Concurrent Objects Companion slides for The Art of Multiprocessor Programming by Maurice Herlihy & Nir Shavit.
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Concurrent Objects
Companion slides forThe Art of Multiprocessor Programming
by Maurice Herlihy & Nir Shavit
Art of Multiprocessor Programming
2
Concurrent Computation
memory
object object
Art of Multiprocessor Programming
3
Objectivism
• What is a concurrent object?– How do we describe one?– How do we implement one?– How do we tell if we’re right?
Art of Multiprocessor Programming
4
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( )
Art of Multiprocessor Programming
6
FIFO Queue: Dequeue Method
q.deq()/
Art of Multiprocessor Programming
7
Lock-Based Queue
headtail0
2
1
5 4
3
yx
capacity = 8
7
6
Art of Multiprocessor Programming
8
Lock-Based Queue
headtail0
2
1
5 4
3
yx
capacity = 8
7
6
Fields protected by single shared lock
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]; }
Art of Multiprocessor Programming
9
A Lock-Based Queue0 1
capacity-12
head tail
y z
Fields protected by single shared lock
Art of Multiprocessor Programming
10
Lock-Based Queue
head
tail
0
2
1
5 4
3
Initially head = tail
7
6
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]; }
Art of Multiprocessor Programming
11
A Lock-Based Queue0 1
capacity-12
head tail
y z
Initially head = tail
Art of Multiprocessor Programming
12
Lock-Based deq()
headtail0
2
5 4
7
36
yx
1
Art of Multiprocessor Programming
13
Acquire Lock
headtail0
2
5 4
7
36
yx
1
Waiting to enqueue…
My turn …
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
14
Implementation: deq()
Acquire lock at method start
0 1capacity-1
2
head tail
y z
Art of Multiprocessor Programming
15
Check if Non-Empty
headtail0
2
5 4
7
36
yx
1
Waiting to enqueue…
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
16
Implementation: deq()
If queue emptythrow exception
0 1capacity-1
2
head tail
y z
Art of Multiprocessor Programming
17
Modify the Queue
headtail0
2
1
5 4
7
36
yx
head
Waiting to enqueue…
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
18
Implementation: deq()
Queue not empty?Remove item and update head
0 1capacity-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
19
Implementation: deq()
Return result
0 1capacity-1
2
head tail
y z
Art of Multiprocessor Programming
20
Release the Lock
tail0
2
1
5 4
7
36
y
x
head
My turn!
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
21
Implementation: deq()
Release lock no matter what!
0 1capacity-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
22
Implementation: deq()
Should be correct because
modifications are mutually exclusive…
Should be correct because
modifications are mutually exclusive…
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
Art of Multiprocessor Programming
24
Wait-free 2-Thread Queue
headtail0
2
1
5 4
7
36
yx
capacity = 8
Art of Multiprocessor Programming
25
Wait-free 2-Thread Queue
tail0
2
5 4
7
36
yx
1
enq(z)deq()
z
head
Art of Multiprocessor Programming
26
Wait-free 2-Thread Queuehead
tail0
2
5 4
7
36
y
1
queue[tail] = z
result = x
z
x
Art of Multiprocessor Programming
27
Wait-free 2-Thread Queue
tail0
2
5 4
7
36
y
1
tail--head++
z
head
x
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;}} Art of Multiprocessor
Programming28
Wait-free 2-Thread Queue
0 1capacity-1
2
head tail
y z
No lock needed !
Wait-free 2-Thread Queue
Art of Multiprocessor Programming
29
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? 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 …
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
Art of Multiprocessor Programming
31
Lets begin with correctness
Art of Multiprocessor Programming
32
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
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
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Pre and PostConditions for Dequeue
• Precondition:– Queue is non-empty
• Postcondition:– Returns first item in queue
• Postcondition:– Removes first item in queue
Art of Multiprocessor Programming
35
Pre and PostConditions for Dequeue
• Precondition:– Queue is empty
• Postcondition:– Throws Empty exception
• Postcondition:– 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
Methods Take Time
timetime
Art of Multiprocessor Programming
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Methods Take Time
time
invocation 12:00
q.enq(...)
time
Art of Multiprocessor Programming
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Methods Take Time
time
Method call
invocation 12:00
time
q.enq(...)
Art of Multiprocessor Programming
41
Methods Take Time
time
Method call
invocation 12:00
time
q.enq(...)
