Analyse de la complexit© et transformation de programmes

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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T (|ω|)!

F 3$. #6C 2 3 Z 8 VI 3(. V V (UC !$TG `(. .$. 8 ( 0C 4 2S& ;-,%= Z'cN'c 0-

f

f -,

g = z !

g = O(f)!GF 5 N9 z 2.E

U 2. 0 0S f C K-, D; _ /1 !A / C ; f

1 U .!

8 / -,-, (. . 4xfp U 2. 8 KH 7 B g 2# f % B-, f 2.B fSLK 0 U /1B-! EC f." U 2.!@T .8-, 0C f / ; !

F 6 6) !M0 ) M0 AB>/+ % 2S& `- 4C . 7 `(.71 2 54C f /

-, = 4* 8 VC # 5I f

; 2 - 2 ;H !W U8 %

n100 %$Y-, . ;SLK % ,G = ; B-,nlog(n) K 8

n 0 (C .

2100 ≈ 1030 1 ! N,0HKYL#.70 E T 6 X Xg ,G / D =U;X/ .

3C

7( 0 6 h1C q0

Z $U;# 2/%H 8(U f#84C

q1 =E(.8$ ;# 0C

q2 1 !

?

Q U . n 7→ 3× n

!/? 7 ;# f % U . $. !

nP #%O%7[fQ 1L#46"503#03"B46.7"B #.Q

MX X g

ωh !jT!uvwzo/p M_v v M_pNxCM l(x!

M2

ω NU f$

. Z B 8(U f#8 $ e 5 (ω, ε, ε, ε, q0)

!FD Z'cN'c NU ;X$ -, $.

(ω1, ω2, ν1, ν2, q) 1 :

ν1

ν2

!

nP #%O%7[ ) F [KYL#.705N,%'Pn0/f0H1L 46"B0 Q

M g< !T Mn NojxfvPOs6H'JEvr v w*o/p M_v"

M f C

E V-

ω% * 7 <

MZ

ω (C . 5.

E(|ω|)!

T B1I8 8 _-K-, U 2. 8 8 `- 0- C !

N,0HKYL#.70 V T U 2. : 2. . . .8 86

n 7→ n!8i A@ ]%: U 8

= Z B 8(U f#8$5-, 7 * .!

! (5 G_ ,V

021 130 5 #"( %3T 2S # . . 2. 2. 2 K ((. C ( 8

_ !AT f 0 7(. D * .2/1 6> !j? N 2\ 7! ' W !j('/8) ?05A5c-+ !e 2 * 0 F ! K^ !/P ) P7P A5A/+ !nP #%O%7[ )() ,^,4RN,0 Q

X%F

%C

8X G = % 2S, C ZS&X !T %( fC # (. =6 Z

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

W Z % ZS&X :C 2S 2.8 `-, !. 1C (' ) " +*-, #"/. 2S&X 2[4Z1!nP #%O%7[ ) E ( T2[T46KYKY0/1 6 Y-, RNpNsur

= 〈X , C,F , E〉-

E 8X7 2# ! F -, = AK.X KH # ,G = AK2. = B

16I # ! b U 7 2S&X 44C A. ;. ZS& -, 2. " !

W ^ L#4 2(0 #0/1IL!T!2ZT46KYK 0H1nP #%O%7[ ) V W ^ L 0/1Q

A = a1, . . . , an 8X6 RS `-,!T! 8X1 H o$vw 4 7

A Types 3 τ ::= a→ τ |a

!

8 -,K-, -, RS 0 6C 2 a1 → (· · · (an → a) · · · )

2. B BC !A V. f ;

(a1, . . . , an)→ a

!

nP #%O%7[ ) d .,%OT[##0HK 0H]#0` ^ L 4 2(0 v r/t]sum2n=r/rAv Nvr=H8v.H o/p :,v 7C

ΓX

⋃

C⋃

F

Types- 7 -, 2S,

@

0 2 8 E7C =1f

H 2.n > 0

RS Γ(f) = (a1, . . . , an) → a

% 0 -, 2S&X H .

