# 1 Finite Automata and Regular Expressions

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1 1 Fini Auom nd Rgulr Exprion Moivion: Givn prn (rgulr xprion) for ring rching, w migh wn o convr i ino drminiic fini uomon or nondrminiic fini uomon o mk ring rching mor fficin; drminiic uomon only h o cn ch inpu ymol onc. Cn hi lwy don? Thorm 1.1 If L 1 = L(M 1 ) nd L 2 = L(M 2 ) for lngug L i Σ hn 1. hr i n uomon M rcognizing L 1 L 2 2. hr i n uomon M rcognizing L 1 L 2 3. hr i n uomon rcognizing L 1 4. hr i n uomon rcognizing Σ L 1 5. hr i n uomon rcognizing L 1 L 2 6. if Σ hn hr i n uomon rcognizing {} 7. hr i n uomon rcognizing From ll of h hing i follow h if A i rgulr lngug hn hr i fini uomon rcognizing A. For xmpl, juify why hr would fini uomon rcognizing h lngug rprnd y (). Proof: W will do h proof for nondrminiic uom inc drminiic nd nondrminiic uom r of quivln powr. 1.1 Union For union, uppo M 1 i (K 1, Σ, 1, 1, F 1 ) nd M 2 i (K 2, Σ, 2, 2, F 2 ). Thn l M (K, Σ,,, F ) whr K = K 1 K 2 {} F = F 1 F 2 = 1 2 {(,, 1 ), (,, 2 )} nd i nw. Thn L(M) = L(M 1 ) L(M 2 ). Digrm: 1

2 M1 1 K1 M2 2 K2 M 1 2 K1 K2 No h ϵ rrow r convnin for hi conrucion Exmpl p Rcogniz * q Rcogniz * 2

3 p Rcogniz * U * q 1.2 Concnion M1 1 K1 F1 M2 2 K2 F2 M 1 K1 F1 2 K2 F2 3

4 Th in F 1 r no longr ccping. Thn L(M) = L(M 1 ) L(M 2 ) Exmpl p Rcogniz * q Rcogniz * p q Rcogniz ** 1.3 Kln r M1 1 K1 F1 4

5 K M F Thn L(M) = L(M 1 ) Exmpl, Rcogniz {,}, Rcogniz {,}* How would you modify hi uomon o rcogniz {, } +? Anohr impl conrucion for Kln r fil for hi uomon: 5

6 1.4 Complmnion L M 1 = (K, Σ, δ,, F ) drminiic fini uomon. L M (K, Σ, δ,, K F ). Thn L(M) = Σ L(M 1 ) Exmpl M1 Rcogniz ring wih vn numr of M Rcogniz ring wih odd numr of Why do h uomon hv o drminiic for hi o work? An xmpl howing h M 1 h o drminiic for hi conrucion o work: 6

7 1.5 Inrcion For hi, no h L 1 L 2 = Σ ((Σ L 1 ) (Σ L 2 )). 1.6 Ohr oprion Pr 6 nd 7 of h horm r rivil. Ak udn o do hm. A conqunc of hi horm, if lngug L i rgulr, hn hr i fini uomon M rcongizing L. 2 Exmpl W conruc nondrminiic fini uomon rcognizing L(() ). Rcogniz {} Rcogniz {} Rcogniz {} 7

8 Rcogniz {}* Rcogniz {}* U {} Of cour, hi uomon i no h impl poil on! Bu om uch conrucion cn ud for ring rching, wih {, } pu on h fron, nd cn hn imuld uing h id. How would you opimiz h ov uomon o rduc h numr of? Wh r h impl nondrminiic nd drminiic uom for hi lngug? W now how h if lngug L i rcognizd y fini uomon M, hn L i rgulr. 3 Convring uom o rgulr xprion Cn ny fini uomon convrd o n quivln rgulr xprion? Would llowing fini uom in rgulr xprion incr h powr of ring rching? Th nwr o h quion r y nd no. For ny fini uomon M hr i rgulr xprion E uch h L(M) = L(E). Givn fini uomon, i cn convrd o rgulr xprion. To do hi, w gnrliz nondrminiic fini uom nd llow rgulr 8

9 xprion on hir rrow. If E whr E i rgulr xpion, hn hi mn h if h uomon i in, i cn rd ring in L(E) nd rniion o. No h ordinry nondrminiic uom do no llow uch rgulr xprion on rrow. Th uomon M cn convrd o rgulr xprion y pplying h following rul. Fir, whnvr poil, h following rnformion hould pplid o M nd o ll ohr uom M, M, cr, oind during hi proc: If for ny nd in M, E 1,..., E n for n > 1 hn ll h rrow r rmovd from M nd r rplcd y h rrow E 1 E 2... E n. Digrm: E1 E2 E3 En Afr procing: E1 U E2 U... U En Nx, ll of M ohr hn h r hould procd, on y on, o limin rrow lving, nd poily o limin. A 9

