robin milner a calculus of communicating systems 1980

8/11/2019 Robin Milner a Calculus of Communicating Systems 1980

1/176

Lecture Notes in

om pu te r Sc ience

Edi ted by G G oos and J Hartmanis

9

Robin M ilner

A Calculus of

Communicating

Systems

Springer Verlag

Berl in Heidelberg

New York 1980


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Editorial Boar d

W . Brauer P. Br inch Ha ns en D. Gr ies C. M oler G. Seegm iJIler

J..q;toer N. Wirth

u tho r

Ro bin Milner

University of Ed inburgh

Dept. of Co m puter Science

James Clerk M axwe ll Bui lding

The K ing s B ui ldings

Mayf ie ld Road

Edinburgh EH9 3JZ

Great Br i tain

AM S Sub ject Classificat ions (1979): 68 -02

CR Subject Classif icat ions (1974): 4.30 , 5.20, 5.2 2, 5.24

ISBN 3-540-10235 -3 Spr inger-Ver lag Ber l in Heidelberg N ew York

ISBN 0 38? 10235 3 Springer-Ver lag New York H eidelberg B er l in

L ib ra ry o f C ong re ss C a ta log ing in Pub l i ca t ion Da ta . M i lne r , Rob in . A ca lcu lus o f

com mu n ica t ing sys tems . Lec tu re no tes in compu te r sc ience ; 92 ) B ib l iog raphy : p .

Inc ludes index, t . Ma ch ine th eory . 2 . Formal langu ages . I. T i tle . 11. Ser ies .

C IA26?.M53 . 511 .3 80 -21 068

T h i s w o r k i s s u b j e c t t o c o p y r i g h t. A l l r i g h ts a r e r e s e rv e d , w h e t h e r th e w h o l e o r p a r t

o f th e m a te r ia l i s conce rned , spec i f i ca l l y thos e o f t rans la t ion , rep r in t ing , re -use o f

i l lus t ra t ions , b roadcas t ing , rep roduc t ion by pho to cop y ing mac h ine o r s im i la r means ,

a n d s t o r a g e in d a t a b an k s. U n d e r 5 4 o f t h e G e r m a n C o p y r i g h t L a w w h e r e c o p i e s

a r e m a d e f o r o t h e r t h a n p r i v a t e u s e, a f e e i s p a y a b l e o t h e p u b l is h e r , t h e a m o u n t o f

t h e f e e t o b e d e t e r m i n e d b y a g r e e m e n t w i t h t h e p u b l is h e r .

b y S p r i n g e r- V e r l a g B e r l in H e i d e l b e r g 1 9 8 0

P r i n te d i n G e r m a n y

Pr in t ing and b ind ing : Be l tz O f fse td ruck , Hemsbach /Be rgs t r .

9 1 4 5 / 3 1 4 0 - 5 4 3 2 1 0


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w o r k w a s m a i n l y d o n e d u ri n g m y s i x -m o n th appo_ n - - t , f r c m

~ t 1 97 9 t o J a n ua r y 1 98 0, a t t h e O m m o ut e r S c ie n ce d e o al b ~ en t i n

A a rh u s U ni ve rs it y, ~ k . A l t ho u gh m u c h o f t h e g r ou n d w o r k h a d b e e n

done previously it was mainly in response to their encouragement (to

make the theory more accessible and related to practice), and to their

informed criticism, that th e materia l reac hed a scmewhat coherent form.

I an deeply grateful to them and their students for allowing me to

lecture once a week on what was, at first, a loosely connected set of

ideas, and for the welccming enviromnent in which I was able to put

the ideas in order. I also thank Edinburg h University for awarding me

five months sabbatical leave subsequently, whi ch helpe d me to cc~plete

the task in a reasonabl e time.

The calculus presented here 9~e w out of work which was inspired

by Dana Scott's theory of computation, though it has since diverged

in some respects. At every stage I have been influenced by Gordon

Plotkin; even where I cannot trace particular ideas to him I have

bee n gre atly illtmlinated by our discussions and by his chance remarks,

and without th em the outccme would certainly be less than it is. I

wou ld also like to thank others with whom I have worked: George Milne,

with whom I worked out the Laws of Flow Algebra; Matthew Hennessy, wi th

wh cm the notion of observation equivalence developed; and Tony Hoare,

whose parallel work cn different but strongly related ideas, expressed

in his Ccamtmicating Se quent ial Processes , has be en a str ong stimulus.

Many peopl e have g iven detailed and helpful criticisms of the manu-

script, and thus improved its final form. In particular I thank Michae l

Gordon and David MacQueem, wh o went through it all in detail in a Seminar

at the Information Sciences Institute, Unive rsity of California; this

not onl y exposed same mistakes and obscurities but ga ve me mor e csnfidence

in the part s they did n't criticise.

Finally, I a m very thank~=ul to Doro thy McKie and Gin a Temple for

their patience an d skill in the long and involved task of typing.


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O.

.

2

3.

I n t r o d u c t i o n

P u r p o s e - C h a r a c t e r - R e l a t e d ~ r k - E v o l u t i o n - O u t l i n e .

E m p e r i m e n t i n 9 o n N o n d e t e r m i n i s t i c M a c h i n e s

T r a d i t i o n a l e q u i v a l e n c e o f f i n it e s t a t e a c c e p to r s - E x p e r ~ t i n g

u p o n a c c e p t o r s - B e h a v i o u r a s a t r e e - A l g e b r a o f R S T s -

U n o b s e r v a b l e a c t i c n s .

S [ n c h r o n i z a t i c n

M u t u a l e x p e r i m e n t a t i o n - C o m p o s i t i o n , r e s t r i c t i o n a n d r e l a b e l l i n g -

E x t e n d i n g t h e A l g e b r a o f S T s - A s i n p l e ~ l e : b i n a r y s e m a p h o r e s -

T h e S T E x p a n s i o n T h e o r e m .

A c a s e s t u d y i n s y n c h r o n iz a t i c n a n d p r o o f t e ~ i q u e s

A s c h e d u l i n g p r o b l e m - B u i l d i n g t h e s c h e d u l e r a s a P e t r i N e t -

O b s e r v a t i o n e q u i v a l e n c e - P r o v i n g t h e s c h e d ul e r .

4. Ca se st ud ie s in value-cc~6~1~dnication

5.

6

7.

R e v i e w - P a s s i n g v a l u e s - A n e x a n p l e :

A n e x a m p l e : Z e r o s e a r c h i n g .

D a t a F l o w - D e r i v a t i o n s -

S y n t a x a n d S e m a n t i c s o f C C S

I n t r o d u c t i o n - S y n t a x - S e m a n t i c s b y d e r i v a t i o n s - D e f i n i n g b e h a v i o u r

i d e n t i f i e r s - S o r t s a n d p r o g r a m s - D i r e c t e q u i v a l e n c e o f b e h a v i o u r

p r o g r a m s - C o n g r u e n c e o f b e h a v i o u r p r o g r a m s - C o n g r u e n c e o f b e h a v i o u r

e x p r e s s i o n s a n d t h e E x p a n s i o n ~ n e o r e m .

C u , , ~ ~ c a t i o n T r e e s (C T s) a s a m o d e l o f C C S

C T s a n d t h e d y n a m i c o p e r a t i c n s - C 'f s a n d t h e s t a t i c o p e r a t i o n s -

C T s d e f i n e d b y r e c u r s i o n - A ~ c a c t i o n s a n d d e r i v a t i o n s o f C T s -

S t ro ng e qu i va l en c e o f C Ts - E q ua l it y i n t he C T m o de l - S ~ .

C b ~ t i c n ~ _ i i v a l e nc e a n d i t s p r o p e r ti e s

R e v i e w - O b s e r v a t i o n e q ui % ra l en ce i n C C S - O b s e r v a t i o n o o n g r u e n c e -

L a w s o f o b s e r v a t i o n c o n g r u e n c e - P r o o f t e d % n i q u e s - P r o o f o f T h e o r ~ n

7 . 7 - F u r t h e r e x e r c i s e s .

1 9

3 3

4 7

6 5

8 4

9 8


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8

9

i0.

II.

V J

S o m e p r o o f s a b o u t D a t a S t r u c t u r e s

I n t r o d u c t i o n R e g i s t e r s a n d m a m o r i e s C h a i n i n g o p e r a t i o n s

P u s h d c ~ s a n d q u e ue s .

T r a n s l a t i c n i n t o C C S

D i s c u s s i o n T h e l a n g u a g e P S o r t s a n d a u x i l i a r y d e f i n i t i c n s

T r a n s l a t i o n o f P A d d i n g p z o c e d u r e s t o P P r o t e c t i o n o f r e s o u r ce s .

~ t e z m i n a c y a n d C o n fl u e nc e

D is c u ss io n S tr on g c on fl ue nc e ~ i t e g ua rd s a nd t he u se o f

c c n f l u e n c e S t r c n g d e t e r m i n a c y: C o n f l u e n t d e t e z m i n a t e C C S

P r o o f i n D C C S: t h e s c h e d u l e r a g a i n a b s e r v a t i c n c o n f l u e n c e a n d

d e t e z m i n a c y .

C o n c l u s i o n

W h a t h a s b e e n a c h i e v e d ? I s O C S a p r o g r a m m i n g l a n g u a g e ?

T h e q u e s t i o n o f f a i r n e s s T h e n o t i o n o f b e h a v i o u r D i r e c t i o n s

f o r f u r t h e r w o r k .

~ : P ro pe rt ie s o f c c~ gr ue nc e a nd e qu iv a le nc e.

1 1 1

1 2 6

1 3 8

1 5 8

1 6 6

R e f e r e n c e s 1 6 9


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CHAPTER 0

Introduction

O .i P u ~ s e

These notes present a calculus of concurrent systems. The presentation

is part ly infozrflal, and aime d at practice; we unfold th e calculus throu gh

the med ium of examples each of which illustrates first its expressive power,

and second the techniques which it offers fo r verifying properties of a

system.

A useful calculus, of cGmputing systems as of anything else, must have

a high level of articulacy in a full sense of the word implying not only

rid ness in expression but also flexibility in manipulation. It should be

possible to describe existing systems, to specify an d progr ~n new systems,

and to argue mathematically about them, all witho ut leaving the notational

frame ~rk of the calculus.

These are demanding criteria, and it may be impossible to mee t them

even for the full range of concurrent syst~ns which are the proper concern

of a ccmputer scientist, let alone for syste~ns in general. But the attempt

n~st be made. We believe that our calculus succeeds at least to this extent:

the sane notations are used both in defining a nd in reasoning about systems,

a n d - as our examples will s how - it appears to be applicable not only to

prograns (e.g. operating systems or parts of them) but als o to data struc-

tures and, at a certa in level of abstraction, to hard ware systems. Fo r

the latter however, w e d o not claim to reach the detailed level at which

the correct functicr/ng of a sy s t ~ depends on timing considerations.

Apa rt from artioulacy, we aim at an underlying, theory whose b asis is

a small well-k nit collection of ideas and whic h justifies the manipulations

of the calculus. ~nis is as important as generality - perhaps even mere

in~portant. An y theory wil l be superseded sooner or latem; during its life,

understanding it and assessing it are only possible and worthwhile if it

is seen as a logical 9~owth f rcm rather fe wb as ic assumptions and concepts.

We take this further in the next section, where we introduce our chosen

conoeptual basis.

One purpose of th ese notes is to provide mater ial for a graduate course.

With this in mind (indeed, the notes grew as a graduate course at Aarhus

University in A u t ~ 1979) we have tried to find a good expository sequence,


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a n d h a v e c m i t t e d s c m e p a r t s o f t h e t h e o r y - w h i c h w i l l a p p e a r i n t e c h n i c a l

p u b l i c a t i o n s - i n f a v o u r o f c a s e s t u d i e s . T h e r e a r e p l e n t y o f e x e r c i s e s ,

a n d a n y o n e w h o b a s e s a c o u r s e o n t h e n o t e s s h o u l d b e a b l e t o t h i n k o f o t h e r s ;

o n e p l e a s a n t f e a t u r e o f c o n c u r r e n t s y s t e m s i s t h e w e a l t h a n d v a r i e t y o f

s m a l l b u t n o n - t r i v i al e ~ m ap l e s ' W e c o u l d h a v e i n c l u de d m a n y m o r e e x a m p l e s

i n t h e t e x t , a n d t h e r e b y g i v e n g r e a t e r e v i d e n c e f o r t h e f a i r l y w i d e

a p p l i c a b i l i t y o f t h e c a l c u l us ; b u t , s i n c e o u r m a i n a i m i s t o p r e s e n t

i t as a c a lc u l us , i t s e e m e d a g o o d r u l e t h a t e v e r y e x a m p l e p r o g r a m o r

s y s t e ~ s h o u l d b e s u b j e c t e d t o s c r ne p r o o f o r t o s a m e m a n i p u l a t i o n .

O . 2 C h a r a c t e r

O u r c a l c u l u s i f f o u n d e d o n t w o c e n t r a l i de a s . T h e f i r s t i s O b s e r v a t i o n ;

w e a i m t o d e s c r i b e a o o n c u r r e n t s y s t e m f u l l y e n o u g h t o d e t e r m i n e e ~ m c t l y

w h a t b e h a v i o u r w i l l b e s e e n o r e x p e r i e n c e d b y a n e x t e r n a l o bs e r v er . T h u s

t h e a p p r o a c h i s t h o ~ o u g h l y e x t e n s i o n a l ; t w o s y s t e m s a r e i n d i s t i n g u i s h a b l e

i f w e c a n n o t t e l l t h e ~ a p a r t w i t h o u t p u l l i n g t h e gn a p a r t . W e t h e r e f o r e

g i v e a f o m a a l d e f i n i t i o n o f o b s e r v a t i o n e q u i v a l e n c e ( in C h a p t e r 7 ) a n d

i n v e s t i g a t e i t s p r o p e r t i e s .

T h i s b y n o m e a D ~ p r e v e n t s u s f r c m s t u d y i n g t h e s t r u c t u r e o f s y s t e ms .

E v e r y i n t e r e s t i n g c o n c u r r e n t s y s t e m i s b u i l t f rc rn i n d e p e n d e n t a g e n t s w h i c h

c c m m u n i c a t e , a n d s ~ n c h r o n i z e d c o m m u n i c a t i o n i s o u r s e c o n d c e n t r a l i de a.

W e r e g a r d a c u t~ t~ ni ca ti on b e t w e e n t w o c o m p o n e n t a g e n t s a s a n i n d i v i s i b l e

a c t i o n o f t h e c c r~ 0o si te s y s te m , a n d t h e h e a r t o f o u r a l g e b r a o f s y s t e m s

i s c o n c ur r e n t c c ~ s i t i o n , a b i n a r y o p e r at i o n w h i c h c c ~ e s t w o i n d e-

p e n d e n t a g e n t s , a l l o w i n g t h e g n t o co m a~ a ni c at e . I t i s a s e e n t r a l f o r u s

a s s e q u e n t i a l c cm io os it io n i s f o r s e q u e n t i a l p r o g r a m m i n g , a n d i n d e e d s u b s u m e s

t h e l a t t e r a s a s p e c i a l c a s e . S i n c e f o r u s a p r o g r a m o r s y s t e m d e s c r i p t i o n

i s j u s t a t el ~n o f t h e c a l c u l u s , t h e s t r u c t u r e o f t h e p r o g r a m o r s y s t e m

( i ts i n t e n s i o n ) i s r e f l e c t e d i n t h e s t r u c t u r e o f t h e t e r m . O u r ~ n an i pu l at i on s

o f t e n c o n s i s t o f t r a n s f o z n d n g a te z m, y i e l d i n g a t e r m w i t h d i f f e r e n t i n t e n -

s i o n b u t i d e n t i c a l b e h a v i o u r ( e x t en s i on ) . S u c h t r a n s f o z m a t i o n s a z ~ f a n i l i a r

i n s e q u e n t i a l p ro gr an mi in g, w h e r e t h e e x t e n s i o n m a y j u s t b e a m a t h e m a t i c a l

f u n c t i o n ( t he i n p u t / o u t p u t b e /q a vi o ur ) ; f o r c o n c u r r e n t s y s t e m s h o w e v e r ,

i t s ee r, c l e a r t h a t f u n c t i o n s a r e i n a d e q u a t e a s e x t e n s i o n s .

T h e s e t w o c e n t r a l i d e a s a r e r e a l l y o ne . F o r w e s u p p o s e t h a t t h e o n l y

w a y t o o b s e r v e a s y s t e m i s t o o.xL municate w i t h i t, w h i c h r 0 a ke s t h e o b s e r v e r


8/176

a n d s y s t e m t o g e t h e r a l a r g e r s y s t e m. T h e o t h e r s i d e o f t h i s o o i n i s

t h a t t o p l a c e t w o c cr ~p on en ts i n ~ c a t i o n ( i . e . t o c c ~ p o s e t he m)

i s j u s t t o l e t t h e m o b s e r v e e a c h o t h er . I f o b s e r v i n g a n d c u t ~ u ~ c a t i n g

a r e t h e s~ n e , i t f o l l o w s t h a t o n e c a n n o t O b s e r v e a s y s t e ~ w i t h o u t i t s

p a r t i ci p a t i on . T h e a na l o g y w i t h q u a n t u m p h y si c s m a y o r m a y n o t b e s u p e r-

f i c i a l, b u t t h e a p p r o a c h i s u n i f y i n g a n d a p p e a r s n a t u r a l .

W e c a l l t h e c a l c u l u s C C S ( C a lc u l u s o f C ~ m . ~ n i c a t i n g S y s t e m s ) . T h e

t e m p s o f C C S s t a n d f o r b e h a v i o u r s ( e x t e n s i o n s ) o f s y s t e m s a n d a r e s u b j e c t

t o e q u a t i o n a l l aw s. T h i s g i v e s u s a n a l g e b r a , a n d w e a r e i n a g r e e m e n t

w i t h v a n E ro de B o a s a n d J a n s s e n [ EH J] w h o a r g u e t h a t ~ r e g e ' s p r i n c i p l e

o f o c m p o s i t i o n a l i t y o f m e a n i n g r e q u i r e s a n a l g e b r a i c f r a m ew o r k. B u t C C S

i s s ~ a t m o r e t h a n a l g eb r a ; f o r e x a m pl e , d e r i v a t i v e s a n d d e r i v a t i o ns

o f t e r m s p l a y a n i n p o r t a n t p a r t i n d e s c r i b i n g t h e d y n a m i c s o f b e h a v i o u r s .

T h e v a r i e t y o f s y s t e m s w h i c h c a n b e e x p r e s s e d a n d d i s c u s s e d i n C C S

i s i l l u s t r a t e d b y t h e e x a n p l e s i n t h e t e x t: a n a g e n t f o r s c h e d u l i n g

t a s k p e r f o m a a n c e b y s e v e r a l o t h e r a g e n t s ( C h a p te r 3) ; ' d a ta f lo w'

c c ~ p u t a t i o n s a n d a c o n c u r r e n t n% ~ re r ic a l a l g o r i t h m ( C h a p t e r 4 ) ; n ~ mn o ry

d e v i c e s a n d d a t a s t r u c t u r e s ( C h a p t e r 8 ); s e m a n t i c d e s c r i p t i o n o f a

p a r a l l e l p r o gr a nl m in g l a n g u a g e ( C h a p t e r 9 ) . I n a d d i t i o n , G . M i l n e [ M l n 3 3

m o d e l l e d a n d v e r i f i e d a p e r i p h e r a l h a r d w a r e d e v i c e - a c a r d r e a d e r - u s in g

a n e a r l i e r v e r s i o n o f t h e p r e s e n t i d ea s .

A f t e r t h e s e r e ma r k s, t h e c h a r a c t e r o f t h e c a l cu l u s i s b e s t d i s c o v e r e d

b y a q u i c k l o o k t h r o u g h C h a p t e r s 1 - 4 , i g n o r i n g t e d a n i c a l d e t a i l s . O . 5

( Ou tl in e) m a y a l s o h e l p , b u t t h e n e x t t w o s e c t i o n s a r e n o t e s s e n t i a l f o r

a q u i c k a p p r a i s a l .

O . 3 R e l a t e d W o r k

A t p r e s e n t, t h e m o s t f u l l y d e v e l o p e d t h e o r y o f c o n c u r r e n c y i s t h a t

o f P e t r i a n d h i s c o l l e a g u e s . ( Se e f o r e x a m p l e C . A . P e t r i , I n t r o d u c t i o n

t o G e n e r a l N e t ~ h e o r y [ Pe t3 , a n d H . J . G e n r ic h , K . L a u t e n b a c h , P . S .

T h i a g a r a j a n , A n O v e r v i e w o f N e t Y ~l eo ry [ GL T] .) I t i s i ~ p o r t a n t t o

c o n t r a s t o u r c a l c u l u s w i t h N e t T h e o r y , i n te z~ as o f u n d e r l y i n g c o n c e p t s .

F o r N e t T h eo r y , a ( p e rh a p s th e) b a s i c n o t i o n i s t h e c o n c u r r e n c ~

r e l a t i o n o v e r t h e p l a c e s ( co nd it io ns ) a n d t r a n s i t i o n s ( ev en ts ) o f a

s y s t e m ; i f t w o e v e n t s ( s a y ) a r e i n t h i s r e l at i o n , i t i n d i c a t e s t h a t


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t h e y a ~ e c a u s a l l y i n d e p e n d e n t a n d m a y o c c u r i n e i t h e r o r d e r o r s i m u l -

t a n e o u s l y . T h i s r e l a t i o n i s c o n s p i c u o u s l y a b s e n t i n o u r t h e o r y , a t

l e a s t a s a b a s i c n o t i o n. H e n w e o ~ p o s e t w o a g e n t s it i s t h e s y n c h r o n i -

z a t i o n o f t h e i r m u t u a l c c~ am ln ic at io ns w h i c h d e t ~ e s t h e c c ~ o o s i t e ; w e

t r e a t t h e i r i n d e p e n d e n t a c t i o ns a s o c c u r r i n g i n a r b it r a r y o r d e r b u t n o t

s i m u l t a n e o u s l y . T h e r e a s o n i s t h a t w e a s sL lr e o f o u r e x t e r n a l o b s e r v e r

t h a t h e c a n m a k e o n l y o n e o b s e r v a t io n a t a ti m e; t h i s i ~ p l ie s t h a t h e

i s b l i n d t o t h e p o s s i b i l i t y t h a t t h e s y s t ~ n c a n s u p p o r t t w o o b s er v a t i o n s

s i m u l t a n e o u s l y , s o t h i s p o s s i b i l i t y i s i r r e l e v a n t t o t h e e x t e n s i o n o f

t h e s y s t e a i n o u r s e n s e. T h i s a s s u m p t i o n i s c e r t a i n l y o p e n t o ( ex te ns iv e )

d e b a t e , b u t g i v e s o u r c a l c u l u s a s i n p l i c i t y w h i c h w o u l d b e a b s e n t o t h e r -

w i s e. T o a n s w e r th e n a t u r a l o b j e c t i o n t h a t i t is u n w i e l d y t o c o n s i d e r a l l

p o s s i b l e s e q u e n c e s ( i n t er l e a vi n g s ) o f a s e t o f c a u s a l l y i n d e p e n d e n t e v e n t s ,

w e r e f e r t h e r e a d e r t o o u r c a s e s t u d i e s , f o r e x a n p l e i n C h a p t e r s 3 a n d 8 ,

t o s a t i s f y h i m s e l f t h a t o u r m e t h o d s c a n a v o i d t h i s u n wi e l d i n e s s a l m o s t

~Zetely

C n t h e o t h e r h a n d, N e t T h e o r y p r o v i d e s m a n y s t r o n g a n a l y t i c t e c h ni q u e s;

w e m u s t j u s t i f y t h e p r o p o s a l o f a n o t h e r th e o ry . T h e e m p ha s i s i n o u r c a l c u l u s

i s u p o n s y n t h e s i s a n d u p o n e x t e n si o n ; a l g e b r a a p p e a r s t o b e a n a t u r a l t o o l

f o r e x p r e s s i n g h o w s y s t e m s a r e b u i l t , a n d i n s h o w i n g t h a t a s y s t e m m e e t s i t s

s p e c i f i c a t i o n w e a r e d e m a n d i n g p r o p e r t i e s o f i t s e x t e n s i o n . ~ a c t i v i t y

o f p r o g r a m m i n g - m o r e g e n e r a l l y , o f s y s t e m s y n t h e s i s - f a l l s n a t u r a l l y

i n t o C C S , a n d w e b e l i e v e o u r a p p r o a c h t o b e m o r e a r t i c u l a t e i n t h i s r e s p e c t

t h a n N e t T h e o r y , a t le a_ st o n p r e s e n t e v i d e n c e . I t r e m a i n s f o r u s t o

d e v e l o p a n a l y t i c t e c h n i q u e s t o m a t c h t h e s e o f ~ N e t T h e o r y , w h o s e a c h i e v e -

m e n t s w i l l b e a v a l u a b l e g u i de .

