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2. ӍH ·ú˜ Ï@s“¦,

3. S ∈ VH rŒ•ús“¦_,

4. PH Òqt$í½©gË:[þt_ Ä»ôÇ |9½+Ës_. yŒ• Òqt$í½©gË:“Ér _{x → y}_ +þA I

\¦ tHX<, #Œl\"f x∈ (V ∪ Σ)^∗V(V ∪ Σ)^∗ s“¦ y∈ (V ∪ Σ)^∗ s_.

\

V\¦ [þt€ A, B ∈ V ü< a, b ∈ Σ\ @/ #Œ aAb→ bBa



H Òqt$í½©gË:s.

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˜+ 2 ëHZO G= (V, Σ, S, P )_ —¸ŽH Òqt$í½©gË:s A∈ Vü<

& % x∈ (V ∪ Σ)^∗\ @/ #Œ

A → x +þ

AIs€ GH ëHАoÁº›'a ëHZO(context-free grammar)s“¦ ÂҏÉr. A→ xü< A → y\¦ צ#Œ"f

A→ x | y

–

Ð ³ðrôÇ.

ë

HZO“Ér ëH©œ`¦ ³ð‰&³ H X< Fg#30A > 6 x)a. úH {9ìøÍ&hܼ

–

Ð ëH©œ $íìr`¦
·p. \V\¦ [þt#Q, ð½¨2£§s íߖ`¦ ōHñH ëH©œ

`

¦ 6£§ ëHZO`¦ s6 x #Œ ëߖ[þt ú e”.

hëH©œi → hÅÒ#Qih3lq&h#Qih"fÕüt#Qi hÅÒ#Qi → h"îihÅҏ›¸i h3lq&h#Qi → h"îih3lq&h›¸i h"fÕüt#Qi → h1lxi

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h3lq&h›¸i → `¦ h1lxi → ōH s

]j ëH©œ`¦ ëߖ[þtl 0AK"fH rŒ•ú S\"fÂÒ' Òqt$í½©gË:`¦  Y

V–Ð &h6 xôÇ. {9ìøÍ&hܼ–Ð u, v ∈ (V ∪ Σ)^∗ü< A ∈ V \ @/ #Œ uAv\ Òqt$í½©gË: A → w\¦ &h6 x € uwv\¦ %3HX<, s õ&ñ`¦ Ä»

•¸(derivation)“¦ ÂÒØԓ¦

uAv ⇒ uwv

–

Ð ³ðrôÇ. "f

hëH©œi ⇒ hÅÒ#Qih3lq&h#Qih"fÕüt#Qi

⇒ h"îihÅҏ›¸ih3lq&h#Qih"fÕüt#Qi

⇒ ½¨2£§hÅҏ›¸ih3lq&h#Qih"fÕüt#Qi

& %

⇒ ½¨2£§sh3lq&h#Qih"fÕüt#Qi

⇒ ½¨2£§sh"îih3lq&h›¸ih"fÕüt#Qi

⇒ ½¨2£§s íߖh3lq&h›¸ih"fÕüt#Qi

⇒ ½¨2£§s íߖ`¦h"fÕüt#Qi

⇒ ½¨2£§s íߖ`¦h1lxi

⇒ ½¨2£§s íߖ`¦ ōH

x\ Òqt$í½©gË:`¦ 0 s©œ &h6 x #Œ y\¦ %3ܼ€, s\¦ x⇒ y–^∗ Ð ³ðr ô

Ç.

• S ⇒ x (x ∈ (V ∪ Σ)^{∗ ∗})s€ x\¦ ëHZO G_ ëH©œ+þAI“¦ ÂҏÉr.

• S ⇒ w (w ∈ Σ^{∗ ∗})s€ w\¦ ëHZO G_ ëH©œ(¢¸H G Òqt$í H Û

¼àÔaA)s“¦ ÂҏÉr.

0

A_ \V\"f “½¨2£§sh3lq&h#Qih"fÕüt#Qi”H ëH©œ+þAIs“¦ “½¨2£§s í

ß

–`¦ ōH”H ëH©œs. ÕªQ
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•

¸ ëߖ[þt ú e”6£§`¦ Ä»_ .

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HZO G_ ƒ#Q L(G)H G Òqt$í H ëH©œ_ |9½+Ës. 7£¤ L(G) = {w ∈ Σ^∗ : S ⇒ w}.^∗

U

dV 1 G = ({S}, {0, 1}, S, P ) s“¦ _P S → 0S1 | ǫ s

€ GH ëHАoÁº›'a ëHZOs_.

S ⇒ 0S1 ⇒ 00S11 ⇒ 0011

\

"f ˜ÐH כ %ƒ!3 _{L(G) = {0}ⁿ₁ⁿ _{: n ≥ 0}}e”`¦ ·ú˜ ú e”_. Ç

a

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#

Q(context-free language)“¦ ÂҏÉr. U

dV 2 G = ({S}, {(, )}, S, P )s“¦P S → SS | (S) | ǫ

& % s

€_,

S ⇒ SS ⇒ (S)(S)^∗ ⇒ ()()^∗ S ⇒ SS ⇒ ((S))(S)^∗ ⇒ (())()^∗ s

_.7£¤ GH #ŒH F‹c ñü< {Œ—H F‹c ñ `¦–Ð Š©œ`¦ sÀҍH Û¼àÔaA

`

¦ Òqt$íôÇ. U

dV 3 6£§ ëHАoÁº›'a ëHZO“Ér úd”`¦ Òqt$íôÇ_. E → E + T | T

T → T ∗ F | F F → (E) | id 7

£

¤

E ⇒ E + T

⇒ E + T ∗ F

⇒ E + T ∗ (E)

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⇒ E + T ∗ (E + T )

⇒ F + F ∗ (F + F )∗

⇒ id + id ∗ (id + id)∗

U

dV 4 L = {0^m1ⁿ : m 6= n}`¦ Òqt$í H ëHАoÁº›'a ëHZO`¦ ½¨ _. m > nõ m < n_ âĺ–Ð
¾º#Q"f ÒqtyŒ•ôÇ_. ëHАoÁº›'a ëHZO`¦ ëߖ [

þ

tl 0AK"fH #Q‹" ú Áº’H Û¼àÔaA`¦ ëߖ[þt כ “t\¦ &ñK ô

Ç. m > n“ âĺ, ú A0õ 1_ >hú °ú “Ér Û¼àÔaA`¦ ëߖ[þt

•

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A → 0A1 | ǫ B → 0B | 0

m < n“ âĺ_, ú _AH Õª@/–Ð s6 x € ÷&“¦ ú _C ₁>h s©œ

& % _

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S → AC C → 1C | 1 U

dV 5 L = {aⁿb^mc^k : n + m = k}\¦ Òqt$í H ëHАoÁº›'a ëHZO`¦ ½¨

. U

dV 6 L = {0ⁿ1^m : n ≤ m ≤ 2n}`¦ Òqt$í H ëHАoÁº›'a ëHZO`¦ ½¨

_. U

dV 7 L = {a^mbⁿc^ud^v : m + n = u + v}\¦ Òqt$í H ëHАoÁº›'a ëH ZO

`¦ ½¨ _. U

dV 8 Lab = {w ∈ {a, b}^∗ : Na(w) = Nb(w)}\¦ Òqt$í H ëHАoÁº›'a ë

HZO`¦ ½¨ _. ç

ß

–éߖôÇ ²úš“Ér

S → aSb | bSa | SS | ǫ

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_.¢¸ Ér ²úš“Ér

S → aB | bA | ǫ A → a | aS | bAA B → b | bS | aBB s

_.#Œl\"f AH a b˜Ð 
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ë

HАoÁº›'a ëHZOs #Q‹" ƒ#Q\¦ ëߖ׼Ht ~1> SX‰“½+É Ãº \O`¦ M: 7£x

"

î

`¦ _K s\¦ µ1߁n= ú e”.

