<s>
The	O
denotational	B-Application
semantics	I-Application
of	I-Application
the	I-Application
Actor	I-Application
model	I-Application
is	O
the	O
subject	O
of	O
denotational	O
domain	O
theory	O
for	O
Actors	B-Application
.	O
</s>
<s>
The	O
Actor	B-Application
uses	O
the	O
domain	O
of	O
event	O
diagram	O
scenarios	O
.	O
</s>
<s>
A	O
central	O
question	O
of	O
denotational	B-Application
semantics	I-Application
is	O
to	O
characterize	O
when	O
it	O
is	O
possible	O
to	O
create	O
denotations	O
(	O
meanings	O
)	O
according	O
to	O
the	O
equation	O
for	O
.	O
</s>
<s>
An	O
important	O
aspect	O
of	O
denotational	B-Application
semantics	I-Application
of	O
programming	O
languages	O
is	O
compositionality	O
,	O
by	O
which	O
the	O
denotation	O
of	O
a	O
program	O
is	O
constructed	O
from	O
denotations	O
of	O
its	O
parts	O
.	O
</s>
<s>
The	O
Actor	B-Application
model	I-Application
provides	O
a	O
modern	O
and	O
very	O
general	O
way	O
the	O
compositionality	O
of	O
programs	O
can	O
be	O
analyzed	O
.	O
</s>
<s>
Scott	O
and	O
Strachey	O
 [ 1971 ] 	O
proposed	O
that	O
the	O
semantics	O
of	O
programming	O
languages	O
be	O
reduced	O
to	O
the	O
semantics	O
of	O
the	O
lambda	B-Language
calculus	I-Language
and	O
thus	O
inherit	O
the	O
denotational	B-Application
semantics	I-Application
of	O
the	O
lambda	B-Language
calculus	I-Language
.	O
</s>
<s>
However	O
,	O
it	O
turned	O
out	O
that	O
concurrent	O
computation	O
could	O
not	O
be	O
implemented	O
in	O
the	O
lambda	B-Language
calculus	I-Language
(	O
see	O
Indeterminacy	B-Application
in	I-Application
concurrent	I-Application
computation	I-Application
)	O
.	O
</s>
<s>
Thus	O
there	O
arose	O
the	O
problem	O
of	O
how	O
to	O
provide	O
modular	O
denotational	B-Application
semantics	I-Application
for	O
concurrent	O
programming	O
languages	O
.	O
</s>
<s>
One	O
solution	O
to	O
this	O
problem	O
is	O
to	O
use	O
the	O
Actor	B-Application
model	I-Application
of	O
computation	O
.	O
</s>
<s>
In	O
Actor	B-Application
model	I-Application
,	O
programs	O
are	O
Actors	B-Application
that	O
are	O
sent	O
messages	O
with	O
the	O
address	O
of	O
an	O
environment	O
(	O
explained	O
below	O
)	O
so	O
that	O
programs	O
inherit	O
their	O
denotational	B-Application
semantics	I-Application
from	O
the	O
denotational	B-Application
semantics	I-Application
of	I-Application
the	I-Application
Actor	I-Application
model	I-Application
(	O
an	O
idea	O
published	O
in	O
Hewitt	O
 [ 2006 ] 	O
)	O
.	O
</s>
<s>
However	O
,	O
responds	O
to	O
the	O
message	O
by	O
creating	O
a	O
closure	B-Language
Actor	B-Application
(	O
process	O
)	O
C	O
that	O
has	O
an	O
address	O
(	O
called	O
body	O
)	O
for	O
and	O
an	O
address	O
(	O
called	O
environment	O
)	O
for	O
E	O
.	O
The	O
Actor	B-Application
then	O
sends	O
C	O
the	O
message	O
[	O
1	O
2 ]	O
.	O
</s>
<s>
When	O
C	O
receives	O
the	O
message	O
[	O
1	O
2 ]	O
,	O
it	O
creates	O
a	O
new	O
environment	O
Actor	B-Application
F	O
which	O
behaves	O
as	O
follows	O
:	O
</s>
<s>
The	O
Actor	B-Application
(	O
process	O
)	O
C	O
then	O
sends	O
an	O
message	O
with	O
environment	O
F	O
to	O
the	O
following	O
actor	B-Application
(	O
process	O
)	O
:	O
</s>
<s>
For	O
another	O
example	O
