Reducing a DFA to a Minimal DFA

Transcription

1 Lexicl Anlysis - Prt 4 Reducing DFA to Miniml DFA Input: DFA IN Assume DFA IN never gets stuck (dd ded stte if necessry) Output: DFA MIN An equivlent DFA with the minimum numer of sttes. Hrry H. Porter,

2 Lexicl Anlysis - Prt 4 Reducing DFA to Miniml DFA Input: DFA IN Assume DFA IN never gets stuck (dd ded stte if necessry) Output: DFA MIN An equivlent DFA with the minimum numer of sttes. 1 Ded Stte , Hrry H. Porter,

3 Lexicl Anlysis - Prt 4 Reducing DFA to Miniml DFA Input: DFA IN Assume DFA IN never gets stuck (dd ded stte if necessry) Output: DFA MIN An equivlent DFA with the minimum numer of sttes. 1 Ded Stte , Approch: Merge two sttes if the effectively do the sme thing. Do the sme thing? At EOF, is DFA IN in n ccepting stte or not? Hrry H. Porter,

4 Lexicl Anlysis - Prt 4 Sufficiently Different Sttes Merge sttes, if t ll possile. Are two sttes sufficiently different... tht they cnnot e merged? Hrry H. Porter,

5 Lexicl Anlysis - Prt 4 Sufficiently Different Sttes Merge sttes, if t ll possile. Are two sttes sufficiently different... tht they cnnot e merged? Stte s is distinguished from stte t y some string w iff: strting t s, given chrcters w, the DFA ends up ccepting,... ut strting t t, the DFA does not ccept. Hrry H. Porter,

6 Lexicl Anlysis - Prt 4 Sufficiently Different Sttes Merge sttes, if t ll possile. Are two sttes sufficiently different... tht they cnnot e merged? Stte s is distinguished from stte t y some string w iff: strting t s, given chrcters w, the DFA ends up ccepting,... ut strting t t, the DFA does not ccept. Exmple: s c t c does not distinguish s nd t. But c distinguishes s nd t. Therefore, s nd t cnnot e merged. Hrry H. Porter,

7 Lexicl Anlysis - Prt 4 Prtitioning Set A prtitioning of set......reks the set into non-overlpping susets. (The prtition reks the set into groups ) Exmple: S = {A, B, C, D, E, F, G} Π = {(A B) (C D E F) (G) } Π 2 = {(A) (B C) (D E F G) } Hrry H. Porter,

8 Lexicl Anlysis - Prt 4 Prtitioning Set A prtitioning of set......reks the set into non-overlpping susets. (The prtition reks the set into groups ) Exmple: S = {A, B, C, D, E, F, G} Π = {(A B) (C D E F) (G) } Π 2 = {(A) (B C) (D E F G) } We cn refine prtition... Π i = { (A B C) (D E) (F G) } Π i+1 = { (A C) (B) (D) (E) (F G) } Note: { (...) (...) (...) } mens {{...}, {...}, {...} } Hrry H. Porter,

9 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Consider the set of sttes. Initilly, we will prtition it into two groups... Finl Sttes All Other Sttes c e d Hrry H. Porter,

10 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Consider the set of sttes. Initilly, we will prtition it into two groups... Finl Sttes All Other Sttes c e d (A B D) (C E) Hrry H. Porter,

11 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Consider the set of sttes. Initilly, we will prtition it into two groups... Finl Sttes All Other Sttes c e d (A B D) (C E) Repetedly refine the prtioning. Two sttes must e plced in different groups... if they cn e distinguished Repet until no group contins sttes tht cn e distinguished. Ech group in the prtitioning ecomes one stte in newly constructed DFA DFA MIN = The miniml DFA Hrry H. Porter,

12 Lexicl Anlysis - Prt 4 How to Refine Prtitioning? Π i = { ( A B D ) ( C E ) } Consider one group... (A B D) Look t output edges on some symol (e.g., x ) P 1 P 2 x x c d x e Hrry H. Porter,

13 Lexicl Anlysis - Prt 4 How to Refine Prtitioning? Π i = { ( A B D ) ( C E ) } Consider one group... (A B D) Look t output edges on some symol (e.g., x ) P 1 P 2 x x c d On x, ll sttes in P 1 go to sttes elonging to the sme group. x e Hrry H. Porter,

14 Lexicl Anlysis - Prt 4 How to Refine Prtitioning? Π i = { ( A B D ) ( C E ) } Consider one group... (A B D) Look t output edges on some symol (e.g., x ) P 1 P 2 x x c d On x, ll sttes in P 1 go to sttes elonging to the sme group. x e y c y e Now consider nother symol (e.g., y ) y d Hrry H. Porter,