0 RS ;-,!

nP #%O%7[ )68 W ^ L#4 2(0 "5ZT0/"$Q

Γ D RS !A

t ∈ T (C,F ,X ) 7 RS

τ% 2.

t : τ%

2 t

2S, 7H 2.0

Γ(t) = τ

! t = f(t1, . . . , tn)

%ti : ai

Γ(f) = τ = (a1, . . . , an)→ a

!

nP #%O%7[ ) % ( T2[T46KYKY0 ^ L P BoAm2nP:m2p,NFN6v H oLJ U N6p=sur

= 〈X , C,F , E ,A, Γ〉 (-,

〈X , C,F , E〉 =6 0

RSe.-1 %A

X 4 RS Γ

8 RS -, # (p1, . . . , pn)→ t%

RS (p1, . . . , pn)

t 6I R'cN'

Γ!

021 1 5 !$"L nP #%O%7[ ) a [ #1 0HTKY0 Q

t

s 8(U !e 5 D-,

t w n=l&w'Hcv m N6v7

s%

t E s2

t = s

s = f(s1, . . . , sn)

0f ∈ F

⋃

C1 U

i<-,

t E si!

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v w Hl(rAvBm2vx;p,H*s nNr nNm =m2vXw l(mT (C,F ,X )

FJANnNr/w2H*m2p Hzs n=r i . , U % ' ZS&6. `-, !

nP #%O%7[ ) * 1 1_&%O& &%'( 6w_l w Hzs6H*l(H*s nNr

σ 4C Y

X

T (C,F ,X )!A 6w l w2H*s6H*l(Hzs n=rMx;nOw v 4C Y

X T (C,F)

!$ 6w_l w Hzs6H*l(H*s nNr M_n=r$w Hzm lIM Hcvl(m 0C EX

T (C)

!Q

σ 2$ 2%

σx ← t Z/ -,

t

x

σ(y) -,=

!T Z/ .e( `-, =$U 2 1 ;# Z σ( (t1, . . . , tn)) = (σ(t1), . . . , σ(tn))

! σ(c(t1, . . . , tn)) = c(σ(t1), . . . , σ(tn))

!+ E . $# % ZC

tσ= Y

σ(t)!

nP #%O%7[ )RQ & P/#0HN Q BNpNsur 6

t !$ mJ vPOY

t 2

u<-,

u E t!

b U 72# d = (p1, . . . , pn)→ v

2$ 2σ

-,σ( (p1, . . . , pn)) = u

! 8 0mJ Nl(sum2v

t

s%& d

u

σ(v)

t!e

t→ s!

nP #%O%7[ E & P/# !"B%7[##[K 46.'%714R&%'( T +

→ K2 7

→!

T ∗→

K2 4→

%= Z'cN'c 4-,t∗→s

t = s

t

+→s

! -, = 2.] UH% 8 4I .8( 8% X nNm NvrjnNm NpNxfv!T !

→ 51rjnNm NpNxys`w p Hzs n=r! = Z'cN'c D-,

t!→s

t∗→s

s

"C 2 2 !

i ./. .% 5C 6 /. 7;-,!@F 6 79 G . B 7C 8%(9 (U 8 .2 -, .!<i %<-,`-, 2;. 2 ZS 2#8 . . . \ e(# 8 C ( _ F0 6 7 E F = EQ 3T = 8 _ -1 !nP #%O%7[ E ) F [ !0/#"B0 ZS 2#8 . . M_nNrl/vr=H 4 $U OS " I 6 VC . V / 6X<% = Z'cN'c -, U

t, s

u7-,

t∗→s

t∗→u

% U v

!-s

∗→v

u

∗→v

!

A

0/KYK 0 E s:l(rhw Ow HANv vmJ2JPM m s6H*l(m2vXvw H M_nNrl/v rIHpNxfn=m_w.M/p l/v.Hcv m N6vpVp=lXojxyl&w7l(rAv nNm NvXrjnNm NpNxfv FJANnNr/w2H*m2p Hzs n=r Q U 1 2

t-, V 8(UhC 2 2

s

u% ,/ U 1 2

v

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u

∗→v

! s

u

4 KC 7 .8$ ! V C .8

s = v = u C `-,!

W 2 =(C: U ( % ;(#7;-,8 D 2S 2#8 K 2.8.f'2 ,/ !M!H^ ) ^K@5c-+G 6 2# 4$. 2 @ 2S # 1 . . ,/ !

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( "

) = (τ1, . . . , τn) → τ!T nNr4M H*s nNr Mp=xCM l(xCJ_v 7NpNsur

[ "

] : [τ1]× . . .× [τn] 7→ [τ ](. 6 Z !

