w = baa tš, w L µ? q 0 K : δ(q 2, b) = q 2 (1)

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1 J S "!$#"%"#"&"'(#$) 26 * 2 +, 3 * -/ /798 :. ; <>="?A@CBED, F$G, H$I JLKNMPO"Q"RCS5T$U"VCWAX"YNS5Z\[P]"O ^`_ Σ = {A, B,, Z, a, b,, z, (acb ),d, e, f } (gihkjlmcnc ) Σ + = { abc de, this is, aa, aaa, aabc, } (oqpsrtuv ) Σ = {ε} Σ + (aqospr (ε) wxyoprtuv ) L wz {}~otuvc ƒc, L Σ ƒ L = { T his is a pen., I am a boy., }. o pr x t w x ˆ..2 this is = 7, aaa = 3, ε = Σ AŠ( "Œ$ H"I>Ž O ^ _ ghjlmcnc (Σ) (L) š, w L w œk ž M wÿ U9 ª«}. a,b Q2 b Q a b a Q. ± ²«t³ Q µ s. 2. ¹ºt»s¼oprt t opt½¾ ÀÁÂÄÃÅ. 3. Æt³s ÇÈ, Yes, ²È, No wäé.

2 Σ = {a, b} L = {a, aba, ababa, abababa, } = {a(ba) n n } w = aba tš, w L µ? w = baa tš, w L µ? J X(Y M S ( S CO ) UC VCW. _ K = {q, q, q 2 } : suv F = {q } (F K) : uv q K : δ : K Σ K : δ(q, a) = q δ(q, b) = q 2 δ(q, a) = q 2 δ(q, b) = q δ(q 2, a) = q 2 δ(q 2, b) = q 2 ª«¼ M = (K, Σ, δ, q, F ) t!qiw" (FA) }". #c{, K, Σ ± a v, q K, F K, δ : K Σ K. δ ±$&%t&'& δ : K Σ K (c¹*) +&, -.k. { () δ (q, ε) = q +&, δ t/ (2) δ (q, ax) = δ (δ(q, a), x) (a Σ, x Σ ) δ (q, aba) = δ (δ(q, a), ba) = δ (q, ba) = δ (δ(q, b), a) = δ (q, a) = δ (δ(q, a), ε) = δ (q, ε) = q F Èi}q u () w M &'ÁÂ76c c }8. L(M) = {w Σ δ (q, w) F } (2) 2

3 ` 4 5 L, L(M) = L Ä ª«M 4 5 (RL) 5 L( Σ ) { Â, L = L(M) ièq iq q q iš L w"z (RL) }8. K"! ' ( ) * H"I $#&% L z tš, L = Σ L ±z sƒ. ) * (+-, t-. ". L w 6 P P 9L w M = (K, Σ, δ, q, F ) `P P š, M = ( K, Σ, δ, q, F ), L = L( M) È Ÿ /, L z s ƒ &.) 2 L, L 2 wz sc š, (i) L L 2 ±z ƒ. (ii) L L 2 ±z s ƒ. 2 3* Σ = {, }, M = {{p, p }, Σ, δ, p, {p }}, M 2 = {{q, q }, Σ, δ 2, q, {q }}. {, δ i : K i Σ K i (K = {p, p }, K 2 = {q, q }) ±,. δ (p, ) = p, δ 2 (q, ) = q δ (p, ) = p, δ 2 (q, ) = q δ (p, ) = p, δ 2 (q, ) = q δ (p, ) = p, δ 2 (q, ) = q (i) M, M 2 t6 k±, 4 5, 6t&'& Èkµ? (ii) L(M ) L(M 2 ) w6 M wÿ 7 8 '. (iii) L(M ) L(M 2 ) w 6 M wÿ 7 8 '..4 Haskell H"I :9 LKM N.4. G H IJ import Data.List W<;>="?:@ DBA DDC "E:F ˆTS Haskell s ± usv ±PORQ. Ususv±7 union WVYX s, Z\[\] È\^ /_ORQ t v& (++a concat) b c -.. duv± intersect WVeX -., Z [ ]wf g ± nub wh8. 3

