b uchi complementation via alternating automata
play

B uchi Complementation via Alternating Automata Fabian Reiter - PowerPoint PPT Presentation

Jan 16, 2023 •126 likes •1.59k views

Automata Theory Seminar B uchi Complementation via Alternating Automata Fabian Reiter July 16, 2012 B uchi Complementation BA B BA B BA: B uchi Automaton 1 / 33 B uchi Complementation 2 ( n log n ) BA B BA B BA: B uchi

  1. � Run Graph (2) Definition (Run) A run of a WAPA A = � Q , Σ , δ, q in , π � on a word a 0 a 1 a 2 . . . ∈ Σ ω is a directed acyclic graph R := � V , E � where V ⊆ Q × N with � q in , 0 � ∈ V V contains only vertices reachable from � q in , 0 � . 11 / 33

  2. � Run Graph (2) Definition (Run) A run of a WAPA A = � Q , Σ , δ, q in , π � on a word a 0 a 1 a 2 . . . ∈ Σ ω is a directed acyclic graph R := � V , E � where V ⊆ Q × N with � q in , 0 � ∈ V V contains only vertices reachable from � q in , 0 � . � � � p , i � , � q , i + 1 � E contains only edges of the form . 11 / 33

  3. � Run Graph (2) Definition (Run) A run of a WAPA A = � Q , Σ , δ, q in , π � on a word a 0 a 1 a 2 . . . ∈ Σ ω is a directed acyclic graph R := � V , E � where V ⊆ Q × N with � q in , 0 � ∈ V V contains only vertices reachable from � q in , 0 � . � � � p , i � , � q , i + 1 � E contains only edges of the form . For every vertex � p , i � ∈ V the set of successors is a minimal model of δ ( p , a i ) � � � � q ∈ Q | � p , i � , � q , i + 1 � ∈ E ∈ Mod ↓ ( δ ( p , a i )) 11 / 33

  4. Acceptance � Definition (Acceptance) Let A be a WAPA, w ∈ Σ ω and R = � V , E � a run of A on w . An infinite path ρ in R satisfies the acceptance condition of A iff the smallest occurring parity is even, i.e. min { π ( q ) | ∃ i ∈ N : � q , i � occurs in ρ } is even. 12 / 33

  5. Acceptance � Definition (Acceptance) Let A be a WAPA, w ∈ Σ ω and R = � V , E � a run of A on w . An infinite path ρ in R satisfies the acceptance condition of A iff the smallest occurring parity is even, i.e. min { π ( q ) | ∃ i ∈ N : � q , i � occurs in ρ } is even. R is an accepting run iff every infinite path ρ in R satisfies the acceptance condition. 12 / 33

  6. Acceptance � Definition (Acceptance) Let A be a WAPA, w ∈ Σ ω and R = � V , E � a run of A on w . An infinite path ρ in R satisfies the acceptance condition of A iff the smallest occurring parity is even, i.e. min { π ( q ) | ∃ i ∈ N : � q , i � occurs in ρ } is even. R is an accepting run iff every infinite path ρ in R satisfies the acceptance condition. A accepts w iff there is some accepting run of A on w . 12 / 33

  7. Acceptance � Example ( a ω ) a a a q 1 • • 1 q 0 • 2 a q 2 a 0 q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 · · · q 0 , 0 q 0 , 1 q 0 , 4 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 · · · 12 / 33

  8. Acceptance � Example ( a ω ) a a a q 1 • • 1 q 0 • 2 a q 2 a 0 Accepting run: q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 · · · q 0 , 0 q 0 , 1 q 0 , 4 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 · · · 12 / 33

  9. Acceptance � Example ( a ω ) a a a q 1 • • 1 q 0 • 2 a q 2 a 0 Accepting run: q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 · · · Rejecting run: q 0 , 0 q 0 , 1 q 0 , 4 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 · · · 12 / 33

