EVENT STRUCTURE SEMANTICS FOR CCS AND RELATED ...

~ = 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,

T.

et

d'Etat,

"Sequential

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Math.

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de Doctorat

Curien,

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1979.

G. Berry, data

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Coll.

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R.I.,

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K.

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Fogh,

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an Sernantics

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part

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af

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Springer-Verlag, "Petri

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nets,

TCS 13,

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Systems",

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G. Winskel,

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w. Brauer

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af

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M. Nielsen, current

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