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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’