  10. Acceptance Alternating : in some set of runs every run is accepting · · · q 0 q 1 a q 2 a q 3 a q 4 a q 5 a q 0 q 1 b q 2 b q 3 b q 4 b q 5 b · · · q 0 q 1 c q 2 c q 3 c q 4 c q 5 c · · · · · · q 0 q 1 d q 2 d q 3 d q 4 d q 5 d · · · q 0 q 1 e q 2 e q 3 e q 4 e q 5 e · · · q 0 q 1 f q 2 f q 3 f q 4 f q 5 f · · · q 0 q 1 g q 2 g q 3 g q 4 g q 5 g · · · q 0 q 1 h q 2 h q 3 h q 4 h q 5 h · · · q 0 q 1 i q 2 i q 3 i q 4 i q 5 i 12 / 33

  11. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  12. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  13. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  14. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  15. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  16. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 Run on ( ba ) ω : b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  17. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 Run on ( ba ) ω : b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  18. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 Run on ( ba ) ω : b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  19. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 Run on ( ba ) ω : b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  20. Infinitely many a ’s Example ( ( b ∗ a ) ω ) a , b a b b q 0 q 1 q 2 a • 2 1 0 Run on b ω : b b b b b b b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · · · · q 1 , 1 q 1 , 2 q 1 , 3 q 1 , 4 q 1 , 5 q 1 , 6 Run on ( ba ) ω : b a b a b a b q 0 , 0 q 0 , 1 q 0 , 2 q 0 , 3 q 0 , 4 q 0 , 5 q 0 , 6 · · · q 1 , 1 q 1 , 3 q 1 , 5 · · · q 2 , 2 q 2 , 3 q 2 , 4 q 2 , 5 q 2 , 6 · · · 13 / 33

  21. � Dual Automaton (1) Definition (Dual Automaton) The dual of a WAPA A = � Q , Σ , δ, q in , π � is A := � Q , Σ , δ, q in , π � 14 / 33

  22. � Dual Automaton (1) Definition (Dual Automaton) The dual of a WAPA A = � Q , Σ , δ, q in , π � is A := � Q , Σ , δ, q in , π � where δ ( q , a ) is obtained from δ ( q , a ) by exchanging ∧ , ∨ and ⊤ , ⊥ for all q ∈ Q and a ∈ Σ 14 / 33

  23. � Dual Automaton (1) Definition (Dual Automaton) The dual of a WAPA A = � Q , Σ , δ, q in , π � is A := � Q , Σ , δ, q in , π � where δ ( q , a ) is obtained from δ ( q , a ) by exchanging ∧ , ∨ and ⊤ , ⊥ π ( q ) := π ( q ) + 1 for all q ∈ Q and a ∈ Σ 14 / 33

  24. Dual Automaton (2) Example ( ( b ∗ a ) ω ) WAPA A : δ ( q 0 , a ) = q 0 δ ( q 0 , b ) = q 0 ∧ q 1 a , b a b δ ( q 1 , a ) = q 2 b δ ( q 1 , b ) = q 1 q 0 q 1 q 2 a • 2 1 0 δ ( q 2 , a ) = q 2 δ ( q 2 , b ) = q 2 15 / 33

  25. Dual Automaton (2) Example ( ( b ∗ a ) ω ) WAPA A : δ ( q 0 , a ) = q 0 δ ( q 0 , b ) = q 0 ∧ q 1 a , b a b δ ( q 1 , a ) = q 2 b δ ( q 1 , b ) = q 1 q 0 q 1 q 2 a • 2 1 0 δ ( q 2 , a ) = q 2 δ ( q 2 , b ) = q 2 Dual A : δ ( q 0 , a ) = q 0 δ ( q 0 , b ) = q 0 ∨ q 1 a , b a , b b δ ( q 1 , a ) = q 2 δ ( q 1 , b ) = q 1 q 0 b q 1 a q 2 3 2 1 δ ( q 2 , a ) = q 2 δ ( q 2 , b ) = q 2 15 / 33

