Reachability and Büchi Games - Rich Model Toolkit

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Exercise 1. Given a reachability game (G,F) with G = (S,S 0 ,E) <strong>and</strong> F ⊆ Q, give an algorithm that computes the 0-Attractor(F) in time O(|E|).
Exercise 1. Given a reachability game (G,F) with G = (S,S 0 ,E) <strong>and</strong> Solution: F ⊆ Q, give an algorithm that computes the 0-Attractor(F) in time O(|E|). 1. Preprocessing: Compute for every state s ∈ S 1 outdegree out(s) 2. Set n(s) := out(s) for each s ∈ S 1 3. To breadth-first search backwards from F with the following conventions: ◮ mark all s ∈ F ◮ mark s ∈ S0 if reached from marked state ◮ mark s ∈ S1 if n(s) = 0, other set n(s) := n(s)−1. The marked vertices are the ones of Attr 0 (F).
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: Summary We know how to solve reacha
Page 25 and 26: Büchi and co-Büchi Games
Page 27 and 28: Idea Compute for i ≥ 1 the set Re
Page 29 and 30: One-Step Attractor We count revisit
Page 31 and 32: Recurrence Set We define Recur 0 0(
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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