Communicating
Sequential Processes
C. A. R. Hoare

June 21, 2004

© C. A. R. Hoare, 1985–2004
This document is an electronic version of Communicating Sequential
Processes, first published in 1985 by Prentice Hall International. It may be
copied, printed, and distributed free of charge. However, such copying,
printing, or distribution may not:
− be carried out for commercial gain; or
− take place within India, Pakistan, Bangladesh, Sri Lanka, or the Maldives;
or
− involve any modification to the document itself.
Questions and comments are welcome, and should be sent to the editor of
this version: Jim.Davies@comlab.ox.ac.uk.

Foreword

For a variety of reasons, this is a book eagerly awaited by all who knew it
was in the making; to say that their patience has been rewarded would be an
understatement.
A simple reason was that it is Tony Hoare’s first book. Many know him
from the lectures he has untiringly given all over the world; many more know
him as the articulate and careful author of a number of articles (of great variety!) that became classics almost before the printer’s ink had dried. But a
book is a different medium: here the author can express himself without the
usually stringent limitations of time and space; it gives him the opportunity
of revealing himself more intimately and of covering a topic of wider span,
opportunities of which Tony Hoare has made the best use we could hope for.
A more solid reason was derived from the direct contents of the book.
When concurrency confronted the computing community about a quarter of
a century ago, it caused an endless confusion, partly by the technically very
different circumstances in which it emerged, partly by the accident of history
that it introduced non-determinism at the same time. The disentanglement of
that confusion required the hard work of a mature and devoted scientist who,
with luck, would clarify the situation. Tony Hoare has devoted a major part
of his scientific endeavours to that challenge, and we have every reason to be
grateful for that.
The most profound reason, however, was keenly felt by those who had
seen earlier drafts of his manuscript, which shed with surprising clarity new
light on what computing science could—or even should—be. To say or feel
that the computing scientist’s main challenge is not to get confused by the
complexities of his own making is one thing; it is quite a different matter to
discover and show how a strict adherence to the tangible and quite explicit
elegance of a few mathematical laws can achieve this lofty goal. It is here
that we, the grateful readers, reap to my taste the greatest benefits from the
scientific wisdom, the notational intrepidity, and the manipulative agility of
Charles Antony Richard Hoare.
Edsger W. Dijkstra

Preface

This is a book for the aspiring programmer, the programmer who aspires to
greater understanding and skill in the practice of an intellectually demanding
profession. It is designed to appeal first to a natural sense of curiosity, which
is aroused by a new approach to a familiar topic. The approach is illustrated
by a host of examples drawn from a wide range of applications, from vending
machines through fairy stories and games to computer operating systems. The
treatment is based on a mathematical theory, which is described by a systematic collection of algebraic laws.
The ultimate objective of the book is to convey an insight which will enable
the reader to see both current and future problems in a fresh light, in which
they can be more efficiently and more reliably solved; and even better, they can
sometimes be avoided.
The most obvious application of the new ideas is to the specification,
design, and implementation of computer systems which continuously act and
interact with their environment. The basic idea is that these systems can be
readily decomposed into subsystems which operate concurrently and interact
with each other as well as with their common environment. The parallel composition of subsystems is as simple as the sequential composition of lines or
statements in a conventional programming language.
This insight brings practical benefits. Firstly, it avoids many of the traditional problems of parallelism in programming—interference, mutual exclusion, interrupts, multithreading, semaphores, etc. Secondly, it includes
as special cases many of the advanced structuring ideas which have been
explored in recent research into programming languages and programming
methodology—the monitor, class, module, package, critical region, envelope,
form, and even the humble subroutine. Finally, it provides a secure mathematical foundation for avoidance of errors such as divergence, deadlock and
non-termination, and for achievement of provable correctness in the design
and implementation of computer systems.
I have tried hard to present the ideas in a logically and psychologically wellordered sequence, starting with the simple basic operators, and progressing
towards their more elaborate applications. An assiduous reader may study the
book from cover to cover. But many readers will start with greater interest in

vi

Preface

some topics than others; and for their benefit each chapter of the book has
been structured to permit judicious selection.
1. Each new idea is introduced by an informal description and illuminated
by a number of small examples, which will probably be helpful to all
readers.
2. The algebraic laws which describe the essential properties of the various
operations will be of interest to those with a taste for mathematical
elegance. They will also be of benefit for those who wish to optimise their
system designs by means of correctness-preserving transformations.
3. The proposed implementations are unusual in that they use a very simple
purely functional subset of the well-known programming language LISP.
This will afford additional excitement to those who have access to a LISP
implementation on which to exercise and demonstrate their designs.
4. The definitions of traces and specifications will be of interest to systems
analysts, who need to specify a client’s requirements before undertaking
an implementation. They will also be of interest to senior programmers,
who need to design a system by splitting it into subsystems with clearly
specified interfaces.
5. The proof rules will be of interest to implementors who take seriously
their obligation to produce reliable programs to a known specification, to
a fixed schedule, and at a fixed cost.
6. Finally, the mathematical theory gives a rigorous definition of the
concept of a process, and the operators in terms of which processes are
constructed. These definitions are a basis for the algebraic laws, the
implementations and the proof rules.
A reader may consistently or intermittently omit or postpone any of these
topics which are of lesser interest, or which present greater difficulty of understanding.
The succession of chapters in the book has also been organised to permit judicious browsing, selection, or rearrangement. The earlier sections of
Chapter 1 and Chapter 2 will be a necessary introduction for all readers, but
later sections may be more lightly skimmed or postponed to a second pass.
Chapters 3, 4 and 5 are independent of each other, and may be started in
any combination or in any order, according to the interest and inclination of the
reader. So if at any stage there is any difficulty of understanding, it is advisable
to continue reading at the next section or even the next chapter, since there
is a reasonable expectation that the omitted material will not be immediately
required again. When such a requirement arises, there will often be an explicit
backward reference, which can be followed when there is sufficient motivation
to do so.
I hope everything in the book will in the end be found interesting and rewarding; but not everyone will wish to read and master it in the order presented.

Preface

vii

The examples chosen to illustrate the ideas of this book will all seem very
small. This is deliberate. The early examples chosen to illustrate each new
idea must be so simple that the idea cannot be obscured by the complexity or
unfamiliarity of the example. Some of the later examples are more subtle; the
problems themselves are of the kind that could generate much confusion and
complexity; and the simplicity of their solution could be due to the power of the
concepts used and the elegance of the notations in which they are expressed.
Nevertheless, each reader will be familiar, perhaps painfully familiar, with
problems of far greater scope, complexity and importance than the examples
appropriate for an introductory text. Such problems may seem to be intractable by any mathematical theory. Please do not give way to irritation or disillusion, but rather accept the challenge of trying to apply these new methods
to existing problems.
Start with some grossly over-simplified version of some selected aspect
of the problem, and gradually add the complexities which appear to be necessary. It is surprising how often the initial over-simplified model will convey
additional insight, to assist in the solution of the problem as a whole. Perhaps
the model can serve as a structure on which complex detail can later be safely
superimposed. And the final surprise is that perhaps some of the additional
complexity turns out to be unnecessary after all. In such cases, the effort of
mastering a new method receives it most abundant reward.
Notations are a frequent complaint. A student setting out to learn the
Russian language often complains at the initial hurdle of learning the unfamiliar letters of the Cyrillic alphabet, especially since many of them have strange
pronunciations. If it is any consolation, this should be the least of your worries. After learning the script, you must learn the grammar and the vocabulary,
and after that you must master the idiom and style, and after that you must
develop fluency in the use of the language to express your own ideas. All this
requires study and exercise and time, and cannot be hurried.
So it is with mathematics. The symbols may initially appear to be a serious hurdle; but the real problem is to understand the meaning and properties
of the symbols and how they may and may not be manipulated, and to gain
fluency in using them to express new problems, solutions, and proofs. Finally,
you will cultivate an appreciation of mathematical elegance and style. By that
time, the symbols will be invisible; you will see straight through them to what
they mean. The great advantage of mathematics is that the rules are much
simpler than those of a natural language, and the vocabulary is much smaller. Consequently, when presented with something unfamiliar it is possible
to work out a solution for yourself, by logical deduction and invention rather
than by consulting books or experts.
That is why mathematics, like programming, can be so enjoyable. But it is
not always easy. Even mathematicians find it difficult to study new branches
of their subject. The theory of communicating processes is a new branch of
mathematics; programmers who study it start with no disadvantage over mathematicians; but they will end with the distinct advantage of putting their knowledge to practical use.

viii

Preface

The material of this book has been tested by presentation in informal workshops as well as on formal academic courses. It was first designed for a onesemester Master’s course in software engineering, though most of it could be
presented in the final or even the second year of a Bachelor’s degree in computing science. The main prerequisite is some acquaintance with high-school
algebra, the concepts of set theory, and the notations of the predicate calculus.
These are summarised on the first page of the glossary of symbols just after
this preface. The book is also a suitable basis for an intensive one-week course
for experienced programmers. In such a course, the lecturer would concentrate on examples and definitions, leaving the more mathematical material for
later private study. If even less time is available, a course which ends after
Chapter 2 is quite worthwhile; and even in a single hour’s seminar it is possible by careful selection, to get as far as the edifying tale of the five dining
philosophers.
It is great fun to present lectures and seminars on communicating sequential processes, since the examples give scope for exercise of the histrionic skills
of the lecturer. Each example presents a little drama which can be acted with
due emphasis on the feelings of the human participants. An audience usually finds something particularly farcical about deadlock. But they should be
constantly warned about the dangers of anthropomorphism. The mathematical formulae have deliberately abstracted from the motives, preferences, and
emotional responses by which the lecturer “lends an air of verisimilitude to
an otherwise bald and unconvincing tale”. So one must learn to concentrate
attention on the cold dry text of the mathematical formulae, and cultivate an
appreciation for their elegant abstraction. In particular, some of the recursively defined algorithms have something of the breathtaking beauty of a fugue
composed by J. S. Bach.

Summary

Chapter 1 introduces the basic concept of a process as a mathematical abstraction of the interactions between a system and its environment. It shows how
the familiar technique of recursion may be used to describe processes that last
a long time, or forever. The concepts are explained first by example and then
by pictures; a more complete explanation is given by algebraic laws, and by an
implementation on a computer in a functional programming language.
The second part of the chapter explains how the behaviour of a process
can be recorded as a trace of the sequence of actions in which it engages. Many
useful operations on traces are defined. A process can be specified in advance
of implementation by describing the properties of its traces. Rules are given
to help in implementation of processes which can be proved to meet their
specifications.
The second chapter describes how processes can be assembled together
into systems, in which the components interact with each other and with their
external environment. The introduction of concurrency does not by itself introduce any element of nondeterminism. The main example of this chapter is
a treatment of the traditional tale of the five dining philosophers.
The second part of Chapter 2 shows how processes can be conveniently
adapted to new purposes by changing the names of the events in which they
engage. The chapter concludes with an explanation of the mathematical theory
of deterministic processes, including a simple account of the domain theory
of recursion.
The third chapter gives one of the simplest known solutions to the vexed
problem of nondeterminism. Nondeterminism is shown to be a valuable technique for achieving abstraction, since it arises naturally from the decision to
ignore or conceal those aspects of the behaviour of a systems in which we are
no longer interested. It also preserves certain symmetries in the definition of
the operators of the mathematical theory.
Proof methods for nondeterministic processes are slightly more complicated than those for deterministic processes, since it is necessary to demonstrate that every possible nondeterministic choice will result in a behaviour
which meets the given specification. Fortunately, there are techniques for

x

Summary

avoiding nondeterminism, and these are used extensively in Chapters 4 and 5.
Consequently the study or mastery of Chapter 3 can be postponed until just
before Chapter 6, in which the introduction of nondeterminism can no longer
be avoided.
In the later sections of Chapter 3, there is given a complete mathematical
definition of the concept of a nondeterministic process. This definition will
be of interest to the pure mathematician, who wishes to explore the foundations of the subject, or to verify by proof the validity of the algebraic laws and
other properties of processes. Applied mathematicians (including programmers) may choose to regard the laws as self-evident or justified by their utility;
and they may safely omit the more theoretical sections.
Chapter 4 at last introduces communication: it is a special case of interaction between two processes, one of which outputs a message at the same time
as the other one inputs it. Thus communication is synchronised; if buffering is
required on a channel, this is achieved by interposing a buffer process between
the two processes.
An important objective in the design of concurrent systems is to achieve
greater speed of computation in the solution of practical problems. This is illustrated by the design of some simple systolic (or iterative) array algorithms.
A simple case is a pipe, defined as a sequence of processes in which each process inputs only from its predecessor and outputs only to its successor. Pipes
are useful for the implementation of a single direction of a communications
protocol, structured as a hierarchy of layers. Finally, the important concept
of an abstract data type is modelled a a subordinate process, each instance of
which communicates only with the block in which it is declared.
Chapter 5 shows how the conventional operators of sequential programming can be integrated within the framework of communicating sequential
processes. It may be surprising to experienced programmers that these operators enjoy the same kind of elegant algebraic properties as the operators of
familiar mathematical theories; and that sequential programs can be proved
to meet their specifications in much the same way as concurrent programs.
Even the externally triggered interrupt is defined and shown to be useful, and
subject to elegant laws.
Chapter 6 describes how to structure and implement a system in which
a limited number of physical resources such as discs and line printers can be
shared among a greater number of processes, whose resource requirements
vary with time. Each resource is represented as a single process. On each
occasion that a resource is required by a user process, a new virtual resource
is created.
A virtual resource is a process which behaves as if it were subordinate to
the user process; but it also communicates with the real resource whenever required. Such communications are interleaved with those of other concurrently
active virtual processes. So the real and virtual processes play the same roles
as the monitors and envelopes of PASCAL PLUS. The chapter is illustrated by
the modular development of a series of complete but very simple operating
systems, which are the largest examples given in this book.

Summary

xi

Chapter 7 describes a number of alternative approaches to concurrency
and communication, and explains the technical, historical, and personal motives
which led to the theory expounded in the preceding chapters. Here I acknowledge my great debt to other authors, and give recommendations and an introduction to further reading in the field.

Acknowledgements

It is a great pleasure to acknowledge the profound and original work of Robin
Milner, expounded in his seminal work on a Calculus for Communicating Systems. His original insights, his personal friendship and his professional rivalry
have been a constant source of inspiration and encouragement for the work
which culminated in the publication of this book.
For the last twenty years I have been considering the problems of programming for concurrent computations, and the design of a programming language
to ease those problems. During that period I have profited greatly by collaboration with many scientists, including Per Brinch Hansen, Stephen Brookes,
Dave Bustard, Zhou Chao Chen, Ole-Johan Dahl, Edsger W. Dijkstra, John Elder,
Jeremy Jacob, Ian Hayes, Jim Kaubisch, John Kennaway, T. Y. Kong, Peter Lauer,
Mike McKeag, Carroll Morgan, Ernst-Rudiger Olderog, Rudi Reinecke, Bill Roscoe, Alex Teruel, Alastair Tocher and Jim Welsh.
Finally, my special thanks for to O.-J. Dahl, E. W. Dijkstra, Leslie M. Goldschlager, Jeff Sanders and others who have carefully studied an earlier draft
of this text, and who have pointed out its errors and obscurities; and in particular to the participants in the Wollongong Summer School on the Science of
Computer Programming in January 1983, the attendants at my seminar in the
Graduate School of the Chinese Academy of Science, April 1983, and students
of the M.Sc. in Computation at Oxford University in the years 1979 to 1984.

Glossary of Symbols

Logic
Notation

Meaning

Example

=

equals

x=x

≠

is distinct from

x ≠x +1



end of an example
or proof

P ∧Q

P and Q (both true)

x ≤x +1∧x ≠x +1

P ∨Q

P or Q (one or both true)

x≤y ∨y ≤x

¬P

not P (P is not true)

¬3≥5

P ⇒Q

if P then Q

x<y ⇒x≤y

P ≡Q

P if and only if Q

x<y ≡y >x

∃x • P

there exists an x
such that P

∃x • x > y

∀x • P

forall x, P

∀x • x < x + 1

∃x : A • P

there exists an x
in set A such that P

∀x : A • P

for all x in set A, P

Sets
Notation

Meaning

Example

∈

is a member of

Napoleon ∈ mankind

∉

is not a member of

Napoleon ∉ Russians

{}

the empty set (with
no members)

¬ Napoleon ∈ {}

xvi

Glossary of Symbols

{a}

the singleton set of a;
a is the only member

x ∈ {a} ≡ x = a

{a, b, c}

the set with members a,
b, and c

c ∈ {a, b, c}

{ x | P (x) }

the set of all x
such that P (x)

{a} = { x | x = a }

A∪B

A union B

A ∪ B = {x | x ∈ A ∨ x ∈ B }

A∩B

A intersect B

A ∩ B = {x | x ∈ A ∧ x ∈ B }

A−B

A minus B

A − B = {x | x ∈ A ∧ ¬ x ∈ B }

A⊆B

A is contained in B

A ⊆ B ≡ ∀x : A • x ∈ B

A⊇B

A contains B

A⊇B≡B⊆A

{ x : A | P (x) }

the set of x in A
such that P (x)

N

the set of natural numbers

{0, 1, 2, . . .}

PA
S

the power set of A

PA = {X | X ⊆ A}
S
n≥0 An = { x | ∃ n ≥ 0 • x ∈ A }
T
n≥0 An = { x | ∀ n ≥ 0 • x ∈ A }

T

n≥0

An

union of a family of sets

n≥0

An

intersection of a family
of sets

Functions
Notation

Meaning

Example

f :A→B

f is a function which maps
each member of A to a
member of B

square : N → N

f (x)

that member of B to which
f maps x (in A)

injection

a function f which maps
each member of A to a
distinct member of B

x ≠ y ⇒ f (x) ≠ f (y)

f −1

inverse of an injection f

x = f (y) ≡ y = f −1 (x)

{ f (x) | P (x) }

the set formed by applying
f to all x such that P (x)

f (C)

the image of C under f

{ y | ∃ x • y = f (x) ∧ x ∈ C }
square({3, 5}) = {9, 15}

f ◦g

f composed with g

f ◦ g(x) = f (g(x))

λ x • f (x)

the function which maps
each value of x to f (x)

(λ x • f (x))(3) = f (3)

Glossary of Symbols

xvii

Traces
Section

Notation

Meaning

Example

1.5

hi

the empty trace

1.5

hai

the trace containing only
a (singleton sequence)

1.5

ha, b, ci

the trace with three symbols, a then b, then c

1.6.1

_

(between traces)
followed by

ha, b, ci = ha, bi _ hi _ hci

1.6.1

sn

s repeated n times

ha, bi2 = ha, b, a, bi

1.6.2

suA

s restricted to A

hb, c, d, ai u {a, c} = hc, ai

1.6.5

s≤t

s is a prefix of t

ha, bi ≤ ha, b, ci

4.2.2

s ≤n t

s is like t with up to n
symbols removed

ha, bi ≤2 ha, b, c, di

1.6.5

s in t

s is in t

hc, di in hb, c, d, a, bi

1.6.6

#s

the length of s

#hb, c, b, ai = 4

1.6.6

s↓b

the count of b in s

hb, c, b, ai ↓ b = 2

1.9.6

s↓c

the communications on
channel c recorded in s

1.9.2

_/ s

flatten s

hc.1, a.4, c.3, d.1i ↓ c =
h1, 3i
_/hha, bi, hi

1.9.7

s;t

s successfully followed
by t

(s _ h✓i) ; t = s _ t

1.6.4

A∗

set of sequences with
elements in A

A∗ = { s | s u A = s }

1.6.3

s0

the head of s

ha, b, ci0 = a

1.6.3

s

0

the tail of s

ha, b, ci0 = hb, ci

1.9.4

s[i]

the ith element of s

ha, b, ci[1] = b

1.9.1

∗

f (s)

f star of s

square∗ (h1, 5, 3i) =
h1, 25, 9i

1.9.5

s

reverse of s

ha, b, c, i = hc, b, ai

Special Events
Section

Notation

Meaning

1.9.7

✓

success (successful termination)

