Consensus Hierarchy Part 1

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Consensus HierarchyPart 1
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Consensus in Shared Memory
Consider processors in shared memory
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Local memory
Local memory
Shared memory
Every process starts with an initial value stored
in local memory (0 or 1)
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communication through shared memory
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At the end of execution, every process has
decided the same value (0 or 1)
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Validity condition
If every process starts with the same value, the
every process should decide that value
Additional condition The decided value is one of
the initial values
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Wait-freedom in asynchronous systems
A process should be able to finish execution of
an algorithm even if all other processes fail
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Read/Write
Shared Memory
Suppose that the shared Memory can only be
accessed through Read or Write operations
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Theorem
Wait-free consensus cannot be solved using only
read/write objects for processors
Proof of Theorem
We will show that any algorithm that solves
consensus, has an execution that never terminates
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System configuration

Is the set of all variables in the system,
including local and shared
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A distributed system execution can be always be
viewed a sequence of configurations
Initial configuration
Final configuration
Read or Write
Processor action
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Valence of system configurations
bivalent
univalent
bivalent
univalent
0-valent
1-valent
Consensus value at possible execution paths
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To prove the theorem, we will show that there is
always an execution where every configuration is
bivalent
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Similar configurations for processor
Same shared variables Local variables of others
may differ
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Lemma
If there exist univalent configurations
and such that
then if is -valent then is
-valent too
Proof of Lemma
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Univalent
All possible executions from
final decision for each Possible execution
Execution with only taking actions
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Univalent
Univalent
Execution with only taking actions
Execution with only taking actions
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Univalent
Univalent
Execution with only taking actions
Execution with only taking actions
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Univalent
Univalent
Execution with only taking actions
Execution with only taking actions
End of Lemma Proof
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Lemma
There exists a bivalent initial configuration
Proof of Lemma
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Possible Initial Configurations
Local Memory
Shared Memory
Initial Configuration
Empty
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Possible Initial Configurations
Local Memory
Shared Memory
Initial Configuration
?
Empty
0-valent
1-valent
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Possible Initial Configurations
Local Memory
Shared Memory
Initial Configuration
Empty
0-valent
1-valent
1-valent?
No, because
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Possible Initial Configurations
Local Memory
Shared Memory
Initial Configuration
Empty
0-valent
1-valent
0-valent?
No, because
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Possible Initial Configurations
Local Memory
Shared Memory
Initial Configuration
Empty
0-valent
1-valent
bivalent
End of Lemma Proof
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Critical processor for a configuration
the configuration is bivalent, and after the
processor takes step the configuration becomes
univalent
Bivalent
Univalent
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Lemma
If is a bivalent configuration then, there
is at least one processor which is not critical
Proof of Lemma
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Assume for contradiction that all processors are
critical
univalent
Possible executions
bivalent
univalent
univalent
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It cannot be that all have the same valence
valent
bivalent
valent
valent
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It cannot be that all have the same valence
Contradiction
valent
bivalent
valent
valent
valent
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There must exist two processors with different
valences
bivalent
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Case 1 suppose that they access different
shared variables
bivalent
x
Read x
y
Read y
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two possible executions
bivalent
Read y
Read x
different valence
impossible since
Read y
Read x
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same result holds for any kind of operation
(Read or Write) that the processors apply to x
and y
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Case 2 suppose that they access the same
shared variable
subcase read/read
bivalent
x
Read x
Read x
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two possible executions
bivalent
Read x
Read x
different valence
impossible since
Read x
Read x
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subcase read/write
bivalent
Read x
Write x
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two possible executions
bivalent
Write x
Read x
different valence
impossible since
Write x
Read x
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subcase write/write
bivalent
Write x
Write x
different valence
impossible since
Write x
Write x
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In all cases we obtained contradiction
Therefore, there exists a processor which is not
critical
univalent
bivalent
univalent
bivalent
(not critical)
univalent
End of Lemma Proof
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Therefore, we can construct an execution
bivalent
bivalent
bivalent
Never ends
Initial configuration
Consensus can never be reached
End of Theorem Proof