<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
process	O
calculi	O
(	O
or	O
process	O
algebras	O
)	O
are	O
a	O
diverse	O
family	O
of	O
related	O
approaches	O
for	O
formally	O
modelling	O
concurrent	B-Operating_System
systems	I-Operating_System
.	O
</s>
<s>
They	O
also	O
provide	O
algebraic	O
laws	O
that	O
allow	O
process	O
descriptions	O
to	O
be	O
manipulated	O
and	O
analyzed	O
,	O
and	O
permit	O
formal	O
reasoning	O
about	O
equivalences	O
between	O
processes	O
(	O
e.g.	O
,	O
using	O
bisimulation	B-Application
)	O
.	O
</s>
<s>
Leading	O
examples	O
of	O
process	O
calculi	O
include	O
CSP	O
,	O
CCS	O
,	O
ACP	O
,	O
and	O
LOTOS	B-Application
.	O
</s>
<s>
More	O
recent	O
additions	O
to	O
the	O
family	O
include	O
the	O
π-calculus	O
,	O
the	O
ambient	O
calculus	O
,	O
PEPA	O
,	O
the	O
fusion	O
calculus	O
and	O
the	O
join-calculus	B-Language
.	O
</s>
<s>
Representing	O
interactions	O
between	O
independent	O
processes	O
as	O
communication	O
(	O
message-passing	B-Architecture
)	O
,	O
rather	O
than	O
as	O
modification	O
of	O
shared	O
variables	O
.	O
</s>
<s>
To	O
define	O
a	O
process	O
calculus	O
,	O
one	O
starts	O
with	O
a	O
set	O
of	O
names	O
(	O
or	O
channels	B-Operating_System
)	O
whose	O
purpose	O
is	O
to	O
provide	O
means	O
of	O
communication	O
.	O
</s>
<s>
In	O
many	O
implementations	O
,	O
channels	B-Operating_System
have	O
rich	O
internal	O
structure	O
to	O
improve	O
efficiency	O
,	O
but	O
this	O
is	O
abstracted	O
away	O
in	O
most	O
theoretic	O
models	O
.	O
</s>
<s>
Channels	B-Operating_System
may	O
be	O
synchronous	O
or	O
asynchronous	O
.	O
</s>
<s>
Asynchronous	O
channels	B-Operating_System
do	O
not	O
require	O
any	O
such	O
synchronization	O
.	O
</s>
<s>
In	O
some	O
process	O
calculi	O
(	O
notably	O
the	O
π-calculus	O
)	O
channels	B-Operating_System
themselves	O
can	O
be	O
sent	O
in	O
messages	O
through	O
(	O
other	O
)	O
channels	B-Operating_System
,	O
allowing	O
the	O
topology	O
of	O
process	O
interconnections	O
to	O
change	O
.	O
</s>
<s>
Some	O
process	O
calculi	O
also	O
allow	O
channels	B-Operating_System
to	O
be	O
created	O
during	O
the	O
execution	O
of	O
a	O
computation	O
.	O
</s>
<s>
Sequential	B-Algorithm
composition	I-Algorithm
can	O
be	O
used	O
for	O
such	O
purposes	O
.	O
</s>
<s>
Recursion	O
and	O
replication	B-Architecture
are	O
operations	O
that	O
allow	O
finite	O
descriptions	O
of	O
infinite	O
behaviour	O
.	O
</s>
<s>
Replication	B-Architecture
can	O
be	O
understood	O
as	O
abbreviating	O
the	O
parallel	O
composition	O
of	O
a	O
countably	O
infinite	O
number	O
of	O
processes	O
:	O
</s>
<s>
In	O
the	O
first	O
half	O
of	O
the	O
20th	O
century	O
,	O
various	O
formalisms	O
were	O
proposed	O
to	O
capture	O
the	O
informal	O
concept	O
of	O
a	O
computable	O
function	O
,	O
with	O
μ-recursive	O
functions	O
,	O
Turing	B-Architecture
machines	I-Architecture
and	O
the	O
lambda	B-Language
calculus	I-Language
possibly	O
being	O
the	O
best-known	O
examples	O
today	O
.	O
</s>
<s>
The	O
subsequent	O
consolidation	O
of	O
computer	B-General_Concept
science	I-General_Concept
required	O
a	O
more	O
subtle	O
formulation	O
of	O
the	O
notion	O
of	O
computation	O
,	O
in	O
particular	O
explicit	O
representations	O
of	O
concurrency	B-Architecture
and	O
communication	O
.	O
</s>
<s>
Models	O
of	O
concurrency	B-Architecture
such	O
as	O
the	O
process	O
calculi	O
,	O
Petri	B-Operating_System
nets	I-Operating_System
in	O
1962	O
,	O
and	O
the	O
actor	B-Application
model	I-Application
in	O
1973	O
emerged	O
from	O
this	O
line	O
of	O
inquiry	O
.	O
</s>
<s>
Bisimulations	B-Application
are	O
a	O
technical	O
tool	O
that	O
aids	O
reasoning	O
about	O
process	O
equivalences	O
.	O
</s>
<s>
The	O
use	O
of	O
channels	B-Operating_System
for	O
communication	O
is	O
one	O
of	O
the	O
features	O
distinguishing	O
the	O
process	O
calculi	O
from	O
other	O
models	O
of	O
concurrency	B-Architecture
,	O
such	O
as	O
Petri	B-Operating_System
nets	I-Operating_System
and	O
the	O
actor	B-Application
model	I-Application
(	O
see	O
Actor	B-Application
model	I-Application
and	I-Application
process	I-Application
calculi	I-Application
)	O
.	O
</s>
<s>
One	O
of	O
the	O
fundamental	O
motivations	O
for	O
including	O
channels	B-Operating_System
in	O
the	O
process	O
calculi	O
was	O
to	O
enable	O
certain	O
algebraic	O
techniques	O
,	O
thereby	O
making	O
it	O
easier	O
to	O
reason	O
about	O
processes	O
algebraically	O
.	O
</s>
