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Part 3: Safety and liveness. Safety vs. liveness Safety: something “bad” will never happen Liveness: something “good” will happen (but we don’t know when)

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Presentation on theme: "Part 3: Safety and liveness. Safety vs. liveness Safety: something “bad” will never happen Liveness: something “good” will happen (but we don’t know when)"— Presentation transcript:

1 Part 3: Safety and liveness

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

3 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”)

4 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”)

5 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”)

6 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) This is much easier.

7 Example: Mutual exclusion It cannot happen that both processes are in their critical sections simultaneously.

8 Example: Mutual exclusion It cannot happen that both processes are in their critical sections simultaneously. Safety

9 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.

10 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

11 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.

12 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

13 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

14 LTL (Linear Temporal Logic) -safety & liveness -linear time [Pnueli 1977; Lichtenstein & Pnueli 1982]

15 LTL Syntax  ::= a |    |   |   |  U 

16 LTL Model infinite trace t = t 0 t 1 t 2... (sequence of observations)

17 a a,bb q1q1 q3q3 q2q2 Run: q 1  q 3  q 1  q 3  q 1  q 2  q 2  Trace: a  b  a  b  a  a,b  a,b 

18 (K,q) |=   iff for all t  L(K,q), t |=  (K,q) |=   iff exists t  L(K,q), t |=  Language of deadlock-free state-transition graph K at state q : L(K,q) = set of infinite traces of K starting at q

19 LTL Semantics t |= aiff a  t 0 t |=    iff t |=  and t |=  t |=  iff not t |=  t |=   iff t 1 t 2... |=  t |=  U  iff exists n  0 s.t. 1. for all 0  i < n, t i t i+1... |=  2. t n t n+1... |=  (K,q) |=   iff  (K,q) |=   

20  X next U U until   = true U  Feventually   =    G always  W  = (  U  )    W waiting-for (weak-until) Defined modalities

21 Important properties Invariance  asafety   (pc1=in  pc2=in) Sequencing a W b W c W dsafety  (pc1=req  (pc2  in) W (pc2=in) W (pc2  in) W (pc1=in)) Response  (a   b)liveness  (pc1=req   (pc1=in))

22 Composed modalities  ainfinitely often a  aalmost always a

23 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))

24 State-transition graph Q set of states{q 1,q 2,q 3 } A set of atomic observations{a,b}   Q  Q transition relation q 1  q 2 [ ]: Q  2 A observation function [q 1 ] = {a}

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

26 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 for each successor t of s if (t  ndfsTable) then ndfs(t) else if (t = seed) then report error }


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