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8Q 1  Y

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(1)

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Ç a

Ä ª¥ U ># a

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 1 ⠉ Ç a Ä ªÐ Ï 

úd”(arithmetic expression)õ Ä» > &ñ½©d”(regular expression)`¦ &ñ_ôÇ.

Ç a

˜+ 1 ·ú˜  Ï@ Σ ©œ_ &ñ½©d”õ Õªכ s ³ðr H |9½+˓Ér 6£§õ

° ú  

1. ∅H &ñ½©d”s“¦ /BN|9½+Ë`¦ ³ðrôÇ. 2. ǫH &ñ½©d”s“¦ {ǫ}`¦ ³ðrôÇ.

3. yŒ• a ∈ ӍH &ñ½©d”s“¦ {a}\¦ ³ðrôÇ.

4. rõ s &ñ½©d”s“¦ yŒ•yŒ• Rõ S |9½+Ë`¦ ³ðrôÇ€, (r + s), (rs), (r)H &ñ½©d”s“¦ yŒ•yŒ• R∪ S, RS, R |9½+Ë`¦ ³ðrôÇ.

&

ñ

½©d” r_ ƒ#Q L(r)“Ér rs ³ðr H |9½+Ës.

&

ñ

½©d” aü< /åJ aH "f–Ð Ér כ `¦  ?/, °ú “Ér +þAIe”`¦ Ä»_

. 7£¤ L(a) = {a}s“¦ L(ǫ) = {ǫ}s.

(2)

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ñ

½©d”\"f ƒíߖ_ íH"f\¦ ∗ · +–Ð &ñ € ´ú§“Ér F‹c ñ\¦ Òqt|Ä̽+É Ãº e”

. 7£¤ ((1(0)) + 0) = 10+ 0s“¦ s &ñ½©d”“Ér {0, 1, 10, 100, . . .}`¦



 ·p. ¢¸ôÇ 6£§õ °ú “Ér ƒíߖ_ ZOgË:`¦ &h6 x½+É Ãº e”.

• + ƒíߖ“Ér ½+Ë|9½+Ë`¦  ?/Ù¼–Ð, “§¨8ŠZOgË:õ ½+ËZOgË:s $íwn ô

Ç. 7£¤ r + s = s + rs“¦ (r + s) + t = r + (s + t)s.

• · ƒíߖ“Ér ]X½+Ë`¦  ?/Ù¼–Ð, “§¨8ŠZOgË:“Ér $íwn t ·ú§“¦ ½+ËZO g

Ë:“Ér $íwnôÇ. 7£¤ (rs)t = r(st)s.

• +ü< · s\ ìrCZOgË:s $íwnôÇ. 7£¤ r(s + t) = rs + rts“¦ (r + s)t = rt + sts.

Ça

˜+ 2 &ñ½©d”ܼ–Ð ³ðr÷&H ƒ#Q\¦ &ñ½©ƒ#Q“¦ ÂҏÉr. U

dV 1 &ñ½©d”

(00)(11)1

“ É

r ‹Œ•Ãº >h_ 0 6£§\ f.Ëú >h_ 1s š¸H Û¼àÔaA_ |9½+Ë`¦ ³ð r

ôÇ.

3

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dV 2 {0n1m : (n + m)“Ér ‹Œ•Ãº}\¦  ?/H &ñ½©d”`¦ ½¨ .

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½©d”_ ƒíߖ +H ¢¸H_ _psÙ¼–Ð,&ñ½©d”`¦ ½¨½+É M: #ŒQ

ĺ–Ð ¾º#Q"f ÒqtyŒ• H כ s a%~. 0s ‹Œ•Ãº >h š¸“¦ 1s ‹Œ• Ã

º >h š¸ŽHt 0s f.Ëú >h š¸“¦ 1s f.Ëú >h ü< ÷&Ù¼–Ð, ²úš

“ É

r (00)(11)+ 0(00)1(11)s.

· ú

¡Ü¼–Ð \V]j\"f ·ú˜  Ï@\ @/ôÇ Z>Ér ƒ/åLs \Oܼ€ Σ = {0, 1}s.

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dV 3 1s ôÇ >h ¢¸H ¿º >h e”H Û¼àÔaA_ |9½+Ë`¦ ³ðr H &ñ½© d

”

`¦ ½¨ .

1s ôÇ >h e”H Û¼àÔaA_ |9½+˓Ér 010s“¦, 1s ¿º >h e”H Û¼àÔaA _ |9½+˓Ér 01010sÙ¼–Ð Ñüt`¦ ½+Ëu€ 010(ǫ + 10)s.

U

dV 4 U´s 3_ Cú“ Û¼àÔaA_ |9½+Ë`¦ ³ðr H &ñ½©d”`¦ ½¨

. U´s 3“ Û¼àÔaA`¦ ³ðr H &ñ½©d”s

(0 + 1)(0 + 1)(0 + 1)sÙ¼–Ð ²úš“Ér ((0 + 1)(0 + 1)(0 + 1))s. (0 + 1)(0 + 1)(0 + 1)`¦ צ#Œ"f (0 + 1)3ܼ–Ð æ¼l–Ð ôÇ.

(3)

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dV 5 11`¦ ÂÒìrÛ¼àÔaAܼ–Ð °úH Û¼àÔaA_ |9½+Ë`¦ ³ðr H &ñ½© d

”

“Ér (0 + 1)11(0 + 1)s. U

dV 6 111s ˆÃÌ ôÇ   H Û¼àÔaA_ |9½+Ë`¦ ³ðr H &ñ½© d

”

`¦ ½¨ . U

dV 7 11`¦ ÂÒìrÛ¼àÔaAܼ–Ð °út ·ú§H Û¼àÔaA_ |9½+Ë`¦ ³ðr H

&

ñ

½©d”`¦ ½¨ . U

dV 8 Σ = {a, b, c}{9 M:, 'Í /åJ r  t ·ú§H Û¼àÔaA_

| 9

½+Ë`¦ ³ðr H &ñ½©d”`¦ ½¨ .

ƒ



#Q\¦ ³ð‰&³ H ~½Ód”\H &ñ½©d”, š¸žÐ , ëHZO 1pxs e”.

Ça

˜+ 3 ƒ#Q_ ³ð‰&³ ~½Ód” A, B L(A) = L(B)\¦ ëߖ7ᤠ€, Aü<

BH 1lx1px(equivalent) .

5

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 2 ⠉ Ë Âø 5  › ïŸ ×  

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Ž

ÉÓ'H {9ìøÍ&hܼ–Ð ×怩œ%ƒo©œu, Bj—¸o, {9ئ§4©œu–Ð ½¨$í÷&

#Q e”. ·ú¡Ü¼–Ð (ŽÉÓ'_ s:r&h“ —¸4Sq[þt`¦ Cĺ> ÷&HX<, Õª ×æ

\

"f ©œ çߖéߖôÇ כ s Ä»ôÇ š¸žÐ s. Ä»ôÇ š¸žÐ H ×怩œ

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ƒo©œu\ K{©œ H Ä»ôÇ]j#Qlü< {9§4©œu“{9§4_…{9ܼ–Ð ½¨$í

)

a. Ä»ôÇ š¸žÐ _ :£¤fç“Ér Bj—¸o\¦ t“¦ e”t ·ú§H כ s

. {9§4_…{9“Ér ñߖ[þt–Ð ¾º#Qt“¦ yŒ• ñߖ“Ér _ /åJ\¦ |9 ú e

”

. {9§4_…{9\H K‰×¼(head) e”#Q"f K‰×¼ ovH ñߖ\ e”



H /åJ\¦ {9`¦ ú e”. Ä»ôÇ]j#QlH ©œI[þt–Ð ½¨$í)a. Õª a

Ë> 1\¦ ˜Ð.

(4)

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1 0 1 1 0

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íl\ K‰×¼H {9§4_…{9_ ±p¢,aAᤠñߖ`¦ ov“¦, Ä»ôÇ]j#QlH œí l

 ©œI\ Z~#Œ e”. yŒ• éߖ>\"f ‰&³F ©œIü< _…{9_ /åJ\ _K

"

f 6£§ ©œI &ñ÷&“¦, K‰×¼H ôÇ ñߖ š¸ÉrAá¤Ü¼–Ð ”'ŸôÇ.

