8Q 1  Y

& %

N

ñ 5 Ñ  ·

1

' $

V

 1 ⠉ È Áø 5  ™ T Ò ¼˜ + æ «e 

Á

ºôÇ|9½+˓Ér ß¼l\ _K ¿º t–Ð
Ðüt ú e”. ƒú_ |9½+Ë N = {0, 1, 2, . . .}õ 1@/1 @/6£x›'a>\¦ tH |9½+Ë`¦ !sq ú e”H Áº ô

Ç|9½+Ës“¦ ÂҏÉr. Õª ü@_ ÁºôÇ|9½+Ë`¦ !sq ú \OH |9½+Ës“¦ Â

ҏÉr. |9½+Ë A !sq ú e”H ÁºôÇ|9½+Ës€, N õ 1@/1 @/6£x›'a>\¦

tÙ¼–Ð s @/6£x›'a>\ _K A_ "鶙è\¦ a0, a1, a2, . . . YV–Ð


\P

½+É Ãº e”.

U

dV 1 A, B !sq ú e”H ÁºôÇ|9½+Ës€_{, A ∪ B}•¸ !sq ú e”H ÁºôÇ

| 9

½+Ës_. é

ß

–íH > l 0A #Œ A∩ B = ∅s“¦ &ñ _{. A, B} !sq ú e”



H |9½+ËsÙ¼–Ð Õª "鶙è\¦ YV–Ð
\P½+É Ãº e”_. 7£¤

A = {a0, a1, . . .}, B = {b0, b1, . . .}. s]j A∪ B_ "鶙è\¦ YV–Ð


\P

 H ~½ÓZO`¦ ˜Ðs€ )a. A ∪ B = {a0, b0, a1, b1, . . .}õ °ú s Aü< B_ "鶙è\¦ °ú˜ €"f
\P € )a_.

& % U

dV 2 N × N“Ér !sq ú e”H ÁºôÇ|9½+Ës.

N × N_ —¸ŽH "鶙è\¦ Ä»ôÇ  s?/\ !sq ú e”H íH"f\¦ ¹1Ô ôÇ



. (0, 0), (0, 1), (0, 2), . . . íH"f–Ð
\P € Ä»ôÇ  s?/\ _{(1, 0)}`¦

\P½+É Ãº \Oܼټ–Ð îߖ )a_{. (i, j)}\¦ i+ j 7£x H íH"f₍7£¤ (0, 0), (0, 1), (1, 0), (0, 2), (1, 1), (2, 0), (0, 3), . . .)–Ð
\P € )a_. Ça

h 1 2^N“Ér !sq ú \OH |9½+Ës_.

¤



ÃZ. 2^Ns !sq ú e”H ÁºôÇ|9½+Ës“¦ &ñ . ÕªQ€ 2^N_ "é¶

™è(N _ ÂÒìr|9½+Ë)\¦
\P H ~½ÓZOs ”>rFôÇ.

2^N = {A0, A1, A2, . . .}

y

Œ• |9½+Ë Ai\¦ ³ð–Ð
?/€ 6£§õ °ú . s ³ð_ 'Ÿ“Ér |9½+Ë Ai\¦,

\P

“Ér "鶙è a\¦
?/“¦ a ∈ Ais€ ³ð_ °úכs 1s“¦, a 6∈ Ais€

3

' $

0s.

0 1 2 3 · · · A0 0 0 0 1 A1 0 1 1 0 A2 1 0 1 0 A3 1 0 0 0

· · · s

]j |9½+Ë D = {i ∈ N : i 6∈ Ai}\¦ “¦9 . 7£¤ DH 0A_ ³ð\"f Å

Ò@/yŒ•‚s 0“ "鶙è[þt_ |9½+Ës. DH ƒú_ |9½+ËsÙ¼–Ð 2^N_ "鶙ès#Q ôÇ. ÕªQ
 DH #QÖ¼ |9½+Ë Aiü<•¸ °ú t ·ú§ (ƒú i Ai\ 5Åq € D\ 5Åq t ·ú§“¦, Ai\ 5Åq € D\ 5Åq  t

 ·ú§Ü¼Ù¼–Ð). 7£¤ —¸íH\ •¸²ú˜ Ù¼–Ð, 2^NH !sq ú \OH |9½+Ës.

2

· ú

˜ Ï@ Σ ©œ_ —¸ŽH Û¼àÔaA_ |9½+Ë Σ^∗H !sq ú e”H ÁºôÇ|9½+Ës.

