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úd(arithmetic expression)õ Ä» > &ñ½©d(regular expression)`¦ &ñ_ôÇ.
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+ 1 ·ú Ï@ Σ ©_ &ñ½©dõ Õªכ s ³ðr H |9½+ËÉr 6£§õ
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1. ∅H &ñ½©ds¦ /BN|9½+Ë`¦ ³ðrôÇ. 2. ǫH &ñ½©ds¦ {ǫ}`¦ ³ðrôÇ.
3. y a ∈ ΣH &ñ½©ds¦ {a}\¦ ³ðrôÇ.
4. rõ s &ñ½©ds¦ yy Rõ S |9½+Ë`¦ ³ðrôÇ, (r + s), (rs), (r∗)H &ñ½©ds¦ yy R∪ S, RS, R∗ |9½+Ë`¦ ³ðrôÇ.
&
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½©d aü< /åJ aH "fÐ Ér כ `¦ ?/, °ú Ér +þAIe`¦ Ä»_
. 7£¤ L(a) = {a}s¦ L(ǫ) = {ǫ}s.
& %
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½©d\"f íß_ íH"f\¦ ∗ · +Ð &ñ ´ú§Ér Fc ñ\¦ Òqt|Ä̽+É Ãº e
. 7£¤ ((1(0∗)) + 0) = 10∗+ 0s¦ s &ñ½©dÉr {0, 1, 10, 100, . . .}`¦
·p. ¢¸ôÇ 6£§õ °ú Ér íß_ ZOgË:`¦ &h6 x½+É Ãº e.
• + íßÉr ½+Ë|9½+Ë`¦ ?/Ù¼Ð, §¨8ZOgË:õ ½+ËZOgË:s $íwn ô
Ç. 7£¤ r + s = s + rs¦ (r + s) + t = r + (s + t)s.
• · íßÉr ]X½+Ë`¦ ?/Ù¼Ð, §¨8ZOgË: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.
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+ 2 &ñ½©dܼР³ðr÷&H #Q\¦ &ñ½©#Q¦ ÂÒÉr. U
dV 1 &ñ½©d
(00)∗(11)∗1
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r ú >h_ 0 6£§\ f.Ëú >h_ 1s ¸H Û¼àÔaA_ |9½+Ë`¦ ³ð r
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dV 2 {0n1m : (n + m)Ér ú}\¦ ?/H &ñ½©d`¦ ½¨ .
&
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â
ĺР¾º#Q"f Òqty H כ s a%~. 0s ú >h ¸¦ 1s Ã
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É
r (00)∗(11)∗+ 0(00)∗1(11)∗s.
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U
dV 3 1s ôÇ >h ¢¸H ¿º >h eH Û¼àÔaA_ |9½+Ë`¦ ³ðr H &ñ½© d
`¦ ½¨ .
1s ôÇ >h eH Û¼àÔaA_ |9½+ËÉr 0∗10∗s¦, 1s ¿º >h eH Û¼àÔaA _ |9½+ËÉr 0∗10∗10∗sټРÑüt`¦ ½+Ëu 0∗10∗(ǫ + 10∗)s.
U
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. U´s 3 Û¼àÔaA`¦ ³ðr H &ñ½©ds
(0 + 1)(0 + 1)(0 + 1)sټР²úÉr ((0 + 1)(0 + 1)(0 + 1))∗s. (0 + 1)(0 + 1)(0 + 1)`¦ צ#"f (0 + 1)3ܼРæ¼lÐ ôÇ.
& % U
dV 5 11`¦ ÂÒìrÛ¼àÔaAܼР°úH Û¼àÔaA_ |9½+Ë`¦ ³ðr H &ñ½© d
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dV 6 111s ÃÌ ôÇ H Û¼àÔaA_ |9½+Ë`¦ ³ðr H &ñ½© d
`¦ ½¨ . U
dV 7 11`¦ ÂÒìrÛ¼àÔaAܼР°út ·ú§H Û¼àÔaA_ |9½+Ë`¦ ³ðr H
&
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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¨ .
#Q\¦ ³ð&³ H ~½Ód\H &ñ½©d, ¸Ð, ëHZO 1pxs e.
Ça
+ 3 #Q_ ³ð&³ ~½Ód A, B L(A) = L(B)\¦ ëß7ᤠ, Aü<
BH 1lx1px(equivalent) .
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4. q0 ∈ QH íl©Is¦
5. F ⊆ QH þj7áx©I[þt_ |9½+Ës.
&ñ Ä»ôÇ ¸Ð(deterministic finite automata)\¦ צ#"f DFA
Ð ÂÒØÔlÐ .
s <Êú δ\¦ SX© # Q × Σ∗\"f QÐ_ <Êú δ∗\¦ 6£§õ °ú s
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_ôÇ.
