Promising Formula

# Temporal Verification of Reactive Systems: Safety

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern