Nested Waiting-for Formulas

# Temporal Verification of Reactive Systems: Safety

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

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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 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 Basic Verification Diagrams
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