Ç a

˜+ 4 &ñ Ä»ôÇ š¸žÐ  M“Ér $Á t כ¹™è (Q, Σ, δ, q0, F)–Ð

½

¨$í)a. #Œl\"f

1. QH ©œI[þt_ Ä»ôÇ |9½+Ës“¦ 2. ӍH ·ú˜  Ï@s“¦

7

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% 3. „s†<Êú δH Q× Σ\"f Q–Ð_ †<Êús“¦

4. q0 ∈ QH œíl©œIs“¦

5. F ⊆ QH þj7áx©œI[þt_ |9½+Ës.



&ñ Ä»ôÇ š¸žÐ (deterministic finite automata)\¦ צ#Œ"f DFA

–

Ð ÂÒØÔl–Ð .

„



s †<Êú δ\¦ SX‰©œ #Œ Q × Σ\"f Q–Ð_ †<Êú δ\¦ 6£§õ °ú s

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ñ

_ôÇ.

• δ(q, ǫ) = q

• —¸ŽH w ∈ Σü< a ∈ Σ\ @/ #Œ, δ(q, wa) = δ(δ(q, w), a) Û

¼àÔaA w ∈ Σ\ @/ #Œ δ(q0, w)s DFA M _ þj7áx©œIs€, M s w\¦ ~ÃÎ[þt““¦ ´ú˜ôÇ. M _ ƒ#Q L(M )“Ér M s ~ÃÎ[þts



H —¸ŽH Û¼àÔaA_ |9½+Ës. 7£¤

L(M ) = {w ∈ Σ : δ(q0, w) ∈ F }.

(5)

& % DFA_ ©œS!(configuration)“Ér ‰&³F ©œIü< {9§4 Û¼àÔaA_ {9t ·ú§

“ É

r ÂÒìrܼ–Ð &ñ)a. {9§4 Û¼àÔaAs w{9 M:, DFA_ rŒ• ©œS!“Ér (q0, w) s“¦, 7áx«Ñ ©œS!“Ér #Q‹" q ∈ Q\ @/ #Œ (q, ǫ) s.

ô

Ç _ „s\ _K"f DFA M _ ©œS!s o H õ&ñ`¦ ⊢Mܼ–Ð

³

ðrôÇ. 7£¤ ‰&³F ©œS!s (q, 10110)s“¦ δ(q, 1) = qs€, (q, 10110) ⊢M (q,0110)

s

. DFA M s "îSX‰ôÇ âĺ ⊢M @/’ ⊢\¦ 6 x½+É Ãº e”. 0 s



©

œ_ „s\¦ ⊢–Ð ³ðrôÇ.

U

dV 9 M = ({q0, q1}, {0, 1}, δ, q0,{q0}) s“¦, δH 6£§õ °ú . 0 1

q0 q0 q1 q1 q1 q0

L(M )“Ér ‹Œ•Ãº >h_ 1`¦ ” Û¼àÔaA_ |9½+Ës.

9

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»ôÇ š¸žÐ \¦ sK l ~1•¸2Ÿ¤ FAÕªaË>ܼ–Ð  ·p. ©œIH

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ñ

&hܼ–Ð, „s δ(p, a) = qH ©œI p\"f q–Ð H 6\š a ·¡­“Ér çߖ

‚



ܼ–Ð, þj7áx©œIH s×æ"é¶Ü¼–Ð, œíl©œIH o¶ú˜³ð–Ð ³ðrôÇ.

\

V]j 9_ DFA\¦ ÕªaË>ܼ–Ð  ?/€ ÕªaË> 2ü< °ú .

q

0

q

1

1

1 0 0

ÕªaË> 2:

{ 9

ìøÍ&hܼ–Ð 6£§ ¿º éߖ>\ _K DFA\¦ ëߖŽH.

1. ‰&³Ft {9“Ér ÂÒìr\"f #Q‹" &ñ˜Ð\¦ l%3K Ht\¦ &ñ 

“

¦ s &ñ˜Ð\¦ ©œI–Ð ³ðrôÇ.

2. Dh–Ðîr /åJ\¦ {9%3`¦ M: l%3K H &ñ˜Ð #QbG> 7 H t

\¦ ˜Ð“¦ „s†<Êú\¦ &ñôÇ.

(6)

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 ×æ\"f ×æכ¹ôÇ כ “Ér l%3K H &ñ˜Ð\¦ &ñ H כ s. \V ]j 9\"fH 1_ >hú ‹Œ•Ãº“t f.Ëú“t\¦ l%3K ôÇ. 

"

f ©œI q0“Ér 1_ >hú ‹Œ•Ãº, q1“Ér 1_ >hú f.Ëúe”`¦  ·p.

U

dV 10 L = {w : N0(w) mod 2 > N1(w) mod 2}`¦ ~ÃÎ[þtsH DFA\¦ ½¨ .#Œl\"f Na(w)H w\ e”H a_ >hú\¦  ·p. 7

£

¤, s DFAH 0_ >húH f.Ëús“¦ 1_ >húH ‹Œ•Ãº“ Û¼àÔaA`¦

~ Ã

Î[þt#Œ )a. s

 âĺ N0(w) mod 2ü< N1(w) mod 2\¦ l%3 € )a. ÕªaË> 3.

11

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1 1

1 1

10

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11

ÕªaË> 3:

U

dV 11 0101`¦ ÂÒìrÛ¼àÔaAܼ–Ð °úH Û¼àÔaA`¦ ~ÃÎ[þtsH DFA

\



¦ ½¨ . s

 ëH]j\"fH0101_ #Q¿º ×æ\"f ‰&³Ft {9“Ér ÂÒìr`¦ l%3K ô

Ç. "f 0101_ #Q¿º\¦ ©œI–Ð ³ðr H DFA\¦ ëߖ[þt€ )a. (ÕªaË> 4) s]j „s†<Êú\¦ &ñôÇ. \V\¦ [þt€, ‰&³Ft {9“Ér #Q¿º

 0s“¦, Õª 6£§\ 1`¦ {9ܼ€ {9“Ér #Q¿º 01s ÷&“¦, 0`¦ {9ܼ

(7)

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 {9“Ér ÂÒìrs 00s ÷&HX< 0101_ #Q¿º\ K{©œ÷&H כ “Ér 0s.

1 1

0 0

0 0 0,1

1 1

0 01 010 0101

ε

ÕªaË> 4:

U

dV 12 0101`¦ ÂÒìrÛ¼àÔaAܼ–Ð °út ·ú§H Û¼àÔaA`¦ ~ÃÎ[þtsH DFA\¦ ½¨ .

ÕªaË> 4_ DFA\"f þj7áx©œIü< Õª ü@_ ©œI\¦ ´ú˨€ )a. (s ü

< °ú s #Œ|9½+Ë`¦ ³ðr H ëH]j &ñ½©d”\"fH #Q9°?HX<, DFA\"fH /'îr כ `¦ ÅÒ3lq .)

U

dV 13 Û¼àÔaA\"f ƒ5Åq(run)sêøÍ °ú “Ér /åJ–Ðëߖ ½¨$í)a (8 s

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©

œ Zþtwn= ú \OH) ÂÒìrÛ¼àÔaA`¦ ´ú˜ôÇ. \V\¦ [þt€, 0110001“Ér U´ s

 2“ 1_ ƒ5Åqõ U´s 3“ 0_ ƒ5Åq`¦ t“¦ e”.U´s

2“ ƒ5Åq`¦ °út ·ú§H Û¼àÔaA`¦ ~ÃÎ[þtsH DFA\¦ ½¨ . 7£¤ Û¼ àÔaAs U´s 2“ ƒ5Åq“Ér °út ·ú§ “¦ U´s 1s 3s©œ“