  26. Complementation Theorem Main statement of this talk: Theorem (Complementation) The dual A of a WAPA A accepts its complement, i.e. L ( A ) = Σ ω \ L ( A ) (Thomas and L¨ oding, ∼ 2000) 16 / 33

  27. Outline Weak Alternating Parity Automata 1 Infinite Parity Games 2 Proof of the Complementation Theorem 3 B¨ uchi Complementation Algorithm 4 17 / 33

  28. c b a Automaton vs. Pathfinder 18 / 33

  29. Automaton vs. Pathfinder c b a player A 18 / 33

  30. Automaton vs. Pathfinder c b a player A find accepting run R 18 / 33

  31. Automaton vs. Pathfinder c b a player A player P find accepting run R 18 / 33

  32. Automaton vs. Pathfinder c b a player A player P find accepting run R find rejecting path in R 18 / 33

  33. � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 19 / 33

  34. � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 Game G A , w : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 19 / 33

  35. � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 Game G A , w : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 19 / 33

  36. � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 Game G A , w : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 19 / 33

  37. � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 Game G A , w : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 19 / 33

  38. � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 Game G A , w : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 19 / 33

  39. � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 Game G A , w : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 19 / 33

  40. � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 Game G A , w : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 19 / 33

  41. � Infinite Parity Game (1) Example ( a ω ) a a a q 1 • • 1 q 0 • w = a ω A : 2 a q 2 a 0 Game G A , w : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 19 / 33

  42. � Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � (Thomas and L¨ oding, ∼ 2000) 20 / 33

  43. � Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) (Thomas and L¨ oding, ∼ 2000) 20 / 33

  44. � Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) V P := 2 Q × N (decision nodes of player P ) (Thomas and L¨ oding, ∼ 2000) 20 / 33

  45. � Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) V P := 2 Q × N (decision nodes of player P ) E ⊆ ( V A × V P ) ∪ ( V P × V A ) (Thomas and L¨ oding, ∼ 2000) 20 / 33

  46. � Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) V P := 2 Q × N (decision nodes of player P ) E ⊆ ( V A × V P ) ∪ ( V P × V A ) s.t. the only contained edges are • � � � q , i � , � M , i � M ∈ Mod ↓ ( δ ( q , a i )) iff for q ∈ Q , M ⊆ Q , i ∈ N (Thomas and L¨ oding, ∼ 2000) 20 / 33

  47. � Infinite Parity Game (2) Definition (Game) A game for a WAPA A = � Q , Σ , δ, q in , π � and w = a 0 a 1 a 2 . . . ∈ Σ ω is a directed graph G A , w := � V A ˙ ∪ V P , E � where V A := Q × N (decision nodes of player A ) V P := 2 Q × N (decision nodes of player P ) E ⊆ ( V A × V P ) ∪ ( V P × V A ) s.t. the only contained edges are • � � � q , i � , � M , i � M ∈ Mod ↓ ( δ ( q , a i )) iff • � � � M , i � , � q , i + 1 � iff q ∈ M for q ∈ Q , M ⊆ Q , i ∈ N (Thomas and L¨ oding, ∼ 2000) 20 / 33

  48. Playing a Game � Definition (Play) A play γ in a game G A , w is an infinite path starting with � q in , 0 � . Example q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 21 / 33

  49. Playing a Game � Definition (Play) A play γ in a game G A , w is an infinite path starting with � q in , 0 � . Definition (Winner) The winner of a play γ is player A iff the smallest parity of occurring V A -nodes is even player P · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · odd Example q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · { q 1 , q 2 } , 0 · · · q 2 , 1 { q 2 } , 1 q 2 , 2 21 / 33

  50. Playing a Game � Definition (Play) A play γ in a game G A , w is an infinite path starting with � q in , 0 � . Definition (Winner) The winner of a play γ is player A iff the smallest parity of occurring V A -nodes is even player P · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · odd X ∈ { A , P } : a player, X : its opponent Definition (Strategy) A strategy f X : V X → V X for player X selects for every decision node of player X one of its successor nodes in G A , w . 21 / 33

