1.  -Automata Ekaterina Mineev

2. Today: 1 Introduction - notation -  -Automata overview

3. Today: 1 Introduction - notation -  -Automata overview 2 Nondeterministic models - B ü chi acceptance - Muller acceptance - Rabin acceptance - Streett acceptance - parity condition

4. Today(cont.): 2.1 Equivalency of nondeterministic models

5. Today(cont.): 3 Deterministic models - B ü chi condition - equivalency of deterministic* models

6. Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - B ü chi condition - equivalency of deterministic* models 4 Some lower bound for transformations

7. Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - B ü chi condition - equivalency of deterministic* models 4 Some lower bound for transformations 5 Weak acceptance conditions - Staiger-Wagner acceptance

8. Today(cont.): 2.1 Equivalency of nondeterministic models 3 Deterministic models - B ü chi condition - equivalency of deterministic* models 4 Some lower bound for transformations 5 Weak acceptance conditions - Staiger-Wagner acceptance 6 Conclusion

9. Notation

10. Notation  := {0, 1, 2, 3, … }  := {0, 1, 2, 3, … }

11. Notation  - finite alphabet  - finite alphabet

12. Notation  := {0, 1, 2, 3, … }  := {0, 1, 2, 3, … }  - finite alphabet  - finite alphabet  * - set of finite words over   * - set of finite words over 

13. Notation  := {0, 1, 2, 3, … }  := {0, 1, 2, 3, … }  - finite alphabet  - finite alphabet  * - set of finite words over   * - set of finite words over    - set of infinite words (  -words) over    - set of infinite words (  -words) over 

14. Notation  := {0, 1, 2, 3, … }  := {0, 1, 2, 3, … }  - finite alphabet  - finite alphabet  * - set of finite words over   * - set of finite words over    - set of infinite words (  -words) over    - set of infinite words (  -words) over  u, v, w – finite wordsu, v, w – finite words

15. Notation  := {0, 1, 2, 3, … }  := {0, 1, 2, 3, … }  - finite alphabet  - finite alphabet  * - set of finite words over   * - set of finite words over    - set of infinite words (  -words) over    - set of infinite words (  -words) over  u, v, w – finite wordsu, v, w – finite words , ,  - infinite words , ,  - infinite words

16. Notation  := {0, 1, 2, 3, … }  := {0, 1, 2, 3, … }  - finite alphabet  - finite alphabet  * - set of finite words over   * - set of finite words over    - set of infinite words (  -words) over    - set of infinite words (  -words) over  u, v, w – finite wordsu, v, w – finite words , ,  - infinite words , ,  - infinite words  =  (0)  (1)  (2) … with  (i)   =  (0)  (1)  (2) … with  (i) 

17. Notation  := {0, 1, 2, 3, … }  := {0, 1, 2, 3, … }  - finite alphabet  - finite alphabet  * - set of finite words over   * - set of finite words over    - set of infinite words (  -words) over    - set of infinite words (  -words) over  u, v, w – finite wordsu, v, w – finite words , ,  - infinite words , ,  - infinite words  =  (0)  (1)  (2) … with  (i)   =  (0)  (1)  (2) … with  (i)  ,  - runs of automata ,  - runs of automata

18. Notation  := {0, 1, 2, 3, … }  := {0, 1, 2, 3, … }  - finite alphabet  - finite alphabet  * - set of finite words over   * - set of finite words over    - set of infinite words (  -words) over    - set of infinite words (  -words) over  u, v, w – finite wordsu, v, w – finite words , ,  - infinite words , ,  - infinite words  =  (0)  (1)  (2) … with  (i)   =  (0)  (1)  (2) … with  (i)  ,  - runs of automata ,  - runs of automata  -language – set of  -words  -language – set of  -words

19. Notation(cont.)

20. Notation(cont.) |  | a – number of occurrences of a in |  | a – number of occurrences of a in 

21. Notation(cont.) Occ(  ) := {a  i  (i)=a}Occ(  ) := {a  i  (i)=a}

22. Notation(cont.) |  | a – number of occurrences of a in |  | a – number of occurrences of a in  Occ(  ) := {a  i  (i)=a}Occ(  ) := {a  i  (i)=a} Inf (  ) := {a  i  j>i  (j)=a}Inf (  ) := {a  i  j>i  (j)=a}

23. Notation(cont.) |  | a – number of occurrences of a in |  | a – number of occurrences of a in  Occ(  ) := {a  i  (i)=a}Occ(  ) := {a  i  (i)=a} Inf (  ) := {a  i  j>i  (j)=a}Inf (  ) := {a  i  j>i  (j)=a} 2 M – powerset of a set M2 M – powerset of a set M

24. Notation(cont.) |  | a – number of occurrences of a in |  | a – number of occurrences of a in  Occ(  ) := {a  i  (i)=a}Occ(  ) := {a  i  (i)=a} Inf (  ) := {a  i  j>i  (j)=a}Inf (  ) := {a  i  j>i  (j)=a} 2 M – powerset of a set M2 M – powerset of a set M REG – class of regular languagesREG – class of regular languages

25. Notation(cont.) |  | a – number of occurrences of a in |  | a – number of occurrences of a in  Occ(  ) := {a  i  (i)=a}Occ(  ) := {a  i  (i)=a} Inf (  ) := {a  i  j>i  (j)=a}Inf (  ) := {a  i  j>i  (j)=a} 2 M – powerset of a set M2 M – powerset of a set M REG – class of regular languagesREG – class of regular languages L(A) := {  *  A accepts  } -  -languageL(A) := {  *  A accepts  } -  -language recognized by A

26.  -Automata  -Automaton is (Q, , , q I, Acc)

27.  -Automata  -Automaton is (Q, , , q I, Acc) Q – finite set of statesQ – finite set of states

28.  -Automata  -Automaton is (Q, , , q I, Acc) Q – finite set of statesQ – finite set of states  - finite alphabet  - finite alphabet

29.  -Automata  -Automaton is (Q, , , q I, Acc) Q – finite set of statesQ – finite set of states  - finite alphabet  - finite alphabet  : Q    2 Q /Q – state transition function  : Q    2 Q /Q – state transition function

30.  -Automata  -Automaton is (Q, , , q I, Acc) Q – finite set of statesQ – finite set of states  - finite alphabet  - finite alphabet  : Q    2 Q /Q – state transition function  : Q    2 Q /Q – state transition function q I  Q – initial stateq I  Q – initial state

31.  -Automata  -Automaton is (Q, , , q I, Acc) Q – finite set of statesQ – finite set of states  - finite alphabet  - finite alphabet  : Q    2 Q /Q – state transition function  : Q    2 Q /Q – state transition function q I  Q – initial stateq I  Q – initial state Acc – acceptance componentAcc – acceptance component

32.  -Automata  -Automaton is (Q, , , q I, Acc) Q – finite set of statesQ – finite set of states  - finite alphabet  - finite alphabet  : Q    2 Q /Q – state transition function  : Q    2 Q /Q – state transition function q I  Q – initial stateq I  Q – initial state Acc – acceptance componentAcc – acceptance component can be given in different way!!!

33.  -Automata  -Automaton is (Q, , , q I, Acc) Q – finite set of statesQ – finite set of states  - finite alphabet  - finite alphabet  : Q    2 Q /Q – state transition function  : Q    2 Q /Q – state transition function q I  Q – initial stateq I  Q – initial state Acc – acceptance componentAcc – acceptance component can be given in different way!!! |A| = |Q| - size of automaton Acc sometimes used too

34. B ü chi acceptance

35.  -Automaton (Q, , , q I, F  Q) is B ü chi if  -Automaton (Q, , , q I, F  Q) is B ü chi if Acc is B ü chi acceptance:

36. B ü chi acceptance  -Automaton (Q, , , q I, F  Q) is B ü chi if  -Automaton (Q, , , q I, F  Q) is B ü chi if Acc is B ü chi acceptance: A word  * is accepted by A iff there exists a run  of A on  satisfying the condition: Inf(  )  F  

37. Example 1 L := {  {a, b}  |  ends with a  or with (ab)  }

38. B ü chi acceptance(cont.)  is accepted by A iff some run of A on  visit some final state q  F infinitely often, i.e.  W(q 0, q)  W(q, q) 

39. B ü chi acceptance(cont.)  is accepted by A iff some run of A on  visit some final state q  F infinitely often, i.e.  W(q 0, q)  W(q, q)  The B ü chi recognizable  -languages are the  -languages of the form: L=  k i=1 U i V i  with k  and U i, V i  REG for i=1, 2, 3, …

40. B ü chi acceptance(cont.) The family of  -languages is also called the  -Kleene closure of the class of regular languages denoted  -KC(REG) The family of  -languages is also called the  -Kleene closure of the class of regular languages denoted  -KC(REG)

