I a m greatly h o n o r e d to receive this award, b e a r i n g the n a m e of Pdan Turing. Perhaps Turing would be pleased that it should go to someone e d u c a t e d at his old college, K i n g ' s College at C a m b r i d g e . W h i l e there in 19561 wrote m y first c o m p u t e r p r o g r a m ; it was on the E D S A C . O f course E D S A C m a d e history. But I a m a s h a m e d to say it did not lure me into c o m p u t i n g , a n d I ignored c o m p u t e r s for four years. I n 1960 I thought that computers might be m o r e peaceful to h a n d l e t h a n s c h o o l c h i l d r e n - - I was then a t e a c h e r - - s o I a p p l i e d for a j o b at Ferranti in L o n d o n , at the time of Pegasus. I was asked at the interview w h e t h e r I would like to devote m y life to computers. This d a u n t i n g notion had never crossed m y mind. Well, here I a m still, and I have h a d the lucky chance to grow alongside com- p u t e r science.
This award gives an u n u s u a l opportunity, a n d I hope a license, to reflect on a line of research from a personal point of view. I thought I should seize the opportunity, because a m o n g m y interests there is one t h r e a d which has p r e o c c u p i e d m e for 20 years. D e s c r i b i n g this kind of experience can surely yield insight, provided one r e m e m b e r s that it is a personal thread; science is woven from m a n y such threads and is all the stronger when each t h r e a d is h a r d to trace in the finished fabric.
T h e t h r e a d which I want to pick u p is the semantic basis of concur- rent computation. I shall begin by explaining how I c a m e to see that concurrency requires a fresh ap- proach, not m e r e l y an extension of the repertoire of entities and con-
structions which explain sequential computing. T h e n I shall talk about m y efforts to find basic construc- tions for concurrency, guided by experience with sequential s e m a n - tics. T h i s is the work which led to a Calculus for C o m m u n i c a t i n g Systems (CCS). At that point I shall briefly discuss the extent to which these constructions m a y be u n d e r s t o o d mathematically, in the way that sequential c o m p u t i n g m a y be u n d e r s t o o d in terms of functions.
Finally, I shall outline a new basic calculus for concurrency; it gives p r o m i n e n c e to the old i d e a of naming or reference, which has hitherto been treated as a second-class citizen by theories of computing.
I m a k e a disclaimer. I reject the idea that there can be a u n i q u e con- ceptual model, or one preferred for- malism, for all aspects of s o m e t h i n g as large as concurrent c o m p u t a t i o n , which is in a sense the whole of o u r s u b j e c t - - c o n t a i n i n g sequential com- p u t i n g as a well-behaved special area.
We need m a n y levels of explanation:
m a n y different languages, calculi, a n d theories for the different spe- cialisms. T h e applications are various: the flow of i n f o r m a t i o n in an insurance company, network com- munications, the real-time com- m u n i c a t i o n a m o n g in-flight control computers, c o n c u r r e n c y control in a database, the behavior of parallel object-oriented p r o g r a m s , the semantic analysis of variables in concurrent logic p r o g r a m m i n g . We surely do not expect the terms of discussion and analysis to be the same for all of these.
But there is a c o m p l e m e n t a r y claim to make, a n d it is this: C o m p u t e r
scientists, as all scientists, seek a com- m o n framework in which to link and to organize m a n y levels of expla- nation; moreover, this c o m m o n f r a m e w o r k m u s t be semantic, since o u r explanations (including pro- grams) are typically in formal l a n g u a g e - - a n d often in a m i x t u r e of formalisms, to deal with the large heterogeneous systems which are o u r business. For the m u c h smaller world of sequential c o m p u t a t i o n , a com- m o n semantic f r a m e w o r k is f o u n d e d on the central notion of a mathematical function a n d is formally expressed
in a functional c a l c u l u s - - o f which A l o n z o C h u r c h ' s l - c a l c u l u s is the famous prototype. F u n c t i o n s are an essential ingredient of the air we breathe, so to speak, when we discuss the semantics of sequential p r o g r a m - ming. But for c o n c u r r e n t p r o g r a m - m i n g and interactive systems in general, we have n o t h i n g c o m p a r a b l e .
So where do we find the semantic ingredients for concurrency, or how can we distill them? It is an a m b i t i o u s goal because, as I said earlier, con- currency is ubiquitous. I believe that the right ideas to explain concurrent c o m p u t i n g will only come from a dialectic between models from logic and m a t h e m a t i c s a n d a p r o p e r distil- lation of a practical experience.
I conduct a piece of the dialectic.
I try to reconcile the a n t i t h e s i s - - f o r it does seem to be o n e - - b e t w e e n two things: on the one hand, the p u r i t y a n d simplicity exemplified by the calculus of functions and, on the other hand, some very concrete ideas a b o u t c o n c u r r e n c y a n d interaction suggested by p r o g r a m m i n g a n d the realities of c o m m u n i c a t i o n .