& % Σ = {0, 1}“ âĺ, Û¼àÔaA_ U´s\ _K ǫ, 0, 1, 00, 01, 10, 11, . . ._

í

H"f–Ð
\P½+É Ãº e”.

ƒ



#QH Σ^∗_ ÂÒìr|9½+ËsÙ¼–Ð, ƒ#Q_ „^‰ |9½+˓Ér 2^Σ∗s. 7£¤ ƒ

#

Q_ „^‰ |9½+˓Ér !sq ú \OH |9½+Ës. 6£§ ©œ\"f F) \P ƒ#Q 7

á

xÀӍH !sq ú e”H ÁºôÇ|9½+Ëe”`¦ ˜Ð“. "f 2^Σ∗\H F) \P

  ƒ#Q(7£¤ ÈÓaAl>–Ð ³ðr½+É Ãº \OH ƒ#Q) ÁºôÇy ´ú§s

”

>rFôÇ. sÊê\ F) \P  ƒ#Q_ \V\¦ ˜Ð> |¨c כ s.

V

 2 ⠉ ƒ 7 £  Ï Úÿ b e N 

· ú

¡Ü¼–Ð ƒ#Q_ ·ú˜ Ï@ Σ\¦ {0, 1}–Ð, TM_ _…{9 ·ú˜ Ï@ Γ\¦ {0, 1, #}–Ð ]jôÇôÇ. ]jôǝ)a ëH]j(ƒ#Q) >íߖÔ¦e”`¦ ˜Ðs€,

8 {9ìøÍ&h“ ëH]jH {©œƒy >íߖÔ¦ )a.

€



$ ÈÓaAl>\¦ Û¼àÔaAܼ–Ð ˨H ~½ÓZO`¦ ˜Ð. TM

M = (Q, {0, 1}, {0, 1, #}, δ, q1, H)\¦ 6£§õ °ú s {0, 1} ©œ_ Û¼àÔ

5

' $

a

Aܼ–Ð ÂÒ ñoôÇ.

1. Q = {q1, . . . , qn}{9 M:, qiH 0ⁱܼ–Ð ³ðrôÇ.

2. Σ = {0, 1}\"f 0õ 1“Ér yŒ•yŒ• 0õ 00ܼ–Ð ³ðrôÇ.

3. Γ = {0, 1, #}\"f 0, 1, #H yŒ•yŒ• 0, 00, 000ܼ–Ð ³ðrôÇ.

4. ~½Ó†¾Ó`¦
?/H L, R, SH yŒ•yŒ• 0, 00, 000ܼ–Ð ³ðrôÇ.

5. „s δ(q, a) = (p, b, d)“Ér q, a, p, b, d_ ÂÒ ñ\¦ s\ 1`¦ ¿º“¦

YV–Ð
\PôÇ. 7£¤ δ(qi,0) = (qj,1, L)“Ér 0ⁱ1010^j10010ܼ–Ð ³ð r

)a.

6. TM M _ ÂÒ ñH „s_ ÂÒ ñ\¦ s\ 11`¦ ¿º“¦ „ íH"f–Ð

\P “¦, ·ú¡+'\ 111`¦ ·¡­“ כ s.

7. œíl ©œIH †½Ó©œ q1s.

8. &ñt ©œIH „s_ 'Í P: "鶙è\

t ·ú§H ©œIs. ƒ#Q

\

¦ &ñ½+É âĺ H = {y, n}“X<, sM: yH ¿º &ñt©œI ×æ „ í

H"f Œ•“Ér כ s“¦ n“Ér H כ s.

& % M _ ÂÒ ñ\¦ hM iܼ–Ð ³ðrôÇ.

TM M _ Σü< Γ {9ìøÍ&h“ âĺ\•¸ s[þts Ä»ôÇ|9½+ËsÙ¼–Ð M `¦ {0, 1} ©œ_ Û¼àÔaAܼ–Ð ³ðr½+É Ãº e”. "f —¸ŽH TM_ |9

½+

Ë(7£¤ F) \P ƒ#Q_ |9½+Ë)“Ér {0, 1}^∗_ ÂÒìr|9½+ËsÙ¼–Ð !sq ú e”



H ÁºôÇ|9½+Ës.

U

dV 3 TM M = ({q1, q2, q3}, {0, 1}, {0, 1, #}, δ, q1,{q3}) s“¦ δ(q1,#) = (q2,#, R)

δ(q2,0) = (q2,1, R) δ(q2,1) = (q2,0, R) δ(q2,#) = (q3,#, S)

s

€ hM i“Ér

11101000100100010011001010010010011

7

' $

001001001010011001000100010001000111 s

_.

