Nested Waiting-for Formulas

Temporal Verification of Reactive Systems: Safety

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Unformatted text preview: CS256/Winter 2007 – Lecture #10 Zohar Manna 10-1 Nested Waiting-for Formulas q m q m- 1 q 1 interval interval • • • interval q [ )[ )[ )[ ) • p ϕ m ϕ m- 1 ϕ 1 ϕ Rule nwait (nested waiting-for) For assertions p , q , q 1 , . . . , q m and ϕ , 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 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 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 = q : 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 .. 2 , 5 ∨ ( at- m 3 ∧ s = 2)) ϕ = q : at- ‘ 4 10-5 Concatenation of waiting-for formulas Rule conc-w p ⇒ q m W ··· q 1 W q q ⇒ r n W ··· W r p ⇒ q m W ··· W q 1 W r n W ··· W r q m ··· q 1 [ ) [ ) • p q r n ··· r 1 [ ) [ ) • q r 10-6 Collapsing of waiting-for formulas Rule coll-w For i > p ⇒ q m W ··· W q i +1 W q i W ··· W q p ⇒ q m W ··· W ( q i +1 ∨ q i ) W ··· W q q m ···...
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Nested Waiting-for Formulas - CS256/Winter 2007 –...

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