ d Q & P/"5 T1&%'(\L#T%'KY%'% .[0 2S # 7 2.8. 2 oAm s6N1s6Hzs 7mPJPM l(m_w s 2# 1C 2
(p1, . . . , pn)→ t
t -,7 ZS&X 4C (. ..] 6 !
(0, p1, . . . , pn)→&(p1, . . . , pn)
& 4C Y-, 6.2.7(. ! (S(x), p1, . . . , pn)→
)(x, (x, p1, . . . , pn), p1, . . . , pn)
) 5C E(. 2. = ! :C J2M l(m_w_sutNv<oAm s6N1s6H*sutOv < ZS 2#8 . . ; C. fC ! % 7 U 2.7-, C ; 2. !
;$>
W 3#P/ZT-/KY0 d l/vx l/vBwn=s6H
k - k
⊂%
FJANnNr/w2H*m2p Hzs n=r T! `(. VY V KH 3 h hg NS H ! . Z 42S 2#8 0 . . f fC . C2.20C 7-, (t, ω, ω′, ν, ν′)
2. Z ( 6 2 $ /;-,$. =E K
t !
b #@C (t+1, . . .) : (t, . . .) 0 DC )B Z. !(. $.8
. C ; C ;X 61 !(T A$ 2S 2#8 D 2.8. 2 /. 8 B *
t!
W @-, * ( 2S # ; C. fCR% Z(> < 8 * ! K ! $. ! f# . ; !(F = ;# 0 2 2. 7 4C z / %1X <%z U %$ H U f ; P1 !
T @ * &$ @C f 2. % 26 :- C =0. ; . !e !+ _ ) + _6;$@/+ . Uf 4C E-, ; 2. K-, ;. 2S # B 2.8. 2 Z !
(0, n) → S(n)
(S(m),0) →
(m,S(0))
(S(m),S(n)) →
(m,
(S(m), n))
T 6(. Z 8 2.B6^ !/i !/J ) J @$C-+ !nP #%O%7[ 8 l*+ #.O&%OL .'0/12TP/"B !T1% .[0/1 2S # 7 2.8. 2
kcmJ2Ml(m_w s K(. 2# C
φ(x, y1, . . . , yk)→ 0 * E
yi & !
φ(x, y1 + 1, . . . , yk + 1)→&(x, y1, . . . , yk, φ1, . . . , φk)
φi = φ(x, y1 + 1, . . . , yi−1 + 1, yi, i1(x, y1, . . . , yk, φ(x, y1 + 1, . . . , yk−1 + 1, yk)), . . . , i
k−i(x, y1, . . . , yk, φ(x, y1 + 1, . . . , yk−1 + 1, yk)) & ij
. 2.7(. . .8 81 ! C
kcmJ2Ml(m_w sutNv ZS 2# . . 2
k' 2. fC !N C
BN1l(x Hzs ojxfvXmJ2Ml(m_w sutNv k' 2. F0z-`-
k1 !
T & '$)+% k
C k' . !$T &('*)+%
C X . !
W 3#P/ZT-/KY038 %
=& '$)+%
1 & '$)+% k ∈
& '$)+% k+1
$ , $ _
nP #%O%7[ 8,) T % XC mJ2M l(m_w_sutNvw 7C ! = Z'cN'c C 2-, U 4 $ 7-, !
W 3#P/ZT-/KY0`!0 .'4 3#46.'0 + W !T%'421936
W ! V% x<vPOs`w H vvw2n=r4M Hzs n=r$w l(s:rjv0w nNr=HHo/pOw p=r$w % kro$p=m HzsQM l(xys v m Gxfp n=r4M Hzs n=r $) l(s/ojm2v rv r v rIH*mJ_vl(r p=x :,nNm s6H Nv l/v x l/vXw nNs6Hxfv Nn" xfvv M_pNxCM l(x w M $n=s`w_s YvAH =s6Hw_sM_vAHpNx :n=m s6H (N6v6tNp H v m N1surAv mnNl _n=lIMxfv msur Gr/s6Nvr=H :r vw2Ho$pNw M_pNxCM l(xfp xfv
8 (.; < =6 I 5 ; !
FJANnNr/w2H*m2p Hzs n=r Q& -, U 4 $ A
-, C $) !F 2 * $
B 6 ;# 2
;TC
B/ .4 /
a!
B;X
AZ /!
B
a !
Qa
%B
5 !eQ<%B
D .2 -,/-,!D
BZ ff' 1I8 0
a = B16D
QB
%A
B(. % B
K ]%/ 2 K1 Z 2.!
QB
2 8%A
(. % 7 V (U ;# B
% - = = 1 = Z_ 2.!
? z /B
U !HT ,S $#1-, V. .XC 0 2 * U 7
A!
Y(.8$ K4-,A
U ]%( C ) 0/1 L 461 "B4Z.'"B #.'4 1#.70 !F 2. Z U 2I88 6$. ( &E * ;S 5 U 2.!:i @ 8%
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f(x) ∈ Io%A(x)
2 K 0 .2 ! W .
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; !
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T . !.,/1. ("C -,
f 8f' % U K $
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Io!
( TZL [1&%O&%'( % $)
∈ RE FJANnNr/w2H*m2p Hzs n=r FD V Z.]- f66.8 (. C ) !Gb Z(> E
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W 3#P/ZT-/KY0 a + W T%7 21936
W V!% x@vOs`w H vvw nNr MAH*s nNr/w l(s:rAvBw nNr=HHo/pOw4w vAN1s M_pNxCM l(xfp xfvw
b U 7 C f' =!
FJANnNr/w2H*m2p Hzs n=r T! U Y 5C f' =8 8\"C 8 6 $ =' .!!i @ 8% z E6C (' $.X = 1-,* h / (. !
!C 8f' -& ,C /. % 2' =' 2 RHcnNl(Hcv w 2.!
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;B:
* & 7(5 B 5 $ _J Z $.2#8 1 6J 2 2.8-, z ;S 6!HF 6
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min%A-, /
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(x, y))
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;$=
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XF !/Q !$T
!) T<,L+ c 9 + . 2.f _ ! # 2. B ;-, X-, 2
" B U .
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T ;7 D 2. D * .2/ 2g CJ 8 B> !/? N 2\F) ? @H?/+ !
" (V@ (V : $ X
$&*I'($nP #%O%7[ 8 d ]T!T0H1 #0`0HTKY%7!46%71(Q
F , C 8(U 8X 2S&X
≺ Z *
T (C,F) !
≺
nNm Nm2v v Hcvm N1surAp=s`wn=r 2#8 8(U ;. . 2 ≺ s v r n=r J% = Z'cN'c 4-< U 2 B(. 4 ! ≺ &NnNrjn,HcnNrAv % 2' =' - / (U2 C@A#- ! Z' % ;
;A- ; Z' B0 U K 8(U t
sG-,
t ≺ s (. . . , t, . . .) ≺
(. . . , s, . . .) 1 !W 3#P/ZT-/KY0 * / 46 ZT
1977 46 a[a
n=s6H"NpNsurhl(rEw Ow HANv v7mJ2J2Mm s6H*l(m2v N6p=sur7H v m N1surAv0w s!vAHGw vl(xfv Nvr=H:w s@sux@vOs`w H v7l(r3nNm =m2v vBHcvm N1surAp=s`w nNr≺Hcvx l/vo/nNl(mRHcnNl(Hcv4w l w Hzs6H*l(Hzs n=r
σv HHo/nNl(mRHcnNl(Hcv7m ;:xfv
l→ rrσ ≺ lσ r Ns6HpNx;nNm_w l/vRNpNsur
≺ F $.# C ; \B ! Q !&> 5) h> ?Tc-+ VQ !$P
! 'c!$T<.&S7) P7T@5c/+ !FJANnNr/w2H*m2p Hzs n=r D -, 2.]( %,V Z' 0 2.] UI1 . Z' ; 0 * YH 2# 4C D(. ,G z _ /1 !/W B * % (. GD -, .8( !/F 1 z _ VC /$.% S Z 7 2.]( 8%( 2S 2#8 N9 !
nP #%O%7[ 8(8 ( TP/"BP/!0/#"B0 Q
F , C (U X 2S, X VC 6 ! ojmJ2MPJ vr Mv
Σ 1
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2F
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c ∈ C ∈ F
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;5?
nP #%O%7[ 8 % ]T!T0H1 1& ! .'0/1]""3#0/KY%'#1Q
F2. .8
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•∗
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j-,t = (t1, . . . , tn) J
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* K / 2 b U
j-
t J sj!
, & ∈ C, m = n U X X
π -,
j%tj J sπ(j)
tj = sπ(j)
U
i <-,
ti J sπ(i)!
≺F&
ti J s
i!
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(t1, . . . , tn) J∗ (s1, . . . , sm)
j%tj J s
!
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81304130 %& L# ! *I nP #%O%7[ 8 a ]T!T03.'0HN,%'"52[T4RL 3!% !0 Q
≺ Z 8X
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(a1, . . . , an) ≺l (b1, . . . , bm) * 4 / Z
a1 ≺ b1!
a1 = b1
(a2, . . . , an) ≺l (b2, . . . , bm)!
0/KYK 0nQ n=s vr=H
≺l(r nNm =m2v s v r n=r Jv H
nl(r vr=Hzs v m l(m4x v r$w v N xfvv w7xys`w Hcv wvXx;nNr :l/v l(m7sur"AJ m s vl(m2v6n=l J :,pNx;v
n!xfvoAm2n=xfnNr:v Nvr=HxfvPOsQM_n:m2po &s l/vv
≺v w H s v r nNr J
Q = < 2 8% 8 <C 8 : Z (. 6 U8 0 2 _ $. 3 ;# YZ -1
(1) (0, 1) (0, 0, 1) . . . (0, 0, . . . , 0, 1) . . .
FJANnNr/w2H*m2p Hzs n=r W . 2n!AQ
n = 1%≺l=≺
('zC /(.!AQ<%≺
. ('*C /(. 6 (. 2 8 .;. $8% &S $#B . %&X (. Z
n− 1 .f.8 !
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t = (t1, . . . , tn) ≺lpo&(s1, . . . , sm) = s0 , & ∈ F ⋃
C1 * 4 / 2
b U j
-t lpo sj
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U B X π
-, j%tj lpo sπ(j)
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& ti ≺lpo s
i!
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j%tj ≺lpo s
!
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(0, n) → S(n)
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(m,S(0))
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(m,
(S(m), n))
;$@
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TW≡&('*)+% !
813041 'L ! &*= nP #%O%7[ 8[Q ]T!T03Kh #.O&%'1&0HQ ≺
2 X E
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E
<-,A = (a1, . . . , an) m (b1, . . . , bm) = B
z B / 2 A = B
A
` ! b U
aj ≡ bi
A− aj m B − bi!
b U 1 ≤ i ≤ m
1 ≤ j1 < j2 < . . . < jk ≤ n
G-,bi aj1 , . . . , ajk
A−aj1 , . . . , ajk
m
B − bi!
N,0HKYL#.70nQ Q& 2 % Z (3, 4) <m (3, 3, 4, 0)
(3) <m (3, 3, 0)
0z (U ;# 2# /1 () <m (3, 0)
0z (U ;# 2# /1 f KX ; !
(3, 2, 2, 2, 1, 1, 1, 4, 0) <m (3, 3, 4, 0)
(3, 2, 2, 2, 1, 1, 1, 0) <m (3, 3, 0)0z $U;# 2# /1
(3) <m
(3, 0)0 f#8 # -1
() <m (0)0 X ` /1 !
(3, 3, 3, 3, 2, 2) <m (3, 3, 4, 0)!
XC d * X 2 / 4 (U X *
≺0 X(. 1 Z 90 U ;-,B z _
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N,0HKYL#.70 ) T 2. .8 (3, 4) <m (3, 3, 4, 0)
(1, 1, 0, 0, 0) <l (1, 2, 0, 0, 1)
! (3, 2, 2, 2, 1, 1, 1, 4, 0) <m (3, 3, 4, 0)
(1, 1, 3, 3, 1) <l (1, 2, 0, 0, 1)
! (3, 3, 3, 3, 2, 2) <m (3, 3, 4, 0)
(0, 4, 2, 0, 0) <l (1, 2, 0, 0, 1)
!
0/KYK 0 ) E s≺
vw HGl(r n=m Nm2v s vr nNr J !w nNrojm2nNx;nNr :,v N6v rIH Nl(x?Hzs`w v H≺m vw H8 s v r nNr J
FD f.2.-, 2 B^ K * &S( K-,A- 2. 2.!i A@ 8%* &S( C I ; 2# $ G-, D-, ^ A. z _ KX Z 7 I = (. G -, !
nP #%O%7[ % l* ( T! _ X 2 F0 %El(x?Hzs`w v H p,H m vm sur:(1 * 2 f.= 7X ! = Z'cN'c -,
t = (t1, . . . , tn) ≺mpo&(s1, . . . , sm) = s
0 , & ∈ F ⋃
C1
* B / Z b U
j-
t mpo sj!
, & ∈ C, m = n U 7 X
π B-,
j%tj mpo sπ(j)
U i
-ti ≺mpo sπ(i)
! ≺F
& ti ≺mpo s
i!
≈F& % , & ∈ F
(t1, . . . , tn) ≺mmpo (s1, . . . , sm)
!
Kv NpNm l/v? 8% YW (. 6W
z _ 6$. ;.JW
0 J DW _ 1 !(F 1 * U `-, K> !$? N \ ! ' W !$ /7) ?05A5c-+*%eJW $. _ 4-,6. _ U ;-, X .0 ZS&X 7 0C =1 !W 3#P/ZT-/KY0 ) V n0H1"3#]%'
1982 n05 *E
v w Hl(r nNm =m2v v.H v m N1surApNs`w n=r ;$A
W 3#P/ZT-/KY0 ) d + F %'" 3!(1990
#0H J 1 46 !0H 1992
F %'"HQ J Q E $pl/v nNr MAH*s nNr"oAm s6N1s6H*sutOvmJ2M l(m_w_sutNv1v w H M_p=xCMl(x;p xfvKo$p=m7l(rYw Ow H N6v vXmJ2J2Mm s6H*l(m2v l(s Hcv m Nsurjvo/pNm v HM $p l/vXw Ow2H N6vvmJ2J2Mm s6H*l(m2v l(s Hcvm N1surAvo$p=m M_p=xCMl(x;vXl(rjv4n=r4M Hzs n=rojm s6Ns6H*sutOvmJ2M l(m_w sutOv
YW≡% !
b U = ,H = _ < 2 !N 8 YWN-, YW -,@9 . = 2 ]%=JW 0 J W _ 1-, :66. TW K YW 0 ZS&X 2I 6I8 2. .8% Z' 2. * U U `-,% z U ]% YC =1 P7L 0zP,('cL / fUE _ 1 (( ? !/i !/P& W !/L!/L e U) P7L ?Tc,+ !
5 $ B 5 X $
T . ;S KH _ 4 2 !$i @ 8% 6 RC 2 2 2 2 8 Z V (2S 2#8 4 (. 7 .8( .0 * 2 2K. YC e(. -G 7 .8( !/1 !
T 2. ;S . . (( "? !Q !T $,C _ ) T (?$A/+ !GL -,* UZ' 4H 2. ;S F0 2 P1 e(.; 0 2.8-, D$
10 ;# D^ f 21% X8 ;-, H G ! 2. Z !=W ; 8 ZS Z'2#8 G . fS /.2. 8% 6 U 2S 2#8 $ )yFT@/? + % ) M &A5:/+ )7) T + !
? f8%/? !/^ C = 0F !$T = 8) ^KT!@B@-+ .0-, U 2.7 ZS 2#8 2.8.f'2 2 . ;S f K $ U % -, 4 2K K U 2.B ; F0 0 7 ;C = ;-,-1-, X-, _ 2 !eM!jL (C %/+ !HF %e ! '*) ! 4^ !eg $'\ ) LF hg A5@(%LF hg c5c/+ > $. .2 2. .] B-, 5C 6 z . ;S 9 4K #; U .B(ZS 2# !
nP #%O%7[ % )l&05L#TP5&4R%7[ L [.'^,(KY%746.'0 T!;surIHcv m*oAmJ H p Hzs n=rho/nNx NrAn,N1s pNx;vYH ZS&
a ∈ F⋃
C 2.
n X ;SLK "
n =
JaK
-, 9 !
JaK(X1, . . . , Xn) >∑n
i=1 Xi!
;!JaK
7 !>!T! 2. V R0 H .
01 !
T 2. . /( =$U 2 7 ;# `-,
Ja(t1, . . . , tn)K = JaK(Jt1K, . . . , JtnK)
ZS 2#81 2.8. 2 Vp N6v HK . fS -, 2# r → l
-2$ 2
σ%JrσK > JlσK
!
( TZL [1&%O&%'()68 n=s6H NpNsur l(r w Ow H Nv v3mPJPJ2M m s6Hzl(mv l(sXp,Nv H1l(rjvYsur=H v m*oAmJ Hcp,H*s nNr o$n=x NrAn Ns p=xfv n=m Nm2v surNl(s6Hw l(m5x;v wHcvm Nv w
t ≺ s ⇐⇒ JtK < JsKvw2Hl(r nNm Nm2v v.Hcvm N1surAp=s`wn=r FJANnNr/w2H*m2p Hzs n=r T! _ KEC /(.
JtK N9 fC z _ Z
N D EC /$.!eT B$.
D . !
F ZT(.'.746%OT0 ) % qrhw Ow HANv vXmJ2JPM m s6H*l(m2v l(sGp N6v Hl(rjv7sur=H v m*oAmJ Hcp,H*s nNro/nNx NrAn,N1s pNx;vFH v m Nsurjv
>Bc
nP #%O%7[ %E n0/T03#0/1IL [.'^,(KY0/120H #0H1]1^,1_&-/KY0/1 ;SLK
P (X1, . . . , Xn)
2(0H!T0 P (X1, . . . , Xn) =
∑ni=1 Xi + b
%b
2(0H!T0 ) P
fSLK 44 .1 E =$
2(0H!T0 E P
fSLK 4-,/-,! ZS 2#86 2.8. 2 ,e 0 ! (' ,/ 21 ( 4 . fS
K B:,vrem2vΠ(i)
0 !∆(i)
1 z . E K i!
W 3#P/ZT-/KY0 ) a 0I( 46&0 + F %'" 3#[ * 4RT%'( W ( 5051998
0 F * W Q * v w7w Ow H Nv w v1mPJPJ2M m s6Hzl(mvp ,NvAH H p=rIHl(rAvXsurIHcvm*ojmJ H p Hzs n=ro/nNx Nrjn,N1s pNxfv Mp=m2p M H'Jm s`w v rIH Ns J m2v rIHcv w M xfpNww vw vM_n,NKoAxfvPOs6H;JBw vx;nNr xfvl(m&:,vrem2v
v r/m2v NlEw Ow H Nv xfpOw_w v Mp=m2p M H'Jm s`w J_vΠ(0)
Π(1) - 1
Π(2) - 2
v r/mv=lVw Ow HANv xfpOw_w v M_pNm2p MAH'Jm s`w J_v∆(0)
, ∆(1)
, - 1
∆(2), - 2
> 9
f
`(..B0 U Q !(F ) F 5A 9 +61 1 . :C A@ 8 30 =/1 : _ 8 ! hI *%6 ; 8 8% =9 U 30 1 2N9 ) M0 A5>-+61% . .. / ;.K 2# f 2. $ ;-, & 9 !T 8;# H (K7+ !,F = ) F Z=5;/+ ! KC U
! U 2.
C f.!=T! . 2e > !=? ! Z =6 <C . ) Zc 9 +D .-, 1 @ *
0 . -, _ ZA. 8
. - k !
= ;$. # 2. D 2. $. 0 Z 1 ;S 2. '
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" $ V (5 $ B
T! ;S .8 H 2S # . . 2 2. $ D DC 2f' 0 (U =(U0
1% <F0 2
S =C
1 !/? 1 ;# # (C % $
2 =$U I 0 2. PKG . D $ < !,T 8 $'NS. 0C E(. 2. 0 N9 I 4 2. < K K . !/ = 7C d 3 0 `(. 0-, $ 0 * . %A-< <C 7 * . KC 2. 0 =h D $ OS 4 * . 7 (. G -1 K-,z . 66. . !
i N9 * fS .8 _ <% ;. . - . .7. 2e f. ! 'z)! K+ !AF ) cBc&%eF 3cBc,+ !
1304130 & I"( &
nP #%O%7[ %(V ]T!T03!03L 0HK !4R&%'(!1Q
≺ 2K X
E!HQVojm2nNxfn=r :,vANv rIH@o/pNmo/vm N1l(Hcp,H*s nNr$w
≺m′ D_ Z H .;. V
E 36I8 4-,
A = (a1, . . . , an) m′
(b1, . . . , bn) = B U 3
X π
B- W
i%ai bπ(i)
! b U
j-
aj ≺ bπ(j)
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Bs6NKoAxys l/vA ≺m B
F ZT(.'.746%OT0 )RQ s≺
vw HGl(r n=m Nm2v s vr nNr J !w nNrojm2nNx;nNr :,v N6v rIH<o$p=mo$v m N1l(H p,H*s nNr/w≺m′ v w H8s v r n=r J
>B>
nP #%O%7[ %[d l* ( T! _ * 2 8 f. X ! = Z'cN'c B-,
t = (t1, . . . , tn) ≺mpo′
&(s1, . . . , sn) = s
* B / Z b U
j-
t mpo′ sj!
≺F&
ti ≺mpo′ s
i!
≈F&
(t1, . . . , tn) ≺m′
mpo′ (s1, . . . , sn)!
0/KYK 0 E *+4RT%'(f0H *+/^(0H2000
*+* /vw2Hl(rAvBm2v w H*m sQMAH*s nNr-v F ZT(.'.746%OT0 E ) /vw2Hl(r nNm Nm2v v.H v m N1surApNs`w n=r W 3#P/ZT-/KY0 E[E * 4RT%'( 0H *+/^[0/
2000 *+*
$pl/v nNr4M H*s nNr6ojm s6Ns6H*sutOvXmJ2M l(m_w_sutNvBvw H M_pNxCM l(xfp xfvo$p=mDl(rEw Ow HANv!v7mJ2JPM m s6H*l(m2v l(s Hcvm N1surAvo$p=m (v HM $p l/vXw Ow2H N6vvmJ2J2Mm s6H*l(m2v l(s Hcvm N1surAvo$p=m =Mp=xCM l(xfvBl(rjv4n=r4M Hzs n=rojm s6Ns6H*sutOvmJ2M l(m_w sutOv FJANnNr/w2H*m2p Hzs n=r YW. G 1 YW% ZS 2# K 2.8. 2 -, YW2 = YW 6 XXC ; . !jJ. (-, ]% 0C 1 0-, # 2. $. f fC . fC YW!
13041 'L ! &*= T hW fS 2.] 0 hW !
nP #%O%7[ % 8 46.'0/#"B0 T 5tNpNxfv r4M_vH ZS&X 7 0C H 2.n
4C ν( ) : 1, · · · , n 7→ 0, 1
!
