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ÈÓaA l>(Turing machine)\¦ צ#"f TMܼРÂÒØÔlÐ . TM
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r ]j#Qlü< _ {9ܼР½¨$í÷&HX<, _ {9 K×¼H ýaĺР¹¡§f{9 ú e
¦ {9¦ æ¼H כ s 0px . ÕªaË> 1`¦ Ð.
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. #l\"f
1. QH ©I[þt_ Ä»ôÇ |9½+Ës¦ 2. ΣH {9§4 ·ú Ï@s¦
3. ΓH _ {9 ·ú Ï@s¦ /BNÑþ /åJ #\¦ í<ÊôÇ.
4. δH Q′× Γ\"f Q× Γ × {L, R, S}ܼÐ_ <Êús.#l\"f Q′ = Q − Hs.
5. q0 ∈ QH íl ©Is¦
6. H ⊆ QH &ñt©I[þt_ |9½+Ës.
TMÉr δ s <ÊúsټР&ñ ¸Ðs. δ_ &ñ_\"f LÉr K
×¼ ¢,aAá¤Ü¼Ð RÉr K×¼ ¸ÉrAá¤Ü¼Ð ôÇ ñß ¹¡§fsH כ `¦, SH Õ
ª o\ QÁºH כ `¦ _pôÇ. Ä»ôÇ ¸Ð ?/o¾º2£§ ¸Ð
H {9§4`¦ {9¦ &ñt tëß, ÈÓaAl>H {9§4`¦ {9
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r Êê\¸ >5Åq '½+É Ãº eܼټР&ñt\¦ "îr H &ñt©I
3
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9
כ¹ . TMÉr ±p ¢,aAᤠñß\"f ¢,aAá¤Ü¼Ð ¹¡§fst ·ú§H¦ &ñ ô
Ç.
TM_ ©S!(configuration)Ér &³F ©I q, _ {9_ ?/6 xõ K×¼_ 0A u
\ _K &ñ)a. K×¼ ¢,aAá¤_ Û¼àÔaAs us¦ K×¼ ©_ /åJ
as¦ K×¼ ¸ÉrAá¤_ Û¼àÔaAs vs (vH ¸ÉrAᤠ=åQ_ /BNÑþ
/åJt í<ÊôÇ), &³F ©S!Ér (q, uav)Ð ³ðr)a. {9§4 Û¼ àÔaAs w{9 M:, r ©S!Ér (q0,#w)s.
U
dV 1 _ {9 ½¨$ís #aaaaa\"f K×¼ ¸ÉrAá¤Ü¼Ð
#aaaaa ÷&¦, s#Q"f ¢,aAá¤Ü¼Ð "é¶A_ ¸_þvs )a.
#aaa\"f ¸ÉrAá¤Ü¼Ð #aaa# ÷&¦, r ¸ÉrAá¤Ü¼Ð
#aaa##s )a. s#Q"f ¢,aAá¤Ü¼Ð ¿º "é¶A ¸_þvs )a. Ça
+ 2 TM M = (Q, Σ, Γ, δ, q0, H)s &ñ_ H #Q L(M )Ér 6£§ õ
°ú .
L(M ) = {w ∈ Σ∗ : (q0,#w) ⊢∗M (h, ∗)}
#
l\"f h∈ Hs¦ ∗H e__ _ {9 ½¨$í`¦ ³ðrôÇ.7£¤
& % L(M )Ér M`¦ &ñt > ëß×¼H Û¼àÔaA_ |9½+Ës. L`¦ &ñ_ H
TM M (7£¤ L= L(M ))s >rF H #Q L`¦ F) \P #Q¦ ÂÒ
É r. U
dV 2 L = {0n1n : n ≥ 1}`¦ &ñ_ H TM M`¦ ½¨ .