W -,ui ∈ [τi], [

" ](u1, . . . , un) = v

8 " (u1, . . . , un)

!→v

v ∈ [τ ]

!j? %

[ "

](u1, . . . , un)< (. !

N,0HKYL#.70 d T ZS 2# 0 . . 2 2 * $ ( & -,. %.;. 2.. !jT RS

, , " % 2 8 0 RS /1

W T #0

: , 46.'1&0 : ,0 : ,S : → , J%7. : " , F [#1 : × " → " !&T RS GC $. # G 2.8. 2 / !T ZS& DC

≤.2. $U(. ; ; 2.!

≤ : × → !

0 ≤ y →W T !0

S(x) ≤ 0→ 4Z.'1&0S(x) ≤ S(y)→ x ≤ y

" : × " → " !

" (x,

J%7.)→

F (!1(x,

J%'.)

" (x,F (!1

(y,.))→

" (x ≤ y, x, y,

.)

" : × × × " → " !

" (W T #0

, x, y,.)→

F (!1(x,F [#1

(y,.))

" ( 46.'1&0 , x, y,

.)→

F (!1(y,

" (x,.))

" =

: " → " !

(J%'.

)→J%7.

(F (#1

(x,.))→

" (x,

(.))

9 c

021 1 8"!$#%3(nP #%O%7[ EZd W 0HKYL#1I!0 "B46.7"B #.Q NpNsur =6

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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i 2 %0(. Z. ]%_9R ]%=! J!res6Hzs n=r" U 2. :1 &$# $. = f$ Nv w l(m2vX(8 D K-,7 K 8 ;X 5 ((#!!T "C -, ;X H &$#5 6 = 80 ZS 2# 2.8. 2 _ 8 P1 A> B-, * & (.A.f = G 6 !Ni 2.8-, % S ;# 7(. U . 2S # G 2.8. 2 !( .8$ f#8 `-,-, = .8( 2 !

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|!

F 6 z = G 8 -,$.% 6 =(U C@j. &$# %6 U % 2 % YH . X .;. 0 D .8$ =1 % !!!7F 4 D-, f D - &$#%<6I8 U Y6I 5(. -6C !@i ;-,% U V $. 2 77-,: ((# B 6 -, (. !

nP #%O%7[ E% F [KYL#.705N,%'Pn0/ 0/KML 1Q N6p=sur42S # 2.8. 2 !OT M_n,NKoAxfvPOs6H;JDvr Hcv NoAw 5NpNsur f C

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

= "

(S(S(0)))→ ( (S(S(0))))∗→ (S(S(S(S(0))))) → 0

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!

nP #%O%7[ E* F [KYL#.705N,%'Pn0/f0H1L 46"B0 Q N6p=surDZS 2#8 . . !8T M_n,NKoAx;vOs6H'JKv r"v w*o/p M_v 5N6p=sur f C

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% +→

CK2 . , U " →

∗→

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 [email protected] 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 $

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i / : f ZC X =6 -G ]%N. $. /% 1 = $ % 6I85C %: V U . ; !

9 ?

$. " =9 . " h f# . ; ,/ % $ RC 2 ( A> & ( A> _ 7 ; &$ !

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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

$ !

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$ C !

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> 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

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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

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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

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(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 ;# ;. !

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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 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

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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

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(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

⊂%

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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%

2 V $ * X 0 Y 4;. .D.f.8 0-, U 2I8 ' 0 ,G !& 9 -G 8 ! eS !&T "3!P/[T-HK 0 !0& %7"B0 ) J &:$>-+( Z$ G-, f.2. 2. &0 = @ (' /1 2 / = /$. ` !

nP #%O%7[ 8 E 0HK % "546.'"5 !.741 .'0 7C

f Bw vAN1s M_pNxCM l(xfp xfv U B E f h 8(U

Io

In

/A

<-, W .

x B-,

f(x) ∈ Io%A(x)

2 K 0 .2 ! W .

y4-,

f(y) ∈ In%A(y)

; !

nP #%O%7[ 8 V & P/"5 T1&%/.[0/KY0/IPH #KY0HT41 .'0 T 7Ji C J2M l(m_w_sutNv N6v rIH r/l(N Jmp xfv w C 8f' !

