?-Automata

1
?-Automata

• Ekaterina Mineev

2
Today

• 1 Introduction
• - notation
• - ?-Automata overview

3
Today

• 1 Introduction
• - notation
• - ?-Automata overview
• 2 Nondeterministic models
• - Büchi acceptance
• - Muller acceptance
• - Rabin acceptance
• - Streett acceptance
• - parity condition

4
Today(cont.)

• 2.1 Equivalency of nondeterministic models

5
Today(cont.)

• 2.1 Equivalency of nondeterministic models
• 3 Deterministic models
• - Büchi condition
• - equivalency of deterministic models

6
Today(cont.)

• 2.1 Equivalency of nondeterministic models
• 3 Deterministic models
• - Büchi condition
• - equivalency of deterministic models
• 4 Some lower bound for transformations

7
Today(cont.)

• 2.1 Equivalency of nondeterministic models
• 3 Deterministic models
• - Büchi 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
• - Büchi 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,

11
Notation

• ? 0, 1, 2, 3,
• ? - finite alphabet

12
Notation

• ? 0, 1, 2, 3,
• ? - finite alphabet
• ? - set of finite words over ?

13
Notation

• ? 0, 1, 2, 3,
• ? - finite alphabet
• ? - set of finite words over ?
• ?? - set of infinite words (?-words) over ?

14
Notation

• ? 0, 1, 2, 3,
• ? - finite alphabet
• ? - set of finite words over ?
• ?? - set of infinite words (?-words) over ?
• u, v, w finite words

15
Notation

• ? 0, 1, 2, 3,
• ? - finite alphabet
• ? - set of finite words over ?
• ?? - set of infinite words (?-words) over ?
• u, v, w finite words
• ?, ?, ? - infinite words

16
Notation

• ? 0, 1, 2, 3,
• ? - finite alphabet
• ? - set of finite words over ?
• ?? - set of infinite words (?-words) over ?
• u, v, w finite words
• ?, ?, ? - infinite words
• ? ?(0)?(1)?(2) with ?(i)??

17
Notation

• ? 0, 1, 2, 3,
• ? - finite alphabet
• ? - set of finite words over ?
• ?? - set of infinite words (?-words) over ?
• u, v, w finite words
• ?, ?, ? - infinite words
• ? ?(0)?(1)?(2) with ?(i)??
• ?, ? - runs of automata

18
Notation

• ? 0, 1, 2, 3,
• ? - finite alphabet
• ? - set of finite words over ?
• ?? - set of infinite words (?-words) over ?
• u, v, w finite words
• ?, ?, ? - infinite words
• ? ?(0)?(1)?(2) with ?(i)??
• ?, ? - runs of automata
• ?-language set of ?-words

19
Notation(cont.)
20
Notation(cont.)

• ?a number of occurrences of a in ?

21
Notation(cont.)

• ?a number of occurrences of a in ?
• Occ(?) a????i ?(i)a

22
Notation(cont.)

• ?a number of occurrences of a in ?
• Occ(?) a????i ?(i)a
• Inf (?) a????i ?jgti ?(j)a

23
Notation(cont.)

• ?a number of occurrences of a in ?
• Occ(?) a????i ?(i)a
• Inf (?) a????i ?jgti ?(j)a
• 2M powerset of a set M

24
Notation(cont.)

• ?a number of occurrences of a in ?
• Occ(?) a????i ?(i)a
• Inf (?) a????i ?jgti ?(j)a
• 2M powerset of a set M
• REG class of regular languages

25
Notation(cont.)

• ?a number of occurrences of a in ?
• Occ(?) a????i ?(i)a
• Inf (?) a????i ?jgti ?(j)a
• 2M powerset of a set M
• REG class of regular languages
• L(A) ????A accepts ? - ?-language
• recognized by A

26
?-Automata

• ?-Automaton is (Q, ?, ?, qI, Acc)

27
?-Automata

• ?-Automaton is (Q, ?, ?, qI, Acc)
• Q finite set of states

28
?-Automata

• ?-Automaton is (Q, ?, ?, qI, Acc)
• Q finite set of states
• ? - finite alphabet

29
?-Automata

• ?-Automaton is (Q, ?, ?, qI, Acc)
• Q finite set of states
• ? - finite alphabet
• ? Q?? ? 2Q/Q state transition function

30
?-Automata

• ?-Automaton is (Q, ?, ?, qI, Acc)
• Q finite set of states
• ? - finite alphabet
• ? Q?? ? 2Q/Q state transition function
• qI?Q initial state

31
?-Automata

• ?-Automaton is (Q, ?, ?, qI, Acc)
• Q finite set of states
• ? - finite alphabet
• ? Q?? ? 2Q/Q state transition function
• qI?Q initial state
• Acc acceptance component

32
?-Automata

• ?-Automaton is (Q, ?, ?, qI, Acc)
• Q finite set of states
• ? - finite alphabet
• ? Q?? ? 2Q/Q state transition function
• qI?Q initial state
• Acc acceptance component
• can be given in different way!!!

