1. Concurrent Objects Companion slides for
The Art of Multiprocessor Programming
by Maurice Herlihy & Nir Shavit

2. Concurrent Computation
memory
object
object
Art of Multiprocessor Programming

3. Art of Multiprocessor Programming
Objectivism
What is a concurrent object?
How do we describe one?
How do we implement one?
How do we tell if we’re right?
In this lecture we will focus on the first and last items.
Art of Multiprocessor Programming

4. Art of Multiprocessor Programming
Objectivism
What is a concurrent object?
How do we describe one?
How do we tell if we’re right?
Art of Multiprocessor Programming

5. FIFO Queue: Enqueue Method
q.enq( )
Consider a FIFO queue.
Art of Multiprocessor Programming

6. FIFO Queue: Dequeue Method
q.deq()/
Art of Multiprocessor Programming

7. Art of Multiprocessor Programming
Lock-Based Queue 1
tail
head x y 2 7 3 6 5 4
capacity = 8
Art of Multiprocessor Programming

8. Art of Multiprocessor Programming
Lock-Based Queue 1
tail
head x y 2 7 3 6
Fields protected by single shared lock 5 4
capacity = 8
Art of Multiprocessor Programming

9. Art of Multiprocessor Programming
A Lock-Based Queue 1
capacity-1 2
head
tail y z
class LockBasedQueue<T> {
int head, tail;
T[] items;
Lock lock;
public LockBasedQueue(int capacity) {
head = 0; tail = 0;
lock = new ReentrantLock();
items = (T[]) new Object[capacity]; }
Here is code for implementing a concurrent FIFO queue. Explain it IN DETAIL to the students. It’s the first algorithm they see which is not a mutual exclusion algorithm.
The queue is implemented in an array.
Initially the Head and Tail fields are equal and the queue is empty. All operations are synchronized
Via the objects lock.
If the Head and Tail differ by the queue size, then the
queue is full. The Enq() method reads the Head()
field, and if the queue is full, it waits until the queue is no longer full. It then stores
the object in the array, and increments the Tail() field.
Fields protected by single shared lock
Art of Multiprocessor Programming

10. Art of Multiprocessor Programming
Lock-Based Queue
Initially head = tail
tail 1
head 2 7 3 6 5 4
Art of Multiprocessor Programming

11. Art of Multiprocessor Programming
A Lock-Based Queue 1
capacity-1 2
head
tail y z
class LockBasedQueue<T> {
int head, tail;
T[] items;
Lock lock;
public LockBasedQueue(int capacity) {
head = 0; tail = 0;
lock = new ReentrantLock();
items = (T[]) new Object[capacity]; }
Here is code for implementing a concurrent FIFO queue. Explain it IN DETAIL to the students. It’s the first algorithm they see which is not a mutual exclusion algorithm.
The queue is implemented in an array.
Initially the Head and Tail fields are equal and the queue is empty. All operations are synchronized
Via the objects lock.
If the Head and Tail differ by the queue size, then the
queue is full. The Enq() method reads the Head()
field, and if the queue is full, it waits until the queue is no longer full. It then stores
the object in the array, and increments the Tail() field.
Initially head = tail
Art of Multiprocessor Programming

12. Art of Multiprocessor Programming
Lock-Based deq() 1
tail
head x y 7 2 6 3 5 4
Art of Multiprocessor Programming

13. Art of Multiprocessor Programming
Acquire Lock 1
tail
head x y
My turn …
Waiting to enqueue… 7 2 6 3 5 4
Art of Multiprocessor Programming

14. Implementation: deq()
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); }
Acquire lock at method start 1
capacity-1 2
head
tail y z
Art of Multiprocessor Programming

15. Art of Multiprocessor Programming
Check if Non-Empty 1
tail
head x y
Waiting to enqueue… 7 2 6 3 5 4
Art of Multiprocessor Programming

16. Implementation: deq()
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); } 1
capacity-1 2
head
tail y z
If queue empty
throw exception
Art of Multiprocessor Programming

17. Art of Multiprocessor Programming
Modify the Queue
head 1
tail
head x y
Waiting to enqueue… 7 2 6 3 5 4
Art of Multiprocessor Programming

18. Implementation: deq()
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); } 1
capacity-1 2
head
tail y z
Queue not empty?
Remove item and update head
Art of Multiprocessor Programming

19. Implementation: deq()
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); } 1
capacity-1 2
head
tail y z
Return result
Art of Multiprocessor Programming

20. Art of Multiprocessor Programming
Release the Lock
head 1
tail y
My turn! 7 2 6 3 x 5 4
Art of Multiprocessor Programming

21. Implementation: deq()
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); } 1
capacity-1 2
head
tail y z
Release lock no matter what!
Art of Multiprocessor Programming

22. Implementation: deq()
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); }
modifications are mutually exclusive…
Should be correct because
Art of Multiprocessor Programming

23. Now consider the following implementation
The same thing without mutual exclusion
For simplicity, only two threads
One thread enq only
The other deq only
This is sometimes called a producer-consumer channel or buffer.
Art of Multiprocessor Programming

24. Wait-free 2-Thread Queue1
tail
head x y 7 2 6 3 5 4
capacity = 8
Art of Multiprocessor Programming

25. Wait-free 2-Thread Queue
head 1
tail x y
deq()
enq(z) 7 2 6 3 5 4 z
Art of Multiprocessor Programming

26. Wait-free 2-Thread Queue
head 1
tail x y
result = x
queue[tail] = z 7 2 6 3 5 4 z
Art of Multiprocessor Programming

27. Wait-free 2-Thread Queue
head 1
tail y
head++
tail-- 7 2 z 6 3 x 5 4
Art of Multiprocessor Programming

28. Wait-free 2-Thread Queue
public class WaitFreeQueue {
int head = 0, tail = 0;
items = (T[]) new Object[capacity];
public void enq(Item x) {
if (tail-head == capacity) throw
new FullException();
items[tail % capacity] = x; tail++; }
public Item deq() {
if (tail == head) throw
new EmptyException();
Item item = items[head % capacity]; head++;
return item; }} 1
capacity-1 2
head
tail y z
No lock needed !
Art of Multiprocessor Programming

29. Wait-free 2-Thread Queue
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); }
How do we define “correct” when modifications are not mutually exclusive?
Art of Multiprocessor Programming

30. What is a Concurrent Queue?
Need a way to specify a concurrent queue object
Need a way to prove that an algorithm implements the object’s specification
Lets talk about object specifications …
In general, to make our intuitive understanding that the algorithm is correct we need a way to specify a concurrent queue object. Need a way to prove that the algorithm implements the object’s specification.
We saw two types of object implementations, one that used locks and intuitively felt like a queue, and one that did not use locks, it was infact, wait-free. What properties are they actually providing, are they both implementations of a concurrent queue? In what ways are they similar and in what ways do they differ?
Lets try and answer these questions.
Art of Multiprocessor Programming

31. Correctness and Progress
In a concurrent setting, we need to specify both the safety and the liveness properties of an object
Need a way to define
when an implementation is correct
the conditions under which it guarantees progress
There are two elements in which a concurrent specification imposes terms on the implementation: safety and liveness, or, in our case, correctness and progress.
Lets begin with correctness
Art of Multiprocessor Programming

32. Art of Multiprocessor Programming
Sequential Objects
Each object has a state
Usually given by a set of fields
Queue example: sequence of items
Each object has a set of methods
Only way to manipulate state
Queue example: enq and deq methods
Keep going back to the queue example throughout this lecture. Later, if you need to talk about multiple objects, give as an example a program that uses a stack and a queue.
Art of Multiprocessor Programming

33. Sequential Specifications
If (precondition)
the object is in such-and-such a state
before you call the method,
Then (postcondition)
the method will return a particular value
or throw a particular exception.
and (postcondition, con’t)
the object will be in some other state
when the method returns,
Art of Multiprocessor Programming

34. Pre and PostConditions for Dequeue
Precondition:
Queue is non-empty
Postcondition:
Returns first item in queue
Removes first item in queue
Art of Multiprocessor Programming

35. Pre and PostConditions for Dequeue
Precondition:
Queue is empty
Postcondition:
Throws Empty exception
Queue state unchanged
Art of Multiprocessor Programming

36. Why Sequential Specifications Totally Rock
Interactions among methods captured by side-effects on object state
State meaningful between method calls
Documentation size linear in number of methods
Each method described in isolation
Can add new methods
Without changing descriptions of old methods
Art of Multiprocessor Programming

37. What About Concurrent Specifications ?
Methods?
Documentation?
Adding new methods?
Art of Multiprocessor Programming

38. Art of Multiprocessor Programming
Methods Take Time
In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time.
time
time
Art of Multiprocessor Programming

39. Art of Multiprocessor Programming
Methods Take Time
invocation 12:00
q.enq(...)
In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time.
time
time
Art of Multiprocessor Programming

40. Art of Multiprocessor Programming
Methods Take Time
invocation 12:00
q.enq(...)
Method call
In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time.
time
time
Art of Multiprocessor Programming

41. Art of Multiprocessor Programming
Methods Take Time
invocation 12:00
q.enq(...)
Method call
In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time.
time
time
Art of Multiprocessor Programming

42. Art of Multiprocessor Programming
Methods Take Time
invocation 12:00
void
response 12:01
q.enq(...)
Method call
In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time.
time
time
Art of Multiprocessor Programming

43. Sequential vs Concurrent
Methods take time? Who knew?
Concurrent
Method call is not an event
Method call is an interval.
Art of Multiprocessor Programming

44. Concurrent Methods Take Overlapping Time
Art of Multiprocessor Programming

45. Concurrent Methods Take Overlapping Time
Method call
time
time
Art of Multiprocessor Programming

46. Concurrent Methods Take Overlapping Time
Method call
Method call
time
time
Art of Multiprocessor Programming

47. Concurrent Methods Take Overlapping Time
Method call
Method call
Method call
time
time
Art of Multiprocessor Programming

48. Sequential vs Concurrent
Object needs meaningful state only between method calls
Concurrent
Because method calls overlap, object might never be between method calls
Art of Multiprocessor Programming

49. Sequential vs Concurrent
Each method described in isolation
Concurrent
Must characterize all possible interactions with concurrent calls
What if two enqs overlap?
Two deqs? enq and deq? …
Art of Multiprocessor Programming