4 *Main> [,2,3] ++ [4,5,6] [,2,3,4,5,6] *Main> concat [[,2,3],[4,5],[6,7,8,9]] [,2,3,4,5,6,7,8,9] *Main> intersect [..6] [2,4..] [2,4,6] *Main> [..6] ++ [2,4..] [,2,3,4,5,6,2,4,6,8,] *Main> nub ([..6] ++ [2,4..]) [,2,3,4,5,6,8,] *Main> union [..6] [2,4..] [,2,3,4,5,6,8,].4.2 G H concat a& union ± tpoq«tæ t OQ«w ({, s tpoq«po Q tsv, t ORQ $s±s È }. u v twususvkwä k±_ö Q st usvkt ty^ V! wu Â, 4tÆ, *$t tuv tuuvw/. "±, u vtuuv#*/&-. &'& s ƒ. 4 t ƒ union. union :: Eq a =>[[a]]->[a] union [] = [] union (x:xs) x == [] = union xs otherwise = [head x]++(union (xs++[tail x])) 2 *Main> union ["abc","2","456","789"] "a47b258c69" *Main> union [[,2,3], [4,5], [6,7,8,9]] [,4,6,2,5,7,3,8,9] *Main> take [..] [,,2,3,4,5,6,7,8,9] *Main> take $ union [[..], [2..]] [,2,,2,2,22,3,23,4,24].4.3 $ % & 'k G H (Σ ) uv Σ c{â, Σ ± t&'& œ (c. (i) ε Σ (ii) x Σ, w Σ t š, xw Σ (iii) (i),(ii) ^ Σ ±œ (c. tœ (±, Σ t ) *uv +, - : S = {ε} {xw w S} (3) t. / S ƒ &w { Â}. x Σ 4

5 ˆ S ˆ c 3 Σ = {a, b} t š, Σ t Äusv +, - (3) wuu v&7 union w t %t ƒ abstar. }Â Haskell {Ä abstar::[[char]] abstar = [""] ++ (union [["a"++ w w <-abstar], ["b"++ w w <-abstar ]]) *Main> take abstar ["","a","b","aa","ab","ba","bb","aaa","aab","aba"] Haskell t œ (±, Èš È e^/ c t,t&'& uqvw Haskell & -7.. ` w ±, È w «take &' -8T/ }. t w {, t5gh jl mcnc uqv Σ c{iâ, Σ w q sstar w q.. uv s = Σ c{ Â, sstar(s) = Σ È 7œ (±, - (??) ' %t '& È. sstar :: [Char] -> [[Char]] sstar [] = [] sstar s = [""] ++ (union [ [[x] ++ w w <- (sstar s)] x <- s ]) 4 Σ = {a, b, c} t vt Σ tw7$&%. *Main> take 3 $ sstar [ a, b, c ] ["","a","b","c","aa","ab","ac","ba","bb","bc","ca","cb","cc", "aaa","aab","aac","aba","abb","abc","aca","acb","acc","baa", "bab","bac","bba","bbb","bbc","bca","bcb"].4.4 ˆ S ' ª«g hkjl m n t± sƒ Char, o psrtk± ƒ String s, susv K wktk u v {, 4t w State. δ c{â, δ wÿ dstar ± - () ' %t&'& È. type State = Int type States = [State] -- type String = [Char] dstar::(state->char->state)->state->string->state dstar d s [] = s dstar d s (a:w) = dstar d (d s a) w $%st ' Σ tsºsusv A { Â, 6s ssusv L(M) A w7/sk 7 accepts ±- (??) ' œ (. A = Σ s{ sv, kè, Σ s tk š, accepts d s f (sstar s), L(M) w. s m {Â, L(m) w/ w language k. type Automaton = (States, [Char], State->Char->State, State, States) accepts::automaton->[string]->[string] accepts (q, s, d, s, f) ss = [w w <- ss, (dstar d s w) elem f] language::automaton ->[String] language (q, s, d, s, f) = accepts (q, s, d, s, f) (sstar s) 5 (Pœ ) A P M = (Σ, Q, δ, s, F ) "{ Â, Σ = {a, b}, Q = {,, 2}, t œ (k. s =, F = {} {, δ : Q Σ Q ± % 5