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Entitles
T h r o u g h the seventies, I became convinced that a t h e o r y o f concur- rency and interaction requires a new conceptual framework, not j u s t a re- f i n e m e n t o f what we find n a t u r a l for sequential c o m p u t i n g .
Often, the experiences which give one conviction are not p l a n n e d a n d not p r o f o u n d . But I want to recall one o f mine, because it serves the t h e m e h e r e in m o r e than one way.
It arose when I was trying to e x t e n d the S c o t t - S t r a c h e y a p p r o a c h to p r o g r a m m i n g - l a n g u a g e semantics, which deals beautifully with the most sophisticated sequential languages, to h a n d l e c o n c u r r e n t languages as well. T h e a t t e m p t had to be made, a n d I was optimistic a b o u t success.
I n that a p p r o a c h a sequential p r o - gram, assuming no i n t e r m e d i a t e i n p u t / o u t p u t , is perfectly r e p r e - sented by a function f r o m m e m o r i e s to memories. (I use the t e r m "mem- ory" to m e a n a m e m o r y state, con- taining values for all the p r o g r a m variables.) But Dana Scott d e v e l o p e d a t h e o r y o f domains--partially or- d e r e d sets o f a special n a t u r e - - w h i c h provides m e a n i n g for the A-calculus, the p r i m e functional calculus. So in the S c o t t - S t r a c h e y a p p r o a c h , the m e a n i n g o f an imperative p r o g r a m lies in the d o m a i n given by the equa- tion
P r o g r a m Meanings =
Memories ~ Memories.
Everything works well with this do- main, a n d the reason is: that to every syntactic construction in any sequen- tial language, t h e r e c o r r e s p o n d s an abstract o p e r a t i o n which builds the m e a n i n g o f a composite p r o g r a m f r o m the meanings o f its c o m p o n e n t p r o g r a m s . T h a t is, the semantics is
compositional--an essential p r o p e r t y . Now, one o f the things that con- c u r r e n c y introduces is n o n d e t e r - minism. (Of course you can also have n o n d e t e r m i n i s m without concur- rency, but in my o p i n i o n it is concur- rency which inflicts n o n d e t e r m i n i s m on you.) Plotkin dealt with n o n d e t e r - minism by means o f his power- d o m a i n construction, a tour de force o f d o m a i n theory. It provides, for any suitable d o m a i n D, the p o w e r d o m a i n T(D) whose elements are subsets o f D.
So with n o n d e t e r m i n i s m in m i n d we can r e d e f i n e the meanings o f pro- g r a m s as
P r o g r a m Meanings =
Memories --> P(Memories) - - e s s e n t i a l l y relations over m e m o - ries. This semantics is perfectly com- positional for the kind o f n o n d e t e r - ministic l a n g u a g e which you get by a d d i n g " d o n ' t care" b r a n c h i n g to a sequential language.
But c o n c u r r e n c y has a shock in store; the compositionality /s lost if you can combine s u b p r o g r a m s to r u n in parallel, because they can i n t e r f e r e with one another. To be precise, t h e r e are p r o g r a m s P1 a n d P2 which have the same relational meaning, but which behave differently when each runs in parallel with a third pro- g r a m Q. A simple e x a m p l e is this:
P r o g r a m P ~ : x : = 1 ; x : = x + 1 P r o g r a m P2 : x := 2
I n the absence o f interference, P l a n d P2 both t r a n s f o r m the initial m e m o r y by replacing the value o f x by 2, so they have the same meaning.
But if you take the p r o g r a m P r o g r a m Q : x := 3
a n d r u n it in parallel with P l a n d P2 in turn:
P r o g r a m R1 : P1 p a r Q P r o g r a m R2 : P2 p a r Q
then the p r o g r a m s Rl a n d Rz have d i f f e r e n t meaning. (Even if an as- s i g n m e n t statement is executed indi- visibly, R1 can end u p with x equal to 2, 3, o r 4, while R2 can only e n d u p with x equal to 2 o r 3.) So a composi- tional semantics must be m o r e re- fined; it has to take account o f the way that a p r o g r a m interacts with the m e m o r y .
This p h e n o m e n o n is h a r d l y a sur-
prise, with hindsight. But if we cannot use functions o r relations over m e m o r i e s to i n t e r p r e t concur- r e n t p r o g r a m s , then what can we use? Well, one can quite naturally give the relational m e a n i n g a finer granularity, so that it records every step which a p r o g r a m makes f r o m one m e m o r y access to the n e x t - - a n d this can be d o n e without leaving d o m a i n theory. But the p h e n o m e - non t a u g h t m e a m o r e radical lesson:
Once the m e m o r y is no l o n g e r at the behest o f a single master, t h e n the master-to-slave (or: function-to- value) view o f the p r o g r a m - t o - m e m o r y relationship becomes a bit o f a fiction. A n old p r o v e r b states:
H e who serves two masters serves none. It is better to d e v e l o p a general m o d e l o f interactive systems in which the p r o g r a m - t o - m e m o r y interaction is j u s t a special case o f interaction a m o n g peers.