ÈÓaAl> M _ {9§4 Û¼àÔaA w\ @/K"f•¸ 0õ 1`¦ yŒ•yŒ• 0õ 00ܼ–Ð

³

ðr “¦ /åJ s\ 1`¦ V,#Q ÂÒ ñ hwi\¦ ëߖŽH. 7£¤

h1001i = 001010100. hM iõ hwi\¦ ]X½+ËôÇ כ `¦ hM, wi–Ð ³ðrôÇ.

t

FKt “¦9ôÇ ÈÓaAl>H {0ⁿ1ⁿ : n ≥ 1}`¦ &ñ H TM, » !lr

H TM, YL!lr H TM 1px :£¤&ñôÇ l0px`¦ ú'Ÿ H TMs%3. Õª





X< tFK ĺo 6 x H (ŽÉÓ'H #ŒQ t l0px`¦ ú'Ÿ½+É Ãº e

”

H #36 x(ŽÉÓ'(general-purpose computer)s. #36 x (ŽÉÓ'\¦  t

“¦ :£¤&ñôÇ l0px`¦ ú'Ÿ “¦ ½+É M:H Õª l0px`¦ ú'Ÿ H áԖРÕ

ªÏþ›`¦ Œ•$í #Œ s áԖÐÕªÏþ›`¦ #36 x(ŽÉÓ' ú'Ÿ € )a. 7£¤ t

FKt “¦9ôÇ TM“Ér áԖÐÕªÏþ›\ K{©œôÇ. #36 x(ŽÉÓ'\ K{©œ

H כ s #36 xÈÓaAl>s.

# 3

6 xÈÓaAl> MuH hM, wi\¦ {9§4ܼ–Ð ~ÃÎ"f M `¦ {9§4 w\¦  t

“¦ [t2; Êê Õª õ\¦ ئ§4 H TMs (7£¤ M s &ñt €

& % Mu•¸ &ñtôÇ). MuH 3_…{9 TMs. MuH 6£§õ °ú s 'Í P: _…

{ 9

\ hM i`¦, ÑütP: _…{9\ M _ _…{9 ?/6 x`¦, !Ó P: _…{9\ M _ ©œ I

\¦ $©œ “¦ M `¦ w\ @/ #Œ Œ•1lxr†.

1. 'Í P: _…{9\ ÅÒ#Q” {9§4 hM, wi ×æ\"f hM i\ K{©œ H ÂÒìr s

 ]j@/–Ð )aTM_ ÂÒ ñ“t &hŽôÇ (syntax error &hŽ). 7£¤ 111–Ð rŒ• “¦ 0\ _K ½¨ìr)a 1s 4>h
š¸“¦ 11s
š¸H

כ

s ìøÍ4Ÿ¤)a Êê\ 111s
š¸Ht, „s 0^a10^b10^c10^d10^e\"f 1 ≤ b, d, e ≤ 3“t, „s ×æ\"f °ú “Ér 0^a10^b1–Ð rŒ•÷&H כ s

\O

Ht &hŽôÇ.

2. 'Í P: _…{9\ hM i`¦ zŒ™l“¦ hwi\¦ ÑütP: _…{9\ `… &hH. !Ó P

: _…{9\ M _ rŒ•©œI q1\ K{©œ H 0`¦ &hH. —¸ŽH _…{9 _

 K‰×¼H ±p ¢,aAá¤\ 0AuôÇ.

3. !Ó P: _…{9_ ?/6 xs 0^as“¦ ÑütP: _…{9_ ‰&³F /åJ 0^bs€ 'Í P

: _…{9\"f 110^a10^b1–Ð rŒ•÷&H „s e”Ht ¹1ԍH. Õª

„



s \Oܼ€ (0^a M _ &ñt©œIs) MuH &ñtôÇ. Õª

„



s e”ܼ€ s\¦ !Ó P: _…{9õ ÑütP: _…{9\ ìøÍ%òôÇ.

9

' $

V

 3 ⠉ Ç a m % K V 

L1s F) \P ƒ#Q{9 M: L1`¦ &ñ_ H TM M1õ L2 F) ƒ#Q {

9

 M: L2\¦ &ñ H TM M2\¦ 6£§ ÕªaË>õ °ú s ³ðrôÇ.

M ₁ h

w w M ₂

y n

ÕªaË> 1:

Ç a

h 2 F) ƒ#Q L_ #Œ|9½+˓Ér F) s_.

