Fina lnotoriousfa1ver3

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JNTU QUALITATIVE VISUAL ALGORITHMS

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Fina lnotoriousfa1ver3

  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

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