It helps to visualize. F i g u r e l shows the s h a r e d - m e m o r y model, very informally. It j u s t r e p r e s e n t s the active/passive distinction between c o m p o n e n t s , by using d i f f e r e n t l y s h a p e d nodes. (I shall consistently use squares for active processes in my pictures a n d circles for passive things.) O f course, in g e n e r a l the p r o g r a m s use several variables, all s t o r e d in M.
To r e m o v e the active/passive dis- tinction, we shall elevate M to the sta- tus o f a process; then we r e g a r d p r o - g r a m variables x, y , . . . as the names o f channels of interaction between pro- g r a m a n d m e m o r y , as shown in Fig- ure 2.
Now, thinking m o r e generally, let us use m e m o r i e s to illustrate the idea that p r o c e s s e s - - o f any k i n d - - c a n be c o m p o s e d to m a k e l a r g e r ones.
I n the sequential world one can maintain the convenient fiction that a m e m o r y is monolithic; but this is quite unrealistic in c o n c u r r e n t p r o - g r a m m i n g , because d i f f e r e n t parts o f m e m o r y may be accessed simultane- ously. So we go one step f u r t h e r , as shown in F i g u r e 3, a n d r e g a r d each cell o f m e m o r y as a process, X say, linked to one o r m o r e p r o g r a m s (themselves processes) by an a p p r o - priately n a m e d channel.
Software engineers may well resist this h o m o g e n e o u s t r e a t m e n t a n d firmly a d h e r e to the s h a r e d - m e m o r y
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model; it is i m p o r t a n t for them, be- cause it admits a m e t h o d o l o g y which can help in writing correct p r o g r a m s . Theoreticians may reply that to tol- erate two kinds o f entity in a basic model, where one kind will do, is sci- entific a n a t h e m a ; they may also point out that the active/passive distinction o f the s h a r e d - m e m o r y m o d e l does not easily a c c o m m o d a t e hybrids, such as a database which reorganizes itself while you are not using it. A n d both these attitudes are right.
So let us recall the need for m a n y levels o f explanation. William o f Occam o p p o s e d the proliferation o f entities, but only when carried be- y o n d what is n e e d e d --procter neces- sitatem! C o m p u t e r systems engineers have a pressing n e e d for a rich ontol- ogy; they welcome the ability to use d i f f e r e n t concepts and models for d i f f e r e n t purposes. For example, the s h a r e d - m e m o r y m o d e l is a n a t u r a l p a r t o f their r e p e r t o i r e . But com- p u t e r scientists must also look for s o m e t h i n g basic which underlies the various models; they are interested not only in individual designs and systems, but also in a unified t h e o r y o f their ingredients. T o attain unity in a basic model o f concurrency, all i n t e r a c t i o n s - - a n d t h e r e f o r e all inter- a c t o r s - - m u s t be treated alike; that is why I have called this work "Ele- ments o f Interaction."
To avoid the impression that the only interactors I am thinking o f are p r o g r a m s , o r memories, o r c o m p u t e r systems, I show in F i g u r e 4 a mobile t e l e p h o n e network in which the channels are r a d i o channels. T h e c o m m u n i c a t i o n protocol allows a car to switch channels to whichever base station is nearest, the whole system being m o n i t o r e d a n d controlled cen- trally. Now, we want o u r construc- tions to describe such systems per- fectly well, at a discrete level; the elements o f interaction must not be specific to c o m p u t e r systems.
Much o f what I have been saying was a l r e a d y well u n d e r s t o o d in the sixties by C a r l - A d a m Petri, who pio- n e e r e d the scientific m o d e l i n g o f dis- crete c o n c u r r e n t systems. Petri's work has a secure place at the r o o t o f c o n c u r r e n c y theory. H e declared the aim that his t h e o r y o f nets s h o u l d - - at its lowest l e v e l s - - s e r v e impartially as a m o d e l o f the physical world a n d
as a m o d e l o f computation. Already, for him, a m e m o r y register and a p r o g r a m are m o d e l e d by the same kind of object--namely a n e t - - a n d this breaks down the active/passive di- chotomy. T h e conceptual f r a m e w o r k o f net t h e o r y is as spare as one can imagine. This has i n d e e d p a i d o f f in clarity a n d d e p t h , both for the analy- sis o f individual systems a n d for the classification o f systems.
Static C o n s t r u c t i o n s
Besides calling the question the ac- tive/passive d i c h o t o m y for the entities o f which a system is composed, con- currency d e m a n d s a fresh a p p r o a c h
F i g u r e I . T h e s h a r e d m e m o r y m o d e l
F i g u r e 2. M e m o r y a s a n
Interactive p r o c e s s F i g u r e 3. M e m o r y a s a
distributed p r o c e s s F i g u r e 4. A m o b i l e
telephone network
D Y x rq J
D x ... .... i
M
Fq J
STATION STATION . . . I STATION I "'"
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l e m e n t s o f I n t e r a c t i o n
in terms o f its primitive constructions.