A s a b r i d g e b e t w e e n N e t T h e o r y a n d p r o g r a m m i n g l a n g ua g e s f o r c o n c u r -

r e n cy , w e s h o u l d m e n t i o n t h e e a r l y w o r k o f K a r p a n d M i l l e r [ KM ] o n p a r a l l e l

p r o g r a m s c h e m at a . T h i s w o r k b e a r s a r e l a t i o n t o N e t T h e o r y i n y i e l d i n g a n

a n a l y s i s o f p r o p e r t i e s o f c o n c u r r e n t s y s t e m s , s u c h a s d e a d l o c k a n d l iv e n e s s ;

i t a l s o e s m e s c l o s e r t o p r o g r a m m i n g ( in t h e c c n v e n t i o n a l s e n s e ) , b e i n g a

g e n e r a l i s a t i c n o f t h e f a m i l i a r n o t i o n o f a s e q u e n t i a l f l o w c h a r t .

I n r e c e n t p r o p o s a l s f o r c o n c u r r e n t p r o gr a m m i n g l a n g u ag e s t h e r e i s a

t r e n d t o w a r d s d i r e c t c c r m ~ n i c a t i o n b e t w e e n c c ~ i0 on en ts o r m o d u l e s , a n d a w a y

f r o m c ~ i ~ u n i c a t i o n t h r o u g h s h a r e d r e g i s t e r s o r v a r i a b l e s . E x a m p l e s a re :


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N. Wirt h MCDUIA: A language for modul ar multiprogramling , [Wir];

P. Brinch Hanse n Distributed Processes; a concurrent programming ccnoept ,

[Bri 2]; C.A.R. Hoar e C~ti~nicating Sequential Processes , [Hoa 3].

Hoare's monitors [Hoa 2] gave a discipline for the administrat ion of

shared resources in concurren t programming. Thes e papers have help ed

towards understanding the art of concurrent programming. Our calculus

differs frcm all of them in two ways: first, it is not in the accepted

sense an imperative l anguage - ther e are no commands, only expressions;

second, it has evolve d as part of a mathemati cal study. In the author's

v i ~ it is hard to do mathematics with imperative languages, though one

may add mathematics (or logic) to them to get a proof methodology, as in

the well-k nown assertion met hod or Hoare's axicmatic method.

One of the main encumbrances to proof i n imperative languages is the

presence of a more-or-le ss global m~mo ry (th e assignable variables). ~his

was recogniz ed by Hoare in Communicating Sequential Processes ; although

CSP is imperative Hoare avoids one aspect of global m~nory whic h makes

concarrent programs hard to analyse, by forbidding any member o f a set of

concurrent programs to alter the value of a variable mentioned by another

m ~ . This significant step brings CSP quite close to our calculus, the

more so because the treatment of eutmunication is similar and expressed in

similar notation. Indeed, algorithms can often be translated easily from

one to the other, and it is reasonable to hope that a semantics and proof

theory for CSP can be developed frGm CCS. Hoare, in his paper and more

recently, gives encouragin g evidencs for the expressivene ss of CSP.

We now turn to two models ba sed on non-synchronized o~t,manication.

One, wit h strong expressive power, is Hewitt's Acto r Systems; a recent

reference is [HAL]. Here the ~ c a t i o n discipline is that each

message sent by an actor will, after finite time, arrive at its destination

actor ; no structure over wait ing messages (e.g. ordering by send-time)

is imposed. This, t ogether wit h the dynamic creation of actors, yields

an interesting programming method. Hc~ever, it see~ns to the author that

the fluidity of structure in the model, and the need to handle the

collection of waiting messages, poses difficulties for a tractable

extensional theory.

Anot her non-synch ronized model, deliberately less expressive, was

first studied by Kah n and relx)rted by h im and MacQueen [KMQ]. Here the

interc~Li~t.~nication f agent s is vi a un bou nde d buffers and queues, the


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w h o l e b e i n g d e te r m i n a t e . P r o b l e m s h a v e a r i s e n i n e x t e n d i n g i t t o n o n -

d e t e r m i n a t e s y s t e m s , b u t m a n y n o n - t r i v i a l a l g o r i t h m s f i n d t h e i r b e s t

e x p r e s s i o n i n t h i s m ed it ~n , a n d i t i s a n e x a m p l e o f a p p l i c a t i v e ( i . e.

n o n - i m p e r a t iv e ) p ro gr ar ml in g w h i c h y i e l d s t o e x t e n s i o n a l t r e a t m e n t b y

t h e s e m a n t i c t e c h n i q u e s o f S c o t t. M o r e o v e r , W a d g e [ Wa d] h as r e c e n t l y

s h o w n h o w s i m p l e c a l c u l a t i o n s c a n d e m o n s t r a t e t h e l i v e m e s s o f s u c h

s y s t e m s .

A l u c i d o c ~ p a r a t i v e a c c o u n t o f t h r e e a p p r o a c h e s - H e w i t t , K a h n /

M a c Q u e e n a n d M i l n e r - i s g i v e n i n [ M Q] .

I n C h a p t e r 9 of t h e s e n o t e s w e s h o w h o w o n e t y p e o f c o n c u r r e n t

l a n g u ag e - w h e r e c c m m m n i c a t io n i s v i a s h a r e d v a r i a b l e s - m a y b e d e r i v e d

f r c ~ o r e x p r e s s e d i n t e r m s o f C C S . T h i s p r o v i d e s s a n e e v i d e n c e t h a t o u r

c a l c u l u s i s r i c h i n e x p r e s s i o n , b u t w e c e r t a i n l y d o n o t c l a i m t o b e a b l e

t o d e r i v e e v e r y l a n g u a g e f o r c o n c u r r e nc y .

A r a t h e r d i f f e r e n t s t y l e o f p r e s e n t i n g a c o n c u r r e n t s y s t e m i s

e xe r~ ol if ie d b y t h e p a t h e x p r e s s i o n s o f C a m p b e l l a n d H a b e r m a n n [ C a H ] .

H e r e t h e a c t i v e p a r t s o f t h e s y s t e m a r e d e f i n e d s e p a r a t el y f r o m t h e

c o n s t ra i n t s ( e . g . t h e p a t h e x pr e ss i o ns ) w h i c h d i c t a t e h o w t h e y m u s t

s y n c h r o n i z e . M o r e r e c e n t w o r k b y L a u e r , S h i e l d s a n d o t h e r s - m a i n l y

a t N e w c a s t l e - s h o ws t h a t t h i s m o d e l i n d e e d y i e l ds t o m a t h e m a t i c a l

a n a l y s i s . A v e r y d i f f e r e n t e x a m p l e o f t h i s s e p a r a t i o n i s t h e e l e g a n t

w o r k o f M a g g i o l o - S c h e t t i ni e t a l [ ~ ] ; h e r e t h e c o n s t r ai n t s a re

p r e s e n t e d n e g a t i v e l y , b y s t a t i n g w h a t c o n j u n c t i o n s o f s t a t e s ( of s e p a r a t e

c c n p o n e n t a g en ts ) m a y n o t o c c u r . T h i s a p p r o a c h h a s a n a d v a n t a g e f o r

s y s t e m s w h o s e c r m p o n e n t s a r e l a r g e l y i n d e p e n d e n t ( th e a u t h o r s c a l l i t

l o o s e c o u p l i n g ) , s i n c e t h e n o n l y f e w c o n s t r a i n t s n e e d t o b e e x p r e s s e d .

T h i s s e c t i o n h a s s h o w n t h e s u r p r i s i n g v a r i e t y o f p o s s i b l e t r e a t m e n t s

o f c o n c u r r e n t s y s t e m s . I t i s n o t h i n g l i k e a c c ~ p r e h e n s i v e s u r v e y , a n d

t h e a u t h or i s a w a re t h a t i m p o r t a n t w o r k h a s n o t b e e n me n t i o n ed , b u t i t

w i l l s e r v e t o p r o v i d e s c ~ e p e r s p e c t i v e o n t h e w o r k p r e s e n t e d h e r e.

O . 4 E v o l u t i o n

% Tu is w o r k e v o l v e d f rc ~n a n a t t er ~o t t o t r e a t c ~ L ~ n i c a t i o n m a t h e m a t i -

c a l ly . I n M i l n e r : P r oc e ss e s: a m a t h e m a t i c a l m o d e l o f ~ t i n 9 a g e n ts

[ M i l 1 1 a m o d e l o f i n t e r a c t i n g a g e n t s w a s c o n s t r u c t e d , u s i n g S c o t t ' s


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t h e o r y o f d c r m~ m s . ~ h i s w a s r e f i n e d a n d g r e w m o r e a l g e b r a i c i n G . M i l n e

a n d M i l n e r : " C o n c u r r e n t P r o o e s s e s a n d t h e i r s y n t a x " E M M 3 A t t h i s

p o i n t w e p r o p o s e d n o p r o g r a m m i n g l a n gu a g e, b u t w e r e a b l e t o p r o v e

p r o p e r t i e s o f d e f i n e d c o n ~ n t b e h a v i o u rs . F o r e x a mp l e , M i l n e i n h i s

P h . D . T h e s i s " A m a t h e m a t i c a l m o d e l o f c o n c u r r e n t c o m p u t a t i o n " [ Ml n]

p r o v e d p a r t i a l c o r r e c t n e s s o f a p i e c e o f h a r d w a r e , a c a r d - r e a d er , b u i l t

f r o m f o u r s e p a r a t e c c ~ p o n e n t s a s d e t a i l e d i n i t s h a r d w a r e d e s c r i p t i o n .

O u r m o de l a t t hi s st ag e dr e w u p on Pl ot ki n' s a nd Sa~yth's P ~ e r ~

c o n s t r u c t i o n s , [ P I o i , S a ~ y ] . w h i c h e x t e n d e d S c o t t ' s t h e o r y t o a d m i t

n o n - d e t e n m i n i s m . P a r t o f o u r a l g e b r a i s s t u d i e d i n d e p t h i n [ M i l 2 ].