33
?-Automata

• ?-Automaton is (Q, ?, ?, qI, Acc)
• Q finite set of states
• ? - finite alphabet
• ? Q?? ? 2Q/Q state transition function
• qI?Q initial state
• Acc acceptance component
• can be given in different way!!!
• A Q - size of automaton
• Acc sometimes used too

34
Büchi acceptance
35
Büchi acceptance

• ?-Automaton (Q, ?, ?, qI, F?Q) is Büchi if
• Acc is Büchi acceptance

36
Büchi acceptance

• ?-Automaton (Q, ?, ?, qI, F?Q) is Büchi if
• Acc is Büchi 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
Büchi acceptance(cont.)

• is accepted by A iff some run of A on ? visit
  some final state q?F infinitely often, i.e.
  ??W(q0, q)?W(q, q)?

39
Büchi acceptance(cont.)

• is accepted by A iff some run of A on ? visit
  some final state q?F infinitely often, i.e.
  ??W(q0, q)?W(q, q)?
• The Büchi recognizable ?-languages are the
  ?-languages of the form
• L?ki1 UiVi? with k??
• and Ui , Vi ? REG for i1, 2, 3,

40
Büchi acceptance(cont.)

• The family of ?-languages is also called the
  ?-Kleene closure of the class of regular
  languages denoted ?-KC(REG)

41
Muller acceptance
42
Muller acceptance

• ?-Automaton (Q, ?, ?, qI, F? 2Q) is Muller if
• Acc is Muller acceptance

43
Muller acceptance

• ?-Automaton (Q, ?, ?, qI, F? 2Q) 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 qa, qa,qb

45
Büchi and Muller automata

• Nondeterministic Büchi automata and
  nondeterministic Muller automata are equivalent
  in expressive power

46
Büchi and Muller automata

• Nondeterministic Büchi automata and
  nondeterministic Muller automata are equivalent
  in expressive power
• One direction is simple
• F K?Q K?F ? ?

47
Büchi and Muller automata

• Nondeterministic Büchi 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
Rabin acceptance

• ?-Automaton (Q, ?, ?, qI, ?),
• ? (E1, F1),,(Ek, Fk) with Ei, Fi ? Q
• is Rabin if Acc is Rabin acceptance

50
Rabin acceptance

• ?-Automaton (Q, ?, ?, qI, ?),
• ? (E1, F1),,(Ek, Fk) with Ei, Fi ? 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 as but
  only finitely many bs
• ? (qb, qa)

52
Example 4

• L words that contain infinitely many bs only
  if they also contain infinitely many as
• ? (?, qa)

53
Streett acceptance
54
Streett acceptance

• ?-Automaton (Q, ?, ?, qI, ?),
• ? (E1, F1),,(Ek, Fk) with Ei, Fi ? Q
• is Streett if Acc is Streett acceptance

55
Streett acceptance

• ?-Automaton (Q, ?, ?, qI, ?),
• ? (E1, F1),,(Ek, Fk) with Ei, Fi ? 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 bs only
  if they also contain infinitely many as
• ? (qa, qb)

57
Transformation Rabin or Streett automaton to
Muller automaton

• Let A (Q, ?, ?, qI, ?) be a Rabin/Streett
  automaton.

58
Transformation Rabin or Streett automaton to
Muller automaton

• Let A (Q, ?, ?, qI, ?) be a Rabin/Streett
  automaton.
• Define A (Q, ?, ?, qI, F) with
• F G? 2Q ?(E,F)?? . G?E ? ? G?F ? ?
• F G? 2Q ?(E,F)?? . G?E ? ? ? G?F ?

