Generalized Buchi automaton 1 Reminder Buchi automata n

Generalized Buchi automaton 1

Reminder: Buchi automata n A=< , S, , I, F> : Alphabet (finite). n S: States (finite). n : S x x S ) S is the transition relation. n I µ S are the Initial states. n 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 n n A S 0 A B S 1 B 2

Generalized Buchi automata n n A=< , S, , I, F> n : Alphabet (finite). n S: States (finite). n : S x x S ) S is the transition relation. n I µ S are the Initial states. S n F µ 2 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 = {S 0} F 2 = {S 0, S 1} A S 0 A B S 1 B 3

Generalized Buchi automata n n 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 = {S 0} F 2 = {S 0, S 1} A S 0 A B S 1 B 4

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

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

Example S 0 What is the language of A ? a c b S 1 1 S 2 2 S 3 1, 2 correspond to F 1 and F 2, the accepting sets 7

Example S 0 a a b S 1 S 2 S 3 S 2' c b S 1' S 0' c S 3' Two copies, because we have two accepting sets. 8

Example S 0 a a b S 1 S 2 S 3 c b S 2' S 1' S 0' c S 3' Choose (arbitrarily) one copy as the initial one 9

Example S 0 a b S 1 S 2 S 3 S 0' c a c b S 2' S 1' S 3' Redirect outgoing edges from accepting states. 10

Example S 0 a S 0' c a b S 1 S 2 S 3 c b S 2' S 1' S 3' Only one copy is accepting 11

Example S 0 a c b S 1 S 2 S 3' Remove unreachable states 12

Example S 0 What is the language of A’ ? a c b S 1 S 2 S 3' And here is our beautiful Buchi automaton 13

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

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

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

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

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

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