W h a t I always w a n t e d to advance, to c o m p l e m e n t Petri net theory, is the synthetic o r compositional view o f systems which is familiar f r o m p r o - g r a m m i n g . T h i s is essentially an al- gebraic view, for a l g e b r a is about constructions a n d their meaning. F o r sequential c o m p u t a t i o n this view is manifest in the A-calculus, in contrast w i t h - - s a y - - t h e classical t h e o r y o f automata.
To h a n d l e concurrency, we should not m e r e l y add extra material to the languages a n d theories o f sequential c o m p u t i n g - - i n particular, e x t r a con- structions for b u i l d i n g b i g g e r sys- tems f r o m smaller ones. I f we do, t h e n o f course we get a l a r g e r r e p e r - toire o f primitive constructions t h a n we h a d before. This is a fine way to fulfill the p r o p h e c y that c o n c u r r e n c y is m o r e c o m p l e x than sequentiality.
A n d it h a s often been done. Well, c o n c u r r e n c y may be m o r e complex, but we should not give in so easily.
We should limit ourselves to con- structions which are essential for c o n c u r r e n c y in its own terms; t h e n i n d e e d we can see sequential com- p u t i n g as a higher, a n d m o r e spe- cific, level o f explanation.
C o n s i d e r sequential composition P; Q
- - t h e familiar semicolon, the essen- tial glue o f sequential imperative p r o g r a m m i n g . To get concurrency, s h o u l d we k e e p sequential composi- tion a n d j u s t add parallel composi- tion? Well, we might want to d o that for a p r o g r a m m i n g language, be- cause we m u s t give p r o g r a m m e r s their familiar tools as well as newer things. But should we do it in a basic calculus? I believe not; for sequential composition is i n d e e d a special case
o f parallel composition
PIQ
when this construction is p r o p e r l y u n d e r s t o o d . I u n d e r s t a n d it to m e a n that P a n d Q are acting side by side, interacting in whatever way we have d e s i g n e d t h e m to interact. So se- quential composition is the special case in which the only interaction occurs when P finishes a n d Q begins.
T o allow a special kind o f interaction h e r e would violate o u r principle that, in the basic model, all interactions are o f the same kind.
It was this sort o f m u n d a n e obser- vation which p r e v e n t e d m e f r o m try- ing to extrapolate f r o m sequentiality a n d led me to try to capture, in a new calculus, a set o f constructions basic to concurrency. This is what I un- d e r s t a n d C h u r c h to have d o n e for sequential c o m p u t i n g , with the A- calculus. We wish to match the func- tional calculus not by c o p y i n g its con- structions, but by e m u l a t i n g two o f its attributes: It is synthetic--we build systems in it, because the structure o f terms r e p r e s e n t s the structure o f processes; a n d it is computational--its basic semantic notion is a step o f c o m p u t a t i o n . Its f u r t h e r attribute, that it has an a g r e e d mathematical i n t e r p r e t a t i o n , we cannot yet match ( t h o u g h g o o d progress is being made). But C h u r c h himself u n d e r - stood his A-calculus terms as func- tions in a c o m p u t a t i o n a l sense o f that word; he did not yet have Scott's denotations.
T o summarize: F o r me, the func- tional calculus was a paradigm--but not a platform--for b u i l d i n g a calcu- lus for c o m m u n i c a t i n g systems.
I p o i n t e d o u t j u s t now that se- quential composition o f processes is a special case o f parallel composition.
I n d e e d , in designing CCS I insisted that t h e r e be only a single c o m b i n a t o r for c o m b i n i n g processes which inter- act o r which coexist. This may seem a tall o r d e r , for I also insisted that m e m o r y registers be m o d e l e d as p r o - cesses, so this same c o m b i n a t o r must be able to assemble t h e m into a m e m - ory, to c o m p o s e the processes which use them, a n d to combine processes with m e m o r y . But one c o m b i n a t o r does i n d e e d suffice, a n d this is be- cause all interactions can i n d e e d be t r e a t e d in the same way. F o r exam-
ple, we can write the system o f F i g u r e 3 as
PIMIQ where M--XIYIZ
o r a s
vlxBYiZlQ
T h e very same expression will be used even when the p r o g r a m s P a n d Q interact in some o t h e r way, over a n d above their interaction via m e m - ory, o r when X, Y, a n d Z a r e not simply storage registers, b u t p e r h a p s processes that are i n t e r m e d i a r y be- tween the p r o g r a m s a n d a r e m o t e m e m o r y . T h e f o r m o f the expression is i n d e p e n d e n t o f the n a t u r e o f these five processes.