15 Lexicl Anlysis - Prt 4 How to Refine Prtitioning? Π i = { ( A B D ) ( C E ) } Consider one group... (A B D) Look t output edges on some symol (e.g., x ) P 1 P 2 x x c d On x, ll sttes in P 1 go to sttes elonging to the sme group. x e y c y e Now consider nother symol (e.g., y ) D is distinguished from A nd B! y d Hrry H. Porter,

16 Lexicl Anlysis - Prt 4 How to Refine Prtitioning? Π i = { ( A B D ) ( C E ) } Consider one group... (A B D) Look t output edges on some symol (e.g., x ) P 1 P 2 x x c d On x, ll sttes in P 1 go to sttes elonging to the sme group. x e y c y y d e Now consider nother symol (e.g., y ) D is distinguished from A nd B! So refine the prtition! Π i+1 = { ( A B ) ( D ) ( C E ) } P 3 P 4 P 2 Hrry H. Porter,

17 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) C A B D E Hrry H. Porter,

18 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) A C B D E Consider (E) Hrry H. Porter,

19 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Consider A C B D E Consider (E) Hrry H. Porter,

20 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? Consider A C B D E Consider (E) Hrry H. Porter,

21 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? Consider (E) A C B D E Hrry H. Porter,

22 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) A C B D E Hrry H. Porter,

23 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) Not possile to rek prt. A C B D E Hrry H. Porter,

24 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) Not possile to rek prt. New Prtitioning: Π 2 = (A B C) (D) (E) A C B D E Hrry H. Porter,

25 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) Not possile to rek prt. New Prtitioning: Π 2 = (A B C) (D) (E) Consider Brek prt? Consider A C B D E Hrry H. Porter,

26 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) Not possile to rek prt. New Prtitioning: Π 2 = (A B C) (D) (E) Consider Brek prt? No Consider Brek prt? A C B D E Hrry H. Porter,

27 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) Not possile to rek prt. New Prtitioning: Π 2 = (A B C) (D) (E) Consider Brek prt? No Consider Brek prt? (A C) (B) A C B D E Hrry H. Porter,

28 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) Not possile to rek prt. New Prtitioning: Π 2 = (A B C) (D) (E) Consider Brek prt? No Consider Brek prt? (A C) (B) New Prtitioning: Π 3 = (A C) (B) (D) (E) Consider Brek prt? Consider Brek prt? A C B D E Hrry H. Porter,

29 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) Not possile to rek prt. New Prtitioning: Π 2 = (A B C) (D) (E) Consider Brek prt? No Consider Brek prt? (A C) (B) New Prtitioning: Π 3 = (A C) (B) (D) (E) Consider Brek prt? No Consider Brek prt? A C B D E Hrry H. Porter,

30 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) Not possile to rek prt. New Prtitioning: Π 2 = (A B C) (D) (E) Consider Brek prt? No Consider Brek prt? (A C) (B) New Prtitioning: Π 3 = (A C) (B) (D) (E) Consider Brek prt? No Consider Brek prt? No A C B D E Hrry H. Porter,

31 Lexicl Anlysis - Prt 4 Exmple DFA IN Initil Prtitioning: Π 1 = (A B C D) (E) Consider (A B C D) Consider Brek prt? No Consider Brek prt? (A B C) (D) Consider (E) Not possile to rek prt. New Prtitioning: Π 2 = (A B C) (D) (E) Consider Brek prt? No Consider Brek prt? (A C) (B) New Prtitioning: Π 3 = (A C) (B) (D) (E) Consider Brek prt? No Consider Brek prt? No A C B DFA MIN D AC B D E E Hrry H. Porter,

32 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Add ded stte nd trnsitions to it if necessry. (Now, every stte hs n outgoing edge on every symol.) Π = initil prtitioning loop Π NEW = Refine(Π) if (Π NEW = Π) then rek Π = Π NEW endloop Hrry H. Porter,

33 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Add ded stte nd trnsitions to it if necessry. (Now, every stte hs n outgoing edge on every symol.) Π = initil prtitioning loop Π NEW = Refine(Π) if (Π NEW = Π) then rek Π = Π NEW endloop Construct DFA MIN Ech group in Π ecomes stte Hrry H. Porter,