consider	O
the	O
Actor	B-Application
for	O
the	O
expression	O
""	O
which	O
has	O
addresses	O
for	O
two	O
other	O
actors	B-Application
(	O
processes	O
)	O
and	O
.	O
</s>
<s>
When	O
the	O
composite	O
expression	O
Actor	B-Application
(	O
process	O
)	O
receives	O
an	O
message	O
with	O
addresses	O
for	O
an	O
environment	O
Actor	B-Application
E	O
and	O
a	O
customer	O
C	O
,	O
it	O
sends	O
messages	O
to	O
and	O
with	O
environment	O
E	O
and	O
sends	O
C	O
a	O
new	O
Actor	B-Application
(	O
process	O
)	O
C0	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
the	O
denotational	B-Application
semantics	I-Application
for	O
process	O
calculi	O
and	O
the	O
Actor	B-Application
model	I-Application
provide	O
a	O
denotational	B-Application
semantics	I-Application
for	O
""	O
in	O
terms	O
of	O
the	O
semantics	O
for	O
and	O
.	O
</s>
<s>
The	O
denotational	O
compositional	O
semantics	O
presented	O
above	O
is	O
very	O
general	O
and	O
can	O
be	O
used	O
for	O
functional	B-Language
,	O
imperative	B-Application
,	O
concurrent	O
,	O
logic	B-Language
,	O
etc	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
easily	O
provides	O
denotation	O
semantics	O
for	O
constructs	O
that	O
are	O
difficult	O
to	O
formalize	O
using	O
other	O
approaches	O
such	O
as	O
delays	B-Operating_System
and	O
futures	B-Operating_System
.	O
</s>
<s>
In	O
his	O
doctoral	O
dissertation	O
,	O
Will	O
Clinger	O
developed	O
the	O
first	O
denotation	O
semantics	O
for	O
the	O
Actor	B-Application
model	I-Application
.	O
</s>
<s>
Clinger	O
 [ 1981 ] 	O
explained	O
the	O
domain	O
of	O
Actor	B-Application
computations	O
as	O
follows	O
:	O
</s>
<s>
The	O
augmented	O
Actor	B-Application
event	O
diagrams	O
[	O
see	O
Actor	B-Application
model	I-Application
theory ]	O
form	O
a	O
partially	O
ordered	O
set	O
< , >	O
from	O
which	O
to	O
construct	O
the	O
power	O
domain	O
(	O
see	O
the	O
section	O
on	O
Denotations	O
below	O
)	O
.	O
</s>
<s>
To	O
repeat	O
,	O
the	O
actor	B-Application
event	O
diagram	O
domain	O
is	O
incomplete	O
because	O
of	O
the	O
requirement	O
of	O
finite	O
arrival	O
delay	O
,	O
which	O
allows	O
any	O
finite	O
delay	O
between	O
an	O
event	O
and	O
an	O
event	O
it	O
activates	O
but	O
rules	O
out	O
infinite	O
delay	O
.	O
</s>
<s>
For	O
the	O
actor	B-Application
event	O
diagram	O
domain	O
,	O
an	O
element	O
of	O
represents	O
a	O
list	O
of	O
possible	O
initial	O
histories	O
of	O
a	O
computation	O
.	O
</s>
<s>
It	O
happens	O
that	O
defining	O
Actor	B-Application
semantics	O
as	O
in	O
Clinger	O
 [ 1981 ] 	O
does	O
not	O
require	O
solving	O
any	O
recursive	O
equations	O
involving	O
the	O
power	O
domain	O
.	O
</s>
<s>
The	O
computations	O
are	O
represented	O
formally	O
by	O
Actor	B-Application
event	O
diagrams	O
.	O
</s>
<s>
Greif	O
specified	O
the	O
event	O
diagrams	O
by	O
means	O
of	O
causal	O
axioms	O
governing	O
the	O
behaviors	O
of	O
individual	O
Actors	B-Application
[	O
Greif	O
1975 ]	O
.	O
</s>
<s>
A	O
program	O
denotes	O
a	O
set	O
of	O
Actor	B-Application
event	O
diagrams	O
.	O
</s>
<s>
The	O
behaviors	O
of	O
individual	O
Actors	B-Application
is	O
defined	O
functionally	O
.	O
</s>
<s>
It	O
is	O
shown	O
,	O
however	O
,	O
that	O
the	O
resulting	O
set	O
of	O
Actor	B-Application
event	O
diagrams	O
consists	O
of	O
exactly	O
those	O
diagrams	O
that	O
satisfy	O
causal	O
axioms	O
expressing	O
the	O
functional	B-Language
behaviors	O
of	O
Actors	B-Application
.	O
</s>
<s>
Augmenting	O
Actor	B-Application
event	O
diagrams	O
with	O
sets	O
of	O
pending	O
events	O
helps	O
to	O
express	O
the	O
finite	O
delay	O
property	O
,	O
which	O
is	O
characteristic	O
of	O
true	O
concurrency	O
[	O
Schwartz	O
1979 ]	O
.	O
</s>
<s>
Instead	O
of	O
beginning	O
with	O
a	O
semantics	O
for	O
sequential	O
programs	O
and	O
then	O
trying	O
to	O
extend	O
it	O
for	O
concurrency	O
,	O
Actor	B-Application
semantics	O
views	O
concurrency	O
as	O
primary	O
and	O
obtains	O
the	O
semantics	O
of	O
sequential	O
programs	O
as	O
a	O
special	O
case	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
partial	O
computations	O
that	O
can	O
be	O
produced	O
by	O
sequential	O
computation	O
form	O
an	O
ω-complete	O
subdomain	O
of	O
the	O
domain	O
of	O
Actor	B-Application
computations	O
.	O
</s>
<s>
From	O
the	O
Actor	B-Application
point	O
of	O
view	O
,	O
sequential	O
computations	O
are	O
a	O
special	O
case	O
of	O
concurrent	O
computations	O
,	O
distinguishable	O
by	O
their	O
event	O
diagrams	O
.	O
</s>
<s>
The	O
above	O
proof	O
applies	O
to	O
all	O
sequential	O
programs	O
,	O
even	O
those	O
with	O
choice	O
points	O
such	O
as	O
guarded	B-Application
commands	I-Application
.	O
</s>
<s>
Thus	O
Actor	B-Application
semantics	O
includes	O
sequential	O
programs	O
as	O
a	O
special	O
case	O
,	O
and	O
agrees	O
with	O
conventional	O
semantics	O
of	O
such	O
programs	O
.	O
</s>
<s>
Hewitt	O
 [ 2006b ] 	O
published	O
a	O
new	O
denotational	B-Application
semantics	I-Application
for	O
Actors	B-Application
based	O
on	O
Timed	O
Diagrams	O
.	O
</s>
<s>
domain	O
for	O
Actor	B-Application
computations	O
.	O
</s>
<s>
The	O
Actor	B-Application
event	O
timed	O
diagrams	O
form	O
a	O
partially	O
ordered	O
set	O
<TimedDiagrams, ≤>	O
.	O
</s>
<s>
Theorem	O
:	O
TimedDiagrams	O
is	O
an	O
ω-complete	O
domain	O
of	O
Actor	B-Application
computations	O
i.e.	O
,	O
</s>
<s>
If	O
D⊆TimedDiagrams	O
is	O
directed	O
,	O
the	O
least	O
upper	O
bound	O
⊔D	O
exists	O
;	O
furthermore	O
⊔D	O
obeys	O
all	O
the	O
laws	O
of	O
Actor	B-Application
model	I-Application
theory	O
.	O
</s>
<s>
An	O
Actor	B-Application
computation	O
can	O
progress	O
in	O
many	O
ways	O
.	O
</s>
<s>
X	O
≡	O
{	O
e’	O
|	O
e	O
─≈	O
→	O
1-message	O
e’}	O
(	O
see	O
Actor	B-Application
model	I-Application
theory	O
)	O
,	O
</s>
<s>
The	O
denotation	O
DenoteS	O
of	O
an	O
Actor	B-Application
system	I-Application
S	O
is	O
the	O
set	O
of	O
all	O
computations	O
of	O
S	O
.	O
</s>
<s>
of	O
Actor	B-Application
systems	I-Application
by	O
directly	O
constructing	O
a	O
minimal	O
fixed	O
point	O
.	O
</s>
<s>
The	O
criterion	O
of	O
continuity	O
for	O
the	O
graphs	O
of	O
functions	O
that	O
Scott	O
used	O
to	O
initially	O
develop	O
the	O
denotational	B-Application
semantics	I-Application
of	O
functions	O
can	O
be	O
derived	O
as	O
a	O
consequence	O
of	O
the	O
Actor	B-Application
laws	O
for	O
computation	O
as	O
shown	O
in	O
the	O
next	O
section	O
.	O
</s>
