Generalized Buchi automaton

1. Generalized Buchi automaton

2. A S0 S1 B A B 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

3. 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 2S 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 S0 S1 B A B F1 = {S0} F2 = {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 S0 S1 B A B F1 = {S0} F2 = {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 nextcopy

7. Example S0 c What is the language of A ? a b S2 S1 2 1   S3  1,2 1,2 correspond to F1 and F2, the accepting sets

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

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

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

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

12. Example S0 c a b S1 S2 S1   S3 S3' S3   Remove unreachable states

13. Example S0 c What is the language of A’ ? a b S1 S2 S1    S3 S3' S3  And here is our beautiful Buchi automaton

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

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

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

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

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

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