2.6.2

l.a

participation in event a by a process named l

xviii

Glossary of Symbols

4.1

c.v

communication of value v on channel c

4.5

l.c

channel c of a process named l

4.5

l.c.v

communication of a message v on channel l.c

5.4.1

“

catastrophe (lightning)

5.4.3

x

exchange

5.4.4

c

checkpoint for later recovery

6.2

acquire

acquisition of a resource

6.2

release

release of a resource

Processes
Section

Notation

Meaning

1.1

αP

the alphabet of process P

4.1

αc

the set of messages communicable
on channel c

1.1.1

a→P

a then P

1.1.3

(a → P | b → Q )

a then P choice b then Q (provided a ≠ b)

1.1.3

(x : A → P (x))

(choice of) x from A then P (x)

1.1.2

µ X : A • F (X )

the process X with alphabet A
such that X = F (X )

1.8

P /s

P after (engaging in events of trace) s

2.3

P || Q

P in parallel with Q

2.6.2

l:P

P with name l

2.6.4

L:P

P with names from set L

3.2

P uQ

P or Q (non-deterministic)

3.3

P Q

P choice Q

3.5

P\C

P without C (hiding)

3.6

P ||| Q

P interleave Q

4.4

P >>Q

P chained to Q

4.5

P // Q

P subordinate to Q

6.4

l :: P // Q

remote subordination

5.1

P ;Q

P (successfully) followed by Q

5.4

P 4Q

P interrupted by Q

5.4.1

ˆQ
P“

P but on catastrophe Q

5.4.2

P̂

restartable P

Glossary of Symbols

xix

5.4.3

P x Q

P alternating with Q

5.5

P <| b >
| Q

P if b else Q

5.1

∗

repeat P

5.5

b∗P

while b repeat P

5.5

x := e

x becomes (value of) e

4.2

b!e

on (channel) b output (value of) e

4.2

b?x

on (channel) b input to x

6.2

l!e?x

call of shared subroutine named l
with value parameter e and results to x

1.10.1

P sat S

(process) P satisfies (specification) S

1.10.1

tr

an arbitrary trace of the specified process

3.7

ref

an arbitrary refusal of the specified process

5.5.2

x✓

the final value of x
produced by the specified process

5.5.1

var (P )

set of variables assignable by P

5.5.1

acc(P )

set of variables accessible by P

2.8.2

P vQ

(deterministic) Q can do
at least as much as P

3.9

P vQ

(nondeterministic) Q is
as good as P or better

5.5.1

De

expression e is defined

P

Algebra
Term

Meaning

reflexive

a relation R such that x R x

antisymmetric

a relation R such that x R y ∧ y R x ⇒ x = y

transitive

a relation R such that x R y ∧ y R z ⇒ x R z

partial order

a relation ≤ that is reflexive, antisymmetric, and transitive

bottom

a least element ⊥ such that ⊥≤ x

monotonic

a function f that respects a partial order: x ≤ y ⇒ f (x) ≤ f (y)

strict

a function f that preserves bottom: f (⊥) =⊥

idempotent

a binary operator f such that x f x = x

symmetric

a binary operator f such that x f y = y f x

associative

a binary operator f such that x f (y f z) = (x f y) f z

xx

Glossary of Symbols

distributive

f distributes through g if x f (y g z) = (x f y) g (x f z) and
(y g z) f x = (y f x) g (z f x)

unit

of f is an element 1 such that x f 1 = 1 f x = x

zero

of f is an element 0 such that x f 0 = 0 f x = 0

Graphs
Term

Meaning

graph

a relation drawn as a picture

node

a circle in a graph representing an element in the domain
or range of a relation

arc

a line or arrow in a graph connecting nodes between which
the pictured relation holds

undirected
graph

graph of a symmetric relation

directed
graph

graph of an asymmetric relation often drawn
with arrows

directed
cycle

a set of nodes connected in a cycle by arrows
all in the same direction

undirected
cycle

a set of nodes connected in a cycle by arcs or
arrows in either direction

Contents

Foreword
Preface
Summary

iii
v
ix

Acknowledgements

xiii

Glossary of Symbols

xv

1

Processes
1.1
Introduction
1.2
Pictures
1.3
Laws
1.4
Implementation of processes
1.5
Traces
1.6
Operations on traces
1.7
Implementation of traces
1.8
Traces of a process
1.9
More operations on traces
1.10 Specifications

1
1
12
14
16
19
21
26
27
34
37

2

Concurrency
2.1
Introduction
2.2
Interaction
2.3
Concurrency
2.4
Pictures
2.5
Example: The Dining Philosophers
2.6
Change of symbol
2.7
Specifications
2.8
Mathematical theory of deterministic processes

45
45
45
48
54
55
61
71
72

xxii

Contents

3

Nondeterminism
3.1
Introduction
3.2
Nondeterministic or
3.3
General choice
3.4
Refusals
3.5
Concealment
3.6
Interleaving
3.7
Specifications
3.8
Divergence
3.9
Mathematical theory of non-deterministic processes

81
81
82
86
88
90
99
101
105
108

4

Communication
4.1
Introduction
4.2
Input and output
4.3
Communications
4.4
Pipes
4.5
Subordination

113
113
113
122
131
142

5

Sequential Processes
5.1
Introduction
5.2
Laws
5.3
Mathematical treatment
5.4
Interrupts
5.5
Assignment

153
153
157
158
161
167

6

Shared Resources
6.1
Introduction
6.2
Sharing by interleaving
6.3
Shared storage
6.4
Multiple resources
6.5
Operating systems
6.6
Scheduling

181
181
182
187
189
198
204

7

Discussion
7.1
Introduction
7.2
Shared storage
7.3
Communication
7.4
Mathematical models

207
207
207
218
226

Select Bibliography

233

Index

235

Processes

1.1

1

Introduction

Forget for a while about computers and computer programming, and think
instead about objects in the world around us, which act and interact with us and
with each other in accordance with some characteristic pattern of behaviour.
Think of clocks and counters and telephones and board games and vending
machines. To describe their patterns of behaviour, first decide what kinds of
event or action will be of interest; and choose a different name for each kind.
In the case of a simple vending machine, there are two kinds of event:
coin—the insertion of a coin in the slot of a vending machine;
choc—the extraction of a chocolate from the dispenser of the machine.
In the case of a more complex vending machine, there may be a greater variety
of events:
in1p—the insertion of one penny;
in2p—the insertion of a two penny coin;
small—the extraction of a small biscuit or cookie;
large—the extraction of a large biscuit or cookie;
out1p—the extraction of one penny in change.
Note that each event name denotes an event class; there may be many occurrences of events in a single class, separated in time. A similar distinction
between a class and an occurrence should be made in the case of the letter
‘h’, of which there are many occurrences spatially separated in the text of this
book.
The set of names of events which are considered relevant for a particular
description of an object is called its alphabet. The alphabet is a permanent
predefined property of an object. It is logically impossible for an object to
engage in an event outside its alphabet; for example, a machine designed to
sell chocolates could not suddenly deliver a toy battleship. But the converse

2

1

Processes

does not hold. A machine designed to sell chocolates may actually never do
so—perhaps because it has not been filled, or it is broken, or nobody wants
chocolates. But once it is decided that choc is in the alphabet of the machine,
it remains so, even if that event never actually occurs.
The choice of an alphabet usually involves a deliberate simplification, a
decision to ignore many other properties and actions which are considered to
be of lesser interest. For example, the colour, weight, and shape of a vending
machine are not described, and certain very necessary events in its life, such as
replenishing the stack of chocolates or emptying the coin box, are deliberately
ignored—perhaps on the grounds that they are not (or should not be) of any
concern to the customers of the machine.
The actual occurrence of each event in the life of an object should be
regarded as an instantaneous or an atomic action without duration. Extended
or time-consuming actions should be represented by a pair of events, the first
denoting its start and the second denoting its finish. The duration of an action
is represented by the internal between the occurrence of its start event and the
occurrence of its finish event; during such an internal, other events may occur.
Two extended actions may overlap in time if the start of each one precedes the
finish of the other.
Another detail which we have deliberately chosen to ignore is the exact
timing of occurrences of events. The advantage of this is that designs and
reasoning about them are simplified, and furthermore can be applied to physical and computing systems of any speed and performance. In cases where
timing of responses is critical, these concerns can be treated independently of
the logical correctness of the design. Independence of timing has always been
a necessary condition to the success of high-level programming languages.
A consequence of ignoring time is that we refuse to answer of even to ask
whether one event occurs simultaneously with another. When simultaneity of a
pair of events is important (e.g. in synchronisation) we represent it as a singleevent occurrence; and when it is not, we allow two potentially simultaneous
event occurrences to be recorded in either order.
In choosing an alphabet, there is no need to make a distinction between
events which are initiated by the object (perhaps choc) and those which are
initiated by some agent outside the object (for example, coin). The avoidance
of the concept of causality leads to considerable simplification in the theory
and its application.
Let us now begin to use the word process to stand for the behaviour pattern
of an object, insofar as it can be described in terms of the limited set of events
selected as its alphabet. We shall use the following conventions.
1. Words in lower-case letters denote distinct events, e.g.,
coin, choc, in2p, out1p
and so also do the letters, a, b, c, d, e.
2. Words in upper-case letters denote specific defined processes, e.g.,

1.1

Introduction

3

VMS —the simple vending machine
VMC—the complex vending machine
and the letters P , Q , R (occurring in laws) stand for arbitrary processes.
3. The letters x, y, z are variables denoting events.
4. The letters A, B, C stand for sets of events.
5. The letters X , Y are variables denoting processes.
6. The alphabet of process P is denoted αP , e.g.,
αVMS = {coin, choc}
αVMC = {in1p, in2p, small, large, out1p}
The process with alphabet A which never actually engages in any of the
events of A is called STOPA . This describes the behaviour of a broken object:
although it is equipped with the physical capabilities to engage in the events
of A, it never exercises those capabilities. Objects with different alphabets are
distinguished, even if they never do anything. So STOPαVMS might have given
out a chocolate, whereas STOPαVMC could never give out a chocolate, only a
biscuit. A customer for either machine knows these facts, even if he does not
know that both machines are broken.
In the remainder of this introduction, we shall define some simple notations to aid in the description of objects which actually succeed in doing something.
1.1.1

Prefix

Let x be an event and let P be a process. Then
(x → P )

(pronounced “x then P ”)

describes an object which first engages in the event x and then behaves exactly
as described by P . The process (x → P ) is defined to have the same alphabet as
P , so this notation must not be used unless x is in that alphabet; more formally,
α(x → P ) = αP

provided x ∈ αP

Examples
X1 A simple vending machine which consumes one coin before breaking
(coin → STOPαVMS )

X2 A simple vending machine that successfully serves two customers before
breaking
(coin → (choc → (coin → (choc → STOPαVMS ))))

4

1

Processes

Initially, this machine will accept insertion of a coin in its slot, but will not allow
a chocolate to be extracted. But after the first coin is inserted, the coin slot
closes until a chocolate is extracted. This machine will not accept two coins in
a row, nor will it give out two consecutive chocolates.

In future, we shall omit brackets in the case of linear sequences of events,
like those in X2, on the convention that → is right associative.
X3 A counter starts on the bottom left square of a board, and can move only
up or right to an adjacent white square

αCTR = {up, right }
CTR = (right → up → right → right → STOPαCTR )

Note that the → operator always takes a process on the right and a single event
on the left. If P and Q are processes, it is syntactically incorrect to write
P →Q
The correct method of describing a process which behaves first like P and
then like Q is described in Chapter 5. Similarly, if x and y are events, it is
syntactically incorrect to write
x→y
Such a process could be correctly described
x → (y → STOP )
Thus we carefully distinguish the concept of an event from that of a process
which engages in events—maybe many events or even none.

1.1.2

Recursion

The prefix notation can be used to describe the entire behaviour of a process
that eventually stops. But it would be extremely tedious to write out the full
behaviour of a vending machine for its maximum design life; so we need a

1.1

Introduction

5

method of describing repetitive behaviours by much shorter notations. Preferably these notations should not require a prior decision on the length of life
of an object; this will permit description of objects which will continue to act
and interact with their environment for as long as they are needed.
Consider the simplest possible everlasting object, a clock which never does
anything but tick (the act of winding it is deliberately ignored)
αCLOCK = {tick}
Consider next an object that behaves exactly like the clock, except that it first
emits a single tick
(tick → CLOCK )
The behaviour of this object is indistinguishable from that of the original clock.
This reasoning leads to formulation of the equation
CLOCK = (tick → CLOCK )
This can be regarded as an implicit definition of the behaviour of the clock,
in the same way that the square root of two might be defined as the positive
solution for x in the equation
x = x2 + x − 2
The equation for the clock has some obvious consequences, which are
derived by simply substituting equals for equals
CLOCK
= (tick → CLOCK )
= (tick → (tick → CLOCK ))
= (tick → (tick → (tick → CLOCK )))

[original equation]
[by substitution]
[similarly]

The equation can be unfolded as many times as required, and the possibility of
further unfolding will still be preserved. The potentially unbounded behaviour
of the CLOCK has been effectively defined as
tick → tick → tick → · · ·
in the same way as the square root of two can be thought of as the limit of a
series of decimals
1.414 . . .
This method of self-referential recursive definition of processes will work only
if the right-hand side of the equation starts with at least one event prefixed
to all recursive occurrences of the process name. For example, the recursive

6

1

Processes

equation
X =X
does not succeed in defining anything: everything is a solution to this equation.
A process description which begins with a prefix is said to be guarded. If F (X )
is a guarded expression containing the process name X , and A is the alphabet
of X , then we claim that the equation
X = F (X )
has a unique solution with alphabet A. It is sometimes convenient to denote
this solution by the expression
µ X : A • F (X )
Here X is a local name (bound variable), and can be changed at will, since
µ X : A • F (X ) = µ Y : A • F (Y )
This equality is justified by the fact that a solution for X in the equation
X = F (X )
is also a solution for Y in the equation
Y = F (Y )
In future, we will give recursive definitions of processes either by equations,
or by use of µ, whichever is more convenient. In the case of µ X : A • F (X ), we
shall often omit explicit mention of the alphabet A, where this is obvious from
the content of context of the process.
Examples
X1 A perpetual clock
CLOCK = µ X : {tick} • (tick → X )

X2 At last, a simple vending machine which serves as many chocs as required
VMS = (coin → (choc → VMS ))
As explained above, this equation is just an alternative for the more formal
definition
VMS = µ X : {coin, choc} • (coin → (choc → X ))


1.1

Introduction

7

X3 A machine that gives change for 5p repeatedly
αCH5A = {in5p, out2p, out1p}
CH5A = (in5p → out2p → out1p → out2p → CH5A)

X4 A different change-giving machine with the same alphabet
CH5B = (in5p → out1p → out1p → out1p → out2p → CH5B)

The claim that guarded equations have a solution, and that this solution
may be unique, may be informally justified by the method of substitution.
Each time that the right-hand side of the equation is substituted for every
occurrence of the process name, the expression defining the behaviour of the
process gets longer, and so describes a longer initial segment of behaviour. Any
finite amount of behaviour can be determined in this way. Two objects which
behave the same up to every moment in time have the same behaviour, i.e.,
they are the same process. Those who find this reasoning incomprehensible or
unconvincing should accept this claim as an axiom, whose value and relevance
will gradually become more apparent. A more formal proof cannot be given
without some mathematical definition of exactly what a process is. This will
be done in Section 2.8.3. The account of recursion given here relies heavily on
guardedness of recursive equations. A meaning for unguarded recursions will
be discussed in Section 3.8.
1.1.3

Choice

By means of prefixing and recursion it is possible to describe objects with a
single possible stream of behaviour. However, many objects allow their behaviour to be influenced by interaction with the environment within which they
are placed. For example, a vending machine may offer a choice of slots for
inserting a 2p or a 1p coin; and it is the customer that decides between these
two events. If x and y are distinct events
(x → P | y → Q )
describes an object which initially engages in either of the events x or y. After
the first event has occurred, the subsequent behaviour of the object is described by P if the first event was x, or by Q if the first event was y. Since x
and y are different events, the choice between P and Q is determined by the
first event that actually occurs. As before, we insist on constancy of alphabets,
i.e.,
α(x → P | y → Q ) = αP

provided {x, y} ⊆ αP and αP = αQ

The bar | should be pronounced “choice”: “x then P choice y then Q ”

8

1

Processes

Examples
X1 The possible movements of a counter on the board

are defined by the process
(up → STOP | right → right → up → STOP )

X2 A machine which offers a choice of two combinations of change for 5p
(compare 1.1.2 X3 and X4, which offer no choice).
CH5C = in5p → (out1p → out1p → out1p → out2p → CH5C
| out2p → out1p → out2p → CH5C)
The choice is exercised by the customer of the machine.



X3 A machine that serves either chocolate or toffee on each transaction
VMCT = µ X • coin → (choc → X | toffee → X )

X4 A more complicated vending machine, which offers a choice of coins and
a choice of goods and change
VMC = (in2p → (large → VMC
| small → out1p → VMC)
| in1p → (small → VMC
| in1p → (large → VMC
| in1p → STOP )))
Like many complicated machines, this has a design flaw. It is often easier to
change the user manual than correct the design, so we write a notice on the
machine
“WARNING: do not insert three pennies in a row.”


1.1

Introduction

9

X5 A machine that allows its customer to sample a chocolate, and trusts him
to pay after. The normal sequence of events is also allowed
VMCRED = µ X • (coin → choc → X
| choc → coin → X )

X6 To prevent loss, an initial payment is extracted for the privilege of using
VMCRED
VMS2 = (coin → VMCRED)
This machine will allow insertion of up to two consecutive coins before extraction of up to two consecutive chocolates; but it will never give out more
chocolates than have been previously paid for.

X7 A copying process engages in the following events
in.0—input of zero on its input channel
in.1—input of one on its input channel
out .0—output of zero on its output channel
out .1—output of one on its output channel
Its behaviour consists of a repetition of pairs of events. On each cycle, it inputs
a bit and outputs the same bit
COPYBIT = µ X • (in.0 → out .0 → X
| in.1 → out .1 → X )
Note how this process allows its environment to choose which value should
be input, but no choice is offered in the case of output. That will be the main
difference between input and output in our treatment of communication in
Chapter 4.

The definition of choice can readily be extended to more than two alternatives, e.g.,
(x → P | y → Q | . . . | z → R)
Note that the choice symbol | is not an operator on processes; it would be
syntactically incorrect to write P | Q , for processes P and Q . The reason for
this rule is that we want to avoid giving a meaning to
(x → P | x → Q )
which appears to offer a choice of first event, but actually fails to do so.
This problem is solved, at the expense of introducing nondeterminism, in Sec-

10

1

Processes

tion 3.3. Meanwhile, if x, y, z are distinct events,
(x → P | y → Q | z → R)
should be regarded as a single operator with three arguments P , Q , R. It cannot
be regarded as equal to
(x → P | (y → Q | z → R))
which is syntactically incorrect.
In general, if B is any set of events and P (x) is an expression defining a
process for each different x in B, then
(x : B → P (x))
defines a process which first offers a choice of any event y in B, and then
behaves like P (y). It should be pronounced “x from B then P of x”. In this
construction, x is a local variable, so
(x : B → P (x)) = (y : B → P (y))
The set B defines the initial menu of the process, since it gives the set of actions
between which a choice is to be made at the start.
Examples
X8 A process which at all times can engage in any event of its alphabet A
αRUNA = A
RUNA = (x : A → RUNA )

In the special case that the menu contains only one event e,
(x : {e} → P (x)) = (e → P (e))
since e is the only possible initial event. In the even more special case that the
initial menu is empty, nothing at all can happen, so
(x : {} → P (x)) = (y : {} → Q (y)) = STOP
The binary choice operator | can also be defined using the more general notation
(a → P | b → Q ) = (x : B → R(x))
where B = {a, b} and R(x) =if x = a then P else Q

1.1

Introduction

11

Choice between three or more alternatives can be similarly expressed.
Thus choice, prefixing and STOP are defined as just special cases of the general choice notation. This will be a great advantage in the formulation of general laws governing processes (Section 1.3), and in their implementation (Section 1.4).

