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**Unformatted text preview: **CS256/Winter 2007 — Lecture #13 Zohar Manna 13-1 Example : ϕ : 1 p Tableau T ϕ : ? 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 ∧ ... ¬ ¬ r | {z } ≈ 1 r ¬ (( ¬ r ) W p ) | {z } ≈ 1 r ∧ ... 13-3 Example: ϕ 1 : p ∧ 1 ¬ p Φ ϕ 1 : ϕ 1 , p, 1 ¬ p , 2 0 p, 2 1 p, p ¬ ϕ 1 , ¬ p , ¬ 1 ¬ p, ¬ 2 0 p, ¬ 2 1 p, ¬ p Only 2 promising formulas in Φ ϕ ψ 1 : ¬ 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 : ¬ p ψ 2 : 1 q promises r 2 : q 13-4 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 ≥ 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 , 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 , . . ....

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