<s>
A	O
fundamental	O
problem	O
in	O
distributed	B-Architecture
computing	I-Architecture
and	O
multi-agent	O
systems	O
is	O
to	O
achieve	O
overall	O
system	O
reliability	O
in	O
the	O
presence	O
of	O
a	O
number	O
of	O
faulty	O
processes	B-Operating_System
.	O
</s>
<s>
This	O
often	O
requires	O
coordinating	O
processes	B-Operating_System
to	O
reach	O
consensus	O
,	O
or	O
agree	O
on	O
some	O
data	O
value	O
that	O
is	O
needed	O
during	O
computation	O
.	O
</s>
<s>
Example	O
applications	O
of	O
consensus	O
include	O
agreeing	O
on	O
what	O
transactions	O
to	O
commit	O
to	O
a	O
database	O
in	O
which	O
order	O
,	O
state	B-Operating_System
machine	I-Operating_System
replication	I-Operating_System
,	O
and	O
atomic	B-Operating_System
broadcasts	I-Operating_System
.	O
</s>
<s>
Real-world	O
applications	O
often	O
requiring	O
consensus	O
include	O
cloud	B-Architecture
computing	I-Architecture
,	O
clock	O
synchronization	O
,	O
PageRank	B-Algorithm
,	O
opinion	O
formation	O
,	O
smart	B-Architecture
power	I-Architecture
grids	I-Architecture
,	O
state	O
estimation	O
,	O
control	O
of	O
UAVs	O
(	O
and	O
multiple	O
robots/agents	O
in	O
general	O
)	O
,	O
load	B-Application
balancing	I-Application
,	O
blockchain	B-Application
,	O
and	O
others	O
.	O
</s>
<s>
The	O
consensus	B-Operating_System
problem	I-Operating_System
requires	O
agreement	O
among	O
a	O
number	O
of	O
processes	B-Operating_System
(	O
or	O
agents	O
)	O
for	O
a	O
single	O
data	O
value	O
.	O
</s>
<s>
Some	O
of	O
the	O
processes	B-Operating_System
(	O
agents	O
)	O
may	O
fail	O
or	O
be	O
unreliable	O
in	O
other	O
ways	O
,	O
so	O
consensus	O
protocols	O
must	O
be	O
fault	B-General_Concept
tolerant	I-General_Concept
or	O
resilient	O
.	O
</s>
<s>
The	O
processes	B-Operating_System
must	O
somehow	O
put	O
forth	O
their	O
candidate	O
values	O
,	O
communicate	O
with	O
one	O
another	O
,	O
and	O
agree	O
on	O
a	O
single	O
consensus	O
value	O
.	O
</s>
<s>
The	O
consensus	B-Operating_System
problem	I-Operating_System
is	O
a	O
fundamental	O
problem	O
in	O
control	O
of	O
multi-agent	O
systems	O
.	O
</s>
<s>
One	O
approach	O
to	O
generating	O
consensus	O
is	O
for	O
all	O
processes	B-Operating_System
(	O
agents	O
)	O
to	O
agree	O
on	O
a	O
majority	O
value	O
.	O
</s>
<s>
However	O
,	O
one	O
or	O
more	O
faulty	O
processes	B-Operating_System
may	O
skew	O
the	O
resultant	O
outcome	O
such	O
that	O
consensus	O
may	O
not	O
be	O
reached	O
or	O
reached	O
incorrectly	O
.	O
</s>
<s>
Protocols	O
that	O
solve	O
consensus	B-Operating_System
problems	I-Operating_System
are	O
designed	O
to	O
deal	O
with	O
limited	O
numbers	O
of	O
faulty	O
processes	B-Operating_System
.	O
</s>
<s>
For	O
instance	O
,	O
a	O
trivial	O
protocol	O
could	O
have	O
all	O
processes	B-Operating_System
output	O
binary	O
value	O
1	O
.	O
</s>
<s>
Integrity	O
If	O
all	O
the	O
correct	O
processes	B-Operating_System
proposed	O
the	O
same	O
value	O
,	O
then	O
any	O
correct	O
process	O
must	O
decide	O
.	O
</s>
<s>
A	O
protocol	O
that	O
can	O
correctly	O
guarantee	O
consensus	O
amongst	O
n	O
processes	B-Operating_System
of	O
which	O
at	O
most	O
t	O
fail	O
is	O
said	O