50. Sequential vs Concurrent
Can add new methods without affecting older methods
Concurrent:
Everything can potentially interact with everything else
Art of Multiprocessor Programming

51. Sequential vs Concurrent
Can add new methods without affecting older methods
Concurrent:
Everything can potentially interact with everything else
Panic!
Art of Multiprocessor Programming

52. Art of Multiprocessor Programming
The Big Question
What does it mean for a concurrent object to be correct?
What is a concurrent FIFO queue?
FIFO means strict temporal order
Concurrent means ambiguous temporal order
Art of Multiprocessor Programming

53. Art of Multiprocessor Programming
Intuitively…
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); }
Art of Multiprocessor Programming

54. Intuitively… All queue modifications are mutually exclusive
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); }
All queue modifications
are mutually exclusive
Art of Multiprocessor Programming

55. Art of Multiprocessor Programming
Intuitively
Lets capture the idea of describing
the concurrent via the sequential
q.deq
lock()
unlock()
enq
deq
deq
q.enq
enq
Behavior is “Sequential”
On an intuitive level, since we use mutual exclusion locks, then each of the actual updates happens in a non-overlapping interval so the behavior as a whole looks sequential. This fits our intuition, and the question is if we can generalize this intuition. Stress the fact that we understand the concurrent execution because we have a way of mapping it to the sequential one, in which we know how a FIFO queue behaves.
time
Art of Multiprocessor Programming

56. Art of Multiprocessor Programming
Linearizability
Each method should
“take effect”
Instantaneously
Between invocation and response events
Object is correct if this “sequential” behavior is correct
Any such concurrent object is
Linearizable™
Linearizability is the generalization of this intuition to general objects, with our without mutual exclusion.
Art of Multiprocessor Programming

57. Is it really about the object?
Each method should
“take effect”
Instantaneously
Between invocation and response events
Sounds like a property of an execution…
A linearizable object: one all of whose possible executions are linearizable
Art of Multiprocessor Programming

58. Art of Multiprocessor Programming
Example
Take you time with this example…don’t let them guess this one, show them, and let them guess the linearizability of the subsequent ones.
time
time
Art of Multiprocessor Programming

59. Art of Multiprocessor Programming
Example
q.enq(x)
time
time
Art of Multiprocessor Programming

60. Art of Multiprocessor Programming
Example
q.enq(x)
q.enq(y)
time
time
Art of Multiprocessor Programming

61. Art of Multiprocessor Programming
Example
q.enq(x)
q.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

62. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
q.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

63. Art of Multiprocessor Programming
Example
linearizable
q.enq(x)
q.enq(x)
q.enq(y)
q.deq(x)
q.deq(y)
q.deq(y)
q.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

64. Art of Multiprocessor Programming
Example
q.enq(x)
q.enq(x)
Valid?
q.deq(y)
q.deq(y)
q.enq(y)
q.enq(y)
q.deq(x)
q.deq(x)
time
time
Art of Multiprocessor Programming

65. Art of Multiprocessor Programming
Example
time
Art of Multiprocessor Programming

66. Art of Multiprocessor Programming
Example
q.enq(x)
time
Art of Multiprocessor Programming

67. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
time
Art of Multiprocessor Programming

68. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
q.enq(y)
time
Art of Multiprocessor Programming

69. Art of Multiprocessor Programming
Example
q.enq(x)
q.enq(y)
q.enq(x)
q.deq(y)
q.enq(y)
time
Art of Multiprocessor Programming

70. Art of Multiprocessor Programming
Example
not linearizable
q.enq(x)
q.enq(x)
q.deq(y)
q.enq(y)
q.enq(y)
time
(5)
Art of Multiprocessor Programming

71. Art of Multiprocessor Programming
Example
time
time
Art of Multiprocessor Programming

72. Art of Multiprocessor Programming
Example
q.enq(x)
time
time
Art of Multiprocessor Programming

73. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(x)
time
time
Art of Multiprocessor Programming

74. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(x)
q.enq(x)
q.deq(x)
time
time
Art of Multiprocessor Programming

75. Art of Multiprocessor Programming
Example
linearizable
q.enq(x)
q.deq(x)
q.enq(x)
q.deq(x)
time
time
Art of Multiprocessor Programming

76. Art of Multiprocessor Programming
Example
q.enq(x)
time
time
Art of Multiprocessor Programming

77. Art of Multiprocessor Programming
Example
q.enq(x)
q.enq(y)
time
time
Art of Multiprocessor Programming

78. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
q.enq(y)
time
time
Art of Multiprocessor Programming

79. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
q.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

80. Art of Multiprocessor Programming
Example
Comme ci
Comme ça
multiple orders OK
q.enq(x)
q.deq(y)
linearizable
q.enq(y)
q.deq(x)
time
Art of Multiprocessor Programming

81. Read/Write Register Example
time
time
Art of Multiprocessor Programming

82. Read/Write Register Example
write(1) already happened
time
time
Art of Multiprocessor Programming

83. Read/Write Register Example
write(1) already happened
time
time
Art of Multiprocessor Programming

84. Read/Write Register Example
not linearizable
write(0)
read(1)
write(2)
write(1)
write(1)
read(0)
write(1) already happened
time
time
Art of Multiprocessor Programming

