Constructing Stronger Registers from Weaker Ones

1
Constructing Stronger Registers from Weaker Ones
Yuliang Bao Adesola
Omotayo Instructor Dr. Lisa Higham
December 11, 2003
2
Outline

• Definitions
• Views of a system
• Registers
• Constructions
• Conclusions

3
Definition Atomic operation

• Execution performed as an indivisible action
• Example Transfer money from a bank account to
  another
• Get the balance of source account
• Subtract the amount to transfer
• Write the new balance of source account
• Get the balance of destination account
• Add the amount to transfer
• Write the new balance of destination account

4
Definition Operation Execution

• A specific instance of executing an operation
• Example Transfer 20 from account 1 to account
  2
• Get the balance of account 1
• Subtract the amount to transfer
• Write the new balance of account 1
• Get the balance of account 2
• Add the amount to transfer
• Write the new balance of account 2

5
Definition Serializable Execution Order

• Scheduling of events so that the net effect is
  the same as if the events had been executed in
  some serial order
• Example Transfer 20 from account 1 to account
  2
• Get the balance of account 1
• Get the balance of account 2
• Subtract the amount to transfer from account 1
• Add the amount to transfer to account 2
• Write the new balance of account 1
• Write the new balance of account 2

Operation A
Operation B
Operation C
6
Definition Precedence Relations

• Precedence Relations
• Precede A B ?a ? µ(A) ?b ?
  µ(B) a b
• Can affect A B ?a ? µ(A) ?b ?
  µ(B) a b
• Example Transfer 20 from account 1 to account
  2

Operation A
Operation B
a1 Get the balance of account 1 a2 Get the
balance of account 2
b1 Subtract transfer amount from 1 b2 Add the
amount to transfer to 2
7
Definition System Execution

• A set of operation executions plus temporal
  precedence relations on those operations
• A triple ltS, , gt where S is a finite
  or countably infinite set whose elements are
  called operation executions, and and
  are precedence relations on S

8
Axioms

• A1 The relation is an irreflexive
  partial ordering
• A2 If A B then A B and B
  A
• A3 If A B C or A B
  C then A C
• A4 If A B C D then A
  D
• A5 For any A, the set of all B such that A
  B is finite

Time
A B C D
9
Views of a system

• Lower Level
• A system execution implementing a higher level
  system execution e.g ltS, , gt
• High Level
• A set of lower level operations that implement it
  e.g ltH, , gt
• Let E, , µ be a model for ltS, ,
  gt, and let the mapping of µ on H be µ(H)
  Uµ(A) A ? H then
• G H ?g ? µ(G) ?h ? µ(H) g
  h
• G H ?g ? µ(G) ?h ? µ(H) g
  h or g h

10
Views of a system

• Higher Level
• A system execution ltS, , gt
  implementing a system execution ltH, ,
  gt
• H1 Each element of H is a finite, nonempty set
  of elements of S
• H2 Each element of S belongs to a finite,
  nonzero number of elements of H
• H3 For any G, H ? H if G H then G
  H

11
Registers Space

• Usually specified in the following format x,y,z

12
Registers Access

• x Coordinate
• Single-Writer Single-Reader (SWSR)
• Multi-Writer Single-Reader (MWSR)
• Single-Writer Multi-Reader (SWMR)
• Multi-Writer Multi-Reader (MWMR)

13
Registers Values

• y Coordinate
• Boolean 0, 1
• Multi-valued 1n

14
Registers Strength

• z Coordinate
• Safe
• Regular
• Atomic

15
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
O(n)
O(K)
O(K)
O(1)
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(1)
O(K)
SRSW,B,R
SRSW,MV,S
MRSW,B,S
SRSW,B,S
16
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
O(n)
O(K)
O(K)
O(1)
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(1)
O(K)
SRSW,B,R
SRSW,MV,S
MRSW,B,S
SRSW,B,S
17
Construction 1SRSW,MV,S/R ? MRSW,MV,S/R

• Let ?1,,?m be single-reader, n-valued registers,
  where each ?i can be written by the same writer
  and read by process i, and construct a single
  n-valued register ? in which the operation ? µ
  is performed as follows
• for all i in 1,,m do ?i µ od
• and process i reads ? by reading the value of
  ?i. If the ?i are safe or regular registers, then
  ? is a safe or regular register, respectively

18
SRSW,MV,S/R ? MRSW,MV,S/R
19
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
O(n)
O(K)
O(K)
O(1)
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(1)
O(K)
SRSW,B,R
SRSW,MV,S
MRSW,B,S
SRSW,B,S
20
Construction 2MRSW,B,S ? MRSW,MV,S