T -,`-,1 .$. 3 2 4 Q !HP = B ! 'c !jT &S ) P7T!@Bc-+ / `- 6 I . / ZS&X 4C !
T $ ; =(U =(U BQ !(L KQ !/F !( $
1 6rjnNm NpNxy% 8 BI Z. PK4. ! $ 7
0
w(m % GI D 2. K ! 2. % 2' =' - !H . j N9 KS =3 $ D
1-,(. ,G!eT .2 4 KC 0I
2.B-, < 0z 01 !
FD K-, 0I ;;.B z . &X $ !(W C . % /C = 1 ! * . B0uC K(. ,G $ G
11 C
' % * . 1. 2. 66. .2 K 0!
nP #%O%7[ %!% Q
ν"C 2
F!: 2
πm2v w*o/v2MAHcvExfvwVtNpNxfv r4M_vw5 & 1 (U
2S, 0C 16I 2.n
K- i%ν(&, i) = ν( , π(i))
!
T 8 2 D ( /. ;# X e !Ab C = - z .;. -,<(. E.;. K
1!
nP #%O%7[ %a ]T!T0JL#46 L 0HTKh !46%'(#1 0H .646.'0/#"B0 Q
≺0≺1
(U 2 !F _ . ≺ν 2 X
2 - 44C 5 ν
(. ZF
!
(s1, . . . , sn) ≺ν, (t1, . . . , tn)
8 U 7 X π
-, & K-, . W
i%si ν(
,i) tπ(i)
! b U
j-
ν(&, j) = 1
sj ≺1 tπ(j)
!
T! _ (. 5( X !
>$C
nP #%O%7[ %!* * ( Q
≺F 2. .] 2
F!:TG`n=m Nm2v*Nl(x H*s`w vAH4w l(mVxfv w*M/vAN1sur$wYpNxuxCJ;: J 0 %!T Y 6W
_ 1 0 (≺k)k=0,1
$. 6 2 !9 !
s ≺k t U 4Z'
u / t<-,
s k u!
;! &(s1, . . . , sn) ≺k (t1, . . . , tm)
(&≺F )
si ≺max(k,ν(,i)) (t1, . . . , tm)
-,i ≤ n
!>! &
(s1, . . . , sn) ≺0 (t1, . . . , tn) &≈F
s1, . . . , sn ≺ν, t1, . . . , tn0 , & ∈ F ⋃
C1 !
T f#8 2# $"-, 7H . 90 2-, 6 (U 6I 2S, 2I /1% 0C = =;$ E-< $ 5
1(. !? ; % 6
;4-, z _ ≺0
% 8 8h-< E 0!$+ z%
2. Z . C . 5 0!
T 7H _ $. C 4 7 2 !
nP #%O%7[ % Q ZS 2#80 2.8. 2 BHcvm N1surAvo/pNm U K. .8 C V -, -, #
l→ r -, Z$
σ%rσ ≺0 lσ
!
W 3#P/ZT-/KY0 EZV * 4RT%'( 2000
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T YW≡ !
b @ 4C =K ; !ei A@ ]% < 4-, 2S # DT YW -, KC 3 !A+ *% 8 ; A. K DT YW
6=, !Ai @ 8% @ 2S # X 2.8. 2
4C 5 , T YW% 8 7- C ;. !
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X
Y f 2 4-
U 6C 4 X
Y
!AW U % X = abaabb
Y = aababa
% ; R' 2 6X
abab%$
4!
W G 2.%C !.2. RSA. / `-/. RS !? f % $ (U0z
11 e. I X(2 Z..K $ < B0
01 D 2' !jT 0 K .$.2. D z ` 6
ε :
,4
: →
,1
: →
!
:
1,
0,
0 → !
(x; n,0)→ n
(x;0, m)→ m
(%(x);S(n),S(m)) → S(
(x; n, m))
%∈ 4,1
" =
:
1,
1 → !
(ε, ε; )→ 0
(ε, y; )→ 0
(x, ε; )→ 0
(%(x),
%(y); )→ S(
(x, y; ))
(%(x), (y); )→
(%(x);
(x, (y); ),
(%(x), y; ))
%6=
>5:
T ZS 2# & DT YW Y ≺F
!jT 1C f. K B !
F U8 7 8(U _-, U 2I8 f !g 6H Z !:T $ < = 8 !Q
4 2. KDC 7(. G 0 / 16 ;# 2# 0 !/i A@ 8%
. % jC =/ -n
m
1!/ ; ; 4= 4 ;# 2#
!m
n
.2 H C 0 1 1C .8 0!:F
I !@i A@ 8% =6 = 2 8 /53. 8(U $ !T 6 & X . !?D 1 &5 * 2.% 2. #C f !AL YH $ 2
2. Z ?> B ;-,!F e 6I8 $ 2;. C V z .% =
I , K ;# f 1 G U 2.-,
4&X (C * _ 84 0-, * U&. Z KC T YW 0 D 07 Z U 2.7 * $ /1 !
T $U;# 8 _-%, = f % Z U 2.f !(i A@ ]% z =' 2# < . . (. 'c 2 8 e( ,G 2.]( @ 8U !HF 0(< " 0- @ $.; 8 KC F0 U8 %A
(4(4(ε)),
1(1(ε)))
% (4(ε),
1(ε))
f.6 8(UhC 21 !<? B , $ ;-,% . V f# # `-, ;-,5 5ojm2n:m2p N Np,H*s nNr NrAp,N1s l/v0 U 8
16 ) FTJ A$C +61 !
+ z%G - * $ U 2.U U % T YW B 1$. 7-, "C % % 4 U . ;S !<F * $ -, fS 2.%& C !e 5 H 46. &4 ;S 4 U .s6NKoAxysQM s6H v1 * $ !
L <8% ;S 5 U 2. ; 2# 2. . f Z'fBH / 6 &!$[ 8 ]% 8 C ZC 2S 2#8 5= -,*% ;*% 8 ;SLK !eb @C `-,1 N JANn`w p,H*s nNr (( BQ !F )yF /? 9 + 7 2. 7> !+ / B> !? ! F) +D5ABC$%ZAH? + + !<T ; Q !/? !eQ R) T@QRABA,+ !
T! `(. Z _ 2. Z 7 -, : 1C 0 !HD /3 C ]% 6 _ !:Qz YC 8 S 5(.R9Z/% 4 C ,GX$.R9Z 2. Z !Q % 3C _ &% &XH . !jF &h & Z.8 `-,A. ZS 2# @ . . !N 6 @H 2. @-,&C 6 (. C ! 9 !$T H 2. ; = f [$ hM
7> !/? !/ ) [AH? + !
b = # . 7 4-,% .$. z% 51.8 ; = f ;G 2. Z 2.8 2. !8b *%* ! 4 U . ;S ( .2 4H = 0 Z(> =6 0 (C 2 2S # 2.8. 2 _7 $. & B 90 ) cBc + 2N9 1 !
, $ / V $ B V
T -,f' . # -, 2S&X "EC $. ;SLK !F (U 2. ;S 8% /. (. -, ; =(U ! 2.]( :C . ;!,?K <% 2(> f EH ZS 2# 7 . . 4 K 7 4 2 2.!AF /. 0 <% KH 76I .2 -, _ ;. . !
1 130 - (*L# L( L T X-, f' . 1. 2. (( ! 'z)! 1 ! '*) ! NS) 3cBc +
2.8 2. VM!!L (C 8) L 3c 9 + !!J !:+ .5-, 8 "(. ` 2S #
>B=
| D $.7 8XX # E
K Z/ σ%t
!→v
Y C
C ′ 8 . !E , σ ` 〈C, t〉 → 〈C ′, v〉
i ]%E , ∅ ` 〈∅, (t1, . . . , tn)〉 → 〈C, v〉
[ ](t1, . . . , tn) = v
(1)σ(x) = v 0 = -1
E , σ ` 〈C, x〉 → 〈C, v〉
(2)c ∈ C E , σ ` 〈Ci−1, ti〉 → 〈Ci, vi〉 0 F 1
E , σ ` 〈C0, c(t1, . . . , tn)〉 → 〈Cn, c(v1, . . . , vn)〉
(3) ∈ F E , σ ` 〈Ci−1, ti〉 → 〈Ci, vi〉 ( (v1, . . . , vn), v) ∈ Cn 0 T B /1
E , σ ` 〈C0, (t1, . . . , tn)〉 → 〈Cn, v〉
(4) ∈ F E , σ ` 〈Ci−1, ti〉 → 〈Ci, vi〉 (p1, . . . , pn)→ r ∈ E piσ
′ = vi E , σ′ ` 〈Cn, r〉 → 〈C, v〉
E , σ ` 〈C0, (t1, . . . , tn)〉 → 〈C⋃
( (v1, . . . , vn), v), v〉
0 -1
(C ! 9 | ;EH 2S # B . . 2 K 4 2. Z 26.8 !
2.8. 2 ( 6E- ' 2. a-, 1 ;S IK 2 z #8 (max, +)
) +K Bc5;-+*% ;# 4. , ,' !
nP #%O%7[ a #461&% %'0HL#PH&4R%7[#1IL [.'^,(KY%746.'0/1 l/pOw s csurIHcv m*oAmJ H p Hzs n=rEo/nNx Nrjn,N1s pNxfv"H 2S&X
a ∈ F⋃
CH 2.
n 0XC
LaM : Nn → N B-, LaM
$. ;S IK ! LaM(X1, . . . , Xn) ≥ Xi
1 ≤ i ≤ n
! LaM
0 (' 21 R'cN' 0 =! LcM(X1, . . . , Xn) =
∑ni=1 Xi + b
c ∈ C
0b /1 !
T -, f' 2. VH 0 2 8H .01 K 4 8 '
!T -, ' . . /( =(U 2 4 ;# `-,
L (t1, . . . , tn)M = L M(Lt1M, . . . , LtnM)
8 _-,-, e Z 2 8 " 1h-, ' .
00 04M!$L $C %&+ !eF %( ! '*) ! ^ !$g $\ 1 !eFD 2#
; K fS !nP #%O%7[ a ) 2S # B . . p ,NvAHX-,f' .
L•M - 2#
l → r -
2$ 2 σ%lσ ≥ rσ
!
T BC 0-, 7(. (' 0 (U K2. KH 2# ; `-,-, -,f' . 2(> 2 ( ZS 2#8!=i @ 8% 2S # . . # ; -, #
(x)→ (x)
>H?
( B- ' 2. !F e 8% 0- ' 2. 1 1 8% ;.
H = 6. & %$ H7 2 U 2.B(2S 2#8 !
0/KYK 0 EZd s
s E tpNx;nNm_w
LsM ≤ LtM FJANnNr/w2H*m2p Hzs n=r L M(X1, . . . , Xn) ≥ Xi
1 ≥ i ≥ n
!
0/KYK 0 E 8 n=s vr=H
tv H
sv l O H v m N6vwBMn=r$w H*m lIMAHcvl(m_w7vAH
L•Ml(rjv l/pOw s csurIHcvm*ojmJ H p Hzs n=r s s / t
pNxfn=m_wLsM < LtM
FJANnNr/w2H*m2p Hzs n=r bc66.8 KH 2# 1C 7 -, f' . 2 8 !
0/KYK 0 E% s
t ∈ T (C)p=xfnNm_w
|t| ≤ LtMv HsuxvPOs`w H vBl(rAvFMn=r$w HcpNrIHcv
cHcv xuxfv l/v
LtM ≤ c · |t| FJANnNr/w2H*m2p Hzs n=r W /( 2
|t|!AFD
|t| = 1 4-,B
t = c ∈ C
LtM = a ≥ 1!
Qt = c(t1, . . . , tn)
LtM =
∑ni=1LtiM + a ≥
∑ni=1 |ti|+ 1 = |t|
!? 06I8%
|t| = 1%LtM = a ≤ c
0 ZL>K / c = max(a)
1
LtM =n
∑
i=1
LtiM + a ≤n
∑
i=1
c · |ti|+ a ≤ c · (n
∑
i=1
|ti|+ 1)
0/KYK 0 E(a s
LtM ≤ LsMpNxfn=m_w
L (· · · , t, · · · )M ≤ L (· · · , s, · · · )M FJANnNr/w2H*m2p Hzs n=r bc66.8 -,f' . !
( TZL [1&%O&%'( E* s
t∗→s
p=xfnNm_wLsM ≤ LtM
FJANnNr/w2H*m2p Hzs n=r .8$ 8 ZG 2.] Uu
t!,? U #
l → r 2$ 2
σ
-,lσ = u
!<W X(. a X- ' 2. 7H 2S # . . %LlσM ≥ LrσM
E( 86 'c 28%, .8( 4C 9 = G 1-, f' 2. !
F ZT(.'.746%OT0 E Q n=s6H NpNsur l(rYw Ow H N6v vXmJ2J2Mm s6H*l(m2v l(sGp ,NvAHl(rAv l/pNw_s csur=H v m*oAmPJAHcp,H*s nNro$n=x NrAn Ns p=xfv n=l(m M/p l/vw ,Nn=xfv v nNr MAH*s nNr v H M/p l/v Hcv m N6vMn=r$w H*m lIMAHcvl(m
v0, . . . , vnH v x w l/v (v1, . . . , vn)
!→v0
|v0| ≤ P (maxi(|vi|))
o$n=l(m7l(r6o/nNx Nr,NvP
FJANnNr/w2H*m2p Hzs n=r W 0(. 3 0-,f' . %L (v1, . . . , vn)M
0 $. D ;S IK maxi(LviM)
%j G ;SLK
maxi(|vi|) B-, f' . 6 2 8 ;S 8
$. 0 81 ;$= 1 !H? 6I %|v0| ≤ Lv0M
!T h;B@ 46 1-,Lv0M ≤
L (v1, . . . , vn)M 4 $. fSLK
maxi(|vi|)!
1 1 8 I4(*= ( &W 3#P/ZT-/KY0 V * 4RT%'( 0H *+/^[0/
2000 *+*
p Mx;pOw_w v vw0w Ow HANvwv7mJ2J2Mm s6H*l(m2v Hcv m NsurjpNrIHHo$p=m (v Hp N6v H HcpNr=H!l(rAv l/pNw_s csur=H v m*oAmPJAHcp,H*s nNrM_p=m2p M H;J m s`w v
>B@
v w H 3=sum2v l/vRHcnNl(Hcv2n=r4M Hzs n=r v v w HMp=xCM l(xfp xfvo$p=m4l(rYw Ow2H N6vv7mPJPJ2M m s6Hzl(mv l(s Hcv m Nsurjvo/pNm =vAH@p,Nv Hl(rjv l/pNw s csurIHcvm*ojmJAHcp,H*s nNr5vAH l/v&H n=l(HHw Ow2H N6vvmPJPJ2M m s6Hzl(mv l(sHcv m Nsurjvo$p=m =vAHp ,NvAHl(rAv l/pNw s csurIHcvm*ojmJAHcp,H*s nNr M_pNxCM l(xfvl(rAv nNr4M H*s nNr v
hW+
D ' 2. ≡ !
N,0HKYL#.70 ) E !"5ZT0 T ZS 2# K 2.8. 2 2 ; Z 1K7 (U 2
: , →
(n,0)→ n
(0, m)→ m
(S(n),S(m)) → S(
(n, m))
%∈ 4,1
" =
:
, →
(ε, ε)→ 0
(ε, y)→ 0
(x, ε)→ 0
(%(x),
%(y))→ S(
(x, y))
(%(x), (y))→
(
(x, (y)),
(%(x), y))
%6=
F 1 YW ≺F
!$b ( - ' 2.
L0M = LεM = 1 LSM(X) = L%M(X) = L M(X) = X + 1
L
M(X, Y ) = max(X, Y ) L
M(X, Y ) = max(X, Y )? ZS 2# 7 . . 04C Y !
T f# 8 _-K-, U -% 1 2 T YW% = U 2.Vs6NojxysQMs6Hcv * $ X-, 2. U 2. U !HT< % / ;S % C ;X =6 E$S = ;-, =h NSY 6. <% 2' =' D * . K 0 C! 9 !
T E 8(U f#8 8 -, X-, * $ & E. . Z f6.!T 2S 2#8 2.8. 2 ;S2. ;/-, . 8 : 1 8!O?D G & * 2.% . 8 80-,DT YW $-< f $ 2 !
F e 8% (.2 7 2 U 2. N9 : K-,;-, ' f -, ; =6 !$FD -,f' . V-(9 PK4 U .B * $ !
[ 8% = _-,G-, ( 4pNlIM l(rAv 2. fS !T G-, f' . f -, 2. K f C K 2 U 2.!
@ , B (5 $ $ $ G $ @ B
F G 8K$.K ) 3c5c-+H- RC 2 =6 YW =6 2 T YW% 8"$. 6 !T 8V 8 V $. 4T YW F) cBc + _-, D-,
5 4$
2 ;S IK "-, . $.2#8 29
30
6- X-, f' 2. . 8(U f.2.50 A $.# ;$@ ;$A 1 !,T 5 8 / 2
6 !]? 6I % K-, f' . 2 : * Q !=L
DQ !AF *) LF AH;/+ (. U !
>BA
1 130 $ MPO′ + L•M
1nP #%O%7[ a(E 4 "5.'461&10
B#0 40I0/.'.'4Z(!% 0H F
?F) LF AH;/+ %<Q !L BQ !F Y(. 4 B
2S # 4 . . 2!F f 0-, 0 F [#146&01
0
0 !"5"B0/11&0/ T1-1Si(x)
i ∈ 0, 1
C 0 ( T 0/"$&%'( 1 πn,m
j (x1, . . . , xn; xn+1, . . . , xn+m)→ xj
1 ≤ j ≤ n + m
0 ( TP/!P/"B0/1&10/ ! 1 (;0)→ 0 p(;Si(x))→ x 0 F [##%O%7[##0H.7.'0(1
(;0, x, y)→ x(;S0, x, y)→ x
(;S1, x, y)→ y
0 & P/"5 T1&%'(\L#TP/!%7"B4R&%/.[0(1
(0, x1, . . . , xn; y1, . . . , ym)→&(x1, . . . , xn; y1, . . . , ym)
(Si(z), x1, . . . , xn; y1, . . . , ym)→)
i(z, x1, . . . , xn; y1, . . . , ym, (z, x1, . . . , xn; y1, . . . , ym))
i ∈ 0, 1
& )i
B0z(. . .8 81 1
0 F [KYL [1&%O&%'(Y1T0(1
(x1, . . . , xn; y1, . . . , ym)→&()1(x1, . . . , xn), . . . ,
)p(x1, . . . , xn);
1(x1, . . . , xn; y1, . . . , ym), . . . ,
q(x1, . . . , xn; y1, . . . , ym))
&,)
i
j
B0z(. ..] 6 1 !
b C 0- 2# 2.C 0(. ,G YW0 D6I 6T YW EC K
1 / Z' K
0 21 !
nP #%O%7[ a[V e !0 #461&% %'0HL#PH&4R%7[ L ( ! .'4 "B.'4Z11&0B
T 6 4.1
06I8 0C 7-,f' . ZS 2#8 2.8. 2 0B
L0M = 1
LSiM(X) = X + 1
LπM(X1, . . . , Xn+m) = max(X1, . . . , Xn+m)
LM(X) = X
L
M(X, Y, Z) = max(X, Y, Z)
W C (. 2. .8 <%
L M(X1, . . . , Xn; Y1, . . . , Ym) =q (X1, . . . , Xn) + max(Y1, . . . , Ym)
q
$. . q (A, X1, . . . , Xn) = A(q
0(A, X1, . . . , Xn) + q
1(A, X1, . . . , Xn)) + g (X1, . . . , Xn)
0(. . .8
q (X1, . . . , Xn) = q (q 1(X1, . . . , Xn), . . . , q
p(X1, . . . , Xn)) +
∑
i q i(X1, . . . , Xn)
(. <!
+ z% XC fS = 0 2S # . . 2XB
-, YW $B-, . !
1 1 MPO′ + L•M
$ & 1J. (-,8 8% 2S # . . -2 YW ( GK-,f' .
= 8 ;SLK * 2. G C! 9 !NF 6 /d -, 7( $. ;S IK 7 B 2.!
nP #%O%7[ aZd & 4Z 2M # 1_^,K 1 (.'0`!0 (!"B%7[T . .8 2 4(. 2ZS& DC !(T 1m2p=r :H ZS& f
p- ZS&X $C . 8 R0 2. .] /1 = f
p− 1!
C5c
Cp * X
n' G 2 2 8 :-, $ : C
p!e 6
|C| ≤∑
1≤p≤k
|Cp|
k
; / 1 2S, F
|A|
$. z 8XA
!
0IZT#05 .'4h46%7.'.'0`# f"B46" 3#0 W
|Cp|% 5 $U
C∨p
C∧
p
!$F ;(. 2.]( " r-,
C =XC # aH # l → r
Z$ σ0
4 * . 1 !:
lσ = (v1, . . . , vm)
vi ∈ T (C)
!<Q &(u1, . . . , un) / rσ
& 7 p!<F 1 ZS 2#8
2.8. 2 YW;% $. &(u1, . . . , un) ≺mpo (v1, . . . , vm)
!&b jS X 0 (U ; !9 !KQ V 8 ; /E-,
p0 &≺F 1 % Y
(u′1, . . . , u
′n)
C∨
p
0u′
i C 7 ui1 !$W 2.8-,]% _ H
C∨p
$.
|C∨p | ≤
∑
i<p
|Ci|
;!? % & 16I /%A n = m
(u1, . . . , un) Em′
(v1, . . . , vn)!ji
C 8% B 6C B p
C 7 (v1, . . . , vn)6C
0 $ -, Z' vi
!(b 7
I(v1, . . . , vn) = (u1, . . . , un) : (u1, . . . , un) Em′
(v1, . . . , vn)
8 K D _ H C∧
p
|C∧p | ≤ |Fp| ·
∑
(v1,...,vn)∈C∨p
|I(v1, . . . , vn)|
W 4 ; $ 7 % 7-|I(v1, . . . , vn)| ≤ n! ·
∑
i≤n |vi|!<? ;8%
$.2#8 h;$A ; `-, -, -,(v1, . . . , vn) ∈ C∨
p
% U h fSLKP
4-,∑
i≤n |vi| ≤ n · P (maxi≤r |ai|)
ai . B * $ !<i X
2. Z 8% |C∧
p | ≤ |C∨p | · P
′(maxi≤r|ai|)
r * .K "
P ′(X) = |Fp| ·n! ·n ·P (X) fSLKF0
n z . NU ; DH
ZS&X /1 ! 8 D 2
C∨p
C∧
p
$-
|Cp| ≤ |C∨p |+ |C
∧p |
T _ K Cp
VE $. ;SLK 3 2.!G? < z X( = < $. D ;SLK 0 S "-< @
Cp%
=V NU f$ = -,4 ZS& DC P1 !$F 6 8 _. = $. G fSLK K 2.0 = K 2.8-, D( $.2#8 ;$A 1% ( $. ;SLK 7 . !
W f % Z(> ! _-,-, (C, t, σ)
= @H 2.]( E , σ ` 〈C, t〉 → 〈C ′, v〉
G $. G ;SLK !i @ 8%|C|
G $. G fSLKD U ;SLK ( C@A. =8%
t : R' 2 H .8-,
E <
U 6-, ;S IK σ
6 -, 6 87- 7 R' 2 H
C!
F 6 S-, ;SLK @- ZS 2# G 2.8. 2 2 0 YW 1 % 0C . 8 ;S IK !
C 9
Q 2S 2#8 B 2.8. 2 YW 0 YTW 1 K$. ` !eFD f#8 B- K#, ..] & 0 K &-1 !
? 7-K9 . &0 2S&X 0 1C 7 KC 8(U 8X 9 21 % 2. .8 C 7 !&b 2(> A@ K 1 K 8XK ! BC &
F & ,GG D XK # 7 2.8. 2 4 !Q 2S # $7-,f' . Z' I 7(.;% Y-, f' . C % ;B .-< 8 3H - ' 2. !(? ;8% 64-, ' . 2# f.4 6 8% =6 = 7 0C / ` =5 . =6 !
FD 2 h-,69 . 2S # 0 bc EF U RS + ;S8J1 ) NSRc 9 + !GT 1 /E ZS 2# X 2.8. 2 0 / ` I X-, f' 2. . 2S 2#8 hWL0 1T<W 1 e ; -, f' 2. ! (U # F 6 ed =6 z ;; !
"
/ ! "
/ #
-, - =6 E C # V . . -,f' . ! . U 2.7( 6 .!
) ) & " $ ! &%(') "*" " %,+-' ) #" " #" 'D/ 1 .% 8 * ` 7 * 2. 2
5
* ; 7 * 2. D !
" " " . " / 0 #" 0 0 0 #" # " 0 #" # "
CH;
/ # / *
#"/ /
) ) & " $ ! &%(') "*" " %,+-' ) #" " #" ' & *$ #" + & $
& ! " " #" + &
$ z 2.8 2 0 1 56. ! & * 2. 2. !
T 1 2 7 / = 7 .8( @ $. * U&. !D;-, 2 5
(4 n(ε),
1 n(ε)) 8 2
n+ Y.8
0 4 41 17 172 63 483 219 974 771 1645 2775 2496 10169 352
8 8 _--, D6. A@ -, `-!
C5>
f
YI8 ##$" ! f f /E-, j% ;(.2. 6 $ > 8% 8 I 2. . !i A@ 8%; 8 f#8 !j%6 U ` .7 C 2 ;. -, /.8% H U . f# f 1 f. . W T #0 !/? f8%AF !eQ !/T %$> !e? !A K+ ! !$L E+K .7-, $ 2Q f\'ZF 6W
) T<,L+ c 9 + X(. =6
"! !? . " #$" ! . $. \ + !DF = ) F Z=5; +
? !$L !$g 8) gR?$; + !(W ; . 86 8%(> !/? !$ 1 2.4-, 0 =6 C 0 2. C =(U B. /1 . " !T<= z% _ 2e. . - k
$" ! !@F . 8X 0 7 !eQ =\ 9) Q =\&@Bc-+ %()!eM0 8) M0 @5>/+ E+ !AM B) MAH;/+ !
Q& ((# VT YW%@+ !:F ! '*) ! (( * ;S 2.] 6TW (U !Hb T BT U BW a 80 T<T<W 14-, ; TW 4$. Z $
1!(b .7-4TTW . ##$" ! !
" $ $ ! @ $ X (5
? B ;# f BT hW% 8 D K-, f' . fS' !e K . Z W 3#P/ZT-/KY0 V ) 0I( 46&0 *+4RT%'( 0H *+/^(0H
2001 0I*+* )
p7Mx;pOw_w v v w1w Ow2H N6vwvmPJPJ2M m s6Hzl(mv H v m N1surApNr=Ho/pNm v HDp N6v H H p=rIHKl(rjv l/pOw s csurIHcvm*ojmJ H p Hzs n=rYo/nNx Nrjn N1s pNxfv M_pNm2p MAH'J m s`wv #$" ! v w H 3=sum2v l/v5HcnNl(Hcv nNr4M H*s nNr v !3vw H Mp=xCM l(xfp xfvo/pNm0l(rw Ow H Nvv0mJ2J2Mm s6H*l(m2v l(s Hcvm N1surAv!o/pNm vAHBp N6v H7l(rAv l/pOw s csurIHcv m*oAmJ H p Hzs n=r o$n=x NrAn Ns p=xfvhv H l/v H n=l(H0w Ow HANv v5mJ2J2M m s6Hzl(m2v l(s Hcvm N1surAvBo/pNm v Hp ,NvAHl(rAv l/pNw s csurIHcvm*ojmJAHcp,H*s nNr"o/nNx Nrjn,N1s pNxfv M_pNx Ml(xfvXl(rAv nNr MAH*s nNr v"
T<W+
Df' . ≡ ! !
N,0HKYL#.70 ) V !46&% 0/ 0 ,(.'0/46 [TKh #.'4 T =6 2 @C XB ;.-, e. = 0 KC X Z'
6=,∨, ∃-, 2(> . /1
(W T #0
) → 46.710 0 = 0 →W T #0
( 46.'1&0 ) →
W T #0S(x) = 0 → 46.710
(W #0
, x) →W T #0
0 = S(y) → 46.'1&0 ( 46.710 , x) → x S(x) = S(y) → x = y
" (φ) →
(φ,
J%'.)"
(x,J%7.
) → 4Z.'1&0" (x,F [#1
(a, l)) →
(x = a,"
(x, l))
CH:
?C Z %t * B 8 W T #0 !
( 4R (x), t) →
" (x, t)
(]
(φ1, φ2), t) →
(
(φ1, t),
(φ2, t))(J[
(φ), t) →
(
(φ, t))( N,%'11
(n, φ), t) →
(
(φ,F (#1
(n, t)),
(φ, t))
F # $. T<W 2. .8
,
, ≺F" ≺F
≺F
" !i ( 6- ' 2. ;S 2 LcM(X1, . . . , Xn) = 1 +
∑ni=1 Xi
% -, n' !
L
M(Φ, T ) = Φ + T%L "
M(Φ) = Φ + 1!
L M(X1, . . . , Xn) = maxni=1 Xi
% = ZS& 0C !T C ;SLK !
T<= z%-, f' 2. 8 :9 PKD U . ( ( !
@ , B (/, & 7 81 130
LPO + L•M $ & 1
W 2. Z 8% C 2 2 7C !i @ 8% G 1
(0, y) →
(y, y) (S(x), y) → (x,
(y, y))
-, 2 4TW ( 46-, 2. 0 4$. B V6I f#1-, * U8 0 1 !/F / 8% fU 2.] U -,K. # ; z 2.!Q 2.]( 2.] U5- 6 0 2. 2
. . 2%. f j,S = 7 1 =1% = 1 2
U 0 ; f`-,$.21 !@i % 1 .8$ 8 .8 U5-, 6 0 .
Y. f &S f
1 % = 1
8 ;SLK !
+ 1#T0H1 P .646.7 !46%'( 6. 2. 2. fC :-, I C : z . f
H 2 !
nP #%O%7[ a 8 4R&1 JAHcp,H 1 ('
〈 , t1, . . . , tn〉 1 ZS& 5 C H .
n
t1, . . . , tn 1
2 2 8 !State(
" )
z E . 2S, 6H =6 " !StateA(
" ) = 〈 , t1, . . . , tn〉 ∈ State(
" ) / |ti| < A
* X 1 0. D R' 2 4 $.
A!
nP #%O%7[ a% W 4Z!1&%O&%'( Q
main 6%
η1 = 〈 , t1, . . . , tn〉
η2 = 〈&, s1, . . . , sm〉
(Uh. DState(
" )!j
H*m2pNr/w_s6H*s nNr η1
e η2
<-, 9 !
e ≡ (p1, . . . , pn)→ t!
;!b U 7 Z$ σ
B-piσ = ti
1 ≤ i ≤ n
!
>!b U 7 Z' &(u1, . . . , um) / t
<-,uiσ
!→si
1 ≤ i ≤ n
!
Transition( "
) z 4 .;.
State( "
)!
∗
CK2 2.$,U 7 ∪e∈E
e
!
C5=
nP #%O%7[ a(a + 1#T0 P .646.' #4R&%'( Q NpNsurE 2S 2#8 1 . . 6-, 2 7T<W
〈 , t1, . . . , tn〉 . 0z
State( "
)1 !
ap=m m2v J tNpNxyl/p Hzs n=rΩ
〈 , t1, . . . , tn〉
K(. . 4
Ω 〈 , t1, . . . , tn〉
!
T (C < -, 7/η1
% H.;. η2 ∈ State(
" ) / η1
e η2 ∈ Transition(
" )
e ∈ E!
CT (〈 , t1, . . . , tn〉) z 8X7 H . ; 5
〈 , t1, . . . , tn〉!
CT A(〈 , t1, . . . , tn〉) = τ ∈ CT (〈 , t1, . . . , tn〉) / ∀η ∈ τ, η ∈ StateA( "
)
* 8X7 2 -,7 . 4 $.!
N,0HKYL#.70 ) d Q / 2S # B . . 2 C YH + _ <% 4H . ;
CT (〈
,S(0),S(0)〉) !
〈Ack,S(0),S(0)〉
〈Ack,S(0),0〉 〈Ack,0,S(S(0))〉
〈Ack,0,S(0)〉
0IZT#05 .'4 3!46 !0/ ! #0H1]4R 1#T0H1 P .64Z.' #4R%7[ 0/KYK 0 VE n=s vr=HNpNsurl(r6w Ow H Nv vmJ2JPM m s6H*l(m2v l(s=Hcvm N1surAvo$p=m
αxfvm2pNr:RNp Os6Nl(N l(r6w ,N _n=xfv v nNr MAH*s nNr
dx pNm s6H'J&Np Os6NpNxfv l(r"w ,N _nNx;vDv H
Al(r"v rIH*s v m l(s _nNm rAvKxfpBH p=suxuxfvv w J H p H w vKx p=m m2v JtNpNxyl/p Hzs n=r n=l(m
HcnNl(Hp=m mv JtNpNxyl/p Hzs n=rΩ ∈ CT A(〈 , t1, . . . , tn〉)
nNr p s
〈 , t1, . . . , tn〉 ∗ 〈 , s1, . . . , sm〉
pNxfn=m_wcp ≺F
nNl ≈F
v H(s1, . . . , sm) ≺l
lpo (t1, . . . , tn) s
〈 , t1, . . . , tn〉 ∗ 〈 , s1, . . . , sm〉 ∈ ΩvAH
≈F p=xfnNm_wExfvErjn,Nmv J Hcp,HzwEv rIH*m2v〈 , t1, . . . , tn〉
v H〈 , s1, . . . , sm〉
vw2H8_nNm r4Jo/pNmAd
<p6x;nNr :l/v l(m vFM/p l/v m2p=r4M $v vΩ
v w H n=m r4J_vo$p=mα×Ad
FJANnNr/w2H*m2p Hzs n=r 9 !W 0-, =6 TW%/ -,2# KC K$. ,G 4TW !;!KQ
〈), u1, . . . , up〉
(C X〈 , t1, . . . , tn〉
!F 6ti
uj
X 2 2 8 8% U X 6-,
(|u1|, · · · , |up|) <l (|tπ(1)|, · · · , |tπ(p)|)!F 6 S ;
Ad
d' : - ! /.
A% 8: -G (. $.
Ad !
>!? -0 % <S =E NU ;X$Ad . 4 ZS& 7C OS Y6I
/% $Ad B f66.8 (C . 8%:vAH'M F 6 S V-,
α K C@A. 8%
H 7 $.α×Ad !
0/KYK 0 V(V n=s vr=H NpNsur l(r w Ow H Nv v3mPJPJ2M m s6Hzl(mv l(sBH v m N1surAvo$p=m v H1p ,NvAH6l(rAv l/pOw s csurIHcvm*ojmJ H p Hzs n=r l(rw ,N _n=xfv v nNr4M H*s nNr v H
t1, . . . , tnvwBH v m Nv wBM_nNr$w H*m lIMAHcv l(m_w
CT ( , t1, . . . , tn) = CT L (t1,...,tn)M( , t1, . . . , tn)
FJANnNr/w2H*m2p Hzs n=r Q
Ω ∈ CT ( , t1, . . . , tn) 〈&, s1, . . . , sm〉
. VΩ
!F 6 si
" 2 8%|si| ≤ LsiM ≤ L
&(s1, . . . , sm)M ≤ L (t1, . . . , tn)M
(. Y -, ' . !
C/?
0/KYK 0 V[d n=s vr=H
t1, . . . , tnvw Hcv m N6vw.Mn=r$w H*m lIMAHcvl(m_w x<vPOs`w H vXl(r6o/nNx Nr,Nv
PHcvx l/v
Lmain(t1, . . . , tn)M ≤ P (n
maxi=1|ti|)
FJANnNr/w2H*m2p Hzs n=r F
LtiM ≤ c · |ti|!
+ L#L 0/. L#4R .646.'0/ !T1 W D fSZ * 2.% ZL>! * 2 . f$'
5 * & C / 8 !$F ;-, 2. DH 8 50
7 . . /1 . f !
W 3#P/ZT-/KY0 V 8 n=s vr=H NpNsurVl(rw Ow H N6vvKmJ2J2Mm s6H*l(m2v l(sHcvm N1surAv:o$p=m v H@p ,NvAH@l(rAv l/pNw_s csur=H v m*oAmPJAHcp,H*s nNrEvAH
t1, . . . , tnvw H v m N6vw M_n=r$w Hzm lIM Hcvl(m_w (t1, . . . , tn)o/vl(H AH*m2v6mJ =l(s6HDvr v w*o/p M_v0o/nNx Nr,N1s pNx Yx pNs v/vxfpEw2H*m2p H;J;:s v
w l(sutOp=rIHcv M/p l/v J H p_o$v nNr M $nNs`w s6Hl(rmJ vOnNr=HH n=l&wGxfvwGw nNl&w 'Hcvm Nvw!w nNrIH v w H v m Nv w M_n=r$w Hzm lIM Hcvl(m_wv H<nNr"p_ooAx s l/vxfvXrAn N m2v J H p_o$vwBr4J2M_v w_w pNsumvo/nNl(m7xfvXrAn=m N6p=xys`w vm FJANnNr/w2H*m2p Hzs n=r F 2. U f#8 4 G * DH . ; "H 7 Z !T & 7 2.]( X1 $. H 4 ; 0( U 0 ]- jC 4 !,T 0 D $. ;S 8
ti0 =6 $D1-, f' . =1 !/T
D( U /. DH K z z 2. NU ; KH 7e * !? H , / 2.]( 7 $. ;S
2 =$U !eT C ;SLK !
81 1 $ LPO + L•M
1W 0 7-,6TW 4- ' 2. 2 7 DC
#$" !j%:9 a-, 2S # Y . . hT<W ( h -, f' . h ;X ((# 5 ZW 0J
((
? !T 6! '*) ! ) T 3A5:-+-, X 2" Vg + !jF / /%&? !eP \ KT !/Q _&8SR) FPBQR@ 9 + K-, ! !
( 464Z.'.'0/. & 0 2(%'10H *+46" 3#%'#0H1nP #%O%7[ a* ( & * 3p M &surjvp=m2pNxux xfv Dv :s`w Hzmv w 0 RW J
0 WGJ 11Z K 1'
2.7 9 ! <H6JAHcp,Hzw
S = s0, s1, . . . , sk-, 6 z . ! !
;! Ymv :s`w Hzmv wΠ = π1, . . . , πm
%πm
2.$' ' 8 Z z
0,1!
>!T! _ U `-,<l 2
W = 0,1∗ε <l y
% (x) <l ) (y)
% %(x) <l % (y)
x <l y!
C ! C com
5. M_n,NFN6p=rv w - [ !"5"
(π′ =%(π), s′)]
%[( 0/
(π′ =p(π), s′)]
%[02T46#" 3
(π, s′, s′′)]%[ Z (s′, s′′)]
%[ [ (s′, s′′)]
%[ !
]!
nP #%O%7[ a Q F [ 2[ !T4R%7[ M_nNr :=l(mp,H*s nNr H "WGJ
M B
(s, F )
s ∈ S .
F X Z/
Π
W
![u1, . . . , um]
2. C - ui
πi
πi ← a[u1, . . . , um]
2. [u1, . . . , ui−1, a, ui+1, . . . , um]
!
C5@
nP #%O%7[ * PHK 46% !0J#0/1 ( & *+1Q
M1WGJ !HT C 1 2. ;-,
: N × S ×Wm 7→ W%j .2 D$
M
. K /.0 2. . $ !
(0, s, F )
E(. ! Q
com(s) = !"5"
(π′ =%(π), s′)
(t + 1, s, F ) =
(t, s′, π′ ←
%(π)F )
0z X ,/# %
π 2. &E( -1 !
Qcom(s) =
( 0/(π′ = p(π), s′)]
% (t + 1, s, F ) =
(t, s′, π′ ← p(π)F )
p(ε) = ε
p(%(x)) = x
! Q
com(s) =02T46#" 3
(π, s′, s′′)
(t + 1, s, F ) =
(t, r, F )%
r = s′
π = 0(w)
r = s′′
π = 1(w)!
Qcom(s) = Z min(s
′, s′′) (t + 1, s, F ) = min<l(
(t, s′, F ),
(t, s′′, F ))
! Q
com(s) = Z (s′, s′′) (t + 1, s, F ) = max<l(
(t, s′, F ),
(t, s′′, F ))
! Q
com(s) = !
(t + 1, s, F ) = F ( '!'
)!
nP #%O%7[ * ) F 46.'"5 !. Q
TC 6
N
N!=T 0WGJ
MM_pNxCM l(xfv
f : Wk →Wv r HcvANKoew
T
(w1, . . . , wk) ∈Wk %/
(T (
kmaxi=1|wi|),
! , [w1, . . . , wk, ε, . . . , ε]) = f(w1, . . . , wk)
W 3#P/ZT-/KY0 V% + F 3!46#!4 B0/f0H &," ,KY0H^[0H 1981
F * ) n=s6H
f : W → W fvw2H3M_pNxCM l(xfp xfv"vr v w*o/p M_vKo$n=x Nr Ns p=x@w_svAHw vl(x;vANvr=HGw s!sux:vPOs`w Hcvl(rAv l(s M_pNxCM l(xfv
fvrH v NKoewo$n=x Nr Ns p=x
%'K !.74R%7[ !0/1( & * 1 L 4R #0/1 1_^,1-/KY0/1 !0`TP/P/"$T%O& T0 T ;X GC D D ;# # D 2# A
RC 2 5 # 7 . . 2!
0/KYK 0 Va 0/KYK 0 ( .7 2 46# ( .'45^ n=s6H
f : W → Wl(rAvn=r4M Hzs n=r M_pNxCM l(xfp xfvo/pNmEl(rAv
Mv rH v NoAw
T pnNr MAH*s nNr f ′ w l(sutNpNrIHcvhvw HM_pNxCM l(xfp xfvo/pNmBl(rEw Ow HANvvBmPJPJ2M m s6Hzl(mv l(s H v m Nsurjvo/pNm vAHp N6v Hl(rAv l/pOw s csur=H v m*oAmJ Hcp,H*s nNr
f ′ : N×W → W
(n, w) 7→ f(w)w s
n > T (|w|)(n, w) 7→ ⊥
w surAnNr
FJANnNr/w2H*m2p Hzs n=r T 2S 2#8 0 . . X 4 2 8 C = 0,1,
1, , ε
⋃
S
S * 4
. ! M
!T 2S, : C " , -, / !
min<l , max<l
% 4-,;X# Z.8 `-,7 WJ ′ !$T 2# . . Z
" (ε, w) → ε
(ε, w) → w "
(w, ε) → ε
(w, ε) → w " (0(w),1(w′)) → 0(w)
(0(w),1(w′)) → 1(w′) "
(1(w),0(w′)) → 0(w′)
(1(w),0(w′)) → 1(w) " (i(w), i(w′)) → i(
" (w, w′))
(i(w), i(w′)) → i(
(w, w′))
i ∈ 0,1
!/ [ "
] = min<l
[
] = max<l
!
9 ! ( 1 (t), s, π1, . . . , πm)→ (t, s′, π1, . . . , πj−1,%(πk), πj+1, . . . , πm)
com(s) = !"5"
(πj =%(πk), s′)
%
;! ( 1 (t), s, π1, . . . ,%(π′
j), . . . , πm)→ (t, s′, π1, . . . , π′j , . . . ,
%(π′
j), . . . , πm)com(s) =
( T0/(πk = p(πj), s
′)%
C5A
>! ( 1 (t), s, π1, . . . , πj−1,%(πj), πj+1, . . . , πm)→ (t, r, π1, . . . , πm)
com(s) =0 4Z!" 3
(πj , s′, s′′)
r = s′
,%= 0
r = s′′
%= 1
%
C ! ( 1 (t), s, π1, . . . , πm)→ "
(
(t, s′, π1, . . . , πm),
(t, s′′, π1, . . . , πm))com(s) = [ (s′, s′′)
:! ( 1 (t), s, π1, . . . , πm)→
(
(t, s′, π1, . . . , πm),
(t, s′′, π1, . . . , πm))com(s) = [ (s′, s′′)
=! ( 1 (t), s, π1, . . . , πm)→ πmcom(s) =
!+ 2. .8
"
, ≺F &% # C $. ,G TW -, z B0 8
/ 21(. G !$T 2S 2#8 ( 6- ' 2. 2 LεM = 1 L0M(X) = X + 1 L
" M(W, W ′) = max(W, W ′)
LM = 1 L1M(X) = X + 1 L
M(W, W ′) = max(W, W ′)L1M(X) = X + 1
∀s ∈ S, LsM = 1 L
M(T, S, Π1, . . . , Πm) = maxΠ1, . . . , Πm+ T + Sb f @-, . C ′ $. ′(n, w)→ (n,
, w, ε, . . . , ε)
!]b 2. -
f ′ = [ ′] !=F 2# !(. G T<W ≺F ′ $ -,f' . L ′M(N, X) = N + X + 1
!
0/KYK 0 V!* n=s vr=H
w ∈WvAH
Pl(r7o/nNx Nr ,Nv @xfp nNr MAH*s nNr
w ∈W 7→ P (|w|)vw HM_p=xCMl(x;p xfvGo$p=ml(rw Ow HANv vDmJ2J2Mm s6H*l(m2v
Hcvm N1surAp=rIHHo/pNm vAHp N6v H H p=rIHGl(rjv l/pOw s csurIHcv m*oAmJ H p Hzs n=r FJANnNr/w2H*m2p Hzs n=r g ;SLK 8 I ;. V !
(, y) → y
(, y) →
(1(x), y) →
1(
(x, y))
(1(x), y) →
(y,
(x, y))
F 2# _ $. T<W 2. .] ≺F
!8i ( -,f' . L
M(X, Y ) = X + Y
L
M(X, Y ) = X × Y
? 71I8 % BH 2 I f.4* ;4 C " (ε)→ 0
" ( % (x)) → S(" (x))
Kv NpNm l/vT 5-,f' . H ;SLK (C / 2. `-,!<FD BV D f.2. 2#8-,1 0-, ' . 8%f.1 1 ;.!F 7 , 4(.A. ; E(. 'c !
