Glynn Winskel. Ab s tr c3;:ct. We give denotational sernantics to a wide range of parallel pro- gramming languages based on the idea of Milner's ccs [M1], that.
~ = I ~ C3t CC
ISSN 0105-8517
~ . ~ "". = ~ ~ -.. ~ < ~ =
~ ~ = I") = ~ ~ ~ ~ e ~ = "". I") rIJ ~ Q ~
~ ~ ~ ~ = Q.
Glynn Winskel
~ ~ ~ Q. ~ ~ = ~ ~ ~ rIJ
DAIMI pB-159 April 1983
Computer
Science
AARHUS Ny Munkegade
-DK
r-phone:
Department
UNIVERSITY ~
Aarhus C -DENMARK æ -7283
55
EVENT STRUCTURE SEMANTICS FOR CCS AND RELATED LANGUAGES
Glynn
Winskel
Ab s tr c3;:ct
We give
denotational
gramming
sernantics
languages
based
to
on the
processes
communicate
by events
Processes
are
by labelled
tures
denoted
represent
concurrency
The sernantics
does
not
a wide
idea of
of
range Milner's
rnutual event
rather
sirnulate
of
parallel ccs
pro-
[M1],
that
synchronisation. structures.
directly
Event
as in
concurrency
net
struc-
theory
[NT]
by non-deterrninistic
interleaving. We first
define
a category
to
synchronised
appropriate natural
relation
functor; the
concept
cesses
of
synchronise for
parallel
obtain
Synchronisation handling
semantics as it
which
basic
level
event
of
is
However
Event
abstraction
but
of
equivalence
(see
[M1]
for
take
concurrency
into
account.
in
by S.A.,
with In
L,
natural
particular
product
in
$.
We
by varying
So the
the
class
is
very
a comrnon framework. for
ccs.
When
synchronisation/communication
semantics
concurrency.
the
semantics
Milner's our
From each
general
structure
a
hoW tWO pro-
languages
very
bears
on labeIs
structures
from
[W])
an interleaving
structures.
CCS-like
are
an event
of
events.
and asynchrony
semantics
reflects
specifies
derived
a class
synchrony
[M1].
An S.A. event
2],
Then we introduce
(S.A.)
labelled
algebras
we get our
a rather notions
for
neatly.
on their
L is
through
algebra
labelled
Composition
interleaved terms
~L of
([NPW1,
The category
trees
trees
[M2].
labels
Composing
As a corollary tree
via
sernantics
broad,
of
to
Milner
a category
operations
S.A.
transfer of
structures
comrnunication.
a synchronisation
an idea
we derive
event
a subcategory
So results
adopting
the
to
~ of
distinguishes structure should
examples)
more
semantics support , including
all
is
at
abstract those
TABLE OF CONTENTS
o.
Introduction
1
1.
Event
7
2.
A
"cpo"
3.
A
category
4.
Two subcategories,
5.
A
structures of
event of
sernantics
structures event
for
17
structures
prime
25
event
communicating
structures processes
Conclusion
and trees
35 42 54
Appendix
A,
Sets
and partial
Appendix
B,
Domains
of
functions
configurations
56 57
Acknowledgements
66
References
66
1
o.
INTRODUCTION
We consider "Calculus
languages of
important
parallel
composition. of
a simple our
approach
is
serialise,
or
connection
with
two of
not
event
eO -the
one
the
item
all
the
We call
coin
af ter
so and
of
each
set
of
the
delivers
is
one
a coin.
events
it
[M1]
can
comrnunicate by
is
warned
trees
performing
a coin what
-and others
quite
We can
represent
the
Ordered
up by
event call
an
short-lived; it
a configuration.
-the
might
Naturally
performed
only
e1
item.
have
a neat
later.)
of
really
that
we do not We promise
M capable
machine
of
we illustrate
occurrences.
we mean
[M1].
form
two processes with
accepting
in
the
same as Milner's;
By event
accepting
sets
figurations
item.
is
which
machine
event
Milner's
described
synchronisation
reading
an
that
familiar
the
Robin
languages
is
event
Milner's
occurrence,
accepts
quite
to
-CCS,
synchronisation
interleave,
delivering
the
(The reader
a simple
events
of
The idea
mutual
related
Systems"
feature
example.
Consider
are
Comrnunicating
The most by events
which
to
it
only
delivers
machine
various
inclusion
as
stages.
the
con-
are:
{eO'e1} Uf {eO} Uf Ø
Initially tion
M has performed Ø; then
it
and afterwards how the occurred
later
diagrams
can be simplified
are
distinct
realise
the
behaviour
on configurations; include
order
eO to
realise
relation
a partial
of
can perform e1 to
machine's
no events
in
~ one point
time
and no point
which
is have
by using x is
the
the
the
covered
can be inserted
configqra~
configuration
configuration
configurations those
interest,
{eO'e1}.
reflected of
occurred
inclusion
which
earlier.
"covering"
have Such
relation.
by another in
Notice
by the events
{eO}
between.
y if
In x and y
Formally,
2
x is
covered
x
For
by
- e#e"
the
be
e'~e
a partial
and
(E,F)
the
x~E
& Ve,e'Ex.,(e#e'
Proposition
precisely
partial
the
& Ve,e'.e'~eEx
1.13
then
In
~ and #) .
a set
#.
iff
are
events.
{e'EEle'~e}
is
symmetric
relation
Purther subsets
the
finite
configurations L(E,~,#}.
for s.t. f
are
13
Conversely,
suppose
binary
symmetric
e,e',e".
Then
Proof
Let
defined
and
~ and
#.
by
the
Let
be
From
any Also
and
veriIied
any
are
I [e]
event
clearly
event
order
e#e'~e"
~ e#e"
Take
satisfy
the
closed
consistent
~ and for
all
structure.
structure.
a left
left-closed
a partial
I«X) and
they is
~ and
properties
consistent
subset
# as
is
stated subset
w.r.t.
a configuration
(E,F).
of
e#e'~e"
~ e#e"
a p.o.
(E,L(E,~,#»
We show
its
of
a prime
a prime
1.8
of
Example
events
is
consist
that
alongside
# s.t.
configuration
coherence
I [e]!
as
structure.
tF~P.
