1. CIS 720
Lecture 4

2. Concurrency rule for all i, { Pi } Si { Qi }
{ P1 /\ …. /\ Pn } co S1 // …. // Sn oc { Q1 /\ …. /\ Qn}

3. {true}
{true}
x = 0; y = 1; x = 0; co co x = 1 x = x + 1 // // y = y + 1 x = x + 2 oc oc
{ x = 0 /\ y = 1}
{ x = 0}
{x = 0}
{x = 0}
{x = 1}
{x = 1}
{y = 1}
{x = 0}
{y = 2}
{x = 2}
{ x = 1 /\ y = 2}
{ x = 0 /\ x = 2}

4. Interference freedom Let a be a statement and C be an assertion.
NI(a, C) iff { pre(a) /\ C } a { C }
Execution of a does not invalidates (or interferes) with C.
{pre(a) /\ C a C

5. { Pi } Si { Qi }, for all i, are interference free if
for all assertions C in proof outline of Si,
for all actions a in Sj, i != j
NI(a, C) holds

6. Concurrency rule { Pi } Si { Qi } are interference free
{ P1 /\ …. /\ Pn } co S1 // …. // Sn { Q1 /\ …. /\ Qn}

7. x= 0; co x = x + 1 // x = x + 2 od

8. Techniques to avoid interference
Disjoint variables
If the write set of each process is disjoint from the read and write set of other processes, then the proof outlines will be interference free.

9. Avoiding interference
Global Invariants: Assume that every assertion in the proof outline of Pi is of the form I /\ L, where L refers to local variables of Pi or to variables that only Pi writes.
- Then the proof outlines will be interference free.

10. Concurrency rule
for all i, { I} Si { I }
{ I } co S1 // …. // Sn { I}

11. x = 0 ; y = 0; z = 0 co
x = 1
y := 1 //
if y = 1  z = 1 oc

12. x = 0 ; y = 0; z = 0 co
x = 1
y := 1 //
if y = 1  z = 1 oc

13. Avoiding interference
Weakened assertions
x= 0; co
x = x + 1 //
x = x + 2 od
{x = 0}
{x = 0 \/ x = 2}

14. Avoiding interference
Synchronization

15. Bank transactions
co // Transfer(x,y,am): Auditor: ac[x] = ac[x] – am; total = 0; i = 0; ac[y] = ac[y] + am do i < n total = total + ac[i] i = i + 1 od oc

16. Avoiding interference
Synchronization
Await Statement rule
{P /\ B } S {Q}
{P} < await(B)  S > {Q}
{ y > 0 } y := x { x > 0 }
{ true } await(y > 0)  y := x { x > 0 }

17. b = false
x = 0 ; y = 0; z = 0 co
x = x + 1
b := true
z = x + 3 //
< await b  y = x > oc

18. b = false x = 0 ; y = 0; z = 0 co x = x + 1 b := true z = x + 3 //
< await b  y = x > oc
(I /\ x = 1 /\ not b) /\ (I /\ b)
false

19. Bank transactions
total = 0; i = 0; co // Transfer(x,y,am): Auditor: < await( (i < x /\ i < y) \/ {A1: Total = ac[1]+….+ac[i] } (i > x /\ i > y)) do i < n  ac[x] = ac[x] – am; i = i + 1 ac[y] = ac[y] + am > total = total + ac[i] od oc

20. Bank transactions
co // Transfer(x,y,am): Auditor: if < await(!auditon)  ac[x] = ac[x] – am; auditon = 1; total = 0; i = 0; ac[y] = ac[y] + am > [] do i < n <await (auditon && ( (x < i && y < i) ||(x > i && y > i))) total = total + ac[i]  ac[x] = ac[x] – am; ac[y] = ac[y] + am> i = i + 1 fi od auditon = 0 oc