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dV 9 \V]j ₈_ çߖéߖôÇ ëHZO`¦ G“¦ ½+É M:<sub>, L(G) = Lab</sub>e”`¦ 7£x

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Lab ⊆ L(G): w ∈ Lab_ U´s\ @/ôÇ ) ±úšZOܼ–Ð 7£x"î _.

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“

¦ (7£¤ S ⇒ w)^∗ &ñ . U



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S ⇒ aSb⇒ axb^∗ sÙ¼–Ð_{, G} w\¦ Òqt$íôÇ_{. (w = bxa}“ â啸



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á

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Ç. w\"f s 0_ ¢,aAᤠÂÒìr`¦ u, š¸ÉrAᤠÂÒìr`¦ v“¦ € (7£¤ _{w = uv), u}ü< _vH yŒ•yŒ• °ú “Ér ú_ _aü< _b\¦ °ú“¦_,Õª U´sH k pëߖs_. ) ±úš [O\ _K S⇒ u^∗ ü< S ⇒ v^∗  $íwnôÇ_.

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º•¸ ðøÍt₎ 2

0 0 −1 0

u v

x w

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& % w∈ Σ \ @/ #Œ

A → wB A → w +þ

AIs€ GH &ñ½©ëHZOs“¦ ÂҏÉr_.

&

ñ

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ºëߖ`¦ °ú> )a.

S ⇒ w1A

⇒ w1w2B

⇒ w1w2w3C

⇒ w1w2w3w4

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ñ

½©ëHZOs Òqt$í H ƒ#Q_ |9½+Ës &ñ½©ƒ#Q 7áxÀÓe”`¦ ˜Ðs“¦ ô

Ç.

Ç a

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ÃZ. Ls &ñ½©ƒ#QsÙ¼–Ð L`¦ ~ÃÎ[þtsH DFA

M = (Q, Σ, δ, q0, F)s ”>rFôÇ. M _ „sõ&ñ`¦ ¿9ª?/?/H &ñ½©ëH ZO

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€, ëHZO“Ér q0 ∗

⇒ wq\ sØԕ¸2Ÿ¤ ëHZO G\¦ ëߖŽH.

=>

q

wq

=

<>

q

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P H 6£§õ °ú s &ñ_)a.

1. yŒ• „s δ(p, a) = q\ @/ #Œ p → aq\¦ P \ 8ôÇ.

2. yŒ• þj7áx ©œI q ∈ F \ @/ #Œ q → ǫ\¦ P \ 8ôÇ.

& % δ^∗(p, w) = q“ âĺ\ëߖ p⇒ wq^∗ e”`¦ ˜Ð{9 ú e”. "f

δ^∗(q0, w) ∈ F“ âĺ\ëߖ q0 ∗

⇒ ws. 7£¤ L(M ) = L(G). 2 U

dV 10 \V]j _??(ÕªaË> ??)_ _DFAü< 1lx1pxôÇ &ñ½©ëHZO`¦ ½¨ _. (º¡¤“Ér ©œIH Òqt|ÄÌ₎

q1 → 0q2 | ǫ q2 → 1q3

q3 → 0q4 | ǫ q4 → 0q2 | 1q3 | ǫ Ç

a

h 2 &ñ½©ëHZO G Òqt$í H ƒ#Q _L(G)H &ñ½©ƒ#Qs_.

¤

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1. yŒ• Òqt$í½©gË: A → wB\ @/ #Œ, ∆^∗(A, w)\ B\¦ 8K ïr.

2. yŒ• Òqt$í½©gË: A → w\ @/ #Œ, ∆^∗(A, w)\ qf\¦ 8K ïr.

s

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dV 11 &ñ½©ëHZO

S → aA A → abS | a ü

< 1lx1pxôÇ _NFA\¦ ½¨ _. ÕªaË> ₃\¦ ˜Ð_.

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a

a

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aƒ#Qs. 7£¤, &ñ½©ƒ#Q 7áxÀӍH ëHАoÁº›'aƒ#Q 7áxÀÓ_ ÂÒìr|9½+Ës

.

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6£§õ °ú .

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2. ?/ÂÒ ”¸×¼_ s2£§“Ér V _ "鶙ès“¦, A s2£§_ ?/ÂÒ ”¸×¼

u1, . . . , ur s2£§_ d” ”¸×¼[þt`¦ t“¦ e”ܼ€ A→ u1· · · ur“Ér P _ Òqt$í½©gË:s#Q ôÇ.

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é ß

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dV 12 \V]j 2\"f Ä»•¸õ&ñ

S ⇒ SS ⇒ (S)S ⇒ ()S ⇒ ()(S) ⇒ ()()



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£

¤ Ä»•¸(leftmost derivation)“¦ ÂҏÉr. ôÇ >h_ Û¼ àÔo\ @/ 

#

Œ #ŒQ t Ä»•¸ õ&ñs e”`¦ ú e”tëߖ ýa8£¤ Ä»•¸H ôÇ tëߖ

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dV 13 ·ú¡ \V]j\"f Ä»•¸õ&ñ

S ⇒ SS ⇒ S(S) ⇒ S() ⇒ (S)() ⇒ ()()

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¸ Û¼ àÔo–Ð
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Û¼ àÔo\¦ t€_{, G}H EB (ambiguous)“¦ ´ú˜ôÇ_. U

dV 14 6£§ ëHZO G\¦ “¦9 . E → E + E E → E ∗ E E → a | b | c

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_.ÕªQ
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E E

E E

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E E

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S → if C then S else S S → if C then S

S → x | y

& % C → a | b

if a then if b then x else yH ¿º >h_ Û¼ àÔo\¦ ”.s

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·

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C then if

y

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if C then

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b x y

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dV 16 L = {aⁿbⁿc^m} ∪ {a^mbⁿcⁿ}“Ér ‘:r|9&hܼ–Ð EB _. (7£x"îs m“¦ [O"î) L`¦ Òqt$í H #Q‹" ëHZOsŽHt {a<sup>n</sup>b<sup>n</sup>c<sup>m</sup>}õ {a<sup>m</sup>b<sup>n</sup>c<sup>n</sup>}`¦ "f–Ð Ér ~½ÓZOܼ–Ð ëߖ[þt#Q Ù¼–Ð aⁿbⁿcⁿ`¦ ëߖ׼



H Û¼ àÔo ₂>h s©œ ”>rF > )a_.

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HАoÁº›'aƒ#Q Ls ǫ`¦ Ÿí†<Ê €, L`¦ Òqt$í H ëHZO“Ér ìøÍ×¼r A→ ǫ +þAI_ Òqt$í½©gË:`¦ 4R ôÇ. s âĺ rŒ•ú S0\"f

S0 → ǫ | S _

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ê_ ëHАoÁº›'a ƒ#QH ǫ`¦ Ÿí†<Ê t ·ú§H“¦ &ñôÇ.

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Ž

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AIs€_{, G}\¦ _Chomsky ³ðïr+þAs“¦ ÂҏÉr_. Ç

a

˜+ 8 (Greibach ¸Ø)KÌfC) ëHАoÁº›'a ëHZO G= (V, Σ, S, P )_ —¸

Ž

H Òqt$í½©gË:s A∈ V , a ∈ Σ, x ∈ V^∗\ @/ #Œ A → ax

+þ

AIs€, G\¦ Greibach ³ðïr+þAs“¦ ÂҏÉr.

\

V]j 8_ ¿º P: ²úš“Ér Greibach ³ðïr+þAs.

Ç a

h 3 —¸ŽH ëHАoÁº›'a ƒ#QH Chomsky ³ðïr+þA_ ëHZO\ _K Òqt

$í

)a_.

& %

¤

ÃZ. (**) e”__ ëHАoÁº›'a ëHZO G = (V, Σ, S, P )\¦ 6£§ õ&ñ\ _ K Chomsky ³ðïr+þAܼ–Ð õH.