10 of M cn procd o oin nohr uomon M. Th uomon M i iniilly o qul o M. Thn rrow nd r ddd o M nd rmovd from M follow: If in M w hv E F G u nd i no h r, hn in M w dd h rrow E(F )G u.. Arrow lik hi r ddd for ll nd u h r no idnicl o No h nd u my idnicl. If hr r no rrow from o, hn h xprion EG i ud ind of E(F )G. Digrm: F E G u Afr procing: E(F*)G u 10

11 Thn, if i no n ccping, nd ll rrow nring or lving i r rmovd from M. If i n ccping, hn if in M w hv hn in M w hv E F E(F ). Thi i don for ll no idnicl o. Th rmin in M, u ll rrow lving r rmovd from M, nd only uch ddd rrow E(F ) nr in M. If hr r no rrow from o in M, hn h xprion E i ud ind of E(F ). Simpl xmpl: F E G u Afr procing: E(F*)G u E(F*) Mor complx xmpl: 11

12 1 E1 F G1 u1 2 E2 E3 G2 G3 u2 3 u3 If i no n ccping hn fr procing w hv hi: E1(F*)G1 E1(F*)G2 E1(F*)G3 E3(F*)G1 E3(F*)G2 E3(F*)G3 u1 u2 u3 If i n ccping hn in ddiion o ll h rrow w hv hi: 1 E1(F*) 2 E2(F*) 3 E3(F*) 12

13 Thn if hr i nohr in M ohr hn h r h h rrow lving i, hn om uch i procd in M o oin M. Thi procing of,, cr, coninu, rpdly pplying h rul, unil n uomon N i oind in which only h r h rrow lving i. Th i, N only h r nd om ccping 1, 2,..., n nd only h rrow from h r o ilf nd o h 1, 2,..., n. Th r my or my no n ccping. Thr will no rrow lving h of N ohr hn h r. Thu w my only hv h following kind of rrow in N: A B i i, 1 i n Thu N my look lik hi, if i no n ccping : A B1 B2 1 2 Bn n Th finl rgulr xprion i oind in h following wy. 13

14 If h r i no n ccping, hn h finl rgulr xprion E i A (B 1 B 2... B n ). If hr i no rrow from o hn E i (B 1 B 2... B n ). If h r i n ccping, hn h finl rgulr xprion E i A ( B 1 B 2... B n ). Thi cn lo wrin A A (B 1 B 2... B n ). If hr i no rrow from o hn E i ( B 1 B 2... B n ). Thu from M w oin rgulr xprion E, nd on cn how h L(M) = L(E), h i, E rprn h lngug rcognizd y M. Th ook giv nohr mhod o convr uom o rgulr xprion, u i i much hrdr o do on xmpl. 3.1 Exmpl Hr r om xmpl of h mhod. Sring uomon: u c 14

15 Afr limining : (*) c u Afr collping rrow: (*) U c u Th finl rgulr xprion i c. Now uppo h i n ccping in hi uomon: u c Afr procing : (*) (*) c u Afr collping rrow: 15

16 (*) (*) U c u Th finl rgulr xprion i c. Now conidr n xmpl in which hr r wo o limin. r u Afr limining : (*) r u Afr limining : r ** u Th finl rgulr xprion i. Now conidr n xmpl in which h nd u r h m: 16

17 Afr procing : (*) Th finl rgulr xprion i ( ). Now conidr n xmpl wih wo hving rrow from : c u1 u2 Afr procing, w hv hi uomon: 17

18 u1 c u2 Th finl rgulr xprion i c. Now uppo hr r mor lf-loop: d d u1 c u2 Afr procing, w hv hi uomon: 18

19 d (d*) u1 (d*)c u2 Th finl rgulr xprion i d ((d ) (d )c). Now uppo h r i n ccping : d d u1 c u2 Afr procing, w hv hi uomon: 19

20 d (d*) u1 (d*)c u2 Th finl rgulr xprion i d d ((d ) (d )c). Do hi conrucion on h following uomon: M Puing ll h rul oghr, lngug L i rgulr if nd only if hr i fini uomon M uch h L = L(M). Exrci: Find rgulr xprion for h of ring hving n odd numr of nd n vn numr of. Exrci: Find rgulr xprion for h inrviw uomon. Exrci: Suppo L 1 i rgulr nd L 2 i non-rgulr. I h concnion L 1 L 2 of L 1 nd L 2 ncrily non-rgulr? Exrci: Suppo L Σ i rgulr. Suppo x Σ. L L {y : xy L}. Show h L i rgulr. 20

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