W 3#P/ZT-/KY0 V Q qrjv n=r4M Hzs n=r M_pNxCM l(xfp xfv5v r H v NKoewDo$n=x Nr Ns p=xo/pNm6l(rAv v w H Mp=xCM l(xfp xfv7o$p=ml(r w Ow2H N6v vmPJPJ2M m s6Hzl(mv l(s H v m Nsurjvo/pNm vAHp N6v Hl(rjv l/pOw s csurIHcv m*oAmJ H p Hzs n=r FJANnNr/w2H*m2p Hzs n=r F (.B 81 >H? >5@!
g YC ! V = WGJ 8 ;S IK z% "2S 2#8 4 2.8. 2 0-, TW ( B- ' 2. !
:$c
, $ / V $ B V $
F 6 z &% fSLK B-,f' . Z.8 `-, (C / !i C 8% = -,!-
t ! . *
nR0 = Z'cN'c
7 0
S1
P ;SLK *0 ZS 2# . . 2 21%
n = |t| = LtM
P (n) = |P (t)| = LP (t)M!& K-, =;$ : ` G- C
Nm
N Z.8 `-KC B- ' 2. !
i 3C 8% ;. 2. 8 8%" % .$. K ; 1-, 2S # `(. B !
+MTKY46%'( d n=s6H&NpNsural(r w Ow H Nv vmJ2JPM m s6H*l(m2v s<o/nNl(m H n=l&wXxfvwFHcv m N6vw M_nNr/w Hzm lIM H v l(m_w
t, t1, . . . , tnH v x w l/v
tw nNs6HKx;p
nNm Nv1rAn=m N6p=xfvv
(t1, . . . , tn)
nNr p|t| = f(|t1|, . . . , |tn|)
p=xfnNm_wsuxvPOs`w2HcvXl(r3w Ow H Nv vmJ2JPM m s6H*l(m2v l(sM_pNxCM l(xfvXxfp N N6v nNr MAH*s nNr vAH l(s:p ,NvAH
fM_n,NFN6v l/pOw s csurIHcvm*ojmJ H p Hzs n=r
F 4C / 2.I =(U (U . !e^ C %e^ !AQ , (' 4P !e+ 7) ^ =C'A5A(%e+0QRcBc/%e^ =C'c5;-+ 2 C Z> ('RQ;\Bb
!(i @ 8% `(#
DC 4 1$ .2 0 $. 6 2. 7 0-, f$' 8 f.2. . 4- * . 4 X 8 D U !Ni 8% C . I vrBoAxfp Mv %= Z'cN'c D H 4 56. 0 * U(. !< 8 B % U 8 % ! 1 #6-, / 2.8 6. e (. / !(F K-, UI 2. G * `-,K ( G
NS. 2 U(. . 2G : : <G-, ( ]d .8 6 \6. = Z.!
=(U-, ' . !QADC > ('RQf\b e% D( 2. Z K $. D . !,i = 2S # D 2.8. 2 K-, $ - ' 2. $. 3 . % 2' =' Y 8 #3-, f' . 5 z #8
(max, +)!e = 9R 0 B-, f' 2. - 6. 2.
. / ;. 4J !<+ 8) +K Bc5;-+ B-, 2 X 1 7 = 2. h0C > ('RQf\Xb 1- ' 2. ; ZS 2#8 K RS 6 C !/^ =C !
.I < :C @-< 2'*I ! - ' 2. !_i A@ ]% /(. `= ZS 2# . . ( @ 2. fS! -,f' . Z' I 2(> =6 C@A. G 2. I K$. ` 0 1 -, < ( = . ( 7-,f' . 41 !
| / B ( 2$ " X ; 1. 2e (. `= .B -, f' . !,i 7. / K(.Z9Z .!$FD @ Z 2# D-,BF !$Q !$T %(> !(? !$ + ! !&L (' +K Z. 2Q f\'ZF KW
) T<,L+ c 9 +
-,% ! '*) ! % !(.e. #Zg .=$U
7W 0 2 Z=9 21 !
: 9
T .=$UV 7W . . (( 0F !e+ !/W 1970
!(b \ =6 2. ((. K 2S # $S = ;-, -,< 8% 6 U B (U2. .0 U ; 3 2 /1 !Hb 4 : 4 (U 2. 0 B =6 !/i * %=9 K ((. * * Y h * .2/3( 2. = hW h (. (U f.2. 2
!
F . ` 0 2 .=$U6 0W !,T f 2. = m2v_v M $n=sQM_v vAH*m s:rAv H w7 7 !/? K !$i \ 0) ?Di A5:,+61 !
" $ ,nP #%O%7[ *E & P/1&0/46 N EJ. =V0W
N
(S, T, F )
S
T (U X 9
F
5 2S
⋃
T B-,
F⋂
(S × S) = F⋂
(T × T ) = ∅!
T .;. S
;.ojxfp M_vw `- . 2. E f' !=T .;. :
T ;.H*m2pNr/w s6Hzs n=r$w `-, . 2. : !
F Ym2v xfp Hzs n=r v n,H( 2. = `-, 2. . D ,e#
!$T .f.8 S
⋃
T%&-,/ ;
V % ;. 4rl Ow !? K ,B $. 7 8%
((S⋃
T ), F ) 0 2.!
W fG 1 ( 2.% ! G @. !z . 8!H ! . `-!
N,0HKYL#.70 )68T =! 9 . .= VW z 8X
S = p1, p2, p3, p4, p5%
* 8XT = t1, t2, t3, t4, t5
E #,
F = (p1, t2), (t2, p2), (p2, t1), (t1, p1), (t2, p3), (p3, t4), (t4, p5), (p5, t3), (t3, p3), (p5, t5), (t5, p4), (p4, t2)!
t1
p1
p2
t2
p3
p4
t3
t4
t5
p5
(=! 9 .V 7W !
:$>
nP #%O%7[ *(V ( TP/10H #L (1_&10HQ
N = (S, T, F ) 2. = 0W
a 7/"
N!,TG 8X •a = b/(b, a) ∈ F
ojmJ w v H
a!$T! 8X
a• = b/(a, b) ∈ F 0o/nOw H w v H
a!
T 2. 0 ! 1 ;. vr=HzmJ_vR0 !vw n=m Hzs v-1 !HT D . 0 ! 21KH ;. vr=HzmPJv0 ! vBw n=m Hzs v-1 !
Q A
X 6 7/!HT 2. ' 0.e( =$U 4 7/7 C d;-, •A =
⋃
a∈A•a
A• =
⋃
a∈A a• !
Kv NpNm l/v? 3 ;# ; ./. ;-,% . = .' f " 8X: GX 0Rs v -, 7/ 8 ,G ; 8 :C G .' H = 7eI1 !F -, I C . 2. 4 ; 0 / 0 X 2./1 -, 0 E #,!
N,0HKYL#.70 ) % ? .B $. ..] 6 /% % •p1 = t1
%p•5 = t3, t5
%t•2 = p2, p3
% •t1, p3 =p2, t2, t3
!
nP #%O%7[ *[d F 3!0/KY%7!1Q
N = (S, T, F ) 2. = W ! M/vAN1sur
N E Z ; 7/
a1, . . . , ak+1 B-,
1 ≤ i ≤ k%(ai, ai+1) ∈ F
!
W 2 % 2. `-, =$U 2. =(U U % 2' =' 8(U - 0 Z';9 U !
#%$ , $ .B V Y ,@ B 5
E _-, K K9 !
nP #%O%7[ * 8 l*+4R #4 2(0H1Q
N = (S, T, F ) ." 4W !& NpNml/p :vX
N 0C
M
C !Q
M5 -, K
N!(T
s -,$.D
M
M(s) > 0!&T 0 _- ` ;-,
c6 -, % 2.0!
Q N
2. =%M
E _- s !e 5 K-,
s
M(s) vAHcnNr/w !
M `-, ]%e3 _-, 2. . 3 7 -, 4 4- X 7 9 , !
T X 9 & B 6 B 9 & -,
M(R) =∑
s∈R M(s)!
T _-, 8 _ $. C ( X 3 "9 5 -, != Z'cN'c 4-,
M EM ′ -, s%M(s) ≤M ′(s)
!
5 _-, 0 H =" 9 E -, = ! <%0-, f#DB9 E -, = N9 KX9 5 -, !
nP #%O%7[ *!% W 4Z!1&%O&%'(!1]46"B%/.(0/1Q
N 2. =%
MY -,
t !j Y 0-,
Mp MAH*sutOv
t0 h-,
t .B
M1
M _- 2.
t%= Z'cN'c
∀s ∈• t, M(s) > 0!
Qt =" -,
M% 8 I HzsumJ_v!,T
t=6# =V _- 2
M ′ - 0$. : 0
M 09 -, 1 . 4
t N9 K
-,7( t%= Z'cN'c
M ′(s) =
M(s)− 1
s ∈• t
s /∈ t•
M(s) + 1
s ∈ t•
s /∈• tM(s)
:TC
M
t→M ′ !
Kv NpNm l/v Y -, 8 ; 8 !/? % UE Y C 0 ;# E(. !
T X 1D9 . 4 .= 8 U 4-, 6I X 0 H . K !
nP #%O%7[ *a P #0/!"503#0J%'1 4R_&0H% 2[#41 %'.7%OP Q
N. !/Q U 7 Z 0 -1
ti 0 7 _-
Mi D-,
i%Mi
ti
Mi
ti→Mi+1 a 6-,
ω = t0, . . . Xhw J l/v r4M_v v H*sum_w 0
/1 !Q
ω % E K-, D6#4
M0
Mn
M0ω→Mn
!/Qω
% M0
ω→
!ε 2.8-, 7 ;% -,
M%
Mε
→M!
ω Xp M H*sutOvX
M
ω U 45 _-
M ′ -M
ω→M ′ ω
M
ω→
!
M ′ p H Hcvs?:rAp x;v M
U 7 Z.]-,7 ω
X-,M
ω→M ′ !j
M∗→M ′ !
* $ B
nP #%O%7[ *!* lKML 461&10/1 0HN,P/"B !%'(#1Q
N.=
ME -, !
M s6No/pOw_w v = !
Q N
.= M
6 -, !O XvPO J2M l(Hzs n=rω
! M
! Z.]- NU ;
M!(FD Z'cN'c D-
ω
Mω→
% ω
%M
ω→M ′
M ′ ; !
nP #%O%7[ * Q W 05TKY%7!4Z%'1&[ .=V DW H v m Nsurjv U = U(. 0 -,`-, 4 _-, 1 !/Q U 4E -, 4-, 0 U&. % . BrjnNr 'Hcv m NsurjpNrIH !
. W n=m H v Nvr=HDrjnNr 'Hcv m NsurjpNrIH U _-, 6-, U(.B - - -G 2.B 4C !
nP #%O%7[fQ % .646"B%'P Q N
.= M0
_-, !N
t8sutNpNr=H4M0
- N9 4I .!<FD Z'cN'c X-, -, _-
Mi 1 /
M0 -,
t%
U 4 2.8-, 7 ω
B-,Mi
ω→M ′
M ′ t!
? ;# ; .$. % X-, 2. = U 6 _- M
!-, .=
M!
0/KYK 0 d ) pt]sutOp Ms6H'J6v w H<oAxyl&w2nNm Hcv l/vBx pw vr Mv s6No/pOw_w v w FJANnNr/w2H*m2p Hzs n=r T .= =!<; ; 0 !(T 2. = f X
t2-@ ! S ;4 D9 4
p1%t1
;I .!
p1 t1 p2 t2
(=!<; 2. = $' f !
0/KYK 0 d E qr3mJ w v_pNl v vAH*m s:t8sutNpNr=Hvw Hn=m H v Nvr=HGrAn=r 'Hcvm N1surAp=rIH
:B:
FJANnNr/w2H*m2p Hzs n=r Q M0
_- !-N
4M0
t1, . . . , tn
1.&$6. !eW (. 2.% U 2.8-, 1
ω1 4 -,
M1 0-,
M0ω1→M1
M1
t1!j Z.]-,
ω′1 = ω1t1
-, e( M0
M ′
1
!W 0(. 3 2.% U X 2.8-,
ω2 Dh _-
M20-,
M ′1
ω2→M2
M2 t′!A Z.]-,
ω′2 = ω′
1ω2t2-, /( K
M0
M ′2
! & @ 2.8-, -
t3% /!!!8% /
tn% / =
t1% !!!T! U(. & -, " ! 2.X 0C !
0/KYK 0 dV pmJ2M s oAm2n l/vXr v w Ho$pNw4t]m2p=s v FJANnNr/w2H*m2p Hzs n=r T .=V DW j $.4 =! > < !&i A@ 8% H -,C -, B9
p1 t2
% G -, j9 p3
-, G ; !F e 8% . <C 8 (' !]i @ 8% 2.8-,
t1t2t3%
6I _-, ! Z.]-, % $ Z.]-, !-, 7 2.XDC !
p1
t1 t2
p2 t3 p3
[=! > .V 7W C (' !
$ G $ . @ ,5
+ V $ ; 2. 2. =(U "W f.2. % (( C 2 . `-,0-, K = = 5 f#8 6 !
T ;B 4 0 * -Zg =C ;\8"W
4)!$i !/T 0) T T?T=-+ 1 !
nP #%O%7[fQ) 0/"$0/ !T1 05 KY46_T%'"50H1I"B464Z"$PH%71_&% #0/1Q N = (S, T, F )
2. =E 4W !A 7 47.,/1. E 7.,/1. E !
Q M
_- !, . G 8N|S| -, X
9 1 e 70 _ * .,/1. /1 !G = M
]% U 2 KB(.; 8 H 5 -, E( 8 . !
Q t !N 2. < 8
Z|S| -, X 9 $.30 -1 $ 0 /. -1 e 0 _ * .,/1. ='6 P1 2-, !O? K6I f#% =
t 2.
!T N6p Hzm sQM_vFM_pNm2p MAH'Jm s`w H*s l/vB
N 1 0
Z|S|×|T | -, 82. ' D 0 z _ X * .&$6. /1 !H 6
ΓN f
Γ U !Q
ω Z.]-0 % tNvPM Hcvl(mRM_pNm2p MAH'Jm s`w H*s l/v 0 0W ;=1 8
N|T | -, / `- X YH E
ω / 0
* .,/1. -1 !e Fω
!
:$=
N,0HKYL#.70 ) a T 1 . `-,4( .50W. < =! 9 Z
t1 t2 t3 t4 t5p1 1 −1 0 0 0p2 −1 1 0 0 0p3 0 1 1 −1 0p4 0 −1 0 0 1p5 0 0 −1 1 −1
0/KYK 0 d(d n=s vr=H
Nl(r6mJwv_p=lv v H*m s
M, M ′ v l ORNpNml/p :vwv Htl(rAv3Hzm2p=r$w s6H*s nNr Hcvxuxfv l/v
Mt→M ′ r"p7x J l/p Hzs n=r
Np,H*m sQMs v xuxfvM + t = M ′
FJANnNr/w2H*m2p Hzs n=r | ` 2 E !
0/KYK 0 d 8 n=s vr=H
Nl(r5mPJ w v_pNl
M, M ′ vl O N6p=ml/p :vw4v Hω
l(rAvDw J l/v r4M_v!v Hzsum_w5H v xuxfv l/vM
ω→M ′ rhp1x J l/p Hzs n=r
Np,H*m sQMs v xuxfvM + ΓN × Fω = M ′
FJANnNr/w2H*m2p Hzs n=r W . Z ω!
Qω = t
7-, <%ΓN × Fω = t
z .8-, & 862. .] !
Qω = ω′t
(. Z.]-E 8% U Y _-, A
K-,M
ω′
→A A
t→M ′ !W ( . `-,%
Fω = Fω′ + Ft!:W &S $#E
2. %M + ΓN × Fω′ = A
A + ΓN × Ft = M ′ !e?
M + ΓN × Fω = M ′ !
:5?
f
T :. / D D ;# 2. z G * U(.H =6 z .; V D =!TG `(. D( Qf\'ZF W
6F !HQ !HT %j> !j? !H 0+ ! !jL ('c+ = ) T<,L+ c 9 +: 9 / `(.
!
" & $ .X 5
T Q;\'ZF W 0zQ/FW 1".2/ 2 ZS 2# V . . 2 E( ./. 2 .8( a 2 % (( 2.EH 6 C 2.]( !G+ *% .2/ -G V EC =f 0 . 2./1% G-, = DH I % E(2S 2#8 4. . !
W d/% jC 6 ` % = Z'cN'c e "0C
K % &$6. !i 2 % % GQf\'ZF M
-, / `- 6 =7 1C 7 =7 1C
f.!(b S ;! Q B = 2. "-, 5C ;.70 1 U 8 " # (x)→
&(x)
1 !/? % 0#,/# ;-,2 1 −→
&1
-, 7-, ;# = 4 & . 4 ;# =7 ! Q @ = 2. !(. G KC ;. 0 6 U
2# (x+1)→&(x)
1 !(? 8% /;-, 0(. 2 ,/# % Y`-, 1 ↓
−→&1
/ `-, -, 8;# & f B-, 8;# = 7 ! ? = 0 = -, ,G 6V (x) →
&(x + 1)
V-(. / f a( .2 1H 6 " (x, y) →
&()(x, y))
% = ,/# 2 = " & 2 !: h`-, 1
&1
0 D 8(U f#8 % AS = ,e# GH . .K 7 ;-,D(. * j ) 1 !F `(. U 8 ZS 2#8 . . 2 0 B 8(U
7. 2. ; /.
&$6. .21 (x,0) → x
(0,S(y)) → 1 : (S(y), y) (S(x),S(y)) → 2 : (S(y), x)
8 _-,-, ZS 2# YW% VTW !GT (U Qf\'ZF M & 2S # $. 2 2?! 9 !
i 2 % /C = -G :H !=W !d/% ;-, ' <C ;. % @ 8 $ !]+ *%N * U f'c Z %
112. -G H f.
-, $ . 2.R0
2# 1% f#
01 (U ;# 2# 0 . .
:$A
appel 1 appel 2
f
f f
f
f
f
f
f2
11
2 2
1 1
2
,? ! 9 T K (UEQf\'ZF BM (2S 2#8 0.2/f.!
I ;-,$.0= 11 !/F e 8%`(.
-G 0H K;-,8 2 C 2 0 K DC !
? * U8 'c 28% ,G ! C 26. 1
2 ; - _
0 /-, < S 1-, 0C 41 !$i 4I . 2. * U 1 + 2ω
!W -G XH % 8 B Z
2$. -,e#
.: Qf\'ZF M !,F GI !,Q U A-,z% ;8% K 2.7 4(. 0Rs v7 ↓
−→1 ,G0H I U(. 2.
1 !Ai @ 8% U(. 0 ! 2. (. 2 4 % 7-, ; f 0 z _ D. VC e(./1 !
+ z% @ - -G BH ! U B K . 1(. % 5$.8( B- 1 =9 !
W * U 2. .] 8% -, -G 5C 12∗
0 BE 1
; 8
21 Gj- B 2
2 = GK$. !, 8 `(. (U
Q ,G G ! 2.D
1/-, U D .0 ↓
−→Z
2!
Q % ASB (e- ,G ;-,K 2
;6 $U -, K-, (U5 2.X ↓
−→!
? - -G YH V 2.3E$. % / X -G D .8( 7(2S 2#8 D . . 2!,? 2S # 2 N9 !
b \XC 2 B = 6L 0 = ZD P1D-, -G0 0(. 0 -G 6H ;. !Ab = f 1 1L -, ,G (. ,G .7 ↓
−→0
G ;(. #! C> -1 !,g / 8(U : KL 6I DC ;S !
F !/Q !/T ! = .4-, 86. ( A@ #$" !,' ;# !T . " 6. & V 2 h-, V(.
H =/ 2 51 e !(+ 2S 2#8 4 2.8. 2 2 2(> KXC 4. XQ;\'ZF 0W
(x) → 1 :&(S(x))&
(0) → 0&(S(0)) → 0&
(S(S(x))) → 2 : (x)
i A@ ]%= 8(U5Q;\ 'RF BM 1
&1
!
&1
↓−→ 1
7 $U;# !T ,G 2. H
1 0H
2 !A -G
= ! * 1
=7 !
W 8% ZS 2# 2 N9 8 x
= 0 1
$ f &02%
A@ Bf ,Y 1!
i % G6. ( # ! GC ;X ;-,$. !]+ z% ( ` 0 8% U %A ; D $ 2? .
1 C = K
=Bc
U 2 ;. 2. " 0z- V -, V 8(U 9 C (. ,G 3 $ 3-, =6#h YC d 2. ;# h C > ('RQf\7b 1 !/ .2. =9 . = ;K 1 ( 2.% 2. ;-, e(!(b 8\C 4 -< 1 $ U 80 = UV 2S # 7 2.8. 2 f `-,$./1 !
(J%'.
) →J%'.
(F [#1
(x,.)) →
" (x,
"(.)) "
(x,( 4Z%O
(.,. ′)) →
(
(.), x,
(. ′))
" (x,
J%'.) →
( 46%'(J%'.
,J%'.
) "(x,F [#1
(y,.)) →
" x > y ) &(y,
"(x,.)) (y,
"(x,.))
(x,
( 4Z%O(.,. ′)) →
( 46%'(F [#1
(x,.),. ′) (x,
( 4Z%O(.,. ′)) →
( 46%'(.,F [#1
(x,. ′))
(J%'.
, y,. ′) →
F [#1(y,. ′)
(F (#1
(x,.), y,
. ′) →F [#1
(x,
(., y,. ′))
? D6I % U 6 * $ D i! ; F) i!_c5c,+ %, 8 2 ;. . <% X 5H 8 U -, 2 (. 2. BQ;\ 'RF 4W !
& (x,0) → x& (x, y) →
" x < y ) & (y − x, x) & (x− y, y)
x− 0 → xS(x)− S(y) → x− y
& $ .X <@
T ;# $ Q;\ 'RF XW 4 6-, = K C d 2# !:T 2 V _ 1 5C d ; 2 ; "-, !<W C % ;X 8 X X B .=$U hW !F -, = 2. ." % h X7 9 0C 8 K 2. =(U" 7W 2 K Z C> 2. -, != f 7(.Z9Z4. 2.C % A9 N9 A :- 2 / % `(. 8\ `-E Y =(U 2 U .; 7 -, = 8% U 8 %$ e( 0 ![ %Nb !,M8 ^ !P 3) [M4P cH; +*% ! $. / D Kg!+ !M) +M cBc + Z 0 U $# 6 6!AW !<Q 4- .2$ 1 f.8 `-, `-, 8 NS. ) Q _c5;-+61 !
T U3 0 4 ; KX-1 ZS 2#8 4 . . 6 (9 BH . / (< YC 0-, 8 D.;. 0 z U(. H =1 0 = ff -, fDH 2S # Y 2.8. 2 0 2.]( =1 I E $U &$. 2. =(UY W !j? ;8% X ((. V(.R9Z". .X 2.% U !Ai 2\ 7) i ZAB@-+ !ji = H .$. h 4> :? ! KE ;. 0 2.R9 6F 4z . .