6 S δ (s, c) c = a c = b s = 2 s = 2 s = a,b Q2 b Q a b a Q m::automaton m = ([,,2], [ a, b ], d,, []) where d a = d b = 2 d a = 2 d b = d 2 a = 2 d 2 b = 2 L(M) È, language m, È, L(M) {b} L(M) {a} w/. *Main> take 5 $ language m ["a","aba","ababa","abababa","ababababa"] *Main> accepts m $ take 5 (sstar [ b ]) [] *Main> accepts m $ take 5 (sstar [ a ]) ["a"] L(M) = {a(ba) n n }, L(M) {b} = φ, Èk k, L(M) {a} = {a} sƒ s -.k..4.5 G H 273 ª m_complement::automaton -> Automaton m_complement (k, s, d, s, f) = (k, s, d, s, fc) where fc = [x x<-k, not (x elem f) ] 6 5 t 5N N m ${ Â, L(m) tpupv L(m) w 6PPA P `5w 5. L(m), L(m) {b}, È, L(m) {a} w/ c{ ÂÅ. *Main> take $ language $ m_complement m ["","b","aa","ab","ba","bb","aaa","aab","abb","baa"] *Main> accepts (m_complement m) $ take 5 (sstar [ b ]) ["","b","bb","bbb","bbbb"] *Main> accepts (m_complement m) $ take 5 (sstar [ a ]) ["","aa","aaa","aaaa"] ` 5 t Â, µs uv 6c Â} & KM natpairs::[(int,int)] natpairs = [(x,z-x) z <- [..], x<- [..z]] pxy::(int,int)->int pxy (i,j) = (div ((i+j) * (i+j+)) 2) + i px::int->int px z = fst (natpairs!!z) py::int->int py z = snd (natpairs!!z) 6

7 ! ƒ ˆ S natpairs ±¹ ) t tuv w. pxy (i,j) ± (i, j) natpairs t µw/c{â}. natpairs t z it x w qt px, y w qt ƒ py ƒ. pxy t & px 7 py t. 7 *Main> take natpairs [(,),(,),(,),(,2),(,),(2,),(,3),(,2),(2,),(3,)] *Main> take [pxy (i,j) (i,j)<-natpairs] [,,2,3,4,5,6,7,8,9] *Main> pxy (4,2) 25 *Main> px $ pxy (4,2) 4 2 *Main> py $ pxy (4,2).5 Œ" Ž Œ$ S"VCW<;>=$?:@ DBA DDC m_join::automaton -> Automaton -> Automaton m_join (kp, sp, dp, sp, fp) (kq, sq, dq, sq, fq) = (kpq, spq, dpq, spq, fpq) where kpq = [pxy (i,j) i<-kp, j<-kq] spq = sp dpq z c = pxy ((dp (px z) c),(dq (py z) c)) spq = pxy (sp,sq) fpq = [pxy (i,j) i<-fp, j<-kq] ++ [pxy (i,j) i<-kp, j<-fq] m_meet::automaton -> Automaton -> Automaton m_meet (kp, sp, dp, sp, fp) (kq, sq, dq, sq, fq) = (kpq, spq, dpq, spq, fpq) where kpq = [pxy (i,j) i<-kp, j<-kq] spq = sp dpq z c = pxy ((dp (px z) c),(dq (py z) c)) spq = pxy (sp,sq) fpq = [pxy (i,j) i<-fp, j<-fq] 8 ¼kt, m, m2 w k, 6s vw6 swÿ 7k. L(m), L(m2) tususv a\du P P PQ PQ PQ PQ Q Q PQ PQ PQ PQ mp::automaton mp = ([,], [, ], dp,, []) where dp = dp = dp = dp = mq::automaton mq = ([,], [, ], dq,, []) where dq = 7

8 @ X dq = dq = dq = *Main> take $ language mp ["","","","","","","","","",""] *Main> take $ language mq ["","","","","","","","","",""] *Main> take 5 $ language $ m_meet mp mq ["","","","",""] *Main> take $ language $ m_join mp mq ["","","","","","","","","",""] tµ ƒ L(m) = {w w t t } L(m2) = {w w t t } L(m) L(m2) = {w w t t # } L(m) L(m2) = {w w t, ş ±, t } = {, } {w w t t # } & &-...6 D Haskell t œ ( m_complement, m_join, m_meet w {Â,, 2 t+,w }. - ^ È, +, š&t w &. }È [], : */,!"# (986). [2] The Grasgow Haskell Compiler, [3] Graphviz Graph Visualization Software, 8

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