ƒ



5Åq“Ér |9 ú e”. s

 ëH]j\"fH ©œ þjH_ ƒ5Åq`¦ l%3K “¦ Õª ×æ\"f U´s

 3s “ ƒ5Åq`¦ l%3 € )a. U

dV 14 010`¦ ÂÒìrÛ¼àÔaAܼ–Ð t“¦, 100`¦ ÂÒìrÛ¼àÔaAܼ–Ð °ú t

 ·ú§H Û¼àÔaA`¦ ~ÃÎ[þtsH DFA\¦ ½¨ . s

 ëH]jH €•çߖ #Q§>. 010s 100s ÂÒìrÛ¼àÔaAܼ–Ð e”Ht \O





Ht\¦ ·ú˜l 0AK"fH ·ú¡_ \V]j\"f%ƒ!3 010õ 100_ #Q¿º\¦ l

% 3

K ôÇ.1lxr\ ‰&³F 100`¦ {9“¦ e”€ s„\ 010s e”%3H t

 \O%3Ht\¦ ·ú˜“¦ e”#Q ôÇ. ìøÍ@/–Ð ‰&³F 010`¦ {9“¦ e”`¦ M: s

„\ 100s e”%3Ht\¦ l%3½+É €9כ¹H \O. = € 100s e”

% 3

€ s Û¼àÔaA“Ér ~ÃÎ[þt#Œtt ·ú§l M:ëHs.

(8)

& % δ ÂÒìr†<Êú{9 M:•¸ DFA–Ð ^¦ ú e”. s âĺ 0A_ &ñ_\ _ôÇ

DFA–Ð ˨9€, Dh–Ðîr ©œI q 6∈ F `¦ •¸{9 #Œ

• &ñ_÷&t ·ú§“Ér —¸ŽH „sH qܼ–Ð •¸2Ÿ¤ “¦,

• —¸ŽH a ∈ Σ\ @/ #Œ δ(q, a) = qs ÷&•¸2Ÿ¤ ôÇ.

qõ °ú s ôÇ  [þt#Q€ 4R `¦ ú \OH ©œI\¦ º¡¤“Ér ©œI(dead state)“¦ ÂҏÉr. \V]j 12\"f ©œI 0101•¸ º¡¤“Ér ©œIs.

V

 3 ⠉ j Ú r Ç a  Ë Âø 5  › ïŸ ×  

q

&ñ Ä»ôÇ š¸žÐ (nondeterministic finite automata)\¦ צ#Œ"f NFA–Ð ÂÒØÔl–Ð . NFAH 6£§ ¿º t €\"f DFA\¦ {9ìøÍo ô

Ç כ s.

1. yŒ• ©œIü< /åJ ÅÒ#Q&’`¦ M:, 6£§ ©œI \O  s©œ {

9

 ú e”.

15

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% 2. yŒ• ©œI\"f /BNÛ¼àÔaA ǫ\ _ôÇ „s e”`¦ ú e”.

Ça

˜+ 5 q&ñ Ä»ôÇ š¸žÐ  M“Ér $Á t כ¹™è

(Q, Σ, ∆, q0, F)–Ð ½¨$í)a.#Œl\"f Q,Σ, q0, FH DFA\"fü< °ú  s

 &ñ_)a. ∆H Q× (Σ ∪ {ǫ}) × Q_ ÂÒìr|9½+˓ „s ›'a>–Ð,

¢

¸H Q× (Σ ∪ {ǫ})\"f 2Q–Ð_ „s †<Êú–Ð &ñ_½+É Ãº e”. NFA\¦ &ñ_½+É M:H ∆\¦ DFA_ „s †<Êú\"f {9ìøÍo)a„s ›'a

>

–Ð €Œ• H כ s ƒÛ¼Qĺ, [O"î\"fH ∆\¦ „s †<Êú–Ð

˜

ЍH כ s ¼#o Ù¼–Ð sÊê\H ∆\¦ 2Qܼ–Ð_ †<Êú–Ð çߖÅÒôÇ.