1. P \"f A → ǫ +þAI_ ǫ Òqt$í½©gË:`¦ \O±p.

(a) A⇒ ǫ`<sup>∗</sup> ¦ ëߖ7ᤠH —¸ŽH ú_ |9½+Ë Vǫ`¦ ½¨ôÇ.

i. A → ǫ“ A\¦ Vǫ\ V,H.

ii. A → B1· · · Br\"f —¸ŽH Bi (1 ≤ i ≤ r)s Vǫ _ "鶙ès€ A\¦ Vǫ\ V,H.

(b) Dh–Ðîr Òqt$í½©gË:_ |9½+Ë P1`¦ ëߖŽH. P _ yŒ• Òqt$í½©gË: A→ x1· · · xt (xi ∈ V ∪ Σ)\ @/ #Œ xi Vǫ_ "鶙ès€ Õª



o\ xiü< ǫ`¦ °ú˜ V,#Q ëߖ[þt#QtH —¸ŽH Òqt$í½©gË:`¦ P1\ V,H. éߖ A → ǫs ëߖ[þt#QtH âĺ sכ “Ér P1\ V, t

 ·ú§H.

(c) G1 = (V, Σ, S, P1)H Gü< 1lx1px .

2. P1\"f A → B +þAI_ éߖ0A Òqt$í½©gË:`¦ \O±p.

(a) A⇒ B\^∗ ¦ ëߖ7ᤠH ú[þt`¦ ½¨ôÇ. éߖ0A Òqt$í½©gË:

27

' $

A→ B\ @/ #Œ A &ñ&h\"f B &ñ&hܼ–Ð H çߖ‚`¦ t



H ÕªAáÔ\¦ ëߖ[þt€ A ⇒ B ›^∗ 'a>\¦ €Œ•½+É Ãº e”.

(b) P1\"f éߖ0A Òqt$í½©gË:s  כ `¦ P2\ V,H.

(c) A⇒ Bs^∗ “¦ P1_ "鶙è B → xs éߖ0A Òqt$í½©gË:s m€ A→ x\¦ P2\ V,H.

(d) G2 = (V, Σ, S, P2)“Ér G1õ 1lx1px .

3. P2_ —¸ŽH Òqt$í½©gË:`¦ Chomsky ³ðïr+þAܼ–Ð õH.

(a) A → a“ Òqt$í½©gË:“Ér sp Chomsky ³ðïr+þAsÙ¼–Ð P3\ V,



H.

(b) r ≥ 2“ yŒ• Òqt$í½©gË: A → x1· · · xr (xi ∈ V ∪ Σ)\"f xi ·ú˜  Ï@ /åJ as€ Dh–Ðîr ú Ca\¦ ëߖ[þt“¦ Ca → a\¦ P3\ V,

“¦ xi\¦ Ca–Ð @/uôÇ. ÕªQ€ Òqt$í½©gË:_ ĺ\ ú ë

ß

–s
èߖ.

(c) Òqt$í½©gË: A → B1· · · Br\ @/ #Œ Dh–Ðîr ú

& % D1, . . . , D_r−2`¦ •¸{9 “¦

A → B1D1

D1 → B2D2

· · ·

D_r−3 → B_r−2D_r−2 D_r−2 → B_r−1Br

`

¦ P3\ V,H.

(d) V ü< Dh ú[þt`¦ ½+Ë #Œ V^′ €, G3 = (V^′,Σ, S, P3)“Ér G2ü< 1lx1px .

2 U

dV 17 6£§ ëHАoÁº›'a ëHZO`¦ _Chomsky ³ðïr+þAܼ–Ð ˨_. S → aB | AAa

29

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% A → B | S

B → b | ǫ

Vǫ = {B, A}sÙ¼–Ð ₁ éߖ> Êê _G1“Ér 6£§õ °ú  S → a | aB | AAa | Aa A → B | S

B → b

A ⇒ B, A^∗ ⇒ S^∗ sÙ¼–Ð 2 éߖ> Êê G2H 6£§õ °ú  S → a | aB | AAa | Aa A → b | a | aB | AAa | Aa B → b

3 éߖ> Êê _Chomsky ³ðïr+þA“Ér 6£§õ °ú _. S → a | CB | AD | AC

& % A → b | a | CB | AD | AC

B → b C → a D → AC Ç

a

h 4 —¸ŽH ëHАoÁº›'a ƒ#QH _Greibach ³ðïr+þA_ ëHZO\ _K Òqt

$í

)a_.

V

 5 ⠉ 6 h Å ÁŸ  ¸ › ïŸ ×  

?

/o¾º2£§ š¸žÐ(pushdown automata)\¦ צ#Œ"f PA–Ð ÂÒØÔl–Ð

. PAH {9§4 _…{9õ ]j#Q©œuü@\ ?/o¾º2£§ $©œ©œu“ Û¼×þ˜

`

¦ t“¦ e”. ?/o¾º2£§ š¸žÐ\¦ Û¼×þ˜ š¸žÐ“¦ ÂÒØÔl•¸ ô

Ç. ÕªaË> 8\¦ ˜Ð.

31

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1 0 1 1 0

B A

#

ÕªaË> 8:

Ç a

˜+ 9 ?/o¾º2£§ š¸žÐ M“Ér #Œ$Á t כ¹™è (Q, Σ, Γ, ∆, q0, F)–Ð ½¨$í)a. #Œl\"f

1. QH ©œI[þt_ Ä»ôÇ |9½+Ës“¦ 2. ӍH {9§4 ·ú˜ Ï@s“¦

3. ÍH Û¼×þ˜ ·ú˜ Ï@s“¦ rŒ• /åJ _#\¦ Ÿí†<ÊôÇ_.

& % 4. ∆H (Q × (Σ ∪ {ǫ}) × (Γ ∪ {ǫ})) × (Q × Γ )_ ÂÒìr|9½+˓ „s

›'

a>s“¦

5. q0 ∈ QH œíl ©œIs“¦

6. F ⊆ QH þj7áx©œI[þt_ |9½+Ës_.

PAH ∆ „s ›'a>sÙ¼–Ð q&ñ š¸žÐs. "f Ä»ôÇ š¸

ž

Ð M:ü< ðøÍt–Ð þj7áx©œIH ôÇ >hëߖ e”“¦ &ñ½+É Ãº e”

. ((p, a, A), (q, u)) ∈ ∆H PA_ ©œI ps“¦ {9§4 /åJ as“¦ Û

¼×þ˜_ ±p0A "鶙è A{9 M:, a\¦ {9“¦ Û¼×þ˜_ A\¦ u–Ð ˨“¦ q ©œ I

 ÷&H „s\¦
·p. "f ((p, 0, 0), (q, 10))“Ér Û¼×þ˜\ 1`¦ V,

H(push) ƒíߖs“¦, ((p, 0, 0), (q, ǫ))H Û¼×þ˜_ ±p0A "鶙è 0`¦ NS



H(pop) ƒíߖs. „s ›'a>\"f Σ ∪ {ǫ}_ ǫH {9§4`¦ {9t ·ú§“¦

„



s {9#Q±ú˜ ú e”6£§`¦
?/“¦, Γ ∪ {ǫ}_ ǫH Û¼×þ˜`¦ ˜Ðt ·ú§“¦

„



s {9#Q±ú˜ ú e”6£§`¦
·p.

PAH rŒ•½+É M:, Û¼×þ˜\ #ëߖ`¦ t“¦ e”. #H Û¼×þ˜_ x9{Œ•`¦

?/H l0px`¦ ôÇ. PA_ „s •¸×æ #\¦ Û¼×þ˜\"f NSH ƒíߖ“Ér

t ·ú§H“¦ &ñôÇ.