A t t h i s p o i n t t h e r e w e r e t w o c r u c i a l d e v e l o ~ a e n t s . F i r s t - a s w e

h a d h o p e d - o u r b e h a v i o u r d e f i n i t io n s l o o k e d c o n s i d e r a b ly l i k e p r o g ra m s ,

a n d t h e r es cm %b la nc e w a s i n c r e a s e d b y m e r e l y i m p r o v i n g n o t a t i o n . T h e

r e s u l t , e s s e n t i a l l y t h e l a n g u a g e o f C C S , i s r e p o r t e d i n [ M i i 3 ] a n d

w a s p a r t l y p ~ t e d b y d i s c u s s io n s w i t h H o a r e a n d S c ot t . ( Fo r c o m pl e t e ne s s ,

t w o o t h e r p a p e r s [ M i l 4 , 5] b y t h e a u t h o r a r e i n c l u d e d i n t h e r e f e r e n c e

l i s t. E a c h g i v e s a s l i g h t l y d i f f e r e n t p e r s p e c t i v e f r o m [ M i l 3 ], a n d

d i f f e r e n t e x a m pl e s . ) T h e s e c o n d d e v e l o l m ~ n t w a s t o r e a l i s e t h a t t h e

r es ul ti ng l an gu ag e h as m a n y i nt er pr et at io ns ; a n d t ha t t he P c ~ r ~

m o d e l , a n d v a r i a n t s o f i t, m a y n o t b e t h e c o r r e c t o n e s. A c r i t e r i o n w a s

n e e d e d , t o r e j e c t t h e w r o n g i n t e r p r e t a t i o n s . F o r t h i s p u r p o s e , w e t u r n e d

t o o b s e r v a t i o n e q u iv a l e n ce ; t w o b e h a v i o u r e x p r e s s i on s s h o u l d h a v e t h e

s a n e i n t e r p r e t a t i o n i n t h e m o d e l i f f i n a l l c o n t ex t s t h e y a r e i n d i s ti n g u i s h -

a b l e b y o b s e r v a t i o n .

I t n o w t u r n s o u t t h a t a d e f i n i t i o n o f o b s e r v a t i o n e q u i v a l e n c e ( fo r

w h i c h a d m i t t e d l y t h e r e a r e a f e w a l t e r n a t iv e s ) d e t e m n i n e s a m o d e l , u p

t o i s c~ o r p hi s m , a n d m o r e o v e r y i e l d s a l g e b r a i c l a w s w h i c h a r e o f p r a c t i c a l

u s e i n a r g u i n g a b o u t be 2~ ir io ur s. W e h a v e s t r o n g h o p e f o r a s e t o f l a w s

w h i c h a r e i n s o m e s e n s e co m p l e t e ; i n f a c t t h e l a w s g i v e n i n C h a p t e r s 5

a n d 7 h a v e b e e n s h o w n c c ~ o l et e f o r a s i n p l i f i e d c l a s s o f f i n i t e ( t en ~ in a ti n g)

b e h a v i o u r s . I n t h i s ca s e , " c o m p l e t e " m e a n s t h a t i f t w o b e h a v i o u r e x p r e s s i o n s

a r e o b s e r v a t i o n - e q ui v a l e n t i n al l c o n t e x t s t h e n t h e y m a y b e p r o v e d e q u a l

b y t h e l a w s; t h i s c o m p l e t e n e s s i s s h o w n i n [H M] .

0 . 5 O u t l i n e

I n C h a p t e r 1 w e d i s c u s s i n f o r m al l y t h e i d e a o f e xp e r i m e n t i n g o n , o r

o b s e r v i n g , a n o n - de t e 2 xn i n i st i c a g e n t ; t h i s l e a d s t o t h e n o t i o n o f


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s y n d 0 m o n i s a t i o n t r e e ( S T ) a s t h e b e h a v i o u r o f a n a g e n t . C h a p t e r 2 d i s -

c t ~ s e s m t u a l e x p e ri m e n t, o r c ~ m L ~n i c a ti c n , b e t w e e n a g e n ts , a n d d ev e l o p s

a n a l g e b r a o f S T s . I n C h a p t e r 3 w e d o a s m a l l c a s e - s t u d y ( a s c h e d u l i n g

s y st e m) a n d p r o v e s c x ne t hi n g a b o u t i t, a n t i c i p a t i n g t h e f o r m a l d e f i n i t i o n

o f o b s e r v a t i on e q u i v a l e n c e a n d i t s p r o p e r t ie s t o b e d e a l t w i t h f u l l y i n

C h a p t e r 7 .

C h a p t e r 4 e n r i c h e s o u r c u l, ud ni ca ti on s - u p t o n o w t h e y h a v e b e e n j u s t

s y n d l r o n i z a t i o n s - t o a l l o w t h e p a s s i n g o f v a l u e s f r c m o n e a g e n t t o a n o t h er ,

a n d i l l u s t ra t e s t h e g r e a t e r e x p r e s s i v e p ~ e r i n t w o m o r e e x ar ~ l es ; c n e i s

a k i n t o D a t a F l o w , a n d t h e o t h e r i s a c o n c u r r e n t a l g o r i t h m f o r f i n d i n g a

z e r o o f a c o n t i n u o u s f u n c t i o n. T h e n o t i o n o f d e r i v a t i v e o f a b e h a v i o u r

i s i nt r o d u c e d , a n d u s e d i n t h e s e c c n d e x a m p l e .

I n C h a p t e r 5 w e d e f i n e C C S f o r ma l l y, g i v i n g i t s d y n am i c s i n t e r ms

o f d e r i v a t i o n s ( d e r i v a t iv e s e q u e n c e s ) . ~ h i s y i e l d s o u r s t r o n g c o n g r u e n c e

r e l a ti o n , u n d e r w h i c h t w o p ro g r a m s a r e c c n g r u e n t i f f t h e y h a v e e s s e n t i a l l y

t h e s a m e d e r i v a t i o n s , a n d w e e s t a b l i s h s e v e r a l l a w s o b e y e d b y t h e co n g r u e n c e .

I n C h a p t e r 6 w e p r e s e n t c ~ m l u n i c a t i c n t r e e s ( C T s , a g e n e r a l i s a t i o n o f ST s)

a s a m o d e l w h i c h o b e y s t h e s e l a ws ; t h i s m o d e l i s n o t n e c e s s a r y f o r t h e

f u r t h e r d e v e l o p m e n t , b u t m e a n t a s a n a i d t o u n d e r s t a n d i n g .

C h a p t e r 7 i s t h e c o r e o f t h e t h e o r y; o b s e r v a t i o n e q u i v a l e n c e i s

t r e a t e d i n d e p th , a n d f r o m i t w e g a i n o u r m a i n c o n g r u e n c e r e l at i o n ,

o b s e r v a t i o n c o n g ru e n ce , u n d e r w h i c h t w o p r o g r a n s a r e c o n g r u e n t ~ i ff t h e y

c a n n o t b e d i s t in g u i s h e d b y o b s e r v at i o n i n a n y c on t ex t . H a v i n g d e r i v e d

s c m e p r op e r t i e s o f t h e c o n g r u en c e , w e u s e t h e m i n C h a p t e r 8 t o p r o ve t h e

c o r r e c t b e h a v i o u r o f t w o f u r t h e r s y st e m s, b o t h t o d o w i t h d a t a s t r uc t u r es .

I n C h a p t e r s 9 a n d I O w e l o o k a t s a n e d e r i v e d A lg e b r a s. O n e t a k es t h e

f o r m o f a n i m p er a t i v e c o n c u r r e n t p r o g r a m m i n g l a n g u ag e , w i t h a s s i s t

s t a t e m e n t s , c o b e g i n - c o e n d s t a t e m e n t s , a n d p r o c e d u r e s . I n e f f e c t , w e

s h o w h o w t o t r a n s l a t e t h i s l a n g u a g e d i r e c t l y i n t o C C S . T h e o t h e r i s a

r e s t r i c t i o n o f C C S i n w h i c h d e t e r m i n a c y i s g ua r a nt e e d , a n d w e i n d i c a te

h o w p r o o f s a b o u t s u c h p r o g r a m s c a n b e s i r ~ l e r t h a n i n th e g e n e r a l c a s e .

F i n a ll y , i n C h a p t e r i i w e t r y t o e v a l u a t e w h a t h a s b e e n a c h i ev e d ,

a n d i n d i c a t e d i r e c t i o n s f o r f u t u r e r e s e a r c h .


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CHAPTER

E x p e ri m e n ti n g o n n o n d e t e n n i n i s t i c m a c h i n e s

i o i

T a k e a p a i r

= { a , b , c , d } :

T r a d i t i o n a l e q u i v a l e n c e o f f i n i t e s t a t e a c c e p t o r s

S , T o f n o n d e t e z a d n i s t i c a c c e p t o rs o v e r t h e a l p h a b e t

T

T h e a c c e p t i n g s t a t e s o f S a n d T a r e s 2 a n d t 2 r e s p e c t i v e l y ; i n n o n -

d e t e r m i n i s t i c a c c e p t o r s w e c a n a l w a y s m a k e d o , a s h e r e , w i t h a s i n g l e d ea d

a c c e p t i n g s t a t e .

A s t an d ar d ~ t t h at S a n d T a re e qu iv al en t, m e a ni n g t h at t h ey

a c c e p t t h e s a m e l a n g u a g e ( s et o f s t r i n g s ), r u n ~ a s f ol l o w s .

t i) t o r e p r e s e n t t h e l a n g u a g e a c c e p t e d s t a r t i n g f r c m s t a t e s i ( r es p t i) , w e

g e t a s e t o f e q u a t i o n s f o r

S , a n d f o r T :

s O = as I t O = at 1

s I = bs 2 + cs 3 t I = bt 2

S 2 = ~ t I = ct 3

S 3 = ds 0 t 2 =

t 3 = d t 0

+ a t ~

H e r e a s u s u a l + s t a n d s fo r u n i o n o f l a n g u a g e s, e f o r t h e l a n g u a g e

c o n t a i n i n g ~ n l y t h e 6 ~ p t y s t r i n g , a n d w e c a n t h i n k o f t h e s y m b o l a

f o r a f u n c t i o n o v e r l a n g u a g e s : a s = a ( s ) = { a ~ ; ~ E s } .

N o w b y s in lo le s u b s t i t u t i o n w e d e d u c e

s O = a (b c + cds0) .

B y a p p l y i n g t h e d i s t r i b u t i v e l a w a ( s + s ) = a s + a s w e d e d u c e

s O = a b ~ + a c d s 0

T a k i n g s i ( r e s p

s t a n d i n g


15/176

1 0

a n d w e c a n g o f u r t h e r, u s i n g a s t a n d a r d r u l e f o r s o l v i n g s u c h e q u a t i o n s k n o w n

as Arden s rule, to get

s = (acd)*abe .

F o r T i t i s e v e n s i m p l er ; w e g e t d i r e c t l y ( w i th o u t u s i n g d i s t r i b u t iv i t y )

t = ab~ + acd t

a n d t h e u n i q u e s o l v a b i l i t y o f s u c h e q u a t i o n s t e l l s u s t h a t s = t O , s o S

a n d T a r e e q u i v a l e n t a c c e p t o r s .

B u t a r e t h e y e q u i v a l en t , i n a l l u s e f u l s e n s e s ?

1 . 2 E x p e r i m e n t i n u p o n a c c e p t o r s

T h i n k d i f f e r e n t l y a b o u t a n a c c e p t o r o v e r { a , b , c , d > . I t i s a b l a c k

b o x, w h o s e b e h a v i o u r y o u w a n t t o i n v e s t i g at e b y a s k i n g i t t o a c c e p t s y m b o l s

o n e a t a t i m e. S o e a c h b o x h a s f o u r b u t t o n s , o n e f o r e a c h s y mb o l :

S b

a a

i

s d T b t

t

T h e r e a r e f o u r a t o m i c e x p e r ~ t s y o u c a n d o, c n e f o r e a c h s y mb o l. D o i n g a n

a - e x p e r i m e n t o n S ( s e c re t l y i n s t a t e s , b u t y o u d o n t k n o w t ha t) c o n -

s i s t s i n t r y i n g t o p r e s s t h e a - b u t t o n , w i t h t w o p o s s i b l e o u t c c m e s i n g e n er a l :

(i) Fail ure - the but ton is locked;

( i i ) S u c c e s s - t h e b u t t o n i s u n l o c k ed , a n d g o e s d o w n ( an d

s e c r e t l y a s t a t e t r a n s i t i c n o c c u r s ) .