T . !.,/1. ("C -,

f 8f' % U K $

A′ -, / . 2 -, .;. Io

!$+ *% Jrel(N J m2v m = NS5H 0C mPJPM l(m_w sutOv .;.

Io!

( TZL [1&%O&%'( % $)

∈ RE FJANnNr/w2H*m2p Hzs n=r FD V Z.]- f66.8 (. C ) !Gb Z(> E

Io* 8X .2 / `- B-* $ 6 /. .2 In

* X .2 /;-,-, < %/ ;X * $ !

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.!

T /.# 7H ;. / 7 BM( ) M(6> 9 + / 7-, ; f 7H . = NU `-,-, 2 0 H n=l&w 0.. D z $6. ;-,6 XW !AT C 3-, / 6 .Va. .V * $6.`- 6 C 90 =1 V * =U `- ' =!jT 2 U 7$. !ji 4(. 1 f# 8\ .

6$ !n,NKoAl(Hcp,H*s nNrAp=x !n,NKoAx;vOs6H " F !(W ; 8) W [A$C,+ !

;B:

* & 7(5 B 5 $ _J Z $.2#8 1 6J 2 2.8-, z ;S 6!HF 6

f ;D 4(.; ;. . 2. * X 0 / % BC 7H $ ; 2 - 7 ;. . 2. I 7(. `(. !/W U % -, ZS 2#8 . . 2 f fC 2. C N9 !

z . / 4 0 !,i @ 8% 5 ;. 2. 2. 27 X 0 $ D-, !eW U 8 % 2 K-, * C

min%A-, /

2 2.

f$ : (U !NT 2S # 2.8. 2 2 :S 2 "

(0, y) → y " (x,0) → x "

(S(x),S(y)) → S( "

(x, y))

F ZS 2#8B. . 1 U 2. O(min(x, y))

!eT C min

f . F0 KI1 !

<T !OF .3)yF AB@-+&- ZS 2#8! . . 2 'c 2 . f fC2. fC @-, U 46 #"B # 2S 2#8 1 . . 2 ; C 2. CG-,

min

O(min(x, y))!+ *%

$ ; C . C #Z / C !ei

2.8-, % f.2.-, 2 $ ; fC . fC 8 2.I !ji A@ 8%5-,@ C 6 27 / D-, G9 2. `-,!D

T $ * ;S =6 G / 8 G(. !=b (C H $ 2(> 1 7 2. Z8%( C = = H-, 2(> =6 / % 2(> 6 ;# 4surIHcv rIH*s nNr/rjvxux;vANvr=H / - Z. ;-,!

;$=

L "-,% 6 * & . .8 81 ]% VH 2S 2#8 D 2.8. 2 /(.;= .$. z% U 4 `-, 2 2 " 2S 2#8 !e 2. 0 $UE `-, _ . ;S !( = `- U 2I8 . % Q;\'ZF DW

XF !/Q !$T

!) T<,L+ c 9 + . 2.f _ ! # 2. B ;-, X-, 2

" B U .

(. = . .8 !]FD Z'cN'c :-, -,@C : U 2.% U 0 $ 8 `-B /.!

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

_ 2Σ = F

⋃

C!

_ F

2F

;$..6 D2. .8 c ≺F

c ∈ C ∈ F

c ≈F c′!

;5?

nP #%O%7[ 8 % ]T!T0H1 1& ! .'0/1]""3#0/KY%'#1Q

F2. .8

•∗0 f#7 _ 2 _ 2

2 0 = Z'cN'c K-, ≺

_ Z 2 %≺∗ _ 2 21 !/

•∗

*unNm =m2vDw l(mxfvw5M/vAN1sur$wJ

j-,t = (t1, . . . , tn) J

&(s1, . . . , sm) = s

* 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!

≈F& % , & ∈ F %

(t1, . . . , tn) J∗ (s1, . . . , sm)

j%tj J s

!

? (U Z -,G9 . %&TW YW%89 G 6 T<W Z.Z * U ;-,K-, ;C 0 XC / U ;-,-,* X Z ]-, Z. hW !&F 2 ]% 9 z e#% Z f#8 A. e-, =$U _ Z I X. f#8 A. A-z _ 4X !

81304130 %& L# ! *I nP #%O%7[ 8 a ]T!T03.'0HN,%'"52[T4RL 3!% !0 Q

≺ Z 8X

E!GQ oAmnNxfn=r :,vANvr=HXxfvPOsQM_n:m2po &s l/v

≺l _ 2 H .;.

