Nested Waiting-for Formulas

# Temporal Verification of Reactive Systems: Safety

• Notes
• davidvictor
• 26

This preview shows pages 1–9. Sign up to view the full content.

CS256/Winter 2007 – Lecture #10 Zohar Manna 10-1

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

Nested Waiting-for Formulas q m q m - 1 q 1 interval interval interval q 0 [ )[ )[ )[ ) p ϕ m ϕ m - 1 ϕ 1 ϕ 0 Rule nwait (nested waiting-for) For assertions p , q 0 , q 1 , . . . , q m and ϕ 0 , ϕ 1 , . . . , ϕ m N1. p m _ j =0 ϕ j N2. ϕ i q i for i = 0 , 1 , . . . , m N3. { ϕ i }T _ j i ϕ j for i = 1 , . . . , m p q m W q m - 1 · · · q 1 W q 0 10-2
Nested Waiting-for Formulas (Cont’d) ϕ i -interval ϕ j -interval p p p p τ τ where j < i Premise N3 states that for each assertion ϕ i , each tran- sition τ ∈ T either preserves ϕ i or leads to some ϕ j , with j < i . 10-3

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

Example: Program mux-pet1 (Fig. 3.4) An example of a nested waiting-for formula is 1-bounded overtaking for mux-pet1 : at - 3 | {z } p ¬ at - m 4 | {z } q 3 W at - m 4 | {z } q 2 W ¬ at - m 4 | {z } q 1 W at - 4 | {z } q 0 It states that when process P 1 is at 3 , process P 2 can enter its critical section at most once ahead of process P 1 . 10-4
With the following strengthenings all premises of rule nwait become state-valid. p : at - 3 ϕ 3 : at - 3 ∧ ¬ at - m 4 at - m 3 s = 1 P 2 has priority over P 1 ϕ 2 : at - 3 at - m 4 ϕ 1 : at - 3 ∧ ¬ at - m 4 ( at - m 3 s = 2) P 1 has priority over P 2 ϕ 0 = q 0 : at - 4 or equivalently, p : at - 3 ϕ 3 : at - 3 at - m 3 s = 1 ϕ 2 : at - 3 at - m 4 ϕ 1 : at - 3 ( at - m 0 .. 2 , 5 ( at - m 3 s = 2)) ϕ 0 = q 0 : at - 4 10-5

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

Concatenation of waiting-for formulas Rule conc-w p q m W · · · q 1 W q 0 q 0 r n W · · · W r 0 p q m W · · · W q 1 W r n W · · · W r 0 q m · · · q 1 [ ) [ ) p q 0 r n · · · r 1 [ ) [ ) q 0 r 0 10-6
Collapsing of waiting-for formulas Rule coll-w For i > 0 p q m W · · · W q i +1 W q i W · · · W q 0 p q m W · · · W ( q i +1 q i ) W · · · W q 0 q m · · · q i +1 q i · · · q 1 [ ) [ )[ ) [ ) p q 0 q m · · · q i +1 q i · · · q 1 [ ) [ ) [ ) p q 0 10-7

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

Basic Verification Diagrams
This is the end of the preview. Sign up to access the rest of the document.
• '
• NoProfessor
• 37 mm Gun M3, M3 Lee, M3 Scout Car, M4 cannon, waiting-for formulas

{[ 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