59
Transformation Rabin or Streett automaton to
Muller automaton

• Let A (Q, ?, ?, qI, ?) be a Rabin/Streett
  automaton.
• Define A (Q, ?, ?, qI, F) with
• F G? 2Q ?(E,F)?? . G?E ? ? G?F ? ?
• F G? 2Q ?(E,F)?? . G?E ? ? ? G?F ?
• Then L(A) L(A)

60
Transformation Büchi automaton to Rabin or
Streett automaton

• Let A (Q, ?, ?, qI, F?Q) is Büchi automaton.

61
Transformation Büchi automaton to Rabin or
Streett automaton

• Let A (Q, ?, ?, qI, F?Q) is Büchi automaton.
• Define A (Q, ?, ?, qI, ?) with
• (?, F)
• (F, Q)

62
Transformation Büchi automaton to Rabin or
Streett automaton

• Let A (Q, ?, ?, qI, F?Q) is Büchi automaton.
• Define A (Q, ?, ?, qI, ?) 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
• E1 ? F1 ? E2 ? Em ? Fm

64
Parity condition

• Parity condition amounts to the Rabin condition
  for the special case
• E1 ? F1 ? E2 ? Em ? Fm
• State of E1 receive color(index) 1,
• State Fi \ Ei have color 2i,
• State Ei \ Fi-1 have color 2i-1

65
Parity condition

• ?-Automaton (Q, ?, ?, qI, c),
• c Q ? 1, , k, k ? ?
• is parity if Acc is parity acceptance

66
Parity condition

• ?-Automaton (Q, ?, ?, qI, 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
• Minc(q) q ? Inf(?) is even

67
interim conclusion

• Nondeterministic Büchi 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 Büchi 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 languages

69
Deterministic models
70
Deterministic models

• Deterministic 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 power
• They all recognize the regular ?
  ?-languages

72
Deterministic models

• Deterministic Muller automata, Rabin automata,
  Streett automata, and parity automata are all
  equivalent in expressive power
• They all recognize the regular ?
  ?-languages
• Büchi deterministic automata is too weak

73
Büchi deterministic automata is too weak

• L words that consist of infinitely many as but
  only finitely many bs

74
Büchi deterministic automata is too weak

• L words that consist of infinitely many as but
  only finitely many bs
• F qa Muller automata

75
Transformation Muller automation to Rabin
automation
76
Transformation Muller automation to Rabin
automation

• Let A (Q, ?, ?, qI, F) be a deterministic
  Muller automation. Assume w.l.o.g. that Q1, ,
  k and qI1. Let ??Q. Define A as following

77
Transformation Muller automation to Rabin
automation

• Let A (Q, ?, ?, qI, F) be a deterministic
  Muller automation. Assume w.l.o.g. that Q1, ,
  k and qI1. Let ??Q. Define A as following
• - Q w?(Q??) ?q?Q?? . wq 1

78
Transformation Muller automation to Rabin
automation

• Let A (Q, ?, ?, qI, F) be a deterministic
  Muller automation. Assume w.l.o.g. that Q1, ,
  k and qI1. Let ??Q. Define A as following
• - Q w?(Q??) ?q?Q?? . wq 1
• - qI ?k1

79
Transformation Muller automation to Rabin
automation

• Let A (Q, ?, ?, qI, F) be a deterministic
  Muller automation. Assume w.l.o.g. that Q1, ,
  k and qI1. Let ??Q. Define A as following
• - Q w?(Q??) ?q?Q?? . wq 1
• - qI ?k1
• - for i, i?Q, a??, and ?(i, a)i for any word
  m1mr? mr1mk ? Q with mki , ims
• ?(m1mr? mr1mk,a) (m1ms-1?
  ms1mki)

80
Transformation Muller automation to Rabin
automation

• - ? (E1, F1), , (Ek, Fk)
• define as following

81
Transformation Muller automation to Rabin
automation

• - ? (E1, F1), , (Ek, Fk)
• define as following
• - Ej u?v u lt j

82
Transformation Muller automation to Rabin
automation

• - ? (E1, F1), , (Ek, Fk)
• define as following
• - Ej u?v u lt j
• - Fj u?v u lt j ?
• u?v uj ? m?Q m?v ? F
• where m?v means m occurs in v

83
Transformation Muller automation to Rabin
automation

• - ? (E1, F1), , (Ek, Fk)
• define as following
• - Ej u?v u lt j
• - Fj u?v u lt j ?
• u?v uj ? 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
• E1 ? F1 ? E2 ? Ek ? Fk
• Delete all pair where Ej Fj 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 nn! states and
  n accepting pairs
• It works analogously for nondeterministic automata

86
Complement of L(A) by Muller automata

• Let A (Q, ?, ?, qI, F) be a deterministic
  Muller automata.

