LEX5: Regexps to NFA. Lexical Analysis. CMPT 379: Compilers Instructor: Anoop Sarkar. anoopsarkar.github.io/compilers-class

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1 LEX5: Regexps to NFA Lexicl Anlysis CMPT 379: Compilers Instructor: Anoop Srkr noopsrkr.github.io/compilers-clss

2 Building Lexicl Anlyzer Token POern POern Regulr Expression Regulr Expression NFA NFA DFA DFA Tble-driven implementron of DFA 2

3 Converts regexps to equivlent NFA Six simple rules Empty lnguge Symbols (Σ) Empty String () AlternRon (r 1 or r 2 ) ConctenRon (r 1 followed by r 2 ) RepeRRon (r 1 *) Used by Ken Thompson for poern-bsed serch in text editor QED (1968) 3

4 Thompson Rule 0 For the empty lnguge φ (opronlly include sinkhole stte) Σ Σ 4

5 Thompson Rule 1 For ech symbol x of the lphbet, there is NFA tht ccepts it x 5

6 Thompson Rule 2 There is n NFA tht ccepts only 6

7 Thompson Rule 3 Given two NFAs for r 1, r 2, there is NFA tht ccepts r 1 r 2 r1 r2 7

8 Thompson Rule 3 Given two NFAs for r 1, r 2, there is NFA tht ccepts r 1 r 2 r1 r2 8

9 Thompson Rule 4 Given two NFAs for r 1, r 2, there is NFA tht ccepts r 1 r 2 r1 r2 9

10 Thompson Rule 4 Given two NFAs for r 1, r 2, there is NFA tht ccepts r 1 r 2 r1 r2 10

11 Thompson Rule 5 Given NFA for r, there is n NFA tht ccepts r* r 11

12 Thompson Rule 5 Given NFA for r, there is n NFA tht ccepts r* r 12

13 Exmple Set of ll binry strings tht re divisible by four (include 0 in this set) Defined by the regexp: ((0 1)*00) 0 Apply Thompson s Rules to crete n NFA 13

14 Bsic Blocks 0 nd 1 ((0 1)*00) 0 NFA for 0 NFA for

15 ((0 1)*00) 0 15

16 0 1 (0 1)* ((0 1)*00) 0 16

17 (0 1)*00 ((0 1)*00) 0 17

18 (0 1)*00 ((0 1)*00) 0 18

19 ((0 1)*00) 0 19

20 ((0 1)*00) 0 20

21 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 21

22 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 22

23 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. push stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 23

24 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 24

25 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. push n2 stck n2, = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 25

26 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. stck n2, = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 26

27 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. push n3 stck n3, n2, = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 27

28 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. stck n3, n2, = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 28

29 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. pop n3,n2 stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 29

30 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. push n4 stck n4, = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 30

31 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. stck n4, = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 31

32 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. pop n4, stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 32

33 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. push stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 33

34 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 34

35 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. push n6 stck n6, = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 35

36 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. stck n6, = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 36

37 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. pop n6, stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 37

38 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. push n7 n7 stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 38

39 Converts regexps to NFA Build NFA recursively from regexp tree (( b))c Post-order trversl of regexp tree n2 n4 n7 n3 n6 b c b.c. pop n7 stck = nf() n2= nf() n3= nf(b) n4= nf(n2, n3, ) = nf(, n4,. ) n6= nf(c) n7= nf(, n6,. ) 39