I n f a c t w e c a n n o t d i s t i n g u i s h b e t w e e n S a n d T , i n t h e i r i n i t i a l s t a t e s,

b y a n y s i n g l e a t c n d c e x p e r i m e n t ; t h e a - e x p e r i m e n t s u c c e e d s i n e a c h c as e , a n d

the other th ree fail.

A f t e r a s u c c e s s f u l a - e x p e r i m e n t o n e a c h m a c h i ne , w h i c h m a y y i e l d

a

S b l s 1 ~ d T

b

t t

t

a

i

d

w e m a y t r y a n o t h e r a tc ~t ic e x p e r i m e n t , i n o u r a i m t o s e e i f t h e m a c h i n e s a r e

e q u i v a l e n t o r n o t . C l e a r l y a b - e x p e r i m e n t n o w s u c c e e d s f o r S a n d f a i l s


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11

f o r T , t h o u g h t h e o t h e r t h r e e e x p e r ~ t s f a i l t o d i s t i n g u i s h t h e~ . A f t e r

t r y i n g t h e b - e x p e r i m e n t , t he n , c a n w e c o n c l u d e t h a t S a n d T a r e n o t

e q u i v a l e n t ?

N o , b e c a u s e S s r e s p o n s e t o th e a - e xp e r i m e n t c o u l d h a v e b e e n d if f e r e n t

( fo r a l l w e k n ow ) a n d l o c k e d t h e b - b u t t o n , w h i l e T s r e s p o n s e c o u l d h a v e

b e e n d i f f e r e n t ( fo r a l l w e k n o w - a n d i t c o u l d i nd e e d ) a n d u n l o c k e d t h e

b - b u t t o n . F o l l o w i n g t h i s a r g L m ~ n t f u r t h e r , w e m a y f e e l f o r c e d t o a d m i t t h a t

n o f i n i t e a m o u n t o f e x p e r i m e n t c o u l d p r o v e t o u s t h a t S a n d T a r e, o r a r e

not, equivalent .

B u t s u p p o s e

(i) I t i s t h e w e a t h e r a t a n y m c n ~ n t w h i c h d e t e r m i n e s t h e c h o i c e o f

t r a n s i t i o n ( in c a s e o f a m b i g u i t y, e . g . T a t t O u n d e r a n

a-experiment) ;

( i i ) T h e w e a t h e r h a s o n l y f i n i t e l y m a n y s t a t e s - a t l e a s t a s f a r

a s c h o i c e - r e s o l u t i o n i s c o n c e r n e d ;

( ii i) W e c a n c o n t r o l t h e w e a t h e r .

F o r s a n e m a c h i n e s t h e s e a s s u n p t i o n s a r e n o t s o o u t r a g e o u s ; f o r e x a m p l e , o n e

o f t w o p u l s e s m a y a l w a y s a r r i v e f i r s t w i t h i n a c e r t a i n t e m p e r a t u re r a n ge , a n d

o u t s i d e t h i s r a n g e t h e o t h e r m a y a l w a y s a r r i v e f i r st . ( At t h e b o u n d a r y o f

t h e r a n g e w e h a v e t h e w e l l - k n o w n g l i t c h p r o b l e m , w h i c h w e s h a l l i g n o r e h e re .)

N c ~, b y c o n d u c t i n g a n a - e x p e r i m e n t a n S a n d T u n d e r a l l w e a t h e r c o n -

d i t i o n s ( al wa ys i n t h e i r s t a r t s t at e s , w h i c h w e h a v e t o a s s u m e a r e r e c o v e r -

a b l e ) , w e c a n f i n d t h a t S s b - b u t t o n i s a l w a y s u n l o c k e d , b u t t h a t T s

b - b u t t o n i s s c ~ e t i m e s l o c k e d, a n d w e c a n c o n c l u d e t h a t t h e m a c h i n e s a r e n o t

e q u i v a l e n t .

I s t h i s s e n s e o f e q u i v a l e n c e , i n w h i c h S a n d T a r e n o t e q u i v a l e n t ,

a m e a n i n g f u l o n e ? W e s h a l l f i n d t h a t w e c a n m a k e i t p r e c i s e a n d s h al l a d o p t

i t - p a r t l y b e c a u s e i t y i e l d s a n i c e t h e or y , p a r t l y b e c a u s e i t i s a f i n e r

( s~ al le r) e q u i v a l e n c e r e l a t i o n t h a n t h e s t a n d a r d c n e ( wh ic h w e c a n a l w a y s

r e c o v e r b y i n t r o d u c i n g t h e d i s t r i b u t i v e l a w u s e d i n i. i) , b u t m o r e f o r t h e

f o l l o w i n g r e a s o n . I m a g i n e t h a t t h e b - b u t t o n s o n S a n d T a r e h i d d e n .

T h e n i n a l l w e a t h e r s e v e r y s u c c e s s f u l e x p e r ~ t u p o n S u n l o c k s s a m e

v i s i b l e b u t t o n :

S ( w it h b h i d d e n) i s n o t d e a d l o c k a b l e


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whil e in sc~e weathers, and after same experiments, all of T s visible

buttons will be lo~ked:

T (with b hidden) is deadlockable.

We wish to think of a nondeterministic choice in such machines as being

resolved irreversibly, at a particular manent, by information flowing into

the system from an unseen source; if a deadlock can thus arise in one machine

b u t n o t i n a n o t he r , w e d o n o t r e g a r d t h e m a s b e h a v i ou r a l l y e qu i v a le n t .

1.3 Behaviou r as a tree

Because we r eject the distributive law a(x + y) = ax + ay , we can no

longer take language s (sets of strings) as the behav iour s of our machines .

W e p r o c e e d t o a n a l t er n a ti v e . F r c m n o w o n w e w i l l u s e N I L i n s t e a d o f e

to stand for a behavi our whic h can do nothing (= admits no experiment) ; we

s h al l a l so u s e G r e e k l e tt e rs f or o u r ~ i s - i.e . na m e s o f b u t to n s - s o y o u

should consider ~,~,y ,5 as replace~_nts for a,b,c,d in our sinple example.

First, take the transition graph for S and unfold it into a tree wit h

states as node labels and symbols as arc labels:

B

Because state na~es are pre sent we hav e lost no information; the state trans-

ition graph can be recovered frcm such a tree. But the experimenter cannot

see the state - he can only see the transitions. This leads us to drop the

node labels, an d take the infinite tree

a s t h e b e h a v i o u r o f S . a - -

I Definiti on A label is a n~m ber of a given (fixed) label set A .

We are using ~,B,Y,.. to stand for labels. (The use of the wor d label in

place of symbol will be further motiv ated later.)


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3

I Definiti on A sort is a subset of A

We sh all usually use L,M,N, .. to stand for sorts. We shall also ofte n use

t h e w o r d a ~ e n t i n p l a c e o f m a d l i n e o r a cc e pt or , s o

S is an acceptor over the alphabet Z

b e c o m e s

S is an agent of sort L

Definit ion A Rigi d Synchroniza tion Tree (RST) of sort L is a rooted,

u no r de re d , f i n i t e l y b r a n ~ t r e e e a c h o f w h o s e a r c s i s l a b e l l e d b y a

m e m b e r o f L .

Thus the tr ee in th e las t diagr an is an RS T of so rt {~, B, ~, 8} . (It is also

a n R S T o f a n y l a r g e r s or t. )

W h y r ig id ? B e c a u s e i t i s t h e b e h a v i o u r o f a r i g i d a g e n t - o n e w h i c h

c a n m a k e n o t r a n s i t i o n e x c e p t t h a t c o r re s p o n d i ng t o a n a t o m i c e x pe ri m en t. W e

shall soon meet other transitions.

Why synchronizaticn ? Because we shall later see how the oammunication

of two age nts c an be rep resented in fozrcing their joint tree frcm the ir

separate trees. Then the joint tree wil l not be rigid, in general, since

intercuf, dnication betw een cc~po nent agen ts i s not observable.

Notice that finit e RSTs can be represente d as expressions:

B / / ~ i s ( BN I L + ~ NI L )

~ ~ is ~ B N IL + ~ N I L

and usuall y there is more than one natur al expression:

(~NIL 7NI L) , or

s ~ r I n

+

+

c~LL + BNIL) + yNIL .

Indeed, + is both oactuutative and associative, since we decla red RSTs to

be unordered trees - and NIL is easily seen to be a zero for summation.

T o j u s t i f y t h e s e r e m a rk s w e n c ~ d e f i n e t h e a l g e b r a o f R S Ts .


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i . 4 A l g e b r a o f R S T s

I g n o r i n g s o r t s f o r a m c n ~ n t , w e h a v e a n e l e m e n t a r y a l g e b r a o v e r R S T s,

w h o s e o p e r a t i o n s a r e :

N I L n u l l a ry o p e r a t io n )

N I L i s t h e t r e e ;

+ b i n a r y o p e r a t i o n )

+ A i s t h e t r e e ~ i d e n t i f y r o o t s ) ~

u n a r y o p e r a t i o n , f o r e a c h ), ~ A)

T h e y o b e y t h e f o l l c ~ i n g l a w s , a s y o u c a n e a s i l y s e e :

A s s o c i a t i v i t y x + y + z ) = x + y ) + z

Ccurnu tativ ity x + y = y + x

N u l l i t y x + N I L = x

I n f a ct , t h e s e l a w s a r e c ca lo le te : a n y t r u e e q u a t i c n b e t w e e n R S T e x p r e s s i o n s

c a n b e d e d u c e d f r c m t h em .

I f w e c o n s i d e r s o r ts , a n d l e t R S T b e t h e s e t o f R S T s o f s o r t L ,

t h e n N I L i s o f s o r t L f o r a n y L :

N I L ~ R S T .

F u r t h e r , + t a k e s t r e e s o f s o r t L , M r e s p e c t i v e l y t o a t r e e o f s o r t L u M :

+ E R S T R S T R S T L u ,

a n d ~ t a k e s a t r e e o f s o r t L t o a t r e e o f s o r t L u { l } :

W e s h a l l u s u a l l y f o r g e t a b o u t s o r t s f o r t h e p r e s e n t , b u t t h e r e a r e t i m e s

l a t e r w h e n t h e y w i l l b e e s s e n ti a l .

C o n s i d e r n o w s o l v i n g r e c u r s i v e e q u a t i o n s o v e r R S T s . W e w i s h t h e e q u -

a t i o n s f o r o u r a g e n t S o f l . 1

s O = ~s I s I = ~s + ys 3

s 2 = NI L s 3 = 6s

t o d e f i n e t h e i n f in i t e) b e h a v i o u r o f S a s a n R S T o f s o r t { ~ , 8 , Y , 8 } -


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T h i s s e t o f e q u a t i o n s h a s a u n i q ue , s o l u t i o n f o r t h e v a r i a b l e s s 0 ' s 3 ;

y o u c a n s e e t h i s b y t h e f a c t t h a t t h e e n t i r e t r e e c a n b e d e v e l o p e d t o p - d o w n

t o a n y d e p t h :

s O

E x e r c i s e i . 1

= = = ... an d so on.

N o t e v e r y s e t o f r e c u r s i v e e q u a t i o n s h a s a u n i q u e s o l u t i on ;

c o n s i d e r t h e s i m p l e e q u a t i o n

s = s

w h i c h i s s a t i s f i e d b y a n y R S T ( or a n y t h i n g e l s e , f o r t h a t m a t t e r) .

A g a i n , s cr ne s e t s o f e q u a t i o D ~ d e f i n e n o R S T a t a l l. C o n s i d e r t h e e q u a t i o n

s = s + ~ N I L ;

a s o l u t i o n w o u l d h a v e t o b e i n f i n i t e ly b r a n c h i n g a t t h e ro o t. E v e n i f w e

a l l o w e d i n f i n i t e l y b r a n c h i n g R S T s, s o t h a t

w o u l d b e a s o l u t i o n, i t w o u l d n o t b e u n i q u e s i n c e s + t w o u l d a l s o

b e a s o l u t i o n f o r a n y t . W e d e f e r t h i s p r o b l e m t o C h a p t e r 5.

C a n y o u f i n d a c o n d i t i o n o n a s e t o f e q u a t i o n s

Q

i

s =

s i =

o o o

S n

(with RST expres sions invol ving s O ,.. ,s

o n t h e r i g h t - h a n d s id es )

w h i c h e n s u r e s t h a t i t p o s s e s s e s a u n i q u e s o l u t i o n i n R S T s ?

( H i n t : c o n s i d e r c y c l e s o f c - t r a n s i t i o n s i n t r a n s i t i o n g r a p hs . )

1 . 5 U n o b s e r v a b l e a c t i o n s

U n d e r t h e c o n v e n t i o n a l d e f i n i t i o n , a n o n d e t e r m i n i s t i c a c c e p t o r m a y

h a v e t r a n s i t i o n s l a b e l l e d b y ~ ~ t h e n u l l s t r i n g . C o n s i d e r R , a m o d i -

ficati on of our S of i.I (reverting bri efl y to Rcn~ml alphabet):

R a ~ ~ d ~ ~


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6

(The loop fomaed by the d-tr ansi tion is irrel evant to our ccrnparison.)

In the conventional sense, R and S are equivalent. But what does the

c - t r a n s i t io n m e a n, i n o u r m o r e m e c h a n i s t i c i n t e rp r e t a ti o n ? I t m e a n s t h a t

R in state r i (i.e. after the a-button has been pressed) may at any time

' and that if a b-e xpe rim ent is nev er attesloted

ove silently to state r I ,

it wil l do so.

Thus, if we attespt a b-expe riment on R , after the successful a-

e x p e ri m e n t, t h e r e a r e s c ~e w e a t h e r c o n d i t i o n s i n w h i c h w e f i n d t h e b -

b u t t o n p e a m a n e n t l y l o ck e d; i f o n t h e o t h e r h a n d w e a t t ~ a c - e x p e r i me n t

(after the a-experiment) we shall in all weat her conditions find the

c-butto n eventually unlocke d (eventually, becau se although R may take a

little t ~ to decide on its e-transition, it will do so since no b-

experiment is attespted).

E xe rc is e 1 .2 U se t hi s a s t he b as is o f a n ~ t t ha t n o p ai r o f R, S

a n d T a r e e q u iv a l e nt . A r i g o r o u s b a s i s f o r t h e a r ~ t w i l l b e g i v e n

later.

Let us retu rn to our Greek alphabet, and ask how we should write the

equations specifying R's behaviour. We choose the symbol T in place of

e (to avoid confusion with the null string), and use it as a new unary

o p e r at i o n u p o n b e h a v i o u r s .

r0,..,r are:

r 0 = ~r

r 2 = NIL

We are ass~aing that

The equat ions dete_~mining the behavio urs

r i = 8r + xr~

r 3 = ~r

T A (the fixed label set).

r I = yr

Definit ion A Synchroni zation Tree (ST) of sort L is a rooted, unordered,

f in i te ly b r an c hi n g t re e e a c~ o f w h o s e a rc s i s l a be l le d b y a ~ r o f

L u { ~ }

Thus a rigid ST (an RST) is just an ST wit h no arcs labelled T ;

t h e b e h a v i o u r o f a n a g e n t w h i c h c a n m a k e n o s i l e n t t r a ns i t io n s .

S i n c e w e a r e t a k i n g t h e u n a r y o p e r a t i o n

we can of course deduce the ST-behaviour of R . It is

it is

T o v e r S T s t o b e g i v e n b y


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Y

y -

STs are a sinlole and useful notion of behaviour. They are just the

unfoldings of behavio ur equations, whic h in turn follow directly fran

transition graphs. Of course in this way different transition graphs can

y i e l d t h e s a n e S T , f rc rn w h i c h w e c a n b e c e r ta i n t h a t t h e y a r e i n d i s ~ s h -

able by experiment

Exercise 1.3 Convince yourse1_f that the transition graphs

have the same unfolding.

However, differe nt STs or transition graphs yielding different STs)

may be indistinguishable by experiment. This is true even for RSTs;

c o n s i d e r t h e s i m p l e p a i r

each of whic h adaits a single ~-experiment and then nothing else.

But it is even more true in the case of unobservable actions. Later

we shall study an equivalence relation, c~servation equivalence, over STs,

which ca n for finite STs) be axicmatized by a finite set of equations

added to those given in 1.4 above. To get a foretaste of the equivalence

consider the following exercise.

E x e r c i s e 1 .4 E x a m i D ~ e a c h o f th e f o l l o w in g p a i r s o f s ~ p l e S T s a n d t r y t o

d e ci d e b y i nf oz m al ~ t , a s i n E xe r ci se 1. 2 a bo ve , w h i ch a r e o b se r va t io n

equivale nt i.e. indistinguishable by experiment). Yo u may reasonably

conclude that four pairs are equivalent, or that si x pairs are equivalent,

b u t y o u s h o u l d a l s o f i n d t h a t t h e n o t i o n o f e q u i v a le n c e i s n o t y e t p re ci se .

E h e p o i n t o f t h i s e x e r c i s e i s t h a t i t i s n o t t r i v i a l t o c a p t u r e o u r i n f o z m a l

arguments b y a precise notion.


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o

~

o

c

~

o

~

v

O

~


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C h a p t e r 2

S y n c h r o n i z a t i o n

2 . 1 M u t u a l e x p e r i m e n t a t i o n

~ h e s uc c e s s o f an s - e x p e r i m e n t e n a b l es t h e m a c h i n e t o p r o c e e d to

o f f e r f u r t h e r e x p e r i m e n t s ) ; i t a l s o a l l o w s t h e o b s e r v e r t o p r o c e e d t o

a t t e m p t f u r t h e r e x p e r i m e n t s ) . T h i s s u g g e s t s a n ~ b v i o u s s y m m e t r y ; w e

w o u l d l i k e t o r e p re s e n t t h e o b s e r v e r a s a m a c h i ne , t h e n t o r e p r e s e n t t h e

c c m p o s i t e o b s e r v e r / m a c h i n e a s a m a c h i ne , t h e n t o u n d e r s t a n d h o w t h i s

m c h i n e b e h a v e s f o r a n e w o b s er v e r .

H o w s h o u l d t w o m a c h i n e s i n t e r ac t ?

S ~ T 0

W e m u s t s a y w h i c h e x p e r i m e n t s o f f e r e d b y S m a y c o m b i n e w i t h o r c cr ap le me nt )

w h i c h ex p e r i m e n t s o f T t o y i e l d a n i n t er a c ti o n . R a t h e r t h a n s e t u p a

l a b e l c o r r e s p o n d e n c e e . g . a + + ~ , ~ + + n) f o r e a c h m a c h i n e c c m b i n a t i o n ,

w e i n t r o d u c e a l i t tl e s t r u c tu r e o n o u r l a b e l s e t A .

We ass~ne a fixe d set A of names. We use ~, B, y, ... to stand for

names

W e a s s u m e a s e t ~ o f c o - n a m es , d i s j o i n t f r c ~ A a n d i n b i j e c t i o n

w i t h i t; t h e b i j e c t i o n i s - ):

o~ ~A) ~


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2

na~ es L) = {n~ne l) ; 16L} .

N O W c o n s i d e r t h e p a i r o f m ac h i n e s

Y

S : { ~ S ~ }

Y

T h e n a t u r a l c a n d i d a t e , p e r h a p s , f o r t h e cc r ~b i ne d m a c h i n e

p i c t u r e d t h u s :

S l i t

m a y b e

or:

T 1_

\ _/

T h e i n t u i t i o n i s t h a t c c~ o l e m e n t a r y p o r t s , o n e i n e a c h m a c h i n e , a r e

l i n k e d a n d h i d d e n l a b e ls r e m o v e d ) , s i n o e t h e s e l i n k s r e p r e s e n t m u t u a l

o b s e r v a t i o n , w h i l e o t h e r p o r t s s t i l l s u p p o r t e x t e r n a l o b s e r v a t i o n .

B u t u n d e r t h i s s c h e m e t h e r e a r e t w o d i s a d v a n t a g e s . F i r s t , c o n s i d e r

r

R: {8,~}


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21

We can form RII[SIIT) and (RIiS) IT:

each of sort {~,~} bu t clea rly different. S's offer s of 8-experiments

are observe d by T in the first case, but by R in the second case. So

II is no t associ ative.

Second, it is usef ul to allo w that S's 8-experiment-offers

(or 8-capabilities as we shall sometimes call them) may be obs erved by

either R or T (that is, each 8-experiment cn S may be done by either

R or T, but not both) ; this makes S into a resource shared by R

and T.

The solution is to factor cc~biD~tion into two separate operations:

one to link ports, the other to hi__~ them. We shall use the %Dr d

.composition for the first of th ese operations, and the second we shall

call restriction.

2.2 Cc~0osition, restr ictic n and relab ellin g

T h e c c s p o s i te R I S o f o u r t w o m a c h i n e s R a n d S m a y b e p i c t u r e d

i s

r

whil e for (RIS) IT we get


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22

8

~ h a t is , f o r e a c h I , i n f o i ~ r / ng U I V w e l i n k e v e r y p o r t l a b e l l e d

i n U t o e v e r y p o r t l a b e l l e d [ i n V .

E x e r c i s e 2 . 1 F r c m R 1 ( SI T) a s a p i c t u r e , a n d c c n v i n c e y o u r s e l f b y o t h e r

~ ex am pl es t h a t - o n p i c t u r e s - c o m p o s i t i o n i s a n a s s o c i a t i v e a n d c c ~ sa u-

t a t i v e o p e r a t i o n .

B e f o r e d e f i n i n g c c ~ p o s i t i c n o f b e h a v i o u rs , l e t u s l o o k a t t w o o t h e r

o p e r a t i o n s o n p i c t u r e s .

F o r e a c h ~ , w e d e f i n e a p o s t f i x e d r e s t r i c t i o n o p e r a t i o n \ 5,

w h i c h o n p i c t u r e s j u s t m e a n s h i d e t h e p o r t s l a b e l l e d ~ o r ~ , i .e .

i t d r o p s t h e l a b e l s

R J S ) \ ~

I i

a n d ~ f r c m p i c t u r e s , t h u s r e d u c i n g t h e i r s o r t .

( = R N S )

((RIS) IT)\8\Y\~

s T i t

\ _


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23

E x e r c i s e 2 . 2

W h i c h o f t h e f o l l o w i ng a r e i d e n t i c a l a s p i c t u r e s?

(i) (( RI S) IT)\B\\~ (v) (RI (Sl T)\ 6)\ Bky

(ii ) ((RI S)kS IT)k y\: (vi) (RI (SlT)\ x)\B\~

(iii ) ((RI S) yIT) \B\~ (vii) ((RIT) 6 IS) \B\y

( i v ) ( ( RX y IS ) I T )X B \6 ( v i i i ) ( ( R I T ) k 6 1 S \ 6 ) \ B \ y

N o t e: \ a b i n d s t i g h t e r t h a n I , s o t h a t U : V ~ m e a n s

u v ~ ) .

