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Finalstartingending01 minimaldfa

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Finalstartingending01 minimaldfa

  1. 1.  DETERMINE THE MINIMAL FINITE AUTOMATA ACCEPTING THE SET OF ALL STRINGS OVER {0,1} WHICH BEGIN OR END WITH 01.
  2. 2.  THE REGULAR EXPRESSION DENOTING THE SET OF ALL STRINGS BEGINNING WITH A 01 IS R1 = 01(0+1)*
  3. 3.  THE REGULAR EXPRESSION DENOTING THE SET OF ALL STRINGS BEGINNING WITH A 01 IS R1 = 01(0+1)* THE REGULAR EXPRESSION DENOTING THE SET OF ALL STRINGS ENDING WITH A 01 IS R2 = (0+1)*01
  4. 4.  THE REGULAR EXPRESSION DENOTING THE SET OF ALL STRINGS BEGINNING WITH A 01 OR ENDING WITH A 01 IS R = R1+R2+R3 = 01(0+1)*+ (0+1)*01 + 01(0+1)*01 R3 is included in R1 and R2 so we can simplify R= R1 + R2=01(0+1)* +(0+1)*01
  5. 5.  THE REGULAR EXPRESSION DENOTING THE SET OF ALL STRINGS BEGINNING WITH A 01 IS R1 = 01(0+1)* THE REGULAR EXPRESSION DENOTING THE SET OF ALL STRINGS ENDING WITH A 01 IS R2 = (0+1)*01 THE REGULAR EXPRESSION DENOTING THE SET OF ALL STRINGS BEGINNING AND ENDING WITH A 01 IS R3 = 01(0+1)*01
  6. 6.  DETERMINE THE MINIMAL FINITE AUTOMATA ACCEPTING THE SET OF ALL STRINGS OVER {‘0’, ‘1’} WHICH START WITH ‘0’ ‘1’ OR END WITH ‘0’’1’ OR BOTH.
  7. 7.  LET ‘0’ AND ‘1’ REPRESENT THE ATTRIBUTES OF DAYS. WE CONSIDER A SEQUENCE OF DAYS AS INPUT TO OUR FINITE AUTOMATA
  8. 8.  R1 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START WITH ‘0’ ‘1’
  9. 9.  R1 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START WITH ‘0’ ‘1’ R2 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT END WITH ‘0’ ‘1’
  10. 10.  R1 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START WITH ‘0’ ‘1’ R2 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT END WITH ‘0’ ‘1’ R3 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START OR END WITH ‘0’ ‘1’
  11. 11.  R1 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START WITH ‘0’ ‘1’ R2 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT END WITH ‘0’ ‘1’ R3 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START OR END WITH ‘0’ ‘1’ R = ACCEPTABLE STRINGS OR SEQUENCES OF DAYS = R1 + R2 + R3
  12. 12.  R1 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START WITH ‘0’ ‘1’ R2 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT END WITH ‘0’ ‘1’ R3 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START OR END WITH ‘0’ ‘1’ R = ACCEPTABLE STRINGS OR SEQUENCES OF DAYS = R1 + R2 + R3 R3 IS CONTAINED IN BOTH R1 AND R2
  13. 13.  R1 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START WITH ‘0’ ‘1’ R2 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT END WITH ‘0’ ‘1’ R3 = SET OF ALL STRINGS OR SEQUENCE OF DAYS THAT START OR END WITH ‘0’ ‘1’ R = ACCEPTABLE STRINGS OR SEQUENCES OF DAYS = R1
  14. 14.  SUBRPOBLEM 1: THE SET OF STRINGS DENOTED BY R1 THE SET OF ALL STRINGS OR SEQUENCES OF DAYS OVER {‘0’, ‘1’} STARTING WITH ‘0’ ‘1’
  15. 15.  SUBRPOBLEM 1: THE SET OF STRINGS DENOTED BY R1 THE SET OF ALL STRINGS OR SEQUENCES OF DAYS OVER {‘0’, ‘1’} STARTING WITH ‘0’ ‘1’ SUBPROBLEM 2: THE SET OF STRINGS DENOTED BY R2 THE SET OF ALL STRINGS OR SEQUENCES OF DAYS OVER {‘0’, ‘1’} ENDING WITH ‘0’ ‘1’
  16. 16.  SUBRPOBLEM 1: THE SET OF STRINGS DENOTED BY R1 THE SET OF ALL STRINGS OR SEQUENCES OF DAYS OVER {‘0’, ‘1’} STARTING WITH ‘0’ ‘1’ SUBPROBLEM 2: THE SET OF STRINGS DENOTED BY R2 THE SET OF ALL STRINGS OR SEQUENCES OF DAYS OVER {‘0’, ‘1’} ENDING WITH ‘0’ ‘1’ SUBPROBLEM 3: THE SET OF STRINGS DENOTED BY R= R1 + R2 THE SET OF ALL STRINGS OR SEQUENCE OF DAYS OVER {‘0’, ‘1’} THAT BEGIN OR END WITH ‘0’ ‘1’
  17. 17.  R1= THE SET OF ALL STRINGS OVE {‘0’, ‘1’} THAT START WITH ‘0’ ‘1’ R1 = ‘0’ ‘1’ {‘0’ + ‘1’}*
  18. 18.  R1= THE SET OF ALL STRINGS OVE {‘0’, ‘1’} THAT START WITH ‘0’ ‘1’ R1 = ‘0’ ‘1’ {‘0’ + ‘1’}* THE DFA FOR THE SAME IS GIVEN BELOW‘AA’
  19. 19.  R1= THE SET OF ALL STRINGS OVE {‘0’, ‘1’} THAT START WITH ‘0’ ‘1’ R1 = ‘0’ ‘1’ {‘0’ + ‘1’}* THE DFA FOR THE SAME IS GIVEN BELOW START‘AA’
  20. 20.  R1= THE SET OF ALL STRINGS OVE {‘0’, ‘1’} THAT START WITH ‘0’ ‘1’ R1 = ‘0’ ‘1’ {‘0’ + ‘1’}* THE DFA FOR THE SAME IS GIVEN BELOW START‘AA’ ‘BB’
  21. 21.  R1= THE SET OF ALL STRINGS OVE {‘0’, ‘1’} THAT START WITH ‘0’ ‘1’ R1 = ‘0’ ‘1’ {‘0’ + ‘1’}* THE DFA FOR THE SAME IS GIVEN BELOW START‘AA’ ‘BB’ ‘CC’
  22. 22.  R1= THE SET OF ALL STRINGS OVE {‘0’, ‘1’} THAT START WITH ‘0’ ‘1’ R1 = ‘0’ ‘1’ {‘0’ + ‘1’}* THE DFA FOR THE SAME IS GIVEN BELOW START FINAL STATE‘AA’ ‘BB’ ‘CC’
  23. 23.  R1= THE SET OF ALL STRINGS OVE {‘0’, ‘1’} THAT START WITH ‘0’ ‘1’ R1 = ‘0’ ‘1’ {‘0’ + ‘1’}* THE DFA FOR THE SAME IS GIVEN BELOW START FINAL STATE ‘0’‘AA’ ‘BB’ ‘CC’
  24. 24.  R1= THE SET OF ALL STRINGS OVE {‘0’, ‘1’} THAT START WITH ‘0’ ‘1’ R1 = ‘0’ ‘1’ {‘0’ + ‘1’}* THE DFA FOR THE SAME IS GIVEN BELOW START FINAL STATE ‘0’ ‘1’‘AA’ ‘BB’ ‘CC’
  25. 25.  R1= THE SET OF ALL STRINGS OVE {‘0’, ‘1’} THAT START WITH ‘0’ ‘1’ R1 = ‘0’ ‘1’ {‘0’ + ‘1’}* THE DFA FOR THE SAME IS GIVEN BELOW START ‘0’,’1’ ‘0’ ‘1’‘AA’ ‘BB’ ‘CC’
  26. 26. ‘0’ ‘1’’AA’ ‘BB’ ‘CC’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’
  27. 27. ‘0’ ‘1’’AA’ ‘BB’ ‘BB’ ‘CC’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’
  28. 28. ‘0’ ‘1’’AA’ ‘BB’ Φ ‘BB’ ‘CC’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’
  29. 29. ‘0’ ‘1’’AA’ ‘BB’ Φ ‘BB’ Φ ‘CC’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’
  30. 30. ‘0’ ‘1’’AA’ ‘BB’ Φ ‘BB’ Φ ‘CC’ ‘CC’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’
  31. 31. ‘0’ ‘1’’AA’ ‘BB’ Φ ‘BB’ Φ ‘CC’ ‘CC’ ‘CC’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’
  32. 32. ‘0’ ‘1’’AA’ ‘BB’ Φ ‘BB’ Φ ‘CC’ ‘CC’ ‘CC’ ‘CC’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’
  33. 33. ‘0’ ‘1’’AA’ ‘BB’ Φ ‘BB’ Φ ‘CC’***‘CC’ ‘CC’ ‘CC’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’
  34. 34.  R2=THE SET OF ALL STRINGS OR SEQUENCES OF DAYS OVER {‘0’, ‘1’} THAT END WITH ‘0’ ‘1’
  35. 35.  R2=THE SET OF ALL STRINGS OR SEQUENCES OF DAYS OVER {‘0’, ‘1’} THAT END WITH ‘0’ ‘1’ R2=(‘0’ + ‘1’)* ‘0’ ‘1’
  36. 36.  R2=THE SET OF ALL STRINGS OR SEQUENCES OF DAYS OVER {‘0’, ‘1’} THAT END WITH ‘0’ ‘1’ R2=(‘0’ + ‘1’)* ‘0’ ‘1’ WE CONSTRUCT AN NFA FROM THE ABOVE REGULAR EXPRESSION R2
  37. 37.  R2=(‘0’ + ‘1’)* ‘0’ ‘1’ THE NFA IS GIVEN BELOW‘DD’
  38. 38.  R2=(‘0’ + ‘1’)* ‘0’ ‘1’ST  THE NFA IS GIVEN BELOWART ‘DD’
  39. 39.  R2=(‘0’ + ‘1’)* ‘0’ ‘1’ST  THE NFA IS GIVEN BELOWART ‘EE’ ‘DD’
  40. 40.  R2=(‘0’ + ‘1’)* ‘0’ ‘1’ST  THE NFA IS GIVEN BELOWART ‘EE’ ‘DD’ ‘FF’
  41. 41.  R2=(‘0’ + ‘1’)* ‘0’ ‘1’  THE NFA IS GIVEN BELOW ‘0’,’1’START ‘EE’ ‘DD’ ‘FF’
  42. 42.  R2=(‘0’ + ‘1’)* ‘0’ ‘1’  THE NFA IS GIVEN BELOW ‘0’,’1’STAR ‘0’T ‘EE’ ‘DD’ ‘FF’
  43. 43.  R2=(‘0’ + ‘1’)* ‘0’ ‘1’  THE NFA IS GIVEN BELOW ‘0’,’1’STA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’
  44. 44.  R2=(‘0’ + ‘1’) *’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’STA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’
  45. 45.  R2=(‘0’ + ‘1’)*’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ ’DD’ ‘EE’S ‘FF’TA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’
  46. 46.  R2=(‘0’ + ‘1’) * ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ ’DD’ ‘DD’,’EE’ ‘EE’S ‘FF’TA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’
  47. 47.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’S ‘FF’TA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’
  48. 48.  R2=(‘0’ + ‘1’ *’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ ΦS ‘FF’TA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’
  49. 49.  R2=(‘0’ + ‘1’) * ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’TA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’
  50. 50.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’
  51. 51.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’
  52. 52.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ THE EQUIVAQLENT DFA IS GIVEN BELOW
  53. 53.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’ -> ‘DD’‘DD’
  54. 54.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ->T ‘DD’‘DD’
  55. 55.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ‘1’ ->T ‘DD’‘DD’
  56. 56.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ‘0’ -> ‘DD’T ‘DD’‘DD’
  57. 57.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ‘1’ -> ‘DD’T ‘DD’ ‘DD’’EE’ ‘0’‘DD’ ‘DD’’EE’
  58. 58.  R2=(‘0’ + ‘1’)*’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ‘1’ -> ‘DD’T ‘DD’ ‘DD’’EE’ ‘0’ ‘DD’’EE’‘DD’ ‘DD’’EE’
  59. 59.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ‘1’ ‘0’ -> ‘DD’T ‘DD’ ‘DD’’EE’ ‘0’ ‘DD’’EE’ ‘DD’’EE’‘DD’ ‘DD’’EE’
  60. 60.  R2=(‘0’ + ‘1’) *’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ‘1’ ‘0’ -> ‘DD’T ‘DD’ ‘DD’’EE’ ‘0’ ‘1’ ‘DD’’EE’ ‘DD’’EE’ ‘DD’’FF’‘DD’ ‘DD’’EE’ ‘DD’’FF’
  61. 61.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ‘1’ ‘0’ -> ‘DD’T ‘DD’ ‘DD’’EE’ ‘0’ ‘1’ ‘DD’’EE’ ‘DD’’EE’ ‘DD’’FF’‘DD’ ‘DD’’EE’ ‘DD’’FF’ ‘DD’’FF’
  62. 62.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ‘1’ ‘0’ -> ‘DD’’T ‘DD’ ‘DD’’EE’ ‘0’ ‘1’ ‘DD’’EE’ ‘DD’’EE’ ‘DD’’FF’‘DD’ ‘DD’’EE’ ‘DD’’FF’ *** ‘DD’’EE’ ‘DD’ ‘DD’’FF’ ‘0’ ‘1’
  63. 63.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ ‘0’ ‘1’ ‘0’,’1’ THIS IS AN NFA ’DD’ ‘DD’,’EE’ ‘DD’ ‘EE’ Φ ‘FF’S ‘FF’ Φ ΦTA ‘0’ ‘1’RT ‘EE’ ‘DD’ ‘FF’ST THE EQUIVAQLENT DFA IS GIVEN BELOW ‘0’ ‘1’AR ‘1’ ‘0’ -> ‘DD’T ‘DD’ ‘DD’’EE’ ‘0’ ‘1’ ‘DD’’EE’ ‘DD’’EE’ ‘DD’’FF’‘DD’ ‘DD’’EE’ ‘DD’’FF’ *** ‘DD’’EE’ ‘DD’ ‘DD’’FF’ ‘0’ ‘1’
  64. 64.  R2=(‘0’ + ‘1’)* ’0’ ‘1’ S T A R ‘0’ ‘0’ T ‘0’ ‘1’ ‘DD’ ‘DD’’EE’ ‘DD’’FF’ ‘0’ ‘1’ST THE EQUIVAQLENT DFA IS ABOVE & BELOW ‘0’ ‘1’AR ‘1’ ‘0’ -> ‘DD’T ‘DD’ ‘DD’’EE’ ‘0’ ‘1’ ‘DD’’EE’ ‘DD’’EE’ ‘DD’’FF’‘DD’ ‘DD’’EE’ ‘DD’’FF’ *** ‘DD’’EE’ ‘DD’ ‘DD’’FF’ ‘0’ ‘1’
  65. 65. ‘0’ ‘1’  R2=(‘0’ + ‘1’)*’0’ ‘1’ -> ‘DD’ ‘DD’’EE’ ‘DD’ S T ‘DD’’EE’ ‘DD’’EE’ ‘DD’’FF’ A R ‘1’ ‘0’ *** ‘DD’’FF’ ‘DD’’EE’ ‘DD’ T ‘0’ ‘1’ ‘DD’ ‘DD’’EE’ ‘DD’’FF’ ‘0’ ‘1’ST RENAMING THE STATES AS BELOWA ‘0’ ‘1’R ‘1’ ‘0’T -> ‘GG’ ‘0’ ‘1’ ‘GG’ ‘HH’‘GG’ ‘HH’ ‘II’ ‘HH’ ‘HH’ ‘DD’’FF’ ‘0’ *** ‘DD’’FF’ ‘DD’’EE’ ‘DD’ ‘1’
  66. 66. ‘0’ ‘1’  R2=(‘0’ + ‘1’) *’0’ ‘1’ -> ‘DD’ ‘DD’’EE’ ‘DD’ S T ‘DD’’EE’ ‘DD’’EE’ ‘DD’’FF’ A R ‘1’ ‘0’ *** ‘DD’’FF’ ‘DD’’EE’ ‘DD’ T ‘0’ ‘1’ ‘DD’ ‘DD’’EE’ ‘DD’’FF’ ‘0’ ‘1’ST R2=(‘0’ + ‘1’)*’0’’’1’A ‘0’ ‘1’R ‘1’ ‘0’T -> ‘GG’ ‘0’ ‘1’ ‘GG’ ‘HH’‘GG’ ‘HH’ ‘II’ ‘HH’ ‘HH’ ‘DD’’FF’ ‘0’ *** ‘DD’’FF’ ‘DD’’EE’ ‘DD’ ‘1’
  67. 67. R2=(0+1)* 01 S T A R ‘1’ ‘0’ T ‘1’ ‘0’ ‘DD’ ‘DD’’EE’ ‘DD’’FF’ ‘0’ ‘1’ START ‘0’,’1’ ‘0’ ‘1’‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  68. 68. RENAME THE STATES TO CONTROL ABSTRACTION R2=(0+1)* 01 S T A R ‘1’ ‘0’ T ‘1’ ‘0’ ‘DD’ ‘DD’’EE’ ‘DD’’FF’ ‘0’ ‘1’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  69. 69. RENAME THE STATES TO CONTROL ABSTRACTION R2=(0+1)* 01 S T A R ‘1’ ‘0’ T ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ ‘1’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  70. 70. CREATE NFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 S T A R ‘1’ ‘0’ T ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ ‘1’ START ‘0’,’1’ ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  71. 71. CREATE NFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  72. 72. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  73. 73. ‘AA’’GG’
  74. 74. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  75. 75. START‘AA’’GG’
  76. 76. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  77. 77. START‘AA’’GG’ ‘BB’’HH’
  78. 78. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  79. 79. START ‘0’‘AA’’GG’ ‘BB’’HH’
  80. 80. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  81. 81. START ‘0’ ‘AA’’GG’ ‘BB’’HH’‘1’ ‘GG’
  82. 82. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  83. 83. START ‘0’ ‘AA’’GG’ ‘BB’’HH’‘1’ ‘0’ ‘GG’ ‘HH’
  84. 84. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  85. 85. START ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’
  86. 86. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  87. 87. START ‘CC’’HH’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’
  88. 88. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  89. 89. START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’
  90. 90. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  91. 91. ‘0’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’
  92. 92. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  93. 93. ‘0’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’
  94. 94. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  95. 95. ‘0’ ‘0’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’
  96. 96. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  97. 97. ‘0’ ‘0’ ‘1’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’
  98. 98. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  99. 99. ‘0’ ‘0’ ‘1’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’ ‘0’
  100. 100. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  101. 101. ‘0’ ‘0’ ‘1’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’ ‘0’ ‘1’
  102. 102. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  103. 103. ‘0’ ‘0’ ‘1’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’ ‘0’ ‘1’ ‘0’
  104. 104. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  105. 105. ‘0’ ‘0’ ‘1’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ ‘1’ ‘1’ ‘0’
  106. 106. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  107. 107. ‘0’ ‘0’ ‘1’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ ‘1’ ‘1’ ‘0’
  108. 108. CREATE DFA GIVING UNION OF THE TWO MACHINES BELOW R2=(0+1)* 01 ‘1’ ‘0’ ‘1’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘0’ € ‘1’‘JJ’ € ‘0’,’1’START ‘0’ ‘1’ ‘AA’ ‘BB’ ‘CC’ R1=01(0+1)*
  109. 109. ‘0’ ‘0’ ‘1’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘0’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘1’ ‘1’ ‘1’ ‘0’
  110. 110. ‘0’ ‘0’ ‘1’ START ‘CC’’GG’ ‘CC’’HH’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘AA’’GG’ ‘BB’’HH’ ‘CC’’II’‘1’ ‘0’ ‘0’ ‘0’ ‘GG’ ‘HH’ ‘II’ ‘1’ ‘1’ ‘1’ ‘0’ MARK THE FINAL STATES WITH A DOUBLE CIRCLE
  111. 111. ‘0’ ‘0’ ‘1’ START ‘SS’ ‘RR’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘QQ’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’RENAME THE STATES OF THE FINAL DFA
  112. 112. ‘0’ ‘0’ ‘1’ START ‘SS’ ‘RR’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘QQ’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’RENAME THE STATES OF THE FINAL DFA
  113. 113. ‘0’ ‘0’ ‘1’ START ‘SS’ ‘RR’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘QQ’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’GIVE THE FINAL STATES ANOTHER COLOR
  114. 114. ‘0’ ‘0’ ‘1’ START ‘SS’ ‘RR’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘QQ’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’‘RR’, ‘SS’ AND ‘QQ’ CAN BE MERGED—MERGE THEM
  115. 115. ‘0’ ‘0’ ‘1’ START ‘SS’ ‘RR’ ‘1’ ‘1’ ‘0’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘QQ’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’‘RR’, ‘SS’ AND ‘QQ’ CAN BE MERGED—MERGE THEM
  116. 116. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’‘RR’, ‘SS’ AND ‘QQ’ CAN BE MERGED—MERGE THEM
  117. 117. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’‘TT’ AND ‘MM’ DIFFER ON INPUT ‘0’—DIFFERENT COLORS
  118. 118. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’‘PP’ AND ‘LL’ DIFFER ON INPUT 1---THEY DIFFER FROM ‘OO’ AND ‘KK’
  119. 119. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’‘PP’ AND ‘LL’ DIFFER ON INPUT 1---THEY DIFFER FROM ‘OO’ AND ‘KK’
  120. 120. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’‘OO’ AND ‘KK’ DIFFER ON INPUT ‘0’– NEW COLORS
  121. 121. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’‘OO’ AND ‘KK’ DIFFER ON INPUT ‘0’– NEW COLORS
  122. 122. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’THE MINIMAL DFA HAS 6 STATES AND IS GIVEN ABOVE
  123. 123. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘0’ ‘KK’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’THE MINIMAL DFA HAS 6 STATES AND IS GIVEN ABOVE
  124. 124. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IIIIIIIVVVI
  125. 125. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’I ‘OO’II ‘PP’III ‘TT’IV ‘KK’V ‘LL’VI ‘MM’
  126. 126. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’I ‘OO’ = €II ‘PP’III ‘TT’IV ‘KK’V ‘LL’VI ‘MM’
  127. 127. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’I ‘OO’ = €II ‘PP’ = ‘OO’ ‘0’ = ‘0’III ‘TT’IV ‘KK’V ‘LL’VI ‘MM’
  128. 128. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’I ‘OO’ = €II ‘PP’ = ‘OO’ ‘0’ = ‘0’III ‘TT’ = ‘PP’ ‘1’ + ‘TT’ (‘0’ + ‘1’)IV ‘KK’V ‘LL’VI ‘MM’
  129. 129. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’I ‘OO’ = €II ‘PP’ = ‘OO’ ‘0’ = ‘0’III ‘TT’ = ‘PP’ ‘1’ + ‘TT’ (‘0’ + ‘1’)IV ‘KK’ SUBSTITUTING II IN IIIV ‘LL’VI ‘MM’
  130. 130. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’I ‘OO’ = €II ‘PP’ = ‘OO’ ‘0’ = ‘0’III ‘TT’ = ‘PP’ ‘1’ + ‘TT’ (‘0’ + ‘1’) = ‘0’’1’ + ‘TT’ (‘0’+’1’)V ‘LL’ SUBSTITUTING II IN IIIVI ‘MM’
  131. 131. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’I ‘OO’ = €II ‘PP’ = ‘OO’ ‘0’ = ‘0’III ‘TT’ = ‘PP’ ‘1’ + ‘TT’ (‘0’ + ‘1’) = ‘0’’1’ + ‘TT’ (‘0’+’1’)V ‘LL’ SOLVING III BY ARDEN’S LEMMAVI ‘MM’
  132. 132. START ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’I ‘OO’ = €II ‘PP’ = ‘OO’ ‘0’ = ‘0’III ‘TT’ = ‘PP’ ‘1’ + ‘TT’ (‘0’ + ‘1’) = ‘0’’1’ + ‘TT’ (‘0’+’1’) = ‘0’ ‘1’ (‘0’ + ‘1’)*VI ‘MM’ SOLVING III BY ARDEN’S LEMMA
  133. 133. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’V ‘LL’VI ‘MM’ SOLVING III BY ARDEN’S LEMMA
  134. 134. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘KK’ ‘1’+ ‘MM’ ‘1’V ‘LL’ = ‘KK’ ‘0’+ ‘PP’ ‘0’+ ‘MM’ ‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’ SET UP IV, V AND VI EQUATIONS