2002 . / BQ;\'ZF 0W =$UC
; .2 -,w v_vANw5Hcn Npv H /v s8v $p=r :,v1p=rAp=x Ow s`wvpNw s vmRHcnl(rv m_w2HcpNr!
2. `(. : : `-,H fS !=T f# ! $.# 8%@-,V . 2S # 0H /.8-, 4 ./% \1(. !Ae 2. C .. 5 . NS z ;-, !
= 9
lbl
inc x
lbl′
x
lbl
dec x
lbl′
x
lbl
jmp lbl′
lbl′
lbl
x = 0 x 6= 0
lblz lblnz
0
call 2
1
call 2
end
2
x = 0x 6= 0
3
jmp 4
4
return
return2
stack0
stack1
ret0
ret1
,?!<; Q/FWG> !
1 130 $ !$ & 3% I$ 3%T Q;\'ZF W />
0*Q/FWG>1!H G f U(. :
1 ;# ('c(. !HTG `(. 7- -, 3( Q/FWG> 2. eY * U(.H 5 1 7 !$T BQ/FWG> -,= *%( - h-,;-, 2;. . 80 8% -,Z' -, 1 !$b <S 1 -, C@A. B K (U - !T X fa z U(. H " % < S 2 "-, 89 / =(U !jT X-, 9 / = K 4 * U&. !eF 9 K ; = = 8 _ 0 Rb 5W
1
_ !D/ jS Z' 8%BQ/FWG> 09 E R' E
"-,X % a !@De R' % 2$.2# . 1 e D-0 z / . * < R' !
T I? ! ;1 4Q/FWG> 0 C@A. 2 4 X(. (C 1( Q/FW>H . !T / = _F 0[ N
% / =(U
1 6 V6. 0 z ; 2. 1 !H+# 6. 7 ?@j. &% "C -, !/T .=5 7W e 2 H # =6 Z
0 :
21 :
2
end2 : x 43 : 44 :
W 2 % ,G ; C 6I 0 6 U 2 0 f' 221 % 2 f G?@j. f. (U$-,
2$.!<T 7 .(U 6I8 -, = 8% 2 / 6 7 !
nP #%O%7[fQ E Q & 57 4 !e 5(. 0C = -, = ;16.] 8 # =6 !/FD 2' =' D-, 1 2
0 :"
0; . . . ;
n :"
n
i < n(
(
i) =
i+1!
=5;
nP #%O%7[fQ V F ( Q & 57 7 8 !$T & .=E 4W
N = (S, T, F )<-,
b U B -, z%A x
-, =%e -, R' = $ R' 0 f 8-, : .
21 stack
-, 0 = $. 1 ! b U " - ?@j. % 8(U
x = 0
x 6= 0 -, 2 x
ret
-, 0 2$. 1 !
Q " (
) ="
x
("
x) ∈ • !
("
x) ∈•x!
("
x) ∈•
(
)!
Q " (
) =
x
(
x) ∈ • !
(
x) ∈ x• ! (
x) ∈•
(
)!
Q " (
) = ′ ( ′) ∈ • ! ( ′) ∈• ′ ! Q "
(
) = x ′
(x = 0), (x 6= 0) ∈ • !
(x 6= 0) ∈•
(
) (x = 0) ∈• ′
Q " (
) =
(
) ∈ • !
(
) ∈•
′ ′ = R' 4-, !
Q " (
) = ′
( ′) ∈ • !
( ′) ∈• ′ ! ( ′) ∈• stack
! (ret ) ∈•
(
) (ret ) ∈ stack(
)• (ret ) ∈ •
′
nP #%O%7[fQ d ( .'4Z"B0/1 ZT#%'#46.'0/1Q & 6 8
N 6Q/FW> !T oAx;p M_v wDn=m NsurjpNxfvwK
N
-, / =$U = & !T / ' !nNrIH*m2n=x !x;n v Hzm sXv H ." 0W 2.0 f'
(hQ/FWG> !g A9 @ ;. v H n=rVnNm NsurAp=x !Q S D-, 8]9 z 8X
_ 8% 9 ;. M_n,NKo4Hcvl(mXn=m NsurjpNx !T -, fC _
X( : 8X 8%H-, / `-,-, Z'
"I U(. .!i @ 8% / U (U / "( 1 I /. !(? /. 8% XF[@WG> 2. 2. ! 0/KYK 0 d% %'.646%746vBrjn,Nm2vv v H n=r$w p=r$w7xfvwojxfp M_vwnNm =surAp=xfv wvw H M_nNr/w H p=rIHGx;nNm_w l(rjvvPOJPM l(H*s nNr FJANnNr/w2H*m2p Hzs n=r F - U 8 2. !
F f.2. -, Q&' V$ 2. = W !TG .I V KH e /. 0* ; 2.7H E -, 4 / !F ZT(.'.746%OT0 d a F [#!%'%7[ !P/"50H1&1&46%OT0J 46"5"B0/11&%/1 %'.7%O&P n=s vr=H
l(rAv3Np M &surjv M_n,NKo4Hcv l(m_w
Nwn=r
Mv H
M ′ v l O N6p=ml/p :vw vN s M ′ vw H@p MPM_vw_w_s xfv
vcoAl(s`wM
p=xfnNm_wM
vAHM ′ N6v H Hcv rIHp=l(H p=rIH v v HcnNr$w pNr$wBx;v woAx;p M_v wXn=m NsurjpNxfvw
=B>
0
1
2
3
end
x = 0 x 6= 0
dec x
inc y
jmp 0
x
y
call 1
end 1
0
ret0
x = 0x 6= 0
2
dec x
3
call 1
4
inc y
5
return
return1
ret3
xy
stack0
stack3
? ! > Q/FWG> $U * $ 0 % . /1 !
N,0HKYL#.70 ) * 9 !+I##%O%7[ !
T 8(U =6 Z * $ 5 0 (U !,T 8 0 -1 : f . !C 0 2' =' # YC
(S(x), y) →
(x,S(y))1 !:T / " f
. 0 (S(x), y)→ S(
(x, y))
1 !AT 6 8(U f#8 .ZH
. K f#8 < . -,
` !!T K 0end
1 . .h(.. E 2.2. 6E R' !i .
x, y!
Q y!
0 : x end 0 :
11 :
x end
2 :"
y 1 : x 53 : 0 2 :
x
end 3 :
14 :
" y
5 :
T QeFWG> (U =6 2. .2 ? ! >!
;! F [L %'0/1! < 4 (UY %
copy
dup!jT 8;# %
copy% &hH
= " = 6 8;# =V0!T 3 (U ;# %
dup( `-, &
=$C
0
y = 0 y 6= 0
end 1
dec y
2
inc x
3jmp 0
x
y
y 6= 01dec y2
3inc x
inc y′ 4 jmp 00
y = 0
5y′ 6= 06dec y′
7inc y 8 jmp 5
y′ = 0
end
y′
x
y
lbl
x := copy(y)
lbl′
y
x
lbl
x := dup(y)
lbl′x
y
,?! C Q/FWG> K copy
dup
e 2. !
H 4 & ;# = !
x := copy(y) x := dup(y)0 : y end 0 : y 51 :
y 1 :
y
2 :"
x 2 :"
x3 : 0 3 :
" y′
end 4 : 05 : y′ end6 :
y′
7 :"
y8 : 5
end
T :Q/FW> ! : 8(UX . 2. Z ?! C/% $.@H # / ('Z.
- 2. 2 . ! W QeFWG>
dup% < 8(U Z ,G
. 2. ;$.. 6 D E 2. = f 0 =6 = =E @ 2 =21 !
>!( .' #1 T46# F [K Kh # n%/.,%710/ ! !T =6 2 8 (U Z z /H i! ` !W $ ; <%
a < b ;$..6 C $ !Ob :C !-,
2. ?> .X z ` BH R' !$T 6 2. I . * 7 2 !
Qa ≥ b
& (a, b) =
& (a− b, b)
!/Q<% & (a, b) =
& (b− a, a)
!i .
a, b!
Q a!
=5:
0
b = 0end
b 6= 0
1
a < b a ≥ b
9 2
a = 0 a 6= 0
dec a
dec b
inc tmp
14
b 6= 0
dec b
inc a
b = 0
15
t 6= 0
dec tmp
inc bt = 0
jmp 0
b 6= 0
dec b
dec a
inc tmp
b = 0
7
t 6= 0
dec tmp
inc b
t = 0
jmp 0
a
b
tmp
,?!<: Q/FW> ; /.B * $ 4H iG ;!
0 : b end1 : if a < b 92 : b 73 :
b
4 :
a5 :
" tmp
6 : 27 : b := copy(tmp)8 : 09 : a 14
10 :
a11 :
b
12 :"
tmp13 : 914 : a := copy(b)15 : b := copy(tmp)16 : 0
end
T Q/FW> =6 D 2. I? ! :!eW @ f .% .2.-,;-, 8 ; /. `-, 0 BB"6I f 21 !F ; = Z.]-, Z H(2. = !
]% Y(. D7C E-, 2 0 ZC . E _-, ( Q/FW> !+ *% * U(. H =6 %-, e E Z VH . 8% / VE2 B 4 _- %( 41 U(. 5(hQ/FW> !
=B=
nP #%O%7[fQ8 4R&1 05 KY46 #4 2(0/1Q & =6 ! .
s = (
, σ,
) & (. 5 _-, !
(
, σ,
)(
Q/FWG> !
!(
, σ,
)(
) = 1!(
, σ,
)( ′) = 0
6= ′ !
!(
, σ,
)(x) = σ(x)!(
, σ,
)(stack ′) = # ( ′)(
)
# (
) 4 1 !
i XC 8% !(
, σ,
) K9 1 = (-, /7 U(.
0 8 1% 9 stack
-, AS BH Z' G $ -, $. = 9 - =7-, =!
1 1 *=!$ & # L$ "I $T KQeFWG> H K f D ;# (' $. 2 B0 $- = / f'
. fU('c(. D 8(U G 21 !b G ;1 K 1 ;# 6 D9 4 =
BH (.8(
f.2. 2 1 !
nP #%O%7[fQ % W 4Z"B0 PH&0H## !0 Q & 3 X 8 %
sh. 0 &
ω X( =6 B 0 * .
s!/T H*m2p M_v JAHcvrNl/vX
s%2.
ϕω% 2 7
ω 1 f# 2
Qω
= 1-4ret
ϕ(ω) = ω
! Q
ω = ω1
ω2
ω3
="
( ′) = K -B
ret ′′ω2
ϕ(ω) = ϕ(ω1
ω2
ret ′ ω3)
!
W 3#P/ZT-/KY0 d!* * 4RT%'( 0H *+/^[0/2003
*+* (V n=s vr=H
l(rAv N6p M(surAv FMn No H v l(m_wDv H
Nw n=r nNs v rIH s l(r9JAHcp,H2Nl7oAm2nP:=mp,NFN6v4v H
Ml(r NpNml/p :v
vN
Hcvx l/vM D (s) s ω
vw2H:l(rjv5Hzmp M_vNlXoAm2nP:m2p,NFN6v Ko$p=m Hzsumv4x JAHcp,Hs
pNxfn=m_wϕ(ω)
vw H:l(rAvDw J l/v r4M_vv.H*sum_w4o$p=m Hzsum v
M sω
vw H !res vv H l/vs
ω→s′
p=xfnNm_w7suxvOs`w H vBl(r7NpNm l/p :,vM ′ v N
Hcvx l/vM
ϕ(ω)→ M ′ vAH
M ′ D (s′) FJANnNr/w2H*m2p Hzs n=r W 5$EQ/FW> /$ 2
ω!
L <8% U Z.]-,D D-, / K = U(.h( ' =6 !$FD (< =C -, ;X" B 4 8 2. =" 0W C 4 ;# $' (.2 !
F ZT(.'.746%OT0 d Q #* 4RT%'( 0H *+/^[0/2003
*+* (V sDxfvojm2n:m2p N NvErAv Hcv m Nsurjvo$pNww l(m HcnNl(Hcv w"wvw5v rIH*mJ_v w 7pNxfn=m_wVsuxvOs`w HcvEl(rAv"w J l/v r4M_v v*Hzsum_w/=l sur Gr/s v n=s vr=H
l(rAv.N6p M(surAv M_n No H v l(m_wvAH
Nw nNr s N
Hcv m Nsurjv VrAvFM_n,NKo$n=m H v6pNlIM l(rAv1vPO J2M l(Hzs n=rsur Gr/s v pNxfn=m_w
H v m Nsurjv7w l(m HcnNl(Hcv w0w v wXv rIH*mJ_v w T 2 V$YQeFWG> ; `-, 5( =6 !
?D. ` $ 2. =6 W G ! ! $ 2S # H $.]- $K ;
R!
i 4C ]%=C = ,> 2. Z ! i A@ ]% Z.]-,H 0C = `-, B $4 4 _-, D- e 4V0 _- =(U 0 * U (
dup 'c 21 !Nb (C . / 2 6( 2. =Xo$n=l(ml(r5v r$w v N xfv vNpNm l/p :,vwKsur/s6H*s pNl O
0 V 9 0
E X h-, /-, 9 5 H . P1 !
=H?
? 16I8% ('c(. Z 6 f =6# ZC 4 Z.]-4 7- / D U&. 3( =6 !/i 8% = 0 A@ 8% 6 X =6 /(. `=B -, K 2. E(. `=% S $. f' K =6 -, -, B(. -1 ! <d 8%$d & ;!Q 0 U z $ X i! ; -1 % Z.]-0 / K 2
x = 0 X
x 6= 0% -, X < ; ; V
1 x
< & /. !AF I U ;!
N,0HKYL#.70 )RQ 9 !T 3Q/FWG> * $ 0 . 2 /1% D-, . . h?! > 2
=9 !i @ 8% K U 0 . 0- 0, 1, 2, 3 I 2.B 2. 7C
x 8 D-< X
9 !/? =6 N9 !? 06I %4Q/FW> * $ % . %. .# ?! > N9 !i @ 8% = S
x0z
2
31 7I .
." C 0 S E X 79 X x1 !@T 8 S
;B-, , 4, 5
! 8%f;#
ret3%/C H9
stack3% K-, f ;-,0H
= 2. 1
K = x!/? 1 2 =9 !
;!T VQ/FW> % 2. 2. n?! C2 N9 !<i A@ 8% Y V-,
0, 1, 2, 3 8 I .; ;
y!!? =6
=9 !T Q/FW>
dup% . 2. I?! C N9 !/i A@ ]% 9
0, 5, y, y′ 8 8 ! B D(
Y( * . !F / 8% U(. !$. 5 (UE9 11 _ ! ]%@ z . ( 2. =% S 3-, 9 a 0
01 !:F 6 X B9 X
_ .% U(. /"= 5. <(ZS 2# !$? BQ/FWG> 4$ . *%/ 1 2 =9 !
>!T 3Q/FWG> z /YH i! ` % . 2.h h? ! : !!i A@ ]% U U(. D $9 _ H-,
0, 1, 2, 7!/F / ]%
U(. 2 X b 6= 0
b = 0
B (
b K /. !/Q =(U U(. /. %
Q/FWG> H U(. %/ 1 =9 !
Q U8 % -, ;# 5 U= 0 f /. 2 8 21 T 2 V(.V 7W K-, ! T 2 $ . z% ` .YH Z.]-,Y R'cN' 6
e 2 _ ! T 2 =(U Z.]- $. Z' =' !
W 3#P/ZT-/KY038 n=s vr=H
l(rjv N6p M(surAv M_n,NKo4Hcvl(m_wBv H
Nw nNr s N
r p N6v Hp=lIMl(rjvXvPO J2M l(Hzs n=rYsur Gr/s v.M_n LJ m2vr=H vt]s`wct8s`wv w7tNpNm s p xfvw1v H l(s:r4J2M_v w_w s6Hcv7l(rYw v l(x vAHcnNr nNm =surAp=x pNx;nNm_w
Hcvm N1surAv7w l(m5HcnNl(HcvwDw vwBv rIH*mJ_vw
W 3#P/ZT-/KY038 ) n=s vr=H
l(rjv N6p M(surAv M_n,NKo4Hcv l(m_w7v H
Nw nNr pNtNnNsum4w s:suxvPOs`w2Hcv7l(rjvBvPO J2M l(Hzs n=rhsur Gr/s vv
N l(s
r4J2M_vww s6Hcv4l(rEw vl(x v H n=r p=r$w4xfvwGojxfp M_vw7n=m NsurjpNxfvwXvAH l(s@w nNs6H M_n LJ m2vr=H v7t8s`w ct]s`w vw0tNpNm s p x;v wXvw2H J2Ms p xfvv H p=r$w,
pm2v t]s vr=H wp=tOn=sum7w s:l(rYw Ow H N6v sur J l/p Hzs n=r$w7xysur4J_p=sum2v wp ,NvAHl(rAv7w nNxyl(H*s nNr J_v wo/nNl(m7xfp4oAm2v l(tOv T ;# 4 /.# $.V !&T 2 2. 4 V B 2S 2#8 \XC ` 8 z U & 7 . 6. ( !
=B@
T ;$. 2 T f D$ .=KW KC 2# f G 2.8-, D
- G0 % 2.8-, 0 . (. `-, 2. `-! T! 2.X(19 2.8-, B $. 8 B0 = f#8 B(
-, H ; 1 ! 8 ]% ; 8 2.8-, 8 `- ; K Z 4 ;# ! 6 2. = U -, . 8 S U !$F 0C #,K !
T $. B Z.]-X 8 D X (UY ;# !eT ;# 0H AI x = 0
x 6= 0
I 2. " x
x
.! Y (U ;# E(. fXQ/FWG> 7 ;# 4 4-,B (U (C 2 $. Z76I $ 4I 2. 0 ( /. 8B =!T 8 f# ; 8 U; C 8 @C 2 H /.8-, /. !=T 7 8(U f#8 8 6(.C V 7W !$ R) 6A5c-+ K$. ;. f' !
$ , $ B B < $
? * $ KH i! ` % (' 2 H U(. 7-,
b 6= 0
b = 0 2 % G(
b!,/ f.D D 8(U f
D.f !/T ;# X =9 ZS 2# BH $.8-,D B0 2' =' 4 /.8-, 21G-, AI U&. VH I ; 5( 2S # !$F 8E2. ;-, 0 (U 8 ?@j. E6I 8 P1 2S 2#8 !(T! = f D 2. ;-,4 2 "$ . !
2. =% -, e Xh = f." 8X3 C 2
x = 0
x 6= 0!T ;. =(U "
;# 2# ; / $.
x = 0 4 "
x
x = 0
2$. x 6= 0
x
x 6= 0
!AT -,: ,G B 0 K C@A. 0$;-,$.4 1 ;# GI (C /( !(T -, 2. 0 /. 50 = Z'cN'c C
x = 0
x 6= 01 2. . 0 - 7( .=E-,
$.-1 !&T 0
x = 0
x
-, =D 2.!W U %& G z /DH iG ;%
1
tmp = 00 /. .D
01
b 6= 0
0 # 1 !,? 6I 2! :( %
b = 0 ,G
;2•b6=0,tmp6=0
0b = 0
1 f * 0b 6= 0
1 !(b < S B-, 0
b 6= 01 / ! PK # 2.
b%
b 6= 0 .90z 66I tmp = 0
.2# 2. " tmp
1 ! 6 6 = =
2%& 0C = !b 8 0H %( C@A.
* %$ -, $U I . ! G( 2. = & 0 /<
a ≥ b1 $. Z ? ! =!
T / = .2. ; .!Q& = .% 2 5-, 2.8-, 5 2. .] f
;!=i -, 4 K-, 2. 2. fGI /(. ; ;
a!Q .= 8% -, 2.8-, !!? =6
2 N9 !T! . G 66. ( B.` ( .= X
= 0y9R2-, VI 1X f. DhC 83k
k 0 V /1 !
2 7 5 ; 2S # !i % . 0 1;$.6 6.( ;
7
2 6 OS Q/FW> *% $ Gd N9 6 !< 8 =9 ;B 4-, f (( !$i @ 8%2# "
x K-,
x 6= 0 !
8 6I8 ; !W _- f# x = 02# 7 "
xDE -G " . "
x 6EC X
=BA
0b = 0end
b 6= 0
1
Cas a < ba ≥ b
2b 6= 0
dec b
dec a
inc tmp2′
b 6= 0
dec b
dec a
inc tmp b = 0
7 tmp 6= 0
dec tmp
inc b7′tmp 6= 0
dec tmp
inc bt = 0
jmp 0
,?! = 7(3F[@WG> * $ 4H iG ;!
x = 1
/ !i & -, 8 B ,!<? 6I % # "
x 8 N9
x > 0%$
x > 1% !!!
, 8 %= ;# N9 8 2-,H K( 2. = KW ; /. A6I /. % z ;S K6I 2 F B-, = Y ;# = !
? ]% ( F[@WG> 5H . ; , !/+ *% K(. K-, 2
0 :"
x1 : x
2 :
x
C & 0 XF[@WG> 0 8 = 1
2 " X 2 * V z = /1 !@i 1( Q/FWG> 0 ; /./1 . . K 6 /]%; f e. ; > 0 , P1 !
F 2 <% ; . 2 % 16I f#8E $' RC 2 % 2# f 3-,hC VW ! a(.C ) 6A5c,+ ! -, A@ 8 fH B .2 ; 0 2' =' H . B 6 7 $. 26.] 21 d KF[@WG> !
?Tc
3
e G : (. 1 2> $'cQ;\b 0 >0Q&b1 % 2' =' -,
!(? 7 f#(C % 0-, = 46. 2f.8 $. C !=T! / % (( ;. =6 >DQ&b !=+K1$. $ !( *% X-, .X X6.8 %e /. X .0 ;K /1 $. = (z% =6 < 9 fK 41.8 !
T 1 !>DQ&b . 2. (( !^ =C 7) ^ =C'A5A(%^ =C'c5;-+j-, 7 .K-,K ' =6 :I . * &% = Z'cN'c : 6. 2;. % U . V 16. 0z 8 21 6 !
nP #%O%7[fQ a Q & 1 ! & G>0Q&b 0z> $'cQ;\b 1 U
c -,
. v1, . . . , vm
% & ` 0, i1 ← v1, . . . , im ← vm, []
∗→
, σ, % ∑
x |σ(x)| ≤∑m
i=1 |vi|+ c!
" !,V@ <
T YJW> 0 J EW K> 216 h Y f = Q/FWG> V2;. Y-, . 1.8 !!T JWG> E . 2. 2 J@! 9 6 h$. (C !!T X E 69
mem 2.
; 66 8 0 f 0V1.8 !D -C X-< U&. .% X59 6.8 % -,1C B-, 2 " U(.2.% f#X9 3 6. !jT
x 6= 0 2. . !