(F
complete
For
obvious.
~
-p
event
causal
event
structure.
y = {p
its
an
is
(F,c)
(E,F)
-then
,
rel~tion
isarnarphisrn
Pr(E,F);
prime
(E,F)
conflict
an
-Proof
Let
event
is
a pre-tree
structure
iff which
is
events,
15
1.18
The event
Example
structure
with
events
{0,1,2}
and
configurations
The event
The
structure
reader
may
isomorphic
to
check
for
for
often
"arcs"
a tree which are
it
the
the of
to
built.
introduced;
then
ordered
correspond
to
as
(A,B)
justified and
of
event
extension
maximal arcs.
are -so
branches For
-and
this
of
reason
events
A -
"branches".
By
from
events
away"
tree
a pre-tree
consisting
B -for
is
the
by
and
"abstracted
(This
of
partial
a tree
we have
and configurations
configurations
configurations
prime is
{0,1,2,3}
events
events
write
-and
is
that
correspond a tree
we shall
events
sequences
configurations that
with
the
formally pre-tree
when above
insisting of
morphisms
will
not
be
isomorphic.)
To sum up we have and those finiteness With
event
1.1
to
not
all
infinite weaker
structures
of
structures
[NPW1,
2]
which
which
includes
satisfy
trees
a simple
restriction.
an eye to
axiom
a class
is
possible
rather
model
strong,
processes
infinite behaviour substitute
generalisations too such
cohfigurations
strong
we note: for
as "fair would
of
a fair
merge.
for
which
much of
the merge"
The coherence
event in
structures a natural
correspond Perhaps the
there
following
to is
of way;
a possible an appropriate
work
still
goes
16
through. it
One
allows
advantage
a smooth
[NPW1,2].
The
processes
which
compatible
primes Possibly of
they
course,
Those work
with
Besides
modified
[KP],
is
to
enabled
in
[BC],
"deterministic" 1.2
[BC].
configurations
"deterministic" relation.)
should
morphisms (the
[e]
IS)
a propert
(Note
structures
are
expect
complete a topology. but,
why we do not
a set
of
events, relation.
be interested
nature
of
~ and #.
us an enabling
a configuration
y unfortunately
incidentally
and events event
in
I
dropped.
only give
model
several
a conflict
exact
of
processes
be
E is
will
the
x one because
way,
for
[W] may wonder where
in
to
continuous
and # is
not
special in
or
relations
primes
a unique
model
to
then
role
that
work
wished
caused
those
is
the
examples;
like
axiom
our
be
however, via
one
a similar
(E,~,#)
that
complete
for
are
finiteness
an enabling
a rather
of
1.1
and configurations, the
example
in
if
could
[W]
play
nets
go
which and
structures
reason
relation,
conflict
be
with
is
events
those
[KP]
axiom,
Petri
would
event
axioms
the
event
with
would
can
familiar
The main
is
an
-see
then
~~P(E)xE in
had
The
coherence
axiom
irreducibles here.
the
connection
stability
ways
complete
of
that a bit (E,~,#)
an event called
because more with
of
general
than
a binary
17
2.
A
"CPO"
OF
EVENT
By restricting to
those
the
STRUCTURES
configurations
inside
a subset
Definition
Let
E'
of
of
an event
structure
E
a newevent
(E,F)
structure
is
formed.
2.1 Define {xE
the
F I x~E
restriction
be
F)
(E,F)
an to
~E'
event be
structure. (E',F')
Let where
E' F'
~ E
=
' } .
L emma
2.2
(E,
The restriction
above
(E,F) ~E'
is
an event
struc-
ture Proof (E,F)
All ~E'
to
sponding
Such ~
the
properties
be
(i)
an event
properties
of
restriction
-(iv)
structure (E,F)
accompanies
of follow
1.1
required
directly
for
from
the
corre-
..~
an
idea
of
substructure
-the
relation
below.
(EO'FO) iff
and
(EO'FO)
Proof
2.5 events
Let
Lemma
(EO'FO)
~
iff
Directly Example Eo
Let ={0,1},
,
EO s: E1
from
the (Eo,Fo)
E1
be event
(E1'F1
structures
EO S E1
and
2.4
,
FOS
F1
Vxs
Ea"xE
F1
(E1 ' f1
be
and
(EO'FO)
~
x E Fa"
event =
structures. (E1 ,F1)
Then
~ EO.
definitions. ,
= {0,1,2}
.
(E1'F1) and
be
event
configurations
structures as
with shown:
18
{0,1}
{0,1}
v
~
~ {o}
Fa:
{2,1}
{o}
F1:
{2}
~
u
~
ø
In
(EO'FO)
the
(E1'F1)
the
or
event
after
ø
event
event 2.
1
can
only
1
can
occur
So
(E1'F1)
to
The
occur
even
relation
~
though
another.
ses
on
is
based
almost
form
a
event
structures
ation
occurs
xEF
(EO'FO)
specifies
approximates
in
The
ordering
a class
[S]
and
(x~E
iff
&
sense
in for
is
and
event
either
a new
O.
af ter
way
not
which
one
for
In
event the
event
recursively
~ .Event
form
ways,
the
0
event
(E1'F1).
sernantics
relation
cpo.
two
~
the
Our
the
in
af ter
introduces
..
1
occur
defined
structures not
a
structure
ordered
cpo
a set.
by
merely
(The
proces~
because
same
kind
of
situ-
[BC].)
('v'nEw.x
n e: Fn
& x
=
Ux
in
n' ,
which
nEw
x
(ii
z!:;X}
n
Let
A
structures hottom
(E,F) element
be
a set. with
,, obvious.
"40;" Suppose op
on event
is
(i)
monotonic
and
"(ii)
the
event
n 'F n ).1op(t1(E UOp(E n ~ have
the
n events.
same
configurations.
,F
above..
(EO'fO)~...~(En'Fn),s1...
structure
n ».
Now
From
Thus
Let
by
the
Ilop(E 'n'
n
U op (E, F) n n n (ii),
Ilop(E Ir of
definition
,F)
= op(U
n
n
(E
n
exists
,F) n n ~ they ,F
.Thenas
».
n
and
and
op(11
l'r the
have Therefore
(E
,F n n same
op
is
continuous.
