1 Generalized Buchi automaton

2 Reminder: Buchi automata A=  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 A B B S0 S1

3 Generalized Buchi automata A=  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 A A B B S0 S1 F 1 = {S0} F 2 = {S0,S1}

4 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... A A B B S0 S1 F 1 = {S0} F 2 = {S0,S1}

5 De-generalization of GBA Turn a generalized Büchi automaton into a Büchi automaton The idea: Each cycle must go through every copy Each cycle must contain accepting states from each accepting set

6 De-generalization of GBA Algorithm: Duplicate the GBA to as many copies as the number of accepting sets Redirect outgoing edges from accepting states to the next copy

7 Example S2 1 1,2 2 1,2 correspond to F 1 and F 2, the accepting sets S0 S1 S3 a b c    What is the language of A ?

8 Example S0 S1S2 S3 S1 S3 S0' S1'S2' S3' S2' S3' Two copies, because we have two accepting sets. a b c   a b c   

9 Example S0 S1S2 S3 S1 S3 S0' S1'S2' S3' S2' S3' Choose (arbitrarily) one copy as the initial one a b c   a b c   

10 Example S0 S1S2 S3 S1 S3 S0' S1'S2' S3' S2' S3' Redirect outgoing edges from accepting states. a b c   a b c    

11 Example S0 S1S2 S3 S1 S3 S0' S1'S2' S3' S2' S3' Only one copy is accepting a b c   a b c    

12 Example S0 S1S2 S3 S1 S3 S3' Remove unreachable states a b c    

13 Example S0 S1S2 S3' S1 S3 And here is our beautiful Buchi automaton a b c    What is the language of A’ ? S3

14 Another example... b b a c c A generalized Buchi automaton

15 b c c b a b c c b a And now... degeneralization One copy for each accepting set in F

16 b c c b a c c b a And now... de-generalization Redirect outgoing edges from accepting states, to next copy b

17 a b c c b a b c c b and so forth... And now... de-generalization

18 b a b c c b a b c c Remove accepting states from all copies but one Remove initial states from all copies but one Remove unreachable states

19 a b c c b a b c (a small optimization: collapsed states that cannot be distinguished)