T h e algebraic n a t u r e o f the calcu- lus is b e g i n n i n g to e m e r g e , with this single c o m b i n a t o r at its heart. T h e intuition b e h i n d parallel composition is that we a r e simply assembling the c o m p o n e n t s o f a system t o g e t h e r - - so we expect the c o m b i n a t o r to be associative a n d commutative. This is why we have no brackets in o u r ex- pressions. Each d i f f e r e n t o r d e r i n g a n d bracketing o f the m e m b e r s would r e p r e s e n t a d i f f e r e n t partition o f a system into subsystems.
How can o u r algebra reflect m o r e explicitly the structure i n d u c e d by the linkage a m o n g system c o m p o - nents? We note first that the c o m p o - nents P, Q, x . . . . in F i g u r e 3 will themselves be process expressions;
m o r e o v e r , the channel y links only the m e m b e r s P a n d Y, since those will be the only expressions in which the channel n a m e y appears. We d o not give h e r e the process expression for a register like Y; suffice it to say that each such expression will d e t e r m i n e its location as a channel name. T h u s we can say that f r o m P's viewpoint, the n a m e y locates the cell Y. Now Fig- u r e 3 exhibits this i d e a o f location very clearly; we also want o u r algebra to c a p t u r e the idea. F o r this p u r p o s e , we i n t r o d u c e a f u r t h e r c o m b i n a t o r to e n s u r e that the register Y is accessible only to P - - i . e . , that the channel y is local to them. We call this new com- b i n a t o r restriction; for e x a m p l e , in the expression
vy(Y[P)
the channel y is restricted for use be- tween Y a n d P. T h e g r e e k letter v is used partly for the p u n on "new"; in
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S o u r c e s a n d R e l a t e d W o r k
S ome of the important sources and relevant papers are cited here, sec- tion by section.
£ntltle$. Tennent [361 is a classic exposition of the Scott-Strachey approach to semantics. Gunter and Scott [131 is a recent exposition of Scott's domain theory. It contains a section on powerdomains, which were originally published in [32]. Petri's Ph.D. dissertation [31] is the first publicatlon on Petri net theory, and Reisig [33] is an introductory textbook.
• l~tlc constructions. Church [9l is the first publication on the A-calculus;
Barendregt [5] gives a recent and complete exposition of the ~.-calculus. MiIner [21] gives the first exposition of CCS and brings It uP to date In a later text- book [22]. MIIner in [20] gives the algebra of flow graphs, a static algeDra of processes.
Dynamic constntctlons. Hoare [16] introduces the programming language CSP and In a t e x t b o o k [17] gives its algebraic theory. Baeten and Weljland [4] ex- POUnd the process algebra ACP due originally t o Bergstra and Klop. The speci- fication language Lotos was designed by Brinksma [7].
Meaning. Some important domain-theoretic models for concurrency are the failures model of Brookes et al. [81, Hennessy's acceptance trees [14], and Abramsky's domain for bislmulation [1]. The observation equivalence of pro- cesses was introduced by MIIner [21]; Park [30] placed it on a firmer mathemat- Ical footing with the notion of bislmulation. Tile event structures of Nielsen et al. [28] combine domain theory with the causality structure of Petrl nets.
Values. B0hm and Berarduccl [61 show how to encode data structures into the h-CalCUlUS. The second-order (polymorphic) A-calculus was discovered in the early seventies independently by Girard [111 and Reynolds [341. An early presentation of Hewitt's ACtors model is by Hewltt et al. [15]; a recent book on Actors Is by Agha [2]. Smalltalk [12] is probably the first programming language to treat values--even the simplest, like numbers--as objects which receive messages.
Names. Astesiano and Zucca [3] studied a version of CCS in which channels (i.e., names, in the terms of this article) could be parametrized on values, thus allowing some mobility. Kennaway and Sleep [19] built the Idea Of transmitting names as messages into a language for distributed operating systems. In 1980, at Arhus, Mogens Nielsen and I had tried t o treat mobility algebraically In CCS, but failed. Engberg and Nielsen [101 later succeeded, giving the first algebraic treatment of a process calculus with dynamic reconflguration (their report is unpublished). These ideas were refined and strengthened by MIIner et al. lead- Ing to the at-calculus [251. Milner [231 gives a recent tutorial exposition of a more general form, also treating type structure.
As an accessible Illustration of how to apply the ~r-calCulUS, Orava and Parrow [29] made a rigorous study Of a simplified mobile telephone network, which I have used here. A translation of the A-calculus into the ~r-calculus iS given by MIIner [24]. Walker [381 explored first the use of the a-calculus to give seman- tics t o concurrent object-oriented programming. Honda and Tokoro [18] give an interesting variant of the ~r-calculus suitable for asynchronous communica- tion, which brings It closer t o Actors and to object-oriented programming.