87
Complement of L(A) by Muller automata

• Let A (Q, ?, ?, qI, F) be a deterministic
  Muller automata.
• Define A (Q, ?, ?, qI, 2Q \ F) Muller
  automata

88
Complement of L(A) by Muller automata

• Let A (Q, ?, ?, qI, F) be a deterministic
  Muller automata.
• Define A (Q, ?, ?, qI, 2Q \ F) Muller
  automata
• Then L(A) is complement of L(A)

89
Complement of L(A) by Rabin/Streett automata

• Let A (Q, ?, ?, qI, ?)

90
Complement of L(A) by Rabin/Streett automata

• Let A (Q, ?, ?, qI, ?)
• The Rabin condition () is
• ?(E,F)?? . (Inf(?)?E ?) ? (Inf(?)?F ? ?)

91
Complement of L(A) by Rabin/Streett automata

• Let A (Q, ?, ?, qI, ?)
• 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, ?, ?, qI, ?)
• 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, ?, ?, qI, c) be a deterministic
  parity automaton

94
Complement L(A) by parity automaton

• Let A (Q, ?, ?, qI, c) be a deterministic
  parity automaton
• Define A (Q, ?, ?, qI, c) with
• c(q) c(q)1

95
Complement L(A) by parity automaton

• Let A (Q, ?, ?, qI, c) be a deterministic
  parity automaton
• Define A (Q, ?, ?, qI, 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
Some lower bound for transformations

• Lemma 1
• Let A (Q, ?, ?, qI, ?) 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
• Let A (Q, ?, ?, qI, ?) 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
• Let A (Q, ?, ?, qI, ?) 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?(An)ngt1 defined over ?1,,n,

101
Some lower bound for transformations

• Let A?(An)ngt1 defined over ?1,,n,
• ??LnL(A) exist k and j1,,jk?1,,n such that
  each pair jtjt1 for tltk and jkj1 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
• 0i1, if iltn
• 0i01, if in

103
Some lower bound for transformations

• We encode the symbols 1,,n by words over 0, 1
  such that i is encoded by
• 0i1, if iltn
• 0i01, if in
• Lemma 3
• There exist a family of languages (Ln)ngt1 over
  the ? 0, 1, recognizable by
  nondeterministic Büchi automata of size O(n) such
  that any nondeterministic Streett automaton
  accepting the complement language of Ln has at
  least n! states

104
Some lower bound for transformations

• From lemma 3 we conclude
• Lemma 4
• There exist a family of languages (Ln)ngt1 over
  the ? 0, 1, recognizable by
  nondeterministic Büchi 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 (Ln)ngt1 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
Weak acceptance conditions

• ?-Automaton (Q, ?, ?, qI, F? 2Q) is weak if
• Acc is Staiger-Wagner acceptance

108
Weak acceptance conditions

• ?-Automaton (Q, ?, ?, qI, F? 2Q) 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-acceptance
• F X ? 2Q X ? F ? ?

111
Weak acceptance conditions

• There are two special cases used
• Occ(?) ? F ? ? - 1-acceptance
• F X ? 2Q X ? F ? ?
• Occ(?) ? F - 1-acceptance
• F X ? 2Q X ? F

112
Weak acceptance conditions

• Acceptance by occurrence set can be simulated by
  Büchi acceptance

113
Weak acceptance conditions

• Acceptance by occurrence set can be simulated by
  Büchi acceptance
• The 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
  Büchi acceptance
• The transformation need exponential blow-up. It
  can be avoided if only 1-acceptance or
  1-acceptance are involved
• The reverse transformation are not possible

115
Conclusion

• We have shown the expressive equivalence of

116
Conclusion

• We have shown the expressive equivalence of
• Nondeterministic Büchi, Muller, Rabin, Streett,
  and parity automata

117
Conclusion

• We have shown the expressive equivalence of
• Nondeterministic Büchi, Muller, Rabin, Streett,
  and parity automata
• Deterministic Muller, Rabin, Streett, and parity
  automata

118
Conclusion

• Theorem(with no proof)
• Nondeterministic Büchi automata accept the same
  ?-languages as deterministic Muller automata

119
Conclusion

• Theorem(with no proof)
• Nondeterministic Büchi automata accept the same
  ?-languages as deterministic Muller automata
• Conclusion all these automata are equivalent in
  expressive power