  51. Playing a Game � Definition (Play) A play γ in a game G A , w is an infinite path starting with � q in , 0 � . Definition (Winner) The winner of a play γ is player A iff the smallest parity of occurring V A -nodes is even player P · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · odd X ∈ { A , P } : a player, X : its opponent Definition (Strategy) A strategy f X : V X → V X for player X selects for every decision node of player X one of its successor nodes in G A , w . f X is a winning strategy iff player X wins every play γ that is played according to f X . 21 / 33

  52. Strategies Example parities q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · q 0 �→ 2 { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · q 1 �→ 1 { q 1 , q 2 } , 0 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · q 2 �→ 0 22 / 33

  53. Strategies Example Winning strategy for player A (so far): parities q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · q 0 �→ 2 { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · q 1 �→ 1 { q 1 , q 2 } , 0 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · q 2 �→ 0 22 / 33

  54. Strategies Example Winning strategy for player A (so far): parities q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · q 0 �→ 2 { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · q 1 �→ 1 { q 1 , q 2 } , 0 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · q 2 �→ 0 Not a winning strategy for player A : q 0 , 0 { q 0 } , 0 q 0 , 1 { q 0 } , 1 q 0 , 2 · · · { q 1 , q 2 } , 1 · · · q 1 , 1 { q 0 , q 1 } , 1 q 1 , 2 · · · { q 1 , q 2 } , 0 q 2 , 1 { q 2 } , 1 q 2 , 2 · · · 22 / 33

  55. Outline 1 Weak Alternating Parity Automata Infinite Parity Games 2 Proof of the Complementation Theorem 3 Lemma 1 Lemma 2 Lemma 3 Sublemma Putting it All Together B¨ uchi Complementation Algorithm 4 23 / 33

  56. Lemma 1 Let A be a WAPA and w ∈ Σ ω . Lemma 1 Player A has a winning strategy in G A , w iff A accepts w . 24 / 33

  57. Lemma 1 Let A be a WAPA and w ∈ Σ ω . Lemma 1 Player A has a winning strategy in G A , w iff A accepts w . Explanation (oral) : Player A wins every play γ played according to f A . G A , w : · · · q , i + 1 { q , q ′ , q ′′ } , i q ′ , i + 1 p , i · · · q ′′ , i + 1 24 / 33

  58. Lemma 1 Let A be a WAPA and w ∈ Σ ω . Lemma 1 Player A has a winning strategy in G A , w iff A accepts w . Explanation (oral) : Player A wins every play γ There is a run graph R in which played according to f A . every path ρ is accepting. G A , w : · · · q , i + 1 q , i + 1 { q , q ′ , q ′′ } , i q ′ , i + 1 q ′ , i + 1 p , i p , i R : · · · q ′′ , i + 1 q ′′ , i + 1 24 / 33

  59. Lemma 2 Let A be a WAPA and w ∈ Σ ω . Lemma 2 Player P has a winning strategy in G A , w iff A does not accept w . (pointed out by Jan Leike) 25 / 33

  60. Lemma 2 Let A be a WAPA and w ∈ Σ ω . Lemma 2 Player P has a winning strategy in G A , w iff A does not accept w . (pointed out by Jan Leike) Explanation (oral) : Player P wins every play γ played according to f P . · · · G A , w : { .., q , .. } , i q , i + 1 · · · · · · { .., q ′ , .. } , i q ′ , i + 1 p , i · · · · · · { .., q ′′ , .. } , i q ′′ , i + 1 · · · 25 / 33

  61. Lemma 2 Let A be a WAPA and w ∈ Σ ω . Lemma 2 Player P has a winning strategy in G A , w iff A does not accept w . (pointed out by Jan Leike) Explanation (oral) : Player P wins every play γ Every run graph R contains a played according to f P . rejecting path ρ . · · · · · · G A , w : { .., q , .. } , i p , i q , i + 1 q , i + 1 R : · · · · · · · · · · · · { .., q ′ , .. } , i q ′ , i + 1 q ′ , i + 1 p , i R ′ : p , i · · · · · · · · · · · · q ′′ , i + 1 R ′′ : p , i { .., q ′′ , .. } , i q ′′ , i + 1 · · · · · · 25 / 33

  62. Sublemma � Let θ ∈ B + ( Q ) be a formula over Q . Sublemma S ⊆ Q is a model of θ for all M ∈ Mod ↓ ( θ ): S ∩ M � = ∅ . iff 26 / 33

  63. Sublemma � Let θ ∈ B + ( Q ) be a formula over Q . Sublemma S ⊆ Q is a model of θ for all M ∈ Mod ↓ ( θ ): S ∩ M � = ∅ . iff Proof: W.l.o.g. θ is in DNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M 26 / 33

  64. Sublemma � Let θ ∈ B + ( Q ) be a formula over Q . Sublemma S ⊆ Q is a model of θ for all M ∈ Mod ↓ ( θ ): S ∩ M � = ∅ . iff Proof: W.l.o.g. θ is in DNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M Then θ is in CNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M 26 / 33

  65. Sublemma � Let θ ∈ B + ( Q ) be a formula over Q . Sublemma S ⊆ Q is a model of θ for all M ∈ Mod ↓ ( θ ): S ∩ M � = ∅ . iff Proof: W.l.o.g. θ is in DNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M Then θ is in CNF, i.e. � � θ = q M ∈ Mod ↓ ( θ ) q ∈ M Thus S ⊆ Q is a model of θ iff it contains at least one element from each disjunct of θ . 26 / 33

Recommend

B uchi Complementation 2 ( n log n ) BA B BA B Automata Theory Seminar BA: B uchi

B uchi Complementation 2 ( n log n ) BA B BA B Automata Theory Seminar BA: B uchi

B uchi Complementation 2 ( n log n ) BA B BA B Automata Theory Seminar BA: B uchi Automaton B uchi Complementation via AA: Alternating Alternating Automata Automaton AA A AA A Fabian Reiter Expensive: If B has n states, B has 2

333 views • 5 slides
B uchi Complementation via Alternating Automata Fabian Reiter July 16, 2012 B uchi

B uchi Complementation via Alternating Automata Fabian Reiter July 16, 2012 B uchi

Automata Theory Seminar B uchi Complementation via Alternating Automata Fabian Reiter July 16, 2012 B uchi Complementation 2 ( n log n ) BA B BA B BA: B uchi Automaton AA: Alternating Automaton AA A AA A Expensive: If B has n

510 views • 38 slides
Complementing Bchi automata via alternating automata Guillaume Sadegh LRDE EPITA Research

Complementing Bchi automata via alternating automata Guillaume Sadegh LRDE EPITA Research

Introduction Overview on -automata The complementation Implementation Conclusion and Perspectives References Complementing Bchi automata via alternating automata Guillaume Sadegh LRDE EPITA Research and Development Laboratory

512 views • 49 slides
B uchi Automata and their Application to Software Verification Finite Automata Theory and

B uchi Automata and their Application to Software Verification Finite Automata Theory and

B uchi Automata and their Application to Software Verification Finite Automata Theory and Formal Languages Wolfgang Ahrendt 22nd April 2013 B uchi Automata: TMV027/DIT321 / GU 130423 1 / 25 Motivating Temporal Logic? But How to

832 views • 64 slides
Efficient Emptiness Check for Timed B uchi Automata F. Herbreteau, B. Srivathsan and I.

Efficient Emptiness Check for Timed B uchi Automata F. Herbreteau, B. Srivathsan and I.

Efficient Emptiness Check for Timed B uchi Automata F. Herbreteau, B. Srivathsan and I. Walukiewicz Universit e de Bordeaux, LaBRI - CNRS August 2010 Efficient Emptiness Check for Timed B uchi Automata - 1/24 Timed B uchi Automata

653 views • 54 slides
Complexity of B uchi automata minimization Dejan Kostyszyn Chair of Software Engineering

Complexity of B uchi automata minimization Dejan Kostyszyn Chair of Software Engineering

Overview Foundations Definitions & Constructions Main proof Complexity of B uchi automata minimization Dejan Kostyszyn Chair of Software Engineering February 3, 2018 Dejan Kostyszyn Complexity of B uchi automata minimization 1

1k views • 63 slides
Two-Way Alternating Automata and Finite Models Tedious proofs of irrelevant results Mikolaj

Two-Way Alternating Automata and Finite Models Tedious proofs of irrelevant results Mikolaj

Two-Way Alternating Automata and Finite Models Tedious proofs of irrelevant results Mikolaj Bojanczyk Warsaw University Two-Way Alternating Automata and Finite Models p.1/18 Intuition on the automaton A two-way alternating automaton

799 views • 57 slides
Modal -Calculus and Alternating Tree Automata Seminar Automata, Logics, and Infinite

Modal -Calculus and Alternating Tree Automata Seminar Automata, Logics, and Infinite

Department of Computer Science Institute for Theoretical Computer Science Modal -Calculus and Alternating Tree Automata Seminar Automata, Logics, and Infinite Games Patrick Bahr s0404888@inf.tu-dresden.de Dresden, February 1, 2008

461 views • 45 slides
Mechanizing the Minimization of Deterministic Generalized B uchi Automata Souheib Baarir 1 , 2

Mechanizing the Minimization of Deterministic Generalized B uchi Automata Souheib Baarir 1 , 2

Mechanizing the Minimization of Deterministic Generalized B uchi Automata Souheib Baarir 1 , 2 Alexandre Duret-Lutz 3 1 Universit e Paris Ouest Nanterre la D efense, Nanterre, France 2 Sorbonne Universit es, UPMC Univ. Paris 6, UMR

795 views • 43 slides
Ranking based Techniques for Disambiguating B uchi Automata Hrishikesh Karmarkar Supratik

Ranking based Techniques for Disambiguating B uchi Automata Hrishikesh Karmarkar Supratik

Ranking based Techniques for Disambiguating B uchi Automata Hrishikesh Karmarkar Supratik Chakraborty January 29, 2011 Hrishikesh Karmarkar Supratik Chakraborty Ranking based Techniques for Disambiguating B uchi Automata

727 views • 69 slides
Overview Word Systems Regular expressiveness Linear temporal logic B uchi-automata

Overview Word Systems Regular expressiveness Linear temporal logic B uchi-automata

ormal ethods roup Linear Temporal Logics and Grammars Joachim Baran The University of Manchester April 2006 Overview Word Systems Regular expressiveness Linear temporal logic B uchi-automata Right-linear grammars TL,

512 views • 33 slides
3 COMP 1 5 9 3 Algorithmic Verification LTL Model Checking and B uchi Automata Dr. Liam

3 COMP 1 5 9 3 Algorithmic Verification LTL Model Checking and B uchi Automata Dr. Liam

<latexit

1.07k views • 96 slides
Alternating Automata: Checking Truth and Validity for Temporal Logics Moshe Y. Vardi ? Rice

Alternating Automata: Checking Truth and Validity for Temporal Logics Moshe Y. Vardi ? Rice

Alternating Automata: Checking Truth and Validity for Temporal Logics Moshe Y. Vardi ? Rice University Department of Computer Science Houston, TX 77005-1892, U.S.A. Email: vardi@cs.rice.edu URL: http://www.cs.rice.edu/ vardi Abstract. We

297 views • 16 slides
Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1 Definitions A sequence a 1 , a 2 , . . . , a k of distinct integers is alternating if a 1 > a 2 < a 3 > a 4 < , and reverse alternating if

1.14k views • 98 slides
Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. Basic

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. Basic

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. Basic definitions A sequence a 1 , a 2 , . . . , a k of distinct integers is alternating if a 1 > a 2 < a 3 > a 4 < , and reverse alternating

1.4k views • 119 slides
Weak Alternating Timed Automata Pawel Parys and Igor Walukiewicz CNRS, LaBRI, Bordeaux IFIP ,

Weak Alternating Timed Automata Pawel Parys and Igor Walukiewicz CNRS, LaBRI, Bordeaux IFIP ,

Weak Alternating Timed Automata Pawel Parys and Igor Walukiewicz CNRS, LaBRI, Bordeaux IFIP , September 2009 Pawel Parys and Igor Walukiewicz (LaBRI) IFIP09 1 / 22 Timed languages [Alur & Dill94] Infinite sequences abaacb . . .