.
As an example from
we show how the
1.15,16
event will
that
structure
from with
2.10
with
Pr
structure
the
is
continuous.
Pr constructs
domain
of
Recall a
prime
configurations.
operation
of
defining
Pr defined
in
1.15
This
event
struc-
recursively. The operation
Theorem
Proof
We use
We first
show
for
an event
an isomorphic
mean Pr commutes
tures
operation
event
~-continuous.
lemma 2.9.
Pr
is
structures
monotonic (EO'FO)
w.r.t. and
~.
Suppose
(E1'F1).
(Eo,Fo).1(E1'F1)
We requi;X:-e
pr(Eo,Fo)
~
Pr(E.,F.) = (P. ,n.) for i = 0,1, so P. the set of 1 1 1 1 1 complete primes of (Fi'S~F Suppose POEPo. As (Eo,Fo)~(E1'F1,) we have POEF1. Assume YSF,Yt' 1 and PoSVY.Then Po = (Vy)nIU' =yl{yYnpo Pr(E
1
where
'F
1
).
is
ynpoEF1
a complete plete
Let
prime
and
prime of
ynposEo of
F1.
Fo'
so PoSY
Consequently
ynpoEFo for
for
some POSP1.
each
yEY.
yEY.
Therefore
Now from
the
Thus
as
Po is definitions
Po
is
a
comof
24
no
and
This
n1
-see
roeans
We now
a chain
of
Pr(E,F)
fini
Pr
is
of ,
we require is
as
(Eo'Fo)~(E1'F1
pr(Eo,Fo)~pr(E1'F1)
show
be
1.10-
so
p is
that
on
event
structures
with
a complete
prime
p is
get
zEll1'&
ZSPOzEnO.
(EO'FO)
~...~(En'Fd;;)..
.
continuous
event
we
an
event
of
sets.
Let
lub(E,
F).
of
(F,S)
pr(En'Fn)
te
so pC=E for some n (we have' -n (E F )~(E,F). Also as p is a complete n n complete prime of (En'Fn) .Thus p is
Let
p be
.To
for
use
some
an
event
Lemma
n.
2.9
However
E = VE) .Now pE F as n n n prime of (E,F) it must an
event
of
-Pr(En'Fn)
p
be as
a
re-
quired.
Applying
2.9
gives
As a corollary of
define
prime
event
dency
and conflict
trary
event
event
relations.
lub
to
find
the
isomorphism
between
its
image
Pr.
Coro!lary
event
Let
structures.
It
ordered in So,we
a chain
and then
under
characterisation
structures
structures
its
.
another
structures
Pr,
2.11
continuous.
we can give
an w-chain for
Pr is
image
by ~. The lub terms
first
their
convert
back
using
Lemma 1.16 an event
n)~...
lub
(E,F)
where
.x
= lJz
where
structures
be
which
structure
an
E = n"wEn
w-chain
of to
depen-
an w-chainof
event
of
lub
simple
causal
prime
(EO'FO)~...~(En'
xEF4"* 3zEL(P,s;.,t)
of
the
is
of
configurations
has
of
arbiusing shows and
of
and
p =
and
t
=
v nEw
From
Proof From
tFn l'
Lemma
Pr (E, F)
the
1.16
under
V.
~-continuity
we know
F is
of
the
Pr
we have
image
of
the
Pr(E,F)
=
U pr(En'Fn nEw
configurations
of .
25
3.
A CATEGORY OF EVENT
We define They
are
events
basic
partial and
with of
a rather
its
STRUCTURES
class
functions
morphism
whenever
be
event
structure
put
in
event
structure
-see
to the appendix * to represent
for our undefined
operator
extends
which
is
with
product
imagined
the
structures. respect
to
synchronise
One notable
compound
another
of
event which
defined.
from
parallel the
is
this
a projection
on
event-sets
An event
event
will
morphisms
between
configurations.
image
of
process
back
event
to
on
structures
events
to
be
of
the
Refer
-we use the 4
a function
event
an
original
3.4.
treatment of partial functions -and a formal definition of a function
example
on
subsets.
structures.
(EO'FO)~(E1'F1)
is
a partial
(i
and
A
(ii)
rnorphisrn
Note
e
that
is
synchronous
condition
together
for
(E1'F1}'
then
a rather
special
one,
Kahn
Plotkin
[KP].
3.2 EO
and
Example =
Let
{aO'a1'bO'b1}'
~Fo~ V
ø
above
with
(EO'FO}~(E1'F1}' the
iff
(ii}
synchronised
have
f
VxEFOVe,e'Ex.8(e)=8(e'
inclusion so
=
is
says
a common two
event
=>e=e'
a
total
no
two
image
.
function
distinct event.
Notice
if and
map i: EO~E1 is a morphism, A i is a rigid embedding in the
in
(E1'F1) {a,b}
and
structures
events
(EO'FO}
(EO'FO)" E1
e
*
be
event
configurations
structures
sense
with
are we
fact of
26
Then
8 defined
so
8(aO)=8(a1)
(synchronous)
morphism.
total,
be
cannot
on
Petri
It
is
nets
easy
gory
of
to
event
functions
=a and
8(bO)=8(b1)=b
(Incidentally
induced
on
~ see
[NT],
check
that
event
the
by
although
a net
morphism
2].)
morphisms
with
a
morphism,
structures
[NPW1,
structures
and identity
this
is
the
morphisms
defined
usual the
above
composition
identity
give of
a cate-
partial
functions
on sets
of
events. 3.3
Definition
Define
and morphisms functions event
3.4
as defined
=Set *
defined
structures
tion
of
t
to
in in
consist
3.1
the
of
with
event
composition
a pp endix.
and synchronous
objects
Define
morphisms
structures
that t sYI;l
with
of to
the
partial
consist
usual
of
composi-
functions.
Proposition
Both
~
and
~
are
categories
with
identity
syn
morphisms
the
identity
functions.