34 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Add ded stte nd trnsitions to it if necessry. (Now, every stte hs n outgoing edge on every symol.) Π = initil prtitioning loop Π NEW = Refine(Π) if (Π NEW = Π) then rek Π = Π NEW endloop Construct DFA MIN Ech group in Π ecomes stte A C c B D c Hrry H. Porter,

35 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Add ded stte nd trnsitions to it if necessry. (Now, every stte hs n outgoing edge on every symol.) Π = initil prtitioning loop Π NEW = Refine(Π) if (Π NEW = Π) then rek Π = Π NEW endloop Construct DFA MIN Ech group in Π ecomes stte Choose one stte in ech group (throw ll other sttes wy) Preserve the edges out of the chosen stte A C c B D c Hrry H. Porter,

36 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Add ded stte nd trnsitions to it if necessry. (Now, every stte hs n outgoing edge on every symol.) Π = initil prtitioning loop Π NEW = Refine(Π) if (Π NEW = Π) then rek Π = Π NEW endloop Construct DFA MIN Ech group in Π ecomes stte Choose one stte in ech group (throw ll other sttes wy) Preserve the edges out of the chosen stte A D c Hrry H. Porter,

37 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Add ded stte nd trnsitions to it if necessry. (Now, every stte hs n outgoing edge on every symol.) Π = initil prtitioning loop Π NEW = Refine(Π) if (Π NEW = Π) then rek Π = Π NEW endloop Construct DFA MIN Ech group in Π ecomes stte Choose one stte in ech group (throw ll other sttes wy) Preserve the edges out of the chosen stte A D c A D c Hrry H. Porter,

38 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Add ded stte nd trnsitions to it if necessry. (Now, every stte hs n outgoing edge on every symol.) Π = initil prtitioning loop Π NEW = Refine(Π) if (Π NEW = Π) then rek Π = Π NEW endloop Construct DFA MIN Ech group in Π ecomes stte Choose one stte in ech group (throw ll other sttes wy) Preserve the edges out of the chosen stte Del with strt stte nd finl sttes A D c A D c Hrry H. Porter,

39 Lexicl Anlysis - Prt 4 Hopcroft s Algorithm Add ded stte nd trnsitions to it if necessry. (Now, every stte hs n outgoing edge on every symol.) Π = initil prtitioning loop Π NEW = Refine(Π) if (Π NEW = Π) then rek Π = Π NEW endloop Construct DFA MIN Ech group in Π ecomes stte Choose one stte in ech group (throw ll other sttes wy) Preserve the edges out of the chosen stte Del with strt stte nd finl sttes If desired... Remove ded stte Remove ny stte unrechle from the strt stte A A D D c c Hrry H. Porter,

40 Lexicl Anlysis - Prt 4 Π NEW = Refine(Π) Π NEW = {} for ech group G in Π do Exmple: Π = (A B C E) (D F) Brek G into su-groups (A B C E) (A C) (B E) s follows: Put S nd T into different sugroups if... For ny symol Σ, S nd T go to sttes in two different groups in Π A B x x D C Must split A nd B into different groups Add the su-groups to Π NEW endfor return Π NEW Π NEW = { } Π NEW = { (A C) (B E) } Π NEW = { (A C) (B E) (D F) } Hrry H. Porter,

41 Lexicl Anlysis - Prt 4 Summrizing... Hrry H. Porter,

42 Lexicl Anlysis - Prt 4 Summrizing... Regulr Expressions to Descrie Tokens Hrry H. Porter,

43 Lexicl Anlysis - Prt 4 Summrizing... Regulr Expressions to Descrie Tokens Algorithm: Regulr Expression NFA Hrry H. Porter,

44 Lexicl Anlysis - Prt 4 Summrizing... Regulr Expressions to Descrie Tokens Algorithm: Regulr Expression NFA Algorithm for Simulting NFA Hrry H. Porter,

45 Lexicl Anlysis - Prt 4 Summrizing... Regulr Expressions to Descrie Tokens Algorithm: Regulr Expression NFA Algorithm for Simulting NFA Algorithm: NFA DFA Hrry H. Porter,

46 Lexicl Anlysis - Prt 4 Summrizing... Regulr Expressions to Descrie Tokens Algorithm: Regulr Expression NFA Algorithm for Simulting NFA Algorithm: NFA DFA Algorithm: DFA Miniml DFA Hrry H. Porter,

47 Lexicl Anlysis - Prt 4 Summrizing... Regulr Expressions to Descrie Tokens Algorithm: Regulr Expression NFA Algorithm for Simulting NFA Algorithm: NFA DFA Algorithm: DFA Miniml DFA Algorithm for Simulting DFA Hrry H. Porter,

48 Lexicl Anlysis - Prt 4 Summrizing... Regulr Expressions to Descrie Tokens Algorithm: Regulr Expression NFA Algorithm for Simulting NFA Algorithm: NFA DFA Algorithm: DFA Miniml DFA Algorithm for Simulting DFA Fst: Get Next Chr Evlute Move Function e.g., Arry Lookup Chnge Stte Vrile Test for Accepting Stte Test for EOF Repet Hrry H. Porter,

49 Lexicl Anlysis - Prt 4 Summrizing... Regulr Expressions to Descrie Tokens Algorithm: Regulr Expression NFA Algorithm for Simulting NFA Algorithm: NFA DFA Algorithm: DFA Miniml DFA Algorithm for Simulting DFA Fst: Get Next Chr Evlute Move Function e.g., Arry Lookup Chnge Stte Vrile Test for Accepting Stte Test for EOF Repet Scnner Genertors Crete n efficient Lexer from regulr expressions! Hrry H. Porter,

50 Lexicl Anlysis - Prt 4 Scnner Genertor: LEX Input: r 1 { ction 1 } r 2 { ction 2 }... r N { ction N } Requirements: Choose the lrgest lexeme tht mtches. If more thn one r i mtches, choose the first one. DFA Simultor (C-code) Trnsition Tles (initilized rrys) lex-egin-ptr forwrd-ptr Input Buffers Cnned code dded y lex tool Computed y lex tool Hrry H. Porter,

51 Lexicl Anlysis - Prt 4 Input: { Action-1 } { Action-2 } *+ { Action-3 } Hrry H. Porter,

52 Lexicl Anlysis - Prt 4 Input: { Action-1 } { Action-2 } *+ { Action-3 } Crete NFA: *+ 0 ε ε ε Pttern-1 Pttern-3 6 Pttern-2 Hrry H. Porter,

53 Lexicl Anlysis - Prt 4 Input: { Action-1 } { Action-2 } *+ { Action-3 } Crete NFA: *+ 0 ε ε ε Pttern-1 Pttern-3 6 Pttern-2 Exmple Input: c... Strt: { 0, 1, 3, 7 } Input: { 2, 4, 7 } Input: { 7 } Input: { 8 } Input: c { } Mtch! Pttern: 1 Length: 1 Mtch! Pttern: 3 Length: 3 Done! Identify the lst mtch. Execute the corresponding ction & djust pointers Hrry H. Porter,

54 Lexicl Anlysis - Prt 4 Find the NFA for r 1 r 2... r N Convert to DFA. Approch Ech stte of the DFA corresponds to set of NFA sttes. A stte is finl if ny NFA stte in it ws finl stte. If severl, choose the lowest numered pttern to e the one ccepted. During simultion, keep following edges until you get stuck. As the scnning proceeds... Every time you enter finl stte... Rememer: The current vlue of uffer pointers Which pttern ws recognized Upon termintion... Use tht informtion to... Adjust the uffer pointers Execute the desired ction Hrry H. Porter,

55 Lexicl Anlysis - Prt 4 Input: { Action-1 } { Action-2 } *+ { Action-3 } Exmple Hrry H. Porter,

56 Lexicl Anlysis - Prt 4 Input: { Action-1 } { Action-2 } *+ { Action-3 } Crete NFA: *+ Exmple 0 ε ε ε *+ 6 Hrry H. Porter,

57 Lexicl Anlysis - Prt 4 Input: { Action-1 } { Action-2 } *+ { Action-3 } Crete NFA: *+ Exmple 0 ε ε ε *+ 6 Construct Miniml DFA 2,4,7 5,8 0,1,3, ,8 Hrry H. Porter,

58 Lexicl Anlysis - Prt 4 Input: { Action-1 } { Action-2 } *+ { Action-3 } Crete NFA: *+ Exmple 0 ε ε ε *+ 6 Construct Miniml DFA Attch Actions 0,1,3,7 *+ 8 2,4,7 7 5,8 6,8 *+ *+ Accept only first pttern Hrry H. Porter,

59 Lexicl Anlysis - Prt 4 Input: { Action-1 } { Action-2 } *+ { Action-3 } Crete NFA: *+ Exmple 0 ε ε ε *+ 6 Construct Miniml DFA Attch Actions Exmple Strings: 0,1,3,7 *+ 8 2,4,7 7 5,8 6,8 *+ *+ Accept only first pttern Hrry H. Porter,

60 Lexicl Anlysis - Prt 4 Oldest, most well-known For Unix/C Environment In UNIX: %lex lex.l %cc lex.yy.c The Lex Tool File: lex.l Lex Tool Contins severl regulr expressions File: lex.yy.c A progrm in C... Redy to compile nd link with Prser (e.g., YACC output) Hrry H. Porter,

61 Lexicl Anlysis - Prt 4 Oldest, most well-known For Unix/C Environment In UNIX: %lex lex.l %cc lex.yy.c The Lex Tool File: lex.l Lex Tool Contins severl regulr expressions Input File Formt: File: lex.yy.c %{ }% %% %%...Any C Code......Regulr Definitions......Regulr Expressions with Actions......Any C Code... A progrm in C... Redy to compile nd link with Prser (e.g., YACC output) Hrry H. Porter,

62 Lexicl Anlysis - Prt 4 Regulr Expressions in Lex c Conctention; Most chrcters stnd for themselves Met Chrters: Usul menings * Exmple: ( )*c* () + One or more, e.g., +c? Optionl, e.g.,?c [x-y] Chrcter clsses, e.g., [-z][-za-z0-9]* [^x-y] Anything ut [x-y] \x The usul escpe sequences, e.g., \n. Any chrcter except \n ^ Beginning of line $ End of line "..." To use the met chrcters literlly, Exmple: PCAT comments: "(*".*"*)" {...} Defined nmes, e.g., {letter} / Look-hed Exmple: /cd (Mtches, ut only when followed y cd) Hrry H. Porter,

63 Lexicl Anlysis - Prt 4 Look-Ahed Opertor, / /cd Mtches, ut only if followed y cd. Hrry H. Porter,

64 Lexicl Anlysis - Prt 4 Look-Ahed Opertor, / /cd Mtches, ut only if followed y cd. Add specil ε edge for / ε c d Hrry H. Porter,

65 Lexicl Anlysis - Prt 4 Look-Ahed Opertor, / /cd Mtches, ut only if followed y cd. Add specil ε edge for / ε c d Mrk the following stte to mke note of... The pttern in question The current vlue of the uffer pointers...whenever this stte is encountered during scnning. / Encountered Sve uffer pointers Hrry H. Porter,

66 Lexicl Anlysis - Prt 4 Look-Ahed Opertor, / /cd Mtches, ut only if followed y cd. Add specil ε edge for / ε c d Mrk the following stte to mke note of... The pttern in question The current vlue of the uffer pointers...whenever this stte is encountered during scnning. / Encountered Sve uffer pointers When pttern is finlly mtched, check these notes. If we pssed through / stte for the pttern ccepted, Use the stored uffer positions, insted of the finl positions to descrie the lexeme mtched. Hrry H. Porter,

67 Lexicl Anlysis - Prt 4 Lex: Input File Formt %{...Any C Code... }%...Regulr Definitions... %%...Regulr Expressions with Actions... %%...Any C Code... Hrry H. Porter,

68 Lexicl Anlysis - Prt 4 %{ }% %% %%...Any C Code... #define ID 13 #define NUM 14 #define PLUS 15 #define MINUS #define WHILE 37 #define IF Regulr Definitions... Lex: Input File Formt...Regulr Expressions with Actions......Any C Code int lookup (chr * p) {...} int enter (chr * p, int i) {...}... Any C code; Copied without chnges to eginning of the output file Any C code; dded to end of file (typiclly, uxillry support routines) Hrry H. Porter,

69 Lexicl Anlysis - Prt 4 %{ }% %% %% Defined Nmes...Any C Code... Lex: Input File Formt...Regulr Definitions... Defined nmes cn delim [ \t\n] e used in regulr expressions white {delim}+ letter [-za-z] digit [0-9] id {letter}({letter} {digit})* num {digit}+(\.{digit}+)?...regulr Expressions with Actions......Any C Code... Blnk: Every chrcter is Itself literlly Hrry H. Porter,

70 Lexicl Anlysis - Prt 4 %{ }% %%...Any C Code......Regulr Definitions... Lex: Input File Formt...Regulr Expressions with Actions... "+" {return PLUS;} "-" {return MINUS;}... while if... {white} {}... {return WHILE;} {return IF;} Regulr expressions {num} {yylvl =...; return NUM;} {id} {yylvl =...lookup(...)...; return ID;} %%...Any C Code... yylvl is where token ttriute info is stored. Any C code. Include return to give the token to prser. No return mens do nothing. (This token is recognized ut not returned to prser) You my use these vriles to ccess the lexeme: chr * yytext; int yyleng; Hrry H. Porter,