33

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&

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% PA_ ©œS!(configuration)“Ér ‰&³F ©œI, {9§4 Û¼àÔaA_ {9t ·ú§“Ér

Â

Òìrõ ‰&³F Û¼×þ˜_ ?/6 x\ _K &ñ)a. {9§4 Û¼àÔaAs w{9 M:, r

Œ• ©œS!“Ér (q0, w,#)s. ôÇ _ „s\ _K"f PA M _ ©œS! s

 o H õ&ñ`¦ ⊢Mܼ–Ð ³ðrôÇ. ‰&³F ©œS!s (q, 01101, 100#)s“¦ ((q, 0, ǫ), (p, 1)) ∈ ∆s€,

(q, 01101, 100#) ⊢M (p, 1101, 1100#).

PAH {9§4 Û¼àÔaA w\¦ {9“Ér Êê_ ©œI þj7áx©œIs€ (Û¼×þ˜_ ?/ 6

x\ ›'a>\Os) w\¦ ~ÃÎ[þt“. 7£¤,

L(M ) = {w ∈ Σ^∗ : (q0, w,#) ⊢^∗_M (p, ǫ, u), p ∈ F, u ∈ Γ^∗}.

U

dV 18 M = ({q1, q2, q3}, {0, 1}, Γ = {0, #}, ∆, q1,{q3})s“¦ _∆H



A_ „s\¦ t“¦ e”_.

((q1,0, ǫ), (q1,0))

& % ((q1, ǫ,#), (q3,#))

((q1,1, 0), (q2, ǫ)) ((q2,1, 0), (q2, ǫ)) ((q2, ǫ,#), (q3,#))

M`¦ ÕªaË>ܼ–Ð
?/€ ÕªaË> ₉ü< °ú . L(M ) = {0ⁿ1ⁿ : n ≥ 0}.

{ 9

§4 Û¼àÔaAs 01s€ M_ „sõ&ñ“Ér 6£§õ °ú . (q1,01, #) ⊢ (q1,1, 0#) ⊢ (q2, ǫ,#) ⊢ (q3, ǫ,#) (q1,01, #) ⊢ (q3,01, #) „sH 8 s©œ ”'Ÿ t 3lwôÇ.

35

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q

q

1 2

3

q

ε, #/# ε, #/#

1, 0/ε 0, ε/0

ÕªaË> 9:

U

dV 19 {wcw^R : w ∈ {a, b}^∗}`¦ ~ÃÎ[þtsH _PA\¦ ½¨ _. Õª a

Ë

> _10.

U

dV 20 {ww^R : w ∈ {a, b}^∗}`¦ ~ÃÎ[þtsH _PA\¦ ½¨ _.Õª a

Ë

> ₁₀\"f _((q1, c, ǫ), (q2, ǫ))`¦ _((q1, ǫ, ǫ), (q2, ǫ))ܼ–Ð õH_.

& %

1

q

q

q

a, b,

ε/a

ε/b a, a/

b, b/

εε c, ε/ε

ÕªaË> 10:

U

dV 21 {aⁿb^mc^k : n + m = k}\¦ ~ÃÎ[þtsH PA\¦ ½¨ . U

dV 22 {0ⁿ1^m : n ≤ m ≤ 2n}\¦ ~ÃÎ[þtsH _PA\¦ ½¨ _. U

dV 23 {vw : v, w ∈ {0, 1}^∗, w6= v^R,|v| = |w|}\¦ ~ÃÎ[þtsH _PA

\

¦ ½¨ _.

37

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%

V

 6 ⠉ 6 h Å ÁŸ  ¸ › ïŸ ×  ÿ ? % K  _ 9 cÈ Á® ¸ ” ¥ U ># a

Ç a

h 5 ëHАoÁº›'a ƒ#Q L\ @/ #Œ, L = L(M )“ ?/o¾º2£§ š¸žÐ 

 Ms ”>rFôÇ_.

¤

ÃZ. L`¦ Òqt$í H Greibach ³ðïr+þA_ ëHАoÁº›'a ëHZO`¦

G = (V, Σ, S, P ) . G_ ýa8£¤ Ä»•¸\¦ ¿9ª?/?/H PA M `¦ ëߖ[þt

“

¦ ôÇ. G Greibach ³ðïr+þAsÙ¼–Ð ýa8£¤ Ä»•¸\"f ëH©œ +þAIH xu (x ∈ Σ^∗, u ∈ V^∗) s. s íHçߖ M “Ér {9§4 Û¼àÔaA\"f x\¦ {9“¦, Û

¼×þ˜\ u#\¦ t“¦ ôÇ.

Gü< 1lx1pxôÇ PA M “Ér 6£§õ °ú  (ÕªaË> 11).

M = ({p, q, r}, Σ, V ∪ {#}, ∆, p, {r}) 1. ((p, ǫ, #), (q, S#)) ∈ ∆

2. —¸ŽH A → ay\ @/ #Œ ((q, a, A), (q, y)) ∈ ∆ 3. ((q, ǫ, #), (r, #)) ∈ ∆.

& %

q

p

a, A/y

,#/S# ε, #/#

r

ÕªaË> 11:

S ⇒ xu^∗ “ âĺ\ëߖ (q, x, S#) ⊢^∗_M (q, ǫ, u#) (1) e

”

`¦ ) ±úšZOܼ–Ð ˜Ðs“¦ ôÇ.

(1)_ ⇒ âĺ\¦ 7£x"î . Ä»•¸ SÃº 0{9 M:H x = ǫ, u = SsÙ¼

–

Ð {©œƒy $íwnôÇ. Ä»•¸ SÃº i pëߖ{9 M:H (1) $íwnôÇ“¦

&ñôÇ. Ä»•¸ SÃº i{9 M: x = x1a“¦ € S ⇒ x^∗ 1Au1 ⇒ x1au2u1

39

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s

“¦ u = u2u1s. ) ±úšZO &ñ\ _K

(q, x1a, S#) ⊢^∗_M (q, a, Au1#)s. A → au2 Òqt$í ½©gË:sÙ¼–Ð ((q, a, A), (q, u2)) ∈ ∆s. "f,

(q, x1a, S#) ⊢^∗_M (q, a, Au1#) ⊢M (q, ǫ, u2u1#).



ðøÍt–Ð (1)_ ⇐ âĺ\¦ 7£x"î½+É Ãº e”. (1)\ _K"f S ⇒ w^∗ “ âĺ\ëߖ (q, w, S#) ⊢^∗_M (q, ǫ, #)s“¦ ÕªA"f

(p, w, #) ⊢M (q, w, S#) ⊢^∗_M (q, ǫ, #) ⊢M (r, ǫ, #) s

. 7£¤ w ∈ L(G)“ âĺ\ëߖ w ∈ L(M )s.

¢

¸ Ér 7£x"î: L`¦ Òqt$í H ëHАoÁº›'a ëHZO`¦ G = (V, Σ, S, P ) 

. G\¦ ¿9ª?/?/H PAH 6£§õ °ú .

({p, q, r}, Σ, V ∪ Σ ∪ {#}, ∆, p, {r}) 1. ((p, ǫ, #), (q, S#)) ∈ ∆

2. —¸ŽH A → x\ @/ #Œ ((q, ǫ, A), (q, x)) ∈ ∆

& % 3. —¸ŽH a ∈ Σ\ @/ #Œ ((q, a, a), (q, ǫ)) ∈ ∆

4. ((q, ǫ, #), (r, #)) ∈ ∆.

s

 âĺ ëHZO G Greibach ³ðïr+þA{9 €9כ¹ \O. 2

U

dV 24 Greibach³ðïr+þA“ ëHZO

S → 0SA | 0A A → 1

õ

 1lx1pxôÇ ?/o¾º2£§ š¸žÐ\¦ ½¨ . (ÕªaË> 12) S⇒ 0SA ⇒ 00AA ⇒ 001A ⇒ 0011 Ä

»•¸õ&ñ\ K{©œ÷&H PA_ „s\¦ [O"îôÇ.