41. Muller acceptance

42.  -Automaton (Q, , , q I, F  2 Q ) is Muller if  -Automaton (Q, , , q I, F  2 Q ) is Muller if Acc is Muller acceptance:

43. Muller acceptance  -Automaton (Q, , , q I, F  2 Q ) is Muller if  -Automaton (Q, , , q I, F  2 Q ) is Muller if Acc is Muller acceptance: A word  * is accepted by A iff there exists a run  of A on  satisfying the condition: Inf(  )  F

44. Example 2 L := {  {a, b}  |  ends with a  or with (ab)  } F = { {q a }, {q a,q b } }

45. B ü chi and Muller automata Nondeterministic B ü chi automata and nondeterministic Muller automata are equivalent in expressive power

46. B ü chi and Muller automata Nondeterministic B ü chi automata and nondeterministic Muller automata are equivalent in expressive power One direction is simple: F := { K  Q | K  F   }

47. B ü chi and Muller automata Nondeterministic B ü chi automata and nondeterministic Muller automata are equivalent in expressive power One direction is simple: F := { K  Q | K  F   } Second is complex and multiples states number exponentially

48. Rabin acceptance

49.  -Automaton (Q, , , q I,  ),  -Automaton (Q, , , q I,  ),  = {(E 1, F 1 ), …,(E k, F k )} with E i, F i  Q is Rabin if Acc is Rabin acceptance:

50. Rabin acceptance  -Automaton (Q, , , q I,  ),  -Automaton (Q, , , q I,  ),  = {(E 1, F 1 ), …,(E k, F k )} with E i, F i  Q is Rabin if Acc is Rabin acceptance: A word  * is accepted by A iff there exists a run  of A on  satisfying the condition:  (E,F) . (Inf(  )  E =  )  (Inf(  )  F   )

51. Example 3 L – words that consist of infinitely many a ’ s but only finitely many b ’ s  = { ({q b }, {q a }) }

52. Example 4 L – words that contain infinitely many b ’ s only if they also contain infinitely many a ’ s  = { ( , {q a }) }

53. Streett acceptance

54.  -Automaton (Q, , , q I,  ),  -Automaton (Q, , , q I,  ),  = {(E 1, F 1 ), …,(E k, F k )} with E i, F i  Q is Streett if Acc is Streett acceptance:

55. Streett acceptance  -Automaton (Q, , , q I,  ),  -Automaton (Q, , , q I,  ),  = {(E 1, F 1 ), …,(E k, F k )} with E i, F i  Q is Streett if Acc is Streett acceptance: A word  * is accepted by A iff there exists a run  of A on  satisfying the condition:  (E,F) . (Inf(  )  E   )  (Inf(  )  F =  )

56. Example 5 L – words that contain infinitely many b ’ s only if they also contain infinitely many a ’ s  = { ({q a }, {q b }) }

57. Transformation Rabin or Streett automaton to Muller automaton Let A = (Q, , , q I,  ) be a Rabin/Streett automaton.

58. Transformation Rabin or Streett automaton to Muller automaton Let A = (Q, , , q I,  ) be a Rabin/Streett automaton. Define A ’ = (Q, , , q I, F) with F = {G  2 Q |  (E,F) . G  E =   G  F   } F = {G  2 Q |  (E,F) . G  E    G  F =  }

59. Transformation Rabin or Streett automaton to Muller automaton Let A = (Q, , , q I,  ) be a Rabin/Streett automaton. Define A ’ = (Q, , , q I, F) with F = {G  2 Q |  (E,F) . G  E =   G  F   } F = {G  2 Q |  (E,F) . G  E    G  F =  } Then L(A) = L(A ’ )

60. Transformation B ü chi automaton to Rabin or Streett automaton Let A = (Q, , , q I, F  Q) is B ü chi automaton.

61. Transformation B ü chi automaton to Rabin or Streett automaton Let A = (Q, , , q I, F  Q) is B ü chi automaton. Define A ’ = (Q, , , q I,  ) with  = { ( , F) }  = { (F, Q) }

62. Transformation B ü chi automaton to Rabin or Streett automaton Let A = (Q, , , q I, F  Q) is B ü chi automaton. Define A ’ = (Q, , , q I,  ) with  = { ( , F) }  = { (F, Q) } Then A ’ is Rabin/Streett automaton that L(A) = L(A ’ )