Art of Multiprocessor Programming
42
Methods Take Time
time
Method call
invocation 12:00
time
void
response 12:01
q.enq(...)
Art of Multiprocessor Programming
43
Sequential vs Concurrent
• Sequential– Methods take time? Who knew?
• Concurrent– Method call is not an event– Method call is an interval.
Art of Multiprocessor Programming
44
time
Concurrent Methods Take Overlapping Time
time
Art of Multiprocessor Programming
45
time
Concurrent Methods Take Overlapping Time
time
Method call
Art of Multiprocessor Programming
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time
Concurrent Methods Take Overlapping Time
time
Method call
Method call
Art of Multiprocessor Programming
47
time
Concurrent Methods Take Overlapping Time
time
Method call Method call
Method call
Art of Multiprocessor Programming
48
Sequential vs Concurrent
• Sequential:– 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
• Sequential:– 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
• Sequential:– Can add new methods without affecting older
methods
• Concurrent:– Everything can potentially interact with
everything else
Art of Multiprocessor Programming
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Sequential vs Concurrent
• Sequential:– Can add new methods without affecting older
methods
• Concurrent:– Everything can potentially interact with
everything elsePanic!
Art of Multiprocessor Programming
52
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
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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…
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
time
Intuitively
q.deq
q.enq
enq deq
lock() unlock()
lock() unlock() Behavior is “Sequential”
enq
deq
Lets capture the idea of describing the concurrent via the sequential
Art of Multiprocessor Programming
56
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™
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
Example
timetime
Art of Multiprocessor Programming
59
Example
time
q.enq(x)
time
Art of Multiprocessor Programming
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Example
time
q.enq(x)
q.enq(y)
time
Art of Multiprocessor Programming
61
Example
time
q.enq(x)
q.enq(y) q.deq(x)
time
Art of Multiprocessor Programming
62
Example
time
q.enq(x)
q.enq(y) q.deq(x)
q.deq(y)
time
Art of Multiprocessor Programming
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Example
time
q.enq(x)
q.enq(y) q.deq(x)
q.deq(y)
linearizableq.enq(x)
q.enq(y) q.deq(x)
q.deq(y)
time
Art of Multiprocessor Programming
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Example
time
q.enq(x)
q.enq(y) q.deq(x)
q.deq(y)
Valid?q.enq(x)
q.enq(y) q.deq(x)
q.deq(y)
time
Art of Multiprocessor Programming
65
Example
time
Art of Multiprocessor Programming
66
Example
time
q.enq(x)
Art of Multiprocessor Programming
67
Example
time
q.enq(x) q.deq(y)
Art of Multiprocessor Programming
68
Example
time
q.enq(x)
q.enq(y)
q.deq(y)
Art of Multiprocessor Programming
69
Example
time
q.enq(x)
q.enq(y)
q.deq(y)q.enq(x)
q.enq(y)
Art of Multiprocessor Programming
70
Example
time
q.enq(x)
q.enq(y)
q.deq(y)q.enq(x)
q.enq(y)
(5)
not linearizable
Art of Multiprocessor Programming
71
Example
timetime
Art of Multiprocessor Programming
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Example
time
q.enq(x)
time
Art of Multiprocessor Programming
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Example
time
q.enq(x)
q.deq(x)
time
Art of Multiprocessor Programming
74
Example
time
q.enq(x)
q.deq(x)
q.enq(x)
q.deq(x)
time
Art of Multiprocessor Programming
75
Example
time
q.enq(x)
q.deq(x)
q.enq(x)
q.deq(x)
linearizable
time
Art of Multiprocessor Programming
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Example
time
q.enq(x)
time
Art of Multiprocessor Programming
77
Example
time
q.enq(x)
q.enq(y)
time
Art of Multiprocessor Programming
78
Example
time
q.enq(x)
q.enq(y)
q.deq(y)
time
Art of Multiprocessor Programming
79
Example
time
q.enq(x)
q.enq(y)
q.deq(y)
q.deq(x)
time
Art of Multiprocessor Programming
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q.enq(x)
q.enq(y)
q.deq(y)
q.deq(x)
Comme ci Example
time
Comme ça multiple orders OK
linearizable
Art of Multiprocessor Programming
81
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(0)
Art of Multiprocessor Programming
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Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(0)
write(1) already happened
Art of Multiprocessor Programming
83
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(0)write(1)
write(1) already happened
Art of Multiprocessor Programming
84
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(0)write(1)
write(1) already happened
not linearizable
Art of Multiprocessor Programming
85
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(1)
write(1) already happened
Art of Multiprocessor Programming
86
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(1)write(1)
write(2)
write(1) already happened
Art of Multiprocessor Programming
87
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(1)write(1)
write(2)
not linearizable
write(1) already happened
Art of Multiprocessor Programming
88
Read/Write Register Example
time
write(0)
write(1)
write(2)
time
read(1)
Art of Multiprocessor Programming
89
Read/Write Register Example
time
write(0)
write(1)
write(2)
time
read(1)write(1)
write(2)
Art of Multiprocessor Programming
90
Read/Write Register Example
time
write(0)
write(1)
write(2)
time
read(1)write(1)
write(2)
linearizable
Art of Multiprocessor Programming
91
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(1)
Art of Multiprocessor Programming
92
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(1)write(1)
Art of Multiprocessor Programming
93
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(1)write(1)
write(2)
Art of Multiprocessor Programming
94
Read/Write Register Example
time
read(1)write(0)
write(1)
write(2)
time
read(2)write(1)
write(2)
Not linearizable
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!
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
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Invocation Notation
A q.enq(x)
(4)
Art of Multiprocessor Programming
99
Invocation Notation
A q.enq(x)
thread
(4)
Art of Multiprocessor Programming
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Invocation Notation
A q.enq(x)
thread method
(4)
Art of Multiprocessor Programming
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Invocation Notation
A q.enq(x)
thread
object(4)
method
Art of Multiprocessor Programming
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Invocation Notation
A q.enq(x)
thread
object
method
arguments(4)
Art of Multiprocessor Programming
103
Response Notation
A q: void
(2)
Art of Multiprocessor Programming
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Response Notation
A q: void
thread
(2)
Art of Multiprocessor Programming
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Response Notation
A q: void
thread result
(2)
Art of Multiprocessor Programming
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Response Notation
A q: void
thread
object
result
(2)
Art of Multiprocessor Programming
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Response Notation
A q: void
thread
object
result
(2)
Met
hod is im
plicit
Art of Multiprocessor Programming
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Response Notation
A q: empty()
thread
object(2)
Met
hod is im
plicit
exception
Art of Multiprocessor Programming
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History - Describing an Execution
A q.enq(3)A q:voidA q.enq(5)B p.enq(4)B p:voidB q.deq()B q:3
Sequence of invocations and
responses
H =
Art of Multiprocessor Programming
110
Definition
• Invocation & response match if
A q.enq(3)
A q:void
Thread names agree
Object names agree
Method call
(1)
Art of Multiprocessor Programming
111
Object Projections
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3
H =
Art of Multiprocessor Programming
112
Object Projections
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3
H|q =
Art of Multiprocessor Programming
113
Thread Projections
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3
H =
Art of Multiprocessor Programming
114
Thread Projections
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3
H|B =
Art of Multiprocessor Programming
115
Complete Subhistory
A q.enq(3)A q:voidA q.enq(5)B p.enq(4)B p:voidB q.deq()B q:3
An invocation is pending if it has no matching respnse
H =
Art of Multiprocessor Programming
116
Complete Subhistory
A q.enq(3)A q:voidA q.enq(5)B p.enq(4)B p:voidB q.deq()B q:3
May or may not have taken effect
H =
Art of Multiprocessor Programming
117
Complete Subhistory
A q.enq(3)A q:voidA q.enq(5)B p.enq(4)B p:voidB q.deq()B q:3
discard pending invocations
H =
Art of Multiprocessor Programming
118
Complete Subhistory
A q.enq(3)A q:void B p.enq(4)B p:voidB q.deq()B q:3
Complete(H) =
Art of Multiprocessor Programming
119
Sequential Histories
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3A q:enq(5)
(4)
Art of Multiprocessor Programming
120
Sequential Histories
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3A q:enq(5)
match
(4)
Art of Multiprocessor Programming
121
Sequential Histories
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3A q:enq(5)
match
match
(4)
Art of Multiprocessor Programming
122
Sequential Histories
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3A q:enq(5)
match
match
match
(4)
Art of Multiprocessor Programming
123
Sequential Histories
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3A q:enq(5)
match
match
match
Final pending invocation OK
(4)
Art of Multiprocessor Programming
124
Sequential Histories
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3A q:enq(5)
match
match
match
Final pending invocation OK
(4)
Method calls of different
threads do not interleave
Art of Multiprocessor Programming
125
Well-Formed Histories
H=
A q.enq(3)B p.enq(4)B p:voidB q.deq()A q:voidB q:3
Art of Multiprocessor Programming
126
Well-Formed Histories
H=
A q.enq(3)B p.enq(4)B p:voidB q.deq()A q:voidB q:3
H|B=B p.enq(4)B p:voidB q.deq()B q:3
Per-thread projections sequential
Art of Multiprocessor Programming
127
Well-Formed Histories
H=
A q.enq(3)B p.enq(4)B p:voidB q.deq()A q:voidB q:3
H|B=B p.enq(4)B p:voidB q.deq()B q:3
A q.enq(3)A q:void
H|A=
Per-thread projections sequential
Art of Multiprocessor Programming
128
Equivalent Histories
H=
A q.enq(3)B p.enq(4)B p:voidB q.deq()A q:voidB q:3
Threads see the same thing in both
A q.enq(3)A q:voidB p.enq(4)B p:voidB q.deq()B q:3
G=
H|A = G|AH|B = G|B
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
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
Precedence
A q.enq(3)B p.enq(4)B p.voidA q:voidB 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
Non-Precedence
A q.enq(3)B p.enq(4)B p.voidB q.deq()A q:voidB q:3
Some method calls overlap one another
Method call
Method call
(1)
Art of Multiprocessor Programming
133
Notation
• Given – History H– method executions m0 and m1 in H
• We say m0 Hm1, if– m0 precedes m1
• Relation m0 Hm1 is a– Partial order – Total order if H is sequential
m0 m1
Art of Multiprocessor Programming
134
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
Art of Multiprocessor Programming
135
Ensuring G S
time
a
b
time
(8)
G
S
cG
G = {ac,bc}
S = {ab,ac,bc}
A limita
tion on th
e
Choice of S!
Art of Multiprocessor Programming
136
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
A q.enq(3)B q.enq(4)B q:voidB q.deq()B q:4B q:enq(6)
Example
time
B.q.enq(4)
A. q.enq(3)
B.q.deq(4) B. q.enq(6)
Art of Multiprocessor Programming
138
Example
Complete this pending
invocation
time
B.q.enq(4) B.q.deq(3) B. q.enq(6)
A q.enq(3)B q.enq(4)B q:voidB q.deq()B q:4B q:enq(6)
A. q.enq(3)
Art of Multiprocessor Programming
139
Example
Complete this pending
invocation
time
B.q.enq(4) B.q.deq(4) B. q.enq(6)
A.q.enq(3)
A q.enq(3)B q.enq(4)B q:voidB q.deq()B q:4B q:enq(6)A q:void
Art of Multiprocessor Programming
140
Example
time
B.q.enq(4) B.q.deq(4) B. q.enq(6)
A.q.enq(3)
A q.enq(3)B q.enq(4)B q:voidB q.deq()B q:4B q:enq(6)A q:void
discard this one
Art of Multiprocessor Programming
141
Example
time
B.q.enq(4) B.q.deq(4)
A.q.enq(3)
A q.enq(3)B q.enq(4)B q:voidB q.deq()B q:4
A q:void
discard this one
Art of Multiprocessor Programming
142
A q.enq(3)B q.enq(4)B q:voidB q.deq()B q:4A q:void
Example
time
B.q.enq(4) B.q.deq(4)
A.q.enq(3)
Art of Multiprocessor Programming
143
A q.enq(3)B q.enq(4)B q:voidB q.deq()B q:4A q:void
Example
time
B q.enq(4)B q:voidA q.enq(3)A q:voidB q.deq()B q:4
B.q.enq(4) B.q.deq(4)
A.q.enq(3)
Art of Multiprocessor Programming
144
B.q.enq(4) B.q.deq(4)
A.q.enq(3)
A q.enq(3)B q.enq(4)B q:voidB q.deq()B q:4A q:void
Example
time
B q.enq(4)B q:voidA q.enq(3)A q:voidB q.deq()B q:4
Equivalent sequential history
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: 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(); } }
0 1capacity-1
2
head tail
y z
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 pointsare when locks are
released
Art of Multiprocessor Programming
154
More Reasoning: Wait-free
0 1capacity-1
2
head tail
y z
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;}}
0 1capacity-1
2
head tail
y z
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;}} Art of Multiprocessor
Programming155
More Reasoning: Wait-free
0 1capacity-1
2
head tail
y zLinearization order is order head and tail
fields modified
Remember that t
here
is only one enqueuer
and only one dequeuer
Art of Multiprocessor Programming
156
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
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
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
– 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
Example
time
(5)
Art of Multiprocessor Programming
161
Example
time
q.enq(x)
(5)
Art of Multiprocessor Programming
162
Example
time
q.enq(x) q.deq(y)
(5)
Art of Multiprocessor Programming
163
Example
time
q.enq(x)
q.enq(y)
q.deq(y)
(5)
Art of Multiprocessor Programming
164
Example
time
q.enq(x)
q.enq(y)
q.deq(y)q.enq(x)
q.enq(y)
(5)
Art of Multiprocessor Programming
165
Example
time
q.enq(x)
q.enq(y)
q.deq(y)q.enq(x)
q.enq(y)
(5)
not linearizable
Art of Multiprocessor Programming
166
Example
time
q.enq(x)
q.enq(y)
q.deq(y)q.enq(x)
q.enq(y)
(5)
Yet Sequentially
Consistent
Art of Multiprocessor Programming
167
Theorem
Sequential Consistency is not composable
Art of Multiprocessor Programming
168
FIFO Queue Example
time
p.enq(x) p.deq(y)q.enq(x)
time
Art of Multiprocessor Programming
169
FIFO Queue Example
time
p.enq(x) p.deq(y)q.enq(x)
q.enq(y) q.deq(x)p.enq(y)
time
Art of Multiprocessor Programming
170
FIFO Queue Example
time
p.enq(x) p.deq(y)q.enq(x)
q.enq(y) q.deq(x)p.enq(y)
History H
time
Art of Multiprocessor Programming
171
H|p Sequentially Consistent
time
p.enq(x) p.deq(y)
p.enq(y)
q.enq(x)
q.enq(y) q.deq(x)
time
Art of Multiprocessor Programming
172
H|q Sequentially Consistent
time
p.enq(x) p.deq(y)q.enq(x)
q.enq(y) q.deq(x)p.enq(y)
time
Art of Multiprocessor Programming
173
Ordering imposed by p
time
p.enq(x) p.deq(y)q.enq(x)
q.enq(y) q.deq(x)p.enq(y)
time
Art of Multiprocessor Programming
174
Ordering imposed by q
time
p.enq(x) p.deq(y)q.enq(x)
q.enq(y) q.deq(x)p.enq(y)
time
Art of Multiprocessor Programming
175
p.enq(x)
Ordering imposed by both
time
q.enq(x)
q.enq(y) q.deq(x)
time
p.deq(y)
p.enq(y)
Art of Multiprocessor Programming
176
p.enq(x)
Combining orders
time
q.enq(x)
q.enq(y) q.deq(x)
time
p.deq(y)
p.enq(y)
Art of Multiprocessor Programming
177
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
The Flag Example
time
x.write(1) y.read(0)
y.write(1) x.read(0)
time
Art of Multiprocessor Programming
179
The Flag Example
time
x.write(1) y.read(0)
y.write(1) x.read(0)
• Each thread’s view is sequentially consistent– It went first
Art of Multiprocessor Programming
180
The Flag Example
time
x.write(1) y.read(0)
y.write(1) x.read(0)
• Entire history isn’t sequentially consistent– Can’t both go first
Art of Multiprocessor Programming
181
The Flag Example
time
x.write(1) y.read(0)
y.write(1) x.read(0)
• Is this behavior really so wrong?– We can argue either way …
Art of Multiprocessor Programming
182
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
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
Art of Multiprocessor Programming
185
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
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
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
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
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?
Art of Multiprocessor Programming
194
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.
Art of Multiprocessor Programming
195
Progress Conditions
Art of Multiprocessor Programming
196
Wait-free
Lock-free
Starvation-free
Deadlock-free
Everyone makes progress
Non-Blocking Blocking
Someone makes progress
Art of Multiprocessor Programming
197
Summary
• We will look at linearizable blocking and non-blocking implementations of objects.
Art of Multiprocessor Programming
198
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