41

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$

%

p

q

r

1, A/

0, S/A 0, S/SA

ε

ÕªaË> 12:

U

dV 25 ëHZO

S → aSb | bSa | SS | ǫ õ

 1lx1pxôÇ ?/o¾º2£§ š¸žÐ\¦ ½¨ . (ÕªaË> 13) S ⇒ aSb ⇒ abSab ⇒ abab Ä

»•¸õ&ñ\ K{©œ÷&H _PA_ „s\¦ [O"îôÇ_.

& % ,#/S#

q

ε ε, #/#

εε ε

, S/SS , S/bSa ε, S/

r

, S/aSb ε b, b/

a, a/ε ε

p

ÕªaË> 13:

Ç a

h 6 ?/o¾º2£§ š¸žÐ M\ @/ #Œ, L(M )“Ér ëHАoÁº›'a ƒ#Qs



_.

¤

 ÃZ. {9ìøÍ$í`¦ {9t ·ú§“¦ M = (Q, Σ, Γ, ∆, q0, F)\ @/ #Œ 6£§õ

° ú

 s &ñôÇ. a, b ∈ Σ ∪ {ǫ}, A ∈ Γ“¦ .

• M “Ér þj7áx©œI qf ôÇ >hëߖ e”“¦, =åQ±ú˜ M:\ Û¼×þ˜\ #ëߖ zŒ™l

“¦ qf–Ð çߖ. (ÕªXOt ·ú§“Ér âĺ\H Û¼×þ˜\"f pop H „s\¦

8KÅҀ )a.)

43

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• M _ „sH ((p, a, ǫ), (q, A)) (push) ¢¸H ((p, a, A), (q, ǫ)) (pop) +þ

AIëߖ`¦ °úH. ({9ìøÍ&h“ „sH pushü< pop_ ƒíߖ[þt–Ð  Ü

ã

J ú e”.) Å

Òכ¹ >h¥Æ“Ér M s p ©œI\"f rŒ• #Œ 1lx{9ôÇ Û¼×þ˜Ü¼–Ð (Û¼×þ˜\ pushôÇ Êê pop½+É ÃºH e”tëߖ AAá¤Ü¼–ЍH ?/9t ·ú§“¦) q ©œI

\

 •¸²ú˜ H 1lxîߖ {9§4\"f x\¦ {9ܼ€, ëHZO\"fH hpqi⇒ xs^∗  ÷&

•

¸2Ÿ¤ H כ s. (ÕªaË> 14)

0000 0000 1111 1111 0000

0000 1111

1111 q

p

x x

ÕªaË> 14:

M s x\¦ {9H 1lxîߖ Û¼×þ˜_ oH 6£§ ¿º t×æ 
s.

& % (a) %ƒ6£§\ push)a /åJ t}Œ•\ pop÷&H âĺ: s âĺ\¦

hpqi → ahrsib–Ð ¿9ª?/?/9“¦ ôÇ. #Œl\"f aü< r“Ér %ƒ6£§ „s M: {9“Ér /åJü< 6£§ ©œIs“¦, bü< sH t}Œ• „s M: {9“Ér /åJ



ü< s„ ©œIs.

(b) %ƒ6£§\ push)a /åJ ×æçߖ\ pop÷&H âĺ: s âĺ\¦ hpqi → hprihrqi–Ð ¿9ª?/?/9“¦ ôÇ. #Œl\"f r“Ér %ƒ6£§\ push)a /åJ pop÷&%3`¦ M:_ ©œIs.

M õ 1lx1pxôÇ ëHАoÁº›'a ëHZO G = (V, Σ, S, P )H 6£§õ °ú .

1. V H p, q ∈ Q\ @/ #Œ hpqi+þAI_ ú[þt–Ð ½¨$í)a.

2. SH hq0qfis.

3. ((p, a, ǫ), (r, A)), ((s, b, A), (q, ǫ)) ∈ ∆\ @/ #Œ hpqi → ahrsib\¦ ë

ß

–ŽH.

4. —¸ŽH p, q, r ∈ Q\ @/ #Œ hpqi → hprihrqi\¦ ëߖŽH.

5. —¸ŽH q ∈ Q\ @/ #Œ hqqi → ǫ`¦ ëߖŽH.

45

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% (p, x, ǫ) ⊢^∗_M (q, ǫ, ǫ)“ âĺ\ëߖ hpqi ⇒ x^∗ (2)

e

”

`¦ ) ±úšZOܼ–Ð ˜Ðs“¦ ôÇ.

(2)_ ⇒ âĺ\¦ 7£x"î . „s SÃº 0“ âĺ\H p = q, x = ǫs“¦, ½©gË: 5\ _K hppi → ǫsÙ¼–Ð (2)s $íwnôÇ. „s S ú i pëߖ{9 M: (2)s $íwnôÇ“¦ &ñ .

„



s SÃº i ≥ 1{9 M: Û¼×þ˜\"f (a)ü< (b)×æ 
 µ1ÏÒqtôÇ.

(a) µ1ÏÒqt H âĺ, A\¦ %ƒ6£§\ Û¼×þ˜\ push)a /åJ, aü< r“Ér %ƒ 6

£

§ „s M: {9“Ér /åJü< 6£§ ©œI, bü< sH t}Œ• „s M: {9“Ér /åJ



ü< s„ ©œI“¦ . 7£¤ %ƒ6£§ „s ((p, a, ǫ), (r, A)), t}Œ•

„



s ((s, b, A), (q, ǫ))s. ½©gË: 3\ _K GH hpqi → ahrsib\¦ °ú



H. x = ayb“¦ . ) ±úšZO &ñ\ _K (r, y, ǫ) ⊢^∗ (s, ǫ, ǫ) s Ù

¼–Ð hrsi⇒ ys^∗ . "f hpqi ⇒ ahrsib⇒ x.^∗

(b) µ1ÏÒqt H âĺ, r`¦ %ƒ6£§\ push)a /åJ pop÷&%3`¦ M:_ ©œ I

“¦ “¦, yü< z\¦ yŒ•yŒ• r s„õ sÊê\ {9“Ér {9§4 Û¼àÔaAs

“

¦ . ½©gË: 4\ _K GH hpqi → hprihrqi\¦ °úH. ) ±úšZO &ñ

& %

\

 _K hpri⇒ ys^∗ “¦ hrqi⇒ zs^∗ . "f hpqi ⇒ hprihrqi⇒ x.^∗



ðøÍt–Ð (2)_ ⇐ âĺ\¦ 7£x"î½+É Ãº e”. (2)\ _K"f hq0qfi⇒ w^∗ “ âĺ\ëߖ (q0, w,#) ⊢^∗ (qf, ǫ,#).

7

£¤ w ∈ L(G)“ âĺ\ëߖ w ∈ L(M )s. 2

b, b/ε a, a/ε b, /bε

a, /aε

ε , /dε c, /cε ,c/ ε ε , d/ε ε

q₁ q₂ q₃ q₄ q₅

ÕªaË> 15:

U

dV 26 ÕªaË> ₁₅_ _PAü< 1lx1pxôÇ ëHАoÁº›'a ëHZO`¦ ½¨ _.

47

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hq1q1i → ǫ

· · ·

hq5q5i → ǫ

hq1q3i → hq1q2ihq2q3i

· · ·

hq1q5i → hq1q3ihq3q5i

hq1q5i → hq2q4i hq2q4i → chq3q3i hq2q4i → ahq2q4ia hq2q4i → bhq2q4ib

& % s

 ëHZOܼ–Ð Û¼àÔaA abcba\¦ Òqt$í H õ&ñ_: hq1q5i ⇒ hq2q4i

⇒ abhq∗ 2q4iba

⇒ abchq3q3iba

⇒ abcba.

V

 7 ⠉ % K  _ 9 cÈ Á® ¸ ” ¥ U ># a˜ + Å ]  ™

#

Q‹" ƒ#Q ëHАoÁº›'aƒ#Q _”`¦ ˜Ðsl 0AK"fH 6£§ &ño\¦



6 xôÇ.

Ç a

h 7 (Š\è« Çah) L`¦ ëHАoÁº›'a ƒ#Q“¦ €<sub>,</sub> 6£§`¦ ëߖ7ᤠ



H €ªœ_ &ñú t ”>rFôÇ. U´s t s©œ“ e”__ Û¼àÔaA w∈ LH _{w= uvxyz}–Ð ³ð‰&³÷& 9 #Œl"f

1. |vxy| ≤ t,

49

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3. —¸ŽH _{i≥ 0}\ @/ #Œ _uvⁱ_xyⁱ_{z ∈ L} s_.

¤

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w∈ L(G)_ Û¼ àÔo_ Z}s is€ |w| ≤ 2ⁱ⁻¹e”`¦ ) ±úšZOܼ–Ð

˜Ð{9 ú e”.

|V | = k“¦ €, tH 2^ks. w ∈ L(G)s“¦ |w| ≥ ts€, w_ Û¼ à

Ôo_ Z}s k + 1 s©œs. ÀÒàÔ\"f ©œ | â–Ð\¦ P “¦ 

€

, P _ U´sH k + 1 s©œs“¦ Õª â–Ð ©œ\ k + 1 s©œ_ ú

èߖ. GH k >h_ úëߖ`¦ tÙ¼–Ð P \¦ oáÔ\"fÂÒ' `¦

€, ìøÍ4Ÿ¤÷&H ú ”>rFôÇ. %ƒ6£§ ìøÍ4Ÿ¤÷&H ú\¦ A “¦ A s2£§_ ”¸×¼\¦ p, q (p q_ AAá¤) “¦ €, oáÔÂÒ' qt _

 U´sH k + 1 s s. ÕªaË> 16\¦ ˜Ð. P  ©œ | â–ÐsÙ¼

–Ð q_ Z}sH k + 1 s s. "f q\¦ ÀÒàԖР°úH ÂÒìr àÔo_

\P

B Û¼àÔaA vxy_ U´sH 2^k s s (7£¤ |vxy| ≤ t). #Œl\"f xH p\¦ ÀÒàԖР°úH ÂÒìr àÔo_ \PB Û¼àÔaAs. 7£¤ wH

& % uvxyz–Ð ³ðr)a. ¢¸ôÇ q\"f 6 x)aÒqt$í½©gË:“Ér A → BC +þAIs

Ù

¼–Ð v, y Ñüt ×æ_ 
H ǫ |¨c ú \O (7£¤ |vy| ≥ 1).

S

u v x y z

A

A q

p

ÕªaË> 16:

qü< p\"f rŒ• H Ä»•¸ õ&ñ“Ér

A ⇒ vAyü^∗ < A⇒ x^∗

51

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s

Ù¼–Ð —¸ŽH i ≥ 0\ @/ #Œ A⇒ v^{∗ i}xyⁱs. 2

U

dV 27 L = {aⁿbⁿcⁿ : n ≥ 0}H ëHАoÁº›'a ƒ#Q _”`¦ 7£x"î 



_.

Ls ëHАoÁº›'a ƒ#Q“¦ &ñ _. ÕªQ€ &ño ₇\¦ ëߖ7ᤠH _t

”

>rFôÇ.Û¼àÔaA w = a^tb^tc^t\¦ “¦9 . |vxy| ≤ tsÙ¼–Ð vyH a, b, c[j /åJ\¦ —¸¿º Ÿí†<ʽ+É Ãº \O_."f uv²xy²zH °ú “Ér ú_ a, b, c\¦ |9 ú \O_.

U

dV 28 L = {ww : w ∈ {a, b}^∗}H ëHАoÁº›'a ƒ#Q m. Ls ëHАoÁº›'a ƒ#Q“¦ &ñ “¦ w = a^tb^ta^tb^t\¦ ‚×þ˜ôÇ_{. vy}

#

Qn\ 0Au ŽHt _{uxz6∈ L}s_. U

dV 29 L = {aⁿ : n“Ér ™èú}“Ér ëHАoÁº›'a ƒ#Q m.

Ls ëHАoÁº›'a ƒ#Q“¦ &ñ € &ño ₇\¦ ëߖ7ᤠH _t ”>rFôÇ_. p\¦ _t˜Ð H ™èú“¦ _. 7£¤ _a^p ∈ L. q = |vy|, r = |uxz|“¦ 

& %

€



, q ≥ 1, r ≥ 1s. i = q + r + 1`¦ ‚×þ˜ €

iq+ r = (q + r)(q + 1)s“¦ q+ r ≥ 2, q + 1 ≥ 2sÙ¼–Ð uvⁱxyⁱz 6∈ Ls_.

Ç a

h 8 ëHАoÁº›'a ƒ#Q 7áxÀӍH (1) ½+Ë|9½+Ë, (2) ]X½+Ë, (3) Kleene Û¼ 

 ƒíߖ\ @/ #Œ {Œ—)€ e”_.

¤

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s“¦ . {9ìøÍ$í`¦ {9t ·ú§“¦ V1ü< V2H "f–Ð “§  t ·ú§H

| 9

½+Ës“¦ .

(1) Dh–Ðîr ú S\¦ 6 x #Œ

G= (V1∪ V2∪ {S}, Σ1∪ Σ2, S, P1∪ P2∪ {S → S1 | S2})

 L(G1) ∪ L(G2)\¦ Òqt$íôÇ.

(2) ½+Ë|9½+Ë_ âĺü< ðøÍts
, S → S1 | S2 @/’ S → S1S2\¦ '

‘

ôÇ.

(3)

G= (V1∪ {S}, Σ1, S, P1∪ {S → SS1 | ǫ})

53

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% s L(G1)^∗\¦ Òqt$íôÇ. 2

Ç a

h 9 ëHАoÁº›'a ƒ#Q 7áxÀӍH (1) “§|9½+Ë, (2) #Œ|9½+Ë ƒíߖ\ @/ 

#

Œ {Œ—)€ e”t ·ú§_.

¤

 ÃZ. (1) L1 = {aⁿbⁿc^m : n, m ≥ 0}, L2 = {aⁿb^mc^m: n, m ≥ 0}

“

¦ . L1õ L2H ëHАoÁº›'a ƒ#Qs. 7£¤ L1“Ér 6£§ ëHZO\ _K Òq

t$í)a.

S → AB A → aAb | ǫ B → cB | ǫ Õ

ªQ
 L1∩ L2H ëHАoÁº›'a ƒ#Q m.

(2)

L1∩ L2 = L1∪ L2

& % s

Ù¼–Ð #Œ|9½+Ë\ @/K {Œ—)€ e”ܼ€ “§|9½+Ë\ @/K"f•¸ {Œ—)€ e”#Q

)

a. 2

Ça

h 10 Ls ëHАoÁº›'a ƒ#Qs“¦ _Rs &ñ½©ƒ#Qs€_{, L ∩ R}“Ér ëHАo Á

º›'a ƒ#Qs.

¤

 ÃZ. M1 = (Q1,Σ, Γ, ∆1, q1, F1)`¦ L`¦ ~ÃÎ[þtsH PA,

M2 = (Q2,Σ, δ, q2, F2)\¦ R`¦ ~ÃÎ[þtsH DFA“¦ . ÅÒכ¹ >h¥Æ

“

Ér M1õ M2\¦ 1lxr\ [toH PA M `¦ ëߖ׼H כ s. ½¨^‰&hܼ–Ð M = (Q1× Q2,Σ, Γ, ∆, (q1, q2), F1× F2)

“



X<, „s ∆H 6£§õ °ú .

• ((p, a, A), (p^′, u)) ∈ ∆1s“¦ δ(q, a) = q^′s€, (((p, q), a, A), ((p^′, q^′), u) ∈ ∆s.

• ((p, ǫ, A), (p^′, u)) ∈ ∆1s€, —¸ŽH q ∈ Q2\ @/ #Œ ((p, q), ǫ, A), ((p^′, q), u)) ∈ ∆s.

