1. CIS 720
Lecture 5

2. Safety and liveness properties
A safety property states that something bad will not happen
A liveness property states that something good will eventually happen

3. Proving safety property
BAD= predicate characterizing the bad property
GOOD = not BAD
Prove that GOOD is an invariant

4. Liveness property
A statement is eligible if it is the next action that could be executed
Scheduling policy determines the next statement to be executed

5. Weak Fairness: A scheduling policy is weak fair if
x = true co
while (x)  skip []
x = false oc
x = 0; y = 0 co
while (x = 0)  y = y + 1 []
await ( y > 5)  x = 1 oc
Unconditional fairness: A scheduling policy is unconditionally fair if every unconditional atomic action (one which does not have a guard) that is eligible is executed eventually.
Weak Fairness: A scheduling policy is weak fair if
It is unconditionally fair
Every conditional action that is eligible is eventually executed assuming that its guard becomes true and remains true.

6. Strong Fairness: A scheduling policy is strong fair if
It is unconditionally fair
Every conditional action that is eligible is eventually executed assuming that its guard becomes true and becomes true infinitely often
x = 0; y = 0 co
while (x = 0)  y = y + 1 []
await ( y is even)  x = 1 oc

7. Critical Section problem
Process i
do (true)
entry protocol;
critical section;
exit protocol;
non-critical section od

8. Correctness
Mutual exclusion: at most one process at a time is executing its critical section
Absence of deadlock: If two or more processes are trying to enter their critical section, at least one will succeed
Absence of unnecessary delay: If a process is trying to enter its critical section and the other processes are executing their non-critical sections or have terminated then the first process is not prevented from entering its critical section.
Eventual entry: A process that is attempting to enter its critical section will eventually succeed.

9. Invariant based approach
{ in1 = false; in2 = false }
CS1 CS2
do (true) do (true) 
entry protocol; entry protocol;
in1 = true in2 = true
critical section critical section
exit protocol; exit protocol;
in1 = false in2 = false
non-critical section non-critical section od
{ I /\¬ in1}
{ I /\¬ in2}
{ I /\ in1}
{ I /\ in2}
{ I /\¬ in1}
{ I /\¬ in2}
BAD = ¬ in1 /\ ¬ in2
I = ¬ BAD
= ¬ (in1 /\ in2)
= ¬ in1 \/ ¬ in2

10. Invariant based approach
CS1 CS2
do (true) do (true) 
in1 = true in2 = true
critical section critical section
in1 = false in2 = false
non-critical section non-critical section od
{ I /\¬ in1}
<await (¬ in2) 
<await (¬ in1) 
>
>
{ I}
{ I /\ ¬ in1}
Weakest precondition wp(A, action)
{wp(A, action) } action { A } given by the assignment axiom
Wp( x = 5, x = x + 1)
{ x= 4} x = x + 1 {x = 5 }

11. Mutual exclusion:
(¬ in1 \/ ¬ in2) /\ in1 /\ in 2
= false
Absence of unnecessary wait:
¬ in2 /\ ¬ in1

12. Invariant based approach
CS CS2
do (true) do (true) 
<await (¬ lock)  lock = true> <await(¬ lock)  lock = true>
critical section critical section
lock = false lock = false
non-critical section non-critical section
od od
lock = in1 \/ in2
<await (¬ in2)  in1 = true>

13. Test and set instruction
lock = false
CS1 CS2
do (true) do (true) 
while (TS(lock)) {}; while(TS(lock)) {};
critical section critical section
lock = false lock = false
non-critical section non-critical section od
TS(x) = {
temp = x;
x = true;
return temp }

14. Tie Breaker Algorithm in1 = false; in2 = false; last = 1 co CS1: CS2:
do true  do true 
last = 1; in1 = true; last = 2; in2 = true
while(in2 /\ last == 1); while(in1 /\ last == 2);
critical section critical section
in1 = false; in2 = false;
non-critical section non-critical section
od od oc

15. Barrier synchronization
Worker[i]: do
true 
code for task i
wait for all tasks to complete od

16. Barrier synchronization
Worker[i]: do
true 
code for task i
<count = count + 1>
< await( count == n) > od

17. Barrier synchronizationco
worker[i]: Coordinator
do true  do true 
code for task i; for (i = 1 to n)
arrive[i] = await(arrive[i]= 1);
await(continue ==1) continue = 1
od od oc

18. Barrier synchronizationco
worker[i]: Coordinator
do true  do true 
code for task I; for (i = 1 to n)
arrive[i] = { await(arrive[i]= 1);
await(continue[i]==1) arrive[i] = 0; }
continue[i] = 0; for (i = 1 to n) continue[i] = 1
od od oc
Flag rule: A process that waits for the synchronization flags should reset it.