Upcoming SlideShare
×

# Fina lnotoriousfa1ver7

331 views

Published on

jntu qualitative visual algorithms

Published in: Entertainment & Humor
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
331
On SlideShare
0
From Embeds
0
Number of Embeds
90
Actions
Shares
0
0
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Fina lnotoriousfa1ver7

1. 1. PROBLEM 1• DETERMINE THE MINIMAL DETERMINISTIC FINITE AUTOMATA ACCEPTING THE SET OF ALL STRINGS OVER {sun,rain}* WHERE THE FIRST SYMBOL FROM THE RIGHT END IS A rain.
2. 2. What strings are accepted?BEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
3. 3. What strings are accepted? FIRST SYMBOL FROM RIGHT ENDBEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
4. 4. What strings are accepted? REST OF THE STRING FIRST SYMBOL FROM RIGHT ENDBEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
5. 5. What strings are accepted? REST OF THE STRING FIRST SYMBOL FROM RIGHT END ANYTHING HERE IN THE FRONT rainBEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
6. 6. What strings are accepted? REST OF THE STRING FIRST SYMBOL FROM RIGHT ENDBEGINNING OF STRING THIS IS THE INPUT STRING rain END OF STRING
7. 7. What strings are accepted? REST OF THE STRING FIRST SYMBOL FROM RIGHT END ANYTHING HERE IN THE FRONT= (sun + rain)*BEGINNING OF STRING rain END OF STRING THIS IS THE INPUT STRING
8. 8. What strings are accepted? REST OF THE STRING FIRST SYMBOL FROM RIGHT END ANYTHING HERE IN THE FRONT= (sun + rain)*BEGINNING OF STRING rain THIS IS THE INPUT STRING END OF STRING (sun + rain)*rain
9. 9. Construct the finite automata. REST OF THE STRING FIRST SYMBOL FROM RIGHT END rain ANYTHING HERE IN THE FRONT= (sun + rain)* Construct the finite automata.BEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING Hansel (sun + rain)*rain
10. 10. Construct the finite automata. REST OF THE STRING FIRST SYMBOL FROM RIGHT END rain ANYTHING HERE IN THE FRONT= (sun + rain)*BEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING Hansel (sun + rain)*rain
11. 11. Construct the finite automata. REST OF THE STRING FIRST SYMBOL FROM RIGHT END rain ANYTHING HERE IN THE FRONT= (sun + rain)*BEGINNING OF STRING END OF STRING sun,rain THIS IS THE INPUT STRING Hansel (sun + rain)*rain
12. 12. Construct the finite automata. REST OF THE STRING FIRST SYMBOL FROM RIGHT END rain ANYTHING HERE IN THE FRONT= (sun + rain)*BEGINNING OF STRING END OF STRING sun,rain THIS IS THE INPUT STRING Grethel Hansel (sun + rain)*rain
13. 13. Construct the finite automata. REST OF THE STRING FIRST SYMBOL FROM RIGHT END rain ANYTHING HERE IN THE FRONT= (sun + rain)*BEGINNING OF STRING END OF STRING sun,rain THIS IS THE INPUT STRING Grethel Grethel Hansel (sun + rain)*rain
14. 14. Construct the finite automata. REST OF THE STRING FIRST SYMBOL FROM RIGHT END rain ANYTHING HERE IN THE FRONT= (sun + rain)*BEGINNING OF STRING END OF STRING sun,rain THIS IS THE INPUT STRING rain Grethel Grethel Hansel (sun + rain)*rain
15. 15. Construct the finite automata. REST OF THE STRING FIRST SYMBOL FROM RIGHT END rain ANYTHING HERE IN THE FRONT= (sun + rain)*BEGINNING OF STRING END OF STRING sun,rain THIS IS THE INPUT STRINGSTART rain Grethel Grethel Hansel (sun + rain)*rain
16. 16. Construct the finite automata. • IF THE FINITE AUTOMATA DETERMINISTIC OR IS IT NONDETERMINISTIC? sun,rainSTART rain Grethel Grethel Hansel (sun + rain)*rain
17. 17. Construct the finite automata. • IF THE FINITE ATOMATA DETERMINISTIC OR IS IT NONDETERMINISTIC? • TO FIND OUT CONSTRUCT THE STATE TRANSITION TABLE. sun,rainSTART rain Grethel Hansel Grethel (sun + rain)*rain
18. 18. Construct the finite automata. THE STATE TRANSITION TABLE sun,rainSTART rain Grethel Grethel Hansel (sun + rain)*rain
19. 19. Construct the STATE TRANSITION TABLE of thefinite automata. STATES THE STATE TRANSITION Hansel TABLE Grethel sun,rainSTART rain Grethel Grethel Hansel (sun + rain)*rain
20. 20. The state transition table of the finite automata. INPUTS sun rain THE STATE STATES TRANSITION Hansel TABLE Grethel sun,rainSTART rain Grethel Hansel Grethel (sun + rain)*rain
21. 21. STATE TRANSITION TABLE INPUTS sun rain THE STATE STATES TRANSITION Hansel Hansel TABLE Grethel sun,rainSTART rain Hansel Grethel Grethel (sun + rain)*rain
22. 22. STATE TRANSITION TABLE INPUTS sun rain STATES THE STATE Hansel Hansel Hansel,Grethel TRANSITION TABLE Grethel sun,rainSTART rain Grethel Hansel Grethel (sun + rain)*rain
23. 23. STATE TRANSITION TABLE INPUTS sun rain STATES THE STATE TRANSITION TABLE Hansel Hansel Hansel,Grethel Grethel φ Φ Φ Φ Φ sun,rainSTART rain Grethel Grethel Hansel (sun + rain)*rain
24. 24. THE MACHINE IS NONDETERMINISTIC INPUTS sun rain STATES THE STATE TRANSITION Hansel Hansel Hansel,Grethel TABLE Grethel φ Φ Φ Φ Φ sun,rainSTART rain Grethel Grethel Hansel (sun + rain)*rain
25. 25. CONSTRUCT THE DETERMINISTIC MACHINE INPUTS sun rain STATES Hansel Hansel Hansel,Grethel Grethel φ Φ Φ Φ Φ sun,rain NFASTART rain Grethel Grethel Hansel (sun + rain)*rain
26. 26. CONSTRUCT THE DETERMINISTIC MACHINE INPUTS sun rain STATES Hansel Hansel Hansel,Grethel Grethel φ Φ Φ Φ Φ sun,rain NFA THE DFA rain Grethel Hansel GrethelSTART (sun + rain)*rain
27. 27. CONSTRUCT THE DETERMINISTIC MACHINE INPUTS sun rain INPUTS sun rain STATES STATES Hansel Hansel Hansel,Grethel Grethel φ Φ Φ Φ Φ sun,rain NFA THE DFA rain Grethel GrethelSTART Hansel (sun + rain)*rain
28. 28. CONSTRUCT THE DETERMINISTIC MACHINE INPUTS sun rain INPUTS sun rain STATES STATES Hansel Hansel Hansel,Grethel Hansel Grethel φ Φ Φ Φ Φ sun,rain NFA THE DFA rainSTART Hansel Grethel L=(sun + rain)*rain
29. 29. CONSTRUCT THE DETERMINISTIC MACHINE INPUTS sun rain INPUTS sun rain STATES STATES Hansel Hansel Hansel,Grethel Hansel Hansel Grethel φ Φ Φ Φ Φ sun,rain NFA THE DFA rainSTART Grethel Grethel Hansel (sun + rain)*rain
30. 30. CONSTRUCT THE DETERMINISTIC MACHINE INPUTS sun rain INPUTS sun rain STATES STATES Hansel Hansel Hansel,Grethel Hansel Hansel HanselGrethel Grethel φ Φ Φ Φ Φ sun,rain NFA THE DFASTART rain Grethel Hansel Grethel (sun + rain)*rain
31. 31. CONSTRUCT THE DETERMINISTIC INPUTS MACHINE sun rain INPUTS sun rain STATES STATES Hansel Hansel Hansel,Grethel Hansel Hansel HanselGrethel Grethel φ Φ HanselGrethel Hansel HanselGrethel Φ Φ Φ sun,rain NFA THE DFA rainSTART Hansel Grethel Grethel (sun + rain)*rain
32. 32. CONSTRUCT THE DETERMINISTIC MACHINE INPUTS sun rain INPUTS sun rain STATES STATES Hansel Hansel Hansel,Grethel Hansel Hansel HanselGrethel Grethel φ Φ HanselGrethel Hansel HanselGrethel Φ Φ Φ sun,rain NFA THE DFA rainSTART Grethel Grethel Hansel (sun + rain)*rain
33. 33. THE DFA HAS 2 STATES=2^1 STATES INPUTS sun rain INPUTS sun rain STATES STATES Hansel Hansel Hansel,Grethel Hansel Hansel HanselGrethel Grethel φ Φ HanselGrethel Hansel HanselGrethel Φ Φ Φ sun,rain NFA THE DFASTART rain Grethel Grethel Hansel (sun + rain)*rain
34. 34. MINIMISE THE DFAINPUTS sun rainSTATESHansel Hansel HanselGrethelHanselGrethel Hansel HanselGrethel
35. 35. MINIMISE THE DFA CONSTRUCT THE TRANSITIONINPUTS DIAGRAM sun rainSTATESHansel Hansel HanselGrethelHanselGrethel Hansel HanselGrethel Hansel HanselGrethel
36. 36. MINIMISE THE DFA CONSTRUCT THE TRANSITIONINPUTS DIAGRAM sun rainSTATESHansel Hansel HanselGrethelHanselGrethel Hansel HanselGrethel Hansel HanselGrethel
37. 37. MINIMISE THE DFA CONSTRUCT THE TRANSITIONINPUTS DIAGRAM sun rainSTATESHansel Hansel HanselGrethelHanselGrethel Hansel HanselGrethel Hansel HanselGrethel
38. 38. MINIMISE THE DFA CONSTRUCT THE TRANSITIONINPUTS DIAGRAM sun rainSTATESHansel Hansel HanselGrethelHanselGrethel Hansel HanselGrethel sun Hansel HanselGrethel
39. 39. MINIMISE THE DFA CONSTRUCT THE TRANSITIONINPUTS DIAGRAM sun rainSTATESHansel Hansel HanselGrethelHanselGrethel Hansel HanselGrethel sun rain Hansel HanselGrethel
40. 40. MINIMISE THE DFA CONSTRUCT THE TRANSITIONINPUTS DIAGRAM sun rainSTATESHansel Hansel HanselGrethelHanselGrethel Hansel HanselGrethel sun rain rain Hansel HanselGrethel
41. 41. MINIMISING THE DFA BY GRAPH COLORINGINPUTS sun rainSTATESHansel Hansel HanselGrethelHanselGrethel Hansel HanselGrethel sun rain rain Hansel HanselGrethel sun
42. 42. MINIMISING THE DFA BY GRAPH COLORINGINPUTS sun rain CREATE TWO PARTITIONS ONE FOR FINAL AND ONE FORSTATES NONFINALHansel Hansel HanselGrethel SO COLOR Hansel BLUE COLOR HanselGrethel PINKHanselGrethel Hansel HanselGrethel sun rain rain Hansel HanselGrethel sun
43. 43. MINIMISING THE DFA BY GRAPH COLORINGINPUTS sun rain CREATE TWO PARTITIONS ONE FOR FINAL AND ONE FORSTATES NONFINALHansel Hansel HanselGrethel SO COLOR Hansel BLUE COLOR HanselGrethel PINK NOW PINK AND BLUE AREHanselGrethel Hansel HanselGrethel SINGLETONS SO MINIMAL DFA HAS TWO STATES sun rain rain Hansel HanselGrethel sun
44. 44. PROBLEM 2• DETERMINE THE MINIMAL DETERMINISTIC FINITE AUTOMATA ACCEPTING THE SET OF ALL STRINGS OVER {sun,rain}* WHERE THE SECOND SYMBOL FROM THE RIGHT END IS A rain.
45. 45. What strings are accepted?BEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
46. 46. What strings are accepted? FIRST SYMBOL FROM RIGHT ENDBEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
47. 47. What strings are accepted? REST OF THE STRING FIRST SYMBOL FROM RIGHT ENDBEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
48. 48. What strings are accepted? REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END ENDBEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
49. 49. What strings are accepted? REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING rain END OF STRING THIS IS THE INPUT STRING •
50. 50. What strings are accepted? REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING rain END OF STRING THIS IS THE INPUT STRING • anything
51. 51. What strings are accepted? REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING rain END OF STRING THIS IS THE INPUT STRING • anything anything
52. 52. What strings are accepted? REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING rain END OF STRING THIS IS THE INPUT STRING • (sun+rain)* • sun+rain
53. 53. What strings are accepted? REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING • rain END OF STRING THIS IS THE INPUT STRING • (sun+rain)* • sun+rain • INITIAL PART MIDDLE LAST • PART PART
54. 54. Strings accepted=(sun+rain)*rain(sun+rain) REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING • rain END OF STRING THIS IS THE INPUT STRING • (sun+rain)* • sun+rain • INITIAL PART MIDDLE LAST • PART PART
55. 55. CONSTRUCT THE NFA REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING • rain END OF STRING THIS IS THE INPUT STRING • (sun+rain)* • sun+rain • INITIAL PART MIDDLE LAST • PART PART
56. 56. CONSTRUCT THE NFA REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING END OF STRING • rain THIS IS THE INPUT STRING • (sun+rain)* sun,rain • • Witch
57. 57. CONSTRUCT THE NFA REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING END OF STRING • THIS IS THE INPUT STRING rain • anything=(sun+rain)* sun,rain • rain Jack Witch Jill
58. 58. CONSTRUCT THE NFA REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING • rain sun,rain • (sun+rain)* • sun, rain rain • Witch Jack Jill •
59. 59. CONSTRUCT THE NFA REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END END OF STRING •BEGINNING OF STRING THIS IS THE INPUT STRING • sun, rain Jack rain Witch Jill
60. 60. CONSTRUCT THE NFA REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING sun,rain rain sun, rain Jack START Witch Jill
61. 61. CONSTRUCT THE NFA REST OF THE SECOND FIRST STRING SYMBOL SYMBOL FROM FROM RIGHT RIGHT END END •BEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING sun,rain sun, rain rain Jack START Witch Jill Jill
62. 62. CONSTRUCT THE STATE TRANSITION TABLE • NFA sun,rain sun, rain rain JackSTART Witch Jill Jill
63. 63. CONSTRUCT THE STATE TRANSITION TABLE STATES • NFA sun,rain sun, rain Jack rainSTART Witch Jill Jill
64. 64. CONSTRUCT THE STATE TRANSITION TABLE INPUTS-  STATES • NFA sun,rain sun, rain rain JackSTART Witch Jill Jill
65. 65. CONSTRUCT THE STATE TRANSITION TABLE INPUTS- STATES Witch Jack Jill Φ • NFA sun,rain rain sun,rain JackSTART Witch Jill Jill
66. 66. CONSTRUCT THE STATE TRANSITION TABLE INPUTS- sun rain STATES Witch Jack Jill Φ • sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
67. 67. CONSTRUCT THE STATE TRANSITION TABLE INPUTS- sun rain STATES Witch Witch Jack Jill Φ • sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
68. 68. CONSTRUCT THE STATE TRANSITION TABLE INPUTS- sun rain STATES Witch Witch WitchJack Jack Jill Jill Jill Φ Φ Φ Φ Φ • sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
69. 69. CONSTRUCT THE STATE TRANSITION TABLE INPUTS- sun rain STATES Witch Witch WitchJack Jack Jill Jill Jill Φ Φ Φ Φ Φ • sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
70. 70. CONSTRUCT THE STATE TRANSITION TABLE-WE HAVE AN NFA INPUTS- sun rain STATES Witch Witch WitchJack Jack Jill Jill Jill Φ Φ Φ Φ Φ • sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
71. 71. CONSTRUCT THE STATE TRANSITION TABLE-WE HAVE AN NFA INPUTS- sun rain STATES Witch Witch WitchJack Jack Jill Jill Jill Φ Φ Φ Φ Φ • sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
72. 72. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFA INPUTS- sun rain STATES Witch Witch WitchJack Jack Jill Jill Jill Φ Φ Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
73. 73. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFA INPUTS- sun rain STATESSTATES Witch Witch WitchJack Jack Jill Jill Jill Φ Φ Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
74. 74. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS INPUTS- sun rain STATESSTATES Witch Witch WitchJack Jack Jill Jill Jill Φ Φ Φ Φ Φ • THE DFA NFA sun,r sun,rain ain rain JackSTART Witch Jill Jill
75. 75. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS INPUTS- sun rain sun rain STATESSTATES Witch Witch WitchJack Jack Jill Jill Jill Φ Φ Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
76. 76. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS INPUTS- sun rain sun rain STATESSTATES Witch Witch WitchJackWitch Jack Jill Jill Jill Φ Φ Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
77. 77. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS INPUTS- sun rain sun rain STATESSTATES Witch Witch WitchJackWitch Witch Jack Jill Jill Jill Φ Φ Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
78. 78. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS INPUTS- sun rain sun rain STATESSTATES Witch Witch WitchJackWitch Witch WitchJack Jack Jill Jill Jill Φ Φ Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
79. 79. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS INPUTS- sun rain sun rain STATESSTATES Witch Witch WitchJackWitch Witch WitchJack Jack Jill JillWitchJack WitchJill Jill Φ Φ Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
80. 80. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS sun rain INPUTS- sun rain STATESSTATES Witch Witch WitchJackWitch Witch WitchJack Jack Jill JillWitchJack WitchJill WitchJackJill Jill Φ Φ Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
81. 81. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS sun rain INPUTS- sun rain STATESSTATES Witch Witch WitchJackWitch Witch WitchJack Jack Jill JillWitchJack WitchJill WitchJackJill Jill Φ ΦWitchJill Witch Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
82. 82. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS sun rain INPUTS- sun rain STATESSTATES Witch Witch WitchJackWitch Witch WitchJack Jack Jill JillWitchJack WitchJill WitchJackJill Jill Φ ΦWitchJill Witch WitchJack Φ Φ Φ • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
83. 83. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS sun rain INPUTS- sun rain STATESSTATES Witch Witch WitchJackWitch Witch WitchJack Jack Jill JillWitchJack WitchJill WitchJackJill Jill Φ ΦWitchJill Witch WitchJack Φ Φ ΦWitchJackJill WitchJill • THE DFA sun,rain NFA rain sun,rain JackSTART Witch JillJill
84. 84. CONSTRUCT THE STATE TRANSITION TABLE-OBTAIN THE DFAINPUTS sun rain INPUTS- sun rain STATESSTATES Witch Witch WitchJackWitch Witch WitchJack Jack Jill JillWitchJack WitchJill WitchJackJill Jill Φ ΦWitchJill Witch WitchJack Φ Φ ΦWitchJackJill WitchJill WitchJackJill • THE DFA sun,rain NFA rain sun,rain JackSTART Witch Jill Jill
85. 85. CONSTRUCT THE STATE TRANSITION TABLE-MARK THE FINAL STATESINPUTS sun rain INPUTS- sun rain STATESSTATES Witch Witch WitchJackWitch Witch WitchJack Jack Jill JillWitchJack WitchJill WitchJackJill Jill Φ ΦWitchJill Witch WitchJack Φ Φ ΦWitchJackJill WitchJill WitchJackJill • THE DFA sun,rain NFA rain sun,rain JackSTART Witch JillJill
86. 86. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHMINPUTS sun rainSTATESWitch Witch WitchJackWitchJack WitchJill WitchJackJillWitchJill Witch WitchJackWitchJackJill WitchJill WitchJackJill
87. 87. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHMINPUTS sun rainSTATESWitch Witch WitchJackWitchJack WitchJill WitchJackJillWitchJill Witch WitchJackWitchJackJill WitchJill WitchJackJill CONSTRUCT THE STATE TRANSITION DIAGRAM
88. 88. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHMINPUTS sun rainSTATES Witch WitchJillWitch Witch WitchJackWitchJack WitchJill WitchJackJillWitchJill Witch WitchJackWitchJackJill WitchJill WitchJackJill WitchJack WitchJackJill CONSTRUCT THE STATE TRANSITION DIAGRAM
89. 89. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHMINPUTS sun rainSTATES Witch WitchJillWitch Witch WitchJackWitchJack WitchJill WitchJackJillWitchJill Witch WitchJackWitchJackJill WitchJill WitchJackJill WitchJack WitchJackJill CONSTRUCT THE STATE TRANSITION DIAGRAM MARK THE FINAL STATES
90. 90. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHMINPUTS sun rainSTATES Witch WitchJillWitch Witch WitchJackWitchJack WitchJill WitchJackJillWitchJill Witch WitchJackWitchJackJill WitchJill WitchJackJill WitchJack WitchJackJill CONSTRUCT THE STATE TRANSITION DIAGRAM MARK THE FINAL STATES MARK THE START STATE
91. 91. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rainSTATES Witch WitchJillWitch Witch WitchJackWitchJack WitchJill WitchJackJillWitchJill Witch WitchJackWitchJackJill WitchJill WitchJackJill WitchJack WitchJackJill PUT IN THE TRANSITIONS
92. 92. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rainSTATES Witch WitchJillWitch Witch WitchJackWitchJack WitchJill WitchJackJillWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill WitchJack WitchJackJill PUT IN THE TRANSITIONS
93. 93. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rainSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJillWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill WitchJack WitchJackJill PUT IN THE TRANSITIONS
94. 94. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rainSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJillWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill rain WitchJack WitchJackJill PUT IN THE TRANSITIONS
95. 95. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rain sunSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJillWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill rain rain WitchJack WitchJackJill PUT IN THE TRANSITIONS
96. 96. MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rain sunSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJill sunWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill rain rain WitchJack WitchJackJill PUT IN THE TRANSITIONS
97. 97. THE STATE TRANSITION DIAGRAM HAS BEEN CONSTRUCTED MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rain sunSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJill sunWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill rain rain WitchJack WitchJackJill CREATE TWO PARTITIONS— ONE FOR THE FINAL STATES AND ONE FOR THE NONFINAL STATES COLOR FINAL STATES GREEN rain COLOR NONFINAL BLUE
98. 98. THE STATE TRANSITION DIAGRAM HAS BEEN CONSTRUCTED MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rain sunSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJill sunWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill rain rain WitchJack WitchJackJill ON INPUT rain WE FIND THAT Witch WANTS TO STAY IN BLUE BUT WitchJack WANTS TO GO rain TO GREEN THEY QUARREL!!! GIVE ANOTHER COLOR SAY ORANGE TO WitchJack!!
99. 99. THE STATE TRANSITION DIAGRAM HAS BEEN CONSTRUCTED MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rain sunSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJill sunWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill rain rain WitchJack ON INPUT sun WitchJill WitchJackJill AND WitchJackJill QUAREL! THE FORMER WANTS TO GO TO BLUE AND THE LATTER WANTS TO STAY IN GREEN!! rain CHAOS AND NOISE!!!! GIVE WitchJill COLOR PURPLE
100. 100. THE STATE TRANSITION DIAGRAM HAS BEEN CONSTRUCTED MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rain sunSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJill sunWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill rain rain WitchJack WitchJackJill rain
101. 101. THE STATE TRANSITION DIAGRAM HAS BEEN CONSTRUCTED MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rain sunSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJill sunWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill rain rain WitchJack WitchJackJill rain
102. 102. THE STATE TRANSITION DIAGRAM HAS BEEN CONSTRUCTED MINIMISATION OF THE DFA USING THE GRAPH COLORING ALGORITHM sunINPUTS sun rain sunSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJillWitchJill Witch WitchJack rain su nWitchJackJill WitchJill WitchJackJill rain rain WitchJack WitchJackJill rain
103. 103. MINIMAL DFA HAS 4 STATES = 2^2 STATES sunINPUTS sun rain sunSTATES Witch WitchJillWitch Witch WitchJack sunWitchJack WitchJill WitchJackJill sunWitchJill Witch WitchJack rainWitchJackJill WitchJill WitchJackJill rain rain WitchJack WitchJackJill rain
104. 104. PROBLEM 3• DETERMINE THE MINIMAL DETERMINISTIC FINITE AUTOMATA ACCEPTING THE SET OF ALL STRINGS OVER {sun,rain}* WHERE THE THIRD SYMBOL FROM THE RIGHT END IS A rain.
105. 105. What strings are accepted?BEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
106. 106. What strings are accepted? FIRST SYMBOL FROM RIGHT ENDBEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
107. 107. What strings are accepted? REST OF THE STRING FIRST SYMBOL FROM RIGHT ENDBEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
108. 108. What strings are accepted? REST OF THE STRING SECOND FIRST SYMBOL SYMBOL FROM FROM RIGHT RIGHT END ENDBEGINNING OF STRING END OF STRING THIS IS THE INPUT STRING
109. 109. What strings are accepted? REST OF THE STRING THIRD SECOND FIRST SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING THIS IS THE • INPUT STRING
110. 110. What strings are accepted? REST OF THE STRING THIRD SECOND FIRST SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING THIS IS • THE rain INPUT STRING
111. 111. What strings are accepted? REST OF THE STRING THIRD SECOND FIRST SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING THIS IS • THE rain sun+rain INPUT STRING
112. 112. What strings are accepted? REST OF THE STRING THIRD SECOND FIRST SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING THIS IS THE • INPUT rain sun+rainsun+rain STRING
113. 113. What strings are accepted? REST OF THE STRING THIRD SECOND FIRST SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING (sun+rain)* rain sun+rain sun+rain
114. 114. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE STRING THIRD SECOND FIRST SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING (sun+rain)* rain sun+rain sun+rain
115. 115. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE THIRD SECOND FIRST STRING SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING (sun+rain)* rain sun+rain sun+rain
116. 116. • STORY• HANSEL AND GRETHEL ARE IN THEIR FATHER’S HOUSE• THEN THEY GO TO HANSEL’S HOUSE• THEN THEY GO THE GRETHEL’S HOUSE• THEY THEY GO TO THE WITHCH’S HOUSE
117. 117. • STORY IN MORE DETAIL• HANSEL AND GRETHEL STAY IN THEIR FATHER’S HOUSE WHETHER IT IS A sun DAY OR A rain DAY• ON A rain DAY THEY CHOOSE TO GO TO HANSEL’S HOUSE• NEXT DAY THEY GO TO GRETHEL’S HOUSE WHETHER IT IS A sun DAY OR A rain DAY• NEXT DAY THEY GO THE WITHCH’S HOUSE WHETHER IT IS A sun DAY OR A rain DAY• END OF THE STORY
118. 118. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE THIRD SECOND FIRST STRING SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING (sun+rain)* rain sun+rain sun+rain
119. 119. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE THIRD SECOND FIRST STRING SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING (sun+rain)* rain sun+rain sun+rain Father Hansel Grethel Witch
120. 120. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE THIRD SECOND FIRST STRING SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING (sun+rain)* rain sun+rain sun+rain sun, rain Father Hansel Grethel Witch
121. 121. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE THIRD SECOND FIRST STRING SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRINGsun,rain rain Father Hansel Grethel Witch
122. 122. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE THIRD SECOND FIRST STRING SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRINGsun,rain sun,rain rain Father Hansel Grethel Witch
123. 123. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE THIRD SECOND FIRST STRING SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRINGsun,rain sun,rain sun,rain rain Father Hansel Grethel Witch
124. 124. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE THIRD SECOND FIRST STRING SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRING sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
125. 125. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) REST OF THE THIRD SECOND FIRST STRING SYMBOL SYMBOL SYMBOL FROM FROM FROM RIGHT RIGHT RIGHT END END END •BEGINNING OFSTRING END OF STRINGsun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
126. 126. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) • WE HAVE CONSTRUCTED THE NFA OF 4 STATES sun,rain rain sun,rain sun,rain Witch Father Hansel GrethelSTART
127. 127. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) • WE HAVE CONSTRUCTED THE NFA OF 4 STATES • WE HAVE TO SHOW THAT THE DFA WILL HAVE 2^3=8 STATES sun,rain rain sun,rain sun,rain Witch Father Hansel GrethelSTART
128. 128. Strings accepted = (sun+rain)*rain(sun+rain)(sun+rain) • WE HAVE CONSTRUCTED THE NFA OF 4 STATES • WE HAVE TO SHOW THAT THE DFA WILL HAVE 2^3=8 STATES • WE WILL FIRST CONSTRUCT THE STATE TRANSITION TABLE OF THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
129. 129. THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
130. 130. STATES THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
131. 131. INPUTS STATES THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
132. 132. INPUTS STATES sun rain THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
133. 133. INPUTS sun rain STATES Father Hansel Grethel Witch Φ THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
134. 134. INPUTS sun rain STATES Father Father Hansel Grethel Witch Φ THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
135. 135. INPUTS sun rain STATES Father Father FatherHansel Hansel Grethel Witch Φ THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
136. 136. INPUTS sun rain STATES Father Father FatherHansel Hansel Grethel Grethel Grethel Witch Φ THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
137. 137. INPUTS sun rain STATES Father Father FatherHansel Hansel Grethel Grethel Grethel Witch Witch Witch Φ THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
138. 138. INPUTS sun rain STATES Father Father FatherHansel Hansel Grethel Grethel Grethel Witch Witch Witch Φ Φ Φ Φ Φ THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
139. 139. INPUTS sun rain STATES Father Father FatherHansel Hansel Grethel Grethel Grethel Witch Witch Witch Φ Φ Φ Φ Φ THE NFAsun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
140. 140. INPUTS STATES sun rain Father Father FatherHansel Hansel Grethel Grethel Grethel Witch Witch Witch Φ Φ Φ Φ Φ THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART
141. 141. INPUTS sun rain STATES Father Father FatherHansel Hansel Grethel Grethel Grethel Witch Witch Witch Φ Φ Φ Φ Φ THE NFA sun,rain sun,rain rain sun,rain Witch Father Hansel GrethelSTART