55

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w∈ L ∩ R“ âĺ\ëߖ w ∈ L(M )e”`¦ ˜Ð{9 ú e”. 2

U

dV 30 (š¿¢4´%KV) CFG G = (V, Σ, S, P )ü<_{w ∈ Σ}^∗ ÅÒ#Q&’`¦ M

:_{, w} _L(G)_ "鶙è“t t\¦ &ñ _. s õ&ñ\"f

w∈ L(G)“ âĺ Û¼ àÔo\¦ ëߖ[þt> ÷&Ù¼–Ð ™è5ÅqëH]j\¦ ÉҍH õ

&

ñ

`¦ dç(parsing)s“¦ ÂҏÉr.

CYK (Cocke, Younger, Kasami) ·ú˜“¦o7£§“Ér 1lx&h >S\‰ZO(dynamic programming)_ {97áxܼ–Ð 6£§õ °ú _.

G _Chomsky ³ðïr+þAs“¦ _{w = a1a2· · · an}s“¦ _.

Vij = {A ∈ V : A⇒ a^∗ i· · · aj}“¦ . ÕªQ€, S ∈ V1n“ âĺ\ ë

ß

– w∈ L(G)s_. "f Vij\¦ d= j − i Œ•“Ér °úכ\"f H °úכܼ–Ð

€"f >íߖ € )a. G\ Òqt$í ½©gË:s ©œÃº >h e”ܼ€, CYK ·ú˜

“

¦o7£§“Ér _O(n³₎ rçߖs 2;_.

& % for i = 1 to n do

Vii = {A : A → ai} od

for d = 1 to n − 1 do for i = 1 to n − d do

j = i + d Vij = ∅

for k = i to j − 1 do

Vij = Vij ∪ {A : A → BC, B ∈ Vik, C ∈ Vk+1,j} od

od od

ÕªaË> 17:

57

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dV 31 6£§ ëHАoÁº›'a ëHZOs baaba\¦ Òqt$í Ht _CYK ·ú˜“¦o 7

£

§`¦ 6 x #Œ SX‰“ .

S → AB | BC A → BA | a B → CC | b C → AB | a CYK ·ú˜“¦o7£§s ëߖ׼H ³ðH 6£§õ °ú _.

i\j 1 2 3 4 5

1 B SA ∅ ∅ SAC

2 AC B B SAC

3 AC SC B

4 B SA

5 AC

& % S∈ V15sÙ¼–Ð baabaH s ëHZO\ _K Òqt$í)a_.

V

 8 ⠉ Ú r Ç a  6 h Å ÁŸ  ¸ › ïŸ ×  

?

/o¾º2£§ š¸žÐ ôÇ r&h\ ¿º >h s©œ_ „s\¦ |9 ú \Oܼ

€



 s\¦ &ñ ?/o¾º2£§ š¸žÐ“¦ ÂҏÉr. &ñ ?/o¾º2£§ š¸žÐ



(deterministic pushdown automata)\¦ צ#Œ"f DPA–Ð ÂÒØÔl

–

Ð . DPA {9§4s
 Û¼×þ˜`¦ {9t ·ú§“¦ „s½+É Ãº e”•¸2Ÿ¤ H

כ

s ¼#o Ù¼–Ð „s†<Êú δ\¦ 6£§ ›¸| `¦ ëߖ7ᤠH Q× (Σ ∪ {ǫ}) × (Γ ∪ {ǫ})\"f Q × Γ^∗–Ð_ ÂÒìr†<Êú(partial function)–Ð &ñ_ôÇ.

• δ(q, ǫ, A) &ñ_÷&€, —¸ŽH a ∈ Σ\ @/ #Œ δ(q, a, A) &ñ_÷&

t

 ·ú§H.

• δ(q, a, ǫ)s &ñ_÷&€, —¸ŽH A ∈ Γ\ @/ #Œ δ(q, a, A) &ñ_÷&

t

 ·ú§H.

59

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¸ôÇ DPA{9§4_ =åQ`¦ ·ú˜ ú e”•¸2Ÿ¤ {9§4 6£§ ñߖ\ /BNÑþ˜ /åJ

# [þt#Q e”“¦ &ñôÇ. DPA ~ÃÎ[þtsH ƒ#Q\¦ &ñ ëHАo Á

º›'a ƒ#Q“¦ ÂҏÉr.

U

dV 32 {0ⁿ1ⁿ : n ≥ 0}`¦ ~ÃÎ[þtsH DPAH ÕªaË> 18õ °ú .

q

q

1 2

3

q

0, ε/0

#, #/# #, #/#

1, 0/ε

1, 0/ε

ÕªaË> 18:

Ç a

h 11 &ñ ëHАoÁº›'a ƒ#QH #Œ|9½+Ë\ @/ #Œ {Œ—)€ e”_.

& %

¤

ÃZ. ([O"î) DPA\"fH þj7áx©œIü< ÕªXOt ·ú§“Ér ©œI\¦ ´úÀDKܼ

–

Ð #Œ|9½+Ë`¦ ~ÃÎ[þtsH DPA\¦ ëߖ[þt ú \O. =
 € >5Åq {9§4

`

¦ {9t ·ú§“¦ ÁºôÇÀÒáÔ\ [þt#QH âĺ e”l M:ëHs. "f s

 7£x"î`¦ 0AK"fH ÅÒ#Q” DPA\¦ B ôÇ /åJm” {9§4`¦ {9H DPA–Ð õH 6£§\ þj7áx©œIü< ÕªXOt ·ú§“Ér ©œI\¦ 7 ôÇ.

2

Ç a

h 12 &ñ ëHАoÁº›'a ƒ#Q 7áxÀӍH ëHАoÁº›'a ƒ#Q 7áxÀÓ_ ”ÂÒìr

| 9

½+Ës_.

¤

 ÃZ. L = {aⁱb^jc^k: i 6= j ¢¸H j 6= k}\¦ “¦9 . L“Ér ëHАoÁº›'aƒ

# Qs.

Ls &ñ ëHАoÁº›'a ƒ#Q“¦ &ñ . ÕªQ€ L•¸ &ñ ëHАoÁº›'a s

“¦ "f ëHАoÁº›'as. L ∩ a^∗b^∗c^∗H ëHАoÁº›'as ÷&HX<, sכ s {aⁿbⁿcⁿ: n ≥ 0}sÙ¼–Ð —¸íHs. 2

61

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V

 9 ⠉ / ’ ×   b 

{ 9

§4`¦ ~ÃÎ[þt#Œ"f Û¼ àÔo\¦ ëߖ׼H õ&ñ`¦ dç(parsing)s

“

¦ ôÇ. CYK ·ú˜“¦o7£§`¦ 6 x € dç`¦ ½+É Ãº e”ܼ
, H áԖРÕ

ªÏþ›`¦ dç½+É âĺ -Áº ´ú§“Ér rçߖs 2;. {9ìøÍ&hܼ–Ð (Ž{9

QH O(n) rçߖ\ dç H ·ú˜“¦o7£§`¦ €9כ¹–Ð ôÇ.

U

dV 33 \V]j ₂₄_ ëHZO

S → 0SA | 0A A → 1

õ

 Û¼àÔaA 0011s ÅÒ#Q&’`¦ M:, dç`¦ K˜Ð. Û

¼àÔaA_ 'Í P: /åJ 0sÙ¼–Ð rŒ•ú\"f S → 0SA ¢¸H S→ 0A\¦ &h6 xK ôÇ_. #QÖ¼ Òqt$í½©gË:`¦ &h6 xK ½+Ét ·ú˜ ú

\O

tëߖ S → 0SA\¦ &h6 xK˜Ð€_, Ä»•¸õ&ñ S ⇒ 0SA

& %

`

¦ %3H_. ú Ss“¦ ¿º P: /åJ•¸ ₀sÙ¼–Ð ·ú¡_ âĺü<  ð

ø

Ít“X<, S → 0A\¦ &h6 xK˜Ð€ S ⇒ 00AA^∗

`

¦ %3H. ú As“¦ [j P: /åJ 1sÙ¼–Ð A→ 1`¦ &h6 xK



 ôÇ_. W1 P: /åJ_ â啸 ðøÍts_{. (}‚ÃГ¦–Ð_,{9§4 /åJ

\

¦ ¿º >h {9ܼ€ #QÖ¼ Òqt$í½©gË:`¦ &h6 xK ½+Ét –Ð ·ú˜ ú e”_.