to	O
be	O
t-resilient	O
.	O
</s>
<s>
Running	O
time	O
is	O
given	O
in	O
Big	O
O	O
notation	O
in	O
the	O
number	O
of	O
rounds	O
of	O
message	O
exchange	O
as	O
a	O
function	O
of	O
some	O
input	O
parameters	O
(	O
typically	O
the	O
number	O
of	O
processes	B-Operating_System
and/or	O
the	O
size	O
of	O
the	O
input	O
domain	O
)	O
.	O
</s>
<s>
Varying	O
models	O
of	O
computation	O
may	O
define	O
a	O
"	O
consensus	B-Operating_System
problem	I-Operating_System
"	O
.	O
</s>
<s>
In	O
some	O
models	O
message	O
authentication	O
is	O
allowed	O
,	O
whereas	O
in	O
others	O
processes	B-Operating_System
are	O
completely	O
anonymous	O
.	O
</s>
<s>
Shared	O
memory	O
models	O
in	O
which	O
processes	B-Operating_System
communicate	O
by	O
accessing	O
objects	O
in	O
shared	O
memory	O
are	O
also	O
an	O
important	O
area	O
of	O
research	O
.	O
</s>
<s>
In	O
the	O
most	O
traditional	O
single-value	O
consensus	O
protocols	O
such	O
as	O
Paxos	B-Operating_System
,	O
cooperating	O
nodes	O
agree	O
on	O
a	O
single	O
value	O
such	O
as	O
an	O
integer	O
,	O
which	O
may	O
be	O
of	O
variable	O
size	O
so	O
as	O
to	O
encode	O
useful	O
metadata	O
such	O
as	O
a	O
transaction	O
committed	O
to	O
a	O
database	O
.	O
</s>
<s>
A	O
special	O
case	O
of	O
the	O
single-value	O
consensus	B-Operating_System
problem	I-Operating_System
,	O
called	O
binary	O
consensus	O
,	O
restricts	O
the	O
input	O
,	O
and	O
hence	O
the	O
output	O
domain	O
,	O
to	O
a	O
single	O
binary	O
digit	O
 { 0 , 1 } 	O
.	O
</s>
<s>
In	O
multi-valued	O
consensus	O
protocols	O
such	O
as	O
Multi-Paxos	O
and	O
Raft	B-Operating_System
,	O
the	O
goal	O
is	O
to	O
agree	O
on	O
not	O
just	O
a	O
single	O
value	O
but	O
a	O
series	O
of	O
values	O
over	O
time	O
,	O
forming	O
a	O
progressively-growing	O
history	O
.	O
</s>
<s>
There	O
are	O
two	O
types	O
of	O
failures	O
a	O
process	O
may	O
undergo	O
,	O
a	O
crash	O
failure	O
or	O
a	O
Byzantine	B-Operating_System
failure	I-Operating_System
.	O
</s>
<s>
Byzantine	B-Operating_System
failures	I-Operating_System
are	O
failures	O
in	O
which	O
absolutely	O
no	O
conditions	O
are	O
imposed	O
.	O
</s>
<s>
A	O
process	O
that	O
experiences	O
a	O
Byzantine	B-Operating_System
failure	I-Operating_System
may	O
send	O
contradictory	O
or	O
conflicting	O
data	O
to	O
other	O
processes	B-Operating_System
,	O
or	O
it	O
may	O
sleep	O
and	O
then	O
resume	O
activity	O
after	O
a	O
lengthy	O
delay	O
.	O
</s>
<s>
Of	O
the	O
two	O
types	O
of	O
failures	O
,	O
Byzantine	B-Operating_System
failures	I-Operating_System
are	O
far	O
more	O
disruptive	O
.	O
</s>
<s>
Thus	O
,	O
a	O
consensus	O
protocol	O
tolerating	O
Byzantine	B-Operating_System
failures	I-Operating_System
must	O
be	O
resilient	O
to	O
every	O
possible	O
error	O
that	O
can	O
occur	O
.	O
</s>
<s>
A	O
stronger	O
version	O
of	O
consensus	O
tolerating	O
Byzantine	B-Operating_System
failures	I-Operating_System
is	O
given	O
by	O
strengthening	O
the	O
Integrity	O
constraint	O
:	O
</s>
<s>
The	O
consensus	B-Operating_System
problem	I-Operating_System
may	O
be	O
considered	O
in	O
the	O
case	O
of	O
asynchronous	O
or	O
synchronous	O
systems	O
.	O
</s>
<s>
In	O
one	O
round	O
,	O
a	O
process	O
may	O
send	O
all	O
the	O
messages	O
it	O
requires	O
,	O
while	O
receiving	O
all	O
messages	O
from	O
other	O
processes	B-Operating_System
.	O
</s>
<s>
In	O
a	O
fully	O
asynchronous	O
message-passing	O
distributed	B-Architecture
system	I-Architecture
,	O
in	O
which	O
at	O
least	O
one	O
process	O
may	O
have	O
a	O
crash	O
failure	O
,	O
it	O
has	O
been	O
proven	O
in	O
the	O
famous	O
1985	O
FLP	O
impossibility	O
result	O
by	O
Fischer	O
,	O
Lynch	O
and	O
Paterson	O
that	O
a	O
deterministic	B-General_Concept
algorithm	I-General_Concept
for	O
achieving	O
consensus	O
is	O
impossible	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
loss	O
of	O
a	O
communication	O
link	O
may	O
be	O
modeled	O
as	O
a	O
process	O
which	O
has	O
suffered	O
a	O
Byzantine	B-Operating_System
failure	I-Operating_System
.	O
</s>
<s>
Randomized	B-General_Concept
consensus	B-Operating_System
algorithms	I-Operating_System
can	O
circumvent	O
the	O
FLP	O
impossibility	O
result	O
by	O
achieving	O
both	O
safety	O
and	O
liveness	O
with	O
overwhelming	O
probability	O
,	O
even	O
under	O
worst-case	O
scheduling	O
scenarios	O
such	O
as	O
an	O
intelligent	O
denial-of-service	O
attacker	O
in	O
the	O
network	O
.	O
</s>
<s>
Consensus	B-Operating_System
algorithms	I-Operating_System
traditionally	O
assume	O
that	O
the	O
set	O
of	O
participating	O
nodes	O
is	O
fixed	O
and	O
given	O
at	O
the	O
outset	O
:	O
that	O
is	O
,	O
that	O
some	O
prior	O
(	O
manual	O
or	O
automatic	O
)	O
configuration	O
process	O
has	O
permissioned	O
a	O
particular	O
known	O
group	O
of	O
participants	O
who	O
can	O
authenticate	O
each	O
other	O
as	O
members	O
of	O
the	O
group	O
.	O
</s>
<s>
In	O
the	O
absence	O
of	O
such	O
a	O
well-defined	O
,	O
closed	O
group	O
with	O
authenticated	O
members	O
,	O
a	O
Sybil	O
attack	O
against	O
an	O
open	O
consensus	O
group	O
can	O
defeat	O
even	O
a	O
Byzantine	O
consensus	B-Operating_System
algorithm	I-Operating_System
,	O
simply	O
by	O
creating	O
enough	O
virtual	O
participants	O
to	O
overwhelm	O
the	O
fault	B-General_Concept
tolerance	I-General_Concept
threshold	O
.	O
</s>
<s>
Bitcoin	B-Protocol
introduced	O
the	O
first	O
permissionless	O
consensus	O
protocol	O
using	O
proof	O
of	O
work	O
and	O
a	O
difficulty	O
adjustment	O
function	O
,	O
in	O
which	O
participants	O
compete	O
to	O
solve	O
cryptographic	B-Algorithm
hash	I-Algorithm
puzzles	O
,	O
and	O
probabilistically	O
earn	O
the	O
right	O
to	O
commit	O
blocks	O
and	O
earn	O
associated	O
rewards	O
in	O
proportion	O
to	O
their	O
invested	O
computational	O
effort	O
.	O
</s>
<s>
Motivated	O
in	O
part	O
by	O
the	O
high	O
energy	O
cost	O
of	O
this	O
approach	O
,	O
subsequent	O
permissionless	O
consensus	O
protocols	O
have	O
proposed	O
or	O
adopted	O
other	O
alternative	O
participation	O
rules	O
for	O
Sybil	O
attack	O
protection	O
,	O
such	O
as	O
proof	B-Protocol
of	I-Protocol
stake	I-Protocol
,	O
proof	B-Protocol
of	I-Protocol
space	I-Protocol
,	O
and	O
proof	B-Algorithm
of	I-Algorithm
authority	I-Algorithm
.	O
</s>
<s>
A	O
collection	O
of	O
processes	B-Operating_System
,	O
numbered	O
from	O
to	O
communicate	O
by	O
sending	O
messages	O
to	O
one	O
another	O
.	O
</s>
<s>
Process	O
must	O
transmit	O
a	O
value	O
to	O
all	O
processes	B-Operating_System
such	O
that	O
:	O
</s>
<s>
for	O
any	O
two	O
correct	O
processes	B-Operating_System
,	O
each	O
process	O
receives	O
the	O
same	O
value	O
.	O
</s>
<s>
Agreement	O
:	O
All	O
correct	O
processes	B-Operating_System
must	O
agree	O
on	O
the	O
same	O
value	O
.	O
</s>
<s>
Strong	O
validity	O
:	O
If	O
all	O
correct	O
processes	B-Operating_System
receive	O
the	O
same	O
input	O
value	O
,	O
then	O
they	O
must	O
all	O
output	O
that	O
value	O
.	O
</s>
<s>
For	O
n	O
processes	B-Operating_System
in	O
a	O
partially	O
synchronous	O
system	O
(	O
the	O
system	O
alternates	O
between	O
good	O
and	O
bad	O
periods	O
of	O
synchrony	O
)	O
,	O
each	O
process	O
chooses	O
a	O
private	O
value	O
.	O
</s>
<s>
all	O
messages	O
sent	O
in	O
a	O
round	O
by	O
a	O
correct	O
process	O
are	O
received	O
in	O
the	O
same	O
round	O
by	O
all	O
correct	O
processes	B-Operating_System
(	O
consistency	O
property	O
)	O
.	O
</s>
<s>
An	O
interactive	O
consistency	O
algorithm	O
can	O
solve	O
the	O
consensus	B-Operating_System
problem	I-Operating_System
by	O
having	O
each	O
process	O
choose	O
the	O
majority	O
value	O
in	O
its	O
consensus	O
vector	O
as	O
its	O
consensus	O
value	O
.	O
</s>
<s>
There	O
is	O
a	O
t-resilient	O
anonymous	O
synchronous	O
protocol	O
which	O
solves	O
the	O
Byzantine	B-Operating_System
Generals	I-Operating_System
problem	I-Operating_System
,	O
if	O
and	O
the	O
Weak	O
Byzantine	B-Operating_System
Generals	I-Operating_System
case	O
where	O
is	O
the	O
number	O
of	O
failures	O
and	O
is	O
the	O
number	O
of	O
processes	B-Operating_System
.	O
</s>
<s>
For	O
systems	O
with	O
processors	O
,	O
of	O
which	O
are	O
Byzantine	O
,	O
it	O
has	O
been	O
shown	O
that	O
there	O
exists	O
no	O
algorithm	O
that	O
solves	O
the	O
consensus	B-Operating_System
problem	I-Operating_System
for	O
in	O
the	O
oral-messages	O
model	O
.	O
</s>
<s>
The	O
Paxos	B-Operating_System
consensus	I-Operating_System
algorithm	I-Operating_System
by	O
Leslie	O
Lamport	O
,	O
and	O
variants	O
of	O
it	O
such	O
as	O
Raft	B-Operating_System
,	O
are	O
used	O
pervasively	O
in	O
widely	O
deployed	O
distributed	B-Architecture
and	O
cloud	B-Architecture
computing	I-Architecture
systems	O
.	O
</s>
<s>
These	O
algorithms	O
are	O
typically	O
synchronous	O
,	O
dependent	O
on	O
an	O
elected	O
leader	O
to	O
make	O
progress	O
,	O
and	O
tolerate	O
only	O
crashes	O
and	O
not	O
Byzantine	B-Operating_System
failures	I-Operating_System
.	O
</s>
<s>
An	O
example	O
of	O
a	O
polynomial	O
time	O
binary	O
consensus	O
protocol	O
that	O
tolerates	O
Byzantine	B-Operating_System
failures	I-Operating_System
is	O
the	O
Phase	O
King	O
algorithm	O
by	O
Garay	O
and	O
Berman	O
.	O
</s>
<s>
The	O
algorithm	O
solves	O
consensus	O
in	O
a	O
synchronous	O
message	O
passing	O
model	O
with	O
n	O
processes	B-Operating_System
and	O
up	O
to	O
f	O
failures	O
,	O
provided	O
n	O
>	O
4f	O
.	O
</s>
<s>
In	O
the	O
first	O
round	O
of	O
each	O
phase	O
each	O
process	O
broadcasts	O
its	O
own	O
preferred	O
value	O
to	O
all	O
other	O
processes	B-Operating_System
.	O
</s>
<s>
It	O
then	O
receives	O
the	O
values	O
from	O
all	O
processes	B-Operating_System
and	O
determines	O
which	O
value	O
is	O
the	O
majority	O
value	O
and	O
its	O
count	O
.	O
</s>
<s>
At	O
the	O
end	O
of	O
f	O
+	O
1	O
phases	O
the	O
processes	B-Operating_System
output	O
their	O
preferred	O
values	O
.	O
</s>
<s>
Google	O
has	O
implemented	O
a	O
distributed	B-Operating_System
lock	I-Operating_System
service	I-Operating_System
library	O
called	O
Chubby	B-Operating_System
.	O
</s>
<s>
Chubby	B-Operating_System
maintains	O
lock	O
information	O
in	O
small	O
files	O
which	O
are	O
stored	O
in	O
a	O
replicated	O
database	O
to	O
achieve	O
high	O
availability	O
in	O
the	O
face	O
of	O
failures	O
.	O
</s>
<s>
The	O
database	O
is	O
implemented	O
on	O
top	O
of	O
a	O
fault-tolerant	B-General_Concept
log	O
layer	O
which	O
is	O
based	O
on	O
the	O
Paxos	B-Operating_System
consensus	I-Operating_System
algorithm	I-Operating_System
.	O
</s>
<s>
In	O
this	O
scheme	O
,	O
Chubby	B-Operating_System
clients	O
communicate	O
with	O
the	O
Paxos	B-Operating_System
master	O
in	O
order	O
to	O
access/update	O
the	O
replicated	O
log	O
;	O
i.e.	O
,	O
read/write	O
to	O
the	O
files	O
.	O
</s>
<s>
Many	O
peer-to-peer	O
online	O
Real-time	O
strategy	O
games	O
use	O
a	O
modified	O
Lockstep	B-Protocol
protocol	I-Protocol
as	O
a	O
consensus	O
protocol	O
in	O
order	O
to	O
manage	O
game	O
state	O
between	O
players	O
in	O
a	O
game	O
.	O
</s>
<s>
Each	O
game	O
action	O
results	O
in	O
a	O
game	O
state	O
delta	O
broadcast	O
to	O
all	O
other	O
players	O
in	O
the	O
game	O
along	O
with	O
a	O
hash	B-Algorithm
of	O
the	O
total	O
game	O
state	O
.	O
</s>
<s>
Each	O
player	O
validates	O
the	O
change	O
by	O
applying	O
the	O
delta	O
to	O
their	O
own	O
game	O
state	O
and	O
comparing	O
the	O
game	O
state	O
hashes	B-Algorithm
.	O
</s>
<s>
If	O
the	O
hashes	B-Algorithm
do	O
not	O
agree	O
then	O
a	O
vote	O
is	O
cast	O
,	O
and	O
those	O
players	O
whose	O
game	O
state	O
is	O
in	O
the	O
minority	O
are	O
disconnected	O
and	O
removed	O
from	O
the	O
game	O
(	O
known	O
as	O
a	O
desync	O
.	O
)	O
</s>
<s>
Synchronous	O
Oral	O
total	O
communication	O
Dolev-Strong	O
Synchronous	O
Written	O
total	O
communication	O
Dolev-Strong	O
Synchronous	O
Written	O
total	O
communication	O
Feldman-Micali	O
Synchronous	O
Oral	O
(	O
expected	O
)	O
Katz-Koo	O
Synchronous	O
Written	O
(	O
expected	O
)	O
Requires	O
PKI	O
PBFT	B-Operating_System
Asynchronous	O
(	O
safety	O
)	O
--	O
Synchronous	O
(	O
liveness	O
)	O
Oral	O
HoneyBadger	O
Asynchronous	O
Oral	O
(	O
expected	O
)	O
per	O
tx	O
communication	O
-	O
requires	O
public-key	O
encryption	O
Abraham	O
et	O
al	O
.	O
</s>
<s>
Bitcoin	B-Protocol
uses	O
proof	O
of	O
work	O
,	O
a	O
difficulty	O
adjustment	O
function	O
and	O
a	O
reorganization	O
function	O
to	O
achieve	O
permissionless	O
consensus	O
in	O
its	O
open	O
peer-to-peer	O
network	O
.	O
</s>
<s>
To	O
extend	O
Bitcoin	B-Protocol
's	O
blockchain	B-Application
or	O
distributed	B-Architecture
ledger	O
,	O
miners	O
attempt	O
to	O
solve	O
a	O
cryptographic	O
puzzle	O
,	O
where	O
probability	O
of	O
finding	O
a	O
solution	O
is	O
proportional	O
to	O
the	O
computational	O
effort	O
expended	O
in	O
hashes	B-Algorithm
per	O
second	O
.	O
</s>
<s>
NEO	O
,	O
STRATIS	O
,	O
...	O
)	O
use	O
proof	B-Protocol
of	I-Protocol
stake	I-Protocol
,	O
in	O
which	O
nodes	O
compete	O
to	O
append	O
blocks	O
and	O
earn	O
associated	O
rewards	O
in	O
proportion	O
to	O
stake	O
,	O
or	O
existing	O
cryptocurrency	O
allocated	O
and	O
locked	O
or	O
staked	O
for	O
some	O
time	O
period	O
.	O
</s>
<s>
One	O
advantage	O
of	O
a	O
'	O
proof	B-Protocol
of	I-Protocol
stake	I-Protocol
 '	O
over	O
a	O
'	O
proof	O
of	O
work	O
 '	O
system	O
,	O
is	O
the	O
high	O
energy	O
consumption	O
demanded	O
by	O
the	O
latter	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
Bitcoin	B-Protocol
mining	O
(	O
2018	O
)	O
is	O
estimated	O
to	O
consume	O
non-renewable	O
energy	O
sources	O
at	O
an	O
amount	O
similar	O
to	O
the	O
entire	O
nations	O
of	O
Czech	O
Republic	O
or	O
Jordan	O
.	O
</s>
<s>
This	O
system	O
used	O
by	O
Ripple	O
,	O
called	O
Ripple	O
Protocol	O
Consensus	B-Operating_System
Algorithm	I-Operating_System
(	O
RPCA	O
)	O
,	O
works	O
in	O
rounds	O
:	O
</s>
<s>
Other	O
participation	O
rules	O
used	O
in	O
permissionless	O
consensus	O
protocols	O
to	O
impose	O
barriers	O
to	O
entry	O
and	O
resist	O
sybil	O
attacks	O
include	O
proof	B-Algorithm
of	I-Algorithm
authority	I-Algorithm
,	O
proof	B-Protocol
of	I-Protocol
space	I-Protocol
,	O
proof	O
of	O
burn	O
,	O
or	O
proof	O
of	O
elapsed	O
time	O
.	O
</s>
<s>
To	O
solve	O
the	O
consensus	B-Operating_System
problem	I-Operating_System
in	O
a	O
shared-memory	O
system	O
,	O
concurrent	O
objects	O
must	O
be	O
introduced	O
.	O
</s>
<s>
A	O
concurrent	O
object	O
,	O
or	O
shared	O
object	O
,	O
is	O
a	O
data	O
structure	O
which	O
helps	O
concurrent	O
processes	B-Operating_System
communicate	O
to	O
reach	O
an	O
agreement	O
.	O
</s>
<s>
Traditional	O
implementations	O
using	O
critical	B-Operating_System
sections	I-Operating_System
face	O
the	O
risk	O
of	O
crashing	O
if	O
some	O
process	O
dies	O
inside	O
the	O
critical	B-Operating_System
section	I-Operating_System
or	O
sleeps	O
for	O
an	O
intolerably	O
long	O
time	O
.	O
</s>
<s>
The	O
consensus	O
number	O
of	O
a	O
concurrent	O
object	O
is	O
defined	O
to	O
be	O
the	O
maximum	O
number	O
of	O
processes	B-Operating_System
in	O
the	O
system	O
which	O
can	O
reach	O
consensus	O
by	O
the	O
given	O
object	O
in	O
a	O
wait-free	O
implementation	O
.	O
</s>
<s>
According	O
to	O
the	O
hierarchy	O
,	O
read/write	B-Operating_System
registers	I-Operating_System
cannot	O
solve	O
consensus	O
even	O
in	O
a	O
2-process	O
system	O
.	O
</s>
<s>
Data	O
structures	O
like	O
stacks	B-Application
and	O
queues	B-Application
can	O
only	O
solve	O
consensus	O
between	O
two	O
processes	B-Operating_System
.	O
</s>
<s>
However	O
,	O
some	O
concurrent	O
objects	O
are	O
universal	O
(	O
notated	O
in	O
the	O
table	O
with	O
)	O
,	O
which	O
means	O
they	O
can	O
solve	O
consensus	O
among	O
any	O
number	O
of	O
processes	B-Operating_System
and	O
they	O
can	O
simulate	O
any	O
other	O
objects	O
through	O
an	O
operation	O
sequence	O
.	O
</s>
