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. 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)

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 = t0 t1 t2 ... (sequence of observations)

17. 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 

18. 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 |= 

19. 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) |= 

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

21. 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 
(pc2in) W (pc2=in) W (pc2in) W (pc1=in))
Response  (a   b) liveness
 (pc1=req   (pc1=in))

22. Composed modalities
 a infinitely often a
 a almost 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 {q1,q2,q3}
A set of atomic observations {a,b}
 Q  Q transition relation q1  q2
[ ]: Q  2A observation function [q1] = {a}

25. 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’?

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
if (t  ndfsTable) then ndfs(t)
else if (t = seed) then report error