Generalized Buchi automaton
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Reminder: Buchi automata


A=<  , S,  , I, F>




 : Alphabet (finite).
S : States (finite).
 : S x  x S ) S is the transition relation.
I µ S are the Initial states.

F µ S is a set of accepting states.
An infinite word is accepted in A if it passes an infinite no. of times in at least one of the F states
A
S0
A
B
S1
B
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Generalized Buchi automata


A=<  , S,  , I, F>




 : Alphabet (finite).
S : States (finite).
 : S x  x S ) S is the transition relation.
I µ S are the Initial states.

F µ 2 S is a set of sets of accepting states.
An infinite word is accepted in A if it passes an infinite no. of times in at least one state in each element of F
F
1
F
2
= {S0}
= {S0,S1}
A
S0
A
B
S1
B
3

Generalized Buchi automata




An infinite word is accepted in A if it passes an infinite no. of times in at least one state in each element of F
B
!
is ....
A
!
is ...
(AB)
!
is ...
F
1
F
2
= {S0}
= {S0,S1}
A
S0
A
B
S1
B
4

De-generalization of GBA




Each cycle must go through every copy
Each cycle must contain accepting states from each accepting set
5

De-generalization of GBA


Duplicate the GBA to as many copies as the number of accepting sets

Redirect outgoing edges from accepting states to the next copy
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Example
What is the language of A ?
S0 c a b
1
S1


S3
1,2

S2
2
1,2 correspond to F
1 and F
2
, the accepting sets
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Example
 a
S0 b c
 
S2

S1'
 a
S0' b c

Two copies, because we have two accepting sets.
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Example
 a
S0 b
 
S2 c

S1'
 a
S0' b c

Choose (arbitrarily) one copy as the initial one
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Example a
S0 b


S2
 c
S1'
 a
S0' b c


Redirect outgoing edges from accepting states.
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Example a
S0 b c


S2

S1'
 a
S0' b c


Only one copy is accepting
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Example a
S0 b c


S2

Remove unreachable states

S3'
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Example
What is the language of A’ ? S0
 a b
S2



S3' c
And here is our beautiful Buchi automaton
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a b b c c
A generalized Buchi automaton
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And now... degeneralization a b b c c a b b c c
One copy for each accepting set in F
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And now... de-generalization a b b c c a b b c c
Redirect outgoing edges from accepting states, to next copy
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And now... de-generalization a b b c c a b b c c and so forth...
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a b b c c a b b c c
Remove accepting states from all copies but one
Remove initial states from all copies but one
Remove unreachable states
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a b b c c a b c
(a small optimization: collapsed states that cannot be distinguished)
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