7 $. 1H 0"9 x0 = Z'cN'c .
x 6= 01
/ % &" F0z 8x1
( e.!&F D ;# / H . ! :( 2. =6 KW Z! ,% = Z'cN'c =5 Y( =6 !
nP #%O%7[fQ * & ( Q & hB
N = (S, T, F ) 3Q/FWG> !AT XJWG>
N ′ = (S′, T ′, F ′) &
(. S′ = S
⋃
mem!
T ′ = T!
F ⊂ F ′
i =9 7F ′ /.
mem ∈• ("
x) "
x & !
mem ∈ (
x)•
x & !
x ∈• (x 6= 0)
x ∈ (x 6= 0)•
x 6= 0
N ′ !
g -, X 6. 0Rs v mem
-,S 21 . 2.E 2 $. !
? 9
lbl
inc x
lbl′
x mem
lbl
dec x
lbl′
x mem
lbl
jmp lbl′
lbl′
lbl
x = 0 x 6= 0
lblz lblnz
x
@! 9 J 0W K !
N,0HKYL#.70 E T 7JW> * $ 0 . 2 /1 /.4 @!<; !
0/KYK 0`8 E %'.646%746 n=s vr=H
l(rAvNp M &surjv M_n,NKo4Hcv l(m_w vAH
Nw n=r v rAn N m2v-v v H n=r$wM_nNr=H v r/l pNr/w3xfvwoAxfp M_v w
tNpNm s p xfvw x"v H"N JANnNsum2v
memvw H M_nNr/w H p=rIHGx;nNm_w v.H n=l(H vvOJ2M l(Hzs n=r v
N FJANnNr/w2H*m2p Hzs n=r F - U 8 76I X 7 . !
i @ ]% - = 2.
x
-, x
. mem
! "
x-,
x
mem .!
x 6= 0-,
x . !
Kv NpNm l/vi C ]%Q/FWG> # G 8(U RS -, Z.. !? ! =8% . 2. @9 8% KH = 8% . . G9 44
stack !<F 4 8(URS 0 / 8 =
0 41 K 0 _L1 1 " Z!$b z% 71.8 PK-, =8% = Z'cN'c E1.8 6 2. !<b #4C C "1I8 0 5
mem 2.
ret 1 !e PK
* h B6.8 Z % Z !AF ;# K X / . fC 6 ` D-, B "H " $.F0 $ `-, 7 B 21 E 76. 2;. 2 !
(. X B 4 -, =(U =(U -, / = (.8$ * U&. 3H 6!/ _- _ @
0%j !9
K 2. 4mem
!A h _-, 8 @ end K@9 K _ !
nP #%O%7[fQ(Q l*+4R #4 2(0H1 %7!%'%746 N 0H !4Z N 3 -,
M sures6Hzs p=x U B 2$ 2
σi X-,
σi(x) = 0 - = X-,
4H .M(p) =
!( "
, σi, [])(p) -,
p ?@j. B
mem!
_-, M
!rjpNx U E2$ 2 σf
Y-M(p) =
!(end, σf , [])(p)
-
p ?@j. B
mem!
?$;
0
1
2
3
end
x = 0 x 6= 0
dec x
inc y
jmp 0
x
y
mem
(@! ; JWG> * $ !
0/KYK 0`8 V qr N6p=ml/p :v !rjpNxvw HGl(rjv7s6NKo$pNww v FJANnNr/w2H*m2p Hzs n=r F - 1 _ 0?@j. B
end2 H 2.!
G V !.7 $ B 5G $ .nP #%O%7[ ) Q & 3
N 3JWG> !
N Xw pNr/w1pNxux;n M_p,H*s nNr v N JANnNsumv U X
c
X-, -,7 -, Mi
- c9 D
mem0Mi(mem) = c
1 -, 2.8-, K : =U; B0 821
ω
Miω→Mf
Mf
_- !W 3#P/ZT-/KY038 d n=s vr=H
l(rjvRNp M &surjv M_n,NKo4Hcvl(m_wXv H
Nw nNr
v w H 0w sN
vw H!w pNr/wXpNxuxfn/M_p,H*s nNr v.N J N6n=sum2v FJANnNr/w2H*m2p Hzs n=r Q& D-, & 4>DQ&b !Q
c 7- e-
σi 2$ 2
H . 0 = h V-σi(x) = 0
x
B 2./1 !Q s = (
, σ,
)
s′ = ( ′, σ′,
′) 8(U". 6 7 -, (0, σi, [])
ω→s
s( x)→ s′ ∑
x σ′(x) >∑
x σ(x) =∑
x σi(x) + c!
? . ω
U & < >DQ&b !j ω′ = ω(
" x)
!W 2 %
ϕ(ω′) D U&. h( Q/FW> & -, . < ; -,@ 2'
-, M D
!(0, σi, [])
M =
!(0, σi, [])
!Q
NJWG> &
M ′0
-, 6 N
-M ′
0(p) =!(0, σi, [])
p C@A. B
mem!M ′
0
E _-, < N
!Q
ϕ(ω′) 4 2.8-, D
N .
M ′0
% U 13 _-, M ′ :-,
M ′0
ω′
→M ′ !W %
M ′ D!(s′)
∑
x
M ′(x) =∑
x
σ′(x) >∑
x
σi(x) + c
?T>
0
x = 0 x 6= 0
5 1
dec x
2
y = 0y 6= 0
end3
dec y
inc x
x
y
mem
@! > T 7JWG> H 1 0>DQ&b C . V 76. !
W E(hQ,' 2 = 16. %
∑
x
M ′(x) + M ′(mem) =∑
x
M ′0(x) + M ′
0(mem)
i 2 E M ′
0
% ∑
x M ′0(x) =
∑
x σi(x)!
8 K E(.8( M ′
0(mem) > c!
+ z% & < >0Q&b %/-,H-, c 8 2 D2.8-, 4 -, $. f
c9 X
mem XE . !<? JWG> & <
6. !i 8% 7JWG> & E 41.8 % & >0Q&b !
N,0HKYL#.70 E ) T 5JWG> z $ $.V @! ; aV6. !@i @ 8% 9 Y
mem =9 . !(+ z%/-,-, X4 9
mem=5(. / 8%( U&. 5. N9 GV. 0 x
H f =9 .
x 6= 0 24-, S E 19 Y
x / 7 /1 !? * $ > $'cQ;\4b !
Kv NpNm l/v 8 2. (-, = -,d!<T =6 2 U >DQ(b YJW> 0 $. 2 @! > 1 E B6.8 B B
0
5!
2. x, y y
0 : x 51 :
x
2 : y end3 :
y
4 : 05 :
" x
6 : 0end
8 / 8 -,-, Y Y x = 0
9R # .h
?C
0
1
2
3
end
0 0
1
−1
0
(@! C M / =EJWG> z $ !
" x!Hi 2 2. = 1 = 2. .8 8%
(. G-, =6 >DQ&b !b =-, 2 .(U 8%= Z'cN'c 8 -,BU(. -, /. R'cN' %2. (-, !
W 3#P/ZT-/KY038[8 n=s vr=H
l(rAv Np M &surjv M_n,NKo4Hcv l(m_w
Sw nNr vAH
Nw nNr s
SH v m Nsurjv JtNv rIH*l/vxuxfv Nvr=H
x p=s v vw Mn=r=s6Hzs n=r$w l(resQMs6H'J/Nl M_n,NKo4Hcvl(mn=m NsurjpNxv H v M_n LJ m2vr M_v v w H*sum_w Dn=rEo$v l(H J2Ms vmw sN
vw Hw pNr/wBp=xuxfn/Mp Hzs n=r-vBN J N6n=sum2v6vr7Hcv NoAwo/nNx Nr ,N1s pNx2p=r$w7xfp H p=suxuxfv v
N nNr M pNr/w7xfp HcpNsuxuxfv v
FJANnNr/w2H*m2p Hzs n=r 2 2.G = (V, E)
N
!V
z 8X7 HS
u
v S / !(Q( -, .0
`-, = −1
/ " ,%1
0
! U(.V
N0 D 8&9 _ $. R'cN' =21 /
8VG
!(T XKG9 mem
* 2.K( ; 0 `D( 8 / = B9 D-, K. . N9 2.
mem1 !j+ z%
N
5 76.8 8 G
-, K 0
K ` Ze. 5. H
c- B(. eV-,7( !/F 7 ;# / V.]- = 4-, S
4 B ; $. CG
!W U 77 B ` /. fCR%$X1.( K CK2
1 7H O9 7 G% 7-, I 0C 8 ;S IK 0
O(n4)1 !
N,0HKYL#.70 E(E T / =6JWG> z $ G $. @! C!F 6 = S K ; $. fCR%KJWG> * $ G 6. %, z $ >DQ&b !
T! U H ; $. C = $.8-, * U DH ; = 2S # H /.8-,
X × IG ≤W
IG
2. 7 KH ;K G0 6 ]% =%
−1
* K(6 ]%1
S %0
=1 W
K$ ` !e 2
25.5 ) FTJ A$C + 0 7.8-, !
T! 1 ;# 2. XH I 4C 6 $.8-, $. !+ z% BC 8 z N9 =(U = /.8-, , 2. = ./ -, C !
,V ,5 , $ @
-,;-, U 8 H = =6 6-, fS = 31.( !Ge U `-,= 1 .e !
?$:
0
u = 0 u 6= 0
end 1
dec u
2
v′ := dup(v)
3
v′ = 0 v′ 6= 0
4
dec v′
5
inc m
u
m
v′
v
mem
(@!<: JWG> 1X !
'L#5 L T 6Z 1X Y 7 (U !i .
u, v!
Q m
!
0 : u end1 :
u
2 : v′ := v3 : v′ 04 :
v′
5 :"
m6 : 3
end
T JWG> =6 $. 2 @! :!]T .= K. . f /. 8X !
T . K dup
= ! ; z 2.( 8 _ *% S ;H U(. ; 7(hQ/FW> =6 N9 !
i % N9 : 0 dup
% JWG> 16.8 !Ni @ 8%-,e- 9 G
mem U 6 _-, -, 2(> 1 G
9 @u
v - . = G (-, e z U(.
dup0 -, 6
= 76. 7- v1 %e 45. K-, < !
?T=
I # *L( (U" . D 1 0- 4 4 !eF 6 < U
: 8 % (C =e $. (( ( : ;-,8 !=F / 86D 1 (C !FD _-,89 2. C .../ : !F 6 .f.8 0 D h 0(h 2 /1% f#8 4 K.1z ;S!HT 6Q/FW> D -, = G-, 8% 5 X1.8 DC B;-,8 2 %4-, 8 f 4 _ R0 0 ,G$.21 * !<^ C !:F B ;(.B-, 6I E1.8 !
W 4$. 8% C 4 (U 2 % ) %&9 0 PKf " !
(a, l)/B : N9 z I !
a =)
(l)f# 8.f.8 K 8 a!
? 6I % @ ( /. ;# DI =Z ! l 8 8
` !A[ 8 8%9 =9 7 if a < b jmp
- = 8
a f X-, B
b!$b C BH . 4B C
!Hb C =/ " 7 `(. 6 1 2 E( !
T =6 Z 2# ! B (.Z9Z ;.!] $8 $. .;. (5(. / 0DC * A@ $.!
i . a, l
!Q
l!
I0 : l I6I1 : b :=
) (l)
I2 : if a ≤ b I5I3 :
(b, acc)
I4 : I0I5 :
(b, l)
I6 :
(a, l)I7 : acc IendI8 : b :=
) (acc)
I9 :
(b, l)I10 : I7
Iend
T 1JWG> * 0 2. 2 = X 2@! =!eb 2 N9 ;# 7 0
I0, I1, I2, I31 8(U f#8 0
I7, I8, I91 I . ; `
lacc
!&b /$. K8 2 * 2.4( _ *% ='8 7 (U" = KU(. !!$b V 46. = $ Z'f9 E ; $. CR0 6 "1.8 -1D 1 >DQ&b 0 HC = 719 E
mem5(.8$ K * U(. =1 !
T 6 2 8 4 !<T Iend
z 0
( (C e(&0 = Z'cN'c - :
Iend =6 e !I 8 .
01 !,? D " @! =$% (U Z.. ! D ;# D G $U 2.(U2. .!
i . l′!
Q l!
0 : l′ end1 : a :=
) (l′)
2 : I0end
T 4JWG> N9 1 00, 1, I0, . . . , Iend
1 I .; ;
l′!eT 2. = 5 76. 4 =6 K>0Q&b !
?B?
I0
l = [] l 6= []
I1
b := hd(l)
I2
a > b a ≤ b
I3
cons(b, acc)
I5
cons(b, l)
I6
cons(a, l)
I7
acc 6= []
I8
b := hd(acc)
acc = []
I9
cons(b, l)
Iend
l
acc
mem 0
l′ = []l′ 6= []
1
a = hd(l′)
end
l′
@! = JWG> ( !
?T@
I L# !- ;-, 8 = 8 `-= ; B =% 8 8%
/ Z? . !
? X ;# . . fC 1 4 8% $. BH (( U a( !<T YC Y $ !<T V (U ;# f#8 $ X2 0 .f.8 6 N9 -1 !:T 1 - X D Hcp : 2 !& 4- 6 X 8 CD `(.. ; 5 6 & 0 1 1 !T YC &0 ! & 1 =9 .0 !;#/1@ B % 2' =' 2.8 .0 !(.2.8 -1 8 .;. 7( 8 H !HT C ) B f.20 $.*0 6 . ` /1 5C . Z B.f.8 0 ;G-, =9 /1 !T 7$*0 :S E $UhC ;BH 7 _ B 66I 21%H 6 2-, 87 8% & & = = A@ 2 ' = @ Z 9 =!
"B(#"B4RF 6 ed 4.1C -, 2#1 (U 0 =9 4 .f.8 B f =
=1 !i .
l1, a, l2Q l′
C0 : l1 C4C1 : b =
) (l1)
C2 :
(b, acc′)C3 : C0C4 :
(a, acc′)
C5 : l2 C9C6 : b =
) (l2)
C7 :
(b, acc′)C8 : C5C9 : acc′ Cend
C10 : b =)
(acc′)C11 :
(b, l′)
C12 : C9Cend
8 -, =6 N9 >DQ&b !
1L#.7%Oi 2 0DC 5- 2.D V 8(U "C " X
a%& .;.
; -,a
KH = .f.8 f /-,a!
i . a, lQ l1, l2
S0 : l SendS1 : b =
) (l)
S2 : if a < b then S5S3 :
(b, l1)
S4 : S0S5 :
(b, l2)
S6 : S0Send
T< % -,1C >DQ&b !
?TA
1&Z[ 8% z $ Y !T 2.
.2 . .8 5 $. . % " 2. ( " * 2. 1 = B 2. !G f1 V-, _. $. !T -, 4
copy 6C GI D(.A. -, (U1
1 0 ?@j. < 6I = 8 D . 71I8 0 K =6 !
T =6 8% @ (.f. R' .
70 2 !$ X U `- Z.8 7 /-,7H U 2.7( !i .
lQ l′
0 : l 161 : a =
) (l)
2 :
(a, acc)3 : l 144 : S0
5 = Send : &
l26 : l := copy(l1)7 :
0
8 : l1 := copy(l′)9 :
&l1 ;
&l2
10 : l := copy(l2)11 :
0
12 : l2 := copy(l′)13 :
&l1
14 : a =)
(acc)15 : C016 :
end
Q .% -, 2 N9 K e h / * .( 8 _ ! = -, 2. = 1 6. %= 1 K>DQ(b !
Q = 6 K6. 50 G`-, =8% 2 P1% 8 - .= B "H 4- 2. $.D 06. 0 / K-, =6 B R'cN' ( R'cN' !
@Bc
F . 7 1.( .8`-, ;S .(U" 4W *% ; f# ' (. 2 VH YQ/FWG> / .B( K $. 0 Z'cN' = P1 * U&. /.!
" 7 $ X V
nP #%O%7[ ) ) JZ&4R%7[ #0H1 .(0H"B0/ !1 05 #0/1IK 4R_T%'"50/1? %N- / 8 Z.% !I 2. 2. / C@j. 6 2' 82' % UB0 .$. 1 D(. -, 1 . !&+ *% 8 .!
Q 2. Z;. 7C % 8 4 B 2' $. ;# R0 $. 21 !
QV
X 88%V (i)
2. i'*#8
V!? 6I %
M %
M(i, j) . 8
(i, j)
M!
QE
#*% K KK.,/1. 2 x1, . . . , xn
G.f.8 GE
!$QV
n = |E| % 8
V (xi)
i' # 7
V!AFD Z'cN'c 7-,
8 ; /. K4C E E
Z!AF ./( =$UV !
nP #%O%7[ ) (E 0/"B0/ !. 4.(0/"$0/ ! #!%'P 00 !
11 . 1 7
00 !
11 !
nP #%O%7[ ) [V ]TT031& ! .'0/1 .[0/"$&0H !T1Q
A
B 8(U 76I B ; %
A ≥ B
i%A(i) ≥ B(i)
! A B
A ≥ B
A 6= B
! A > B
i%A(i) > B(i)
!
nP #%O%7[ ) Zd , !L!L Z ! .(0H"B0/ ! 1& L#L [] # 0/!1&0/K 1 .'0 Q
V 87 f# & !<Q aw l]oo$n=m H
V1
8 & d -, &
1!eFD Z'cN'c 4-,
V1(x) = min(V (x), 1)!
Q E
8X E′ ⊂ E
Z' 8XB E
!AT w l]oo/nNm HKE′ K 8
E′1
|E|
<-,E′
1(x) = 1
x ∈ E′
0 !
J. (-,8 8% E
: V
8:|E|
8% `(#V1
6 R' 8X0
E-,
x -
V1(x) 6= 0!
0/KYK 0`8 % n=s6H
Vl(rhtOv2M H v l(m M_n No/nOw pNr=H vwo$nNw s6HzsutNv wn=lYr/l(xuxfvw n=rYw l]oo/nNm H
V1
v w HGx;pw nNx l(Hzs n=r Nl5w Ow H Nv X ≤ V0 ≤ X ≤ 1X × V = 1× V
@ 9
FJANnNr/w2H*m2p Hzs n=r V1
; E(2S 2#8 !Q
X f ( ZS 2#8%B 8(U f#8K.8-, f ;-,D- :
X
0
1!
T 8;# ; `-,K-, X
= 0-,/ /
V
0!
T ;# f `-X-, 7 X
= 1-,/ / 4
V =
10 6 ; -1 !
nP #%O%7[ ) 8 *+4R_T%7"B03#%'4 2((!46.70 Q
V 8!(T Np,H*m sQM_v Ns p :n=rAp=xfv
V ;.0
V .0 6
V
> 0FD Z'cN'c 4-
V (i, i) = V (i)
V (i, j) = 0
i 6= j!
0/KYK 0`8 a n=s6H
Vl(rhtOv2M H v l(m p N6p Hzm sQM_v Ns p :n=rApNx;vpOw_w n M sQJ_v1vw HGxfpw nNxyl(H*s nNr-vXx J l/p Hzs n=r
M =∑
i
aiV aibi
nExfvwai
wn=rIH!x;v w4tOv2M H v l(m_w M_nNxfn=r/rAv vXxfp _pOw v.M_p=rAnNres l/vl(r1
v ri N6vo$nNw s6Hzs n=r vAH8vw
0p=suxuxfv l(m_w !xfvw
biwn=rIHxfv w7tNv2MAHcvl(m_w7xys?:rAvw vXxfp_pNwv M_pNrjnNres l/v1vAH
Vvw H M_nNr/w_s JmPJFM_n N NvBl(r3tOv2M H v l(m4xys?:=rjv
FJANnNr/w2H*m2p Hzs n=r T (( aibi
4 4-, 41
Z i' # 4 6
0 !(T
(( aiV
V (i)
0 i' # D
V1 !eF -,4.f.8 1 0
B i' # B =
V (i) =
0!eT 1 D Y. B
4 ;.4V
!
nP #%O%7[ ) % Q
V 8!-,
0 ≤ V ≤ 1!
Λ(V )* X 8
V ′ X-< U -. V (i) = 1 V ′(j) = 0
V (j) = 0 V ′(i) = 1−∑
j V (j) V ′(k) = 1
k 6= i<-,
V (k) = 1
0/KYK 0`8 * n=s6H
Vl(rhtOv2M H v l(mRHcvx l/v
0 ≤ V ≤ 1 Λ(V )v w HGx vr/wv Nxfv v w4w nNxyl(H*s nNr/w
XNl5w Ow2H N6v
Y · 1 = 10 Y VX = V − (V · 1) · Y
FJANnNr/w2H*m2p Hzs n=r Y
B Z.;-,8 X 0% =$C -,
1 X-, /3E
& 7 V
!e? X
(. * X B f!
nP #%O%7[ ) (a Q
Γ7 Z.4
−1%0
1!A
Γ+ 0 !Γ− 1 6 7 -,
= Γ+(i, j) = max(0, Γ(i, j))
0 !Γ−(i, j) = min(0, Γ(i, j))
1 !
* $ B ( ,V $ , (5 <@
W 3#P/ZT-/KY038ZQ %70/1976
%'0 a% n=s vr=H
Nl(r3mJw vp=l vv Hzm svAH
Γw p N6p Hzm sQM_v M_pNm2p MAH'Jm s`w H*s l/v FJ2Ms vmXw_s
Nv w HnNm HcvANvr=HrAnNr 'H v m N1surApNr=H
vw2H pNr/w , !v l(sKm2v t8s v rIH J2Ms vm5w sKsuxvPOs`w Hcv5l(rAvVw nNx l(Hzs n=r v rIH*s m2vYpNl w Ow HANv sur J l/p Hzs n=r$ww l(sutOp=rIH
Γ×X ≥ 0X ≥ 1
@5;
FJANnNr/w2H*m2p Hzs n=r QN
4C (' 8% U 6 _-, M0
-, 1 U(. ω
-, - -G . XC !F B _-, U(. h(. 0 2 1 _-,
(M)n0 . &aX( 2. .8 X" X
ω
* 1 !(? 2 %U 2 (M ′)n
B-, - . =E 4C
M ′i
M ′
i+1
!F 6 z _
E "C /(.% U 4 (UV -,
M ′i
M ′
j
-,i < j
M ′
i EM ′j
!(W
% U 4 Z.]-,7 ω′ B-
M ′i
ω′
→M ′j
-, -G =5 4C
ω′ !+ .8 ;-,8 8%
X = Fω′
% M ′
i +Γ×X = M ′j
0 2.8-, 0 P1%X ≥ 1
0 - ,G =5 0C 21
M ′i ≤M ′
j
0 _ 2 _- P1 !i ;# $.8-,E f#%
M ′i + Γ×X ≥M ′
i
%$ Γ×X ≥ 0
!J. (-,8 8% ZS 2#81H $.]-, ( 4 f % DC 8 X
2.8-, ω
"-,Fω = X
X -, M
- z !? 2# 8 f# $.8-,%M
ω→M ′ D M
%H M ′
ω 2. %
ω 7I .6 2. 1C 8% /$ 7
U(. !!e? # $U;# $.]- <% -, . =E 7C
ω%/ * U&. . .8 86 - 5 ! 2.X 4C 8% 2. =
C 8 (' 2 !