¤

ÃZ. L`¦ &ñ H TM\"f yü< n`¦ "f–Ð ˨#Q ÅҀ ¯L`¦ &ñ 



H TMs )a. 2

& % Ç

a

h 3 ƒ#Q Lõ Õª #Œ|9½+Ë L^¯s Ñüt  F) \Ps€ L“Ér F) s



_.

¤

ÃZ. Lõ ¯Ls F) \PsÙ¼–Ð s\¦ &ñ_ H TM M1õ M2 ”>r F

ôÇ. {9§4 w\ @/ #Œ M1õ M2\¦ 1lxr\ [to€"f M1_ &ñt



©

œI\¦ y–Ð, M2_ &ñt©œI\¦ nܼ–Ð °úH TMs L`¦ &ñ H TM s

. 2

&

ñ

o 2õ 3\ _K"f ôÇ Š©œ_ ƒ#Q Lõ ¯LH 6£§_ âĺ[þtëߖ`¦ 

| 9

 ú e”.

• (Ñüt ×æ_ 
•¸ F) “ âĺ) Lõ ¯L Ñüt  F) s. (
 Qt âĺ ×æ, Ñüt  F)   F) \P{9 ú \Oܼټ–Ð 6£§ _

 ¿º t zŒ™H.)

• Lõ ¯L Ñüt  F) \P m.

• Lõ ¯L Ñüt ×æ_ 
H F)   F) \Ps“¦, Ér 
H F

) \P m.

11

' $

&

ñ

tëH]jêøÍ {9§4ܼ–Ð áԖÐÕªÏþ› P ü< {9§4 w\¦ Åғ¦ P  w\ @/ 

#

Œ &ñt½+Ét îߖ ½+Ét\¦ &ñ H ëH]js. &ñtëH]j\¦ ƒ#Q–Ð  Ë

¨€

Lu = {hM, wi : w ∈ L(M )}

s

 )a. &ñtëH]j >íߖÔ¦e”`¦ ˜Ðsl 0A #Œ Lu F)   _

”

`¦ ˜Ðs“¦ ôÇ.



6£§ ôÇ Š©œ_ ƒ#Q\¦ &ñ_ôÇ.

Ld = {hM i : hM i ∈ L(M )}

7

£¤, w ∈ {0, 1}^∗ #Q‹" ÈÓaAl> M _ ÂÒ ñs“¦ M s hM i\ @/ 

#

Œ &ñt H âĺ\ëߖ wH Ld_ "鶙ès.

L¯dH Ld_ #Œ|9½+Ës. 7£¤, w ∈ {0, 1}^∗ ÈÓaAl>_ ÂÒ ñ m


 ws #Q‹" ÈÓaAl> M _ ÂÒ ñ“X< M s hM i\ @/ #Œ &ñt  t

 ·ú§H âĺ\ëߖ wH ¯Ld_ "鶙ès.

Ç a

h 4 ¯LdH F) \P m_.

& %

¤

ÃZ. {0, 1}^∗_ "鶙è\¦ ³ðïr íH"f–Ð 'Ÿõ \P\
\PôÇ ³ð\¦ ÒqtyŒ• 

. s ³ð_ 'Ÿ“Ér ÈÓaAl>_ ÂÒ ñ hM is“¦, \P“Ér ÈÓaAl>\ @/ôÇ {

9

§4 éߖ#Q ws“¦, w ∈ L(M )s€ ³ð_ °úכ“Ér 1s“¦, m€ ³ð_ °úכ

“

Ér 0s. LdH ÅÒ@/yŒ•‚s 1“ éߖ#Q[þt_ |9½+Ës“¦, ¯LdH ÅÒ@/yŒ•‚ s 0“ éߖ#Q[þt_ |9½+Ës.

L¯d F) \Psl 0AK"fH ¯Ld\¦ &ñ_ H TM(7£¤ L(M ) = ¯Ld“ TM M )s ”>rF #Œ ôÇ. —¸ŽH TM“Ér 0A_ ³ð_ 'Ÿ\
\P÷&#Q e

”

HX< s ×æ #Q‹" TM M •¸ L(M ) 6= ¯Lds. =
 €

hM i ∈ L(M )“ âĺ\ëߖ hM i 6∈ ¯Ldsl M:ëHs. ¯Ld\¦ &ñ_ H ÈÓaAl> \Oܼټ–Ð ¯LdH F) \P m. 2

L¯dH F) \P  ƒ#Q_ 'Í P: \Vs. 0A_ [j t âĺ\ _

K"f LdH F)   F) \Ps
 F) \P m#Q 



HX<, Lu\¦ “¦9†<Êܼ–Ð+‹ Ld F)   F) \Pe”`¦ µ1ßy“¦



 ôÇ.