‰

&

³F ú S{9 M:, 00`¦ {9ܼ€ S → 0SA\¦, 01`¦ {9ܼ€ S→ 0A\¦ &h6 x € )a_.)

‰

&

³F ú Aü< /åJ a\ @/K"f A→ a . . . +þAI_ Òqt$í½©gË:s 
 ë

ߖ e”ܼ€ Ä»•¸õ&ñ`¦ _O(n)\ ¢-a$í½+É Ãº e”_. 7£¤_{, O(n)} rçߖ\  d

ç

½+É Ãº e”.

†¾Ó dç(topdown parsing)“Ér 0A_ >h¥Æ`¦ {9ìøÍor† כ s. 

†

¾

Ó dç“Ér {9§4`¦ ¢,aAá¤\"f š¸ÉrAá¤Ü¼–Ð {9ܼ€"f Û¼ àÔo\¦ 0A\

"

f A–Ð ëߖ[þt#Q H dçs. Û¼ àÔo\¦ 0A\"f A–Ð ëߖ[þt

#

Q H õ&ñ“Ér ýa8£¤ Ä»•¸\¦ ëߖ׼H כ õ 1lx{9 .

63

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6£§ ëHZO G\ @/ #Œ w ∈ L(G)\¦ dç H DPA\¦ ½¨ .

E → E + T | T T → T ∗ F | F F → id

#

Œl\"f idH 
_ /åJ–Ð çߖÅÒôÇ.

†¾Ó dçs 6 xs •¸2Ÿ¤ 6£§ ¿º ½©gË:`¦ 6 x #Œ ëHZO`¦ õH.

1. A → Ax +þAI_ Òqt$í½©gË:`¦ \O±p. A ýa\

H —¸

Ž

H Òqt$í½©gË:s

A → Ax1 | · · · | Axr

A → y1 | · · · | yt

s

€, s\¦

A → y1A^′ | · · · | ytA^′ A^′ → x A^′ | · · · | x A^′ | ǫ

& %

–

Ð õH.

2. Òqt$í½©gË:_ ĺs °ú “Ér Û¼àÔaAܼ–Ð rŒ•÷&t ·ú§•¸2Ÿ¤ “¦•2;.

A→ xy1 | · · · | xyr

(x 6= ǫ, r ≥ 2)s€, s\¦

A → xA^′

A^′ → y1 | · · · | yr

–

Ð õH.

ë

HZO G\ ½©gË: 1`¦ &h6 x €

E → T E^′ E^′ → +T E^′| ǫ

T → F T^′ T^′ → ∗F T^′ | ǫ

F → id

65

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dV 34 0A_ ëHZOܼ–Ð ^id_{+ id}\¦ Òqt$í H õ&ñ“Ér 6£§õ °ú _. E ⇒ T E^′

⇒ T + T E^′

⇒ F T^′+ T E^′

⇒ F + T∗

⇒ F + F∗

⇒ id + id∗

DPA{9§4 /åJ\¦ 
 {9#Q"f ‰&³F ú\"f #Q‹" Òqt$í½©gË:`¦

&

h

6 x½+Ét &ñ •¸2Ÿ¤ ëߖ[þt9“¦ ôÇ. s\¦ 0AK"f ú Aü< {9§4 /

å

J a ÅÒ#Qt€ &h6 x½+É Òqt$í½©gË:`¦ dç ³ð M (A, a)\ $©œôÇ

. yŒ• Òqt$í½©gË: A → x\ @/ #Œ

1. x\"f Ä»•¸÷&H 'Í P: /åJ as€ (7£¤ x ⇒ ay) M (A, a)\^∗  A→ x\¦ V,H.

2. x⇒ ǫ^∗ “ âĺ, A 6£§\
`¦ ú e”H /åJ as€ M (A, a)\

& % A→ x\¦ V,H.

G_ Òqt$í½©gË:`¦ “¦9 €

• E → T E^′: T H F \¦ 5g id\¦ Òqt$í Ù¼–Ð M (E, id)\ s Òqt$í

½©gË:`¦ V,H.

• E^′ → ǫ: E → T E^′\ _K E^′“Ér {9§4_ =åQ\ 0Au Ù¼–Ð Õª  6

£

§\ #s `¦ ú e”#Q"f M (E^′,#)\ V,H.

• T^′ → ǫ: T^′•¸ {9§4_ =åQ\ 0Au½+É Ãº e”ܼټ–Ð M (T^′,#)\ V,

“

¦ ¢¸ôÇ T^′ 6£§\ +s `¦ ú e”ܼټ–Ð M (T^′,+)\ V,H.

s

ü< °ú s dç ³ð\¦ ëߖ[þt€ ÕªaË> 19õ °ú . dç ³ð\¦ ëߖ׼H  [

jôÇ ~½ÓZO“Ér [?]\¦ ‚ÃГ¦ .

67

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id + ∗ #

E E → T E^′

E^′ E^′ → +T E^′ E^′ → ǫ T T → F T^′

T^′ T^′ → ǫ T^′ → ∗F T^′ T^′ → ǫ F F → id

ÕªaË> 19:

s

]j DPAH {9§4 /åJ\¦ {9“Ér Êê 0A_ dç ³ð\ _K Òqt$í½©gË:`¦

&

h

6 xôÇ (ÕªaË> 20).

Figure omitted

ÕªaË> 20:

& % {

9

§4 id + id ∗ id#\ @/ôÇ DPA_ 1lxŒ•õ&ñs ÕªaË> 21\ e”. ©œ I


 {9§4, Û¼×þ˜s  t ·ú§“Ér âĺ\H ––Ð ³ðr %i. sü< °ú  s

 ëߖ[þt#Q” Û¼ àÔoH ÕªaË> 22\ e”.

Table omitted ÕªaË> 21:

69

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%

E’

T +

ε

* F T’

id F

id

ε id ε T’

F T’

E T E’

ÕªaË> 22:

& % ifëH`¦
?/H ëHZO`¦ çߖ|ÄÌy € 6£§õ °ú .

S → iCtSeS | iCtS | x C → a

#

Œl\"f i, t, eH yŒ•yŒ• if, then, else\¦
·p.

½©gË: 2\¦ &h6 x €

S → iCtSS^′ | x S^′ → eS | ǫ

C → a

\

¦ %3H. s]j dç ³ð\¦ ½¨ .

• S^′ → eSH M (S^′, e)\ [þt#Q )a.

• S^′ → ǫ`¦ “¦9 €, iCtSS^′\"f S → iCtSS^′ü< S^′ → eS\¦  6

x € S^′ 6£§\ e
š¸Ù¼–Ð M (S^′, e)\ S^′ → ǫs [þt#Q

) a.

71

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"

é

¶A ëHZOs EB l M:ëH\ M (S^′, e)\ ¿º >h_ Òqt$í½©gË:s [þt#Q

> )a. s âĺ ¿º Òqt$í½©gË: ×æ S^′ → eS\¦ ‚×þ˜ € else\¦ 



©

œ îr if\ ·¡­sH õ )a. dç ³ð\¦ ¢-a$í € ÕªaË> 23õ

° ú  .

i e x a #

S S → iCtSS^′ S → x

S^′ S^′ → eS S^′ → ǫ

C C → a

ÕªaË> 23: