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3. ÍH _…{9 ·ú˜  Ï@s“¦ /BNÑþ˜ /åJ #\¦ Ÿí†<ÊôÇ.

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5. q0 ∈ QH œíl ©œIs“¦

6. H ⊆ QH &ñt©œI[þt_ |9½+Ës.

TM“Ér δ „s †<ÊúsÙ¼–Ð &ñ š¸žÐ s. δ_ &ñ_\"f L“Ér K

‰×¼ ¢,aAá¤Ü¼–Ð R“Ér K‰×¼ š¸ÉrAá¤Ü¼–Ð ôÇ ñߖ ¹¡§f”sH כ `¦, SH Õ

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#aaa\"f š¸ÉrAá¤Ü¼–Ð € #aaa# ÷&“¦, r š¸ÉrAá¤Ü¼–Ð €

#aaa##s )a. s#Q"f ¢,aAá¤Ü¼–Ð ¿º  € "é¶A —¸_þvs )a. Ça

˜+ 2 TM M = (Q, Σ, Γ, δ, q0, H)s &ñ_ H ƒ#Q L(M )“Ér 6£§ õ

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L(M ) = {w ∈ Σ : (q0,#w) ⊢M (h, ∗)}

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& % L(M )“Ér M`¦ &ñt > ëߖ׼H Û¼àÔaA_ |9½+Ës. L`¦ &ñ_ H

TM M (7£¤ L= L(M ))s ”>rF H ƒ#Q L`¦ F) \P ƒ#Q“¦ ÂÒ

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dV 2 L = {0n1n : n ≥ 1}`¦ &ñ_ H TM M`¦ ½¨ .

M“Ér š¸ÉrAá¤Ü¼–Ð ôÇ ñߖ "f, ±p ¢,aAá¤_ 0`¦ x–Ð ˨“¦ (q0), š¸Ér A

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Ç. \V: 000111 . . . x00y11.

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{anbncn : n ≥ 1}`¦ &ñ_ H TM•¸ ðøÍt–Ð ½¨½+É Ãº e”.

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$ {9§4 Û¼àÔaA`¦ ·ú¡ÂÒìrõ z´»ÂÒìrܼ–Ð ¾ºl 0A #Œ ·ú¡ÂÒìr_ 0, 1“Ér yŒ•yŒ• x, y–Ð z´»ÂÒìr_ 0, 1“Ér yŒ•yŒ• x, y–Ð õH.\V:

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Êê ·ú¡ÂÒìrõ z´»ÂÒìr`¦ ´úÆÒ#Q ˜ÐH כ “Ér ·ú¡_ \V]jü< q5pw .

(5)

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¸ôÇ TM“Ér —¸ŽH {9§4\ @/K &ñt €"f &ñt©œI\  ƒ#Q\¦

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• w ∈ Ls€ (q0,#w) ⊢M (y, ∗)

• w 6∈ Ls€ (q0,#w) ⊢M (n, ∗)

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¦ ëߖ7ᤠ€ Ms L`¦ &ñôÇ“¦ ´ú˜ôÇ. L`¦ &ñ H TMs ”>r F

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dV 4 L = {0n1n : n ≥ 0}\ @/ #Œ \V 2_ TM\"f

H = {q4, q5}s“¦ q4 = n, q5 = ys€ sH L`¦ &ñ H TMs. Ça

h 1 F) ƒ#Q 7áxÀӍH F) \Pƒ#Q 7áxÀÓ_ ÂÒìr|9½+Ës.

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ÃZ. L`¦ F) ƒ#Q“¦ €, L`¦ &ñ H TM M s ”>rFôÇ.

M \"f ©œI n`¦ ÁºôÇ íH¨8Š ©œI–Ð ëߖ[þt“¦ yëߖ`¦ &ñt ©œI–Ð 

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, L`¦ &ñ_ H TM`¦ %3H. "f L“Ér F) \P ƒ#Qs.

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dV 6 YL l †<Êú f(n, m) = nm`¦ >íߖ H TM`¦ ½¨ .

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6£§`¦ ˜Ðs“¦ ôÇ. 7£¤ l0px`¦ 8K"f >íߖ0px§4`¦ 7£xrvH

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\"f ôÇ>\ •¸²ú˜Ùþ¡H כ s.

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k_…{9 ÈÓaAl>H k >h_ _…{9`¦ ”. "f ôÇ \ k >h_ /åJ



\¦ {9“¦ ¹¡§f”e”`¦ &ñôÇ. 7£¤ „s†<Êú δH Q× Γk\"f

Q× (Γ × {L, R, S})kܼ–Ð_ †<Êús. k_…{9 TM_ ©œS!“Ér ‰&³F ©œ I

ü< k >h _…{9_ —¸€ªœÜ¼–Ð ½¨$í÷&“¦, (q, u1a1v1, . . . , ukakvk)–Ð ³ð r

)a. rŒ•½+É M:, {9§4“Ér 'Í P: _…{9\ ÅÒ#Qt“¦  Qt _…{9“Ér q

#Q e”. 7£¤ rŒ• ©œS!“Ér (q0,#w, #, . . . , #)s. †<Êú\¦ >íߖ½+É â ĺ, k_…{9 TM_ ئ§4“Ér 'Í P: _…{9\ ÅÒ#Qt“¦  Qt _…{9_ ?/6 x