  135. 135. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘KK’ ‘1’+ ‘MM’ ‘1’V ‘LL’ = ‘KK’ ‘0’+ ‘PP’ ‘0’+ ‘MM’ ‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’ SUBSTITUTE VI IN IV
  136. 136. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘KK’ ‘1’+ ‘LL’ ‘1’‘1’V ‘LL’ = ‘KK’ ‘0’+ ‘PP’ ‘0’+ ‘MM’ ‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’ SUBSTITUTE VI IN IV
  137. 137. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘KK’ ‘1’+ ‘LL’ ‘1’‘1’V ‘LL’ = ‘KK’ ‘0’+ ‘PP’ ‘0’+ ‘MM’ ‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’ SUBSTITUTE VI IN V
  138. 138. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘KK’ ‘1’+ ‘LL’ ‘1’‘1’V ‘LL’ = ‘KK’ ‘0’+ ‘PP’ ‘0’+ ‘LL’ ‘1’‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’ SUBSTITUTE VI IN V
  139. 139. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘LL’ ‘1’‘1’+ ‘KK’ ‘1’V ‘LL’ = ‘KK’ ‘0’+ ‘PP’ ‘0’+ ‘LL’ ‘1’‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’ REWRITING IV FOR ARDEN’S LEMMA
  140. 140. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘LL’ ‘1’‘1’+ ‘KK’ ‘1’V ‘LL’ = ‘KK’ ‘0’+ ‘PP’ ‘0’+ ‘LL’ ‘1’‘0’VI ‘MM’ = ‘LL’ ‘1’ SUBSTITUTE ‘0’ FOR ‘PP’ IN EQUATION V
  141. 141. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘LL’ ‘1’‘1’+ ‘KK’ ‘1’V ‘LL’ = ‘KK’ ‘0’+ ‘0’‘0’+ ‘LL’ ‘1’‘0’VI ‘MM’ = ‘LL’ ‘1’ SUBSTITUTE ‘0’ FOR ‘PP’ IN EQUATION V
  142. 142. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘LL’ ‘1’‘1’+ ‘KK’ ‘1’V ‘LL’ = ‘0’‘0’+ ‘KK’ ‘0’+ ‘LL’ ‘1’‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’ REWRITING EQUATION V
  143. 143. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘OO’ ‘1’+ ‘LL’ ‘1’‘1’+ ‘KK’ ‘1’V ‘LL’ = ‘0’‘0’+ ‘KK’ ‘0’+ ‘LL’ ‘1’‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’REWRITING EQUATION IV BY SUBSTITING FOR ‘OO’
  144. 144. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘1’+ ‘LL’ ‘1’‘1’+ ‘KK’ ‘1’V ‘LL’ = ‘0’‘0’+ ‘KK’ ‘0’+ ‘LL’ ‘1’‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’REWRITING EQUATION IV BY SUBSTITING FOR ‘OO’
  145. 145. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘1’+ ‘LL’ ‘1’‘1’+ ‘KK’ ‘1’ = ‘1’’1’*+ ‘LL’ ‘1’’1’’1’*V ‘LL’ = ‘0’‘0’+ ‘KK’ ‘0’+ ‘LL’ ‘1’‘0’ + ‘LL’ ‘0’VI ‘MM’ = ‘LL’ ‘1’SOLVING IV BY ARDEN’S LEMMA
  146. 146. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’IV ‘KK’ = ‘1’+ ‘LL’ ‘1’‘1’+ ‘KK’ ‘1’ = ‘1’’1’*+ ‘LL’ ‘1’’1’’1’*V ‘LL’ = ‘0’‘0’+ ‘KK’ ‘0’+ ‘LL’ ‘1’‘0’ = ‘0’’0’+ ‘1’’1’*’0’ ‘LL’ (‘1’’1’’1’* + ‘1’’0’ + ’0’ +’0’)VI ‘MM’ = ‘LL’ ‘1’SUBSTITUTNG FOR ‘KK’ IN V
  147. 147. START ‘TT’=‘0’’1’(‘0’ + ‘1’)* ‘0’,’1’ ‘0’ ‘1’ ‘OO’ ‘PP’ ‘TT’‘1’ ‘0’ ‘0’ ‘KK’ ‘0’ ‘LL’ ‘MM’ ‘1’ ‘1’ ‘1’ ‘0’SOLVING V BY ARDEN’S LEMMAIV ‘KK’ = ‘1’+ ‘LL’ ‘1’‘1’+ ‘KK’ ‘1’ = ‘1’’1’*+ ‘LL’ ‘1’’1’’1’*V ‘LL’ = ‘0’‘0’+ ‘KK’ ‘0’+ ‘LL’ ‘1’‘0’ = ‘0’’0’+ ‘1’’1’*’0’ ‘LL’ (‘1’’1’’1’* + ‘1’’0’ + ’0’ +’0’) =(‘0’’0’+’1’’1’*’0’)[‘1’’1’’1’*’0’ + ‘1’’0’]*VI ‘MM’ = ‘LL’ ‘1’

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