E-

(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 !

nP #%O%7[ 8 * ( T! _ U ;-,Z 8B0 % vPOsQM_nP:=mpo (sQM p H m v m sur :(1 * _ 2 f. U `-! = Z'cN'c -,

t = (t1, . . . , tn) ≺lpo&(s1, . . . , sm) = s0 , & ∈ F ⋃

C1 * 4 / 2

b U j

-t lpo sj

! , & ∈ C, m = n

U B X π

-, j%tj lpo sπ(j)

U i

-ti ≺lpo sπ(i)

! ≺F

& ti ≺lpo s

i!

≈F& % , & ∈ F %

(t1, . . . , tn) ≺llpo (s1, . . . , sm)

j%tj ≺lpo s

!

W 3#P/ZT-/KY0 ) J46KY%7 0 . ^ * vw HGl(r n=m Nmvv.Hcv m NsurjpNs`w nNr N,0HKYL#.70 * T 2S # B 2.8. 2 Z 8%e- C EH + _ % TW !

(0, n) → S(n)

(S(m),0) →

(m,S(0))

(S(m),S(n)) →

(m,

(S(m), n))

;$@

W 3#P/ZT-/KY0 )[) + 0/%'0HTKY46# 0/%7Q8 $pl/vhw Ow2H N6v vYmPJPJ2M m s6Hzl(mv l(s Hcvm N1surAv1o$p=m Mp=xCM l(xfvhl(rjvnNr MAH*s nNrNl(x H*s oAx;vYmJ2M l(m_w_sutNv3v HFM $pl/vnNr MAH*s nNrNl(x?Hzs ojxfvBmPJPM l(m_w sutOv1vw2HM_pNxCM l(xfp xfvo/pNm RpNl9NnNsur$wl(rEw Ow H N6v vXmJ2J2Mm s6H*l(m2v l(s Hcvm N1surAvo$p=m

TW≡&('*)+% !

813041 'L ! &*= nP #%O%7[ 8[Q ]T!T03Kh #.O&%'1&0HQ ≺

2 X E

!&Q ojm2nNx;nNr :,v N6v rIH N1l(x H*s`w vAH≺m Z H .;.

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 _

2N1 2. $ rjn,Nmv n M2M l(m m2vr M_v wB -0.f.8 K -,0X !

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. '

!T! ;(. 4. . (( G^ !,Q;6 3) Q; @B@,+ %(. e. Q !L Q !&F ) LF A5;(%L A$C,+ ? !&T ! '*) ! *) T A$C$%&T A5:&%&T 3A/?,%(T 3c5c=H%&T 3c5c&%(T A5A-+ %^ !AQ & & ) L>0QRcBc-+ !/^ C ) ^ =C'A/?,%$L^ c5c-+ !

" $ 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)

0/KYK 0 ) * voAm2n=xfnNr:v Nvr=H(o/pNm<o$v m Nl(Hcp,H*s nNr/wvw Hl(rAvm2vw Hzm sQM H*s nNr =lojm2nNxfn=r :,vANv rIHN1l(x H*s`wv H M v w H 3=sum2v l/v A ≺m′

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

*+46 4 M_po Hzl(m2v vw H 3Nsum2v l/v7M/p l/vYw Ow HANv vYmJ2J2Mm s6H*l(m2v l(s H v m N1surAv1o/pNm M_pNx Ml(xfvl(rAv4nNr4M H*s nNr v v H l/v.M $pl/v nNr MAH*s nNr-v vw H M_pNx Ml(xfp x;vo/pNm7l(rEw Ow H Nv vXmJ2J2Mm s6H*l(m2v l(sHcvm N1surAvo$p=m

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 ;. !

N,0HKYL#.70 )() F (KML#.70HN,%OPn%'KYL#.7%'"B%'0n0HI%70/%7[#!4Z.'%O&P T 2S # 2.8. 2 Z B ; Z' 2 1*0 /14E (U2 !( X 8(U Z

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 .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 =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 !

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 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 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-,

. 

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 ! ;# "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

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) 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!

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T $# 0 $# * ;S `-,K U .K =6 8% 7 U 2.7 XC ;. Y * $ ; ;. 2.!

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$ 8 C ;S = 8VOS!

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Analyse de la complexit© et transformation de programmes