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dV 7 2_…{9 TM\"f „s†<Êú δ(q, a, b) = (q, (a, R), (a, R))H 'Í P

: _…{9\"f {9“Ér /åJ\¦ ÑütP: _…{9\ 漓¦ K‰×¼\¦ š¸ÉrAá¤Ü¼–Ð ¹¡§ f

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sH כ s. U

dV 8 k_…{9 TM“Ér "é¶A TM˜Ð çߖéߖ > >íߖ`¦ ½+É Ãº e”. \V

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¦ [þt#Q #w\¦ #w#w–Ð ˨H >íߖ`¦ “¦9 . "é¶A TM“Ér ýa Ä

º–Ð ¹¡§f”s€"f w\¦ ôÇ /åJm” `… &hH. n = |w|“¦ € "é¶ A

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(9)

& % 2_…{9 TM“Ér s Œ•\O`¦ 6£§õ °ú s O(n)_ „s\ ½+É Ãº e”.

1. ¿º K‰×¼\¦ š¸ÉrAá¤Ü¼–Ð ¹¡§f”s€"f 'Í P: _…{9_ ?/6 x`¦ ÑütP: _… {

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.

3. ¿º K‰×¼\¦ š¸ÉrAá¤Ü¼–Ð ¹¡§f”s€"f ÑütP: _…{9_ ?/6 x`¦ 'Í P: _… {

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\ &hH. 7£¤ (#w#w#, #w#)–Ð õH. 4. 'Í P: _…{9_ K‰×¼\¦ ¢,aAá¤Ü¼–Ð `…|.

s

ü< °ú s :£¤&ñôÇ >íߖ`¦ H ¼#s$í(¢¸H rçߖ)\ e”#Q"fH k_…

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Ö¼_ ›'a&h\"fH ¿º >h 1lx{9 . 7£¤ k_…{9 TMs ½+É Ãº e”H

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íߖ“Ér "é¶A TM•¸ ½+É Ãº e”.

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h 2 k_…{9 TM M = (Q, Σ, Γ, δ, q0, H)ü< 1lx1pxôÇ "é¶A TM Ms

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>rFôÇ (7£¤ L(M ) = L(M)).

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k >h_ àÔÏþ˜`¦ 6 x H כ s "é¶A TM_ &ñ_\¦ #Á #QH כ s  m

“¦, éߖt M_ _…{9 ·ú˜  Ï@ Γܼ–Ð Γ ∪ (Γ ∪ ¯Γ)k`¦ 6 x H כ ÷r

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(10)

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\ K‰×¼ e”ܼ 9 ÑütP: _…{9\H b e”H כ `¦ ·ú˜ ú e”.

{ 9

§4 w ∈ Σ ÅÒ#Qt€ M“Ér 6£§õ °ú s Œ•1lxôÇ.

1. {9§4`¦ m” š¸ÉrAá¤Ü¼–Ð `…| Êê _…{9`¦ M _ k _…{9 +þAI–Ð



õH. w = abaas“¦ M s 2 _…{9 TMs€, M_ _…{9`¦

#abaa\"f ##abaa–Ð, Õªo“¦

#( ¯#, ¯#)(a, #)(b, #)(a, #)(a, #)ܼ–Ð õH.

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£

§_ ¹¡§f”e”`¦ ôÇ.

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M _ ©œI\¦ l%3 “¦ k K‰×¼\ e”H /åJ\¦ {9“¦ €, M _ „s\¦ ·ú˜ ú e”.

2. _…{9`¦ t€"f M _ „s\¦ ìøÍ%òôÇ. ëߖ€• M s #Q‹" K

‰×¼\¦ š¸ÉrAá¤Ü¼–Ð ¹¡§f”#Œ"f Ms #s [þt#Qe”H ñߖܼ–Ð `…



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%ò ôÇ.

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–Ð ˨“¦ &ñtôÇ.

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œAᤠ\P2; _…{9 TM“Ér €ªœ~½Ó†¾Óܼ–Ð \P2; _…{9`¦ ôÇ >h t“¦ e”.

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íl\ {9§4s _…{9\ ÅÒ#Qt“¦ K‰×¼H {9§4 –Ð ¢,aAá¤_ /BNÑþ˜\ Z

~#Œ e”“¦ &ñôÇ. s TM“Ér 2_…{9 TM\ _K ¿9ª?/è­q ú e”.

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/6 x“Ér 'Í P: _…{9\ $©œ “¦ ¢,aAá¤_ ?/6 x“Ér ÑütP: _…{9\ %iܼ–Ð

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©œôÇ Êê ¿9ª?/?/€ )a. Óüt:r s 2_…{9 TM“Ér r "é¶A TMܼ–Ð

¿

9ª?/è­q ú e”.

kK‰×¼ TM“Ér ôÇ >h_ _…{9\ k>h_ K‰×¼\¦ t“¦ e”“¦ k K‰×¼\ e

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#w\¦ #w#w–Ð ˨H Œ•\O`¦ 8 çߖéߖy ½+É Ãº e”.

1. ¿º K‰×¼\¦ #w#_ 0Au\ Z~H.

(11)

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