W 3#P/ZT-/KY0 % FJPM s v m0w s@l(rEmJw v_pNl vvAH*m svw H!rAn=r 'Hcvm N1surAp=rIHGvw H p=r$w !v l(s@m2v t8s v rIH wp=tOn=sum4w ssux<vPOs`w2Hcv4l(rjvw nNxyl(H*s nNr vr=Hzs m2v1pNl5w Ow H N6v sur J l/p Hzs n=r$w
Γ×X ≥ 0X 0
FJANnNr/w2H*m2p Hzs n=r T 8 f " 5( /.# 2. .8 !T h (U ;# $.8-, / `-, C 1-, I 2. X $. fCD "C % -, =5 0C R0 4 z U(. &.A. Z.]-,4 ! 2.B 0C 8% GS ! G- 21 !T 7 $U;# $.8-, / `- = /-, 6 I 2.K R
-, 2 H . ;
X = 0!
F 8 8% X
X ; a( 2S # % c · X
0 c
-, /-,/1 = ( ; !=T! U f!
N .8-, * U ; :
Q!
? f8% XC 2 7(ZS 2#8 K -,z 4 fC 2 0 K7 * !&i 2.8-, % U f
R% U = $ f
Q!&F 6 8 z U
H ; *0 ; K (' U -1 0 8 U f f -1 % ZL>6 U f a
R!! " ;#
0 .; 5 2S # BH $.]-, $. R1 & I 7 !
N,0HKYL#.70 E[V 9 !T 1 2. `-D(hQ/FWG> z $ 0 . -1 2
x = 0 x 6= 0
x"
y 00 −1 −1 0 0 11 0 1 −1 0 02 0 0 1 −1 03 0 0 0 1 −1
end 1 0 0 0 0x 0 0 −1 0 0y 0 0 0 1 0
T f ( 2S # Γ ·X ≥ 0
X = 0
!&T 7Q/FWG> H U(. !%( 1 2 =9 !T 2. ;-,0(EQeFWG> z $ 80z (U ;# =1 Z 0
0 . .
@B>
8. ` 21
1 (1) x = 0 x 6= 0
x
1 (3) ret3"
y
ret00 −11 1 −1 −1 12 1 −13 1 −14 1 −15 1 1 −1
end 1 1 −1 1 −1
stack0 1 −1stack3 1 −1
x −1y 1
T< % f$ 2S # Γ ·X ≥ 0
X = 0
Q/FWG> 2 % D H U(. !
;!T 1 2. `-D 1 x := copy(y)
!
y = 0 y 6= 0
y"
x 00 −1 −1 11 1 −12 1 −13 1 −1
end 1x 1y −1
T ; 3( 2S # Γ ·X ≥ 0
. X = 0
% .=hXW 2 1 N9 !
5 (/, X"@ ," 7 (5 $ N,0HKYL#.70 EZd W "
x := dup(y)% V 2 0 4 $U R
2. 2. 0 8(U
/1y 6= 0
" y′ y = 0 y′ 6= 0
" y y′ = 0
0 −1 1 −12 1 −15 1 −1 1 −17 1 −1
end 1x 1y −1 1y′ 1 −1
T</% 2S 2#8 Γ ·X ≥ 0
$D ; 8%e 6C 2 (k, k, 0, k, k, 0)
k
!Hi @ 8% Z.]-7 / 8 BI . 2. = 2 !$W % -, Z.]-, .-, <S *0 =E 21"9
0v H
5 6-, eY = h. ;. ( 2S # 0 S
8(UD9 (UI1 !? .6W o$n=l(m&H n=l(HJ Hcp,Hsur/s6H*s pNxy%( =6 2 N9 !
F ;# =6#6C 8 f(a -, f$ -, %<. $.
X Ba -, f$ -, Z.]-"
ω 5-,
Fω = X!
@$C
F f#8 .2.h 2. , ,' ?> "g! = %:)! ;\ VP ! ) @5A-+ ! 8 ]% 56. ( -, 4 2. / = !Hi A@ ]% 2 KC -, -, 2.8-, 0 50 8
X1 G-,
S 1 2.B 8 X
0 ; (ZS 2# -1 !^ 8 ]% / !-, GH = `-,!=i A@ 8%
BC 2 4( ." 4W ;-,8 Z7F[@WG> & 6 0 = f#8 .K( _ (4 ;# U .%=-, ;f'*6I8 $.1 -,!
6 . 16. ( (. H 2. U!
81 130 8 %&'( QL# R & nP #%O%7[ ) * F (#!0HN,%O&P !"B(#!0HN,%O&P Z&0 Q
G = (V, E) 2.!
G M_n=r/rAvPO&v U 1 1
sG-, 6
u 6= s U 7 8Y-,
s
u!G
nNm HcvANv rIH3Mn=r/rjvO&v B 1 u, v U 0 Y-,
u
v!
Kv NpNm l/vF (. U .< E(. 2 !Hi A@ ]% 6
a, b, cK (U
(a, b), (c, b) U Z<( 2 4 (. f' Z!
T 5 H ` "H . B -, B B-, 6 = K K- "H 2I !/T A>
−1 * 2I K$
6 8%1
7S 0
!
nP #%O%7[ ) Q *+4R_T%7"B03 %7!"5%'#0H#"B0 Q
G = (V, E) 2.!/Q N6p Hzm sQM_v sur4M s v r4M_v
IG 4
|V | × |E|
IG(u, (u, v)) = −1 IG(v, (u, v)) = 1 IG(a, (u, v)) = 0
a
u
v!
nP #%O%7[ )() #[Q
G = (V, E) 2.!/ ,
f 4C h-, 7 -6 D -
0 B-, 1 u%f(u) =
∑
v/(u,v)∈E f(u, v)−∑
v/(v,u)∈E = f(v, u)!
FD Z'cN'c -, , 7 -, 6 7. : 5 C@A. 6 4 #, 2 K , 2 !
, 8 0I . 2. 6 (U % z |V |
e ;-, 0( , -, 1 z = 4
|E| / `-, ( , 2 - !
0/KYK 0 % ) n=s vr=H
G = (V, E)l(r.:=mpo /vDv H
Xl(rtNv2MAHcvl(m
|E|Mn No/nOw p=rIHcv w x/vPOs`w H vl(r n H
fX J!res,o$p=m
fX(u, v) =X(u, v)
v HfX(u) = (IG ×X)(u)
FJANnNr/w2H*m2p Hzs n=r fX
. 4 / 6(. . .8 !
0/KYK 0 %E n=s vr=H
Gl(r :=mpo /vVn=m s vr=H;J
IGw p9Np,H*m sQMv sur Ms vr Mv
svAH
tv l OEw n N NvAHzw x!vOs`w Hcvl(r M/vAN1sur v
stNvm_wtw s!v H:w vl(xfv Nvr=H:w s@sux<vPOs`w H v4l(r n H
fHcvx l/v
f(s) = −1v H
f(t) = 1vAH
f(u) = 0o$n=l(mHcnNl&w0xfvw7p=l(Hzm2vw
w n,NFN6v Hzw FJANnNr/w2H*m2p Hzs n=r Q U % /
f(u, v) = 12 -, 4(
0 !
Q #, U % 0 80 C . ; 61- f(u, v)
C 2 -,
(u, v)!
F ZT(.'.746%OT0 %(V n=s vr=H
G = (V, E)l(r :=m2p_o /vnNm s v rIH'J
IGw p Np,H*m sQM_v sur4M s v r4M_v
sv H
tv l Own N Nv H w n=s6H A
xfvXtOv2M H v l(m@5:
|V |
M_n,NKo$nNw pNrIHcvw Hcv x l/vA(s) = −1
A(t) = 1
v HA(u) = 0
o/nNl(m7xfvwXpNl(H*m2vwDw n,NFNvAHzw xvPOs`w HcvVl(r M $v Nsur vs
tNvm_wtw_s7v HDw v l(xfv N6v rIH0w sDx;vVw Ow HANv J l/p,H*s nNr/w
IG × X = Ap N6v H7l(rAv
w nNxyl(H*s nNr FJANnNr/w2H*m2p Hzs n=r FD 0 2.8-, 0 (U 6 . .8 !
X
A 0 (U 0(.
,!
W 3#P/ZT-/KY0 %Zd n=s vr=H
G = (V, E)l(r :=m2p_o /vBnNm s v rIH'J
IGw p N6p Hzm sQM_v sur4M s v r4M_vXv H
sl(r"w n N NvAH nNs6H As xfv0tNv2MAHcvl(m
|V |M_n,NKo$nNw pNrIHcvw Hcv x l/vAs(s) = 1− |V |
v HAs(u) = 1
o$n=l(m.M/p l/v1p=l(Hzmv4w n,NFN6v H enNl&wxfvwwn N Nv H w6w nNr=H4p MPM_vw_w_s xfvw/vcoAl(s`ws
w sDv HDwv l(xfv N6v rIHDw sxfvw Ow2H N6vIG × X = As p ,NvAH4l(rjv
w nNxyl(H*s nNr FJANnNr/w2H*m2p Hzs n=r F K X-, U 7 ,D B $
s Y = 1 K$ D
X , E 8 !
F ZT(.'.746%OT0 % 8 n=s vr=H
Gl(r :=m2p_o /v1n=m s vr=H;J1v H
IGw p N6p Hzm sQM_v sur4M s v r4M_v
Gv w HMn=r/rjvO&vw sv Hw vl(x;vANvr=HHw sjsuxevOs`w Hcvl(r6w n,NFN6v H
sH v x l/vxfvw Ow HANv
IG×X = As p N6v H@l(rAvw nNx l(Hzs n=r G
vw H n=m H v N6v rIHRM_nNrerAvPO,vw sDv HKw v l(xfvANv rIHDw_s:o$n=l(m H n=l(HKw n,NFN6v Huxfv"w Ow H Nv
IG × X = Au p ,NvAH4l(rjvw nNxyl(H*s nNr W 3#P/ZT-/KY0 %% n=s6H
Gl(r :=mpo /vnNm s v rIH'JBHcv x l/vFM $p l/v6pNm Hcv1p_oo/pNm H*s vr=H "pNl*NnNsur/wBl(rM M xfv G
vw Hn=m H v N6v rIH3M_nNrerAvPO,vw sGvAHw vl(x;vANvr=H!w s:suxvw H M_nNrerAvPO,v FJANnNr/w2H*m2p Hzs n=r Q
G U % U 4 6
s -, 6
u U 0 8 5
s
u!&W
. 2 8 <% 8 0 Eu
s!
i @ 8% ;$# v
6 9R u
Z !W &S $#%z (v, u)
C 4H S !&b U 4 K V
u
v!(W &S $# 4 2. % U 0
v
s!
Q ;(#7 D (U 6 D- e-u
v% U B h
u
s K
8Y s
v!/? U 7 5
u
v% C U!
T . (-,8 7-C U .; ;-, N9 U 2.!
81 1 5 " L# $ 2 Z' 6 BH 8X"H I 7-, I
6 -, U. 2. !
nP #%O%7[ )()[) ,( !1 2[T4RL 3!0 Q
G = (V, E) %
IG X 0H `
E′ ⊂ E 8X4H 2I !(T 1w nNl&w Q:m2po $v
G|E′
U G
E′ <-,
T! 7 6 G|E′
u ∈ V/(u, v) ∈ E′
(v, u) ∈ E′!
T! 7 G|E′
E′ !
0/KYK 0 %a n=s vr=H
G = (V, E)l(rR:m2p_o $vDnNm s v rIH'J
IGw p5N6p Hzm sQM_v sur4M s v r4M_v4vAH
E′ ⊂ El(rvr/w vANx;v p=mM w p5N6p Hzm sQM_v
sur Ms vr M_v=lVw n=l&wQ:m2p_o $vvO H*m2pNs6H Ko$p=m Hzsum!vE′ vw H!xfvojm2n Nl(s6H v7xfp N6p Hzm sQM_v sur4M s v r4M_v Nl :=m2p_o /vv H v
xfp N6p Hzm sQM_v Ns p :,nNrjpNxfv"pOw_w n M sQJ_vp=lEw_l]oo$n=m H8vE′
IG|E′ = IG ×E′1
i C ]% & ; f h-, hH ` $ R' - f 0 0 0
0 / (U 1 =(U I D- 0. .
2 f6. !
@B=
FJANnNr/w2H*m2p Hzs n=r T BX E′
1
& !-, e B G E′ !,T
-, / DV 6 4-, 4 G|E′
&;. 8 8 &
IG K- / 6 Z;6.!
81 1 L# & " $ nP #%O%7[ )() E 46L43#03#0JL#.746"B0/1Q
N .= "W -, - U 8 X VH . V
!!Q :=mpo /v voAxfp M_v wG
.5-, 6 5 N
6 !
T h . `-,"( 2. = VW . " "H ` E 5
ΓN = IG!
T XN
(C /$ 1 X (C e( ( 4 !
0/KYK 0 %!* n=s vr=H
Nl(rhmJwv_p=l3vAH
Gw nNr :m2p_o $vvojxfp M_vw n=s6H ω
l(rjv7w J l/v r4M_vv Hzsum_w vN ′ l(sGvw2Hp M H*sut-J_vo$p=m7l(r
NpNm l/p :,vp=tOv2Ml(rYw v l(x vAHcnNr ωv w HGl(r8M/vAN1sur-pNr/w
G FJANnNr/w2H*m2p Hzs n=r W 5$ 7 !
nP #%O%7[ )() V Q & "4 0 8 8%
NVQeFWG>
N ′ EF[@WG>X%Γ
2. ;-,KN
!
Γ′ V 2. ;-,1 N ′ - &1 7
Γ Z;
/ =(U 4-, _ !HF 6F[@WG> / =(U / 0 (. ..] %
Γ′ 1 7H ; B$ 4 N ′ !
0/KYK 0 % Q n=s vr=H
l(rjvBNp M &surjv 9Mn No H v l(m
Nw nNr
N ′ w nNr Mi E Mf
v l O N6p=ml/p :vw vN
v Hω
l(rAv7wAJ l/vr M_vvBH*sum_wRH v xuxfv l/vMi
ω→Mf n=s vr=H
M ′i
vAHM ′
f
x;v wFN6p=ml/p :vw vN ′ l(s M_nur Ms vr=HDpNtNv2M
Miv H
Mfw l(mXxfvwoAx;p M_v wnNm NsurAp=xfvw r p
M ′i = M ′
f
FJANnNr/w2H*m2p Hzs n=r M ′
i E M ′f
8(U -, !F 6 E9 _ ]% -,
M ′i
M ′
f
-, % M ′
i = M ′f
!
F ZT(.'.746%OT0 a n=s vr=H
ΓvAH
Γ′ xfvw Np,H*m sQM_v w.M_pNmp M H;J m s`w Hzs l/v w vN
v HN ′ v H
Xl(rYtNv2MAHcv l(m s
Γ×X ≥ 0p=xfnNm_w
Γ′ ×X = 0 FJANnNr/w2H*m2p Hzs n=r W E .8-, -, . 8 .=$UV 4W !
F ZT(.'.746%OT0 a ) n=s vr=H
N ′ l(r ' vAHΓ′ w p N6p Hzm sQM_v M_pNm2p MAH'J m s`w2H*s l/v n=s6H G
xfv3:=mpo /vvGojxfp M_vw vN ′ nNs6H V
l(r"tNvPM Hcvl(mHcvx l/v
Γ′ × V = 0 G|V1
v w H M_n No/nOw Jl(r/s l/vANv rIH v M M xfv w FJANnNr/w2H*m2p Hzs n=r F -,"9 V
N ′ 3 6I # .
V!/? S 1 8E $ -, 6
G|V1
;f'*6I8!
W 3#P/ZT-/KY0 a[E n=s vr=H
N ′ l(r ' ω
l(rjvXwAJ l/vr M_vvFH*sum_w l(sGxfpNs`w_w vN ′ suretOp=m s pNrIH'M vw2H 3=sum2v l/v
Mω→M
o/nNl(m H n=l(HNpNm l/p :,v l(sGp M H*sutOv
ωvAH
F = Fω nNs6H
Gxfv:m2p_o $v vojxfp M_vwv
N ′ G′ = G|F1
vw H nNm Hcv N6v rIHM_n=r/rAvPO&v1w svAHw v l(xfv N6v rIHw ssuxGvOs`w Hcv6l(rAv1w J l/vr Mvv H*sum_wω′ p M Hzsut,J_v4o/pNm1l(r
NpNm l/p :,vp=tOv2Ml(rYw v l(x vAHcnNr vAH H v xuxfv l/vFω′ = Fω
@H?
FJANnNr/w2H*m2p Hzs n=r QG′ < C U % 1
G′ 2.h`-, E S 0 2# 6 . .8 1 %
G′ U !N? U : 1G′ -, !N?
U 2.8-, K = 9 "-, / =$U D
G′ !<F 6 ω
/ =(U D G′ % ω
.E -, D (9 !
J. (-,8 8% G′ C 8 U% . 2 X"X9
$. 7 2.8-, B !AF `(. 2.8-, 7 ω′ X-,
Fω′ = Fω-,
. <B _-, M0
: $U9 !NQ p
p′ 8(U <
N ′ <-, M0(p)+M0(p′) >
10
p = p′1 !
F 6 G′ C 8 U %
p′ f B $
p . (- !j 8 0 Z'
7 ω′ 2.8-, B-, . 7 _-
M ′0
8H9 !<i A@ ]% /
M ′0(p) = 1
% $. X 6 _-, M ′
1
M ′1(p
′) = 1% / 0-, 2 Z
-, 7
p′% $
-, K- 2 K M ′
0
!
81 1 & L f.2. B( F[@WG> ;# "H 2.5(
_ f#8 U .X !$b ;K-, 8X =$U / C 2 7 2S # 7H $.]- $. !
0/KYK 0 a[V n=s vr=H
G = (V, E)l(r :m2p_o $vKvAH
E′ ⊂ El(r6v r$w v N xfv8v3M Mx;v w 'M vw2H 3=sum2vl(rvr/wv Nxfv8 pNmMwp=s`wp=rIHHcnNl&w
o/pNm H*s v l(rM Mxfv n=s6H G′ = G|E′ = (V ′, E′)xfv :m2po $v6vPO H*m2p=s6H n=s vr=H I
vAHI ′
xfvwRN6p Hzm sQM_vw sur4M s v r4M_vw n=s6HA
l(rhtOv2M H v l(m |V |
Mn No/nOw p=rIHcv wBHcvx l/vXsuxvOs`w HcvBl(ru ∈ V ′ l(st-J m s v
A(u) = 1− |V ′| A(v) = 1
w_sv ∈ V ′ vAH
v 6= u A(v) = 0
w_surjnNrM vw H 3Nsum2v
A ∈ Λ(V ′1)
G′ v w H n=m H v Nvr=HM_nNrerAvPO&v4w sGvAHw vl(x;vANvr=H!w s:xfvBw Ow2H N6vI ′ ×X = A
pl(rAv7w nNxyl(H*s nNr voAxyl&w
V ′1
= I ′+ · 1 = I+ ·E′1
FJANnNr/w2H*m2p Hzs n=r X
A
$. K ,0-, 6 u
6 G′ 2.B;-,8
S % U 2.; `-, C U .!T ;# . 2. &
I ′+ G
1-, /
V ′1
!/T 6 8(U f#8 B(.7(5C K-I ′ · 1 = I · E′
1· 1 = I ·E′
1
0 6 =H? 1 !F ZT(.'.746%OT0 aZd n=s vr=H
Nl(r/
Gw nNrB:m2po $v v<oAxfp Mvw
Γ = IxfpN6p Hzm sQM_v3M_pNmp M H;J m s`w Hzs l/v v
NRvAH Np,H*m sQM_v p p M_v r4M_v
vG n=s vr=H
A 0l(r tOv2M H v l(m H v x l/v
Γ × A ≥ 0vAH
G|A1= (V ′, E′) vwXojm2no/nOw s6H*s nNr$wEw l(sutNpNrIHcvw"w nNr=H
J l(sutNpNx;vr=H vw x:vOs`w H vl(rjvwAJ l/vr M_v v H*sum_w
ωp=r$w
NHcvxuxfv l/v
Fω = AvAH
ωv w Hp M H*sut-J_vDo/pNm1l(rNpNml/p :vv
NpNtNvPMXl(rYw v l(x vAHcnNr G|A1
vw H M_nNrerAvPO&v G|A1
vw H n=m H v N6v rIH3M_nNrerAvPO,v <vw Ow2H N6v
Γ′ · X = A′ p,Nv HDl(rjvw nNx l(Hzs n=r DpNtNv2MΓ′ = Γ · A1
x;p7Np,H*m sQM_v sur4M s v r4M_v vG|A1
v HA′ ∈ Λ(V ′
1)
FJANnNr/w2H*m2p Hzs n=r 2
3
.]-, GA1
2.4`- K S R0 $.2#82? 9 1 !1
3
.]-, H #$.# 2?$; !3
4
.]-, H # 81 . .8 !
@B@
W 3#P/ZT-/KY0 a 8 n=s vr=H
l(rAvNp M &surAv FM_n,NKo4Hcvl(m_w
Nw nNr
N ′ wn=r ' Γ
v HΓ′ xfvw&N6p Hzm sQM_vwM_pNm2p MAH'Jm s`w H*s l/vw
Np ,NvAH4l(rjv5vPOJ2Ml(H*s nNr sur !res v l(sKrAvVr4J2M_v w_w s6Hcv l l(raw v l(x v H n=r n=m NsurjpNxw sxfvVw Ow H Nv"x sur4J_pNsum2vVw l(sutNpNr=H
p N6v Hl(rAv4w n=xyl(Hzs n=r Γ ·X ≥ 0X 0V ∈ Λ(Γ′+ ·X1)Γ′ ·X1 · Y = V
FJANnNr/w2H*m2p Hzs n=r T K 8(U8 f# $.8-, X
-, 2. ;-,7H Z.]-,X D N
- 8 I .; !T (UE ;# .-, U 4 2.8-, 7 = &9 j
8 2. ;-, X
!
Γ%Γ′ Γ′+ G(. / :-,G( . !
X1
%X1
Λ(V ′)
; 2S # H $.]- $. != : z 8X e 2S # $.8-, /. 0 < 2.8-, 7 K-H /1
Γ ·X ≥ 0X 00 <$2 Z.]-/1X1 ≤ X0 ≤ X1 ≤ 1X1 ×X = 1×X0 < , Z _ P1X ′ = Γ′+ ·X1
Y · 1 = 10 Y VV = X ′ − (X ′ · 1) · Y0 < 1 7 f.0
X1
1X1 =
∑
i aiV aibi0 U .B( R' /1Γ′ ·X1 · Z = VT &
X, Y
Z / ; U %
X 2. ;-, Z.]-
/ !T "$ ZS 2#8 ;S V 5( .=<%< " V Y "
8 !Q U K; ;# , !i @ 8% 2(> 2
8
; !
N,0HKYL#.70 E 8Q / * U "(
dup% ; X
Γ · X ≥ 0 X 5C 2
X = (k, k, 0, k, k, 0)!@
Y X1 = (1, 1, 0, 1, 1, 0)
!< X ′ = (1, 1, 1, 1, 0)
% "-, - ; 2.