MÉr ¸ÉrAá¤Ü¼Ð ôÇ ñß "f, ±p ¢,aAá¤_ 0`¦ xР˨¦ (q0), ¸Ér A
á
¤Ü¼Ð ' # ±p ¢,aAá¤_ 1`¦ ¹1Ô yР˨¦ (q1), ¢,aAá¤Ü¼Ð '
# ±p ¸ÉrAá¤_ x\¦ ¹1ÔH (q2) \O`¦ ìøÍ4¤&hܼРú'ôÇ.
¢
¸ôÇ q0Ér 0s \O`¦ M: ¸ÉrAá¤Ü¼Ð ' # 1¸ \Oܼ (q3) &ñt ô
Ç. \V: 000111 . . . x00y11.
½
¨^&hܼРM = ({q0, . . . , q5}, {0, 1}, {0, 1, x, y, #}, δ, q0,{q5})s¦
5
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0 1 x y #
q0 (q1, x, R) (q3, y, R) (q0,#, R) q1 (q1,0, R) (q2, y, L) (q1, y, R)
q2 (q2,0, L) (q0, x, R) (q2, y, L)
q3 (q3, y, R) (q5,#, S) q4
#
l\"f ¸H q∈ Q, a ∈ Γ\ @/ # ñßÉr δ(q, a) = (q4, a, S)\¦ _
pôÇ. 7£¤ q4H ÁºôÇ íH¨8 H ©Is.·ú¡Ü¼Ð ÁºôÇ íH¨8 ©I
H ³ðr t ·ú§lÐ ôÇ. M`¦ ÕªaË>ܼР?/ ÕªaË> 2õ °ú .
6
& %
#/#, R y/y, R y/y, L
y/y, R
#/#, S
x/x, R y/y, R
1/y, L 0/x, R
q q
q q
0/0, L 0/0, R
2
3 5
q
0 1ÕªaË> 2:
s
ü< °ú s TMÉr /åJ\¦ Ér /åJР˨"f >íß½+É Ãº e.
7
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{anbncn : n ≥ 1}`¦ &ñ_ H TM¸ ðøÍtÐ ½¨½+É Ãº e.
U
dV 3 L = {ww : w ∈ {0, 1}+}`¦ &ñ_ H TM`¦ ½¨ .
$ {9§4 Û¼àÔaA`¦ ·ú¡ÂÒìrõ z´»ÂÒìrܼР¾ºl 0A # ·ú¡ÂÒìr_ 0, 1Ér yy x, yÐ z´»ÂÒìr_ 0, 1Ér yy x′, y′Ð õH.\V:
110011 . . . yyxx′y′y′. ÕªaË> 3.
0 1 x y x′ y′ #
q0 (q1, x, R) (q1, y, R) (q4, x′, L) (q4, y′, L) (q0,#, R) q1 (q1,0, R) (q1,1, R) (q2, x′, L) (q2, y′, L) (q2,#, L) q2 (q3, x′, L) (q3, y′, L)
q3 (q3,0, L) (q3,1, L) (q0, x, R) (q0, y, R)
q4 (q4, x, L) (q4, y, L) (q5,#, S) s
Êê ·ú¡ÂÒìrõ z´»ÂÒìr`¦ ´úÆÒ#Q ÐH כ Ér ·ú¡_ \V]jü< q5pw .
& %
#/#, S #/#, R
x/x, R x’/x’, L
0/x, R 0/x’, L
y/y, L
1/1, L 0/0, L 0/0, R
1/1, R
y’/y’, L
1/y’, L
# /# , L
q q
q q
y/y, R x’/x’, L
1/y, R
y’/y’, L
x/x, L
q
2q
34 5 0 1
ÕªaË> 3:
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¸ôÇ TMÉr ¸H {9§4\ @/K &ñt "f &ñt©I\ #Q\¦
9
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%
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ñ
_½+É Ãº e.
Ça
+ 3 #Q L\ @/ # TM M = (Q, Σ, Γ, δ, q0,{y, n})s
• w ∈ Ls (q0,#w) ⊢∗M (y, ∗)
• w 6∈ Ls (q0,#w) ⊢∗M (n, ∗)
\
¦ ëß7ᤠMs L`¦ &ñôǦ ´úôÇ. L`¦ &ñ H TMs >r F
L`¦ F) #Q¦ ÂÒÉr. U
dV 4 L = {0n1n : n ≥ 0}\ @/ # \V 2_ TM\"f
H = {q4, q5}s¦ q4 = n, q5 = ys sH L`¦ &ñ H TMs. Ça
h 1 F) #Q 7áxÀÓH F) \P#Q 7áxÀÓ_ ÂÒìr|9½+Ës.
¤
ÃZ. L`¦ F) #Q¦ , L`¦ &ñ H TM M s >rFôÇ.
M \"f ©I n`¦ ÁºôÇ íH¨8 ©IÐ ëß[þt¦ yëß`¦ &ñt ©IÐ
, L`¦ &ñ_ H TM`¦ %3H. "f LÉr F) \P #Qs.
2
TMÉr {9§4\"f ئ§4`¦ >íß H ¸½¨Ð 6 x|¨c ú¸ e. s â
10
& % ĺ TMÉr #Q" <Êú\¦ >íß > )a. ú\"f úÐ_ <Êú
\
¦ Ò¦ M:, ú\¦ éß{9ZOܼР³ðrôÇ. 7£¤ ú nÉr 1nܼ
Ð ³ðr)a.
Ça
+ 4 f\¦ Σ∗\"f Σ∗Ð_ <Êú¦ . TM M = (Q, Σ, Γ, δ, q0,{h})s
(q0,#w) ⊢∗M (h, #f (w))
`
¦ ëß7ᤠMs f\¦ >íßôǦ ´úôÇ.s âĺ f\¦ >íß0px
<
Êú¦ ÂÒÉr.
<
Êú f {9§4ܼРk >h_ ú\¦ 2[½+É M: 7£¤ f (n1, . . . , nk){9 M:, {
9
§4Ér 1n10 · · · 01nkÐ ³ðr)a.
U
dV 5 8 l <Êú f(n, m) = n + m`¦ >íß H TM`¦ ½¨ . {
9
§4s 1n01msټР0`¦ 1Р˨¦ ±p =åQ_ 1`¦ /BNÑþܼР˨
11
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a. TM_ s<ÊúH 6£§õ °ú . ÕªaË> 4.
0 1 #
q0 (q1,#, R) q1 (q1,1, R) (q1,1, R) (q2,#, L) q2 (q3,#, L)
q3 (q3,1, L) (h, #, S)
0/1, R 1/1, R
#/#, L
#/#, R 1/#, L
1/1, L
#/#, S
q
0q
1 2q
3
h
q
ÕªaË> 4:
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dV 6 YL l <Êú f(n, m) = nm`¦ >íß H TM`¦ ½¨ .
& % {
9
§4 ×æ ns 0s _ {9_ ?/6 x`¦ tĺ¦ =åQèß (7£¤ 0`¦ ئ§4).
ns 2 s©s m`¦ n− 1 4¤ôÇ Êê _ {9\ eH 0`¦ tĺ"f 1`¦ ¢,aAá¤Ü¼Ð s1lxr. ÕªaË> 5.
n= 2, m = 3{9 M: _ {9_ o
#110111#
#100111#
#a001110#
#a0011b01#
#a001110111#
#000111111##
#111111#
13
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when n=0
remove 0
copy m
1/1, R
#/#, L
0/0, R 0/#, L
1/#, L
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#/#, R 1/a, R
0/0, R 1/1, R
1/1, R
#/#, L
1/1, L
#/#, L 0/1, R
1/1, R
#/#, S
1/#, L a/0, L
#/0, L
1/b, R
1/1, R 0/0, R
#/1, L
0/0, L 1/1, L
b/1, L 1/1, R
0/0, L 1/1, L 1/0, L
0/0, R
h
0/0, L
ÕªaË> 5:
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³F 6 x÷&H (ÉÓ' Áº%Á `¦ >íß½+É Ãº eH? (ÉÓ'_ l>
#Q(machine instruction)\¦ Ð load, store ü@\ gË:íßõ ¸| ë
H`¦ ú' l 0AôÇ jump 1pxܼР½¨$í÷&#Q e. ÈÓaAl>¸ {9¦ j
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t ú e¦, gË:íß`¦ ½+É Ãº e¦ ]j#Ql\¦ :xK ¸| ëH`¦ ú'½+É Ãº eܼټР&³F (ÉÓ' H ¸H >íß`¦ ½+É Ãº eH כ `¦ Æ
Ò:r½+É Ãº e.
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6£§`¦ Ðs¦ ôÇ. 7£¤ l0px`¦ 8K"f >íß0px§4`¦ 7£xrvH
\"f ôÇ>\ ¸²úÙþ¡H כ s.
15
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k_ {9 ÈÓaAl>H k >h_ _ {9`¦ . "f ôÇ \ k >h_ /åJ
\¦ {9¦ ¹¡§fe`¦ &ñôÇ. 7£¤ s<Êú δH Q′× Γk\"f
Q× (Γ × {L, R, S})kܼÐ_ <Êús. k_ {9 TM_ ©S!Ér &³F © I
ü< k >h _ {9_ ¸ªÜ¼Ð ½¨$í÷&¦, (q, u1a1v1, . . . , ukakvk)Ð ³ð r
)a. r½+É M:, {9§4Ér 'Í P: _ {9\ ÅÒ#Qt¦ Qt _ {9Ér q
#Q e. 7£¤ r ©S!Ér (q0,#w, #, . . . , #)s. <Êú\¦ >íß½+É â ĺ, k_ {9 TM_ ئ§4Ér 'Í P: _ {9\ ÅÒ#Qt¦ Qt _ {9_ ?/6 x
É
r Áºr)a.
U
dV 7 2_ {9 TM\"f s<Êú δ(q, a, b) = (q, (a, R), (a, R))H 'Í P
: _ {9\"f {9Ér /åJ\¦ ÑütP: _ {9\ 漦 K×¼\¦ ¸ÉrAá¤Ü¼Ð ¹¡§ f
sH כ s. U
dV 8 k_ {9 TMÉr "é¶A TMÐ çßéß > >íß`¦ ½+É Ãº e. \V
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¦ [þt#Q #w\¦ #w#wР˨H >íß`¦ ¦9 . "é¶A TMÉr ýa Ä
ºÐ ¹¡§fs"f w\¦ ôÇ /åJm ` &hH. n = |w|¦ "é¶ A
TMÉr O(n2)_ s 9כ¹ .
& % 2_ {9 TMÉr s \O`¦ 6£§õ °ú s O(n)_ s\ ½+É Ãº e.
1. ¿º K×¼\¦ ¸ÉrAá¤Ü¼Ð ¹¡§fs"f 'Í P: _ {9_ ?/6 x`¦ ÑütP: _ {
9
\ &hH. 7£¤ _ {9_ ©S!`¦ (#w, #)\"f (#w#, #w#)Ð
õH.
2. ÑütP: _ {9_ K×¼\¦ ¢,aAá¤Ü¼Ð ` |. 7£¤ (#w#, #w#)Ð õH
.
3. ¿º K×¼\¦ ¸ÉrAá¤Ü¼Ð ¹¡§fs"f ÑütP: _ {9_ ?/6 x`¦ 'Í P: _ {
9
\ &hH. 7£¤ (#w#w#, #w#)Ð õH. 4. 'Í P: _ {9_ K×¼\¦ ¢,aAá¤Ü¼Ð ` |.
s
ü< °ú s :£¤&ñôÇ >íß`¦ H ¼#s$í(¢¸H rçß)\ e#Q"fH k_
{
9 TMs "é¶A TMÐ `¦ ú etëß, #Q" >íß`¦ ½+É Ãº eÖ¼
\O
Ö¼_ 'a&h\"fH ¿º >h 1lx{9 . 7£¤ k_ {9 TMs ½+É Ãº eH
>
íßÉr "é¶A TM¸ ½+É Ãº e.
Ça
h 2 k_ {9 TM M = (Q, Σ, Γ, δ, q0, H)ü< 1lx1pxôÇ "é¶A TM M′s
>rFôÇ (7£¤ L(M ) = L(M′)).
17
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a
# a b # a b a a
# # #
a b a a #
# a
# a b #
ÕªaË> 6:
¤
ÃZ. "é¶A TM M′s M `¦ ¿9ª?/?/l 0AK"fH k _ {9_ ?/6 x`¦ Ð
'
a ¦ e#Q ôÇ. s\¦ 0AK M′_ _ {9`¦ k >h_ àÔÏþ(track)ܼ
Ð èH. ÕªaË> 6. ¯Γ = {¯a: a ∈ Γ}¦ . ¯aH M \"f K×¼
a/åJ\ e6£§`¦ ·p. ÕªaË>õ °ú s i P: àÔÏþ\H i P: _ {
9
_ ?/6 xõ K×¼_ 0Au\¦ l2¤ôÇ.
k >h_ àÔÏþ`¦ 6 x H כ s "é¶A TM_ &ñ_\¦ #Á #QH כ s m
¦, éßt M′_ _ {9 ·ú Ï@ Γ′Ü¼Ð Γ ∪ (Γ ∪ ¯Γ)k`¦ 6 x H כ ÷r
18
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. 7£¤ M′s (¯a, b)\¦ {9ܼ 'Í P: _ {9_ s 0Au\ a e¦ # l
\ K×¼ eܼ 9 ÑütP: _ {9\H b eH כ `¦ ·ú ú e.
{ 9
§4 w ∈ Σ∗ ÅÒ#Qt M′Ér 6£§õ °ú s 1lxôÇ.
1. {9§4`¦ m ¸ÉrAá¤Ü¼Ð ` | Êê _ {9`¦ M _ k _ {9 +þAIÐ
õH. w = abaas¦ M s 2 _ {9 TMs, M′_ _ {9`¦
#abaa\"f ##abaaÐ, Õªo¦
#( ¯#, ¯#)(a, #)(b, #)(a, #)(a, #)ܼРõH.
2. M _ >íß`¦ ¿9ª?/·p. M _ ôÇ s\¦ ¿9ª?/?/l 0AK M′Ér 6
£
§_ ¹¡§fe`¦ ôÇ.
1. _ {9`¦ t"f M _ k K×¼\ eH /åJ\¦ {9H.
M _ ©I\¦ l%3 ¦ k K×¼\ eH /åJ\¦ {9¦ , M _ s\¦ ·ú ú e.
2. _ {9`¦ t"f M _ s\¦ ìøÍ%òôÇ. ëß M s #Q" K
×¼\¦ ¸ÉrAá¤Ü¼Ð ¹¡§f#"f M′s #s [þt#QeH ñßܼР`
M Ér $ #\¦ (#, #)Ð õH Êê M _ s\¦ ìøÍ
%ò ôÇ.
19
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3. M s &ñt , M′Ér 'Í P: àÔÏþ\ eH ?/6 x`¦ "é¶A _ {9_ +þA I
Р˨¦ &ñtôÇ.
2
ª
Aᤠ\P2; _ {9 TMÉr ª~½Ó¾ÓܼР\P2; _ {9`¦ ôÇ >h t¦ e.
íl\ {9§4s _ {9\ ÅÒ#Qt¦ K×¼H {9§4 Ð ¢,aAá¤_ /BNÑþ\ Z
~# e¦ &ñôÇ. s TMÉr 2_ {9 TM\ _K ¿9ª?/èq ú e.
ª
Aᤠ\P2; _ {9`¦ {9§4_ Ð ¢,aAá¤\"f ]XìøÍܼР]X#Q"f ¸ÉrAá¤_
?
/6 xÉr 'Í P: _ {9\ $© ¦ ¢,aAá¤_ ?/6 xÉr ÑütP: _ {9\ %iܼÐ
$
©ôÇ Êê ¿9ª?/?/ )a. Óüt:r s 2_ {9 TMÉr r "é¶A TMܼÐ
¿
9ª?/èq ú e.
kK×¼ TMÉr ôÇ >h_ _ {9\ k>h_ K×¼\¦ t¦ e¦ k K×¼\ e
H k >h_ /åJ\¦ {9¦ s\¦ &ñôÇ. kK×¼ TM`¦ s6 x
#w\¦ #w#wР˨H \O`¦ 8 çßéßy ½+É Ãº e.