Another way to achieve mobility Is t o allow processes themselves to be trans- mitted in Interaction. Nlelson [271 and Thomsen [371 have studied these so- called seconcl-order processes. Thomsen also gave a translation of second- order processes into the 7r-calculus and showed that It preserves operational behavior. Sanglorgl generalizes this translation to oo-order processes and has proved that a suitable behavioral equivalence Is preserved in both directions by the translation. His results are summarized by MIIner [231 and will appear in Sanglorgi's Ph.D. dissertation [35]. This work on higher order reinforces the claim of exPressive completeness for the st-calculus.
COnclusion. The Turlng award lecture of Newell and Simon I261 examines the nature of empirical enquiry in computer science.
fact, restriction is j u s t a distillation of the notion of local variable declaration in p r o g r a m m i n g . T h u s for the whole system o f Figure 3 we may more ac- curately write
vx(xl y(YIP)l z(zl ) ( ,
This kind o f expression well repre- sents the spatial or static structure of interactive systems in general, t h o u g h we have only illustrated it for programs a n d memory. Diagrams like Figure 3, which we may call flow graphs, are quite formal objects. In- deed, restriction a n d composition obey certain equations which define the algebraic theory of flow graphs.
D y n a m i c C o n s t r u c t i o n s
Now let us t u r n to the dynamic aspect of such systems: their behavior. It is in the dynamics, in fact, that we can contrast the sequential, hierarchical control of the X-calculus with the c o n c u r r e n t , heterarchical control of CCS. I n the X-calculus, all computa- tion comes down to j u s t one thing, called reduction--the act of passing an a r g u m e n t to a function; we may call this the atom of behavior of the X- calculus. I n j u s t the same way, an in- teraction-the passage o f a single d a t u m between processes--is the atom of behavior in CCS. We shall now see how this allows symmetry between partners a n d how it reflects the idea that each process in a com- m u n i t y has persistent identity.
We shall look at this contrast in terms o f the basic computational rule of each calculus. I n each calculus, systems are built using a binary com- binator. I n the X-calculus the com- binator is called application a n d is nei- ther commutative n o r associative.
W h e n one term M is applied to an- other, N,
M(N)
t h e n the first term M - - t h e operator-- holds control. T h e operator is com- mitted to receive N, the operand; if a n d when the operator takes the form of a n abstraction Ax.M[x] (where the square brackets indicate that M may contain the b o u n d variable x), then the symbol A represents the operator's locus of control, a n d the following reduction will occur:
(Xx.M[x])(N) ~ M[N]
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l e m e n t s o f I n t e r a c t i o n
(where the brackets on the right indi- cate that N has r e p l a c e d x in M). This is the basic c o m p u t a t i o n a l rule o f the A-calculus, the dynamics o f function application; for m a n y p e o p l e it is m o r e familiar as the copy rule o f Algol60, which was d e r i v e d f r o m it.
F i g u r e 5 r e p r e s e n t s this o p e r a t i o n pictorially; note that the o p e r a n d N is r e p r e s e n t e d by a circle, being a pas- sive d a t u m in the r e d u c t i o n - - i t will only later assume control, as an o p e r - ator, when M allows it to d o so.
I n CCS on the o t h e r h a n d , when two terms P a n d Q are c o m p o s e d in parallel,
PIQ
then both hold control. P may receive a message f r o m Q, j u s t as Q may re- ceive one f r o m P. T h e r e a r e two so- called actions which a t e r m may take,
allowing it to interact with another.
W h e n one p a r t n e r takes the input f o r m Ax.P[x] while the o t h e r takes the output f o r m ~V.Q, then we have the composition
Ax.PLxJlXV.Q
in which the p a r t n e r s have what may be called complementary loci o f control A a n d A - - t h a t is, the positive a n d negative ends o f a channel n a m e d A.
T h e n the following interaction may
O c c u r :
Xx.P[x]lXV.Q---, P[V]lQ
It is a synchronization o f the part- ners' actions.
But crucially, in c o n c u r r e n c y t h e r e
a r e many channels, not j u s t one; so
the symbol A, which r e p r e s e n t e d the single locus o f control in the A- calculus, now becomes j u s t one o f m a n y channels, p e r h a p s concur- rently active. CCS uses the simple names a, b, c . . . . for such channels.
T h u s we arrive at the basic c o m p u t a - tion rule o f CCS, shown pictorially in F i g u r e 6. C o m p u t a t i o n j u s t consists in the iteration o f this single rule, r e p e a t e d l y t r a n s f o r m i n g an expres- sion; at any stage the structure o f the expression r e p r e s e n t s the spatial c o n f i g u r a t i o n o f the system. T h e dia- g r a m in F i g u r e 6 shows the dynamic b e h a v i o r - - t h e passage o f a passive d a t u m - - s u p e r p o s e d on the flow
- ®
BEFORE
M®
®
AFTER
BEFORE
a .
AFTER Q
F i g u r e S. T h e x-calCulUS r e c l u c t l o n ( . ~ x . M [ x ] ) N ~ M [ N ] F i g u r e 6 . T h e CCS I n t e r a c t i o n a x . P [x] l a V . Q ~ P [ v ] I Q
g r a p h which r e p r e s e n t s the spatial configuration.
Let us look m o r e closely at the dif- ferences here. First, in the A-calculus case the second p a r t n e r was merely a passive d a t u m , but in the CCS case it yields u p its d a t u m V a n d continues an i n d e p e n d e n t existence. It is ex- actly this which admits continual interaction a m o n g c o n c u r r e n t p r o - cesses, each r e t a i n i n g its identity.
Second, parallel composition is com- mutative; PIQ means the same as QIP.
In CCS, in contrast to the A-calculus, e i t h e r p a r t n e r may act as the re- c e i v e r - a n d i n d e e d that p a r t n e r may later b e c o m e the transmitter.
Third, parallel composition is also associative; t o g e t h e r with c o m m u t a - tivity, this frees the interaction disci- pline f r o m the t e r m structure, which r e p r e s e n t s hierarchical control in the A-calculus. As we saw in the e x a m p l e o f p r o g r a m - m e m o r y interaction, the control discipline is now d e t e r m i n e d by which channels a, b . . . . link which processes.
T h e s e details are somewhat tech- nical. But the focal p o i n t is this: T h e d i f f e r e n c e between c o n c u r r e n t a n d sequential c o m p u t a t i o n can be con- c e n t r a t e d in the single basic computa- tion rule o f the respective calculi. By m o d i f y i n g the notion o f reduction f r o m the A-calculus, we attain a rule o f interaction for c o n c u r r e n t pro- cesses, a n d this is the first stage in o u r synthesis between the p u r i t y o f the functional m o d e l a n d the d y n a m - ics o f real c o n c u r r e n t systems.
T h e idea o f synchronized interac- tion as a p r o g r a m m i n g primitive was known to T o n y H o a r e b e f o r e I ex- pressed it in algebraic form. T h e fact that these steps were taken i n d e p e n - dently, with d i f f e r e n t motives, is some evidence that the idea is a natu- ral one. H o a r e i n c o r p o r a t e d the idea in his p r o g r a m m i n g l a n g u a g e CSP a n d later with colleagues d e v e l o p e d a t h e o r y a r o u n d it. It p r o v i d e d the basis for the r e n d e z v o u s mechanism o f Ada, as well as b e c o m i n g the a t o m o f interaction in the p r o g r a m m i n g l a n g u a g e Occam. My c o n t r i b u t i o n was to m a k e interaction the c o r n e r - stone o f an algebraic calculus. T h e process algebras d e v e l o p e d in the eighties, such as CCS, CSP, a n d the ACP o f Bergstra a n d Klop, have b e e n the subject o f m u c h semantic
8 4 January 1993/Vo1.36, No.1 / C O M M U N I C A T I O N S O F T H E A I l l M
research. T h e y have also become, o r have grown into, actively used design tools; in particular, the specification language Lotos has been widely ap- plied to c o m m u n i c a t i o n protocols.
Meaning
I would like to allay the fear that con- currency, at least in the way I am treating it, is d e s p e r a t e l y concrete a n d mechanistic. Are there not ab- stract mathematical things underly- ing it all, for which we should reserve the t e r m " p r o c e s s " - - r a t h e r than dig- nifying o u r algebraic expressions with that name? A f t e r all, we have come to use the t e r m "function" in an abstract sense, so that when we p o i n t to an expression in the A-calculus a n d say "that function," p e o p l e usually take us to m e a n "the mathematical function denoted by that expression." I f
an expression has no a g r e e d denota- tion we may feel uncomfortable.
T h e semantics o f processes has a large a n d growing literature. It is not simple; b u t some exciting progress has been made. I spoke earlier about the possible use o f Scott's domains for c o n c u r r e n t p r o g r a m s , a n d i n d e e d a variety o f interesting d o m a i n s are studied for process algebra. T h e r e has also been m u c h research on the m e a n i n g o f a process as d e t e r m i n e d by its observable behavior. T h e idea is this: To observe a process is exactly to interact with it, and two s y s t e m s - - say two process terms in C C S - - should have the same d e n o t a t i o n if a n d only if we cannot distinguish t h e m by observation. This research is largely d e v o t e d to classifying the s t r o n g e r a n d weaker kinds o f obser- vational distinction which can be m a d e a n d to a mathematical charac- terization in each case. Much has been achieved; much remains to be done.
T h e s e r e m a r k s about observation, so far, apply equally to sequential processes; b u t - - a s might be ex- p e c t e d - c o n c u r r e n c y raises prob- lems all o f its own. O n e i m p o r t a n t topic is the concept o f causal indepen- dence, which is central in Petri net theory. Now, two p r o c e s s e s - - e a c h having c o n c u r r e n t c o m p o n e n t s - - may be indistinguishable to an exter- nal observer, and yet d i f f e r in the causal relationship a m o n g their ob- served actions. I may observe you
falling out o f the tree a n d t h e n ob- serve the a m b u l a n c e arriving, but I still d o not know if one caused the other. H e n c e observational semantics ignores causality. But there are also g o o d reasons for semantic models which respect causal c o n n e c t i o n - - such as the event structures o f Nielsen et al. This topic is too c o m p l e x to tackle here; I m e n t i o n it only to show that c o n c u r r e n c y does i n d e e d raise new semantic questions.
W h a t e v e r mathematical models are studied, I believe that process cal- culi p r o v i d e an essential perspective for the study. Many p e o p l e will only be satisfied with the semantic t h e o r y o f c o n c u r r e n t systems w h e n - - e v e n t u a l l y - - i t becomes an abstract t h e o r y as well as a formal one. But to attain this goal we must first distill the essence o f the dynamics o f inter- action; this is what a f o r m a l process calculus like CCS tries to do.
values
We now e m b a r k on the second stage in a synthesis between the functional p a r a d i g m a n d the realities o f interac- tion. This time I am c o n c e r n e d with m a k i n g a c o n c u r r e n t calculus fully expressive, within its own conceptual frame. First, I need to explain how the A-calculus succeeds in this re- spect, while CCS a n d similar concur- rency calculi fall short. Later we shall find a r e m e d y .
Often p e o p l e use the A-calculus informally, simply to derive new op- erators over familiar data. W h e n dis- cussing arithmetic, they will write f = Ax (x 2 + l)
o r p e r h a p s
F = AfAx (f(x) + 1)
- - i . e . , they mix the constructions o f the A-calculus with those o f arithme- tic, not wishing to code one into the other. T r e a t e d thus, the A-calculus is like some c o m p u t a t i o n a l scaffolding, u p a n d down which the real workers climb: d a t a such as numbers, a n d o p e r a t o r s such as squaring, adding, o r differentiating.
This auxiliary use o f the A-calculus is very natural, but h a r d l y makes for a self-contained m o d e l o f c o m p u t i n g within the functional framework.
Each time we a d d new types o f d a t a s t r u c t u r e - - l i k e arrays, lists, t r e e s - - we are in effect e x t e n d i n g the calcu-
lus. But it is i m p o r t a n t for the u n d e r - standing o f p r o g r a m m i n g languages, for e x a m p l e , that the basic model which we use to explain t h e m should be as h o m o g e n e o u s and complete as possible, r e q u i r i n g no ad hoc exten- sion.
Quite r e m a r k a b l y , the A-calculus does achieve this h o m o g e n e i t y a n d completeness. I n its p u r e form, it does i n d e e d have full p o w e r to rep- resent d a t a structures and c o m p u t e with them; we can d o everything with the scaffolding alone. C h u r c h showed this for arithmetic; B6hm and others have e x t e n d e d it to o t h e r forms o f d a t a structure. Moreover, t h e r e are type disciplines--notably the G i r a r d - R e y n o l d s s e c o n d - o r d e r A- c a l c u l u s - - w h i c h allow this to be d o n e in a controlled way, b u i l d i n g a uni- f o r m f r a m e w o r k for analyzing se- quential p r o g r a m m i n g l a n g u a g e s - - a n d their respective type disciplines.
Now, what about concurrency?
Most c o n c u r r e n c y formalisms also p r o v i d e a scaffolding, o f a d i f f e r e n t kind, on which values a n d o p e r a t o r s can move a r o u n d ; these values a n d o p e r a t o r s are quite distinct from the scaffolding itself. Look again at CCS (Figure 6); in that interaction, the d a t u m V is actually an expression standing for s o m e t h i n g like a number--quite a d i f f e r e n t type o f animal f r o m the process expression P.
To u n d e r l i n e the point, F i g u r e 7 shows a CCS definition for a process Double, which r e p e a t e d l y inputs a n u m b e r along the a channel a n d out- puts twice that n u m b e r on the b channel. R e g a r d i n g the short defini- tion, it contains no less than five dif- f e r e n t kinds o f thing: processes (e.g., Double), channels (a, b), variables (x), operators ( x ) , a n d value expressions (2 × x). A n d f o u r o f these (all except operators) a l r e a d y occur in the basic c o m p u t a t i o n rule o f C C S - - s e e Fig- ure 6. Is not this promiscuity exces- sive?
Well, it is perfectly admissible in a high-level p r o g r a m m i n g language or specification l a n g u a g e such as Lotos, where a user expects a rich and re- d u n d a n t tool kit. But for a truly basic calculus the situation is less comfort- able. A basic calculus should impose as little t a x o n o m y as it can, because d i f f e r e n t h i g h e r levels o f explanation will impose d i f f e r e n t taxonomies.
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