Reachability and Büchi Games - Rich Model Toolkit

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Visit versus Revisit F Attr 0 (F) F Attr + 0 (F)
Recurrence Set We define Recur 0 0(F) := F Recur i+1 0 (F) := F ∩Attr + 0 (Recuri 0 (F)) Recur 0 (F) := ⋂ i≥0 Recuri 0 (F) We show that there exists k such that Recur 0 (F) := ⋂ k i≥0 Recuri 0(F) by proving Recur i+1 0 (F) ⊆ Recur i 0 (F) for all i ≥ 0. Proof. ◮ i = 0: F ∩Attr + 0 (F) ⊆ F ◮ i → i+1: Recur i+1 0 (F) = F ∩Attr + 0 (Recuri 0(F)) ⊆ F ∩Attr + 0 (Recuri−1 0 (F)) = Recur i 0(F) since (i) Recur i 0(F) ⊆ Recur i−1 0 (F) by ind. hyp. <strong>and</strong> (ii) Attr + 0 is monotone.
Page 1 and 2: Reachability and Büchi Games Barba
Page 3 and 4: Reachability and Safety Games
Page 5 and 6: Force Visit in Next Step Given a se
Page 7 and 8: Computing the Attractor Constructio
Page 9 and 10: Example s 5 s 7 Attr 0 0 = {s 3 ,s
Page 11 and 12: Example s 5 s 6 s 7 Attr 0 0 = {s 3
Page 13 and 14: Example s 5 s 3 s 6 s 4 s 7 Attr 0
Page 15 and 16: Computing the Attractor Constructio
Page 17 and 18: 0-Attractor To show W 0 = Attr 0 (F
Page 19 and 20: 0-Attractor cont. Proof cont. S \At
Page 21 and 22: Summary We know how to solve reacha
Page 23 and 24: Exercise 1. Given a reachability ga
Page 25 and 26: Büchi and co-Büchi Games
Page 27 and 28: Idea Compute for i ≥ 1 the set Re
Page 29: One-Step Attractor We count revisit
Page 33 and 34: Recurrence Set cont. We show S \Att
Page 35 and 36: Büchi games We have shown that Pla
Page 37 and 38: Co-Büchi Games Given a Co-Büchi G
Page 39 and 40: Exercise 2. Consider the game graph

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