578 views • 22 slides
Unit 10: Alternating-current circuits Introduction. Alternating current features. Phasor

Unit 10: Alternating-current circuits Introduction. Alternating current features. Phasor

Unit 10: Alternating-current circuits Introduction. Alternating current features. Phasor diagram. Behaviour of basic dipoles (resistor, inductor, capacitor) to an alternating current. RLC series circuit. Impedance and phase lag.

491 views • 18 slides
Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1 Basic

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1 Basic

Alternating Permutations Richard P. Stanley M.I.T. Alternating Permutations p. 1 Basic definitions A sequence a 1 , a 2 , . . . , a k of distinct integers is alternating if a 1 > a 2 < a 3 > a 4 < , and reverse

1.59k views • 119 slides
Parameter-Synthesis Problems for One-Counter Automata Guillermo A. P erez (slides by Ritam

Parameter-Synthesis Problems for One-Counter Automata Guillermo A. P erez (slides by Ritam

Parameter-Synthesis Problems for One-Counter Automata Guillermo A. P erez (slides by Ritam Raha) INFINITY 2020 Outline 1. One-Counter Automata (Parametric) and Synthesis Problem 2. Previous Approach 3. Approach with Alternating Two-Way

1.33k views • 87 slides
Closed under complementation Proposition: Let be some finite alphabet. If

Closed under complementation Proposition: Let be some finite alphabet. If

15-251 Great Ideas in Theoretical Computer Science Lecture 4: Deterministic Finite Automaton (DFA), Part 2 January 25th, 2018 Closure properties of regular languages Closed under complementation Proposition: Let be some finite

268 views • 9 slides
Pushdown Automata 7-0 Pushdown Automata The automata we saw so far were

Pushdown Automata 7-0 Pushdown Automata The automata we saw so far were

Griffith University 3130CIT Theory of Computation (Based on slides by Harald Sndergaard of The University of Melbourne) Pushdown Automata 7-0 Pushdown Automata The automata we saw so far were limited by their

368 views • 15 slides
Alternating Current Slide 2 / 69 Topics to be covered Sources of alternating EMF Transformers

Alternating Current Slide 2 / 69 Topics to be covered Sources of alternating EMF Transformers

Slide 1 / 69 Alternating Current Slide 2 / 69 Topics to be covered Sources of alternating EMF Transformers AC Circuits and Impedance LRC Series AC Circuits Resonance in AC Circuit Oscillations Slide 3 / 69 Sources of Alternating EMF

273 views • 23 slides
Game-Theoretic Semantics for Alternating- Time Temporal Logic Valentin Goranko Stockholm

Game-Theoretic Semantics for Alternating- Time Temporal Logic Valentin Goranko Stockholm

Game-Theoretic Semantics for Alternating- Time Temporal Logic Valentin Goranko Stockholm University Joint work with Antti Kuusisto and Raine Rnnholm Highlights of Logic, Games and Automata 2016 Brussels, September 8, 2016 V Goranko Logic

646 views • 15 slides
Automata Theory Why Study Automata? What the Course is About 1 Why Study Automata? A survey of

Automata Theory Why Study Automata? What the Course is About 1 Why Study Automata? A survey of

Automata Theory Why Study Automata? What the Course is About 1 Why Study Automata? A survey of Stanford grads 5 years out asked which of their courses did they use in their job. Basics like intro-programming took the top spots, of course.

584 views • 32 slides

More recommend