Promising Formula

Temporal Verification of Reactive Systems: Safety

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CS256/Winter 2007 — Lecture #13 Zohar Manna 13-1 Example : ϕ 0 : 1 p Tableau T ϕ 0 : ? 6 A 1 : { p, 2 1 p, 1 p } - ? 6 A 2 : p, 2 1 p, 1 p } @ @ @ @ @ @ R A 3 : { p, ¬ 2 1 p, 1 p } ? A 4 : p, ¬ 2 1 p, ¬ 1 p } 13-2 Promising Formula In T 1 p , a path can start and stay forever in atom A 2 . But A 2 includes 1 p , i.e., A 2 promises that p will eventually happen, but it is never fulfilled in the path. We want to exclude these paths. The idea is that if a path contains an atom that in- cludes a promising formula , then the path should fulfill the promise. A formula ψ Φ ϕ is said to promise the formula r if ψ is one of the forms: 1 r p U r | {z } 1 r ... ¬ 0 ¬ r | {z } 1 r ¬ (( ¬ r ) W p ) | {z } 1 r ... 13-3 Example: ϕ 1 : 0 p 1 ¬ p Φ ϕ 1 : ϕ 1 , 0 p, 1 ¬ p , 2 0 p, 2 1 p, p ¬ ϕ 1 , ¬ 0 p , ¬ 1 ¬ p, ¬ 2 0 p, ¬ 2 1 p, ¬ p Only 2 promising formulas in Φ ϕ ψ 1 : ¬ 0 p promises r 1 : ¬ p ψ 2 : 1 ¬ p promises r 2 : ¬ p Example: ϕ 3 : 1 0 ¬ p 0 1 q ψ 1 : 1 0 ¬ p promises r 1 : 0 ¬ p ψ 2 : 1 q promises r 2 : q 13-4
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Promise Fulfillment Property: Let σ be an arbitrary model of ϕ , and ψ Φ ϕ a formula that promises r . If ( σ, j ) q ψ then ( σ, k ) q r for some k j Proof: Follows from the semantics of temporal formulas. Claim: (promise fulfillment by models) Let σ be an arbitrary model of ϕ , and ψ Φ ϕ a formula that promises r . Then σ contains infinitely many positions j 0 such that ( σ, j ) q ¬ ψ or ( σ, j ) q r Proof: 1. Assume σ contains infinitely many ψ -positions. Then σ must contain infinitely many r -positions, since ψ promises r . 2. Assume σ contains finitely many ψ -positions. Then it contains infinitely many ¬ ψ -positions. 13-5 Fulfilling Atoms Definition: Atom A fulfills ψ Φ ϕ (which promises r ) if ¬ ψ A or r A . Example: In T 1 p , Only one promising formula: ψ : 1 p promises r : p A + 1 : { p, 2 1 p, 1 p } fulfills 1 p since p A 1 A + 3 : { p, ¬ 2 1 p, 1 p } fulfills 1 p since p A 3 A + 4 : p, ¬ 2 1 p, ¬ 1 p } fulfills 1 p since ¬ 1 p A 4 But A - 2 : p, 2 1 p, 1 p } does not fulfill 1 p since 1 p, ¬ p A 2 13-6 Tableau T 1 p @ @ @ R 6 A + 1 : { p, 2 1 p, 1 p } - 6 A - 2 : p, 2 1 p, 1 p } @ @ @ @ @ @ R A + 3 : { p, ¬ 2 1 p, 1 p } ? A + 4 : p, ¬ 2 1 p, ¬ 1 p } 13-7 Fulfilling Paths Definition: A path π : A 0 , A 1 , . . . is fulfilling if for every promising formula ψ Φ ϕ it contains infinitely many A j that fulfill ψ . Example: In T 1 p , A - 2 , A - 2 , A - 2 , A + 3 , A + 4 , A + 4 , . . . A - 2 , A + 1 , A - 2 , A + 1 , A + 1 , A + 1 , . . . are fulfilling paths, but A - 2 , A - 2 , A - 2 , A - 2 , A - 2 , A - 2 , A - 2 , . . . is not a fulfilling path. 13-8
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Fig. 5.3: Tableau T ϕ 1 for formula ϕ 1 : 0 p 1 ¬ p A ++ 2 : n ¬ p, ¬ 2 0 p, 2 1 ¬ p, ¬ 0 p, 1 ¬ p, ¬ ϕ 1 o A -- 3 : n p, ¬ 2 0 p, 2 1 ¬ p, ¬ 0 p, 1 ¬ p, ¬ ϕ 1 o A ++ 0 : n ¬ p, ¬ 2 0 p, ¬ 2 1 ¬ p, ¬ 0 p, 1 ¬ p, ¬ ϕ 1 o A - + 1 : n p, ¬ 2 0 p, ¬ 2 1 ¬ p, ¬ 0 p, ¬ 1 ¬ p, ¬ ϕ 1 o A ++ 4 : n ¬ p, 2 0 p, ¬ 2 1 ¬ p, ¬ 0 p, 1 ¬ p, ¬ ϕ 1 o A ++ 5 : n p, 2 0 p, ¬ 2 1 ¬ p, 0 p, ¬ 1 ¬ p, ¬ ϕ 1 o A ++ 6 : n ¬ p, 2 0 p, 2 1 ¬ p, ¬ 0 p, 1 ¬ p, ¬ ϕ 1 o A + - 7 : n p, 2 0 p, 2 1 ¬ p, 0 p, 1 ¬ p, ϕ 1 o 13-9 Example: ϕ 1 : 0 p 1 ¬ p T ϕ 1 in Fig 5.3 There are two promising formulas in Φ : ψ 1 : ¬ 0 p promises r 1 : ¬ p ψ 2 : 1 ¬ p promises r 2 : ¬ p A ++ 0 : { ¬ p, ¬ 0 p, 1 ¬ p, . . . } A - + 1 : { p, ¬ 0 p, ¬ 1
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