• δ∗(q, ǫ) = q
• ¸H w ∈ Σ∗ü< a ∈ Σ\ @/ #, δ∗(q, wa) = δ(δ∗(q, w), a) Û
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H ¸H Û¼àÔaA_ |9½+Ës. 7£¤
L(M ) = {w ∈ Σ∗ : δ∗(q0, w) ∈ F }.
& % DFA_ ©S!(configuration)Ér &³F ©Iü< {9§4 Û¼àÔaA_ {9t ·ú§
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s
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_ s\¦ ⊢∗Ð ³ðrôÇ.
U
dV 9 M = ({q0, q1}, {0, 1}, δ, q0,{q0}) s¦, δH 6£§õ °ú . 0 1
q0 q0 q1 q1 q1 q0
L(M )Ér ú >h_ 1`¦ Û¼àÔaA_ |9½+Ës.
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ܼÐ, þj7áx©IH s×æ"é¶Ü¼Ð, íl©IH o¶ú³ðÐ ³ðrôÇ.
\
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q
0q
11
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{ 9
ìøÍ&hܼР6£§ ¿º éß>\ _K DFA\¦ ëßH.
1. &³Ft {9Ér ÂÒìr\"f #Q" &ñÐ\¦ l%3K Ht\¦ &ñ
¦ s &ñÐ\¦ ©IÐ ³ðrôÇ.
2. DhÐîr /åJ\¦ {9%3`¦ M: l%3K H &ñÐ #QbG> 7 H t
\¦ Ц s<Êú\¦ &ñôÇ.
& % s
×æ\"f ×æכ¹ôÇ כ Ér l%3K H &ñÐ\¦ &ñ H כ s. \V ]j 9\"fH 1_ >hú út f.Ëút\¦ l%3K ôÇ.
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~ Ã
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0s¦, Õª 6£§\ 1`¦ {9ܼ {9Ér #Q¿º 01s ÷&¦, 0`¦ {9ܼ
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ÕªaË> 4_ DFA\"f þj7áx©Iü< Õª ü@_ ©I\¦ ´ú˨ )a. (s ü
< °ú s #|9½+Ë`¦ ³ðr H ëH]j &ñ½©d\"fH #Q9°?HX<, DFA\"fH /'îr כ `¦ ÅÒ3lq .)
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dV 13 Û¼àÔaA\"f 5Åq(run)sêøÍ °ú Ér /åJÐëß ½¨$í)a (8 s
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\ 100s e%3Ht\¦ l%3½+É 9כ¹H \O. = 100s e
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s Û¼àÔaAÉr ~ÃÎ[þt#tt ·ú§l M:ëHs.
& % δ ÂÒìr<Êú{9 M:¸ DFAÐ ^¦ ú e. s âĺ 0A_ &ñ_\ _ôÇ
DFAР˨9, DhÐîr ©I q′ 6∈ F `¦ ¸{9 #
• &ñ_÷&t ·ú§Ér ¸H sH q′ܼР¸2¤ ¦,
• ¸H a ∈ Σ\ @/ # δ(q′, a) = q′s ÷&¸2¤ ôÇ.
q′õ °ú s ôÇ [þt#Q 4R `¦ ú \OH ©I\¦ º¡¤Ér ©I(dead state)¦ ÂÒÉr. \V]j 12\"f ©I 0101¸ º¡¤Ér ©Is.
V
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Ç כ s.
1. y ©Iü< /åJ ÅÒ#Q&`¦ M:, 6£§ ©I \O s© {
9
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(Q, Σ, ∆, q0, F)Ð ½¨$í)a.#l\"f Q,Σ, q0, FH DFA\"fü< °ú s
&ñ_)a. ∆H Q× (Σ ∪ {ǫ}) × Q_ ÂÒìr|9½+Ë s 'a>Ð,
¢
¸H Q× (Σ ∪ {ǫ})\"f 2QÐ_ s <ÊúР&ñ_½+É Ãº e. NFA\¦ &ñ_½+É M:H ∆\¦ DFA_ s <Êú\"f {9ìøÍo)as 'a
>
Ð H כ s Û¼Qĺ, [O"î\"fH ∆\¦ s <ÊúÐ
ÐH כ s ¼#o ټРsÊê\H ∆\¦ 2QܼÐ_ <ÊúРçßÅÒôÇ.
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°ú s SX©ôÇ. P ⊆ Qü< a ∈ Σ ∪ {ǫ}\ @/ #
∆(P, a) = [
q∈P
∆(q, a).
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. E¸ %ir SX© #, ©I |9½+Ë P \ @/ # E(P ) = Sq∈P E(q).
& %
∆\¦ SX© # Q × Σ∗\"f 2QܼÐ_ <Êú ∆∗\¦ 6£§õ °ú s &ñ_ ô
Ç.