1. ¿º K×¼\¦ #w#_ 0Au\ Z~H.
& % 2. ¿º K×¼\¦ ¸ÉrAá¤Ü¼Ð ¹¡§fs"f 'Í P: _ {9_ ?/6 x`¦ ÑütP: _
{ 9
\ &hH. 7£¤ #w#w#Ð õH.
kK×¼ TM¸ k_ {9 TMõ ðøÍtÐ K×¼_ 0Au\¦ ?/H ¯Γ\¦ s
6 x # "é¶A TMs ¿9ª?/èq ú e.
V
3 â j Ú r Ç a Ï Úÿ b e N
q
&ñ TMÉr (Q, Σ, Γ, ∆, q0, H)Ð ½¨$í÷&¦, &ñ TMõ Ér &hÉr
∆ (Q′× Γ) × (Q × Γ × {L, R, S})_ ÂÒìr|9½+Ë s'a>s.
Ça
+ 5 q&ñ TM M = (Q, Σ, Γ, ∆, q0, H)s &ñ_ H #Q L(M )Ér 6£§õ °ú .
L(M ) = {w ∈ Σ∗ : (q0,#w) ⊢∗M (h, ∗)}
Ms q&ñsټР&ñt ©IÐ [þt#QH >íß õ&ñs ¸ e
ܼ w∈ L(M )s.
21
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h 3 q&ñ TM M = (Q, Σ, Γ, ∆, q0, H)ü< 1lx1pxôÇ "é¶A TMs
>rFôÇ.
¤
ÃZ. M Ér q&ñ TMsټРôÇ ©S! C\"f #Q >h_ ©S! C1, . . . , CiÐ s½+É Ãº e. #l\"f iH q&ñ TM_ &ñ_\ _ K r = |Q| × |Γ| × 3 s _ °úכ`¦ °úH.
M _ >íßõ&ñ`¦, ©S!`¦ &ñ&hܼР(r©S!`¦ ÀÒàÔÐ) s\¦ çß
ܼР³ðr # àÔo T Ð èq ú e. s àÔo\"f ôÇ &ñ&h_
d_ >húH þj@/ rs. "f s àÔo_ ¸H &ñ&h`¦ &ñú 1, . . . , r_ Û¼àÔaAܼР³ðr½+É Ãº e. ÀÒàÔH 1s¦ ÀÒàÔ_ dÉr 11, . . . , 1rs¦ 11_ dÉr 111, . . . , 11rs. Óüt:r 11_ ds rÐ
&hܼ 11r\ K{© H &ñ&h(©S!)Ér \O> )a. ÕªaË> 7.
22
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11r . . .
1r1
1rr . . . 11
1r 1
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. . .
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Ôo T _ &ñ&h\ K{© H ©S!`¦ ¸¿º ëß[þt#Q "f Õª ×æ\"f
&
ñ
t©S!`¦ ëß &ñt 9¦ ôÇ. T Ä»ôÇ àÔo u´ ú e
23
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¼Ù¼Ð T \¦ U·s ĺ ÃÐÒoܼР¹1Ô T \ &ñt©S!s eHX<
¸ Ô¦½¨ ¦ s\¦ ¹1Ôt 3lw½+É Ãº e. "f M′Ér T \¦ -q ĺ
Ã
ÐÒoܼР¹1Ôçß.
T \¦ -q ĺ ÃÐÒoܼР¹1Ôl 0AK"fH {9ìøÍ&hܼÐ
Ç©(queue) 9כ¹ôÇX<, 6£§õ °ú s Ç©\¦ 6 x t ·ú§¦ rçß`¦ 8
6 x # K½+É Ãº e. 7£¤ T _ y &ñ&h`¦ ?/H {1, . . . , r} © _
Û¼àÔaA`¦ -q ĺ ÃÐÒo_ íH"fÐ m ëß[þt"f y Û¼àÔaA
\
K{© H ©S!`¦ r©S!\"fÂÒ' (s ©S!`¦ l%3 t ·ú§¦ e
ܼټÐ) ëß[þt )a.
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