1.1.4

Mutual recursion

Recursion permits the definition of a single process as the solution of a single
equation. The technique is easily generalised to the solution of sets of simultaneous equations in more than one unknown. For this to work properly, all
the right-hand sides must be guarded, and each unknown process must appear
exactly once on the left-hand side of one of the equations.
Example
X1 A drinks dispenser has two buttons labelled ORANGE and LEMON. The
actions of pressing the two buttons are setorange and setlemon. The actions of
dispensing a drink are orange and lemon. The choice of drink that will next be
dispensed is made by pressing the corresponding button. Before any button
is pressed, no drink will be dispensed. Here are the equations defining the
alphabet and behaviour of the process DD. The definition uses two auxiliary
definitions of O and L, which are mutually recursive
αDD = αO = αL = {setorange, setlemon, orange, lemon}
DD = (setorange → O | setlemon → L)
O = (orange → O | setlemon → L | setorange → O)
L = (lemon → O | setorange → O | setlemon → L)
Informally, after the first event (which must be a setting event) the dispenser
is in one of two states O or L. In each state, it may serve the appropriate drink
or be set into the other state. Pressing the button for the same state is allowed,
but has no effect.

By using indexed variables, it is possible to specify infinite sets of equations.
Examples
X2 An object starts on the ground, and may move up. At any time thereafter
it may move up and down, except that when on the ground it cannot move any
further down. But when it is on the ground, it may move around. Let n range
over the natural numbers {0, 1, 2, . . .}. For each n, introduce the indexed name
CTn to describe the behaviour of the object when it is n moves off the ground.
Its initial behaviour is defined as
CT0 = (up → CT1 | around → CT0 )

12

1

Processes

and the remaining infinite set of equations consists of
CTn+1 = (up → CTn+2 | down → CTn )
where n ranges over the natural numbers 0, 1, 2, …
An ordinary inductive definition is one whose validity depends on the fact
that the right-hand side of each equation uses only indices less than that of
the left-hand side. Here, CTn+1 is defined in terms of CTn+2 , and so this can be
regarded only as an infinite set of mutually recursive definitions, whose validity
depends on the fact that the right-hand side of each equation is guarded.


1.2

Pictures

It may be helpful sometimes to make a pictorial representation of a process as
a tree structure, consisting of circles connected by arrows. In the traditional
terminology of state machines, the circles represent states of the process, and
the arrows represent transitions between the states. The single circle at the
root of the tree (usually drawn at the top of the page) is the starting state; and
the process moves downward along the arrows. Each arrow is labelled by the
event which occurs on making that transition. The arrows leading from the
same node must all have different labels.
Examples (1.1.1 X1, X2; 1.1.3 X3)

X1

X2
coin

X3
coin

coin

choc

coin

choc

toffee

choc

coin

coin

choc

toffee


In these three examples, every branch of each tree ends in STOP , represented as a circle with no arrows leading out of it. To represent processes with
unbounded behaviour it is necessary to introduce another convention, namely

1.2

Pictures

13

an unlabelled arrow leading from a leaf circle back to some earlier circle in the
tree. The convention is that when a process reaches the node at the tail of the
arrow, it immediately and imperceptibly goes back to the node to which the
arrow points.

X4

X5
coin

choc

coin

choc

toffee

toffee

coin

choc

toffee


Clearly, these two pictures illustrate exactly the same process (1.1.3 X3). It is
one of the weaknesses of pictures that proofs of such an equality are difficult
to conduct pictorially.
Another problem with pictures is that they cannot illustrate processes with
a very large or infinite number of states, for example CT0

around

up

down

up

down

up

There is never enough room to draw the whole picture. A counter with only
65 536 different states would take a long time to draw.

14

1.3

1

Processes

Laws

Even with the very restricted set of notations introduced so far, there are many
different ways of describing the same behaviour. For example, it obviously
should not matter in which order a choice between events is presented
(x → P | y → Q ) = (y → Q | x → P )
On the other hand, a process that can do something is not the same as one
that cannot do anything
(x → P ) ≠ STOP
In order to understand a notation properly and to use it effectively, we must
learn to recognise which expressions describe the same object and which do
not, just as everyone who understands arithmetic knows that (x + y) is the
same number as (y + x). Identity of processes with the same alphabet may be
proved or disproved by appeal to algebraic laws very like those of arithmetic.
The first law (L1 below) deals with the choice operator (1.1.3). It states that
two processes defined by choice are different if they offer different choices on
the first step, or if after the same first step the behave differently. However,
if the initial choices are the same, and for each initial choice the subsequent
behaviours are the same, then obviously the processes are identical.
L1 (x : A → P (x)) = (y : B → Q (y)) ≡ (A = B ∧ ∀ x : A • P (x) = Q (x))
Here and elsewhere, we assume without stating it that the alphabets of the
processes on each side of an equation are the same.
The law L1 has a number of consequences
L1A

STOP ≠ (d → P )

Proof : LHS = (x : {} → P )

L1B

by definition (1.1.3 end)

≠ (x : {d} → P )

because {} ≠ {d}

= RHS

by definition (1.1.3 end)

(c → P ) ≠ (d → Q )

if c ≠ d

Proof : {c} ≠ {d}
L1C

(c → P | d → Q ) = (d → Q | c → P )

Proof : Define R(x) = P

if x = c

=Q

if x = d

LHS = (x : {c, d} → R(x))

by definition

= (x : {d, c} → R(x))

because {c, d} = {d, c}

= RHS

by definition

1.3

L1D

Laws

15

(c → P ) = (c → Q ) ≡ P = Q

Proof : {c} = {c}
These laws permit proof of simple theorems.
Examples
X1 (coin → choc → coin → choc → STOP ) ≠ (coin → STOP )
Proof : by L1D then L1A.



X2 µ X • (coin → (choc → X | toffee → X ))
= µ X • (coin → (toffee → X | choc → X ))
Proof : by L1C.



To prove more general theorems about recursively-defined processes, it is
necessary to introduce a law which states that every properly guarded recursive
equation has only one solution.
L2 If F (X ) is a guarded expression,
(Y = F (Y )) ≡ (Y = µ X • F (X ))

An immediate but important corollary states that µ X • F (X ) is indeed a solution of the relevant equation
L2A

µ X • F (X ) = F (µ X • F (X ))
Example

X3 Let VM1 = (coin → VM2) and VM2 = (choc → VM1)
Required to prove VM1 = VMS .
Proof : VM1 = (coin → VM2)
= (coin → (choc → VM1))

definition of VM1
definition of VM2

Therefore VM1 is a solution of the same recursive equation as VMS . Since
the equation is guarded, there is only one solution. So VM1 and VMS are just
different names for this unique solution.
This theorem may seem so obviously true that its proof in no way adds
to its credibility. The only purpose of the proof is to show by example that
the laws are powerful enough to establish facts of this kind. When proving
obvious facts from less obvious laws, it is important to justify every line of the
proof in full, as a check that the proof is not circular.


16

1

Processes

The law L2 can be extended to mutual recursion. A set of mutually recursive equations can be written in the general form using subscripts
Xi = F (i, X )

for all i in S

where
S is an indexing set with one member for each equation, and
X is an array of processes with indices ranging over the set S , and
F (i, X ) is a guarded expression.
Under these conditions, the law L3 states that there is only one array X whose
elements satisfy all the equations
L3 Under the conditions explained above,
if (∀ i : S • (Xi = F (i, X ) ∧ Yi = F (i, Y ))) then X = Y

1.4

Implementation of processes

Every process P expressible in the notations introduced so far can be written
in the form
(x : B → F (x))
where F is a function from symbols to processes, and where B may be empty
(in the case of STOP ), or may contain only one member (in the case of prefix),
or may contain more than one member (in the case of choice). In the case of
a recursively defined process, we have insisted that the recursion should be
guarded, so that it may be written
µ X • (x : B → F (x, X ))
and this may be unfolded to the required form using L2A
(x : B → F (x, µ X • (x : B → F (x, X ))))
Thus every process may be regarded as a function F with a domain B, defining
the set of events in which the process is initially prepared to engage; and for
each x in B, F (x) defines the future behaviour of the process if the first event
was x.
This insight permits every process to be represented as a function in some
suitable functional programming language such as LISP. Each event in the alphabet of a process is represented as an atom, for example "COIN , "TOFFEE .
A process is a function which can be applied to such a symbol as an argument.

1.4

Implementation of processes

17

If the symbol is not a possible first event for the process, the function gives
as its result a special symbol "BLEEP , which is used only for this purpose. For
example, since STOP never engages in any event, this is the only result it can
ever give, and so it is defined
STOP = λ x • "BLEEP
But if the actual argument is a possible event for the process, the function gives
back as its result another function, representing the subsequent behaviour of
the process. Thus (coin → STOP) is represented as the function
λ x • if x = "COIN then
STOP
else
"BLEEP
This last example takes advantage of the facility of LISP for returning a
function (e.g., STOP ) as the result of a function. LISP also allows a function
to be passed as an argument to a function, a facility used in representing a
general prefix operation (c → P)
prefix(c, P ) = λ x • if x = c then
P
else
"BLEEP
A function to represent a general binary choice (c → P | d → Q) requires
four parameters
choice2(c, P , d, Q ) = λ x • if x = c then
P
else if x = d then
Q
else
"BLEEP
Recursively defined processes may be represented with the aid of the
LABEL feature of LISP. For example, the simple vending machine process
(µ X • coin → choc → X) is represented as
LABEL X • prefix("COIN , prefix("CHOC, X ))
The LABEL may also be used to represent mutual recursion. For example,
CT (1.1.4 X2) may be regarded as a function from natural numbers to processes

18

1

Processes

(which are themselves functions—but let not that be a worry). So CT may be
defined
CT = LABEL X • (λ n • if n = 0 then
choice2("AROUND, X (0), "UP , X (1))
else
choice2("UP , X (n + 1), "DOWN , X (n − 1)))
The process that starts on the ground is CT (0).
If P is a function representing a process, and A is a list containing the
symbols of its alphabet, the LISP function menu(A, P ) gives a list of all those
symbols of A which can occur as the first event in the life of P
menu(A, P ) = if A = NIL then
NIL
else if P (car (A)) = "BLEEP then
menu(cdr (A), P )
else
cons(car (A), menu(cdr (A), P ))
If x is in menu(A, P ), P (x) is not "BLEEP , and is therefore a function defining the future behaviour of P after engaging in x. Thus if y is in menu(A, P (x)),
then P (x)(y) will give its later behaviour, after both x and y have occurred. This
suggests a useful method of exploring the behaviour of a process. Write a program which first outputs the value of menu(A, P ) on a screen, and then inputs
a symbol from the keyboard. If the symbol is not in the menu, it should be
greeted with an audible bleep and then ignored. Otherwise the symbol is accepted, and the process is repeated with P replaced by the result of applying P
to the accepted symbol. The process is terminated by typing an "END symbol.
Thus if k is the sequence of symbols input from the keyboard, the following
function gives the sequence of outputs required
interact (A, P , k) =
cons(menu(A, P ), if car (k) = "END then
NIL
else if P (car (k)) = "BLEEP then
cons("BLEEP , interact (A, P , cdr (k)))
else
interact (A, P (car (k), cdr (k)))
The notations used above for defining LISP functions are very informal,
and they will need to be translated to the specific conventional S-expression

1.5

Traces

19

form of some particular implementation of LISP. For example in LISPkit, the
prefix function can be defined
(prefix
lambda
(a p)
(lambda (x) (if (eq x a) p (quote BLEEP ))))
Fortunately, we shall use only a very small subset of pure functional LISP, so
there should be little difficulty in translating and running these processes in a
variety of dialects on a variety of machines.
If there are several versions of LISP available, choose one with proper static
binding of variables. A LISP with lazy evaluation is also more convenient, since
it permits direct encoding of recursive equations, without using LABEL, thus
VMS = prefix("COIN , prefix("CHOC, VMS ))
If input and output are implemented by lazy evaluation, the interact function
may be called with the keyboard as its third parameter; and the menu for the
process P will appear as the first output. By selecting and inputting a symbol
from the successive menus, a user can interactively explore the behaviour of
the process P .
In other versions of LISP, the interact function should be rewritten, using
explicit input and output to achieve the same effect. When this has been done,
it is possible to observe the computer executing any process that has been
represented as a LISP function. In this sense, such a LISP function may be
regarded as an implementation of the corresponding process. Furthermore, a
LISP function such as prefix which operates on these representations may be
regarded as the implementation of the corresponding operator on processes.

1.5

Traces

A trace of the behaviour of a process is a finite sequence of symbols recording
the events in which the process has engaged up to some moment in time.
Imagine there is an observer with a notebook who watches the process and
writes down the name of each event as it occurs. We can validly ignore the
possibility that two events occur simultaneously; for if they did, the observer
would still have to record one of them first and then the other, and the order
in which he records them would not matter.
A trace will be denoted as a sequence of symbols, separated by commas
and enclosed in angular brackets
hx, yi consists of two events, x followed by y.
hxi is a sequence containing only the event x.
hi is the empty sequence containing no events.

20

1

Processes

Examples
X1 A trace of the simple vending machine VMS (1.1.2 X2) at the moment it
has completed service of its first two customers
hcoin, choc, coin, choci

X2 A trace of the same machine before the second customer has extracted his
choc
hcoin, choc, coini
Neither the process nor its observer understands the concept of a completed
transaction. The hunger of the expectant customer, and the readiness of the
machine to satisfy it are not in the alphabet of these processes, and cannot be
observed or recorded.

X3 Before a process has engaged in any events, the notebook of the observer
is empty. This is represented by the empty trace
hi
Every process has this as its shortest possible trace.



X4 The complex vending machine VMC (1.1.3 X4) has the following seven
traces of length two or less
hi
hin2pi
hin2p, largei

hin2p, smalli

hin1pi
hin1p, in1pi

hin1p, smalli

Only one of the four traces of length two can actually occur for a given machine.
The choice between them will be determined by the wishes of the first customer
to use the machine.

X5 A trace of VMC if its first customer has ignored the warning is
hin1p, in1p, in1pi
The traces does not actually record the breakage of the machine. Breakage is
only indicated by the fact that among all the possible traces of the machine
there is no trace which extends this one, i.e., there is no event x such that
hin1p, in1p, in1p, xi

1.6

Operations on traces

21

is a possible trace of VMC, The customer may fret and fume; the observer may
watch eagerly with pencil poised; but not another single event can occur, and
not another symbol will ever be written in the notebook. The ultimate disposal
of customer and machine are not in our chosen alphabet.


1.6

Operations on traces

Traces play a central role in recording, describing, and understanding the behaviour of processes. In this section we explore some of the general properties
of traces and of operations on them. We will use the following conventions
s, t , u stand for traces
S , T , U stand for sets of traces
f , g, h stand for functions
1.6.1

Catenation

By far the most important operation on traces is catenation, which constructs
a trace from a pair of operands s and t by simply putting them together in this
order; the result will be denoted
s _t
For example
hcoin, choci _ hcoin, toffeei = hcoin, choc, coin, toffeei
hin1pi _ hin1pi = hin1p, in1pi
hin1p, in1pi _ hi = hin1p, in1pi
The most important properties of catenation are that it is associative, and
has hi as its unit
L1 s _ hi = hi _ s = s
L2 s _ (t _ y) = (s _ t ) _ u
The following laws are both obvious and useful
L3 s _ t = s _ u ≡ t = u
L4 s _ t = u _ t ≡ s = u
L5 s _ t = hi ≡ s = hi ∧ t = hi
Let f stand for a function which maps traces to traces. The function is
said to be strict if it maps the empty trace to the empty trace
f (hi) = hi

22

1

Processes

It is said to be distributive if it distributes through catenation
f (s _ t ) = f (s) _ f (t )
All distributive functions are strict.
If n is a natural number, we define t n as n copies of t catenated with each
other. It is readily defined by induction on n
L6 t 0 = hi
L7 t n+1 = t _ t n
This definition itself gives two useful laws; here are two more which can be
proved from them
L8 t n+1 = t n _ t
L9 (s _ t )n+1 = s _ (t _ s)n _ t

1.6.2

Restriction

The expression (t u A) denotes the trace t when restricted to symbols in the set
A; it is formed from t simply by omitting all symbols outside A. For example
haround, up, down, aroundi u {up, down} = hup, downi
Restriction is distributive and therefore strict
L1 hi u A = hi
L2 (s _ t ) u A = (s u A) _ (t u A)
Its effect on singleton sequences is obvious
L3 hxi u A = hxi

if x ∈ A

L4 hyi u A = hi

if y ∉ A

A distributive function is uniquely defined by defining its effect on singleton
sequences, since its effect on all longer sequences can be calculated by distributing the function to each individual element of the sequence and catenating
the results. For example, if y ≠ x
hx, y, xi u {x}
= (hxi _ hyi _ hxi) u {x}
= (hxi u {x}) _ (hyi u {x}) _ (hxi u {x})
= hxi _ hi _ hxi
= hx, xi

[by L2]
[by L3 and L4]

1.6

Operations on traces

23

The following laws show the relationship between restriction and set operations. A trace restricted to the empty set of symbols leaves nothing; and a
successive restriction by two sets is the same as a single restriction by the intersection of the two sets. These laws can be proved by induction on the length
of s
L5 s u {} = hi
L6 (s u A) u B = s u (A ∩ B)
1.6.3

Head and tail

If s is a nonempty sequence, its first sequence is denoted s0 , and the result of
removing the first symbol is s 0 . For example
hx, y, xi0 = x
hx, y, xi0 = hy, xi
Both of these operations are undefined for the empty sequence.
L1 (hxi _ s)0 = x
L2 (hxi _ s)0 = s
L3 s = (hs0 i _ s 0 )

if s ≠ hi

The following law gives a convenient method of proving whether two traces
are equal
L4 s = t ≡ (s = t = hi ∨ (s0 = t0 ∧ s 0 = t 0 ))
1.6.4

Star

The set A∗ is the set of all finite traces (including hi) which are formed from
symbols in the set A. When such traces are restricted to A, they remain unchanged. This fact permits a simple definition
A∗ = { s | s u A = s }
The following laws are consequences of this definition
L1 hi ∈ A∗
L2 hxi ∈ A∗ ≡ x ∈ A
L3 (s _ t ) ∈ A∗ ≡ s ∈ A∗ ∧ t ∈ A∗
They are sufficiently powerful to determine whether a trace is a member of A∗
or not. For example, if x ∈ A and y ∉ A
hx, yi ∈ A∗

24

1

Processes

≡ (hxi _ hyi) ∈ A∗
≡ (hxi ∈ A∗ ) ∧ (hyi ∈ A∗ )

[by L3]

≡ true ∧ false

[by L2]

≡ false
Another useful law could serve as a recursive definition of A∗
L4 A∗ = { t | t = hi ∨ (t0 ∈ A ∧ t 0 ∈ A∗ ) }
1.6.5

Ordering

If s is a copy of an initial subsequence of t , it is possible to find some extension
u of s such that s _ u = t . We therefore define an ordering relation
s ≤ t = (∃ u • s _ u = t )
and say that s is a prefix of t . For example,
hx, yi ≤ hx, y, x, wi
hx, yi ≤ hz, y, xi ≡ x = z
The ≤ relation is a partial ordering, and its least element is hi, as stated in
laws L1 to L4
L1 hi ≤ s
L2 s ≤ s
L3 s ≤ t ∧ t ≤ s ⇒ s = t
L4 s ≤ t ∧ t ≤ u ⇒ s ≤ u

least element
reflexive
antisymmetric
transitive

The following law, together with L1, gives a method for computing whether
s ≤ t or not
L5 (hxi _ s) ≤ t ≡ t ≠ hi ∧ x = t0 ∧ s ≤ t 0
The prefixes of a given subsequence are totally ordered
L6 s ≤ u ∧ t ≤ u ⇒ s ≤ t ∨ t ≤ s
If s is a subsequence of t (not necessarily initial), we say s is in t ; this may
be defined
L7 s in t = (∃ u, v • t = u _ s _ v)
This relation is also a partial ordering, in that it satisfies laws L1 to L4 above.
It also satisfies
L8 (hxi _ s) in t ≡ t ≠ hi ∧ ((t0 = x ∧ s ≤ t 0 ) ∨ (hxi _ s in t 0 ))

1.6

Operations on traces

25

A function f from traces to traces is said to be monotonic if it respects the
ordering ≤, i.e.,
f (s) ≤ f (t )

whenever s ≤ t

All distributive functions are monotonic, for example
L9 s ≤ t ⇒ (s u A) ≤ (t u A)
A dyadic function may be monotonic in either argument, keeping the other argument constant. For example, catenation is monotonic in its second argument
(but not its first)
t ≤ u ⇒ (s _ t ) ≤ (s _ u)

L10

A function which is monotonic in all its arguments is said simply to be monotonic.