85. Read/Write Register Example
write(1) already happened
time
time
Art of Multiprocessor Programming

86. Read/Write Register Example
write(1) already happened
time
time
Art of Multiprocessor Programming

87. Read/Write Register Example
not linearizable
write(0)
read(1)
write(2)
write(2)
write(1)
write(1)
read(1)
write(1) already happened
time
time
Art of Multiprocessor Programming

88. Read/Write Register Example
time
time
Art of Multiprocessor Programming

89. Read/Write Register Example
time
time
Art of Multiprocessor Programming

90. Read/Write Register Example
linearizable
write(0)
write(1)
write(2)
write(2)
write(1)
read(1)
time
time
Art of Multiprocessor Programming

91. Read/Write Register Example
time
time
Art of Multiprocessor Programming

92. Read/Write Register Example
time
time
Art of Multiprocessor Programming

93. Read/Write Register Example
time
time
Art of Multiprocessor Programming

94. Read/Write Register Example
linearizable
Not
write(0)
read(1)
write(2)
write(2)
write(1)
write(1)
read(2)
time
time
Art of Multiprocessor Programming

95. Talking About Executions
Why?
Can’t we specify the linearization point of each operation without describing an execution?
Not Always
In some cases, linearization point depends on the execution
Art of Multiprocessor Programming

96. Formal Model of Executions
Define precisely what we mean
Ambiguity is bad when intuition is weak
Allow reasoning
Formal
But mostly informal
In the long run, actually more important
Ask me why!
We now define a formal model to allow us to precisely define linearizability and other consistency conditions.
Art of Multiprocessor Programming

97. Split Method Calls into Two Events
Invocation
method name & args
q.enq(x)
Response
result or exception
q.enq(x) returns void
q.deq() returns x
q.deq() throws empty
Art of Multiprocessor Programming

98. Art of Multiprocessor Programming
Invocation Notation
A q.enq(x)
(4)
Art of Multiprocessor Programming

99. Art of Multiprocessor Programming
Invocation Notation
thread
A q.enq(x)
(4)
Art of Multiprocessor Programming

100. Art of Multiprocessor Programming
Invocation Notation
thread
method
A q.enq(x)
(4)
Art of Multiprocessor Programming

101. Art of Multiprocessor Programming
Invocation Notation
thread
object
method
A q.enq(x)
(4)
Art of Multiprocessor Programming

102. Art of Multiprocessor Programming
Invocation Notation
thread
object
method
arguments
A q.enq(x)
(4)
Art of Multiprocessor Programming

103. Art of Multiprocessor Programming
Response Notation
A q: void
(2)
Art of Multiprocessor Programming

104. Art of Multiprocessor Programming
Response Notation
thread
A q: void
(2)
Art of Multiprocessor Programming

105. Art of Multiprocessor Programming
Response Notation
thread
A q: void
result
(2)
Art of Multiprocessor Programming

106. Art of Multiprocessor Programming
Response Notation
thread
object
A q: void
result
(2)
Art of Multiprocessor Programming

107. Art of Multiprocessor Programming
Response Notation
thread
object
A q: void
Method is implicit
result
(2)
Art of Multiprocessor Programming

108. Art of Multiprocessor Programming
Response Notation
thread
object
exception
A q: empty()
Method is implicit
(2)
Art of Multiprocessor Programming

109. History - Describing an Execution
A q.enq(3)
A q:void
A q.enq(5)
B p.enq(4)
B p:void
B q.deq()
B q:3
H =
Sequence of invocations and responses
Art of Multiprocessor Programming

110. Art of Multiprocessor Programming
Definition
Invocation & response match if
Object names agree
Thread names agree
A q.enq(3)
A q:void
Method call
(1)
Art of Multiprocessor Programming

111. Art of Multiprocessor Programming
Object Projections
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
H =
Art of Multiprocessor Programming

112. Art of Multiprocessor Programming
Object Projections
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
H|q =
Art of Multiprocessor Programming

113. Art of Multiprocessor Programming
Thread Projections
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
H =
Art of Multiprocessor Programming

114. Art of Multiprocessor Programming
Thread Projections
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
H|B =
Art of Multiprocessor Programming

115. An invocation is pending if it has no matching respnse
Complete Subhistory
A q.enq(3)
A q:void
A q.enq(5)
B p.enq(4)
B p:void
B q.deq()
B q:3
H =
An invocation is pending if it has no matching respnse
Art of Multiprocessor Programming

116. May or may not have taken effect
Complete Subhistory
A q.enq(3)
A q:void
A q.enq(5)
B p.enq(4)
B p:void
B q.deq()
B q:3
H =
May or may not have taken effect
Art of Multiprocessor Programming

117. discard pending invocations
Complete Subhistory
A q.enq(3)
A q:void
A q.enq(5)
B p.enq(4)
B p:void
B q.deq()
B q:3
H =
discard pending invocations
Art of Multiprocessor Programming

118. Art of Multiprocessor Programming
Complete Subhistory
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
Complete(H) =
Art of Multiprocessor Programming

119. Art of Multiprocessor Programming
Sequential Histories
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
A q:enq(5)
(4)
Art of Multiprocessor Programming

120. Art of Multiprocessor Programming
Sequential Histories
match
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
A q:enq(5)
(4)
Art of Multiprocessor Programming

121. Art of Multiprocessor Programming
Sequential Histories
match
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
A q:enq(5)
match
(4)
Art of Multiprocessor Programming

122. Art of Multiprocessor Programming
Sequential Histories
match
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
A q:enq(5)
match
match
(4)
Art of Multiprocessor Programming

123. Final pending invocation OK
Sequential Histories
match
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
A q:enq(5)
match
match
Final pending invocation OK
(4)
Art of Multiprocessor Programming

124. Method calls of different threads do not interleave
Sequential Histories
match
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3
A q:enq(5)
Method calls of different threads do not interleave
match
match
Final pending invocation OK
(4)
Art of Multiprocessor Programming

125. Well-Formed Histories
A q.enq(3)
B p.enq(4)
B p:void
B q.deq()
A q:void
B q:3 H=
Art of Multiprocessor Programming

126. Well-Formed Histories
Per-thread projections sequential
B p.enq(4)
B p:void
B q.deq()
B q:3
H|B=
A q.enq(3)
B p.enq(4)
B p:void
B q.deq()
A q:void
B q:3 H=
Art of Multiprocessor Programming

127. Well-Formed Histories
Per-thread projections sequential
B p.enq(4)
B p:void
B q.deq()
B q:3
H|B=
A q.enq(3)
B p.enq(4)
B p:void
B q.deq()
A q:void
B q:3 H=
A q.enq(3)
A q:void
H|A=
Art of Multiprocessor Programming

128. Threads see the same thing in both
Equivalent Histories
H|A = G|A
H|B = G|B
Threads see the same thing in both
A q.enq(3)
B p.enq(4)
B p:void
B q.deq()
A q:void
B q:3
A q.enq(3)
A q:void
B p.enq(4)
B p:void
B q.deq()
B q:3 H= G=
Art of Multiprocessor Programming

129. Sequential Specifications
A sequential specification is some way of telling whether a
Single-thread, single-object history
Is legal
For example:
Pre and post-conditions
But plenty of other techniques exist …
Art of Multiprocessor Programming

130. Art of Multiprocessor Programming
Legal Histories
A sequential (multi-object) history H is legal if
For every object x
H|x is in the sequential spec for x
Art of Multiprocessor Programming

131. Art of Multiprocessor Programming
Precedence
A q.enq(3)
B p.enq(4)
B p.void
A q:void
B q.deq()
B q:3
A method call precedes another if response event precedes invocation event
Method call
Method call
(1)
Art of Multiprocessor Programming

132. Some method calls overlap one another
Non-Precedence
A q.enq(3)
B p.enq(4)
B p.void
B q.deq()
A q:void
B q:3
Some method calls overlap one another .
Method call
Method call
(1)
Art of Multiprocessor Programming

133. Art of Multiprocessor Programming
Notation
Given
History H
method executions m0 and m1 in H
We say m0 H m1, if
m0 precedes m1
Relation m0 H m1 is a
Partial order
Total order if H is sequential m0 m1
Maybe the main point to notice is that in general, given a history, precedence only defines a partial order – because we have methods that overlaps. So in a sequential history, this is a total order, because we have no overlapping methods. The fact that this is only partial, is exactly the ambiguity about the order that we do not like, and that linearizability comes to fix. So what we really want, is to take the partial order, and somehow complete it to a total order – and that’s how we’ll define linearizability in the next slide.
Art of Multiprocessor Programming

134. Art of Multiprocessor Programming
Linearizability
History H is linearizable if it can be extended to G by
Appending zero or more responses to pending invocations
Discarding other pending invocations
So that G is equivalent to
Legal sequential history S
where G  S
First part: takes care of pending invocations. If the operation took effect, we want to append a response, otherwise, let’s drop it.
Second part, takes care of completing the missing part in the partial order – it tells us to find an equivalent legal sequential history (in which precedence relation defines total order), such that this total order respects the partial order in the original history. Of course, the correctness part hides in the fact that the sequential history is legal.
Explain that equivalence captures, apart from the ordering, the fact that the input/output relation of the methods must be the same as in the sequential execution. The ordering part is also covered by the G \subset S requirement but the fac that the values must be the same is captured only by the equivalence requirement.
When we later go to seq consistency the only ordering property will come from the equivalence property.
Art of Multiprocessor Programming

135. Art of Multiprocessor Programming
Ensuring G  S
G = {ac,bc}
S = {ab,ac,bc}
A limitation on the
Choice of S! a G b c
The G  S is a limitation on S NOT ON G! Its saying that you cannot just pick any S, you must pick an  S so that it contains G, that is, its saying that when we pick the linearization points, they have got to be within the intervals, and we can only pick an arbitrary order when the intervals are unordered. G
time
time S
(8)
Art of Multiprocessor Programming

136. Art of Multiprocessor Programming
Remarks
Some pending invocations
Took effect, so keep them
Discard the rest
Condition G  S
Means that S respects “real-time order” of G
Art of Multiprocessor Programming

137. Art of Multiprocessor Programming
Example
A q.enq(3)
B q.enq(4)
B q:void
B q.deq()
B q:4
B q:enq(6)
A. q.enq(3)
B.q.enq(4)
B.q.deq(4)
B. q.enq(6)
time
Art of Multiprocessor Programming