We
have
~
is
a
proper
sub-
syn
category
of
structure also
Let
(EO'FO)
~ and
initial of
~
-+ (E"F1)
be
a morphism
a total
a monomorphism
in
1-1
E
(but
and
(E syn
e is
an
epimorphism
in
E
(E syn
The category within basis
E has
isornorphism, for
sernantics
defining, of
ccs
products
have the syn The null event
not
~
onto
variants.
null
event
structure
is
) .
E.
Then
function
e is
such
an
iso-
that
iff
e is
a monomorphism
in
~*
(~)
.
iff
e is
an epimorphism
in
~*
(~)
.
following
and proving
syn
in
and coproducts
by the and its
~
object.
object
iff e is A e(x)EF,.
XEFO ~
categories as
the-terminal
e:
is
Both
(Ø,{Ø})
morphism
e
~.
relations
characterised,
constructions. between,
to They
different
provide
a
27
The parallel
composition
restriction loose
of
the
may not
figuration
of
the
events
of
contain
Definition
Let
(EO'FO)
product
(EO'FO) ~*
x
with
iff
with
of
two event
(Partially
(E1'F1)
to
~
structures.
Define
be
where
E=EO~E1'
F is
the
given
A nO (X)EFO
A n1 (x)
&
(a)
&
(b)
Ve,e.Ex.nO(e)=nO(e')~
&
(c)
ve,e.Ex.e~e'~3ysX.no(y)EFo
and finitary
3.6
Example
A conproject
morphisms.
their
product
product
in
&
EF1
*
ar
n1(e)=n1(e.)~ & w1(Y)EF1
& e'~y)
or
(e~y
*
~
e=el
&
& e'Ey)) I y I
e,e'EEOo
Suppose A 8(X)EF1°
"
~
,e#Oe'o Thus
f(b)
=
commutative
A=
A'
is *
A.O=O and
two
is events
introductio~
containing
a binaryassociative,
VAEL.
event
of
0
labelse
algebra labels,
real
A,A',
synchronisede
operation
a set
no
satisfies:
(i)
it
the event by * -just
label;
A synchronisation
Definition
L which
other.
for
3.5.
labelled
5.1
the
to pretend, mathematically, that with the unreal "event"* labelled
constant
saves
of
on
synchronised
as a binaryoperation
represents
important
Possible
parallel
algebra is not two distinguished
* still
PROCESSES
processes.
events
labels
and present
morphisms
of set
labels,
events
events
processes
chronisation their
FOR COMMUNICATING
= *
a quadruple
and
O with
operation
on
'43
An S.A. one
label
Thus
is
a
"divides"
a divisor,
condition
unique
5.3
determines
in
the
(L,*,O,
.
(ii)
divisor
of
(i) (ii) (iii) (iv)
factor,
as
of
follows.
It
says
when
another.
definition
of
an
be an S.A.
Then
S.A.
says
*
is
the
*
Let
Lemma
or
relation
The
relation
div
For
AEL r if
Adiv*
then
A= * °
For
AELr
OdivA
then
A = O°
Let
aora1rSorS1
if
is
EL.
reflexive
If
and
aOdivSO
transitive.
and
a1divS1
then
aO.a1divSO.S1.
Proof
(i)
byassociativity.
(ii) (iii) (iv)
that
in
labels
a,S,
...,
as
O is
by
commutativity
of
ccs
(ii)
in
the
definition
of
an
S.A.
a zero. and
for
this
[M1]
their
events
produce a T-labelled and division relation identity-
y
associativity.
.
ccs
passing
Recall
pairs
propert
The S.A.
5.4Example Withoutvalue
Only
by
is
there
are
three
complementary with
complementary
kinds
labels
truein
general
(see
(non*,O)
~,S,...
labels
event. Thus we get the for CCS. In this case not
of
can
and the
5.6)
label
synchronize
following * behaves ex.
labels;
.
S.A. like
table an
to
T.
45
s
A
Notice how morphisms in !A may be partial functions as *divT. We get !A=!. However morphisms in !s must be total functions as * does
not
Wenow
divide
T.
define
tures
as
the
parallel
a restriction
(one of which
! s =!
We get
composition of
may be the
.
syn
the
of
product
fictitious
in
event
two ]f;.
labelled
Only
pairs
* ) whose
composition
can
be
synchronised.
(See
product;
TrO'Tr1 below
are
the
projection
morphisms.)
Definition
Define
Let
their
«EO'FO)
parallel
x (E1'F1)
Examp1e
refer
to
{ (eO'*)
5.5.
5.9
is
Then on
with
synchronisation
have
a
3.5
of
.
(EO'FO'lO)'
(E1'F1'11)EtL. (E1'F1'11)
E={eEEO~E1Il0nO(e)
S.A.
product
.11n1
=
(e)~O}
and
to
the
their
parallel
,11 (e1)
l(eO'*)
-
events
(eO'e1)EEOxE1110(eO) labelling
passing
are
= 10(eO)'
= T.
Let of
events used
va1ue Then
restricted
a subsequent
the
in
L be
form
composition
is
CCS without
(E1'F1'11)E!L.
le1EE1}u{
laws
T-labelled
for
in!
"Broadcasting"
parallel
a while
the
1(eO'e1)
satisfying the
Let
events
definition
(EO'FO'l0)@
(EO'FO10)'
U { (*,e1)
= 11 (e1) ,
*,O,a,T
L be
their
leOEE}
Example
where
Suppose
comp1ementary} 1(*,e1)
S.A.
struc-
(e). Let
composition
an
composition
~E,l)
1(e)=lOnO(e).11n1 5.8
L be
of
labels
non-zero
5.7
event
of
[H],
a.a=a,
several
occur [Mi],
the
S.A.
with
a.*=a.T=O processes
asynchronously [LST]
-see
labels and
must (such [M2]
T .*=T.
synchronise multiway too)
.
46
5.10
Example
The
following
chronously
in
all
occur
synchronously)
obvious
projection
events
There
are
which
suggests
@
is
~
It
operation
of
however
least
(i)
Let
The
operation
eo'
e1
tive
in
i.e.
for
Let
in
the
The
in
fact,
a functor
it
ensures
composition the is
behaves
operation not
always
like
the
.
S.A. to
eO@e1
a =
functor
]f:L 2 -+]f:l
eO~e1.
there
is
(E1'F1'11)E~L.
obvious
The a
:
functor
natural
For is
morphisms associa-
isomorphism
Then
projections,
is
(EO'FO'l0>@
their
(E1'F1'11
categorical
product
:IE'.L iff YyEL
(iii)
which
asyn-
~ (EO@E1)@E2.
(EO'FO'lO>'
with
occur
a parallel
(l.c.m)
an
5.6
operation.
EO'E1'E2EJt~
Eo0(E10E2)
(ii)
the
extends
define
for
to
muLtiple
L be
@ ]f:L
extend
events
S of
Although,
when
common
Proposition
functions
does
all
(cf.
a product.
and is
ensures
composition
is
associative
a product.
5.11
a parallel
S.A.
Ya.EIOE&,
parallel
always
gives
Va,S,yEL.
8E11E1.
composition a categorical adivy
and
a.divy @
and
8divy=>
with
the
product
in
Sdivy
~
(a..8)divy.
evident tL
(a.S)divy.
iff
projections
47
Proof for
(i)
Let
i=0,1.
Let
eEEo~E1
6i: e
s.t.
(Ei'F
be
an
101T0(e)
i~1i)
~ (Ei,Fi,1'i)
event
of
.111T1 (e)
1'0601T0(e)
.1'1611T1
(e)*O
so
(EO'Fo1O)@(E1,F1,11)
The functor it
is (ii
of
"if"
part
ensures
ly
the
x"the
is
The
be
Suppose and
where
true
even
are
product
event y.
and
(ii)
must
beThe
having
"onto
Take
where
lO(eO)=a
is
the
be
product the
divide
.The
event
y.
"only
if"
had
to
part be
full.
Eo=({EO}'{Ø,{EO}}'{(f:;0'(1.)I)
{ (E1'13)})
.The
by
morphisms.
]f:.
where
structures
and
above
Bdivy.
@
g+(eO'e1)
Take
{(E:2'Y)})
in
eO:Et-+eO
in
as
and,
not
adivy
and
follow
product
assumption.
Let
80:E2-'+"~and
Because
the
is
EO@E1 E2
=
81:e:2-'+"e:1 -
mediating
morphism
(1..13divy.
.
Examples Let
L be
product: cannot
(ii
if
13 divide
,
80'81
exist
(i)
so
a.B
as
@
x exists
structures
theobviousprojections,
both
of
on ]f:.
Assuming
labelled
follows
({E2}'{Ø,{E:2}}
5.12
the
morphism
part
(1. and
event
E1 = ({e:1}'{Ø,{f:;1}}'
with
event
stated
for
of
l1(e1)=B.
be
y of
condition
morphisms
a mediating
lIif"
the
BE"l1E1
and
must
would
(iv),
(iii),
an
functor
morphism
aE1OEO'
e1:E:~11
(eO'e1)
(iii)
5.3
5.3
is
propert
as
on
({e:},Ø,{(e:,y)})
there
morphisms
Lemma
,611T1 (e)
product
follows
relies
Take
and
by
~L
ieee
are
By
a restriction
part
full:
60.'61
Then
(601T0(e)
mediating
@is
if"
As
and associativity
a restriction
cause
in
.
laws
The
mQrphisms
(EoFo,10)~(E1'F1'11
* O.
1!6.1T.(e)div1.1T(e) for i = 0,1. 1. 1. 1. 1. (1'0601T0(e).11611T1(e})div(101TO(e).111T1(e)).
b~
For
the
the
S.A.
We may divide
for
ccs.
have
T(#
S.A.'s
adivT
Then
C!)
does
and
BdivT
not
and
coincide
a.B=
D so
with a.B
D) .
A:
and
S
of
]f;
and
]f;
it
may
be
checked
syn
that.
does
As we know
satisfy they
do
the
condition
in
have
products
given
proposition by
~
5.11' and
QY
(ii) .
.
48
More generally coincides
Befare sent tures
giving
with
the
a few extra based
an
Definitions an
]f:L.
Lifting
where
E'
L
products:
language
rnuch sirnpler [M1,
based
aperations
an an S.A.
an labelled
we pre-
event
struc-
2]. L
be
an
S.A.
Define
( E. , F. , I .) Et L
(E, F, 1 ) ,
Let
SA's
pragramming
Let
5.13
for
3.
3.
Suppose
the
for
i
following
= O,
operations
1 .
3.
AEL
{*,O}.
Define
= (E',F',l'
A(E,F,l
= { O} U({ 1 } x E) and
xE F' x~ I (e ,
= A
, and if
e=
x=ø
ar
(O Ex and
O,
l(e)
if
af
3.8
and
e'
=
{el
1,e)
1,e)Ex}EF)
for
e'EE' ,1)
where
+ is
1«1,e» 5.15 where
5.16 serves fine
the
capraduct
l«O,e»
= lO(e)
and
=11(e). Restriction E'
Let
= {eEEil(e)#A}
and
Relabelling *"
Let
O and.
( E, F"l)
AEL'{*,O}.
and =
l'
S be VAEL.
(E, F,Sl)
(E,F,I)'A
=
(E, F) ~E' , l'
= I}-E'.
an S(A)
.
Define
endamarphism = O=> A = 0
an & S(A)
L
(i.e.
S pre-
= *=> A = *)
.De-
49
Apart
from
on tL
in
restriction,
an obvious
continuous
with
we can take
5.17
the way.
respect
fixed
define
Sum is to
points
extend
coproduct
in
labelled
them
and their
L be an S.A.
~L(E1'F1'11)
(EO'FO'lO:
operations
~L the
of
Let
Proposition
above
iff
tL.
to
They
version
functors are
all
~.
Thus
of
compositions.
For
(EO'FO'10)'
(EO'Fo)~(E1'F1)
(F1'F1'11)E:lf:L and 10=11~EO.
Then
(i
tL
has
lubs
of
all
Each operation
(ii
they
above
preserve
Let
(iii
r be
lubs
Proof
follows
particular
~,
operations
use
L,
w-chains
Lemma
Pro CL.
(iii)
is
well
we define
a language
Each
of
term
~L i.e.
Let fixr
r(fixr)
to
be
the
lub
= fixr.
property
because
to
by ~L.
t~~tL.
corresponding
continuous
respect
Define
.Then
2.9.
with
on
(Ø, {Ø} ,Ø) ) Et~.
the
by ~L.
ordered
operation
from
+ are
an S.A.,
ses called in
af
ordered
continuous
~~Lr~~L...~Lrn~~L...
(i)
Given
is
a continuous
~ = ( (Ø,{Ø} ,Ø) , ..., of
w-chains
x,+
are.
known
see
for
of For
~.
the
In
remaining
[S].
.
communicating
Pro CL denotes
(ii)
proces-
an event
structure
set
pracess-
JtL.
5.18
The
syntax
variables
x EX.
af
Pro cL
Define
Assume
a term
af
S is
of
The sernantics
of
tion
p:
process-variables
For ture
a term t
duction. their
t
(Note, sernantic
Pro CL
Define
and an environment
denotes
where
with that
respect syntactic
counterparts,
x EX,
AEL'{*,O}
L.
5.19
from
af
by:
isrect,
an endornorphisrn
x~tL
infinite
Pro cL
t::=~lxIAtlt+tlt'AltI~tlx and
an
to p,
to
an environment
p,
define by the
operators operations
labelled
event
[[t]]p,
the
following occur on tL
to
structures. event
struc-
structural
on the occur
be a func-
left
on the
inand right.)
50
[[liIL]]
=
(Ø, {Ø} ,Ø)
[[ x ]] p
=
p (x )
[[ A t ]] p
=
A ( [[ t ]] p )
[[ t 1 + t 2 )] p
=
[[ t 1 )] p +
[[t'A)]
=
[[t)]
[[ t < S > )] p
=
( [[ t )] p) < S >
[[t1@t2)]p
=
([[t1)]P)CY
=
fix
[[x
p
isrect)]
p
P'
[[ t 2 )] p
A
r
([[t2)]P)
where
r:
t: L
+n
is
given
by
L
r ( E) A
structural
above
In
induction
definition
is
a sirnilar
parallel
of
lently
§ 3 and take
oneobtains
sernantics
with
is
indeed
category
environrnent in
in
fL
and
trL;
the
define
as a restriction the
and trL to
so
5.17.
into
fL
I extended
continuous
Proposition
sernantics
either
sernantics Pr and
r
by
way one can obtain in
[[ t )] p [ x+ E ]
that
justified
cornposition
product
5.20
shows
=
cope
o fx
categoriese
Equiva-
by cornposing with
the
the
above
labelse
Definition
(i
Define
prL::lf:L-.:fL
prL(E,F,l}
=
(pr(E,F},levE
F}
-re
fer
, to
4.3.
For
(ii
by
p:
X-.
Define
:J>L and
lL:]f;L-+'!'rL
t
E Fro
by
cL
define
[[ t]]p
IL(E,F,l)
=
p = prL
( [[ t]]
(I(E,F),l1TE,F)
p} .
-re
fer
to
4.3. For
When L is agrees
p: X-+trL
the
with
SoAo for
Milner's
because
of
general
than
branching
and
the
t E Pro CL define
CCS our
[[t]]T.rP
interleaved
= IL([[t]]p)
sernantics
synchronisation/communication
following
Milner's
when recursion
facto so our is
(Our
treatmment
denotations not
"guardedly
of
as trees
in
.
TrL
tree
sernantics
recursion
is
may be Xo-
well-defined"o)
more
51
5.21
Proposition
operation are
in
~.EL'{*,O} J by
Proof
by restricting
in
categary
terros
t t
5.23
p Tr
an
s.A.
Write
each
af
pracL'
the *L'
where
iff
[[ t ]] p
~
[[ t'
]] p
t'
i f f
[[ t ]1p p ~
[[ t'
]]p
t'
iff
[[t]]
Proposition
:l!>L' trL
L is
t'
product
the
closed
,..,c,..,
parallel
composition
an
S.A
' ,
~
.
a cangruence
an
p.
TrP
~ [t'llTrP.
The
relations
Pro cL
trees.
,
~
w.r.t.
@,
T
define
congruences
r
+,
A-,
-'A,
-(s>.
We have
c,.., p-
Tr
Proof
Each operation
Generally currency 5.24 of
terrns
of
induces
p
on
the
as
From 4.7
Isaroarphisro
t
'1'rL
L be
@ so T@ S = IL (T@ S} .suppose T,sE'1'rL Tr Tr of the form T=~A.T. and s=4-~.s. for labels A. , r-3. 3. Jd J 3. indexed by i and j. Then T~S is given recursively Tr
sums
clased
Let
their
terros
the
event
congruences
Exarople
the
tained
because
on ~respects
Let
by restricting
structures are
L be the
a8NIL~aNIL the
isomorphism.
in
strictly
S.A. the
products
of
for
*L
.
and ~L reflect
included
in
CCS. We look
categories drawn
*L' in
'L
Exarople
that at
for
trLo
denotations
and trL 4.8)
con-
.
(ob-
52
s
~
L
a
s In
We
have
labelled
aS NILØ not
for
strict
the
because
other
is
purely
Definition
Let
coverings.
Clearly
in
trL
+ a (C:tSNIL + S'&NIL)
'!'rL
we have
which
does
in
an
interesting
synchronous,
when
synchronous
L be
an
S.A.
special no
case
asynchrony
where
is
allowed
law:
Say
L is
synchronous
iff
= O.
is
chronous,
an event
ii
synchronous
synchronise, is
synchronous
product
inherits
is
parallel to
no event
composition
5.26
In
congruences.
a strict
When an S.A. must
two
fails
L satisfies
VAEL'{*}.A.*
and
JPL
CtaSNIL Tot TSNIL
inclusion
communication
5.25
events
C:tNIL ""'Tr
hold
This
In
]f:L
occur
some nice
of is
Let
a parallel
L be
an
syn-
in
Then
the
it
Then parallel
product
so parallel
product
S.A.
pure ly
composition
synchronous
on ~
from
is
asynchronously.
the
based
properties
Proposition
in
can occur
a restriction ~
composition
'~
.The
composition
~. following
are
equi-
valent (i) (ii) (iii)
L
is
synchronous
NILcis
an@
Parallel to@
-zero
composition (t1+t2)
When L is isomorphic
",tO@t1
synchronous tO'
those
i.e.
t@NIL-"' @
NIL
for
dcistributesov-er
+ tO@t2
for
denotationsø~ intrL'so
all
= "'Tro
terms
sum
i.e.
terms
closed "'p
all
t E Pro CL.
tO,t1't2
terrns
in
E ~
~L
are
53
This
indicates
rules
for
5.27
on
synchronous as the
Abelian
following
adjoining
synchronous the nised when terros
with it
an
event
Pro cL
are
can be based
synchronous
calculi
extends O to
AEL.
E1;
a delay
or
to
proof
on synchronous
*L
simply
composition
so
Pro CL includes
event may
action. and
[M2]) (L,*,O,.)
monoid
event
id le
S.A.
extending
the
every
the
of
language
with
in
an
M and
The
EO@E1
e1 of
pre-trees
and
shows:
associated
composition
represents of
all
calculus
parallel
[M2]
* and
* .A = O for
laws
terros.
(M,.,1)
elements
* .* = * and
L deterroine
proposition (The
monoid
on
calculi
Proposition
Any by
assuroptions
congruences
Milner's S.A.s
how
trees
M in
eO of be
in
'L.
[M2].
EO is
labelled
Denotations
the In synchroby
of
1 closed
54
CONCLUSION Thus to
we have
a wide
a framework
range
framework a special observer" satisfya but
causal
to in
example
Montanari tics
[NPW1, nets.
Net
structure fact
of
sernantics
by the
a more
structures.
which
is
playing
have game.)
Take
occurred Then,
synchronous
the
token
event
structures
and synchronous
tinuous
functions
Clearly
we have
between
game,determines
left
several
works
prime
and
event
[W].
nets
with
initial
marking,
an event
structure
induce
the
most as
which token
and partially special
B.
at
events
playing
domains
appendix like:
of
rather
(In and
occurs
Scott
ends
sernan-
condition
in
-see
and Ugo
and Petri
Petri
infinite
morphisms
because
given
the to
[NT]
a
get
structures
sets
represent
loose
with
[F]
have
be those
possibly
domains
frame-
structures
Aarhus
between every
to
Unlike
important
[NPW1, 2]
system
that
configurations
by some stage
of
syn-
algebras
events
translates
connection
and such
the
[LTS]
event
Then the
because
such
techniques
direct
is
[MS])
and also between
A condition
contact-free
follows:
Fogh of e.g.
the
"sequential
We automatically
and Torben
in
up to
of
event
of
This
that
are
synchrony.
Prime
structures
(see
trees
and asynchrony,
structures. Pisa
sernantics
a synchronous
synchrony
the
of
idea
all
in
algebra.
links
like
is
of
more,
synchronisation
bans
asynchrony
structures,
sernantics
there
in
of
[W] establish
event once
Nielsen
such
once
relation.
these
kinds
operator.)
arise
and simple
of
Mogens
2],
model
But
Milner's
essentially
and conflict
terms
terms
[M2]
a free-mix
and coworkers
in
that
sernantics
Thus we link
interleaving
which
intuitive
languages.
structure.
synchronisation
dependency
sernantics
Milner
allow
on the
correspond
the
denotational
synchronisation
event
we do not
rather
give different
Milner's
in
law,
[M2]
depending
for
of
strict in
between
incidentally
embodied
calculi
to
programming
labelled
(Notice
is
chronous
work
of
[M1].
Milner
parallel
approaches. kind
in
which
makes connections
and different work
of
in
con-
55
l
l
l
l
l
How to go from our semantics to proof rules. How to go from our rather basic semantics to more abstract semantics. How to give an operational semantics which justifies denotations which are sensitive to concurrency (the most philosophical and probably the most difficult loose end to tidy up); How to generalise event structures and still keep a useful category (for example are there more general event structures which model continuous processes or express "fairness"in some way? Then our present definition of morphisms should still be useful.). How to define homomorphisms of synchronisation algebras and use the attendant algebraic constructions.
56
APPENDIX
A
Sets
and
partial
We take
~
functions
to
composition.
be
the
category
To cope
with
of
partial
sets
with
functions,
sets as objects but morphisms are now the value * (representing "undefined") drawn
as
~e(x)=
e:x~*y.
(e(x))
functions) yield
For If form the
*.
The if
morphisms
e(x)~*,
correspond For
e:x
us,
a notable
X and
y are
XxY=æf * obvious
*
and to
y and
fact sets
{(x,*) projections.
their
morphisms
A~X define
about
~*
categorical
I xEX}U{(*,y)
6(A)
is
function-
we take
~*
functions which .A morphism in
X ~ y and * * otherwise.
those
usual
the
~
of
never
~*
I eEA
nature
(x,y)
to
which
= {e(e)
of in
its
~* xEX
have
may take ~* is
y ~ Z compose * Morphisms in
product I yEY}U{
to
(total
& e(e)~*}.
products. takes
the
& yEY}
with
57
APPENDIX
B
Domains
of
Here
show
we
configurations
structures
the
and
A directed 3uES.
(cpa)
iff
S have D is
iff
for
A cpo
all
cpo
upper
said
Let
X~D.
Vx,yEX
consistently
Similarlya finite
of
consistently
of
element
dominates
way: plete
such
the
primes
in
iff
for
to
then
x~s
subsets
be
some
xED the
some
set (An
authors
coherent
in
sES.
S of algebraic
insist
it
structure
iff
every po
a po
number within
as events
is
a cpo.)
iff
every
(D,~)
is
subset
represents special
has
a
of
a domain
properties.
and so that
event
pairwise
completeness.)
(E,F)
algebraic
Then
compatible
rather
to
iff
coherent
D.
consistent
a prime
for
finitelycompatible
finitely
a finite
= compact)
x = US.
every
every
some element
arder
directed
finite
each
is
bound
primes
all
compatible
(D,~)
represented, fact
partial
(=
and
upper
prime
that
below.)
satisfying
complete below
x~US
(Clearly
said
an event
is
if
pairwise
"implies
only
structure, Take
po
iff
coherent,
domain
isolated
-see
The
order.