1.6.6

Length

The length of the trace t is denoted #t . For example
#hx, y, xi = 3
The laws which define # are
L1 #hi = 0
L2 #hxi = 1
L3 #(s _ t ) = (#s) + (#t )
The number of occurrences in t of symbols from A is counted by #(t u A).
L4 #(t u (A ∪ B)) =
#(t u A) + #(t u B) −
#(t u (A ∩ B))
L5 s ≤ t ⇒ #s ≤ #t
L6 #(t n ) = n × (#t )
The number of occurrences of a symbol x in a trace s is defined
s ↓ x = #(s u {x})

26

1

1.7

Processes

Implementation of traces

In order to represent traces in a computer and to implement operations on
them, we need a high-level list-processing language. Fortunately, LISP is very
suitable for this purpose. Traces are represented in the obvious way by lists
of atoms representing their events
hi = NIL
hcoini = (cons("COIN , NIL))
hcoin, choci = " (COIN CHOC)
which means cons("COIN , cons("CHOC, NIL))
Operations on traces can be readily implemented as functions on lists.
For example, the head and tail of a nonempty list are given by the primitive
functions car and cdr

t0 = car (t )
t 0 = cdr (t )
hx i _ s = cons(x, s)
General catenation is implemented as the familiar append function, which is
defined by recursion

s _ t = append(s, t )
where
append(s, t ) =
if s = NIL then
t
else
cons(car (s), append(cdr (s), t ))
The correctness of this definition follows from the laws
hi _ t = t

s _ t = hs0 i _ (s 0 _ t )
whenever s ≠ hi
The termination of the LISP append function is guaranteed by the fact that the
list supplied as the first argument of each recursive call is shorter than it was
at the previous level of recursion. Similar arguments establish the correctness
of the implementations of the other operations defined below.

1.8

Traces of a process

27

To implement restriction, we represent a finite set B as a list of its members. The test (x ∈ B ) is accomplished by a call on the function
ismember (x, B) =
if B = NIL then
false
else if x = car (B) then
true
else
ismember (x, cdr (B))
(s u B ) can now be implemented by the function
restrict (s, B) =
if s = NIL then
NIL
else if ismember (car (s), B) then
cons(car (s), restrict (cdr (s), B))
else
restrict (cdr (s), B)
A test of (s ≤ t ) is implemented as a function which delivers the answer
true or false; it relies on 1.6.5 L1 and L5
isprefix(s, t ) = if s = NIL then
true
else if t = NIL then
false
else
car (s) = car (t ) and
isprefix(cdr (s), cdr (t ))

1.8

Traces of a process

In Section 1.6 a trace of a process was introduced as a sequential record of
the behaviour of a process up to some moment in time. Before the process
starts, it is not known which of its possible traces will actually be recorded:
the choice will depend on environmental factors beyond the control of the
process. However the complete set of all possible traces of a process P can be
known in advance, and we define a function traces(P ) to yield that set.

28

1

Processes

Examples
X1 The only trace of the behaviour of the process STOP is hi. The notebook
of the observer of this process remains forever blank
traces(STOP ) = {hi}

X2 There are only two traces of the machine that ingests a coin before breaking
traces(coin → STOP ) = {hi, hcoini}

X3 A clock that does nothing but tick
traces(µ X • tick → X ) = {hi, hticki, htick, ticki, . . .}
= {tick}∗
As with most interesting processes, the set of traces is infinite, although of
course each individual trace is finite.

X4 A simple vending machine
traces(µ X • coin → choc → X ) = { s | ∃ n • s ≤ hcoin, chocin }

1.8.1

Laws

In this section we show how to calculate the set of traces of any process defined
using the notations introduced so far. As mentioned above, STOP has only one
trace
L1 traces(STOP ) = { t | t = hi } = {hi}
A trace of (c → P ) may be empty, because hi is a trace of the behaviour of
every process up to the moment that it engages in its very first action. Every
nonempty trace begins with c, and its tail must be a possible trace of P
L2 traces(c → P ) = { t | t = hi ∨ (t0 = c ∧ t 0 ∈ traces(P )) }
= {hi} ∪ { hci _ t | t ∈ traces(P ) }
A trace of the behaviour of a process which offers a choice between initial
events must be a trace of one of the alternatives
L3 traces(c → P | d → Q ) =
{ t | t = hi ∨ (t0 = c ∧ t 0 ∈ traces(P )) ∨ (t0 = d ∧ t 0 ∈ traces(Q )) }

1.8

Traces of a process

29

These three laws are summarised in the single general law governing choice
L4 traces(x : B → P (x)) = { t | t = hi ∨ (t0 ∈ B ∧ t 0 ∈ traces(P (t0 ))) }
To discover the set of traces of a recursively defined process is a bit more
difficult. A recursively defined process is the solution of an equation
X = F (X )
First, we define iteration of the function F by induction
F 0 (X ) = X
F n+1 (X ) = F (F n (X ))
= F n (F (X ))
= F (. . . (F (F (X ))) . . .)
| {z }
n times

Then, provided that F is guarded, we can define
L5 traces(µ X : A • F (X )) =

S

n≥0

traces(F n (STOPA ))

Examples
X1 Recall that RUNA was defined in 1.1.3 X8 as
µ X : A • F (X )
where F (X ) = (x : A → X )
We wish to prove that
traces(RUNA ) = A∗
Proof : A∗ =

S

n≥0 { s

| s ∈ A∗ ∧ #s ≤ n }

This is done by induction on n.
1. traces(STOPA )
= {hi}
= { s | s ∈ A∗ ∧ #s ≤ 0 }
2. traces(F n+1 (STOPA ))
= traces(x : A → F n (STOPA ))

[def. F , F n+1 ]

= { t | t = hi ∨ (t0 ∈ A ∧ t 0 ∈ traces(F n (STOPA ))) }
0

∗

0

∗

0

= { t | t = hi ∨ (t0 ∈ A ∧ (t ∈ A ∧ #t ≤ n)) }
= { t | (t = hi ∨ (t0 ∈ A ∧ t ∈ A )) ∧ #t ≤ n + 1 }
∗

= { t | t ∈ A ∧ #t ≤ n + 1 }

[L4]
[ind. hyp.]
[property of #]
[1.6.4 L4]


30

1

Processes

X2 We want to prove 1.5 X4, i.e.,
traces(VMS ) =

S

n≥0 { s

| s ≤ hcoin, chocin }

Proof : The inductive hypothesis is
traces(F n (VMS )) = { t | t ≤ hcoin, chocin }
where F (X ) = coin → choc → X
1. traces(STOP ) = {hi} = { s | s ≤ hcoin, choci0 }

[1.6.1 L6]

2. traces(coin → choc → F n (STOP ))
= {hi, hcoini} ∪

[L2 twice]

{ hcoin, choci _ t | t ∈ traces(F n (STOP )) }
= {hi, hcoini} ∪

[ind. hyp.]

{ hcoin, choci _ t | t ≤ hcoin, chocin }
= { s | s = hi ∨ s = hcoini ∨
∃ t • s = hcoin, choci _ t ∧ t ≤ hcoin, chocin }
= { s | s ≤ hcoin, chocin+1 }
The conclusion follows by L5.



As mentioned in Section 1.5, a trace is a sequence of symbols recording
the events in which a process P has engaged up to some moment in time. From
this it follows that hi is a trace of every process up to the moment in which it
engages in its very first event. Furthermore, if (s _ t ) is a trace of a process up
to some moment, then s must have been a trace of that process up to some
earlier moment. Finally, every event that occurs must be in the alphabet of the
process. These three facts are formalised in the laws
L6 hi ∈ traces(P )
L7 s _ t ∈ traces(P ) ⇒ s ∈ traces(P )
L8 traces(P ) ⊆ (αP )∗
There is a close relationship between the traces of a process and the picture
of its behaviour drawn as a tree. For any node on the tree, the trace of the
behaviour of a process up to the time when it reaches that node is just the
sequence of labels encountered on the path leading from the root of the tree
to that node. For example, in the tree for VMC shown in Figure 1.1, the trace
corresponding to the path from the root to the black node is
hin2p, small, out1pi

1.8

Traces of a process

31
VMC
in1p

in2p

large

in2p

in1p

small

in1p

out1p

in2p

in1p

large

small

in2p

in1p

in1p

Figure 1.1
Clearly, all initial subpaths of a path in a tree are also paths in the same
tree; this is stated more formally in L7 above. The empty trace defines the path
from the root to itself, which justifies the law L6. The traces of a process are
just the set of paths leading from the root to some node in the tree.
Conversely, because the branches leading from each node are all labelled
with different events, each trace of a process uniquely specifies a path leading
from the root of a tree to a particular node. Thus any set of traces satisfying
L6 and L7 constitutes a convenient mathematical representation for a tree with
no duplicate labels on branches emerging from a single node.
1.8.2

Implementation

Suppose a process has been implemented as a LISP function P , and let s be
a trace. Then it is possible to test whether s is a possible trace of P by the
function
istrace(s, P ) = if s = NIL then
true
else if P (car (s)) = "BLEEP then
false
else
istrace(cdr (s), P (car (s)))
Since s is finite, the recursion involved here will terminate, having explored
only a finite initial segment of the behaviour of the process P . It is because
we avoid infinite exploration that we can safely define a process as an infinite
object, i.e., a function whose result is a function whose result is a function
whose result…

32

1

1.8.3

Processes

After

If s ∈ traces(P ) then
P /s

(P after s)

is a process which behaves the same as P behaves from the time after P has
engaged in all the actions recorded in the trace s. If s is not a trace of P , (P / s)
is not defined.
Examples
X1 (VMS / hcoini) = (choc → VMS )



X2 (VMS / hcoin, choci) = VMS



X3 (VMC / hin1pi3 ) = STOP



X4 To avoid loss arising from installation of VMCRED (1.1.3 X5, X6), the owner
decides to eat the first chocolate himself
(VMCRED / hchoci) = VMS2

In a tree picture of P (Figure 1.1), (P / s) denotes the whole subtree whose
root lies at the end of the path labelled by the symbols of s. Thus the subtree
below the black node in the Figure 1.1 is denoted by
VMC / hin2p, small, out1pi
The following laws describe the meaning of the operator /. After doing
nothing, a process remains unchanged
L1 P / hi = P
After engaging in s _ t , the behaviour of P is the same as that of (P / s) after
engaging in t
L2 P / (s _ t ) = (P / s) / t
After engaging in a single event c, the behaviour of a process is as defined by
this initial choice
L3 (x : B → P (x)) / hci = P (c)

provided that c ∈ B

A corollary shows that / hci is the inverse of the prefixing operator c →
L3A

(c → P ) / hci = P

The traces of (P / s) are defined

1.8

Traces of a process

L4 traces(P / hsi) = { t | s _ t ∈ traces(P ) }

33

provided that s ∈ traces(P )

In order to prove that a process P never stops it is sufficient to prove that
∀ s : traces(P ) • P / s ≠ STOP
Another desirable property of a process is cyclicity; a process P is defined as
cyclic if in all circumstances it is possible for it to return to its initial state, i.e.,
∀ s : traces(P ) • ∃ t • (P / (s _ t ) = P )
STOP is trivially cyclic; but if any other process is cyclic, then it also has the
desirable property of never stopping.
Examples
X5 The following processes are cyclic (1.1.3 X8, 1.1.2 X2, 1.1.3 X3, 1.1.4 X2)
RUNA , VMS , (choc → VMS ), VMCT , CT7

X6 The following are not cyclic, because it is not possible to return them to
their initial state ( 1.1.2 X2, 1.1.3 X3, 1.1.4 X2)
(coin → VMS ), (choc → VMCT ), (around → CT7 )
For example, in the initial state of choc → VMCT only a chocolate is obtainable,
but subsequently whenever choc is obtainable a choice of toffee is also possible;
consequently none of these subsequent states is equal to the initial state.

Warning: The use of / in a recursively defined process has the unfortunate consequence of invalidating its guards, thereby introducing the danger of multiple
solutions to the recursive equations. For example
X = (a → (X / hai))
is not guarded, and has as its solution any process of the form
a→P
for any P .
Proof : (a → ((a → P ) / hai)) = (a → P )

by L3A.

For this reason, we will never use the / operator in recursive process definitions.

34

1

1.9

Processes

More operations on traces

This section describes some further operations on traces; it may be skipped
at this stage, since backwards references will be given in later chapters where
the operations are used.
1.9.1

Change of symbol

Let f be a function mapping symbols from a set A to symbols in a set B. From
f we can derive a new function f ∗ which maps a sequence of symbols in A∗ to
a sequence in B ∗ by applying f to each element of the sequence. For example,
if double is a function which doubles its integer argument
double∗ (h1, 5, 3, 1i) = h2, 10, 6, 2i
A starred function is obviously distributive and therefore strict
L1 f ∗ (hi) = hi
L2 f ∗ (hxi) = hf (x)i
L3 f ∗ (s _ t ) = f ∗ (s) _ f ∗ (t )
Other laws are obvious consequences
L4 f ∗ (s)0 = f (s0 )

if s ≠ hi

L5 #f ∗ (s) = #s
But here is an “obvious” law which is unfortunately not generally true
f ∗ (s u A) = f ∗ (s) u f (A)
where f (A) = { f (x) | x ∈ A }.
The simplest counterexample is given by the function f such that
f (b) = f (c) = c

where b ≠ c

Therefore
f ∗ (hbi u {c})
= f ∗ (hi)

[since b ≠ c]

= hi

[L1]

≠ hci
= hci u {c}
= f ∗ (hci) u f ({c})

[since f (c) = c]

However, the law is true if f is a one-one function (injection)
L6 f ∗ (s u A) = f ∗ (s) u f (A)

provided that f is an injection.

1.9

More operations on traces

1.9.2

35

Catenation

Let s be a sequence, each of whose elements is itself a sequence. Then _/ s
is obtained by catenating all the elements together in the original order. For
example
_/h h1, 3i, hi, h7 i i = h1, 3i _ hi _ h7 i
= h1, 3, 7 i
This operator is distributive
L1 _/hi = hi
L2 _/hsi = s
L3 _/(s _ t ) = (_/ s) _ (_/ t )
1.9.3

Interleaving

A sequence s is an interleaving of two sequences t and u if it can be split into
a series of subsequences, with alternate subsequences extracted from t and u.
For example
s = h1, 6, 3, 1, 5, 4, 2, 7 i
is an interleaving of t and u, where
t = h1, 6, 5, 2, 7 i

and

u = h3, 1, 4i

A recursive definition of interleaving can be given by means of the following laws
L1 hi interleaves (t , u) ≡ (t = hi ∧ u = hi)
L2 s interleaves (t , u) ≡ s interleaves (u, t )
L3 (hxi _ s) interleaves (t , u) ≡
(t ≠ hi ∧ t0 = x ∧ s interleaves (t 0 , u)) ∨
(u ≠ hi ∧ u0 = x ∧ s interleaves (t , u 0 ))
1.9.4

Subscription

If 0 ≤ i ≤ #s, we use the conventional notation s[i] to denote the i th element
of the sequence s as described by L1
L1 s[0] = s0 ∧ s[i + 1] = s 0 [i]
L2 (f ∗ (s))[i] = f (s[i])

provided s ≠ hi
for i < #s

36

1

1.9.5

Processes

Reversal

If s is a sequence, s is formed by taking its elements in reverse order. For
example
h3, 5, 37 i = h37 , 5, 3i
Reversal is defined fully by the following laws
L1 hi = hi
L2 hxi = hxi
L3 s _ t = t _ s
Reversal enjoys a number of simple algebraic properties, including
L4 s = s
Exploration of other properties is left to the reader. One of the useful facts
about reversal is that s 0 is the last element of the sequence, and in general
L5 s[i] = s[#s − i − 1]
1.9.6

for i ≤ #s

Selection

If s is a sequence of pairs, we define s ↓ x as the result of selecting from s all
those pairs whose first element is x and then replacing each pair by its second
element. We write a pair with a dot between its two components. Thus if
s = ha.7 , b.9, a.8, c.0i
then s ↓ a = h7 , 8i and s ↓ d = hi
L1 hi ↓ x = hi
L2 (hy.zi _ t ) ↓ x = t ↓ x

if y ≠ x

L3 (hx.zi _ t ) ↓ x = hzi _ (t ↓ x)
If s is not a sequence of pairs, s ↓ a denotes the number of occurrences of a in
s (as defined in Section 1.6.6).
1.9.7

Composition

Let ✓ be a symbol denoting successful termination of the process which engages in it. As a result, this symbol can appear only at the end of a trace. Let t
be a trace recording a sequence of events which start when s has successfully
terminated. The composition of s and t is denoted (s ; t ). If ✓ does not occur
in s, then t cannot start

1.10

Specifications

L1 s ; t = s

37

if ¬ (h✓i in s)

If ✓ does occur at the end of s, it is removed and t is appended to the result
L2 (s _ h✓i) ; t = s _ t

if ¬ (h✓i in s)

The symbol ✓ may be regarded as a sort of glue which sticks s and t together;
in the absence of the glue, t cannot stick (L1). If ✓ occurs (incorrectly) in the
middle of a trace, we stipulate for the sake of completeness that all symbols
after the first occurrence are irrelevant and should be discarded
(s _ h✓i _ u) ; t = s _ t

L2A

if ¬ (h✓i in s)

This unfamiliar operator enjoys a number of familiar algebraic properties.
Like catenation it is associative. Unlike catenation, it is monotonic in its first
as well as its second argument. Also, it is strict in its first argument, and has
✓ as its left unit
L3 s ; (t ; u) = (s ; t ) ; u
L4A

s ≤ t ⇒ ((u ; s) ≤ (u ; t ))

L4B

s ≤ t ⇒ ((s ; u) ≤ (t ; u))

L5 hi ; t = hi
L6 h✓i ; t = t
If ✓ never occurs except at the end of a trace, h✓i is a right unit as well
L7 s ; h✓i = s

1.10

provided ¬ (h✓i in (s)0 )

Specifications

A specification of a product is a description of the way it is intended to behave.
This description is a predicate containing free variables, each of which stands
for some observable aspect of the behaviour of the product. For example, the
specification of an electronic amplifier, with an input range of one volt and
with an approximate gain of 10, is given by the predicate
AMP10 = (0 ≤ v ≤ 1 ⇒ | v 0 − 10 × v | ≤ 1)
In this specification, it is understood that v stands for the input voltage and
v 0 stands for the output voltage. Such an understanding of the meaning of
variables is essential to the use of mathematics in science and engineering.
In the case of a process, the most obviously relevant observation of its
behaviour is the trace of events that occur up to a given moment in time. We
will use the special variable tr to stand for an arbitrary trace of the process
being specified, just as v and v 0 are used for arbitrary observations of voltage
in the previous example.

38

1

Processes

Examples
X1 The owner of a vending machine does not wish to make a loss by installing
it. He therefore specifies that the number of chocolates dispensed must never
exceed the number of coins inserted
NOLOSS = (#(tr u {choc}) ≤ #(tr u {coin}))

In future we will use the abbreviation (introduced in 1.6.6)
tr ↓ c = #(tr u {c})
to stand for the number of occurrences of the symbol c in tr .
X2 The customer of a vending machine wants to ensure that it will not absorb
further coins until it has dispensed the chocolate already paid for
FAIR1 = ((tr ↓ coin) ≤ (tr ↓ choc) + 1)

X3 The manufacturer of a simple vending machine must meet the requirements both of its owner and its customer
VMSPEC = NOLOSS ∧ FAIR1
= (0 ≤ ((tr ↓ coin) = (tr ↓ choc)) ≤ 1)

X4 The specification of a correction to the complex vending machine forbids
it to accept three pennies in a row
VMCFIX = (¬ hin1pi3 in tr )

X5 The specification of a mended machine
MENDVMC = (tr ∈ traces(VMC) ∧ VMCFIX )

X6 The specification of VMS2 (1.1.3 X6)
0 ≤ ((tr ↓ coin) − (tr ↓ choc) ≤ 2


1.10

Specifications

1.10.1

39

Satisfaction

If P is a product which meets a specification S , we say that P satisfies S , abbreviated to
P sat S
This means that every possible observation of the behaviour of P is described
by S ; or in other words, S is true whenever its variables take values observed
from the product P , or more formally, ∀ tr • tr ∈ traces(P ) ⇒ S . For example,
the following table gives some observations of the properties of an amplifier
1 2 34 5
v 0 .5 .5 2 .1
v0 0 5 4 1 3
All observations except the last are described by AMP10. The second and third
columns illustrate the fact that the output of the amplifier is not completely
determined by its input. The fourth column shows that if the input voltage is
outside the specified range, the output voltage can be anything at all, without
violating the specification. (In this simple example we have ignored the possibility that excessive input may break the product.)
The following laws give the most general properties of the satisfies relation.
The specification true which places no constraints whatever on observations
of a product will be satisfied by all products; even a broken product satisfies
such a weak and undemanding specification
L1 P sat true
If a product satisfies two different specification, it also satisfies their conjunction
L2A

If

P sat S

and

P sat T

then

P sat (S ∧ T )

The law L2A generalises to infinite conjunctions, i.e., to universal quantification. Let S (n) be a predicate containing the variable n
L2

If

∀ n • (P sat S (n))

then P sat (∀ n • S (n))
provided that P does not depend on n.
If a specification S logically implies another specification T , then every observation described by S is also described by T . Consequently every product
which satisfies S must also satisfy the weaker specification T

40

1

L3

If

Processes

P sat S

and

S ⇒T

then

P sat T

In the light of this law, we will sometimes lay out proofs as a chain; so if S ⇒ T ,
we write
P sat S
⇒ T
as an abbreviation for the fuller proof
P sat S
S ⇒T
P sat T

[by L3]

The laws and their explanations given above apply to all kinds of products
and all kinds of specifications. In the next section we shall give the additional
laws which apply to processes.
1.10.2

Proofs

In the design of a product, the designer has a responsibility to ensure that it will
satisfy its specification; this responsibility may be discharged by the reasoning
methods of the relevant branches of mathematics, for example, geometry or
the differential and integral calculus. In this section we shall give a collection of
laws which permit the use of mathematical reasoning to ensure that a process
P meets its specification S .
We will sometimes write the specification as S (tr ), suggesting that a specification will normally contain tr as a free variable. However, the real reason
for making tr explicit is to indicate how tr may be substituted by some more
elaborate expression, as for example in S (tr 0 ). It is important to note that both
S and S (tr ) can have other free variables besides tr .
Any observation of the process STOP will always be an empty trace, since
this process never does anything
L4A

STOP sat (tr = hi)

A trace of the process (c → P ) is initially empty. Every subsequent trace begins
with c, and its tail is a trace of P . Consequently its tail must be described by
any specification of P
L4B

If
then

P sat S (tr )
(c → P ) sat (tr = hi ∨ (tr0 = c ∧ S (tr 0 )))

A corollary of this law deals with double prefixing

1.10

Specifications

L4C

If
then

41

P sat S (tr )
(c → d → P ) sat (tr ≤ hc, di ∨ (tr ≥ hc, di ∧ S (tr 00 )))

Binary choice is similar to prefixing, except that the trace may begin with either
of the two alternative events, and its tail must be described by the specification
of the chosen alternative
L4D

If
and
then

P sat S (tr )
Q sat T (tr )
(c → P | d → Q ) sat
(tr = hi ∨ (tr0 = c ∧ S (tr 0 )) ∨ (tr0 = d ∧ T (tr 0 )))

All the laws given above are special cases of the law for general choice
L4

If
then

∀ x : B • (P (x) sat S (tr , x))
(x : B → P (x)) sat (tr = hi ∨ (tr0 ∈ B ∧ S (tr 0 , tr0 )))

The law governing the after operator is surprisingly simple. If tr is a trace
of (P / s), s _tr is a trace of P , and therefore must be described by any specification which P satisfies
L5

If
and
then

P sat S (tr )
s ∈ traces(P )
P / s sat S (s _ tr )

Finally, we need a law to establish the correctness of a recursively defined
process
L6

If

F (x) is guarded

and

STOP sat S

and

((X sat S ) ⇒ (F (X ) sat S ))

then

(µ X • F (X )) sat S

The antecedents of this law ensure (by induction) that
F n (STOP ) sat S

for all n

Since F is guarded, F n (STOP ) fully describes at least the first n steps of the
behaviour of µ X • F (X ). So each trace of µ X • F (X ) is a trace of F n (STOP ) for
some n. This trace must therefore satisfy the same specification as F n (STOP ),
which (for all n) is S . A more formal proof can be given in terms of the mathematical theory of Section 2.8.

42

1

Processes

Example
X1 We want to prove (1.1.2 X2, 1.10 X3) that
VMS sat VMSPEC
Proof :
1. STOP
sat tr = hi
⇒ 0 ≤ (tr ↓ coin = tr ↓ choc) ≤ 1

[L4A]
[since (hi ↓ coin) = (hi ↓ choc) = 0]

The conclusion follows by an (implicit) appeal to L3.
2. Assume X sat (0 ≤ ((tr ↓ coin) − (tr ↓ choc)) ≤ 1), then
(coin → choc → X )
sat (tr ≤ hcoin, choci) ∨

[L4C]

(tr ≥ hcoin, choci ∧
0 ≤ ((tr 00 ↓ coin) − (tr 00 ↓ choc)) ≤ 1))
⇒ 0 ≤ ((tr ↓ coin) − (tr ↓ choc)) ≤ 1
since
hi ↓ coin = hi ↓ choc = hcoini ↓ choc = 0
and
hcoini ↓ coin = (hcoin, choci ↓ coin) = hcoin, choci ↓ choc = 1
and
tr ≥ hcoin, choci ⇒
(tr ↓ coin = tr 00 ↓ coin + 1 ∧ tr ↓ choc = tr 00 ↓ choc + 1)
The conclusion follows by appeal to L3 and L6.

The fact that a process P satisfies its specification does not necessarily
mean that it is going to be satisfactory in use. For example, since
tr = hi ⇒ 0 ≤ (tr ↓ coin − tr ↓ choc) ≤ 1
one can prove by L3 and L4A that
STOP sat 0 ≤ (tr ↓ coin − tr ↓ choc) ≤ 1

1.10

Specifications

43

Yet STOP will not serve as an adequate vending machine, either for its owner
or for the customer. It certainly avoid doing anything wrong; but only by the
lazy expedient of doing nothing at all. For this reason, STOP satisfies every
specification which is satisfiable by any process.
Fortunately, it is obvious by independent reasoning that VMS will never
stop. In fact, any process defined solely by prefixing, choice, and guarded
recursions will never stop. The only way to write a process that can stop is
to include explicitly the process STOP , or the process (x : B → P (x)) where B
is the empty set. By avoiding such elementary mistakes one can guarantee to
write processes that never stop. However, after introduction of concurrency
in the next chapter, such simple precautions are no longer adequate. A more
general method of specifying and proving that a process will never stop is
described in Section 3.7.

Concurrency

2.1

2

Introduction

A process is defined by describing the whole range of its potential behaviour.
Frequently, there will be a choice between several different actions, for example, the insertion of a large coin or a small one into a vending machine VMC
(1.1.3 X4). On each such occasion, the choice of which event will actually occur
can be controlled by the environment within which the process evolves. For
example, it is the customer of the vending machine who may select which coin
to insert. Fortunately, the environment of a process itself may be described
as a process, with its behaviour defined by familiar notations. This permits
investigation of the behaviour of a complete system composed from the process together with its environment, acting and interacting with each other as
they evolve concurrently. The complete system should also be regarded as a
process, whose range of behaviour is definable in terms of the behaviour of
its component processes; and the system may in turn be placed within a yet
wider environment. In fact, it is best to forget the distinction between processes, environments, and systems; they are all of them just processes whose
behaviour may be prescribed, described, recorded and analysed in a simple
and homogeneous fashion.

2.2

Interaction

When two processes are brought together to evolve concurrently, the usual
intention is that they will interact with each other. These interactions may be
regarded as events that require simultaneous participation of both the processes involved. For the time being, let us confine attention to such events,
and ignore all others. Thus we will assume that the alphabets of the two processes are the same. Consequently, each event that actually occurs must be a
possible event in the independent behaviour of each process separately. For
example, a chocolate can be extracted from a vending machine only when its
customer wants it and only when the vending machine is prepared to give it.

46

2

Concurrency

If P and Q are processes with the same alphabet, we introduce the notation
P || Q
to denote the process which behaves like the system composed of processes
P and Q interacting in lock-step synchronisation as described above.
Examples
X1 A greedy customer of a vending machine is perfectly happy to obtain a
toffee or even a chocolate without paying. However, if thwarted in these desires, he is reluctantly prepared to pay a coin, but then he insists on taking a
chocolate
GRCUST = (toffee → GRCUST
| choc → GRCUST
| coin → choc → GRCUST )
When this customer is brought together with the machine VMCT (1.1.3 X3)
his greed is frustrated, since the vending machine does not allow goods to be
extracted before payment. Similarly, VMCT never gives a toffee, because the
customer never wants one after he has paid
(GRCUST || VMCT ) = µ X • (coin → choc → X )
This example shows how a process which has been defined as a composition
of two subprocesses may also be described as a simple single process, without
using the concurrency operator ||.

X2 A foolish customer wants a large biscuit, so he puts his coin in the vending
machine VMC. He does not notice whether he has inserted a large coin or a
small one; nevertheless, he is determined on a large biscuit
FOOLCUST = (in2p → large → FOOLCUST
| in1p → large → FOOLCUST )
Unfortunately, the vending machine is not prepared to yield a large biscuit for
only a small coin
(FOOLCUST || VMC) = µ X • (in2p → large → X | in1p → STOP )
The STOP that supervenes after the first in1p is known as deadlock. Although
each component process is prepared to engage in some further action, these
actions are different; since the processes cannot agree on what the next action
shall be, nothing further can happen.


2.2

Interaction

47

The stories that accompany these examples show a sad betrayal of proper
standards of scientific abstraction and objectivity. It is important to remember
that events are intended to be neutral transitions which could be observed and
recorded by some dispassionate visitor from another planet, who knows nothing of the pleasures of eating biscuits, or of the hunger suffered by the foolish
customer as he vainly tries to obtain sustenance. We have deliberately chosen
the alphabet of relevant events to exclude such internal emotional states; if
and when desired, further events can be introduced to model internal state
changes, as shown in 2.3 X1.

2.2.1

Laws

The laws governing the behaviour of (P || Q ) are exceptionally simple and
regular. The first law expresses the logical symmetry between a process and
its environment
L1 P || Q = Q || P
The next law shows that when three processes are assembled, it does not matter in which order they are put together
L2 P || (Q || R) = (P || Q ) || R
Thirdly, a deadlocked process infects the whole system with deadlock; but
composition with RUNαP (1.1.3 X8) makes no difference
L3A

P || STOPαP = STOPαP

L3B

P || RUNαP = P

The next laws show how a pair of processes either engage simultaneously in
the same action, or deadlock if they disagree on what the first action should
be
L4A

(c → P ) || (c → Q ) = (c → (P || Q ))

L4B

(c → P ) || (d → Q ) = STOP

if c ≠ d

These laws readily generalise to cases when one or both processes offer a choice
of initial event; only events which they both offer will remain possible when
the processes are combined
L4 (x : A → P (x)) || (y : B → Q (y)) = (z : (A ∩ B) → (P (z) || Q (z)))
It is this law which permits a system defined in terms of concurrency to be
given an alternative description without concurrency.

48

2

Concurrency

Example
X1

Let

and
Then

P = (a → b → P | b → P )
Q = (a → (b → Q | c → Q ))

(P || Q ) =
= a → ((b → P ) || (b → Q | c → Q ))

[by L4A]

= a → (b → (P || Q ))

[by L4A]

= µ X • (a → b → X )

[since the recursion is guarded.]


2.2.2

Implementation

The implementation of the || operator is clearly based on L4
intersect (P , Q ) = λ z • if P (z) = "BLEEP or Q (z) = "BLEEP then
"BLEEP
else
intersect (P (z), Q (z))

2.2.3

Traces

Since each action of (P || Q ) requires simultaneous participation of both P and
Q , each sequence of such actions must be possible for both these operands.
For the same reason, / s distributes through ||.
L1 traces(P || Q ) = traces(P ) ∩ traces(Q )
L2 (P || Q ) / s = (P / s) || (Q / s)

2.3

Concurrency

The operator described in the previous section can be generalised to the case
when its operands P and Q have different alphabets
αP ≠ αQ
When such processes are assembled to run concurrently, events that are in both
their alphabets (as explained in the previous section) require simultaneous
participation of both P and Q . However, events in the alphabet of P but not
in the alphabet of Q are of no concern to Q , which is physically incapable of

2.3

Concurrency

49

controlling or even of noticing them. Such events may occur independently
of Q whenever P engages in them. Similarly, Q may engage alone in events
which are in the alphabet of Q but not of P . Thus the set of all events that are
logically possible for the system is simply the union of the alphabets of the
component processes
α(P || Q ) = αP ∪ αQ
This is a rare example of an operator which takes operands with different alphabets, and yields a result with yet a third alphabet. However in the case when
the two operands have the same alphabet, so does the resulting combination,
and (P || Q ) has exactly the meaning described in the previous section.
Examples
X1 Let αNOISYVM = {coin, choc, clink, clunk, toffee}, where clink is the sound
of a coin dropping into the moneybox of a noisy vending machine, and clunk
is the sound made by the vending machine on completion of a transaction.
The noisy vending machine has run out of toffee
NOISYVM =
(coin → clink → choc → clunk → NOISYVM )
The customer of this machine definitely prefers toffee; the curse is what he
utters when he fails to get it; he then has to take a chocolate instead
αCUST =
{coin, choc, curse, toffee}
CUST =
(coin → (toffee → CUST | curse → choc → CUST ))
The result of the concurrent activity of these two processes is
(NOISYVM || CUST ) =
µ X • (coin → (clink → curse → choc → clunk → X
| curse → clink → choc → clunk → X ))
Note that the clink may occur before the curse, or the other way round. They
may even occur simultaneously, and it will not matter in which order they
are recorded. Note also that the mathematical formula in no way represents
the fact that the customer prefers to get a toffee rather than utter a curse. The
formula is an abstraction from reality, which ignores human emotions and concentrates on describing only the possibilities of occurrence and non-occurrence
of events within the alphabet of the processes, whether those events are desired or not.


50

2

X2

Concurrency

i
2

1
1

2

3

j

A counter starts at the middle bottom square of the board, and may move
within the board either up, down, left or right . Let
αP = {up, down}
P = (up → down → P )
αQ = {left , right }
Q = (right → left → Q | left → right → Q )
The behaviour of this counter may be defined P || Q .
In this example, the alphabets αP and αQ have no event in common. Consequently, the movements of the counter are an arbitrary interleaving of actions from the process P with actions from the process Q . Such interleavings
are very laborious to describe without concurrency. For example, let Rij stand
for the behaviour of the counter (X2) when situated in row i and column j of
the board, for i ∈ {1, 2}, j ∈ {1, 2, 3}. Then
(P || Q ) = R12
where
R21 = (down → R11 | right → R22 )
R11 = (up → R21 | right → R12 )
R22 = (down → R12 | left → R21 | right → R23 )
R12 = (up → R22 | left → R11 | right → R13 )
R23 = (down → R13 | left → R22 )
R13 = (up → R23 | left → R12 )

2.3.1

Laws

The first three laws for the extended form of concurrency are similar to those
for interaction (Section 2.2.1)
L1,2 || is symmetric and associative
L3A

P || STOPαP = STOPαP

2.3

Concurrency

L3B

51

P || RUNαP = P

Let a ∈ (αP − αQ ), b ∈ (αQ − αP ) and {c, d} ⊆ (αP ∩ αQ ). The following
laws show the way in which P engages alone in a, Q engages alone in b, but c
and d require simultaneous participation of both P and Q
L4A

(c → P ) || (c → Q ) = c → (P || Q )

L4B

(c → P ) || (d → Q ) = STOP

L5A

(a → P ) || (c → Q ) = a → (P || (c → Q ))

L5B

(c → P ) || (b → Q ) = b → ((c → P ) || Q )

if c ≠ d

L6 (a → P ) || (b → Q ) = (a → (P || (b → Q )) | b → ((a → P ) || Q ))
These laws can be generalised to deal with the general choice operator
L7

Let

P = (x : A → P (x))

and

Q = (y : B → Q (y))

Then
where
and

(P || Q ) = (z : C → P 0 || Q 0 )
C = (A ∩ B) ∪ (A − αQ ) ∪ (B − αP )
P 0 = P (z)
0

P =P
and

Q 0 = Q (z)
0

Q =Q

if z ∈ A
otherwise
if z ∈ B
otherwise.

These laws permit a process defined by concurrency to be redefined without
that operator, as shown in the following example.
Example
X1 Let

αP = {a, c}

and

αQ = {b, c}

P = (a → c → P )
Q = (c → b → Q )

P || Q
= (a → c → P ) || (c → b → Q )
= a → ((c → P ) || (c → b → Q ))
= a → c → (P || (b → Q ))
Also
P || (b → Q )

[by definition]
[by L5A]
[by L4A …‡]

52

2

= (a → (c → P ) || (b → Q )

Concurrency

[by L6]

| b → (P || Q ))
= (a → b → ((c → P ) || Q )

[by L5B]

| b → (P || Q ))
= (a → b → c → (P || (b → Q ))

[by ‡ above]

| b → a → c → (P || (b → Q )))
= µ X • (a → b → c → X

[since this is guarded]

| b → a → c → X)
Therefore
(P || Q ) = (a → c → µ X • (a → b → c → X

by ‡ above

| b → a → c → X ))

2.3.2

Implementation

The implementation of the operator || is derived directly from the law L7. The
alphabets of the operands are represented as finite lists of symbols, A and B.
Test of membership uses the function ismember (x, A) defined in Section 1.7.
P || Q is implemented by calling a function concurrent (P , αP , αQ , Q ),
which is defined as follows
concurrent (P , A, B, Q ) = aux(P , Q )
where
aux(P , Q ) = λ x • if P = "BLEEP or Q = "BLEEP then
"BLEEP
else if ismember (x, A) and ismember (x, B) then
aux(P (x), Q (x))
else if ismember (x, A) then
aux(P (x), Q )
else if ismember (x, B) then
aux(P , Q (x))
else
"BLEEP

2.3

Concurrency

2.3.3

53

Traces

Let t be a trace of (P || Q ). Then every event in t which belongs to the alphabet
of P has been an event in the life of P ; and every event in t which does not
belong to αP has occurred without the participation of P . Thus (t u αP ) is a
trace of all those events in which P has participated, and is therefore a trace
of P . By a similar argument (t u αQ ) is a trace of Q . Furthermore, every event
in t must be in either αP or αQ . This reasoning suggests the law
L1 traces(P || Q ) =
{ t | (t u αP ) ∈ traces(P ) ∧ (t u αQ ) ∈ traces(Q ) ∧ t ∈ (αP ∪ αQ )∗ }
The next law shows how the / s operator distributes through parallel composition
L2 (P || Q ) / s = (P / (s u αP )) || (Q / (s u αQ ))
When αP = αQ , it follows that
s u αP = s u αQ = s
and these laws are then the same as in Section 2.2.3.
Example
X1 (See 2.3 X1.)
Let t1 = hcoin, click, cursei, then
t1 u αNOISYVM = hcoin, clicki

[which is in traces(NOISYVM )]

t1 u αCUST = hcoin, cursei

[which is in traces(CUST )]

therefore
t1 ∈ traces(NOISYVM || CUST )
Similar reasoning shows that
hcoin, curse, clinki ∈ traces(NOISYVM || CUST )
This shows that the curse and the clink may be recorded one after the other
in either order. They may even occur simultaneously, but we have made the
decision to provide no way of recording this.

In summary, a trace of (P || Q ) is a kind of interleaving of a trace of P with
a trace of Q , in which events which are in the alphabet of both of them occur
only once. If αP ∩αQ = {} then the traces are pure interleavings (Section 1.9.3),
as shown in 2.3 X2. At the other extreme, where αP = αQ , every event belongs
to both of the alphabets, and the meaning of (P || Q ) is exactly as defined for
interaction (Section 2.2).

54

L3A

2

Concurrency

If αP ∩ αQ = {}, then
traces(P || Q ) = { s | ∃ t : traces(P ); u : traces(Q ) • s interleaves (t , u) }

L3B

If αP = αQ then
traces(P || Q ) = traces(P ) ∩ traces(Q )

2.4

Pictures

A process P with alphabet {a, b, c} is pictured as a box labelled P , from which
emerge a number of lines, each labelled with a different event from its alphabet (Figure 2.1). Similarly, Q with its alphabet {b, c, d} may be pictured as in
Figure 2.2.

a

b

P

b

d

Q

c

c

Figure 2.1

Figure 2.2

When these two processes are put together to evolve concurrently, the
resulting system may be pictured as a network in which similarly labelled lines
are connected, but lines labelled by events in the alphabet of only one process
are left free (Figure 2.3).

a

b
Q

P

d

c
Figure 2.3

A third process R with αR = {c, e} may be added, as shown in Figure 2.4.
This diagram shows that the event c requires participation of all three processes, b requires participation of P and Q , whereas each remaining event is
the sole concern of a single process. Pictures of this kind will be known as
connection diagrams.

2.5

Example: The Dining Philosophers

a

55

b
Q

P

d

c

R
e
Figure 2.4

b
a

d

( P || Q || R )

c

e
Figure 2.5

But these pictures could be quite misleading. A system constructed from
three processes is still only a single process, and therefore be pictured as a
single box (Figure 2.5). The number 60 can be constructed as the product of
three other numbers (3 × 4 × 5); but after it has been so constructed it is still
only a single number, and the manner of its construction is no longer relevant
or even observable.

2.5

Example: The Dining Philosophers

In ancient times, a wealthy philanthropist endowed a College to accommodate
five eminent philosophers. Each philosopher had a room in which he could engage in his professional activity of thinking; there was also a common dining

56

2

Concurrency

room, furnished with a circular table, surrounded by five chairs, each labelled
by the name of the philosopher who was to sit in it. The names of the philosophers were PHIL0 , PHIL1 , PHIL2 , PHIL3 , PHIL4 , and they were disposed in this
order anticlockwise around the table. To the left of each philosopher there was
laid a golden fork, and in the centre stood a large bowl of spaghetti, which was
constantly replenished.
A philosopher was expected to spend most of his time thinking; but when
he felt hungry, he went to the dining room, sat down in his own chair, picked
up his own fork on his left, and plunged it into the spaghetti. But such is the
tangled nature of spaghetti that a second fork is required to carry it to the
mouth. The philosopher therefore had also to pick up the fork on his right.
When we was finished he would put down both his forks, get up from his chair,
and continue thinking. Of course, a fork can be used by only one philosopher
at a time. If the other philosopher wants it, he just has to wait until the fork
is available again.
2.5.1

Alphabets

We shall now construct a mathematical model of this system. First we must
select the relevant sets of events. For PHILi , the set is defined
αPHILi = {i.sits down, i.gets up,
i.picks up fork.i, i.picks up fork.(i ⊕ 1),
i.puts down fork.i, i.puts down fork.(i ⊕ 1) }
where ⊕ is addition modulo 5, so i ⊕ 1 identifies the right-hand neighbour of
the ith philosopher.
Note that the alphabets of the philosophers are mutually disjoint. There
is no event in which they participate jointly, so there is no way whatsoever in
which they can interact or communicate with each other—a realistic reflection
of the behaviour of philosophers in those days.
The other actors in our little drama are the five forks, each of which bears
the same number as the philosopher who owns it. A fork is picked up and put
down either by this philosopher, or by his neighbour on the other side. The
alphabet of the ith fork is defined
αFORKi = {i.picks up fork.i, (i
i.puts down fork.i, (i

1).picks up fork.i,
1).puts down fork.i}

where denotes subtraction modulo 5.
Thus each event except sitting down and getting up requires participation of exactly two adjacent actors, a philosopher and a fork, as shown in the
connection diagram of Figure 2.6.

2.5

Example: The Dining Philosophers

57

2.sits down
2.puts down
fork.3

2.gets up

2.picks up
fork.3

FORK3

2.puts down
fork.2

PHIL2

2.picks up
fork.2

FORK2
1.puts down
fork.2

3.puts down
fork.3
1.picks up
fork.2

3.picks up
fork.3

3.gets up

1.sits down

PHIL3

3.sits down

3.puts down
fork.4

PHIL1

1.picks up
fork.1

3.picks up
fork.4

1.gets up

1.puts down
fork.1

FORK1

FORK4
5.picks up
fork.1

4.picks up
fork.4

5.puts down
fork.1

4.puts down
fork.4
4.picks up
fork.5

PHIL4

5.picks up
fork.5

4.gets up

PHIL5
5.sits down

FORK5
4.sits down

4.puts down
fork.5

5.gets up
5.puts down
fork.5

Figure 2.6

2.5.2

Behaviour

Apart from thinking and eating which we have chosen to ignore, the life of
each philosopher is described as the repetition of a cycle of six events
PHILi = (i.sits down →
i.picks up fork.i →
i.picks up fork.(i ⊕ 1) →
i.puts down fork.i →
i.puts down fork.(i ⊕ 1) →
i.gets up → PHILi )

58

2

Concurrency

The role of a fork is a simple one; it is repeatedly picked up and put down
by one of its adjacent philosophers (the same one on both occasions)
FORKi = (i.picks up fork.i → i.puts down fork.i → FORKi
| (i

1).picks up fork.i → (i

1).puts down fork.i → FORKi )

The behaviour of the whole College is the concurrent combination of the behaviour of each of these components
PHILOS = (PHIL0 || PHIL1 || PHIL2 || PHIL3 || PHIL4 )
FORKS = (FORK0 || FORK1 || FORK2 || FORK3 || FORK4 )
COLLEGE = PHILOS || FORKS
An interesting variation of this story allows the philosophers to pick up
their two forks in either order, or put them down in either order. Consider
the behaviour of each philosopher’s hand separately. Each hand is capable of
picking up the relevant fork, but both hands are needed for sitting down and
getting up
αLEFTi = {i.picks up fork.i, i.puts down fork.i,
i.sits down, i.gets up}
αRIGHTi = {i.picks up fork.(i ⊕ 1), i.puts down fork.(i ⊕ 1),
i.sits down, i.gets up}

LEFTi = (i.sits down → i.picks up fork.i →
i.puts down fork.i → i.gets up → LEFTi )
RIGHTi = (i.sits down → i.picks up fork.(i ⊕ 1) →
i.puts down fork.(i ⊕ 1) → i.gets up → RIGHTi )
PHILi = LEFTi || RIGHTi
Synchronisation of sitting down and getting up by both LEFTi and RIGHTi ensures that no fork can be raised except when the relevant philosopher is seated.
Apart from this, operations on the two forks are arbitrarily interleaved.
In yet another variation of the story, each fork may be picked up and put
down many times on each occasion that the philosopher sits down. Thus the
behaviour of the hands is modified to contain an iteration, for example
LEFTi = (i.sits down →
µ X • (i.picks up fork.i → i.puts down fork.i → X
| i.gets up → LEFTi ))

2.5

Example: The Dining Philosophers

2.5.3

59

Deadlock!

When a mathematical model had been constructed, it revealed a serious danger.
Suppose all the philosophers get hungry at about the same time; they all sit
down; they all pick up their own forks; and they all reach out for the other
fork—which isn’t there. In this undignified situation, they will all inevitably
starve. Although each actor is capable of further action, there is no action
which any pair of them can agree to do next.
However, our story does not end so sadly. Once the danger was detected,
there were suggested many ways to avert it. For example, one of the philosophers could always pick up the wrong fork first—if only they could agree
which one it should be! The purchase of a single additional fork was ruled
out for similar reasons, whereas the purchase of five more forks was much too
expensive. The solution finally adopted was the appointment of a footman,
whose duty it was to assist each philosopher into and out of his chair. His
alphabet was defined as
S4

i=0

{i.sits down, i.gets up}

This footman was given secret instructions never to allow more than four philosophers to be seated simultaneously. His behaviour is most simply defined by
mutual recursion. Let
U =

S4

i=0

{i.gets up}

D=

S4

i=0

{i.sits down}

FOOTj defines the behaviour of the footman with j philosophers seated
FOOT0 = (x : D → FOOT1 )
FOOTj = (x : D → FOOTj+1 | y : U → FOOTj−1 )

for j ∈ {1, 2, 3}

FOOT4 = (y : U → FOOT3 )
A college free of deadlock is defined
NEWCOLLEGE = (COLLEGE || FOOT0 )
The edifying tale of the dining philosophers is due to Edsger W. Dijkstra. The
footman is due to Carel S. Scholten.

2.5.4

Proof of absence of deadlock

In the original COLLEGE the risk of deadlock was far from obvious; the claim
that NEWCOLLEGE is free from deadlock should therefore be proved with some
care. What we must prove can be stated formally as
(NEWCOLLEGE / s) ≠ STOP

for all s ∈ traces(NEWCOLLEGE )

60

2

Concurrency

The proof proceeds by taking an arbitrary trace s, and showing that in all
cases there is at least one event by which s can be extended and still remain in
traces(NEWCOLLEGE ). First we define the number of seated philosophers
seated(s) = #(s u D) − #(s u U )

where U and D are defined above

Because (by 2.3.3 L1) s u (U ∪ D) ∈ traces(FOOT0 ), we know seated(s) ≤ 4. If
seated(s) ≤ 3, at least one more philosopher can sit down, so that there is no
deadlock. In the remaining case that seated(s) = 4, consider the number of
philosophers who are eating (with both their forks raised). If this is nonzero,
then an eating philosopher can always put down his left fork. In the remaining
case, that no philosopher is eating, consider the number of raised forks. If this
is three or less, then one of the seated philosophers can pick up his left fork. If
there are four raised forks, then the philosopher to the left of the vacant seat
already has raised his left fork and can pick up his right one. If there are five
raised forks, then at least one of the seated philosophers must be eating.
This proof involves analysis of a number of cases, described informally in
terms of the behaviour of this particular example. Let us consider an alternative proof method: program a computer to explore all possible behaviours
of the system to look for deadlock. In general, we could never know whether
such a program had looked far enough to guarantee absence of deadlock. But
in the case of a finite-state system like the COLLEGE is is sufficient to consider
only those traces whose length does not succeed a known upper bound on
the number of states. The number of states of (P || Q ) does not exceed the
product of the number of states of P and the number of states of Q . Since each
philosopher has six states and each fork has three states, the total number of
states of the COLLEGE does not exceed
6 5 × 35 ,

or approximately 1.8 million

Since the alphabet of the footman is contained in that of the COLLEGE , the
NEWCOLLEGE cannot have more states than the COLLEGE . Since in nearly
every state there are two or more possible events, the number of traces that
must be examined will exceed two raised to the power of 1.8 million. There
is no hope that a computer will ever be able to explore all these possibilities.
Proof of the absence of deadlock, even for quite simple finite processes, will
remain the responsibility of the designer of concurrent systems.

2.5.5

Infinite overtaking

Apart from deadlock, there is another danger that faces a dining philosopher—
that of being infinitely overtaken. Suppose that a seated philosopher has an
extremely greedy left neighbour, and a rather slow left arm. Before he can pick
up his left fork, his left neighbour rushes in, sits down, rapidly picks up both
forks, and spends a long time eating. Eventually he puts down both forks,
and leaves his seat. But then the left neighbour instantly gets hungry again,

2.6

Change of symbol

61

rushes in, sits down, and rapidly snatches both forks, before his long-seated
and long-suffering right neighbour gets around to picking up the fork they
share. Since this cycle may be repeated indefinitely, a seated philosopher may
never succeed in eating.
The correct solution to this problem is probably to regard it as insoluble,
because if any philosopher is as greedy as described above, then somebody
(either he or his neighbours) will inevitably spend a long time hungry. There is
no clever way of ensuring general satisfaction, and the only effective solution
is to buy more forks, and plenty of spaghetti. However, if it is important to
guarantee that a seated philosopher will eventually eat, modify the behaviour
of the footman: having helped a philosopher to his seat he waits until that
philosopher has picked up both forks before he allows either of his neighbours
to sit down.
But there remains a more philosophical problem about infinite overtaking.
Suppose the footman conceives an irrational dislike for one of his philosophers, and persistently delays the action of escorting him to his chair, even when
the philosopher is ready to engage in that event. This is a possibility that cannot be described in our conceptual framework, because we cannot distinguish
it from the possibility that the philosopher himself takes an indefinitely long
time to get hungry. So here is a problem, like detailed timing problems, which
we have deliberately decided to ignore, or rather to delegate it to a different
phase of design and implementation. It is an implementor’s responsibility to
ensure that any desirable event that becomes possible will take place within
an acceptable interval. The implementor of a conventional high-level programming language has a similar obligation not to insert arbitrary delays into the
execution of a program, even though the programmer has no way of enforcing
or even describing this obligation.

2.6

Change of symbol

The example of the previous section involved two collections of processes,
philosophers and forks; within each collection the processes have very similar
behaviour, except that the names of the events in which they engage are different. In this section we introduce a convenient method of defining groups of
processes with similar behaviour. Let f be a one-one function (injection) which
maps the alphabet of P onto a set of symbols A
f : αP → A
We define the process f (P ) as one which engages in the event f (c) whenever
P would have engaged in c. It follows that
αf (P ) = f (αP )
traces(f (P )) = { f ∗ (s) | s ∈ traces(P ) }
(For the definition of f ∗ see 1.9.1).

62

2

Concurrency

Examples
X1 After a few years, the price of everything goes up. To represent the effect
of inflation, we define a function f by the following equations
f (in2p) = in10p

f (large) = large

f (in1p) = in5p

f (small) = small

f (out1p) = out5p
The new vending machine is
NEWVMC = f (VMC)

X2 A counter behaves like CT0 (1.1.4 X2), except that it moves right and left
instead of up and down
f (up) = right ,

f (down) = left ,

f (around) = around,

LR0 = f (CT0 )

The main reason for changing event names of processes in this fashion is
to enable them to be composed usefully in concurrent combination.
X3 A counter moves left , right , up or down on an infinite board with boundaries at the left and at the bottom

It starts at the bottom left corner. On this square alone, it can turn around. As
in 2.3 X2, vertical and horizontal movements can be modelled as independent
actions of separate processes; but around requires simultaneous participation
of both
LRUD = LR0 || CT0


2.6

Change of symbol

63

X4 We wish to connect two instances of COPYBIT (1.1.3 X7) in series, so that
each bit output by the first is simultaneously input by the second. First, we
need to change the names of the events used for internal communication; we
therefore introduce two new events mid.0 and mid.1, and define the functions
f and g to change the output of one process and the input of the other
f (out .0) = g(in.0) = mid.0
f (out .1) = g(in.1) = mid.1
f (in.0) = in.0,

f (in.1) = in.1

g(out .0) = out .0,

g(out .1) = out .1

The answer we want is
CHAIN2 = f (COPYBIT ) || g(COPYBIT )
Note that each output of 0 or 1 by the left operand of || is (by the definition of
f and g) the very same event (mid.0 or mid.1) as the input of the same 0 or 1
by the right operand. This models the synchronised communication of binary
digits on a channel which connects the two operands, as shown in Figure 2.7.

in.0

out.0
COPYBIT

COPYBIT
out.1

in.1
mid.0

mid.1
Figure 2.7

The left operand offers no choice of which value is transmitted on the connecting channel, whereas the right operand is prepared to engage in either of
the events mid.0 or mid.1. It is therefore the outputting process that determines on each occasion which of these two events will occur. This method of
communication between concurrent processes will be generalised in Chapter 4.
Note that the internal communications mid.0 and mid.1 are still in the
alphabet of the composite processes, and can be observed (or even perhaps
controlled) by its environment. Sometimes one wishes to ignore or conceal
such internal events; in the general case such concealment may introduce nondeterminism, so this topic is postponed to Section 3.5.

X5 We wish to represent the behaviour of a Boolean variable used by a computer program. The events in its alphabet are
assign0—assignment of value zero to the variable
assign1—assignment of value one to the variable
fetch0—access of the value of the variable at a time when it is zero
fetch1—access of the value of the variable at a time when it is one

64

2

Concurrency

The behaviour of the variable is remarkably similar to that of the drinks dispenser (1.1.4 X1), so we define
BOOL = f (DD)
where the definition of f is a trivial exercise. Note that the Boolean variable
refuses to give its value until after a value has been first assigned. An attempt
to fetch an unassigned value would result in deadlock—which is probably the
kindest failure mode for incorrect programs, because the simplest postmortem
will pinpoint the error.