• ∆∗(q, ǫ) = E(q)
• ¸H w ∈ Σ∗ü< a ∈ Σ\ @/ #, ∆∗(q, wa) = E(∆(∆∗(q, w), a)) Û
¼àÔaA w ∈ Σ∗\ @/ # ∆∗(q0, w)s NFA M _ þj7áx©I\¦ &h#Q
¸ í<Ê , M s w\¦ ~ÃÎ[þt¦ ´úôÇ. M _ #Q L(M )Ér M s ~ÃÎ[þtsH ¸H Û¼àÔaA_ |9½+Ës. 7£¤
L(M ) = {w ∈ Σ∗ : ∆∗(q0, w) ∩ F 6= ∅}.
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L(M ) = {w ∈ Σ∗ : (q0, w) ⊢∗M (q, ǫ), q ∈ F }.
{ 9
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U
dV 15 &ñ½©d (010 + 01)∗ ³ðr H #Q\¦ ~ÃÎ[þtsH NFA
\
¦ ½¨ .
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• ∆∗(q1,0) = {q2},
• ∆∗(q1,01) = {q3, q1},
• ∆∗(q1,010) = {q1, q2} s
¦, #l\ þj7áx©I q1s í<Ê÷&#Q eܼټРs NFAH 010`¦
~ Ã
Î[þt. sכ `¦ s õ&ñܼР?/ 6£§õ °ú . (q1,010) ⊢ (q2,10) ⊢ (q3,0) ⊢ (q1, ǫ)
7
£
¤ (q1,010) ⊢∗ (q1, ǫ)s¦ q1s þj7áx©IsټРs NFAH 010`¦ ~ÃÎ
[þt.
& % 1
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ª /åJ 1s q2Ð s1lxôÇ. q3H s ÆÒ8£¤s ´ú¤Ht SXôÇ
.
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19
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• ∆∗(q1,0) = {q1},
• ∆∗(q1,01) = {q1, q2},
• ∆∗(q1,011) = {q1, q2, q3} s
Ù¼Ð, s NFAH 011`¦ ~ÃÎ[þt. {9§4ۼ߼aAs 100{9 M:,
• ∆∗(q1,1) = {q1, q2},
• ∆∗(q1,10) = {q1, q3},
• ∆∗(q1,100) = {q1} s
Ù¼Ð, s NFAH 100`¦ ~ÃÎ[þtst ·ú§H.
1 0,1
0,1
q
1q
2q
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H q ∈ F \ @/K s (q, ǫ, qf)\¦ 8Kïr.
= q&ñ$í(nondeterminism)s 9כ¹ôÇ? &³F 6 x÷&H nt_O (
ÉÓ'H ¢-ay &ñ&hܼР¹¡§f. ¢¸ pA\ q&ñ&hܼР1
l
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°ú Ér sĻРq&ñ$íÉr ×æכ¹ôÇ %i½+É`¦ ôÇ.
• q&ñ$í`¦ s6 x ¸Ð\¦ ~1> ëß[þt ú e. #Q" #Q
\
¦ ~ÃÎ[þtsH NFA\¦ ëß×¼H כ s {9ìøÍ&hܼРDFA\¦ ëß×¼H
כ
Ð ~1.
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j>H ĺo_ ëH]jK 0px§4õ 'a>\Os \ ÅÒ#Q4ReH
כ
s. s\ qK nt_O (ÉÓ' 1pxÉr ĺo t¦ eH ëH ]
jK¸½¨s. &³F ĺo t¦ eH ëH]jK¸½¨H
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H >h¥Æs 9כ¹ .
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DFA ~ÃÎ[þtsH #Q 7áxÀÓü< NFA ~ÃÎ[þtsH #Q 7áxÀÓ °ú 6
£
§`¦ Ðs¦ ôÇ. DFAH NFA_ {97áxsټРDFA_ #Q 7áxÀÓ
H NFA_ #Q 7áxÀÓ_ ÂÒìr|9½+Ës. s]j ìøÍ@/_ âĺ\¦ 7£x"î
.
Ç a
h 1 e__ NFA\ @/ # sü< 1lx1pxôÇ DFA >rFôÇ.
¤
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l
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É
r #QÖ¼ r&h\"f NFA_ &³F ©I[þt_ |9½+Ë`¦ DFA_ _ © I
Ð çßÅÒ H כ s. 7£¤ DFA D_ ©IH QN_ "4|9½+Ë_ "é¶ès
& %
.
DFA D_ íl ©I q′H E(q0)s. D_ ©I P ⊆ QNü< /åJ a∈ Σ\ @/ #, D_ s <ÊúH δ(P, a) = E(∆(P, a))s. s\¦ s
6 x # 6£§õ °ú s DFA D\¦ ½¨ôÇ.
QD ← {E(q0)}
mark E(q0)
while ∃ marked state P ∈ QD do unmark P
for each a ∈ Σ do R← E(∆(P, a)) if R is not in QD then
add R as marked state to QD fi δ(P, a) ← R
od od
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s
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49