63. Parity condition Parity condition amounts to the Rabin condition for the special case: E 1  F 1  E 2  … E m  F m

64. Parity condition Parity condition amounts to the Rabin condition for the special case: E 1  F 1  E 2  … E m  F m State of E 1 receive color(index) 1, State F i \ E i have color 2i, State E i \ F i-1 have color 2i-1

65. Parity condition  -Automaton (Q, , , q I, c),  -Automaton (Q, , , q I, c), c : Q  { 1, …, k}, k   is parity if Acc is parity acceptance:

66. Parity condition  -Automaton (Q, , , q I, c),  -Automaton (Q, , , q I, c), c : Q  { 1, …, k}, k   is parity if Acc is parity acceptance: A word  * is accepted by A iff there exists a run  of A on  satisfying the condition: Min{c(q) | q  Inf(  )} is even

67. interim conclusion Nondeterministic B ü chi automata, Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive power, i.e. they recognize the same  -languageNondeterministic B ü chi automata, Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive power, i.e. they recognize the same  -language

68. interim conclusion Nondeterministic B ü chi automata, Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive power, i.e. they recognize the same  -languageNondeterministic B ü chi automata, Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive power, i.e. they recognize the same  -language The  -language recognized by these  -automata from class  -KC(REG), i.e. the  -Kleene closure of the class of regular languagesThe  -language recognized by these  -automata from class  -KC(REG), i.e. the  -Kleene closure of the class of regular languages

69. Deterministic models

70. Deterministic Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive powerDeterministic Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive power

71. Deterministic models Deterministic Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive powerDeterministic Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive power They all recognize the regular   -languagesThey all recognize the regular   -languages

72. Deterministic models Deterministic Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive powerDeterministic Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive power They all recognize the regular   -languagesThey all recognize the regular   -languages B ü chi deterministic automata is too weak …B ü chi deterministic automata is too weak …

73. B ü chi deterministic automata is too weak … L – words that consist of infinitely many a ’ s but only finitely many b ’ s

74. B ü chi deterministic automata is too weak … L – words that consist of infinitely many a ’ s but only finitely many b ’ s F = { {q a } } – Muller automata

75. Transformation Muller automation to Rabin automation

76. Let A = (Q, , , q I, F) be a deterministic Muller automation. Assume w.l.o.g. that Q={1, …, k} and q I =1. Let  Q. Define A ’ as following:

77. Transformation Muller automation to Rabin automation Let A = (Q, , , q I, F) be a deterministic Muller automation. Assume w.l.o.g. that Q={1, …, k} and q I =1. Let  Q. Define A ’ as following: - Q ’ := { w  (Q  {  })* |  q  Q  {  }. |w| q = 1}

78. Transformation Muller automation to Rabin automation Let A = (Q, , , q I, F) be a deterministic Muller automation. Assume w.l.o.g. that Q={1, …, k} and q I =1. Let  Q. Define A ’ as following: - Q ’ := { w  (Q  {  })* |  q  Q  {  }. |w| q = 1} - q I ‘ :=  k … 1

79. Transformation Muller automation to Rabin automation Let A = (Q, , , q I, F) be a deterministic Muller automation. Assume w.l.o.g. that Q={1, …, k} and q I =1. Let  Q. Define A ’ as following: - Q ’ := { w  (Q  {  })* |  q  Q  {  }. |w| q = 1} - q I ‘ :=  k … 1 - for i, i ’  Q, a , and  (i, a)=i ’ for any word m 1 … m r  m r+1 … m k  Q with m k =i, i ’ =m s :  ’ (m 1 … m r  m r+1 … m k,a)= (m 1 … m s-1  m s+1 … m k i ’ )  ’ (m 1 … m r  m r+1 … m k,a)= (m 1 … m s-1  m s+1 … m k i ’ )

80. Transformation Muller automation to Rabin automation -  = {(E 1, F 1 ), …, (E k, F k )} define as following:

81. Transformation Muller automation to Rabin automation -  = {(E 1, F 1 ), …, (E k, F k )} define as following: - E j := {u  v | |u| < j}

82. Transformation Muller automation to Rabin automation -  = {(E 1, F 1 ), …, (E k, F k )} define as following: - E j := {u  v | |u| < j} - F j := {u  v | |u| < j}  {u  v | |u|=j  {m  Q | m  v}  F} where m  v means “ m occurs in v ”