¢

¸ôÇ [O"î_ ¼#_\¦ 0AK"f ∆\¦ 2Q × (Σ ∪ {ǫ})\"f_ †<Êú–Ð 6£§ õ

 °ú s SX‰©œôÇ. P ⊆ Qü< a ∈ Σ ∪ {ǫ}\ @/ #Œ

∆(P, a) = [

q∈P

∆(q, a).

e

”

__ ©œI q\ @/ #Œ, E(q)\¦ q\"fÂÒ' {9§4 /åJ\¦ {9t ·ú§“¦ (ǫ 6\šs ·¡­“Ér çߖ‚`¦ ) •¸²ú˜½+É Ãº e”H ©œI_ |9½+Ës“¦ 

. E•¸ %ir SX‰©œ #Œ, ©œI |9½+Ë P \ @/ #Œ E(P ) = Sq∈P E(q).

(9)

& %

∆\¦ SX‰©œ #Œ Q × Σ\"f 2Qܼ–Ð_ †<Êú ∆\¦ 6£§õ °ú s &ñ_ ô

Ç.

• ∆(q, ǫ) = E(q)

• —¸ŽH w ∈ Σü< a ∈ Σ\ @/ #Œ, ∆(q, wa) = E(∆(∆(q, w), a)) Û

¼àÔaA w ∈ Σ\ @/ #Œ ∆(q0, w)s NFA M _ þj7áx©œI\¦ &h#Q

•

¸  Ÿí†<Ê €, M s w\¦ ~ÃÎ[þt““¦ ´ú˜ôÇ. M _ ƒ#Q L(M )“Ér M s ~ÃÎ[þtsH —¸ŽH Û¼àÔaA_ |9½+Ës. 7£¤

L(M ) = {w ∈ Σ : ∆(q0, w) ∩ F 6= ∅}.

¢

¸H „sõ&ñ`¦ 6 x #Œ L(M )`¦ 6£§õ °ú s &ñ_½+É Ãº e”.

L(M ) = {w ∈ Σ : (q0, w) ⊢M (q, ǫ), q ∈ F }.

{ 9

ìøÍ&hܼ–Ð NFAH DFA˜Ð s`› ~1> ƒ#Q\¦ ³ð‰&³½+É Ãº e”.

U

dV 15 &ñ½©d” (010 + 01) ³ðr H ƒ#Q\¦ ~ÃÎ[þtsH NFA

\



¦ ½¨ .

17

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% ÕªaË> 5. {9§4 Û¼àÔaAs 010{9 M:,

• ∆(q1,0) = {q2},

• ∆(q1,01) = {q3, q1},

• ∆(q1,010) = {q1, q2} s

“¦, #Œl\ þj7áx©œI q1s Ÿí†<Ê÷&#Q e”ܼټ–Ð s NFAH 010`¦

~ Ã

Î[þt“. sכ `¦ „s õ&ñܼ–Ð  ?/€ 6£§õ °ú . (q1,010) ⊢ (q2,10) ⊢ (q3,0) ⊢ (q1, ǫ)

7

£

¤ (q1,010) ⊢ (q1, ǫ)s“¦ q1s þj7áx©œIsÙ¼–Ð s NFAH 010`¦ ~ÃÎ



[þt“.

(10)

& % 1

0

0 ε

q

q q

1

2 3

ÕªaË> 5:

U

dV 16 =åQ\"f ¿º P: /åJ 1“ Û¼àÔaA`¦ ~ÃÎ[þtsH NFA\¦

½

¨ . s NFAH q1\ QÓüt =åQ\"f ¿º P: /åJ\¦ ÆÒ8£¤ #Œ Õ

ª /åJ 1s€ q2–Ð s1lxôÇ. q3H s ÆÒ8£¤s ´ú€Œ¤Ht SX‰“ôÇ



.

ÕªaË> 6. {9§4 Û¼àÔaAs 011{9 M:,

19

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%

• ∆(q1,0) = {q1},

• ∆(q1,01) = {q1, q2},

• ∆(q1,011) = {q1, q2, q3} s

Ù¼–Ð, s NFAH 011`¦ ~ÃÎ[þt“. {9§4ۼ߼aAs 100{9 M:,

• ∆(q1,1) = {q1, q2},

• ∆(q1,10) = {q1, q3},

• ∆(q1,100) = {q1} s

Ù¼–Ð, s NFAH 100`¦ ~ÃÎ[þtst ·ú§H.

1 0,1

0,1

q

1

q

2

q

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