The tree picture of f (P ) may be constructed from the tree picture of P by
simply applying the function f to the labels on all the branches. Because f
is a one-one function, this transformation preserves the structure of the tree
and the important distinctness of labels on all branches leading from the same
node. For example, a picture of NEWVMC is shown in Figure 2.8.
NEWVMC

large

in10p

in5p

in10p

small

in5p

in5p

out5p

in10p

in10p

large

small

in10p

in5p

in5p

Figure 2.8

2.6.1

Laws

Change of symbol by application of a one-one function does not change the
structure of the behaviour of a process. This is reflected by the fact that function application distributes through all the other operators, as described in the
following laws. The following auxiliary definitions are used
f (B) = { f (x) | x ∈ B }
f −1

is the inverse of f

f ◦ g is the composition of f and g
f∗

is defined in Section 1.9.1

2.6

Change of symbol

65

(the need for f −1 in the following laws is an important reason for insisting that
f is an injection).
After change of symbol, STOP still performs no event from its changed
alphabet
L1 f (STOPA ) = STOPf (A)
In the case of a choice, the symbols offered for selection are changed, and the
subsequent behaviour is similarly changed
L2 f (x : B → P (x)) = (y : f (B) → f (P (f −1 (y))))
The use of f −1 on the right-hand side may need explanation. Recall that P
is a function delivering a process depending on selection of some x from the
set B. But the variable y on the right-hand side is selected from the set f (B).
The corresponding event for P is f −1 (y), which is in B (since y ∈ f (B)). The
behaviour of P after this event is P (f −1 (y)), and the actions of this process
must continue to be changed by application of f .
Change of symbol simply distributes through parallel composition
L3 f (P || Q ) = f (P ) || f (Q )
Change of symbol distributes in a slightly more complex way over recursion,
changing the alphabet in the appropriate way
L4 f (µ X : A • F (X )) = (µ Y : f (A) • f (F (f −1 (Y ))))
Again, the use of f −1 on the right-hand side may be puzzling. Recall that the
validity of the recursion on the left-hand side requires that F is a function which
takes as argument a process with alphabet A, and delivers a process with the
same alphabet. On the right-hand side, Y is a variable ranging over processes
with alphabet f (A), and cannot be used as an argument to F until its alphabet
has been changed back to A. This is done by applying the inverse function
f −1 . Now F (f −1 (Y )) has alphabet A, so an application of f will transform the
alphabet to f (A), thus ensuring the validity of the recursion on the right-hand
side of the law.
The composition of two changes of symbol is defined by the composition
of the two symbol-changing functions
L5 f (g(P )) = (f ◦ g) (P )
The traces of a process after change of symbol are obtained simply by
changing the individual symbols in every trace of the original process
L6 traces(f (P )) = { f ∗ (s) | s ∈ traces(P ) }
The explanation of the next and final law is similar to that of L6
L7 f (P ) / f ∗ (s) = f (P / s)

66

2

2.6.2

Concurrency

Process labelling

Change of symbol is particularly useful in constructing groups of similar processes which operate concurrently in providing identical services to their common environment, but which do not interact with each other in any way at all.
This means that they must all have different and mutually disjoint alphabets.
To achieve this, each process is labelled by a different name; and each event
of a labelled process is also labelled by its name. A labelled event is a pair l.x
where l is a label, and x is the symbol standing for the event.
A process P labelled by l is denoted by
l:P
It engages in the event l.x whenever P would have engaged in x. The function
required to define l : P is
fl (x) = l.x

for all x in αP

and the definition of labelling is
l : P = fl (P )
Examples
X1 A pair of vending machines standing side by side
(left : VMS ) || (right : VMS )
The alphabets of the two processes are disjoint, and every event that occurs
is labelled by the name of the machine on which it occurred. If the machines
were not named before being placed in parallel, every event would require
participation of both of them, and the pair would be indistinguishable from a
single machine; this is a consequence of the fact that
(VMS || VMS ) = VMS

The labelling of processes permits them to be used in the manner of variables in a high-level programming language, declared locally in the block of
program which uses them.
X2 The behaviour of a Boolean variable is modelled by BOOL (2.6 X5). The
behaviour of a block of program is represented by a process USER. This process
assigns and accesses the values of two Boolean variables named b and c. Thus
αUSER includes such compound events as
b.assign.0—to assign value zero to b
c.fetch.1—to access the current value of c when it is one

2.6

Change of symbol

67

The USER process runs in parallel with its two Boolean variables
b : BOOL || c : BOOL || USER
Inside the USER program, the following effects may be achieved
b := false ; P by (b.assign.0 → P )
b := ¬ c ; P by (c.fetch.0 → b.assign.1 → P
| c.fetch.1 → b.assign.0 → P )
Note how the current value of the variable is discovered by allowing the variable
to make the choice between fetch.0 and fetch.1; and this choice affects in an
appropriate way the subsequent behaviour of the USER.

In X2 and the following examples it would have been more convenient to
define the effect of the single assignment, e.g.,
b := false
rather than the pair of commands
b := false ; P
which explicitly mentions the rest of the program P . The means of doing this
will be introduced in Chapter 5.
X3 A USER process needs two count variables named l and m. They are initialised to 0 and 3 respectively. The USER process increments each variable by
l.up or m.up, and decrements it (when positive) by l.down and m.down. A test
of zero is provided by the events l.around and m.around. Thus the process
CT (1.1.4 X2) can be used after appropriate labelling by l and by m
(l : CT0 || m : CT3 || USER)
Within the USER process the following effects (expressed in conventional notation) can be achieved
(m := m + 1 ; P )

by

(m.up → P )

if l = 0 then P else Q

by

(l.around → P
| l.down → l.up → Q )

Note how the test for zero works: an attempt is made by l.down to reduce
the count by one, at the same time as attempting l.around. The count selects
between these two events: if the value is zero, l.around is selected; if non-zero,
the other. But in the latter case, the value of the count has been decremented,
and it must immediately be restored to its original value by l.up. In the next
example, restoration of the original value is more laborious.

68

2

Concurrency

(m := m + l ; P )
is implemented by ADD, where ADD is defined recursively
ADD = DOWN0
DOWNi = (l.down → DOWNi+1 | l.around → UPi )
UP0 = P
UPi+1 = l.up → m.up → UPi
The DOWNi processes discover the initial value of l by decrementing it to zero.
The UPi processes then add the discovered value to both m and to l, thereby
restoring l to its initial value and adding this value to m.

The effect of an array variable can be achieved by a collection of concurrent
processes, each labelled by its index within the array.
X4 The purpose of the process EL is to record whether the event in has occurred or not. On the first occurrence of in it responds no and on each subsequent occurrence it responds yes
αEL = {in, no, yes}
EL = in → no → µ X • (in → yes → X )
This process can be used in an array to mimic the behaviour of a set of small
integers
SET3 = (0 : EL) || (1 : EL) || (2 : EL) || (3 : EL)
The whole array can be labelled yet again before use
m : SET3 || USER
Each event in α(m : SET3) is a triple, e.g., m.2.in. Within the USER process, the
effect of
if 2 ∈ m then P else (m := m ∪ {2} ; Q )
may be achieved by
m.2.in → (m.2.yes → P | m.2.no → Q )

2.6.3

Implementation

To implement symbol change in general, we need to know the inverse g of
the symbol-changing function f . We also need to ensure that g will give the

2.6

Change of symbol

69

special answer "BLEEP when applied to an argument outside the range of f .
The implementation is based upon 2.6.1 L4.
change(g, P ) = λ x • if g(x) = "BLEEP then
"BLEEP
else if P (g(x)) = "BLEEP then
"BLEEP
else
change(g, P (g(x)))
The special case of process labelling can be implemented more simply.
The compound event l .x is represented as the pair of atoms cons("l, "x). Now
(l : P ) is implemented by
label(l, P ) = λ y • if null(y) or atom(y) then
"BLEEP
else if car (y) ≠ l then
"BLEEP
else if P (cdr (y)) = "BLEEP then
"BLEEP
else
label(l, P (cdr (y)))

2.6.4

Multiple labelling

The definition of labelling can be extended to allow each event to take any label
l from a set L. If P is a process, (L : P ) is defined as a process which behaves
exactly like P , except that it engages in the event l.c (where l ∈ L and c ∈ αP )
whenever P would have done c. The choice of the label l is made independently
on each occasion by the environment of (L : P ).
Example
X1 A lackey is a junior footman, who helps his single master to and from his
seat, and stands behind his chair while he eats
αLACKEY = {sits down, gets up}
LACKEY = (sits down → gets up → LACKEY )
To teach the lackey to share his services among five masters (but serving only
one at a time), we define

70

2

Concurrency

L = {0, 1, 2, 3, 4}
SHARED LACKEY = (L : LACKEY )
The shared lackey could be employed to protect the dining philosophers from
deadlock when the footman (2.5.3) is on holiday. Of course the philosophers
may go hungrier during the holiday, since only one of them is allowed to the
table at a time.

If L contains more than one label, the tree picture of L : P is similar to
that for P ; but it is much more bushy in the sense that there are many more
branches leading from each node. For example, the picture of the LACKEY is
a single trunk with no branches (Figure 2.9).

sits down

gets up

sits down

Figure 2.9
However, the picture of {0, 1} : LACKEY is a complete binary tree (Figure 2.10);
the tree for the SHARED LACKEY is even more bushy.

0.sits down

0.gets up

1.sits down

1.gets up

0.gets up

1.gets up

Figure 2.10
In general, multiple labelling can be used to share the services of a single
process among a number of other labelled processes, provided that the set
of labels is known in advance. This technique will be exploited more fully in
Chapter 6.

2.7

2.7

Specifications

71

Specifications

Let P and Q be processes intended to run concurrently, and suppose we have
proved that P sat S (tr ) and that Q sat T (tr ). Let tr be a trace of (P || Q ). It
follows by 2.3.3 L1 that (tr u αP ) is a trace of P , and consequently it satisfies
S , i.e.,
S (tr u αP )
Similarly, (tr u αQ ) is a trace of Q , so
T (tr u αQ )
This argument holds for every trace of (P || Q ). Consequently we may deduce
(P || Q ) sat (S (tr u αP ) ∧ T (tr u αQ ))
This informal reasoning is summarised in the law
L1

If
and
then

P sat S (tr )
Q sat T (tr )
(P || Q ) sat (S (tr u αP ) ∧ T (tr u αQ ))

Example
X1 (See 2.3.1 X1)
Let

αP = {a, c}
αQ = {b, c}

and

P = (a → c → P )
Q = (c → b → Q )

We wish to prove that
(P || Q ) sat 0 ≤ tr ↓ a − tr ↓ b ≤ 2
The proof of 1.10.2 X1 can obviously be adapted to show that
P sat (0 ≤ tr ↓ a − tr ↓ c ≤ 1)
and
Q sat (0 ≤ tr ↓ c − tr ↓ b ≤ 1)
By L1 it follows that
(P || Q )

72

2

Concurrency

sat (0 ≤ (tr u αP ) ↓ a − (tr u αP ) ↓ c ≤ 1 ∧
0 ≤ (tr u αQ ) ↓ c − (tr u αQ ) ↓ b ≤ 1)
⇒ 0 ≤ tr ↓ a − tr ↓ b ≤ 2

[since (tr u A) ↓ a = tr ↓ a whenever a ∈ A.]


Since the laws for sat allow STOP to satisfy every satisfiable specification,
reasoning based on these laws can never prove absence of deadlock. More
powerful laws will be given in Section 3.7. Meanwhile, one way to eliminate
the risk of stoppage is by careful proof, as in Section 2.5.4. Another method
is to show that a process defined by the parallel combinator is equivalent
to a non-stopping process defined without this combinator, as was done in
2.3.1 X1. However, such proofs involve long and tedious algebraic transformations. Wherever possible, one should appeal to some general law, such as
L2 If P and Q never stop and if (αP ∩ αQ ) contains at most one event, then
(P || Q ) never stops.
Example
X2 The process (P || Q ) defined in X1 will never stop, because
αP ∩ αQ = {c}

The proof rule for change of symbol is
L3

If
then

P sat S (tr )
f (P ) sat S (f −1∗ (tr ))

The use of f −1 in the consequent of this law may need extra explanation. Let
tr be a trace of f (P ). Then f −1 (tr ) is a trace of P . The antecedent of L3 states
that every trace of P satisfies S . It follows that f −1∗ (tr ) satisfies S , which is
exactly what is stated by the consequent of L3.

2.8

Mathematical theory of deterministic processes

In our description of processes, we have stated a large number of laws, and we
have occasionally used them in proofs. The laws have been justified (if at all)
by informal explanations of why we should expect and want them to be true,
For a reader with the instincts of an applied mathematician or engineer, that
may be enough. But the question also arises, are these laws in fact true? Are
they even consistent? Should there be more of them? Or are they complete
in the sense that they permit all true facts about processes to be proved from
them? Could one manage with fewer and simpler laws? These are questions
for which an answer must be sought in a deeper mathematical investigation.

2.8

Mathematical theory of deterministic processes

2.8.1

73

The basic definitions

In constructing a mathematical model of a physical system, it is a good strategy
to define the basic concepts in terms of attributes that can be directly or indirectly observed or measured. For a deterministic process P , we are familiar
with two such attributes
αP —the set of events in which the process is in principle capable of
engaging
traces(P )—the set of all sequences of events in which the process can
actually participate if required.
We have explained how these two sets must satisfy the three laws 1.8.1 L6,
L7, L8. Consider now an arbitrary pair of sets (A, S ) which satisfy these three
laws. This pair uniquely identifies a process P whose traces are S constructed
according to the following definitions. Let
P 0 = { x | hxi ∈ S }
and, for all x in P 0 , let P (x) be the process whose traces are
{ t | hxi _ t ∈ S }
Then
αP = A

and

P = (x : P 0 → P (x))

Furthermore, the pair (A, S ) can be recovered by the equations
A = αP
S = traces(x : P 0 → P (x))
Thus there is a one-one correspondence between each process P and the pairs
of sets (αP , traces(P )). In mathematics, this is a sufficient justification for
identifying the two concepts, by using one of them as the definition of the
other.
D0 A deterministic process is a pair
(A, S )
where A is any set of symbols and S is any subset of A∗ which satisfies the two
conditions
C0 hi ∈ S
C1 ∀ s, t • s _ t ∈ S ⇒ s ∈ S
The simplest example of a process which meets this definition is the one
that does nothing

74

2

Concurrency

D1 STOPA = (A, {hi})
At the other extreme there is the process that will do anything at any time
D2 RUNA = (A, A∗ )
The various operators on processes can now be formally defined by showing how the alphabet and traces of the result are derived from the alphabet
and traces of the operands
D3 (x : B → (A, S (x))) = (A, {hi} ∪ { hxis | x ∈ B ∧ s ∈ S (x) })
provided B ⊆ A
D4 (A, S ) / s = (A, { t | (s _ t ) ∈ S })
D5 µ X : A • F (X ) = (A,

S

n≥0

provided s ∈ S

traces(F n (STOPA )))
provided F is a guarded expression

D6 (A, S ) || (B, T ) = (A ∪ B, { s | s ∈ (A ∪ B)∗ ∧ (s u A) ∈ S ∧ (s u B) ∈ T })
D7 f (A, S ) = (f (A), { f ∗ (s) | s ∈ S })

provided f is one-one

Of course, it is necessary to prove that the right-hand sides of these definitions
are actually processes, i.e., that they satisfy the conditions C0 and C1 of D0.
Fortunately, that is quite easy.
In Chapter 3, it will become apparent that D0 is not a fully adequate definition of the concept of a process, because it does not represent the possibility of nondeterminism. Consequently, a more general and more complicated
definition will be required. All laws for nondeterministic processes are true
for deterministic processes as well. But deterministic processes obey some
additional laws, for example
P || P = P
To avoid confusion, in this book we have avoided quoting such laws; so all
quoted laws may safely be applied to nondeterministic processes as well as
deterministic ones, (except 2.2.1 L3A, 2.2.3 L1, 2.3.1 L3A, 2.3.3 L1, L2, L3A,
L3B, which are false for processes containing CHAOS (3.8)).
2.8.2

Fixed point theory

The purpose of this section is to give an outline of a proof of the fundamental
theorem of recursion, that a recursively defined process (2.8.1 D5) is indeed a
solution of the corresponding recursive equation, i.e.,
µ X • F (X ) = F (µ X • F (X ))
The treatment follows the fixed-point theory of Scott.

2.8

Mathematical theory of deterministic processes

75

First, we need to specify an ordering relationship v among processes
D1 (A, S ) v (B, T ) = (A = B ∧ S ⊆ T )
Two processes are comparable in this ordering if they have the same alphabet,
and one of them can do everything done by the other—and maybe more. This
ordering is a partial order in the sense that
L1 P v P
L2 P v Q ∧ Q v P ⇒ P = Q
L3 P v Q ∧ Q v R ⇒ P v R
A chain in a partial order is an infinite sequence of elements
{ P0 , P1 , P2 , . . . }
such that
Pi v Pi+1

for all i

We define the limit (least upper bound) of such a chain
F

i≥0

Pi = (αP0 ,

S

i≥0

traces(Pi ))

F
In future, we will apply the limit operator
only to sequences of processes
that form a chain.
A partial order is said to be complete if it has a least element, and all chains
have a least upper bound. The set of all processes with a given alphabet A
forms a complete partial order (c.p.o.), since it satisfies the laws
provided αP = A

L4 STOPA v P
F

L5 Pi v

i≥0

Pi

L6 (∀ i ≥ 0 • Pi v Q ) ⇒ (

F

i≥0

Pi ) v Q

Furthermore the definition of µ (2.8.1 D5) can be reformulated in terms of a
limit
L7 µ X : A • F (X ) =

F

i≥0

F i (STOPA )

A function F from one c.p.o. to another one (or the same one) is said to
be continuous if it distributes over the limits of all chains, i.e.,
F(

F

i≥0

Pi ) =

F

i≥0

F (Pi )

if { Pi | i ≥ 0 } is a chain

(All continuous functions are monotonic in the sense that
P v Q ⇒ F (P ) v F (Q )

for all P and Q ,

76

2

Concurrency

so that the right-hand side of the previous equation is also the limit of an
ascending chain.) A function G of several arguments is defined as continuous
if it is continuous in each of its arguments separately, for example
F
F
G(( i≥0 Pi ), Q ) = i≥0 G(Pi , Q )

for all Q

and
G(Q ,

F

i≥0

F

Pi )) =

i≥0

G(Q , Pi )

for all Q

The composition of continuous functions is also continuous; and indeed any
expression constructed by application of any number of continuous functions
to any number and combination of variables is continuous in each of those
variables. For example, if F , G and H are continuous
G(F (X ), H (X , Y ))
is continuous in X , i.e.,
G(F (

F

i≥0

Pi ), H ((

F

i≥0

Pi ), Y )) =

F

i≥0

G(F (Pi ), H (Pi , Y ))

for all Y

All the operators (except /) defined in D3 to D7 are continuous in the sense
defined above
F
F
L8 (x : B → ( i≥0 Pi (x))) = i≥0 (x : B → Pi (x))
F
F
L9 µ X : A • F (X , ( i≥0 Pi )) = i≥0 µ X : A • F (X , Pi )
provided F is continuous
F

L10

(

L11

f(

i≥0

F

Pi ) || Q = Q || (

i≥0

Pi ) =

F

i≥0

F

i≥0

Pi ) =

F

i≥0 (Q

|| Pi )

f (Pi )

Consequently if F (X ) is any expression constructed solely in terms of these
operators, it will be continuous in X . Now it is possible to prove the basic
fixed-point theorem
F (µ X : A • F (X ))
F
= F ( i≥0 F i (STOPA ))
F
= i≥0 F (F i (STOPA ))
F
= i≥1 F i (STOPA )
F
= i≥0 F i (STOPA )
= µ X : A • F (X )

[def. µ]
[continuity F ]
[def. F i+1 ]
[STOPA v F (STOPA )]
[def. µ]

This proof has relied only on the fact that F is continuous. The guardedness
of F is necessary only to establish uniqueness of the solution.

2.8

Mathematical theory of deterministic processes

2.8.3

77

Unique solutions

In this section we treat more formally the reasoning given in Section 1.1.2
to show that an equation defining a process by guarded recursion has only
one solution. In doing so, we shall make explicit more general conditions for
uniqueness of such solutions. For simplicity, we deal only with single equations; the treatment easily extends to sets of simultaneous equations.
If P is a process and n is a natural number, we define (P u n) as a process
which behaves like P for its first n events, and then stops; more formally
(A, S ) u n = (A, { s | s ∈ S ∧ #s ≤ n })
It follows that
L1 P u 0 = STOP
L2 P u n v P u (n + 1) v P
F
L3 P = n≥0 P u n
F
F
L4
n≥0 Pn =
n≥0 (Pn u n)
Let F be a monotonic function from processes to processes. F is said to
be constructive if
F (X ) u (n + 1) = F (X u n) u (n + 1)

for all X

This means that the behaviour of F (X ) on its first n + 1 steps is determined by
the behaviour of X on its first n steps only; so if s ≠ hi
s ∈ traces(F (X )) ≡ s ∈ traces(F (X u (#s − 1)))
Prefixing is the primary example of a constructive function, since
(c → P ) u (n + 1) = (c → (P u n)) u (n + 1)
General choice is also constructive
(x : B → P (x)) u (n + 1) = (x : B → (P (x) u n)) u (n + 1)
The identity function I is not constructive, since
I (c → P ) u 1 = c → STOP
≠ STOP
= I ((c → P ) u 0) u 1
We can now formulate the fundamental theorem
L5 Let F be a constructive function. The equation
X = F (X )
has only one solution for X .