138. Art of Multiprocessor Programming
Example
A q.enq(3)
B q.enq(4)
B q:void
B q.deq()
B q:4
B q:enq(6)
Complete this pending
invocation
A. q.enq(3)
B.q.enq(4)
B.q.deq(3)
B. q.enq(6)
time
Art of Multiprocessor Programming

139. Art of Multiprocessor Programming
Example
A q.enq(3)
B q.enq(4)
B q:void
B q.deq()
B q:4
B q:enq(6)
A q:void
Complete this pending
invocation
A.q.enq(3)
B.q.enq(4)
B.q.deq(4)
B. q.enq(6)
time
Art of Multiprocessor Programming

140. Art of Multiprocessor Programming
Example
A q.enq(3)
B q.enq(4)
B q:void
B q.deq()
B q:4
B q:enq(6)
A q:void
discard this one
A.q.enq(3)
B.q.enq(4)
B.q.deq(4)
B. q.enq(6)
time
Art of Multiprocessor Programming

141. Art of Multiprocessor Programming
Example
A q.enq(3)
B q.enq(4)
B q:void
B q.deq()
B q:4
A q:void
discard this one
A.q.enq(3)
B.q.enq(4)
B.q.deq(4)
time
Art of Multiprocessor Programming

142. Art of Multiprocessor Programming
Example
A q.enq(3)
B q.enq(4)
B q:void
B q.deq()
B q:4
A q:void
A.q.enq(3)
B.q.enq(4)
B.q.deq(4)
time
Art of Multiprocessor Programming

143. Art of Multiprocessor Programming
Example
A q.enq(3)
B q.enq(4)
B q:void
B q.deq()
B q:4
A q:void
B q.enq(4)
B q:void
A q.enq(3)
A q:void
B q.deq()
B q:4
A.q.enq(3)
B.q.enq(4)
B.q.deq(4)
time
Art of Multiprocessor Programming

144. Art of Multiprocessor Programming
Example
Equivalent sequential history
A q.enq(3)
B q.enq(4)
B q:void
B q.deq()
B q:4
A q:void
B q.enq(4)
B q:void
A q.enq(3)
A q:void
B q.deq()
B q:4
A.q.enq(3)
B.q.enq(4)
B.q.deq(4)
time
Art of Multiprocessor Programming

145. Art of Multiprocessor Programming
Concurrency
How much concurrency does linearizability allow?
When must a method invocation block?
The following slides about non-blocking can be presented to graduate student sbut not clear its needed for undergradutes
Art of Multiprocessor Programming

146. Art of Multiprocessor Programming
Concurrency
Focus on total methods
Defined in every state
Example:
deq() that throws Empty exception
Versus deq() that waits …
Why?
Otherwise, blocking unrelated to synchronization
Art of Multiprocessor Programming

147. Art of Multiprocessor Programming
Concurrency
Question: When does linearizability require a method invocation to block?
Answer: never.
Linearizability is non-blocking
Art of Multiprocessor Programming

148. Art of Multiprocessor Programming
Non-Blocking Theorem
If method invocation
A q.inv(…)
is pending in history H, then there exists a response
A q:res(…)
such that
H + A q:res(…)
is linearizable
This is important because consistency conditions such as serializability do not provide it, transactions may have to be rolled back to
work but linearizability means you don’t need to role back.
Art of Multiprocessor Programming

149. Art of Multiprocessor Programming
Proof
Pick linearization S of H
If S already contains
Invocation A q.inv(…) and response,
Then we are done.
Otherwise, pick a response such that
S + A q.inv(…) + A q:res(…)
Possible because object is total.
Since H is linearizable, let’s pick up the sequential history S that defines the linearization of H.
If S contains the pending invocation and a response – then we’re done (I want to prove that we don’t need to throw it away in order to be linearizable, so if the linearization S did not through it away – there is nothing to do).
If it did throw it away – then we need to show we can add it somewhere – so what we’ll do is adding it at the end of S. It is possible because all methods are total – so they are defined in the state at the end of S (meaning that we make the operation take effect last).
I claim that what I ended up with is a valid linearization of H:
It is still equivalent – I only added the invocation as it was in the original history, with the same arguments, and an appropriate response for both the original and the new sequential histories. These invocation and response are the last operation done by thread A in H, and it is the last operation done by thread A in the new sequential history we’ve created. So each thread’s view of the two histories is the same.
It is legal: because we added it up at the end, it cannot effect the returned values of previous calls.
Finally, it does not violate the partial order, because we can linearize that operation anywhere between it’s invocation and response, and in this case there was no response – so we can definitely linearize it last without violating any ordering.
Art of Multiprocessor Programming

150. Composability Theorem
History H is linearizable if and only if
For every object x
H|x is linearizable
We care about objects only!
(Materialism?)
Art of Multiprocessor Programming

151. Why Does Composability Matter?
Modularity
Can prove linearizability of objects in isolation
Can compose independently-implemented objects
Art of Multiprocessor Programming

152. Reasoning About Linearizability: Locking1
capacity-1 2
head
tail y z
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); }
Art of Multiprocessor Programming

153. Reasoning About Linearizability: Locking
public T deq() throws EmptyException {
lock.lock();
try {
if (tail == head)
throw new EmptyException();
T x = items[head % items.length];
head++;
return x;
} finally {
lock.unlock(); }
Linearization points
are when locks are released
Art of Multiprocessor Programming