• Let ?1,,?n be boolean m-reader registers, each
  written by the same writer and read by the same
  set of readers. Let ? be the 2n-valued, m-reader
  register in which the number with binary
  representation µ1 µn is written by
• for all i in 1,,n do ?i µi od
• and in which the value is read by reading all
  the ?i. If each ?i is safe, then ? is safe

21
MRSW,B,S ? MRSW,MV,S
22
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
O(n)
O(K)
O(K)
O(1)
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(1)
O(K)
SRSW,B,R
SRSW,MV,S
MRSW,B,S
SRSW,B,S
23
Construction 3MRSW,B,S ? MRSW,B,R

• Let ? be an m-reader boolean register, and let x
  be a variable internal to the writer (not a
  shared register) initially equal to the initial
  value of ?. Define ? to be the m-reader boolean
  register in which the write operation ? µ
  is performed as follows
• If x ? µ then ? µ
• x µ fi
• and a read of ? is performed by reading ?. If ?
  is a safe then ? is regular register

24
MRSW,B,S ? MRSW,B,R
25
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
O(n)
O(K)
O(K)
O(1)
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(1)
O(K)
SRSW,B,R
SRSW,MV,S
MRSW,B,S
SRSW,B,S
26
Construction 4(MRSW,B,R ? MRSW,MV,R)
Algorithm

• The write ? ? is performed by
• 1 ?? 1
• 2 for i ? -1 step 1 until 1 do ?i 0 od
• A read is performed by
• 1 ? 1
• 2 while ?? 0 do ? ? 1 od
• 3 return ?

• Notation
• ?1,, ?n MRSW,B,R
• ? MRSW,MV,R

27
Construction 4(MRSW,B,R ? MRSW,MV,R)
?1 ?2 ??-1 ?? ?n
0 0 0 1
1

• ?i atomic ?i ? ? atomic

28
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
O(n)
O(K)
O(K)
O(1)
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(1)
O(K)
SRSW,B,R
SRSW,MV,S
MRSW,B,S
SRSW,B,S
29
Construction 5(SRSW,MV,R ? SRSW,MV,A)

• Notation
• w, r write and read processes
• V a finite set
• v SRSW,MV,R written by w and read by r and w
• V ? V ? 1,2,3 ? true, false
• num(v) initially 3
• E.g. for v (x,y,i,k), old(v) x, new(v) y,
  num(v) i, col(v) k
• c SRSW,B,R written by r and read by w
• V SRSW,MV,A written by w and read by r
• nc, x, rv, rv, nuret local variables
• nuret initially, false nuret true if the
  reader returned the y value on the preceding
  read.

30
Construction 5(SRSW,MV,R ? SRSW,MV,A)
Algorithm

• The write v y is performed by
• 1 nc ?c
• 2 x old(v)
• 3 for i 1 until 3 do v (x,y,i,nc) od

31
Construction 5(SRSW,MV,R ? SRSW,MV,A)

• A read is performed by
• 1 rv rv
• 2 rv v
• 3 c col(rv)
• 4 if num(rv) 3
• 5 then nuret true
• 6 return new(rv)
• 7 else if nuret ? col(rv) col(rv) ?
  num(rv) ? num(rv) 1
• 8 then return new(rv)
• 9 else nuret false
• 10 return old(rv)
• fi
• fi

32
Construction 5(SRSW,MV,R ? SRSW,MV,A)
rv
Case 2
(x, y, 2, k)
ret x
33
Construction 5(SRSW,MV,R ? SRSW,MV,A)
Case 2
nuret F
34
Construction 5(SRSW,MV,R ? SRSW,MV,A)
rv
(x, y, 3, k)
Case 3
ret y
col(rv) col(rv) k, 2 ? 3-1
rv
rv'
(y, z, 1, k)
(y, z, 2, k)
ret z
col(rv) col(rv) k, 1 ? 2-1
35
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
O(n)
O(K)
O(K)
O(1)
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(1)
O(K)
SRSW,B,R
SRSW,MV,S
MRSW,B,S
SRSW,B,S
36
Construction 6(SRSW,B,A ? SRSW,MV,A)
Algorithm
when write(R,v) occurs // the writer writes
the value v to R 1 Bv 1 2 for i
v 1 downto 0 do Bi 0 3 ack(R)
Notation B0,Bk-1 SRSW,B,A initially all
0, except Bi 1 (i initial value
of the simulated register R)
37
Construction 6(SRSW,B,A ? SRSW,MV,A)
Algorithm
when read(R) occurs // the reader reads
from R 1 i 0 2 while Bi 0 do i i
1 // forward scan 3 up, v i 4
for i up 1 downto 0 do // backward scan 5
if Bi 1 then v i 6 return (R,v)
38
Proof of algorithm(SRSW,B,A ? SRSW,MV,A)

• No overlap between read and write Every read
  returns the value of the latest preceding write.
• Overlap between reads R1, R2 and write w2(avoid
  new-old value problem)

0 1 5 8 k-1
0 0 1 1 0
B
0 1 5 8 k-1
0 0 1 1 0
B
R
R
R1
R1
R2
R2
r1(R) r2(R)8

• r1(R) r2(R)5

39
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
O(n)
O(K)
O(K)
O(1)
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(1)
O(K)
SRSW,B,R
SRSW,MV,S
MRSW,B,S
SRSW,B,S
40
Construction 7 (SRSW,MV,A ? MRSW,MV,A)

• One writer pw and n readers p1, , pn
• Vali (value, seq) written by pw and read by
  reader pi, 1 ? i ? n
• Reporti,j The value returned by the most
  recent read operation performed by pi written by
  pi, read by pj, 1 ? i,j ? n
• Initially, Reporti,j Vali (v0,0), 1 ? i,j
  ? n, v0 initial value of R

Algorithm
when write(R,v) occurs // the writer writes v
to R 1 seq seq 1 2 for i 1 to n
do Vali (v, seq) 3 ack(R)
41
Construction 6(SRSW,B,A ? SRSW,MV,A)
i 1 2 3 4 5 6
1 1 1 1 1 1
5 5 5 5 5 5
V
Vali
seq
Write(R,10)
10 10 10 10 10 10
6 6 6 6 6 6
V
Vali
seq
42
Construction 7 (SRSW,MV,A ? MRSW,MV,A)
Algorithm

• when readr(R) occurs // the reader
  pr reads from R
• 1 (v0, s0) Valr // value
  reported to pr by writer
• 2 for i 1 to n do
• (vi, si) Reporti,r // value
  reported to pr by reader pi
• 3 let j be such that sj maxs0,s1,,sn
  
• 4 for i 1 to n do
• Reportr,i (vj, sj) // pr
  reports to each pi
• 5 returnr (R,vj)

43
Proof of algorithm (SRSW,MV,A ? MRSW,MV,A)

• P4 read Val4 where s0 3,
• P4 receives Report(i,4), ? i

read4(R)
Seq in Report(i,j)
1 2 3 4 5 6
1 6 6 6 6 6 6
2 8 8 8 8 8 8
3 3 3 3 3 3 3
4 2 2 2 2 2 2
5 4 4 4 4 4 4
6 1 1 1 1 1 1
(2) s2 maxs0,,s6
max3,6,8,3,2,4,1 8
44
Proof of algorithm (SRSW,MV,A ? MRSW,MV,A)
(1) P4 receives Report(i,4), ? i
Seq in Report(i,j)
1 2 3 4 5 6
1 6 6 6 6 6 6
2 8 8 8 8 8 8
3 3 3 3 3 3 3
4 8 8 8 8 8 8
5 4 4 4 4 4 4
6 1 1 1 1 1 1
(2) Assume s0 3 s2 maxs0,,sn
max3,6,8,3,2,4,1 8
45
Proof of algorithm (SRSW,MV,A ? MRSW,MV,A)

• To prove For any admissible execution ?, ? a
  permutation ? of the high-level operations in ?,
  such that ? preserves the order of
  non-overlapping operations, and every read
  operation returns the value of the latest
  preceding write.
• Constructing ? in two steps
• (1) Put in ? all the write operations
  according to their order in ?
• (2) add each read operation to ? according
  to its response in ?
• w(x,T) ? r(x,T) ? w(y,T1)

46
Proof of algorithm (SRSW,MV,A ? MRSW,MV,A)
Lemma Let op1 and op2 be two high-level
operations in ? such that op1 ends before op2
begins. Then op1 precedes op2 in ?.

• Proof For a read operation, r, by pi that
  returns a value with timestamp T,
• A write w follows r in ? (by contradiction,
  suppose w ? r in ?)

47
Proof of algorithm (SRSW,MV,A ? MRSW,MV,A)
Proof For a read operation, r, by pi that
returns a value with timestamp T,

• A write w precedes r in ? (Vali written by
  w or a later write)

• A read r by pj follows r in ? (Report(i)
  written by pi during r or later)

48
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
O(n)
O(K)
O(K)
O(1)
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(1)
O(K)
SRSW,B,R
SRSW,MV,S
MRSW,B,S
SRSW,B,S
49
Construction 8 (MRSW,MV,A ? MRMW,MV,A)

• m writers (p0,, pm-1) and n readers (p0,,
  pn-1)
• TSi The vector timestamp of writer pi,
• written by writer pi and read by
  all writers
• Vali (value, vector timestamp),
• written by writer pi and read by
  all readers
• Initially, TSi ?0,,0? and Vali initial
  value of R, 0 ? i ? m 1

50
Construction 8 (MRSW,MV,A ? MRMW,MV,A)

• // writer pw writes v to register R, 0 ? w ? m
  1
• when writew(R,v) occurs
• 1 ts NewCTS()
• 2 Valw (v, ts)
• 3 ackw(R)
• procedure NewCTSw() // writer pw obtains a
  new TS
• 1 for i 0 to m-1 do // extract the i-th
  entry
• ltsi TSi.i // from TS of
  i-th writer
• 2 ltsw ltsw 1 // increment own
  entry
• 3 TSw lts // write to
  shared register
• 4 return lts

51
Construction 8 (MRSW,MV,A ? MRMW,MV,A)

• // reader pr reads from register R, 0 ? r ? n 1
• when readr(R) occurs
• 1 for i 0 to m - 1 do
• (vi, ti) Vali // v and
  t are local variables
• 2 let j be such that tj
  maxt0,t1,,tm-1
• //
  lexicographic max
• 3 returnr(R,vj)

52
Proof of algorithm (SRSW,MV,A ? MRSW,MV,A)
Write2 (R, 10)
0 1 2
0 3 2 1
1 2 2 1
2 2 1 1
0 1 2
0 3 2 1
1 2 2 1
2 3 2 2
TS
TS
Val2 (10, TS2)
Readr (R)
0 1 2
0 3 2 1
1 2 2 1
2 3 2 2
TS
TS2 maxTS0, TS1, TS2 Return v2

53
Proof of algorithm (MRSW,MV,A ? MRMW,MV,A)

• Lemma The lexicographic order of the timestamps
  is a total order consistent with the partial
  order in which they are generated.

TS
0 1
0 0 0
1 0 0
0 1
0 1 0
1 0 0
0 1
0 1 0
1 1 1
0 1
0 2 1
1 1 1
write0(R)
write1(R)
write0(R)
TS0 ? TS1
54
Proof of algorithm (MRSW,MV,A ? MRMW,MV,A)

• ? an admissible execution
• ? a permutation of the high-level operations in
  ? (same as
• the previous algorithm)
• Constructing ? in two steps
• (1) Put in ? all the write operations
  according to their lexicographic ordering on the
  timestamps in ?
• (2) add each read operation to ? according
  to its response in ?
• w(x,VT) ? r(x,VT) ? w(y,VT)

55
Proof of algorithm (MRSW,MV,A ? MRMW,MV,A)
Lemma Let op1 and op2 be two high-level
operations in ? such that op1 ends before op2
begins. Then op1 precedes op2 in ?.

• Proof For a read operation, r, by pi that
  returns a value with timestamp VT,
• A write w follows r in ? (by contradiction,
  suppose w ? r in ?)

56
Proof of algorithm (MRSW,MV,A ? MRMW,MV,A)
Proof For a read operation, r, by pi that
returns a value with timestamp VT,

• A write w by pj precedes r in ? (Val j
  written by w or a later write)

• A read r by pj follows r in ? (TS
  non-decreasing lexicographic order)

TS(r) ? TS(r)
57
RegistersHasse Diagram
m of writers n of readers k of values
MRMW,MV,A
O(m)
MRSW,MV,A
MRMW,MV,R
O(n)
MRSW,MV,R
MRSW,MV,R
SRSW,MV,A
5
O(n)
O(K)
O(K)
O(1)
4
SRSW,B,A
MRSW,MV,S
SRSW,MV,R
MRSW,B,R
MRSW,MV,S
O(n)
O(K)
5
O(1)
O(K)
1
4
O(1)
3
2
SRSW,B,R
SRSW,MV,S
MRSW,B,S
1
3
O(K)
2
O(n)
O(1)
SRSW,B,S
58
Conclusions

• Lamports paper and other related papers were
  thoroughly reviewed.
• Eight Constructions and their proofs are well
  understood and described in this project.
• Five new constructions were found and presented
  by studying previous constructions.

59
Thank you!

• Dr. Lisa Higham
• Fellow graduate students

60
References

• L. Lamport, On Interprocess Communication,
  Parts 1 and 2, Dec. 25, 1985
• H. Attiya and J. Welch, Distributed Computing