B e s id e s it s u s e w i t h c c ~ i t i c ~ , t h e r e s t ri c t i on o pe r a t i on b y it s el f

c o r r e s p o n d s t o a s i n p l e , r a t h e r c o n c re t e , a c t i o n : - t h a t o f h i d i n g o r

' i n t e rn a l \ s i ng ' c e r t a i n p o r t s o f a m a c h i n e . C o m p a r e t h e r e ~ a r k s o n h i d i n g

t h e b - b u t t o n s o f t w o m a c h i n e s , a t t h e e n d o f 1. 2.

N o t e t h a t w e c a n d e f i n e S l iT , w h e r e S : L a n d T : M , b y

Sll T = ( S I T ) \ ~ I . . . \ ~ n w h e r e { ~ l , . . O , a n } = n a m e s ( L ~ ) .

W e s h a l l h e n c e f o r t h a b a n d o n t h e u s e o f u p p e r c a s e l e t t e r s f o r m a c h i n e s .

T h e r e i s a f i n e d i s t i n c t i o n b e t w e e n t h e i d e a s o f (i) a m a c h i n e w h i c h m a y

m o v e t h r o u g h s t a t e s b u t ~ + _h e s a m e m a c h i n e ( a p h y s i c a l n ot io n) a n d

(ii) a m a c h i n e - s t a t e p a i r , i . e . a w a y o f s p e c i f y i n g a b e h a v i o u r w i t h a

d e f i n i t e s t a r t ( a m o r e m t h e m a t i c a l n o t i on , e x e m p l i f i e d b y t h e n o n n a l

d e f i n i t i o n o f F i n i t e - s t a t e A c c e p t o r a s c o n s i s t i n g o f a s t a t e s et , a t r a n s i -

t i o n r e l a t i o n , a s e t o f a c c e p t i n g s t a t e s a n d a s t a r t st a t e ) o O u r l o w e r

c a s e l e t t er s c o r r e s p o n d t o t he l a t t e r i d e a - i n d ee d , t h e y d e n o te t h e s p e c i -

f i e d b e h a v i o u r s ( h e re a s ST s ) , a n d i t i s t h e s e w h i c h a r e t h e d c ~ a i n o f o u r

a l g e b r a ; w e s h a l l s o o n s e e w h a t r l s e t c . m e a n a s b e h a v i o u r s o

W e a l s o h a v e a n o t h e r u s e f o r u p p e r c a s e l et t er s ; w e s a y t h a t S : L + M

( w h e re L , M a r e s o r ts ) i s a r e a b e l l i n g f r c m L t o M i f

( i) i t i s a b i j e c t i o n ;

(ii) i t r e s p e c t s c c ~ l e m e n t s

( i . e o S ( ~) = S ( ~) f o r ~ , ~ e L ) o

W e d e f i n e t h e p o s t f i x e d r e l a b e l l i n ~ o p e r a t i o n I S ], o v e r ( p i c tu r e s o f)

m a c h i n e s o f s o r t L , a s s i m p l y r e p l a c i n g e a c h l a b e l l e L b y S ( 1 ).

T h u s f o r r , t a s a b o v e w e h a v e

~ I t = I

-

Y


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24

a n d S : { f l, y ,8 , ~ } { S t y , e , [ } , g i v e n b y

s ~ ) = 6 , s ~ ) = y , s ~ ) = ~ , s ~ ) = s

i s a r e l a be l l i n g; w e t h e n h a v e

r ) [ ] =

W e s h a l l u s e c o n v e n i e n t a b b r e v i a t i o n s i n w r i t i n g r e l a b e l l i ng s e x p l i ci t l y .

T h u s

i i / ~ I , . . . , n / ~ n o r 1 1 1 2 i n / ~ l S 2 ~ n

( w he r e e l , . O . , a n a r e d i s t i n c t n a n e s , a n d I i , .. . , I n a r e l a b e l s w i t h

d i s t i n c t n a re s ) s t a n d s f o r t h e r e l a b e l l i n g S : L M g i v e n b y

( i) S ( ~ i ) = I i f e i ~ L

( i i ) S ( ~ i) = I i f m i ~ L

( i i i ) S( 1 ) = I i f n a m e ( 1 ) ~ { e l , . . . , a n }

p r o v i d e d t h a t t h e f u nc t i o n s o d e f i n e d i s a r e l a b e ll i n g . S o i n p l a c e o f

( r l t ) [ ~ a b o v e, w e w r i t e

( r lt ) [~ / 8 , ~ / 8 ] o r ( r ] ) [ ~ e / 8 ~ ] .

W h e n w e s e e t h e l aw s o f th e F l o w A l g e b r a ( la ws f o r t he C ~ i t i o n ,

R e s t r i c t i o n a n d R e l a b e l l i n g o p e ra t i on s ) i n T h e o r e m 5 . 5, w e s h a l l s e e t h a t

r e l a b e l l i n g d i s t r i b u t e s o v e r o c~ i0 o si ti on , s o t h a t w e h a v e

( r I t ) [ ~ / ~ , ~ / 8 ] = r [ ~ / B , e / 6 ] ] t [ ~ / ~ , c / 8 ]

( as y o u c a n r e a d i l y c he ck ) - e v e n t h o u g h i n s t r i c t f o z m a l i t y ~ / B , e / 8

s t a n d s f o r a d i f f e r e n t m e l a b e l l i n g i n e a c h c a s e, b e c a u s e r , t a n d r ] t

p o s s e s s d i f f e r e n t s o r t s .

2 . 3 E x t e m f ~ n @ t h e A l g e b r a o f S y n c h r o n i z a t i o n T r e e s

W e m u s t n o w a d d o u r t h r e e n e w o p e r a ti o n s t o t h e a l g e b r a o f S T s , u s i n g

i n t u i t i o n a b o u t t h e o p e r a t i o n a l m e a n i n g o f t h e s e tr e e s. I n f u t u r e w e

c o n t i n u e t o u s e I t o r a n g e o v e r A , a n d u s e ~ ,~ t o r a n g e o v e r A u{ T} .


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5

Cc~oo sition I : ST x ST STLu

C o n s i d e r t w o S T s

t = ~ u =

F o r t h e i r ~ i t e , f o u r a c t io n s a r e p os si bl e, t l u

(because t does), so one branc h of tlu will be

This branc h represents independ ent actio n by one ccrsponent, and simila r

b r a n c h e s e x i s t f o r a 8 - ex p er i me n t o n t a n d a n m - e x p e r i m e n t o n u . N o n e

of these three branche s represents interaction betwe en t and u; but

there is a possible interaction, sinoe u s m-offer ccmpl~ments t s

e-offer. Since this actio n is internal (no t observable) we use T and

r e p r e s e n t i t i n t h e c o m p o s i t e t r e e b y a b r a n c h

P u t t i n g a l l t h e b r a n c h e s t o g e t h e r y i e l d s

a d m i t s a n e - e x p e r i m e n t

N o w c c ~ p o s i t i o n o f t a n d u h a s b e e n d e f i n e d i n t e zm s o f c c ~ p o s i t i o n

of their sons; clearly this amounts to a recursive definit ion of 1

M o r e p r e ci s el y , s i n c e e v e r y t r e e m a y b e w r i t t e n i n t h e f o z m

t = ~ ~iti , ~i~Au{~}

l_


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26

E x e r c i s e 2 . 3 ( C o n s i d er o n l y f i n i t e S T s ) .

(i) P r o v e b y ~ d u c t i o n o n t h e d e p t h o f t t h a t t l N I L = t .

( i i) W o r k o u t t l u f o r t = a / ~ 8 a n d u = I ~; c h o o s e s a n e

o t h e r e x a m p l e s .

( i i i ) P r o v e b y i n d u c t i o n o n t h e s tw n o f t h e d e p t h s o f t r e e s t h a t

tl u = ul t an d t I(ulv) = (tlu) Iv.

W e s h o u l d c r i t i c i z e t w o a s p e c t s ( at l e a s ~ ) o f o u r d e f i n it i o n .

C o n si d e r in g o u r f i r st e x a m pl e o f S T ~ s i t i o n , i t c a n w e l l b e a r g ue d

t h a t t h e f o r m w e g a v e f o r t l u f a i ls t o r e p r e s e n t t h e p o s s i b l e c o n -

c u r r e n t a c t i v i t y o f t a n d u - f o r e x a m p l e , w e m a y t h i n k t h a t a

g - e xp e r im e nt o n t c a n b e p e r f or m e d s ~ t a n e o u s l y w i t h a n W - e x pe r i m en t

o n u , w h i l e ( l o o k in g a t y o u r r e s u l t f o r E x e r c i s e 2 .3 (i i) a ls o) t h e S T

f o r t l u m e r e l y i n d i c at e s t h a t t h e t w o e x p e r i m e n t s m a y b e p e r f o r m e d i n

e i t h e r o r d e r. I n d e e d, S T s i n n o w a y r e p r e s e n t t r u e c o n c u r r e n c y .

T w o n o t c o m p l e t e l y c o n v i n c i n g d e f e n c e s c a n b e g i v e n . F i r s t , S T s

a r e s i mp l e , a n d t r a c t a b i l i t y i n a m o d e l h a s g r e a t a d v a n t a g e s ; s e c o n d ,

i n s o f a r a s w e w i s h a ' b e h a v io u r - o b j ec t ' t o t e l l u s h o w a s y s t e m m a y

a p p e a r t o a n o b s e r v e r w h o i s o n l y c a p a b l e o f o n e e x p e r i m e n t a t a t i m e ,

w e f i n d i t p o s s i b l e t o i g n o r e t r u e c o n c u r r e n c y . Y o u a r e u r g e d t o

c o n s i d e r t h i s q u e s t i o n i n g r e a t e r d e p t h .

T h e s e c o n d a s p e c t f o r c r i t i c i s ~ i s t h e i n t ro d u c t i o n o f T t o r ep r e s e n t

s u c c e s s f u l ' m u t ua l o b s e r v a t i o n s ' . I f w e h a d n o n e e d f o r i t i n d e f i n i n g I,

w e c o u l d l e a v e i t o u t o f o u r t h e o r y a l t o ge t h e r.

A g a i n , t h e r e a r e t w o d e f e n c e s , b u t t h i s t i m e c o n v i n c i n g o n e s . F i r s t ,

c o n s i d e r r e p l a c i n g t he t h i r d t e r m i n t h e r e c u r s i v e d e f i n i t i o n o f t l u -

n a m e l y t h e t e a m [ T ( t i l u j) - b y j u s t [ ( t i I ) ;

i n t u i t i v e l y , a n i n t e r n a l a c t i o n j u s t v a n i s h e s . I t t u r n s o u t t h a t 1

i s n o l o n g e r a n a s s o c i a ti v e o p e r at i o n , w h i c h c o n f li c t s s t r o n g l y w i t h o u r

a s s u m p t i o n t h a t t h e j o i n t b e h a v i o u r o f t / kr ee a g e n t s s h o u l d i n n o w a y

d e p e n d u p o n t h e o r d e r i n w h i c h w e w i r e t h e m t o g e t h e r b e f o r e t h e y d o a n y -

t h i n g '

E x e r c i s e 2 . 4 W i t h t h i s n e w d e f i n i t i o n w o r k o u t t I u l v ) a n d ( t l u ) I v

f o r t = e , u = a , v = 8 t o j u s t i f y t h e a b o v e a s s e r t i o n .


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27

T h e s e c o n d d e f e n c e i s t h a t w e m u s t s c ~ e h o w e x p r e s s , i n t h e S T t lu )\ ~

w h e n t = a A B , u = I ~, t h e p o s s i b i l i t y t h a t c c m a u n i c a t i o n b e t w e e n t

a n d u c a n p r e v e n t a n y B - e x p e r i m en t .

E x e r c i s e 2. 5 U n d e r t h e n o n a a l d e f i n i t i o n o f

w o r k o u t t h a t

t l u ) \ ~ = T ~ B

i n t h i s c a s e .