S~D such
~ED and
directed
be
an
configurations are
event
X has
1.8
domains
is
a lub.
coherence
By proposition
is S,
to
X is
complete
(Clearly
(F,S)
has
event
a partial
a camplete
a dommain though
& y=d.
subset subset
x
of
US.
algebraic
said
subset
D is
(lub)
complete
X is
3dED.x~d
compatible
any
be
be
subset
element
xED
called
Then
p.a.
baund
below
generally
be
The
subsets
to
(D,~)
a nan-null
element
elements
also
lub.
an
categories
Scott-domains.
a least
directed
D is
is
D is
is
our
Let
& t~u.
there
a cpo,
isolated
af
s~u
a least
If
of
definitions:
subset
Vs,tES
between
categories
Some basic
B1
relation
every
elements.
and all
as configurations.
isolated
Conversely
isomorphism, structure,
Such
in the
byan the sets
following of
com-
58
B2 Definition order
Let
satisfying
onlya
(D,~
the
finite
property
nurnber
of
prirnes
of
B3
In
above
ture X
so
>
~
(D,~) UX
Directly
The concept
of
There
net
Petri
particular prime
with
net.
the the
saying
from
prime
in
coherent
isolated define
3zED.
{pEP
element p(D)
=def
x = {pEP
is
p(D)
partial
with
(P,F'
where
I p~z}.
a prime
I p~x}
dorninates
event
struc-
inverse
of
prime
is
well
(see
known
corollaries follow
isomorphism, orders
isolated
non-null for
all
iff
(nX)uy
[CL])
x,y,zED =
.We
that
domains
dominates
so
formed domain. is
Now it the
only
(D,~)
be
a partial
Say
(D,~)
is
distributive
yt]z for
exists. subsets
non-null
Say
for
are,
number order
rise
out
that,
the
concept
lattices
to
and
within
coherent that of
with
it
par-
every
elements. meets
of
all
iffxn(yuz)=(xny)[J(xnz)
(D,~)
XSD s.t. subsets
ly
property
a finite
giving
turns
algebraic
finiteness
as a
same as saying
proof
configurations
concepts. associated
so real
the
[NPW 1].
itself
a subbasis
just
Let
n xuy x EX = U x ny for xE}{
showed
configurations
distributive, the
in
Scott-domain
a net
present
of
the
satisfy
subsets.
event
algebraic
domains.
B4 Definition
in
primes
for
which
to
and algebraic,
precisely
element
related
algebraic
prime
.
was introduced
were
The complete
concept
definitions.
a dommain of
is
(UX)ny
iff
occurrence
distributive
tial
xEF
x ~
the
concepts
a domain
will
Then
algebraicity
completely
It
every
definition
under
is its
D and
an event
complete to
(F,S)
that
algebraic
.
Proof
In
the
a prime
elernents.
P = cornplete
Lemma
be
is
cbmpletelydiStributive
xuy
exists
for
XSD s.t.
UX
exists.
all
x EX and
59
Proof
It
is
easy
completely
distributive
Conversely
suppose
Algebraicity
show and
(D,~)
expresses
Vx,yED.X~y
(i)
to
a prime
algebraic
algebraic
is
(or
completely
a kind
of
lattice
see
is
[NPW1,
2]).
distributive
and
discreteness,
~ 3z,z'ED.x~z-(F
Dn
structures
o
the
o
.
setting.
represent
o
o
The
categary
damains.
sernantics
for
on ~,
this
be essentially
though
on prime
event
gories.
will
(the ~L to
of
recursion
cursive
of
labelled
the
reader; can
we shall
event
structure
w-chain
of
colimit
(see
learnt labelled tics
to seek
[KP], the
in
way
the
so
not
cartesian
morphisms
are
subcategory [Ber .
[M1,
[P1])
the to
a ~-respecting detailed
PL.
.There
definition
The treatmment
One expects
point". of
Starting
the
functor
fixed
point
used
to
are give
a re-
functor,
for
with
null
will will
closely
the yield
an
be its
may be some lessons they
intervals
to
be
akin
to
operational
sernan-
2]) . ~ and
c.m. Gerard
~
syn however
closed;
of
fixed
often
in
based
cate-
prime
an w-cocontinuous
~L because
categories
merely
to
The least
systems
e.g.
leave
embeddings.
application
categories
By the are
[F],
transition
(used
"least
embeddings.
L but
be equivalent
correspond
the
.We
on rigid
repeated
rigid
from
should
be based
definition
which
it
equivalent
~L by labelling
algebra
~ as above)
directly
same as a sernantics
~ and p are
domains
a synchronisation
communication the
p because
same relation
of
synchronised
structures
One obtains
by elements
full (see
-f o o -} ,C)~}
~).
One can base
way
easily
cancepts
event
:E n
is
cartesian
Berry's
do
not
a larger closed
dI-domains
have
exponentiations
category and with
in
in fact
stable
which is
a
functions
66
Acknowledgernents I am grateful for previously to
given
Gordon
a prime
Plotkin
quarter-baked.
for
to
grant
by Robin
MØller
is
even
to
CCS. Thanks were
Gerard:Berry's
ideas
appear
and Angelika
and Gordon
has
when they
essentially
in
Paysen
was supported
Milner
Mogens
sernantics
Related
The work
Nielsen.
morphisms
[BC].
Karen
paper.
directed
structure axiorn
condition
Many thanks
Royal
event
Mogens
encouraging
a symbol-laden the
with
The stability
"deterministic" [MS].
discussions
in
for
part
Plotkin
[F]and typing
by an SRC
and in
part
by
Society.
References [B]
G. Berry,
"Modeles
A-calculs
types",
Paris
VII,
[Bi
These
P.L.
G. Birkhoff,
to
appear
Soc.,
in
stables
des
Universit.-
Providence,
K.H.
Haffman,
D. Scatt,
"A Campendlum
algorithms
on concrete
TCS.
theory",
G. Gle~z,
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af
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and
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