Ç a

h 5 LuH F) \Ps_.

13

' $

¤



ÃZ. #36 xÈÓaAl> MuH hM, wi_ {9§4\ @/ #Œ w\¦ t“¦ M `¦ [t9˜Ð"f M s &ñt € Mu•¸ &ñtôÇ. 7£¤ L(Mu) = Lus

. Lu\¦ &ñ_ H ÈÓaAl> ”>rF Ù¼–Ð LuH F) \Ps.

2

Ç a

h 6 LdH F) \Ps.

¤

ÃZ. {9§4 w ∈ {0, 1}^∗\¦ whwi–Ð ˨#Q"f #36 xÈÓaAl> Mu\ ˜Ð



·

p Êê Mu_ õ\¦ ئ§4 H ÈÓaAl>\¦ Md“¦ . ÕªaË> 2. Õª

Q€ w #Q‹" ÈÓaAl> M _ ÂÒ ñs“¦ M s w = hM i\ @/ #Œ

&

ñ

t H âĺ\ëߖ MdH &ñtôÇ. 7£¤ L(Md) = LdsÙ¼–Ð LdH F )

\Ps. 2

& %

w h

M_d

M_u h w<w>

ÕªaË> 2:

Ç a

h 7 Luü< LdH F)  m.

¤

ÃZ. Lu F) “¦ &ñ . ÕªQ€ Lu\¦ &ñ H ÈÓaAl> M0s ”>rFôÇ. 7£¤ M0“Ér {9§4 whwi ∈ {0, 1}^∗\ @/ #Œ w #Q‹" ÈÓaAl> M _ ÂÒ ñs“¦ w ∈ L(M )s€ y\"f &ñt “¦

w6∈ L(M )s€ n\"f &ñtôÇ.

15

' $

s

]j M0`¦ s6 x #Œ Ld\¦ &ñ H ÈÓaAl> M1`¦ ëߖ[þt“¦ ôÇ

. M1“Ér {9§4 w ∈ {0, 1}^∗\ @/ #Œ whwi\¦ ëߖ[þt#Q"f M0\> ˜Ð



·

p Êê, M0_ õ\¦ ئ§4ôÇ. ÕªaË> 3. ÕªQ€ M1“Ér Ld\¦ &ñ H ÈÓaAl>s.

M

n

w<w> y y

n w

M₁

0

ÕªaË> 3:

7

£¤ Lu F) s€ Ld•¸ F) s“¦, &ño 2\ _K"f Ld F) s€ L¯d•¸ F) s“¦, "f ¯LdH F) \Ps. sכ “Ér &ño 4\ —¸íH

)

a. 2

& % s

©œõ °ú s F)   F) \P ƒ#Q_ ¿º \V Luü< Ld\¦ %3%3

. Church-Turing "î]j\ _K F) ƒ#Q\¦ (ŽÉÓ'–Ð Û¦ ú e”H ë

H]j“¦ &ñ_ %il M:ëH\, &ñtëH]jH >íߖÔ¦ ëH]js.

V

 4 ⠉ N ñ 5 Ñ  · % K V 

>

íߖÔ¦ ëH]j“ &ñtëH]j\¦ ¹1ԀŒ¤Ü¼Ù¼–Ð s\¦ s6 x #Œ #ŒQ t ë

H]j >íߖÔ¦e”`¦ ˜Ð{9 ú e”. s\¦ 0AK ëH]j_ 

¨8

Š(reduction)sH lZO`¦ 6 x “¦ ôÇ.

Ç a

˜+ 1 ƒ#Q L1\"f ƒ#Q L2–Ð_ ¨8ŠsêøÍ w∈ L1“ âĺ\ëߖ _{f(w) ∈ L2}

\

¦ ëߖ7ᤠH >íߖ0px †<Êú fs_.

L2 >íߖÔ¦e”`¦ ˜Ðsl 0AK"fH sp >íߖÔ¦e”s ·ú˜9” ƒ

#Q L1`¦ ‚×þ˜ #Œ L1`¦ L2–Ð ¨8ŠôÇ. ¨8Š_ _pH L2\¦ &ñ

17

' $

½+

É Ãº e”ܼ€ L2_ ²úšs L1_ ²úšs ÷&Ù¼–Ð L1`¦ &ñ½+É Ãº e”. Õª a

Ë> 4. @/ĺ\ _K L1s >íߖÔ¦sÙ¼–Ð L2•¸ >íߖÔ¦H õ

\

¦ %3H. #Œl\"f ¨8Š_ ~½Ó†¾Ó\ ÅÒ_\¦ lÖ¦#Œ ôÇ. L2\¦ L1–Ð ¨8Š € Áº õ•¸ %3t 3lwôÇ.

w f(w)

M_L2

n y n

y

M_L1

ÕªaË> 4:

Ç a

h 8 “TM Ms ÅÒ#Q&’`¦ M:_{, M}s ǫ\ @/ #Œ &ñt H_” ëH ]

j_, 7£¤ _Lǫ = {hM i : ǫ ∈ L(M )}“Ér >íߖÔ¦s_.

& %

¤



ÃZ. Lǫs F) “¦ &ñ . ÕªQ€ Lǫ`¦ &ñ H TM Mǫs ”>r F

ôÇ. s]j Lu\¦ Lǫܼ–Ð ¨8Š . Lu_ {9§4 hM, wis ÅÒ#Qt€ Lǫ_ {9§4 hMwi–Ð ¨8ŠôÇ. MwH {9§4 ǫs ÅÒ#Qt€ _…{9\ w\¦ æ

¼“¦ M `¦ rŒ•rvH TMs. ÕªaË> 5. hM, wi\"f hMwi\¦ %3l 0

AK"fH _…{9\ w\¦ 漓¦ M _ œíl©œI–Ð [þt#QH „s\¦ hM i\ 8KÅҀ )a. ÕªQ€

• M s w\ @/ #Œ &ñt € MwH ǫ\ @/ #Œ &ñt “¦

• M s w\ @/ #Œ &ñt t ·ú§Ü¼€ MwH ǫ\ @/ #Œ &ñt t

· ú

§H.



"f Mǫs Mw_ ǫ\ @/ôÇ &ñt #ŒÂÒ\¦ ·ú˜9 ÅҀ Õªכ s –Ð M s w\ @/ôÇ &ñt #ŒÂÒ )a. 2

19

' $

M

<M>

n y y

n

<M >_w

ÕªaË> 5:

Ç a

h 9 “TM Ms ÅÒ#Q&’`¦ M:, Ms —¸ŽH {9§4\ @/ #Œ &ñt  t

 ·ú§H_” ëH]j_, 7£¤ _L∅ = {hM i : L(M ) = ∅}“Ér >íߖÔ¦s_.

¤



ÃZ. L∅\¦ F) “¦ &ñ € L∅\¦ &ñ H TM M∅s ”>rFôÇ.

s

]j Lǫ`¦ L∅–Ð ¨8Š . Lǫ_ {9§4 hM is ÅÒ#Qt€ L∅_ {9§4 hM<sup>′</sup>iܼ–Ð ¨8ŠôÇ. M<sup>′</sup>“Ér ÅÒ#Q” {9§4`¦ t哦 ǫ\ @/ #Œ M `¦ r

Œ•rvH TMs. hM i\"f hM^′i`¦ %3l 0AK"fH _…{9_ ?/ 6

 x(ÅÒ#Q” {9§4)`¦ t哦 M _ œíl©œI–Ð [þt#QH „s\¦

& % hM i\ 8KÅҀ )a. ÕªaË> 6. ÕªQ€

• M s ǫ\ @/ #Œ &ñt € M^′“Ér {9§4s Áº%Á sŽHt &ñt >

÷

&Ù¼–Ð L(M^′) = Σ^∗s“¦

• M s ǫ\ @/ #Œ &ñt t ·ú§Ü¼€ L(M^′) = ∅s.

M∅s L(M^′) = ∅“t t ·ú˜9ÅÒÙ¼–Ð M s ǫ\ @/ #Œ &ñt 



Ht t ·ú˜ ú e”. 2

<M> M

n

<M'> y

n y

ÕªaË> 6:

21

' $

Ç a

h 10 “TM Ms ÅÒ#Q&’`¦ M:_{, L(M )}s &ñ½©ƒ#Q“_” ëH]j_, 7£¤ Lr = {hM i : L(M )s &ñ½©ƒ#Q}H >íߖÔ¦s.

