1. CIS 720
Lecture 6

2. Invariant based approach
Identify synchronization regions in your program
Synchronization region: segment of code or control point at which a thread must wait for another thread or signal another thread.

3. Mutual exclusion CS1 CS2 do (true) do (true) 
****start of region **** start of region
critical section critical section
**** end of region **** end of region
non-critical section non-critical section od

4. Mutual exclusion CS1 CS2 Associate in and out counter with each region
do (true) do (true) 
****start of region **** start of region
in1 = in in2 = in2 + 1
critical section critical section
**** end of region **** end of region
out1 = out1+1; out2 = out2+1;
non-critical section non-critical section od
Associate in and out counter with each region

5. Mutual exclusion CS1 CS2 in1 = 0; out1 = 0 in2 = 0; out2 = 0
do (true) do (true) 
****start of region **** start of region
in1 = in in2 = in2 + 1
critical section critical section
**** end of region **** end of region
out1 = out1+1; out2 = out2+1;
non-critical section non-critical section
od od
Associate in and out counter with each region
Write an invariant using in and out counters
I : (in1=out1) \/ (in2=out2)
{ in2 = 0 /\ out2 = 0 }
{ in2 = out2 }
{ in2 = out2 + 1}
{ I }
{ in1 = 0 /\ out1 = 0 }
{ in1 = out1 }
{ I }
{ in1 = out1 + 1}
{ I }
{ in1 = out1 }
{ I }
{ in1 + 1= out1 \/ in2 = out2 } in1 = in1 + 1 { in1 = out1 \/ in2 = out2 }

6. Mutual exclusion CS1 CS2 do (true) do (true) 
****start of region **** start of region
<await(in2=out2) in1 = in1 + 1> <await(in1=out1) in2 = in2 + 1>
critical section critical section
**** end of region **** end of region
out1 = out1+1; out2 = out2+1;
non-critical section non-critical section od
Associate in and out counter with each region
Write an invariant using in and out counters
I = (in1=out1) \/ (in2=out2)
Derive guards for the increment assertions

7. Barrier Synchronization
P1 P2
do (true) do (true) 
task for p1 task for p2
****start of region **** start of region
**** end of region **** end of region od

8. Barrier CS1 CS2 Associate counters with each region
do (true) do (true) 
task for p1 task for p2
****start of region **** start of region
arrive1 = arrive arrive2 = arrive2 + 1
**** end of region **** end of region
depart1 = depart1+1; depart2 = depart2+1; od
Associate counters with each region

9. Barrier CS1 CS2 arrive1 = 0; depart1 = 0 arrive2 = 0; depart2 = 0
do (true) do (true) 
task for p1 task for p2
<await depart2 < arrive1  <await depart1 < arrive2 
arrive1 = arrive1 + 1 > arrive2 = arrive2 + 1 >
<await depart1 < arrive2  <await depart2 < arrive1 
depart1 = depart1+1; > depart2 = depart2+1 >
od od
Associate counters with each region
Write an invariant using the counters
(depart1 <= arrive2 /\ (depart2<= arrive1)
{ depart1 <= arrive2 /\ depart2 <= arrive1 + 1}
arrive1 = arrive1 + 1
{ depart1 <= arrive2 /\ depart2 <= arrive1 }

10. Semaphores Two operations: P(s), V(s) Semaphore invariant:
nP = number of P operations invoked so far
nV = number of V operations invoked so far
init = initial value of s
nP <= nV + init
nV + init – np >= 0
S = nv + init – np
S >= 0

11. Binary semaphore: 0 <= s <= 1
Let s = nV + init – nP
Invariant: s >= 0
P(s): nP = nP + 1
<await s > 0  s = s - 1>
V(s): nV = nV + 1
< s = s + 1>
Binary semaphore: 0 <= s <= 1

12. Mutual exclusion CSi do (true)
in[i] = 1
critical section
in[i] = 0
non-critical section od
I = ( 0 <= (in[1] + in[2] + …. + in[n]) <= 1 )
Let mutex = 1 - (in[1] + in[2] + …. + in[n])
I = ( 0 <= mutex <= 1 )

13. Mutual exclusion CSi do (true) in[i] = 1 mutex++ P(mutex)
critical section
in[i] = mutex V(mutex)
non-critical section od
I = ( 0 <= (in[1] + in[2] + …. + in[n]) <= 1 )
Let mutex = 1 - (in[1] + in[2] + …. + in[n])
I = ( 0 <= mutex <= 1 )

14. Barrier CS1 CS2 do (true) do (true)  task for p1 task for p2
****start of region **** start of region
arrive1 = arrive arrive2 = arrive2 + 1
**** end of region **** end of region
depart1 = depart1+1; depart2 = depart2+1; od
Barrier1 = (arrive1 – depart2)
Barrier2 = (arrive2 – depart1)

15. Barrier CS1 CS2 Barrier1 = (arrive1 – depart2)
do (true) do (true) 
task for p1 task for p2
barrier barrier2++
barrier2--; barrier1--; od
Barrier1 = (arrive1 – depart2)
Barrier2 = (arrive2 – depart1)

16. Producer/consumer problem
do true  do true 
produce pdata
buf = pdata cdata = buf;
consume cdata
od od oc

17. Producer/consumer problem
in1 = out1 = in2 = out2 = 0; co
Producer: Consumer:
do true  {in1=out1} do true  {in2=out2}
produce pdata
in1 = in in2 = in2 + 1
buf = pdata {in1 = out1+1} cdata = buf {in2 = out2+1}
out1 = out out2 = out2 + 1
{in1=out1} consume cdata {in2 = out2}
od od oc
Invariant: in2 <= out1 /\ in1 <= out2 + 1

18. Producer/consumer problem
do true  do true 
produce pdata
<await(in1 <=out2)  in1 = in1 + 1> <await(in2 < out1) in2 = in2 + 1>
buf = pdata cdata = buf
<out1 = out1 + 1> <out2 = out2 + 1>
consume cdata
od od oc
Invariant: (in1 <= out2 + 1) /\ (in2 <= out1)
Empty = out2 + 1 – in1
Full = out1 – in2

19. Producer/consumer problem
do true  do true 
produce pdata
<await(empty > 0)  empty= empty -1> <await(full > 0)  full = full -1 >
buf = pdata cdata = buf
<full = full + 1> <empty = empty + 1>
consume cdata
od od oc
Invariant: (in1 <= out2 + 1) /\ (in2 <= out1)
empty = out2 + 1 in1
full = out1 – in2

20. Producer/consumer problem
do true  do true 
produce pdata
P(empty) P(full)
buf = pdata cdata = buf
V(full) V(empty)
consume cdata
od od oc
Invariant: (in1 <= out2 + 1) /\ (in2 <= out1)
empty = out2 + 1 in1
full = out1 – in2

21. Producer/consumer problem
in1 = out1 = in2 = out2 = 0; co
Producer: Consumer1: Consumer2:
do true  do true  do true  produce pdata
in1 = in in2 = in in3 = in3 + 1
buf = pdata cdata = buf cdata = buf
out1 = out out2 = out out3 = out3 + 1
consume cdata consume cdata
od od od oc
Invariant: