# Regular languages can be expressed as regular expressions.

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1 Regulr lnguges cn e expressed s regulr expressions. A generl nondeterministic finite utomton (GNFA) is kind of NFA such tht: There is unique strt stte nd is unique ccept stte. Every pir of nodes re connected y n rrow in ech direction, ech leled with regulr expression. Exceptions re: The strt stte hs no incoming edges. The ccept stte hs no outgoing edges. The lnguge ccepted y the GNFA is the union of ll L(R) such tht R is the regulr expression constructed y conctenting ll the regulr expressions ppering on the pth from the strt stte to the ccept stte in the order they pper. CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 12

2 An Exmple of GNFA,, Here the dshed rrows represent rrows with s the lel. CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 13

3 Proof Pln 1. Show tht ny NFA cn e converted to n equivlent GNFA. 2. Show tht ny GNFA cn e converted to regulr expression. CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 14

4 From n NFA to GNFA Given n NFA N, construct GNFA G s follows: 1. Add specil strt stte q strt nd connect it to the initil stte of N with ɛ s the lel. Connect q strt to ech remining stte with s the lel. CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 15

5 From n NFA to GNFA Given n NFA N, construct GNFA G s follows: 1. Add specil strt stte q strt nd connect it to the initil stte of N with ɛ s the lel. Connect q strt to ech remining stte with s the lel. 2. Add specil ccept stte q ccept nd connect to it from ech finl stte of N with ɛ s the lel. Connect from ech of the remining stte of N to q ccept with s the lel. CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 16

6 From n NFA to GNFA Given n NFA N, construct GNFA G s follows: 1. Add specil strt stte q strt nd connect it to the initil stte of N with ɛ s the lel. Connect q strt to ech remining stte with s the lel. 2. Add specil ccept stte q ccept nd connect to it from ech finl stte of N with ɛ s the lel. Connect from ech of the remining stte of N to q ccept with s the lel. 3. Add n rrow with s the lel wherever necessry. CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 17

7 Before nd After,, CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 18

8 Before nd After,,,, CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 19

9 Wht Hve We Done? We hve constructed our initil GNFA G so tht: (I) For ech word w in L(N), thereispth[u 0,...,u m ] in G from q strt to q ccept such tht () w is decomposed s w 1 w 2 w m ; () for ll i, 1 i m, w i is memer of the lnguge represented y the regulr expression of the rrow (u i 1,u i ). CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 20

10 Wht Hve We Done? We hve constructed our initil GNA G so tht: (I) For ech word w in L(N), thereispth[u 0,...,u m ] in G from q strt to q ccept such tht () w is decomposed s w 1 w 2 w m ; () for ll i, 1 i m, w i is memer of the lnguge represented y the regulr expression of the rrow (u i 1,u i ). (II) For ech word w tht cn e decomposed s in the ove, w is memer of L(N). CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 21

11 Convert GNFA to Regulr Expression We will remove intermedite nodes one fter the other while preserving the two properties. Repet the following until there is no stte left other thn the strt stte nd the ccept stte. Select n ritrry stte s. Produce from the current GNFA n equivlent GNFA y: For ech rrow (p, q) such tht p, q s, replce its lel with the lel corresponding to the pths [p, q], [p, s, q], [p, s, s, q], [p, s, s, s, q],... Remove s nd ll edges ttched to it. CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 22

12 Stte Elimintion,, ecomes {,}*, CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 24

13 Stte Elimintion {,}*, ecomes {,}* {,}* CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 25

14 Stte Elimintion {,}* {,}* ecomes {,}* {,}* CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 26

15 Stte Elimintion {,}* {,}* ecomes {,}* {,}* CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 27

16 Stte Elimintion {,}* {,}* ecomes {,}* {,}* CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 28

17 Stte Elimintion {,}* {,}* ecomes {,}* {,}* CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 29

18 Stte Elimintion {,}* {,}* ecomes {,}* {,}* CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 30

19 Stte Elimintion {,}* {,}* ecomes {,}* {,}* The conversion hs een completed. CSC527, Chpter 1, Prt 3 c 2012 Mitsunori Ogihr 31

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