¤

ÃZ. Lr`¦ F) “¦ &ñ € Lr`¦ &ñ H TM Mrs ”>rFôÇ.

s

]j Lǫ`¦ Lr–Ð ¨8Š . L = {0<sup>n</sup>1<sup>n</sup>: n ≥ 1}“¦ “¦ ML`¦ L`¦

&

ñ

_ H TMs“¦ . 7£¤ L = L(ML). Lǫ_ {9§4 hM is ÅÒ#Q t

€ Lr_ {9§4 hM^′iܼ–Ð ¨8ŠôÇ. M^′“Ér

• ÅÒ#Q” {9§4 w\¦ –Ð (\V\¦ [þt€, ÑütP: _…{9\) ˜Ð›'a “¦

• ǫ\ @/ #Œ M `¦ [to“¦

• M s &ñt € w\ @/ #Œ ML`¦ [toH

TMs. hM i\"f hM^′i`¦ %3l 0AK"fH hM iõ hMLi\ 6£§_

„



s\¦ 8KÅҀ )a.

• ÅÒ#Q” {9§4 w\¦ {9§4 _…{9\"f tĺ€"f –Ð ˜Ð›'a “¦ M _

œ

íl©œI–Ð [þt#QH „s

• M _ &ñt©œI\"f rŒ• #Œ w\¦ r {9§4 _…{9\ &h“¦ M _ œíl©œI–Ð [þt#QH „s

& % ÕªaË> 7. ÕªQ€

• M s ǫ\ @/ #Œ &ñt € M^′“Ér MLõ 1lx{9ôÇ õ\¦ tÙ¼

–Ð L(M^′) = Ls“¦

• M s ǫ\ @/ #Œ &ñt t ·ú§Ü¼€ M^′“Ér w Áº%Á sŽHt &ñt

t ·ú§Ü¼Ù¼–Ð L(M^′) = ∅s.

∅H &ñ½©ƒ#Qs“¦ L“Ér &ñ½©ƒ#Q mÙ¼–Ð Mrs L(M^′) = ∅“t

t ·ú˜9ïr. ÕªaË> 8. ÕªQÙ¼–Ð M s ǫ\ @/ #Œ &ñt Ht 





t ·ú˜ ú e”. 2

23

' $

h h w

M h

M_L

ε

M’

start

ÕªaË> 7:

& % y

<M>

n n

y

M

ÕªaË> 8:

s

©œõ °ú s TM\ @/ôÇ @/ÂÒìr_ |9ëH“Ér >íߖÔ¦s. s\¦ 


–

Ð כ¹€•ôÇ כ s Rice_ &ños.

25

' $

Σ^*

L 2

C φ r.e.

ÕªaË> 9:

Ç a

h 11 (Rice) C\¦ F) \P ƒ#Q 7áxÀÓ_ /BN|9½+Ës  ”ÂÒìr

| 9

½+Ës“¦ _{. LC} = {hM i : L(M ) ∈ C}H >íߖÔ¦s_.

¤

ÃZ. €$ C —¸ŽH F) \P ƒ#Q_ |9½+Ës
 /BN|9½+Ës€ LCH Û¦ ú e”H ëH]js. ÕªaË> 9.

& %

• C —¸ŽH F) \P ƒ#Q_ |9½+Ës€ LCH

{hM i : L(M )“Ér F) \P}s. ÕªX< F) \P_ &ñ_\ _ K L(M )“Ér †½Ó©œ F) \Ps. "f {9§4Û¼àÔaA w TM_ Â

Ò ñs€ w ∈ LCs“¦ TM_ ÂÒ ñ m€ w 6∈ LCs. w

TM_ ÂÒ ñ“tH ~1> SX‰“½+É Ãº e”ܼټ–Ð LCH Û¦ ú e”H ë

H]js.

• C /BN|9½+Ës€ LCH {hM i : L(M ) ∈ ∅}s. F) \P_ &ñ _

\ _K L(M )s F) \P u´ ú \O. "f {9§4Û¼àÔ a

A w TM_ ÂÒ ñsŽHt mŽHt w 6∈ LCs.

Õ

ªü@_ âĺ\H LC >íߖÔ¦e”`¦ 7£x"î . {9ìøÍ$í`¦ {9t ·ú§

“

¦ ƒ#Q ∅s C\ Ÿí†<Ê÷&t ·ú§H“¦ &ñ . (ÕªXOt ·ú§Ü¼€  6

£

§\"f C@/’ ¯C\¦ “¦9 € )a.) C /BN|9½+Ës mÙ¼–Ð C\ 5

Å

q H ƒ#Q Ls ”>rFôÇ. L`¦ &ñ_ H TM`¦ MLs“¦ .

