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

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

6. Art of Multiprocessor Programming
A Lock-Based Queue
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.
Art of Multiprocessor Programming

7. 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.
Queue fields protected by single shared lock
Art of Multiprocessor Programming

8. 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

9. Art of Multiprocessor Programming
Implementation: Deq 1
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

10. Art of Multiprocessor Programming
Implementation: Deq 1
capacity-1 2
head
tail y z 1
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(); }
Method calls
mutually exclusive
Art of Multiprocessor Programming

11. Art of Multiprocessor Programming
Implementation: Deq 1
capacity-1 2
head
tail y z 1
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(); }
If queue empty
throw exception
Art of Multiprocessor Programming

12. Art of Multiprocessor Programming
Implementation: Deq 1
capacity-1 2
head
tail y z 1
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(); }
Queue not empty:
remove item and update
head
Art of Multiprocessor Programming

13. Art of Multiprocessor Programming
Implementation: Deq 1
capacity-1 2
head
tail y z 1
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(); }
Return result
Art of Multiprocessor Programming

14. Implementation: Deq Release lock no matter what! y z y z1
capacity-1 2
head
tail y z 1
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(); }
Release lock no matter what!
Art of Multiprocessor Programming

15. Art of Multiprocessor Programming
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(); }
Should be correct because
modifications are mutually exclusive…
Art of Multiprocessor Programming

16. 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

17. 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
Art of Multiprocessor Programming

18. 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 1
capacity-1 2
head
tail y z
Art of Multiprocessor Programming

19. 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
How do we define “correct” when modifications are not mutually exclusive?
Queue is updated without a lock!
Art of Multiprocessor Programming

20. Defining concurrent queue implementations
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

21. 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

22. 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

23. 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

24. 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

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

26. 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

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

28. 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

29. 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

30. 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

31. 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

32. 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

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

34. Concurrent Methods Take Overlapping Time
Art of Multiprocessor Programming

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

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

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

38. 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

39. 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

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

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

42. 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

43. 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

44. 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(); }
All modifications
of queue are done
mutually exclusive
Art of Multiprocessor Programming

45. 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

46. 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

47. 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

48. 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
(6)
Art of Multiprocessor Programming

49. Art of Multiprocessor Programming
Example
q.enq(x)
time
time
(6)
Art of Multiprocessor Programming

50. Art of Multiprocessor Programming
Example
q.enq(x)
q.enq(y)
time
time
(6)
Art of Multiprocessor Programming

51. Art of Multiprocessor Programming
Example
q.enq(x)
q.enq(y)
q.deq(x)
time
time
(6)
Art of Multiprocessor Programming

52. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
q.enq(y)
q.deq(x)
time
time
(6)
Art of Multiprocessor Programming

53. 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
(6)
Art of Multiprocessor Programming

54. 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
(6)
Art of Multiprocessor Programming

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

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

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

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

59. 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

60. 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

61. Art of Multiprocessor Programming
Example
time
time
(4)
Art of Multiprocessor Programming

62. Art of Multiprocessor Programming
Example
q.enq(x)
time
time
(4)
Art of Multiprocessor Programming

63. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(x)
time
time
(4)
Art of Multiprocessor Programming

64. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(x)
q.enq(x)
q.deq(x)
time
time
(4)
Art of Multiprocessor Programming

65. Art of Multiprocessor Programming
Example
linearizable
q.enq(x)
q.enq(x)
q.deq(x)
q.deq(x)
time
time
(4)
Art of Multiprocessor Programming

66. Art of Multiprocessor Programming
Example
q.enq(x)
time
time
(8)
Art of Multiprocessor Programming

67. Art of Multiprocessor Programming
Example
q.enq(x)
q.enq(y)
time
time
(8)
Art of Multiprocessor Programming

68. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
q.enq(y)
time
time
(8)
Art of Multiprocessor Programming

69. Art of Multiprocessor Programming
Example
q.enq(x)
q.deq(y)
q.enq(y)
q.deq(x)
time
time
(8)
Art of Multiprocessor Programming

70. 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

71. Read/Write Register Example
time
time
(4)
Art of Multiprocessor Programming

72. Read/Write Register Example
write(1) already happened
time
time
(4)
Art of Multiprocessor Programming

73. Read/Write Register Example
write(1) already happened
time
time
(4)
Art of Multiprocessor Programming

74. Read/Write Register Example
not linearizable
write(0)
read(1)
write(2)
write(1)
write(1)
read(0)
write(1) already happened
time
time
(4)
Art of Multiprocessor Programming

75. Read/Write Register Example
write(1) already happened
time
time
(4)
Art of Multiprocessor Programming

76. Read/Write Register Example
write(1) already happened
time
time
(4)
Art of Multiprocessor Programming

77. 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
(4)
Art of Multiprocessor Programming

78. Read/Write Register Example
time
time
(4)
Art of Multiprocessor Programming

79. Read/Write Register Example
time
time
(4)
Art of Multiprocessor Programming

80. Read/Write Register Example
linearizable
write(0)
write(1)
write(2)
write(2)
write(1)
read(1)
time
time
(4)
Art of Multiprocessor Programming

81. Read/Write Register Example
time
time
(2)
Art of Multiprocessor Programming

82. Read/Write Register Example
time
time
(2)
Art of Multiprocessor Programming

83. Read/Write Register Example
time
time
(2)
Art of Multiprocessor Programming

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

85. 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

86. 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

87. 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

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

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

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

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

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

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

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

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

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

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

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

99. 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

100. 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

101. 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

102. 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

103. 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

104. 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

105. An invocation is pending if it has no matching response
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 response
Art of Multiprocessor Programming

106. 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

107. 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

108. 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

109. 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

110. 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

111. 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

112. 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

113. 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

114. 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

115. 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

116. 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

117. 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

118. 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

119. 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

120. 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

121. 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

122. 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

123. 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

124. 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

125. Art of Multiprocessor Programming
What is 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 can 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

126. 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

127. 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

128. 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

129. 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

130. 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

131. 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

132. 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

133. 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

134. 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

135. 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

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

137. 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

138. 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

139. 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

140. 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

141. 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

142. Linearizability: Summary
Powerful specification tool for shared objects, good for high level objects (composability)
Allows us to capture the notion of objects being “atomic”
Operation takes effect instantaneously between invocation and response
Uses sequential specification
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

143. 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

144. 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

145. 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

146. Art of Multiprocessor Programming
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Art of Multiprocessor Programming