83. Transformation Muller automation to Rabin automation -  = {(E 1, F 1 ), …, (E k, F k )} define as following: - E j := {u  v | |u| < j} - F j := {u  v | |u| < j}  {u  v | |u|=j  {m  Q | m  v}  F} where m  v means “ m occurs in v ” Then L(A) = L(A ’ ) …

84. Transformation Muller automation to parity automation From definition we have: E 1  F 1  E 2 …  E k  F k E 1  F 1  E 2 …  E k  F k Delete all pair where E j = F j and left strictly increasing chain of sets Thus have defined a parity automaton recognize same L(A)

85. Transformation Muller automation to Rabin automation By transformation a deterministic Muller automation with n states is transformed into a deterministic Rabin automata with n · n! states and n accepting pairs It works analogously for nondeterministic automata

86. Complement of L(A) by Muller automata Let A = (Q, , , q I, F) be a deterministic Muller automata.

87. Complement of L(A) by Muller automata Let A = (Q, , , q I, F) be a deterministic Muller automata. Define A ’ = (Q, , , q I, 2 Q \ F) Muller automata

88. Complement of L(A) by Muller automata Let A = (Q, , , q I, F) be a deterministic Muller automata. Define A ’ = (Q, , , q I, 2 Q \ F) Muller automata Then L(A ’ ) is complement of L(A)

89. Complement of L(A) by Rabin/Streett automata Let A = (Q, , , q I,  )

90. Complement of L(A) by Rabin/Streett automata Let A = (Q, , , q I,  ) The Rabin condition (*) is:  (E,F) . (Inf(  )  E =  )  (Inf(  )  F   )

91. Complement of L(A) by Rabin/Streett automata Let A = (Q, , , q I,  ) The Rabin condition (*) is:  (E,F) . (Inf(  )  E =  )  (Inf(  )  F   ) The Streett condition (**) is:  (E,F) . (Inf(  )  E   )  (Inf(  )  F =  )

92. Complement of L(A) by Rabin/Streett automata Let A = (Q, , , q I,  ) The Rabin condition (*) is:  (E,F) . (Inf(  )  E =  )  (Inf(  )  F   ) The Streett condition (**) is:  (E,F) . (Inf(  )  E   )  (Inf(  )  F =  ) Then L(A, (*)) is complement of L(A, (**))

93. Complement L(A) by parity automaton Let A = ( Q, , , q I, c) be a deterministic parity automaton

94. Complement L(A) by parity automaton Let A = ( Q, , , q I, c) be a deterministic parity automaton Define A ’ = ( Q, , , q I, c ’ ) with c ’ (q) = c(q)+1

95. Complement L(A) by parity automaton Let A = ( Q, , , q I, c) be a deterministic parity automaton Define A ’ = ( Q, , , q I, c ’ ) with c ’ (q) = c(q)+1 Then L(A ’ ) is complement of L(A)

96. interim conclusion Deterministic Muller automata, Rabin automata, Streett automata, and parity automata recognize same  -languages, and the class of these  -languages is closed under complementation

97. Some lower bound for transformations

98. Lemma 1:Lemma 1: Let A = (Q, , , q I,  ) be Robin automaton, and assume  1,  2 are two non-accepting runs. Then any run  with Inf(  ) = Inf(  1 )  Inf(  2 ) is also non-accepting

99. Some lower bound for transformations Lemma 1:Lemma 1: Let A = (Q, , , q I,  ) be Robin automaton, and assume  1,  2 are two non-accepting runs. Then any run  with Inf(  ) = Inf(  1 )  Inf(  2 ) is also non-accepting Lemma 2:Lemma 2: Let A = (Q, , , q I,  ) be a Streett automata, and assume  1,  2 are two accepting runs. Let A = (Q, , , q I,  ) be a Streett automata, and assume  1,  2 are two accepting runs. Then any run  with Inf(  ) = Inf(  1 )  Inf(  2 ) is also accepting

100. Some lower bound for transformations Let A  (A n ) n>1 defined over  ={1, …,n,#}

101. Some lower bound for transformations Let A  (A n ) n>1 defined over  ={1, …,n,#}  L n =L(A) – exist k and j 1, …,j k  {1, …,n} such that each pair j t j t+1 for t<k and j k j 1 appears infinitely often in 

102. Some lower bound for transformations We encode the symbols 1, …,n by words over {0, 1}* such that i is encoded by: 0 i 1, if i<n 0 i 0*1, if i=n

