Classi cation Diagram (Fig. 0.18)

Temporal Verification of Reactive Systems: Safety

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CS256/Winter 2007 — Lecture #5 Zohar Manna Announcements Homework 2 due NOW Homework 3 out today, due Tue next week 5-1 Classification Diagram (Fig. 0.18) For each κ ∈ { safety , guarantee , obligation response , persistence , reactivity } the κ class of temporal formulas is characterized by a canonical κ -formula , with p , q , p i , q i – past formulas A formula is a κ -formula if it is equivalent to a canonical κ -formula A property is a κ -property if it is specifiable by a κ -formula 5-2 5-3 Closure of Classes Reactivity : closure under , , ¬ Persistence : closure under , 1 0 p 1 0 q 1 0 ( p q ) 1 0 p 1 0 q 1 0 ( q « ( p S ( p ∧ ¬ q ))) Response : closure under , 0 1 p 0 1 q 0 1 ( p q ) 0 1 p 0 1 q 0 1 ( q « (( ¬ q ) S p )) Obligation : closure under , , ¬ Guarantee : closure under , 1 p 1 q 1 ( p q ) 1 p 1 q 1 ( Q p Q q ) Safety : closure under , 0 p 0 q 0 ( p q ) 0 p 0 q 0 ( p q ) 5-4
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Duality of classes Safety vs. Guarantee ¬ 0 p 1 ¬ p ¬ 1 p 0 ¬ p Response vs. Persistence ¬ 0 1 p 1 0 ¬ p ¬ 1 0 p 0 1 ¬ p 5-5 Classification Diagram strict inclusion between boxes P 1 ⊃ P 2 P 2 P 1 Example: Obligation Persistence ( 0 p i 1 q i ) 1 0 ( p i Q q i ) Theorem: Every quantifier free temporal formula is equivalent to a reactivity formula. 5-6 Classification Diagram Con’t strict inclusion between conjunctions (Obligation and Reactivity ) In Obligation n +1 ^ i =1 [ 0 p i 1 q i ] n ^ i =1 [ 0 p i 1 q i ] In Reactivity n +1 ^ i =1 [ 0 1 p i 1 0 q i ] n ^ i =1 [ 0 1 p i 1 0 q i ] 5-7 Note: Properties specified by state formulas are safety proper- ties and guarantee properties, since p 0 ( first p ) p 1 ( first p ) but also 2 p, 2 2 p, . . . since 2 p 0 ( « first p ) 2 p 1 ( « first p ) 2 2 p 0 ( « « first p ) 2 2 p 1 ( « « first p ) 5-8
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Reactivity n=1 Reactivity n>1 Obligation n>1 Obligation Persistence Response Safety Guarantee n=1 5-9 Example Formulas Safety 0 p conditional safety p 0 q 0 ( Q ( p first ) q ) p 0 q 0 ( Q p q ) waiting-for p W q 0 ( Q ¬ p Q q ) Guarantee 1 p conditional guarantee p 1 q 1 Q ( first p ) q until p U q 1 ( q c p ) 5-10 Example formulas (Con’t) Obligation n +1 ^ i =1 ( 0 p i 1 q i ) p W ( 1 q ) 0 p 1 q Response 0 1 p response p 1 q 0 1 ( ¬ p ) B q justice 0 1 ( ¬ enabled ( τ ) last - taken ( τ )) where enabled ( τ ) : V 0 . ρ τ ( V, V 0 ) 5-11 Example formulas (Con’t) Persistence 1 0 p conditional stabilization p 1 0 q 1 0 ( Q p q ) Reactivity n +1 ^ i =1 ( 1 0 p i 0 1 q i ) compassion
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