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**Part 3: Safety and liveness**

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Safety vs. liveness Safety: something “bad” will never happen Liveness: something “good” will happen (but we don’t know when)

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**Safety vs. liveness for sequential programs**

Safety: the program will never produce a wrong result (“partial correctness”) Liveness: the program will produce a result (“termination”)

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**Safety vs. liveness for sequential programs**

Safety: the program will never produce a wrong result (“partial correctness”) Liveness: the program will produce a result (“termination”)

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**Safety vs. liveness for state-transition graphs**

Safety: those properties whose violation always has a finite witness (“if something bad happens on an infinite run, then it happens already on some finite prefix”) Liveness: those properties whose violation never has a finite witness (“no matter what happens along a finite run, something good could still happen later”)

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This is much easier. Safety: the properties that can be checked on finite executions Liveness: the properties that cannot be checked on finite executions (they need to be checked on infinite executions)

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**Example: Mutual exclusion**

It cannot happen that both processes are in their critical sections simultaneously.

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**Example: Mutual exclusion**

It cannot happen that both processes are in their critical sections simultaneously. Safety

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**Example: Bounded overtaking**

Whenever process P1 wants to enter the critical section, then process P2 gets to enter at most once before process P1 gets to enter.

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**Example: Bounded overtaking**

Whenever process P1 wants to enter the critical section, then process P2 gets to enter at most once before process P1 gets to enter. Safety

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**Example: Starvation freedom**

Whenever process P1 wants to enter the critical section, provided process P2 never stays in the critical section forever, P1 gets to enter eventually.

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**Example: Starvation freedom**

Whenever process P1 wants to enter the critical section, provided process P2 never stays in the critical section forever, P1 gets to enter eventually. Liveness

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**Example: Starvation freedom**

Whenever process P1 wants to enter the critical section, provided process P2 never stays in the critical section forever, P1 gets to enter eventually. Liveness

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**LTL (Linear Temporal Logic)**

-safety & liveness -linear time [Pnueli 1977; Lichtenstein & Pnueli 1982]

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LTL Syntax ::= a | | | | U

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LTL Model infinite trace t = t0 t1 t2 ... (sequence of observations)

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q1 a a,b b q2 q3 Run: q1 q3 q1 q3 q1 q2 q2 Trace: a b a b a a,b a,b

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**Language of deadlock-free state-transition graph K at state q :**

L(K,q) = set of infinite traces of K starting at q (K,q) |= iff for all t L(K,q), t |= (K,q) |= iff exists t L(K,q), t |=

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LTL Semantics t |= a iff a t0 t |= iff t |= and t |= t |= iff not t |= t |= iff t1 t2 ... |= t |= U iff exists n 0 s.t for all 0 i < n, ti ti |= tn tn |= (K,q) |= iff (K,q) |=

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Defined modalities X next U U until = true U F eventually = G always W = ( U ) W waiting-for (weak-until)

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**Sequencing a W b W c W d safety (pc1=req **

Important properties Invariance a safety (pc1=in pc2=in) Sequencing a W b W c W d safety (pc1=req (pc2in) W (pc2=in) W (pc2in) W (pc1=in)) Response (a b) liveness (pc1=req (pc1=in))

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Composed modalities a infinitely often a a almost always a

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**Example: Starvation freedom**

Whenever process P1 wants to enter the critical section, provided process P2 never stays in the critical section forever, P1 gets to enter eventually. (pc2=in (pc2=out)) (pc1=req (pc1=in))

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**State-transition graph**

Q set of states {q1,q2,q3} A set of atomic observations {a,b} Q Q transition relation q1 q2 [ ]: Q 2A observation function [q1] = {a}

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**Is there an infinite path starting from q’ **

(K,q) |= Tableau construction (Vardi-Wolper) (K’, q’, BA) where BA K’ Is there an infinite path starting from q’ that hits BA infinitely often? Is there a path from q’ to p BA such that p is a member of a strongly connnected component of K’?

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dfs(s) { add s to dfsTable for each successor t of s if (t dfsTable) then dfs(t) if (s BA) then { seed := s; ndfs(s) } } ndfs(s) { add s to ndfsTable if (t ndfsTable) then ndfs(t) else if (t = seed) then report error

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ESE601: Hybrid Systems Introduction to verification Spring 2006.

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