103. Some lower bound for transformations We encode the symbols 1, …,n by words over {0, 1}* such that i is encoded by: 0 i 1, if i<n 0 i 0*1, if i=n Lemma 3: There exist a family of languages (L n ) n>1 over the  = {0, 1, #} recognizable by nondeterministic B ü chi automata of size O(n) such that any nondeterministic Streett automaton accepting the complement language of L n has at least n! states

104. Some lower bound for transformations From lemma 3 we conclude: Lemma 4: There exist a family of languages (L n ) n>1 over the  = {0, 1, #} recognizable by nondeterministic B ü chi automata of size O(n) such that any equivalent deterministic Rabin automata must be of size n! or larger There exist a family of languages (L n ) n>1 over the  = {0, 1, #} recognizable by nondeterministic B ü chi automata of size O(n) such that any equivalent deterministic Rabin automata must be of size n! or larger

105. Some lower bound for transformations Lemma 5(with no proof): There exist a family of languages (L n ) n>1 over the  = {0, 1} recognizable by deterministic Streett automata with O(n) states and O(n) pairs of designated state sets such that any equivalent deterministic Rabin automata must be of size n! or larger There exist a family of languages (L n ) n>1 over the  = {0, 1} recognizable by deterministic Streett automata with O(n) states and O(n) pairs of designated state sets such that any equivalent deterministic Rabin automata must be of size n! or larger

106. Weak acceptance conditions

107.  -Automaton (Q, , , q I, F  2 Q ) is weak if  -Automaton (Q, , , q I, F  2 Q ) is weak if Acc is Staiger-Wagner acceptance:

108. Weak acceptance conditions  -Automaton (Q, , , q I, F  2 Q ) is weak if  -Automaton (Q, , , q I, F  2 Q ) is weak if Acc is Staiger-Wagner acceptance: A word  * is accepted by A iff there exists a run  of A on  satisfying the condition: Occ(  )  F

109. Weak acceptance conditions There are two special cases used:

110. Weak acceptance conditions There are two special cases used: Occ(  )  F   - 1-acceptanceOcc(  )  F   - 1-acceptance F = {X  2 Q | X  F   }

111. Weak acceptance conditions There are two special cases used: Occ(  )  F   - 1-acceptanceOcc(  )  F   - 1-acceptance F = {X  2 Q | X  F   } Occ(  )  F- 1 ’ -acceptanceOcc(  )  F- 1 ’ -acceptance F = {X  2 Q | X  F} F = {X  2 Q | X  F}

112. Weak acceptance conditions Acceptance by occurrence set can be simulated by B ü chi acceptanceAcceptance by occurrence set can be simulated by B ü chi acceptance

113. Weak acceptance conditions Acceptance by occurrence set can be simulated by B ü chi acceptanceAcceptance by occurrence set can be simulated by B ü chi acceptance The transformation need exponential blow-up. It can be avoided if only 1-acceptance or 1 ’ -acceptance are involvedThe transformation need exponential blow-up. It can be avoided if only 1-acceptance or 1 ’ -acceptance are involved

114. Weak acceptance conditions Acceptance by occurrence set can be simulated by B ü chi acceptanceAcceptance by occurrence set can be simulated by B ü chi acceptance The transformation need exponential blow-up. It can be avoided if only 1-acceptance or 1 ’ -acceptance are involvedThe transformation need exponential blow-up. It can be avoided if only 1-acceptance or 1 ’ -acceptance are involved The reverse transformation are not possibleThe reverse transformation are not possible

115. Conclusion We have shown the expressive equivalence of:

116. Conclusion Nondeterministic B ü chi, Muller, Rabin, Streett, and parity automataNondeterministic B ü chi, Muller, Rabin, Streett, and parity automata

117. Conclusion We have shown the expressive equivalence of: Nondeterministic B ü chi, Muller, Rabin, Streett, and parity automataNondeterministic B ü chi, Muller, Rabin, Streett, and parity automata Deterministic Muller, Rabin, Streett, and parity automataDeterministic Muller, Rabin, Streett, and parity automata

118. Conclusion Theorem(with no proof):Theorem(with no proof): Nondeterministic B ü chi automata accept the same  -languages as deterministic Muller automata

119. Conclusion Theorem(with no proof):Theorem(with no proof): Nondeterministic B ü chi automata accept the same  -languages as deterministic Muller automata Conclusion: all these automata are equivalent in expressive powerConclusion: all these automata are equivalent in expressive power