I, an d of \~ see below) ,

T h i s S T d o e s i n d e e d r e p r e s e n t p o s s i b l e p r e v e n t i o n o f a 8 - e x p e r i m en t ,

a n d u n l e s s w e l e a v e S T s a n d d e r i v e d m o d e l s ) a l t o g e t h e r i t i s h a r d t o

s ee h o w s uc h d ea dl oc k p h e ~ c an h e r ep re se nt ed w i th o ut z.

R e s t r i c t i o n \ ~ : S T S T L _ { ~ , ~ } ~ cA )

W e w i s h t o d e n y a l l ~ - a n d m - e x p e r i m e n t s , s o t h a t t \ ~ i s f o r m e d

b y p r u n i n g a w a y a l l b r a n c h e s a n d s lA b- br an ch es l a b e l l e d ~ o r e . C o n s i d e r i n g

t =

a g a i n , w e s e e t h a t

tlu) \~ =

M o r e f o r m a l l y , f o r t = ~ . i t i w e h a v e

D e f i n i t i o n t \ ~ = ~ - ~ i t i ~ )

A n o b v i o u s a l t e r n a t i v e t o t he r e s t r i c t i on o p e r a t i o n w o u l d b e t o d e f i n e

\ l f o r e a c h m e ~ b e r i o f A b y

t\l ui~ l~i ti\l) ;

i n o t h e r w o r d s , w e m i g h t c h o o s e t o r e s t r i c t n a ~ e s a n d o o - n a m e s i n d e pe n d e nt l y ,

i n s t e a d o f b o t h a t o n c e . ~ n i s w o u l d , o f c o u r s e , h a v e a c o r r e s p o n d i n g l y


33/176

8

d i f f e r e n t e f f e c t o n p i c t ur e s . T h e r e a s o n f o r o u r c h o i c e i s i n f a c t to

d o w i t h t h e a l g e b r a o f p i c t u r e s ( F l o w A l g e b r a ) u n d e r I, \ e a n d [ S] ;

i t h a s a p a r t i c u l a r l y s i m p l e a l g e b r a i c t h e o r y [ ~M , M i l 2 ] , w h i c h w e

h a v e n o t f o u n d f o r t h e s u g g e s t e d a l t e r n a t i v e .

R e a b e l l i n I S ]: S T + S T ( S : L M a r e l a b e l l i n g )

T h i s o p e r a t i o n i s a s s i m p l e o n S T s a s i t i s o n p i c t u r e s ; i t j u s t

a p p l i e s t h e r e l a b e l l i n g S t o a l l l a b e l s i n t h e t r e e . M o r e f o r m a l l y ,

f o r t = _ u i t i w e h a v e

D e f i n i t i o n t [ S ] = _ ~S (~ )

(ti[S])

1

w h e r e w e n o w a d o p t t h e c o n v e n t i o n t h a t S ( T ) = f o r a n y r e l a b e l l i n g S .

A n i m F ~ r t a n t ( t h o ug h n o t t h e o nl y) u s e o f r e l a b e l l i n g i s i n c a s e s

w h e r e w e h a v e s e v e r a l i n s t a n c e s o f a s i n g l e a g e n t r i n a s y s t e m, b u t

e a c h w i t h d i f f e r e n t l a b e l l i n g, s o t h a t u n d e r e c ~ p o s i t i o n t h e y a r e p r o p e r l y

l i n k e d. W e h a v e o n l y t o d e f i n e s e v e r a l ' o p i e s '

r i = r[S i]

o f t h e g e n e r i c a g e n t r , a n d t h e n c c ~ p o s e t h e r .

O n e m i g h t h a v e a l l o w e d m o r e g e n e r a l r e l a b e l li n g s, u s i n g m a n y - o n e

f u n c t i o n s o v e r A ( so t h a t d i f f e r e n t l y l a b e l l e d p o r t s c c m e t o b e a r t h e

s a n e l a b el ) o r e v e n r e l a t i o n s i n p l a c e o f f u n c t i o n s ( so t h a t o n e p o r t

c o u l d ' sp li t' i n t o t w o d i f f e r e n t l y l a b e l l e d p o r t s ) . S u f f i c e i t t o s a y

t h a t th i s c r e a t e s p r o b l e m s i n t h e a x i c m a t i z a t i o n o f F l o w A l g e b r a . T h e

p r e s e n t c h o i c e a l l o w s p l e n t y o f s c o p e .

2 . 4 A s i m p l e e xa r ~o le : b i n a r y S ~ n a p h o r e s

A b i n a r y s e m a p h o r e s , o f s o r t { ~ ,~ } , m a y b e p i c t u r e d

[

r

TO g a i n t h e s ~ ,a t :t ~ o re ( D i j k s t r a s P O p e r a t i o n ) w e m u s t p e r f o r m a

e x t: ex i m e n t; w e r e l e a s e i t t h e V o p e r a t i o n ) b y a ~ - e x ~ r ~ m e n t .

s ~ s

1[

C l e a r l y


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9

e x p r e s s e s t h e a p p r o p r i a t e b e h a v i o u r ( a l o n g t h i n S T ') . I m a g i n e a

g e n e r i c a g e n t p , w h o s e c r i t i c a l s e c t i o n w e r e p r e s e n t b y a s e q u e n c e

of atomic actions (experiments upo n a resource, say), and who se

n o n - c r i % i c a l s e c t i o n w e i g no r e:

p = ~ 6 ~ p .

W e w i s h t o p l a c e s e v e r a l i n st a nc e s o f p

P i = P [ S i ] = ~ i 6 i ~ P i ( w he r e S i = ~ i 8i / ~6 )

i n ~ c a t i o n w i t h s , a n d d e r i v e t h e o c m p os i t e S T. C o n s i d e r j u s t

t w o c o pi e s o f p ( i = 1 , 2 ) a n d f o r m

q = (pllP21S)\~\

w h i c h m a y b e p i c t u r e d a s s h o wn :

~1

a 2

p s p

61 62

q

We e a s i l y d e r i v e a n e q u a t i o n f o r t h e a z ~ p o s i t e ST q u s i n g th e E x p an s io n

Theorem - g i v e n i n 2 .5 - r e p e a t e d l y . Y ou s h o u l d r e a d t h a t s e c t i o n w i t h

r e f e r e n c e t o t h e e x p a n s io n w h i ah f o l l o w s :

q = ( ~a 8 1 ~p l I ~ 2 6 2 P 2 ~ S ) ~ \

= T ~ I s ~ p l i P 2 1 S s ) ~ \ ) + T p ~ I ~ 2 6 2 ~ P 2 i s )

~ ~

= re161( (#Pl IP2 I~s) ~\~) + ~ 2 8 2 ((Pll#P2 lSs) ~\~)

= T~161T ( Pl IP2 Is) ~\~) + T~262~ ( Pl I 2 Is) \~\# )

= ~e181 Tq + T~262T

S O q i s t h e S T g i v e n r e c u r s i v e l y b y

~I ~2


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30

a n d e x a c t l y e x p r e s s e s t h e f a c t t h a t t h e c r i t i c a l s e c t i o n s o f P l a n d

P 2 c a n n e v e r o v e r l a p i n t i m e , i . e . a s e q u e n c e l i k e ~ I ~ 2 B I . . . i s n o t

p o s s i b l e .

I n f a ct , a n n - b o u n d e d s ~ m a p h o r e ( n_ > i) c a n b e c o n s t r u c t e d a s

s = s l s l I s

n

n t i m e s

t h i s i s a n e x a n p l e o f c c ~I oc si ti on w h i c h e f f e c t s n o l i n k a g e , b u t w i l l y i e l d

a m u l t i - w a y l i n k a ge w i t h u s e r a g e n ts .

T h e 2 - b o u n d e d s e m a p h o r e s ,

B

c~3

P B 3 P

w i t h 3 u s er s , c a n b e p i c t u r e d

i

D i a g r a m f o r ( p i I p 2 1 P 3 1 S 2 ) \ ~ \ ~

s 2

( s2 s b o r d e r , a n d i t s t w o c o l l e c t o r n o d e s , a r e f ic t i t i o u s ; t h e y a r e j u s t

u s e d h e r e t o a v o i d d r a w i n g 1 2 l i n k s i n t h e p i c t u r e ) .

E x e r c i s e 2 . 6 A s p r a c t i c e i n u s i n g t h e E x p a n s i o n T h e o r ~ n , d e v e l o p t h e

e x p r e s s i o n q = ( p l [ p 2 1 P 3 1S l S ) \ ~ \ ~ , a n d d r a w p a r t o f t h e S T t o o o n v i n o e

y o u r s e l f t h a t a t m o s t t w o c r i t i c a l s e c t i o n s c a n b e s i m u l t a n e o u s l y a c t i v e .

C a n y o u e v e n d e r i ve a s e t o f m u t u a l l y r e c u r s i v e b e h a v i o u r e q u a ti o n s , f o r

w h i c h q i s t h e s o l u t i o n ? I t s a b i t l e n g t h y, b u t p o s s i b l e . T h e

d e v e l o p m e n t i s s h o r t e r i f y o u t a k e ~ i = ~ 2 = ~ 3 =~ , B 1 = ~ 2 = B3 = 8 ; i .e . d e a l

w i t h ( p l p l p l sl s ) \ ~ \ ~ i n s t e a d; t h e n t h e S T w i l l n o t d i s t i n g u i s h t h e

c r i t i c a l s e c t i o n s o f e a c h c o p y o f p , b u t y o u s h o u l d b e a b l e t o s h o w

t h a t a t a n y p o i n t i n t i m e t h e e x c e s s o f e s o v e r B s p e r f o r m e d

l i e s i n t h e r a n g e [ 0 ,2 3 .


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2 . 5 T he S T ~ p a n s i o n T h e o r e m

W e co n si d er t re e s e x p r e s s e d i n t h e f o rm

t = [ ~ i t i .

l ~ i ~ n

F o r a s e t { ~ l , . . . , e k } = A o f na m e s , w e a b b r e v i a t e

\ ~ 1 \ ~ 2 . . . \ ~ k b y \ A .

T h e o r e m 2 . 1 ( T h e E x p a n s i o n T he or em _)

L e t t = ( t

I t 1 . . . . I t

\ A , w h e r e e a c h t i s a s ~ n a s a b o v e .

T h e n t = ~ { ~ ( ( t l l ' I t l l ' I t m ) \ A ) ; l _ < i _ < m , ~t.'l a s t l ~m a nd o f

t i ,

n r~ (V) ~A}

+ ~ { T( (t lI .. .I t .' l . .. It .' l . . . I tm ) \ A ); 1 < i < j _ < m ,

i 3

It ' a su~Inand of ti, ~tj' a sun, and of tj }

P r o o f O m i t t e d ; i t u s e s p r o p e r t i e s o f t h e F l o w o p e r a t i o n s $, \ ~ a n d

[ S ], a n d c a n b e d o n e b y i n d u c t i o n o n m .

T h e t h e o r e ~ s t a t e s t h a t e a c h b r a n c h o f t c o r r e s p o n ds e i t h e r t o a n

u n r e s t r i c t e d a c t i o n o f s c ~ e t i , o r t o a n i n t e r n a l c c ~ m u n i c a t i o n b e t w e e n

t . a n d t . ( i < j ). F o r e x a m p l e c o n s i d e r

l 3

( ( a t + ~ t ) I ( ~ u + y u ) I ~ v + ~ v ) ) \ ~ \ ~ ;

t h e t h e o r e m g i v e s u s

( u n r e s t r i c t e d a c t i o n s )

+ ~ ( ( ( ~ t + s t ) i ( ~ u + ~ u ) I v ) \ ~ \ ~ )

+ T ( t l u ]

robin milner a calculus of communicating systems 1980

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