V1 = (−3, 1, 1, 1, 0)%V2 = (1,−3, 1, 1, 0)
%V3 = (1, 1,−3, 1, 0)
V4 = (1, 1, 1,−3, 0)
! 6
y = 0
Γ′ ·X1
8 0% f# $.]-, D ; h--
V -, ;!?
dup N9 !
!& @ 5 )(V $ $
N,0HKYL#.70 E% T " 2. ;-,B( Q/FWG> z $ X i! ; Z 90 R ,G
4. .
16.6 A
G
1 H 5 % b = 0
1. 2.
@BA
16.21 b = 0 b 6= 0 a < b a ≥ b a = 0 b = 0 A B C D E F G b = 0
0 −1 −1 1 11 1 −1 −12 1 −17 −1 19 1 −1
14 1 −115 1 −1
end 1a 1 −1 −1b 1 −1 −1 −1 1
tmp −1 1 1 −1
T 2S 2#8 Γ · X ≥ 0
( B f & 8%H6I 7H . !Hi A@ 8% 80 U /1 z U(. ( 2. =3 6W
b 6= 0, a ≥ b, b = 0, G!HT f#8
C Z' e $. Z' =' ! =b%= Z'cN'c -, 8
b 6= 0$
b = 0
-, 8 b
.2.7 ( /. !
F 6 z & 2. .8 6 8% ;# 2. / = V$ 2. = 4W .f U(. $. !$F /]% U H .8 ;-,4 !
W 3#P/ZT-/KY0 a % n=s vr=H
Nl(r vAH
Γw p9N6p Hzm sQM_v Mp=m2p M H'Jm s`w H*s l/v N
p N6v HKl(rjvvOJ2M l(Hzs n=rasur Gr/s v M_n LJ m2vr=H vt8s`w ct8s`wvw7tOp=m s p xfvwBw s:suxvOs`w HcvBl(r
X 0H v x l/v
Γ×X ≥ 0v H<o$n=l(m.M/p l/vBtNpNm s p xfv
x
((X(x = 0) > 0) ∧ (X(x 6= 0) > 0))⇒ ((X(
x) > 0) ∧ (X(
x) > 0))
Q U ; / ` ;-, R02v;: f b = 0
* $ GH i! ` /1%
X(b = 0). 6
X / <
!
FJANnNr/w2H*m2p Hzs n=r T / Z;. 6.f U(. x = 0
x 6= 0
. 4-, x .E ; &$. !
F / ;-,2;. 8 = <C 2S 2#8 /. !]W /%!C = / = 6 f. $. 2.X z f 0 - 1 f. P1 / C N9 !F -,E 9 C N9 e 1 $.8-,Zf.8 !
W 3#P/ZT-/KY0 a[a n=s vr=H
x, y, z, t l/p,H*m2v1vr=Hzs v m_w7rAp,H*l(m2v x w s6NKoAxysQM_p Hzs n=r
((x > 0) ∧ (y > 0))⇒ ((z > 0) ∧ (t > 0))vw Ht]m2p=s v
w sGvAHw vl(x;vANvr=H!w s:xfvBw Ow2H N6v7w_l(sutNpNr=Hp ,NvAHl(rAv7w nNxyl(H*s nNr cv rIH*s m2v 0 ≤ (a, b, c, d) ≤ 1(1− a) + (1− b) + c + d > 0(1− a) + (1− b) + c + (1− d) > 0(1− a) + (1− b) + (1− c) + d > 0(c, d) ≤ (z, t)a · x = xb · y = y
FJANnNr/w2H*m2p Hzs n=r Qx = 0
% * ; a = 0
% ZS 2#8 1 ; !bR 8
y = 0
b = 0!
Qx, y, z, t
-, (' &8% z ; ZS 2# $ f
a = b = c = d = 1!
ABc
Qx
y
& -,z
&*%* ; C = !x
y
&C a = b = 1
%z
&C
c = 0!(Q
d = 0
(1− a) + (1− b) + c + d = 0
d = 1
(1− a) + (1− b) + c + (1− d) = 0%
2S # ( ; !? 71I8
x
y &
d = 0!
? C $. K * U&. . N9 = ZS 2# 4-, 7 !&T % 2S # 0 fS B(2. = U 7 ; =2S 2#8 K#, !
8 2 E z /4H i! ` !L <8% ;" X B $. 7H 2.5(
8 8 2S 2#8 KH $.]-, %, ;S K D(.= !
A 9
3 f
F $# 2. ." 8(U .2 (U !T 8 2.6 B U .2 0 !L1 B NSB ZS =U`-, Z =6 0 _ f' G-,f' . P1 !NF .2 4 8(U 2.I =(U !=? 6 . " U 2.s6NKoAxysQM s6H v1 * $ U 2.U !/T ZC 2' @ $. 7H K .;. U =6 0 1.8 =1 1 0 K6.8 =1 !e jC = ;-,8 K =6 5$S = ;-,1-,/ 8 # $. !/? = 8% 2. 2 = ; R
$ 1-, . .8 8% f.2/ f
e !=b A ;KH . 2e U 2.DH 7w*oLJ2M s M_p,H*s nNrH (.]( $ A> !
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1 ! =
H OS G A. : = 2e. 8% U &< 2S 2#8 !C 1 F (- X>/W !N+ z% @ 1 ! /. z U 8 U 2. * ;S !]? 1I8 H ;$.8% ; _ 0 f. -,f' . P1 6 _ '*C H ZC E 6 0 U 7 2 41 !$+ z% X Z 1-, ZC =6 6I 1 V Z U 2.B$ =6 # !
T E 8(U f#8.2 ;SV ;. 2. U 1 $ NS .=$U VW !@T< *% ;S 8%Y % . f X -, =';S U 2 . ; `- 0 1 * $ 4 i! ; -1 !/T f.2/ K f f 4-, B `-. .8 !
T f#8 X6. ( D C ; U .X( !$b 0 ; H OSH . / . &0 ;S 1 ;@ 1 !8F 1 H9 z . U 8%< U .B a 7 7 ?> .!T (. -, DC = ;4( _ U %-, 3" 2 X 7 8 7H B R6.8
-, . ! B-, * z =
N9 2 = H I f # 8X &0 &S ( 21 .!b / / ;H fS U .H ( ;.R0 2. ;./1 6. & f#7 !
.I ;S (C -, X ; 8 ;. 2. $ . / f $.8 !(+ z% XC * $ 2 % @ 0 (.C Y 2!jFD `6 2# . !ei 2' ; 6H = ;S ;-, 7 ZC 1 B V6.( !DE 5- !+ 5 $# %/9 8 z C@A. 6. ( ;S E ZC V =6 8%9 # 8 * ; -, 6. & 4 ;E 2 6I K (U `- 7 -,`-, K !H 6-< B ;1 01.( :
= 6 6C fS 1 B ` 8 4
e ;. .4 f.2. ZC =6 !+ z%j9 U Y-< ;YH -, f'
. (U .=$U" 7W !e % E6I K- = ;S %( 7 2 U .7( =6 !
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F z ;S ZC 7 =6 Z' I .=$U
6W *%H 8X 4I 6 /;! 6-, f 7 ,7 . / `-,H ;S K RC 2 V$.R9Z U !
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) + _R;B@,+ + _2 80 ! 1%'aj$^ ; h+K(C =h 3 . % 3p,H r/rjpNxfv r/% !ZABA/%9 AH;$@/% ! 9B9 @ 9 >B>!
) +M cBc,+ + 0 g! 1 M 0 ! 1% 'Eg =C 0 7 e S . % /v_nNm2v HzsQM_p=x!n,NKoAl(Hcvm M s vr Mv % !Z;B>B=/% 9 ;$%Z;$cBc5c$% ! 9 >B> 9 ?$@!
) +KHA$C + + e 0z> ! 1 B 0z> !? ! 1% ' M ;\ YF A ZC f _ . % <P &$ =0 ! 1% "0z^ ! 1 @J \& B0zM! 1 %O.8 % Kvw l(x H wp=r5H*m2v rOwsur8H /vn=m2v HzsQM_pNxM_n,NKoAl(Hcvm4w M s vr Mv % !6@ 9 ; 0 ! <v2M Hzl(mv Xn H vw7sur !n,NKoAl(Hcv m M s vr M_v1% ! 9 9 @$%9 A5A$C !
) + BcH;-+ + 0zJ ! ! 1 % ' NU&' ;-, f' 2. . !aJ ;-, 9 c=' ;BcBcH;(%T<bR[%;$c5c5;!i:U / ]= -G 4gTF++;$c5cB>!
) +DQRc5c,+ + 0zP ! 1 Q & & 0z^ ! 1 % ' + ZS ;S CK> $'cQ;\Vb ;S,' ; / . %A m2n/M_vv =sur :=w1n RH $v Gs 2Hcv_vr=H 2k!kGk ,No/nOw sul(N nNrn:sQM1sur!n,NKoAl(Hcvm M s vr Mv N% !Z@$C A 9 %R;$cBc5c!
) L+ h? + L 80 W ! 1 %D+ OS 0 L! 1 % NS 0z! 'z)! 1 E?D$ - 0 ! 'RF ! 1 % ' i! " @9 . !:+ f7 /) & ' ' !) LF AH;-+ L 0zQ !/ ! 1 0F 70zQ !/+ ! 1 %' + 8 (' ;\ =C ;S,'; 4C . %4!n,NKoAl(Hcp,H*s nNrjpNx !n NojxfvOs6H N% !Z;(% 9 ABA5;$% !ZAH? 9B9 c!
) LF 3g AB@,+7L (C .0*M! 1%(F 70z+ ! 1 % 70 ! 'z)! 1 g $\ 0z^ ! 1 % ' F U RS / 2S 8 D fS . %A :n Nojl(H v m M s vr Mv <nP:=sQM H rIHcvm rjp,H*s nNrAp=x1n=m Ow/no =% ! 9 :B@$C 0 ! v2MAH*l(m2v n,Hcv w4sur !n Nojl(H v m M s vr M_v1% !Z>/?$; >B@BC/% 9 ABA5@!
) LF 3g cBc,+7L (C *0zM! 1 %:F 0z+ ! 1 % 0z! 'z)! 1Xg $\ 0 ^ ! 1 % ' + $ 4 fS 2 =C . %(nNl(m rjpNxn3 Hl(r MAH*s nNrjpNxm2n:m2p N N1sur :% ! 959 %Z;BcBcBc!
) L ABC + L 0*Q ! ! 1%/' W 8 / ;S ; &S . %eBF 90 W ! 1 J 81 0z! 1 %A.8 8 % <v_pOw s xfv 3p,H $v Np,H*sQM w % ! 'aW F / 0Q . !L = 8% 9 A5A$C !
) L^ cBc + L 0*Q !:! 1 ^ C 0 ! 1 % ' + C ; $ . %&n=l(m rAp=xn3w ,N_nNxysQMxfnP:=sQM %(;BcBc5c! D ,G !
) L 3c 9 + L (C 0*M! 1% 0 ! '*) ! 1 NS 0z! '*) ! 1 % ' U _ K / . %,GL90z? ! 1 %$L NS 0 ! 1 = 0z+ !& ! 1 %&.] 8% % !Z;B;TC5C 0 ! <v2M Hzl(mv Xn H vw4sur !n,NKoAl(Hcvm Ms v r4M_vP1 % !6C5@H; CHAB>!&Q 8%]9R ;$c5c 9 !
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)yF ,[>B@,+ F & 0 + ! 1%' + 5 = 8 =C 8 2S &$ 2S . % Nvm sQM_pNr&n=l(m rAp=x<n 3p,H $v N6p HzsQMw % :B@$% 9 A5>B@/% !Z>$C/: >5=B>!
)yF ABc,+ F 0 + ! 1% ' L eE C C . !:J ;-,% RS =CT / <%$J NSj^ NOS eVL 8(C _V> F % 9 ABABc!
)yFPBQR@ 9 + F / 0 + ! 1%/P \70 ? ! 1 Q _&8S50zT ! 1 % ' + . %(nNl(m rjpNxn 5H $v Y% !6;$@/%9 A5@ 9 % ! 9B9 C 9 >B>!
)yFT@H?-+ F fC 0 + !L ! 1 BT 0 W ! 1 % ' g 2 =C 8 ZS 8 0&S ;S 2' / ; 8 . % Ms v r4M_v1n BM_n,NKoAl(Hcvmm2n:m2p N N1sur :% 9 A5@H?(% ! 9 > 9 9 :$A!
A5:
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)yF 3cBc + F 0 + ! 1 0z! 'z)! 1% ' g1T 7T U 4W a _ . !7J $'`-,%/T /%6;$cBc5c!@ 2 5J!
)yF Z=H;-+ F = 0 + ! 1 % ' g $ ! C> RS CC . %XL Z'c^ &0u) ! 1 %.8 ]%mn M_v_v Nsur :=wn BH /v rIHcvm rjp,H*s nNrjpNx !n=r"v m2vr M_v1n=r n:sQM 3vAH $n n=xfnP: pNr (suxfnOw n_o n M s vr Mv % !(;TC >Bc!,> ('c^ /%$+ _=E% 9 AB=5;!
)yF AB@,+ F 0zT ! 1 % ' [$ 2D+ $ . %<k 4 l(xuxfv Hzsure% ! =5:$% 9 ABA5@!7g $ ;$ !
)yF R?Tc + F 0*Q !+ ! 1 % 'VW 2S e . %]4m2n/M_v_v =sur :=wDn H $v r r/rel/pNx ,NKo$nNw_sul(N nNrH /v /vn=m Vn :n Nojl(Hzsur:=% !(?Tc ?B;(% 9 AH?$c!)yF R? 9 + F 80zQ !A+ ! 1 % ' F ;\ =C $/ N E =C ; ' / ] $$' . % (nNl(m rjpNx<n3RH /v Y% ! 9 @$% 9 %O9 9 AH? 9 % !RC 9 @!)yF HA 9 + F *0zQ !&+ ! 1% ' F $ F U RS C^ gS D[@ . %& 8mn M_v_v Nsur :=wBn
r=H v m r !nNr:=mv w_wXn 3p H /vANp,H*sQMs p=r$w n H n (po/nNr/% !Z:B: =BA!$Q 2'* e% 9 ABA 9 !) ?Ki AH:-+ ? "0z! 1 i 2\ 0z! 1 %m2v_v /nNsQM_vv Hzm s vAHzw !eF =X` 7 RSW % 9 ABAH: !) ?X i AB@ + ? i .0 ! 1% p s m2v xfvwxfp :,vm_wXv Hx;p,NsQMw !eg 2. .% 9 ABA5@!) ? @H;-+ ? N 2\80 > ! 1%' _ 4C ' ZS . % $v_n=m2vAH*sQM_pNx!n,NKoAl(Hcv m M s v r4M_v%
! 9 ?&% >/% 9 AB@H;(% ![;H?TA >Bc 9 !) ? @/?/+ ? N 2\50 > ! 1% 'g C8 . %(nNl(m rjpNx$n ,N_nNxysQM!!n,NKoAl(Hcp,H*s nNr/% 9 AB@/?&% !T=5A 9B9 : !) ?DHABc + ? N 2\0 > ! 1 =eF0z! ' W ! 1 %Xp=r n]nBn /v_nNm2v HzsQM_p=x :n Nojl(H v m Ms v r4M_vtNnNx % !
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M s v r4M_v%(;$cBcH; ! D -G !) i [AB@,+ i \0z! 1 % ' ? ` RS / F U RS =C7W D> VW 8 6'1+ b (( . %
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% !DbZb !$+ U / % ' >Bc5c! $ $v&sum Hcv_vr n]nNwn KklIMxys w0kxfv N6v rIHzw g!$T !/^ %/? W!/8% 9 AH:$=!
) [M4P c5;,+ [ 0* ! 1 %<M 8 0 b ! 1 7P F0z^ ! 1 % ' F+KJbRL7j+ b /( L8YW =CgC 7g2 3 Q . %j!mn M_v_v Nsur :=wn FH /vnNl(m H rIHcv m rAp,H*s nNrjpNx !nNr vmvr MvnNr m sur Ms ojxfvw6pNrm2p M H*sQMvn3 1v2Mx;pNm2p HzsutNv!m2nP:=mp,NFNsur: O%jW Z/ %e0Q(+B% ;$c5c5;!&+KF W !
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) M &A5;,+ M_ 0 + ! 1 % ' F ;\ U RSD&S RS ; X f' . % /vn=m2v HzsQM_pNx !n,NKoAl(Hcvm Ms v r4M_v % ! 9 cBc$% 9 % 9 ABAH;(% !6C/: =B=!) M0 @5>,+ M0 0 ) ! 1 %' + CC f 0C . %e Gv rIH <n=l(m H ,NKo$nNw_sul(N nNr <n=l(r p Hzs n=r$wn3!n,NKoAl(Hcvm Ms v r4M_v % !Z; 9 c ; 9 C !bciGi!i F $ KQ RSW % 9 AB@B>!
) M0 A5>,+ M0 0u)! 1% ' [ f 0[ . % nNrnNr 3p,H $v Np,H*sQMp=x n/M s vAH v w xfv H H v m % ;BcB=/%9R9 A5AB>/% !Z= ?!) M0 A5A,+ M0 0u)! 1%' g ]- H+DQ . %ek l(xuxfv Hzsure% !R=H?(%C . 9 A5ABA$% !6A5> 9 ;$C!
) M&Z> 9 + M( 0zP ! 1%'K @C & ;$ Q 2\ W K/ / Q&S . % 3nNrAp H w/v 2Hcv (m hp H /vANp,H*s"l(r Ow sN% !R>5@$% 9 A5> 9 % ! 9 ?$> 9 AB@!) ^KT@5@,+ ^ C = 0z? ! 1 GT = 90 F ! 1% 'Eg C / =CH . %= Kv :m s6H*sur :evPM &res l/vw1p=r ooAxysQM_p Hzs n=r$w % ![>H:B:*0 ! v2MAH*l(m2v Xn H vwXsur !n,NKoAl(Hcvm M s vr MvP1 %9 A5@B@!
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) ZAH?-+ B0 > !/? ! 1 % !n,NKoAl(Hcp suxys6H 5pNr !n,NKoAx;vOs6H _m2n N p m2nP:m2p,NFNsur:vm_w*o/v2MAH*sutOv! hbRg 8%9 A5AH? !
) ZcBc + 70 > !? ! 1% ' gYi!U YW N =C7^ 2' gS ]%!T fC h EF0>DQ . %(nNl(m rjpNxn3 Hl(r MAH*s nNrjpNx mn:m2p,NFN1sur :% ! 959 % 9 %Z;BcBc5c$% ![:5: ABC!) Zc 9 + *0 > !? ! 1 % ' gEi:U EW N =C7^ _ gS 8%!T C Y 5F0>DQ . %(nNl(m rjpNxHn l(r4M H*s nNrAp=x mn:m2p,NFN1sur :% ! 9B9 % 9 % 9 ;$cBc 9 % !/:B: A$C !/>/1. A.Z[/ W 1 /hF $ F U RS !
) P7L ?$c,+ P& 0 ? !$i ! 1 KL / fU0 W !/L! 1 % ' Q; 7 _ 8 E . % !n,NKoAl(Hcp H*s nNrAp=xmn xfvANw7sur w Hzm2p MAH x :,v" m2pN% 9 AH?$c$% ![;B=B> ;$A/?!) P7P ABA,+ P 0*F ! 1 BP 0 ^ ! 1 % ' J 8 e%<Q e%jW . !X 2. ; 6H ;4() ' " ' " ) ' ' & % 9 ABABA!) P7T@Bc + P = 0zQ ! 1 T.,S90 ! ' ! 1%#'3+ C ! . !HJ $' ;-,%$ 2.B* b %/ 2 /% 9 AB@5c!> $$ ;.!
) P :B;-+ P 0*Q !jF ! 1 % rIH*m2n NlIM Hzs n=r7Hcn N6v Hcp,Np,H $v N6p HzsQMw !e> ('c^ /H% 9 A5:5; !) T ?B?/+ T $,C _ 0 ? !Q ! 1% ' Q ]-, RS C $ G+ Z S /
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) T ?TA,+ T $,C _70 ? !$Q ! 1 % ' 2S 8 0> . !<J `-, hgW:' >$%eT 1g RS% 9 AH?$A!
) T + pNmM !/) ' ' ' " ' ) " ' ) !) T A$C + T &0z? ! 1 % '"W 8 e $ U RS7b N _ / ;S' ; . %7F 90 W ! 14J 6 &0 ! 1 %H.8 % v_pOw s x;v 3p,H $v N6p HzsQMw % !>H;$c >$CH>!L = 8% 9 A5A$C !
) T ABA + T 0z? ! 1 % ' J = 8 e $ U RShb2bZb <^ RS 2' / 8 2S U RS . % r/rjpNx wXn3 l(m2v6p=r ooAxys v n:sQM% ![AB=$% 9 ' >$% 9 ABA5A$% ![;BcBA ;B;BA!) T ?T= + T 0 ) !ei ! 1% ' g C ;\]"W > . % nNl(m rjpNx@n3.M_n,NKoAlH*sur :% !Z:$% ;(% 9 A/?T=/% ![;B: 9 ;B=5:!) T<,L+ c 9 + T 0 F !,Q ! 1 %&0 > !,? ! 1 L ('c+ 0z+ ! ! 1%#'hg0Q;\ 'RF W C !W g . %& ,NKo/nOw sul(Nn=r/m sur4M s oAx;v wn3 m2n:m2p N N1sur : <p=r :l/p :,v w % !R;B@$% !6@ 9 AH; !+KF %O9 ;$c5c 9 !
) T 3A5>,+ T 0 ? ! 1 0 ! 'z)! 1%' T =AVF ;4F ;\ CW ;S'cg; . % l(r p N6v rIHcp r"n=m N6p HzsQM_pv% ! 9 A/% 9 %<;(% X 9 A5AB>/% ! 9 =/? 9 @$C !) T 3AH:-+ T 80 ? ! 1 0z! 'z)! 1% ' J = 8 / $ U RS bZb Q&$ e ;S,' . %NW 2 0zT ! 1 !g;2S90 ! 1%N.8 8 % !n,NKoAl(Hcv m M s vr Mvn:sQM H 6nNm Nw$no4 % !&AB>B>.0 ! <v2M Hzl(m2vn,Hcv wsur !n,NKoAl(Hcvm M s v r4M_v1% !CH@B= :$c5c$%P =\f \%(W % 9 ABA5:!$Q !
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) T 3A/?/+ T B0z? ! 1 0z! '*) ! 1 %'3W 8 4[$ J /"W ;S,'cQ . %( L ; F0 ! 1 D?= F0 ! 1 %A.8 8 8% $v_nNm YpNr m2p M HzsQM_vn3 n32H Gp=m2v 1v tNvxfn_o4Nvr=Hz% ! 9 ; 9 C0 ! v2MAH*l(m2v n,Hcv w1sur!n,NKoAl(Hcvm M s v r4M_v1% !,>B=5A >B@Bc!eQ 8% 9 A5AH? !
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