Implementing Sequentially Consistent Programs on Processor Consistent Platforms

1
Implementing Sequentially Consistent Programson
Processor Consistent Platforms

• Lisa Higham and Jalal Kawash
• University of Calgary, Canada
• American University of Sharjah, UAE

2
Outline

• Memory consistency models
• Sequential consistency
• P-RAM
• Coherence
• PC-G
• Compiling from SC to PC-G
• Good news
• Bad news
• Proof

3
Multi-Processors Computation
..
P
P
P
P
o1
o1
o1
o1
o2
o2
o2
o2
O
o3
o3
o3
o3
..
..


4
Sequential Consistency
..
P
P
P
P
Switch
Shared Memory
5
Sequential Consistency Example
P
P
P
w(x, 3)
r(y) 4 r(x) 2
w(x, 2) w(y, 1)
w(x, 3) w(y, 4)
w(y, 4)
r(y) 4
w(x, 2)
w(y, 1)
r(x) 2
0
0
0
0
0
0
3
4
2
1
x
y
6
Sequential Consistency Definition

• A computation is sequentially consistent iff ? a
  valid total order on O such that
• (O, ) ? (O, )

7
P-RAM
x
y
z
x
y
z

P
P
FIFO Channels
Copies of Memory
P
P
x
y
z
x
y
z
8
P-RAM Example
x
y
x
y
0
0
0
0
2
1
0
0
0
0
2
1
y ? 2
P
w(x, 1)
x? 1
P
x? 1
y ? 2
y ? 2
x? 1
w(y, 2)
r(x) 1
r(y) 0
r(y) 2
P
r(x) 0
P
0
0
0
0
2
1
0
0
0
0
2
1
x
y
x
y
P
P
P
P
r(y) 2 r(x) 0
r(x) 1 r(y) 0
w(y, 2)
w(x, 1)
9
P-RAM Definition

• A computation is P-RAM iff for each process p,
  ? a valid total order such that
• (Op Ow, ) ? (Op Ow,
  )

n
n
10
Coherence
...
P
P
P
P
z
y
x
11
Coherence Example
P
P
P
P
P
w(x, 2)
w(y, 3)
x ? 2
y ? 3
r(y) 0
r(x) 0
w(x, 2)
r(y) 0
2
3
0
0
w(y, 3)
r(x) 0
y
x
12
Coherence Definition

• A computation is Coherent iff for each
  variable x, ? a valid total order
  such that
• (Ox , ) ? (Ox , )

13
PC-G P-RAM and Coherence
Coherence
P-RAM
PC-G
14
P-CG Definition

• A computation is P-CG iff for each process p,
  ? a valid total order such that
• (Op Ow, ) ? (Op Ow,
  )
• ? processes q, and ? variable x
• (Ow n Ox, ) (Ow n Ox, )

n
n
15
P-CG vs. SC

• Algorithms are designed for SC machines
• Some of them work directly when run on
• P-CG (e.g. Peterson 2)
• Most of the SC algorithms do not work on P-CG
  machines (e.g. testset and Bakery algorithm)

16

• Can we transform an SC algorithm to an equivalent
  P-CG algorithm?
• Can we find a compiler that transforms any SC
  algorithm to an equivalent P-CG algorithm?

17
Program Transformation and Interpretation
C
E
Program P
Execute P on SC
Transformation a
Interpretation
D
Program a(P)
Execute a(P) on PC-G
C Computations of P on SC machines D
Computations of a(P) on M machines E
Interpretations of D on SC machines
18
Program Implementation
C
E
Program P
Execute P on SC
Transformation a
Interpretation
D
Program a(P)
Execute a(P) on PC-G
If ? program P, a implements P, then a is a
compiler from SC to PC-G
C SC Computations of P D P-CG Computations
of a(P) E Interpretations of D
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Transformation Function a

• m a new multi-writer variable

20
Results

• Claim 1
• ? implements Lamports Bakery algorithm for 2
  processes on PC-G
• Claim 2
• ? is a compiler from SC to PC-G for any program
  provided
• Only 2 processes
• Only single-writer variables

21
Transformation Example
Program
P
P
w(x, 1) r(y)
w(y, 4) w(y, 2) r(x)
Under SC if r(y) returns 4, then r(x) returns 1
Under PC-G r(y) returns 4 and r(x) may return 0
22
Transformation Example
Program
?(Program)
P
P
w1(m, P)
w1(m, P)
w(x, 1)
w(y, 4)
w2(m, P)
w2(m, P)
r(y)
w3(m, P)
w(y, 2)
w4(m, P)
r(x)
23
Transformation Example
P
P
View for
View for
w1(m, P)
w1(m, P)
w1(m, P)
w1(m, P)
w(x, 1)
w(y, 4)
w(x, 1)
w(y, 4)
w2(m, P)
w2(m, P)
w2(m, P)
w2(m, P)
r(y)
w3(m, P)
w3(m, P)
w(y, 2)
w(y, 2)
If r(y) returns 4
w4(m, P)
r(x) must return 1
w4(m, P)
r(x)
24
Proof Sketch
Program
PC-G Views
P
P
P
P
w1
w
w2


25
Proof Sketch

• System view
• Contains all reads and writes by both processes
• Maintains program order
• Is valid

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Summary

• Compiler
• Only one additional variable
• Only writes to that variable
• Provided
• Two processors
• Single writer variables

27
Impossibilities

• For more than 2 processors, there is no compiler
  from SC to PC-G that
• Only adds write instructions (with any number of
  variables)
• nor
• Uses only one additional variable (with any
  number of reads and writes)

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Pros and Cons

• Restricted
• 2 processes
• Only single writer variables
• Valuable
• ME Lamports Bakery algorithm
• Wait-free testset had no known solutions in weak
  memory consistency

29
Wait-Free Testset

• Define testsetlt(O, i)gt
• if (si you and sj ? rst) the return 1
• repeat
• si ? choose
• case sj is
• you, rst
• si ? me
• me
• si ? you
• choose
• si ? random (me, you)
• end case
• until (si ? sj)
• if (si me) then return 0
• else return 1

Define resetlt(O, i)gt si ? rst
m? i
m? i
m? i
m? i
m? i
m? i
m? i
m? i
30
Conclusions

• a works for any two-process program with
• single-writer variables
• a works for particular programs with gt 2
  processes (randomized wait-free n-process
  testset)
• If there is a transformation that work for other
  cases, it must be more complicated
• Cannot be write-adding
• Must use more than one additional variable

31
Thank You?

32
Proof Sketch
Program
PC-G Views
P
P
P
P
w1
w
w2