154. More Reasoning: Wait-free
public class WaitFreeQueue {
int head = 0, tail = 0;
items = (T[]) new Object[capacity];
public void enq(Item x) {
if (tail-head == capacity) throw
new FullException();
items[tail % capacity] = x; tail++; }
public Item deq() {
if (tail == head) throw
new EmptyException();
Item item = items[head % capacity]; head++;
return item; }} 1
capacity-1 2
head
tail y z 1
capacity-1 2
head
tail y z
Art of Multiprocessor Programming

155. More Reasoning: Wait-free
public class WaitFreeQueue {
int head = 0, tail = 0;
items = (T[]) new Object[capacity];
public void enq(Item x) {
if (tail-head == capacity) throw
new FullException();
items[tail % capacity] = x; tail++; }
public Item deq() {
if (tail == head) throw
new EmptyException();
Item item = items[head % capacity]; head++;
return item; }} 1
capacity-1 2
head
tail y z
Linearization order is order head and tail fields modified
Remember that there
is only one enqueuer
and only one dequeuer
Art of Multiprocessor Programming

156. Art of Multiprocessor Programming
Strategy
Identify one atomic step where method “happens”
Critical section
Machine instruction
Doesn’t always work
Might need to define several different steps for a given method
We will see this phenomena that you might need different steps for different executions in lists chapter.
Art of Multiprocessor Programming

157. Linearizability: Summary
Powerful specification tool for shared objects
Allows us to capture the notion of objects being “atomic”
Don’t leave home without it
We will see this phenomena that you might need different steps for different executions in lists chapter.
Art of Multiprocessor Programming

158. Alternative: Sequential Consistency
History H is Sequentially Consistent if it can be extended to G by
Appending zero or more responses to pending invocations
Discarding other pending invocations
So that G is equivalent to a
Legal sequential history S
The difference from linearizability is in the second property, that it preserves the real time order…notice that the fact that G is EQUIVALENT to S implies that S must preserve the PROGRAM ORDER in each thread. So dropping Where G  S only allows reordering operations by different threads in order to establish a global total (sequential) order.
Where G  S
Differs from
linearizability
Art of Multiprocessor Programming

159. Sequential Consistency
No need to preserve real-time order
Cannot re-order operations done by the same thread
Can re-order non-overlapping operations done by different threads
Often used to describe multiprocessor memory architectures
Art of Multiprocessor Programming

160. Art of Multiprocessor Programming
Example
time
(5)
Art of Multiprocessor Programming

161. Art of Multiprocessor Programming
Example
q.enq(x)
time
(5)
Art of Multiprocessor Programming

162. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
time
(5)
Art of Multiprocessor Programming

163. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
q.enq(y)
time
(5)
Art of Multiprocessor Programming

164. Art of Multiprocessor Programming
Example
q.enq(x)
q.enq(y)
q.enq(x)
q.deq(y)
q.enq(y)
time
(5)
Art of Multiprocessor Programming

165. Art of Multiprocessor Programming
Example
not linearizable
q.enq(x)
q.enq(x)
q.deq(y)
q.enq(y)
q.enq(y)
time
(5)
Art of Multiprocessor Programming

166. Yet Sequentially Consistent
Example
Yet Sequentially Consistent
q.enq(x)
q.enq(x)
q.deq(y)
q.enq(y)
q.enq(y)
time
(5)
Art of Multiprocessor Programming

167. Art of Multiprocessor Programming
Theorem
Sequential Consistency is not composable
Art of Multiprocessor Programming

168. Art of Multiprocessor Programming
FIFO Queue Example
p.enq(x)
q.enq(x)
p.deq(y)
time
time
Art of Multiprocessor Programming

169. Art of Multiprocessor Programming
FIFO Queue Example
p.enq(x)
q.enq(x)
p.deq(y)
q.enq(y)
p.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

170. Art of Multiprocessor Programming
FIFO Queue Example
p.enq(x)
q.enq(x)
p.deq(y)
q.enq(y)
p.enq(y)
q.deq(x)
History H
time
time
Art of Multiprocessor Programming

171. H|p Sequentially Consistent
p.enq(x)
q.enq(x)
p.deq(y)
q.enq(y)
p.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

172. H|q Sequentially Consistent
p.enq(x)
q.enq(x)
p.deq(y)
q.enq(y)
p.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

173. Art of Multiprocessor Programming
Ordering imposed by p
p.enq(x)
q.enq(x)
p.deq(y)
q.enq(y)
p.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

174. Art of Multiprocessor Programming
Ordering imposed by q
p.enq(x)
q.enq(x)
p.deq(y)
q.enq(y)
p.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

175. Ordering imposed by both
p.enq(x)
q.enq(x)
p.deq(y)
q.enq(y)
p.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

176. Art of Multiprocessor Programming
Combining orders
p.enq(x)
q.enq(x)
p.deq(y)
q.enq(y)
p.enq(y)
q.deq(x)
time
time
Art of Multiprocessor Programming

177. Art of Multiprocessor Programming
Fact
Most hardware architectures don’t support sequential consistency
Because they think it’s too strong
Here’s another story …
Art of Multiprocessor Programming

178. Art of Multiprocessor Programming
The Flag Example
y.read(0)
x.write(1)
y.write(1)
x.read(0)
time
time
Art of Multiprocessor Programming

179. Art of Multiprocessor Programming
The Flag Example
y.read(0)
x.write(1)
y.write(1)
x.read(0)
Each thread’s view is sequentially consistent
It went first
time
Art of Multiprocessor Programming

180. Art of Multiprocessor Programming
The Flag Example
y.read(0)
x.write(1)
y.write(1)
x.read(0)
Entire history isn’t sequentially consistent
Can’t both go first
time
Art of Multiprocessor Programming

181. Art of Multiprocessor Programming
The Flag Example
y.read(0)
x.write(1)
y.write(1)
x.read(0)
Is this behavior really so wrong?
We can argue either way …
time
Art of Multiprocessor Programming

182. Art of Multiprocessor Programming
Opinion1: It’s Wrong
This pattern
Write mine, read yours
Is exactly the flag principle
Beloved of Alice and Bob
Heart of mutual exclusion
Peterson
Bakery, etc.
It’s non-negotiable!
Art of Multiprocessor Programming

183. Opinion2: But It Feels So Right …
Many hardware architects think that sequential consistency is too strong
Too expensive to implement in modern hardware
OK if flag principle
violated by default
Honored by explicit request
Art of Multiprocessor Programming

184. Art of Multiprocessor Programming
Memory Hierarchy
On modern multiprocessors, processors do not read and write directly to memory.
Memory accesses are very slow compared to processor speeds,
Instead, each processor reads and writes directly to a cache
Read chapter notes. Explain to students that they can raed the hardware chapter in the book for more details.
Art of Multiprocessor Programming

185. Art of Multiprocessor Programming
Memory Operations
To read a memory location,
load data into cache.
To write a memory location
update cached copy,
lazily write cached data back to memory
Art of Multiprocessor Programming

186. While Writing to Memory
A processor can execute hundreds, or even thousands of instructions
Why delay on every memory write?
Instead, write back in parallel with rest of the program.
Art of Multiprocessor Programming

187. Art of Multiprocessor Programming
Revisionist History
Flag violation history is actually OK
processors delay writing to memory
until after reads have been issued.
Otherwise unacceptable delay between read and write instructions.
Who knew you wanted to synchronize?
Art of Multiprocessor Programming

188. Who knew you wanted to synchronize?
Writing to memory = mailing a letter
Vast majority of reads & writes
Not for synchronization
No need to idle waiting for post office
If you want to synchronize
Announce it explicitly
Pay for it only when you need it
Art of Multiprocessor Programming

189. Explicit Synchronization
Memory barrier instruction
Flush unwritten caches
Bring caches up to date
Compilers often do this for you
Entering and leaving critical sections
Expensive
Art of Multiprocessor Programming

190. Art of Multiprocessor Programming
Volatile
In Java, can ask compiler to keep a variable up-to-date with volatile keyword
Also inhibits reordering, removing from loops, & other “optimizations”
Art of Multiprocessor Programming

191. Real-World Hardware Memory
Weaker than sequential consistency
But you can get sequential consistency at a price
OK for expert, tricky stuff
assembly language, device drivers, etc.
Linearizability more appropriate for high-level software
Art of Multiprocessor Programming

192. Art of Multiprocessor Programming
Linearizability
Linearizability
Operation takes effect instantaneously between invocation and response
Uses sequential specification, locality implies composablity
Good for high level objects
Art of Multiprocessor Programming

193. Correctness: Linearizability
Sequential Consistency
Not composable
Harder to work with
Good way to think about hardware models
We will use linearizability as in the remainder of this course unless stated otherwise
Art of Multiprocessor Programming

194. Art of Multiprocessor Programming
Progress
We saw an implementation whose methods were lock-based (deadlock-free)
We saw an implementation whose methods did not use locks (lock-free)
How do they relate?
So far we have talked about correctness. Now its time to discuss progress.
Art of Multiprocessor Programming

195. Art of Multiprocessor Programming
Progress Conditions
Deadlock-free: some thread trying to acquire the lock eventually succeeds.
Starvation-free: every thread trying to acquire the lock eventually succeeds.
Lock-free: some thread calling a method eventually returns.
Wait-free: every thread calling a method eventually returns.
The above are definitions of progress conditions we have used and will use in the coming lectures.
We give here informal definitions of progress conditions…formal ones need to talk about fair histories which is beyond the scope of this lecture…for the above conditions:
A method implementation is deadlock-free if it guarantees min-
imal progress in every fair history, and maximal progress in some fair history.
A method implementation is starvation-free if it guarantees
maximal progress in every fair history.
A method implementation is lock-free if it guarantees minimal
progress in every history and maximal progress in some history.
A method implementation is wait-free if it guarantees maximal
progress in every history.
Art of Multiprocessor Programming

196. Art of Multiprocessor Programming
Progress Conditions
Non-Blocking
Blocking
Everyone
makes
progress
Wait-free
Starvation-free
Someone
makes
progress
Lock-free
Deadlock-free
Art of Multiprocessor Programming

197. Art of Multiprocessor Programming
Summary
We will look at linearizable blocking and non-blocking implementations of objects.
Art of Multiprocessor Programming

198. Art of Multiprocessor Programming
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Art of Multiprocessor Programming