27

' $

<M> M

y n y

n

<M’>

C

ÕªaË> 10:

LC\¦ F) “¦ &ñ € LC\¦ &ñ H TM MC ”>rFôÇ. s ]j Lǫ`¦ LC–Ð ¨8Š . ÕªaË> 10. Lǫ_ {9§4 hM is ÅÒ#Qt€ Lr_ {

9

§4 hM^′iܼ–Ð ¨8ŠôÇ. M^′“Ér ·ú¡_ 7£x"î\"fü< °ú s

• ÅÒ#Q” {9§4 w\¦ –Ð ˜Ð›'a “¦

• ǫ\ @/ #Œ M `¦ [to“¦

• M s &ñt € w\ @/ #Œ ML`¦ [toH

& % TMs. ÕªQ€

• M s ǫ\ @/ #Œ &ñt € M^′“Ér MLõ 1lx{9ôÇ õ\¦ tÙ¼

–Ð L(M^′) = Ls“¦

• M s ǫ\ @/ #Œ &ñt t ·ú§Ü¼€ M^′“Ér w Áº%Á sŽHt &ñt

t ·ú§Ü¼Ù¼–Ð L(M^′) = ∅s.

L∈ Cs“¦ ∅ 6∈ CsÙ¼–Ð MC L(M^′) = L“t t ·ú˜9ïr.

Õ

ªQÙ¼–Ð M s ǫ\ @/ #Œ &ñt Ht t ·ú˜ ú e”. 2

TM\ @/ôÇ |9ëH ü@\ 7᧠8 ½¨^‰&h“ ëH]j ×æ\ >íߖÔ¦“ ëH]j

•

¸ ´ú§s e” (program correctness).

Ça

˜+ 2 Post@/6£xëH]j_{: P = ((u1, v1}), . . . , (un, vn))s ÅÒ#Q&’`¦ M: (ui, vi ∈ Σ<sup>+</sup>), #Q‹" i1, . . . , ir\ @/ #Œ w= ui<sub>1</sub>· · · uir = vi<sub>1</sub>· · · vir`¦ ë

ß

–7ᤠH w\¦ P_ @/6£xs“¦ ÂҏÉr_{. Post} @/6£xëH]jH P_ @/6£x s

 ”>rF H H ëH]js.

29

' $

U

dV 4 P = ((a, ab), (b, ca), (ca, a), (abc, c)){9 M:_, abcaaabc

= u1u2u3u1u4

= v1v2v3v1v4

s

Ù¼–Ð P_ @/6£xs ”>rFôÇ. Ç

a

h 12 Post @/6£xëH]jH >íߖÔ¦s. ë

HАoÁº›'a ëHZO\ @/ôÇ 6£§ |9ëH[þt•¸ >íߖÔ¦s.

Ça

h 13 “ëHАoÁº›'a ëHZO _G ÅÒ#Q&’`¦ M:_{, G} EBôÇ_” ëH]jH

>

íߖÔ¦s_.

¤



ÃZ. Post @/6£x ëH]j\¦ ëHАoÁº›'a ëHZO_ EB$í ëH]j–Ð ¨8ŠôÇ.

Post @/6£x ëH]j_ ·ú˜ Ï@ Σ\ 5Åq t ·ú§H /åJ a1, . . . , an\¦ ‚×þ˜ ô

Ç. ëHZO G1“Ér

S1 → uiS1ai| uiai

& % (1 ≤ i ≤ n)ܼ–Ð ½¨$í÷&“¦, ëHZO G2H

S2 → viS2ai| viai

–

Ð ½¨$í÷&“¦, ëHZO GH 0A_ Òqt$í½©gË:\ S→ S1 | S2

\

¦ 8 #Œ %3H.

ë

HZO G1“Ér EB t ·ú§“¦ G2•¸ ðøÍts. "f ëHZO G E B

ôÇ âĺH G1õ G2 °ú “Ér Û¼àÔaA`¦ ëߖ׼H âĺs. 7£¤ Post

@

/6£xëH]j @/6£x`¦ tH âĺ\ëߖ GH EB . 2 Ç

a

h 14 “ëHАoÁº›'a ëHZO G1, G2 ÅÒ#Q&’`¦ M:_, L(G1) ∩ L(G2) = ∅“” ëH]jH >íߖÔ¦s.

¤

ÃZ. %ir Post @/6£xëH]j\¦ ¨8ŠôÇ. ·ú¡_ 7£x"îõ °ú s Post @/6£x ë

H]j\"f G1õ G2\¦ %3H. ÕªQ€ Post @/6£xëH]j @/6£x`¦ t



H âĺ\ëߖ L(G1) ∩ L(G2) 6= ∅s. 2

31