78

2

Concurrency

Proof : Let X be an arbitrary solution. First by induction we prove the lemma
that
X u n = F n (STOP ) u n
Base case.
X u 0 = STOP = STOP u 0 = F 0 (STOP ) u 0
Induction step.
X u (n + 1)
= F (X ) u (n + 1)

[since X = F (X )]

= F (X u n) u (n + 1)

[F is constructive]

n

= F (F (STOP ) u n) u (n + 1)
= F (F n (STOP )) u (n + 1)

[hypothesis]
[F is constructive]

= F n+1 (STOP ) u (n + 1)

[def. F n ]

Now we go back to the main theorem
X
=

F

=

F

=

F

n≥0 (X
n≥0
n≥0

u n)

[L3]

n

[just proved]

n

[L4]

F (STOP ) u n
F (STOP )

= µ X • F (X )

[2.8.2 L7]

Thus all solutions of X = F (X ) are equal to µ X • F (X ); or in other words,
µ X • F (X ) is the only solution of the equation.

The usefulness of this theorem is much increased if we can recognise
which functions are constructive and which are not. Let us define a nondestructive function G as one which satisfies
G(P ) u n = G(P u n) u n

for all n and P .

Alphabet transformation is nondestructive in this sense, since
f (P ) u n = f (P u n) u n
So is the identity function. Any monotonic function which is constructive is
also nondestructive. But the after operator is destructive, since
((c → c → STOP ) / hci) u 1 = c → STOP
≠ STOP
= (c → STOP ) / hci
= (((c → c → STOP ) u 1) / hci) u 1

2.8

Mathematical theory of deterministic processes

79

Any composition of nondestructive functions (G and H ) is also nondestructive, because
G(H (P )) u n = G(H (P ) u n) u n = G(H (P u n) u n) u n = G(H (P u n)) u n
Even more important, any composition of a constructive function with nondestructive functions is also constructive. So if all of F , G, …, H are nondestructive
and just one of them is constructive, then
F (G(. . . (H (X )) . . .))
is a constructive function of X .
The above reasoning extends readily to functions of more than one argument. For example parallel composition is nondestructive (in both its arguments) because
(P || Q ) u n = ((P u n) || (Q u n)) u n
Let E be an expression containing the process variable X . Then E is said to be
guarded in X if every occurrence of X in E has a constructive function applied to
it, and no destructive function. Thus the following expression is constructive
in X
(c → X | d → f (X || P ) | e → (f (X ) || Q )) || ((d → X ) || R)
The important consequence of this is that constructiveness can be defined
syntactically by the following conditions for guardedness
D1 Expressions constructed solely by means of the operators concurrency,
symbol change, and general choice are said to be guard-preserving.
D2 An expression which does not contain X is said to be guarded in X .
D3 A general choice
(x : B → P (X , x))
is guarded in X if P (X , x) is guard-preserving for all x.
D4 A symbol change f (P (X )) is guarded in X if P (X ) is guarded in X .
D5 A concurrent system P (X ) || Q (X ) is guarded in X if both P (X ) and Q (X )
are guarded in X .
Finally, we reach the conclusion
L6 If E is guarded in X , then the equation
X =E
has a unique solution.

Nondeterminism

3.1

3

Introduction

The choice operator (x : B → P (x)) is used to define a process which exhibits
a range of possible behaviours; and the concurrency operator || permits some
other process to make a selection between the alternatives offered in the set B.
For example, the change-giving machine CH5C (1.1.3 X2) offers its customer
the choice of taking his change as three small coins and one large, or two large
coins and one small.
Such processes are called deterministic, because whenever there is more
than one event possible, the choice between them is determined externally by
the environment of the process. It is determined either in the sense that the
environment can actually make the choice, or in the weaker sense that the
environment can observe which choice has been made at the very moment of
the choice.
Sometimes a process has a range of possible behaviours, but the environment of the process does not have any ability to influence of even observe the
selection between the alternatives. For example, a different change-giving machine may give change in either of the combinations described above; but the
choice between them cannot be controlled or even predicted by its user. The
choice is made, as it were internally, by the machine itself, in an arbitrary or
nondeterministic fashion. The environment cannot control the choice of even
observe it; it cannot find out exactly when the choice was made, although it
may later infer which choice was made from the subsequent behaviour of the
process.
There is nothing mysterious about this kind of nondeterminism: it arises
from a deliberate decision to ignore the factor which influence the selection.
For example, the combination of change given by the machine may depend on
the way in which the machine has been loaded with large and small coins; but
we have excluded these events from the alphabet. Thus nondeterminism is
useful for maintaining a high level of abstraction in descriptions of the behaviour of physical systems and machines.

82

3.2

3

Nondeterminism

Nondeterministic or

If P and Q are processes, then we introduce the notation
P uQ

(P or Q )

to denote a process which behaves either like P or like Q , where the selection
between them is made arbitrarily, without the knowledge of control of the
external environment. The alphabets of the operands are assumed to be the
same
α(P u Q ) = αP = αQ
Examples
X1 A change-giving machine which always gives the right change in one of
two combinations
CH5D = (in5p → ((out1p → out1p → out1p → out2p → CH5D)
u (out2p → out1p → out2p → CH5D)))

X2 CH5D may give a different combination of change on each occasion of use.
Here is a machine that always gives the same combination, but we do not know
initially which it will be (see 1.1.2 X3, X4)
CH5E = CH5A u CH5B
Of course, after this machine gives its first coin in change, its subsequent behaviour is entirely predictable. For this reason
CH5D ≠ CH5E

Nondeterminism has been introduced here in its purest and simplest form
by the binary operator u. Of course, u is not intended as a useful operator for
implementing a process. It would be very foolish to build both P and Q , put
them in a black bag, make an arbitrary choice between them, and then throw
the other one away!
The main advantage of nondeterminism is in specifying a process. A process specified as (P u Q ) can be implemented either by building P or by building Q . The choice can be made in advance by the implementor on grounds not
relevant (and deliberately ignored) in the specification, such as low cost, fast
response times, or early delivery. In fact, the u operator will not often be used
directly even in specifications; nondeterminism arises more naturally from use
of the other operators defined later in this chapter.

3.2

Nondeterministic or

3.2.1

83

Laws

The algebraic laws governing nondeterministic choice are exceptionally simple
and obvious. A choice between P and P is vacuous
L1 P u P = P

(idempotence)

It does not matter in which order the choice is presented
L2 P u Q = Q u P

(symmetry)

A choice between three alternatives can be split into two successive binary
choices. It does not matter in which way this is done
L3 P u (Q u R) = (P u Q ) u R

(associativity)

The occasion on which a nondeterministic choice is made is not significant. A
process which first does x and then makes a choice is indistinguishable from
one which first makes the choice and then does x
L4 x → (P u Q ) = (x → P ) u (x → Q )

(distribution)

The law L4 states that the prefixing operator distributes through nondeterminism. Such operators are said to be distributive. A dyadic operator
is said to be distributive if it distributes through u in both its argument positions independently. Most of the operators defined so far for processes are
distributive in this sense
L5 (x : B → (P (x) u Q (x))) = (x : B → P (x)) u (x : B → Q (x))
L6 P || (Q u R) = (P || Q ) u (P || R)
L7 (P u Q ) || R = (P || R) u (Q || R)
L8 f (P u Q ) = f (P ) u f (Q )
However, the recursion operator is not distributive, except in the trivial
case where the operands of u are identical. This point is simply illustrated by
the difference between the two processes
P = µ X • ((a → X ) u (b → X ))
Q = (µ X • (a → X )) u (µ X • (b → X ))
P can make an independent choice between a and b on each iteration, so its
traces include
ha, b, b, a, bi
Q must make a choice between always doing a and always doing b, so its traces
do not include the one displayed above. However, P may choose always to do

84

3

Nondeterminism

a or always to do b, so
traces(Q ) ⊆ traces(P )
In some theories, nondeterminism is obliged to be fair, in the sense that an
event that infinitely often may happen eventually must happen (though there
is no limit to how long it may be delayed). In our theory, there is no such
concept of fairness. Because we observe only finite traces of the behaviour of
a process, if an event can be postponed indefinitely, we can never tell whether
it is going to happen or not. If we want to insist that the event shall happen
eventually, we must state that there is a number n such that every trace longer
than n contains that event. Then the process must be designed explicitly to
satisfy this constraint. For example, in the process P0 defined below, then
event a must always occur within n steps of its previous occurrence
Pi = (a → P0 ) u (b → Pi+1 )
Pn = (a → P0 )
Later, we will see that both Q and P0 are valid implementations of P .
If fairness of nondeterminism is required, this should be specified and
implemented at a separate stage, for example, by ascribing nonzero probabilities to the alternatives of a nondeterministic choice. It seems highly desirable
to separate complex probabilistic reasoning from concerns about the logical
correctness of the behaviour of a process.
In view of laws L1 to L3 it is useful to introduce a multiple-choice operator.
Let S be a finite nonempty set
S = {i, j, . . . k}
Then we define

ux:S P (x) = P (i) u P (j) u . . . u P (k)
ux:S is meaningless when S is either empty or infinite.
3.2.2

Implementations

As mentioned above, one of the main reasons for the introduction of nondeterminism is to abstract from details of implementation. This means that
there may be many different implementations of a nondeterministic process P ,
each with an observably different pattern of behaviour. The differences arise
from different permitted resolutions of the nondeterminism inherent in P . The
choice involved may be made by the implementor before the process starts, or
it may be postponed until the process is actually running. For example, one
way of implementing (P u Q ) is to select the first operand
or1(P , Q ) = P

3.2

Nondeterministic or

85

Another implementation is obtained by selecting the second operand, perhaps
on the grounds of greater efficiency on a particular machine
or2(P , Q ) = Q
Yet a third implementation postpones the decision until the process is running;
it then allows the environment to make the choice, by selecting an event that
is possible for one process but not the other. If the event is possible for both
processes, the decision is again postponed
or3(P , Q ) = λ x • (if P (x) = "BLEEP then
Q (x)
else if Q (x) = "BLEEP then
P (x)
else
or3(P (x), Q (x))
Here we have given three different possible implementations of the same operator. In fact there are many more: for example, an implementation may behave
like or3 for the first five steps; and if all these steps are possible both for P
and for Q , it then arbitrarily chooses P .
Since the design of the process (P u Q ) has no control over whether P
or Q will be selected, he must ensure that his system will work correctly for
both choices. If there is any risk that either P or Q will deadlock with its
environment, then (P u Q ) also runs that same risk. The implementation or3
is the one which minimises the risk of deadlock by delaying the choice until
the environment makes it, and then selecting whichever of P or Q does not
deadlock. For this reason, the definition of or3 is sometimes known as angelic
nondeterminism. But the price to be paid is high in terms of efficiency: if the
choice between P and Q is not made on the first step, both P and Q have
to be executed concurrently until the environment chooses an event which is
possible for one but not the other. In the simple but extreme case of or3(P , P ),
this will never happen, and the inefficiency will also be extreme.
In contrast to or3, the implementations or1 and or2 are asymmetric:
or1(P , Q ) ≠ or1(Q , P )
This seems to violate law 3.2.1 L2; but this is not so. The laws apply to processes, not to any particular implementation of them. In fact they assert the
identity of the set of all implementations of their left and right hand sides. For
example, since or3 is symmetric,
{ or1(P , Q ), or2(P , Q ), or3(P , Q ) }
= { P , Q , or3(P , Q ) }
= {or2(Q , P ), or1(Q , P ), or3(Q , P ) }

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3

Nondeterminism

One of the advantages of introducing nondeterminism is to avoid the loss of
symmetry that would result from selecting one of the two simple implementations, and yet to avoid the inefficiency of the symmetric implementation or3.
3.2.3

Traces

If s is a trace of P , then s is also a possible trace of (P u Q ), i.e., in the case
that P is selected. Similarly, if s is a trace of Q , it is also a trace of (P u Q ).
Conversely, each trace of (P u Q ) must be a trace of one or both alternatives.
The behaviour of (P u Q ) after s is defined by whichever of P or Q could
engage in s; if both could, the choice remains nondeterministic.
L1 traces(P u Q ) = traces(P ) ∪ traces(Q )
L2 (P u Q ) / s = Q / s

if s ∈ (traces(Q ) − traces(P ))

=P /s

if s ∈ (traces(P ) − traces(Q ))

= (P / s) u (Q / s)

if s ∈ (traces(P ) ∩ traces(Q ))

3.3

General choice

The environment of (P u Q ) has no control or even knowledge of the choice
that is made between P and Q , or even the time at which the choice is made. So
(P u Q ) is not a helpful way of combining processes, because the environment
must be prepared to deal with either P or Q ; and either one of them separately
would be easier to deal with.
We therefore introduce another operation (P  Q ), for which the environment can control which of P and Q will be selected, provided that this control
is exercised on the very first action. If this action is not a possible first action
of P , then Q will be selected; but if Q cannot engage initially in the action,
P will be selected. If, however, the first action is possible for both P and Q ,
then the choice between them is nondeterministic. (Of course, if the event is
impossible for both P and Q , then it just cannot happen.) As usual
α(P  Q ) = αP = αQ
In the case that no initial event of P is also possible for Q , the general
choice operator is the same as the | operator, which has been used hitherto to
represent choice between different events
(c → P  d → Q ) = (c → P | d → Q )

if c ≠ d.

However if the initial events are the same, (P  Q ) degenerates to nondeterministic choice
(c → P  c → Q ) = (c → P u c → Q )
Here we have adopted the convention that → binds more tightly than .

3.3

General choice

3.3.1

87

Laws

The algebraic laws for  are similar to those for u, and for the same reasons.
L1–L3  is idempotent, symmetric, and associative.
L4 P  STOP = P
The following law formalises the informal definition of the operation
L5 (x : A → P (x))  (y : B → Q (y)) =
(z : (A ∪ B) →
(if z ∈ (A − B) then
P (z)
else if z ∈ (B − A) then
Q (z)
else if z ∈ (A ∩ B) then
(P (z) u Q (z))))
Like all other operators introduced so far (apart from recursion),  distributes
through u
L6 P  (Q u R) = (P  Q ) u (P  R)
What may seem more surprising is that u distributes through 
L7 P u (Q  R) = (P u Q )  (P u R)
This law states that choices made nondeterministically and choices made by
the environment are independent, in the sense that the selection made by one
of them does not influence the choice made by the other.
Let John be the agent which makes nondeterministic choices and Mary be
the environment. On the left-hand side of the law, John chooses (u) between
P and letting Mary choose () between Q and R. On the right-hand side, Mary
chooses either (1) to offer John the choice between P and Q or (2) to offer John
the choice between P and R.
On both sides of the equation, if John chooses P , then P will be the overall
outcome. But if John does not select P , the choice between Q and R is made
by Mary. Thus the results of the choice strategies described on the left- and
right-hand sides of the law are always equal. Of course, the same reasoning
applies to L6.
The explanation given above is rather subtle; perhaps it would be better to
explain the law as the unexpected by unavoidable consequence of other more
obvious definitions and laws given later in this chapter.

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3

3.3.2

Nondeterminism

Implementation

The implementation of the choice operator follows closely the law L5. Assuming the symmetry of or , it is also symmetrical
choice(P , Q ) = λ x • if P (x) = "BLEEP then
Q (x)
else if Q (x) = "BLEEP then
P (x)
else
or (P (x), Q (x))

3.3.3

Traces

Every trace of (P  Q ) must be a trace of P or a trace of Q , and conversely
L1 traces(P  Q ) = traces(P ) ∪ traces(Q )
The next law is slightly different from the corresponding law for u
L2 (P  Q ) / s = P / s

if s ∈ traces(P ) − traces(Q )

=Q /s

if s ∈ traces(Q ) − traces(P )

= (P / s) u (Q / s)

if s ∈ traces(P ) ∩ traces(Q )

3.4

Refusals

The distinction between (P u Q ) and (P  Q ) is quite subtle. They cannot
be distinguished by their traces, because each trace of one of them is also a
possible trace of the other. However, it is possible to put them in an environment in which (P u Q ) can deadlock at its first step, but (P  Q ) cannot. For
example let x ≠ y and
P = (x → P ),

Q = (y → Q ),

αP = αQ = {x, y}

Then
(P  Q ) || P = (x → P ) = P
but
(P u Q ) || P = (P || P ) u (Q || P ) = P u STOP
This shows that in environment P , process (P u Q ) may reach deadlock
but process (P  Q ) cannot. Of course, even with (P u Q ) we cannot be sure

3.4

Refusals

89

that deadlock will occur; and if it does not occur, we will never know that it
might have. But the mere possibility of an occurrence of deadlock is enough
to distinguish (P  Q ) from (P u Q ).
In general, let X be a set of events which are offered initially by the environment of a process P , which in this context we take to have the same alphabet
as P . If it is possible for P to deadlock on its first step when placed in this environment, we say that X is a refusal of P . The set of all such refusals of P is
denoted
refusals(P )
Note that the refusals of a process constitute a family of sets of symbols. This
is an unfortunate complexity, but it does seem to be unavoidable in a proper
treatment of nondeterminism. Instead of refusals, it might seem more natural
to use the sets of symbols which a process may be ready to accept; however
the refusals are slightly simpler because they obey laws L9 and L10 of Section 3.4.1 (below), whereas the corresponding laws for ready sets would be
more complicated.
The introduction of the concept of a refusal permits a clear formal distinction to be made between deterministic and nondeterministic processes. A
process is said to be deterministic if it can never refuse any event in which it
can engage. In other words, a set is a refusal of a deterministic process only if
that set contains no event in which that process can initially engage; or more
formally
P is deterministic ⇒ (X ∈ refusals(P ) ≡ (X ∩ P 0 = {}))
where P 0 = { x | hxi ∈ traces(P ) }.
This condition applies not only on the initial step of P but also after any
possible sequence of actions of P . Thus we can define
P is deterministic ≡
∀ s : traces(P ) • (X ∈ refusals(P / s) ≡ (X ∩ (P / s)0 = {}))
A nondeterministic process is one that does not enjoy this property, i.e., there
is at some time some event in which it can engage; but also (as a result of some
internal nondeterministic choice) it may refuse to engage in that event, even
though the environment is ready for it.
3.4.1

Laws

The following laws define the refusals of various simple processes. The process
STOP does nothing and refuses everything
L1 refusals(STOPA ) = all subsets of A (including A itself)
A process c → P refuses every set that does not contain the event c
L2 refusals(c → P ) = { X | X ⊆ (αP − {c}) }

90

3

Nondeterminism

These two laws have a common generalisation
L3 refusals(x : B → P (x)) = { X | X ⊆ (αP − B) }
If P can refuse X , so will P u Q if P is selected. Similarly every refusal of Q is
also a possible refusal of (P u Q ). These are its only refusals, so
L4 refusals(P u Q ) = refusals(P ) ∪ refusals(Q )
A converse argument applies to (P  Q ). If X is not a refusal of P , then P
cannot refuse X , and neither can (P  Q ). Similarly, if X is not a refusal of Q ,
then it is not a refusal of (P  Q ). However, if both P and Q can refuse X , so
can (P  Q )
L5 refusals(P  Q ) = refusals(P ) ∩ refusals(Q )
Comparison of L5 with L4 shows the distinction between  and u.
If P can refuse X and Q can refuse Y , then their combination (P || Q ) can
refuse all events refused by P as well as all events refused by Q , i.e., it can
refuse the union of the two sets X and Y
L6 refusals(P || Q ) = { X ∪ Y | X ∈ refusals(P ) ∧ Y ∈ refusals(Q ) }
For symbol change, the relevant law is clear
L7 refusals(f (P )) = { f (x) | X ∈ refusals(P ) }
There are a number of general laws about refusals. A process can refuse
only events in its own alphabet. A process deadlocks when the environment
offers no events; and if a process refuses a nonempty set, it can also refuse
any subset of that set. Finally, any event x which cannot occur initially may be
added to any set X already refused.
L8 X ∈ refusals(P ) ⇒ X ⊆ αP
L9 {} ∈ refusals(P )
L10

(X ∪ Y ) ∈ refusals(P ) ⇒ X ∈ refusals(P )

L11

X ∈ refusals(P ) ⇒ (X ∪ {x}) ∈ refusals(P ) ∨ hxi ∈ traces(P )

3.5

Concealment

In general, the alphabet of a process contains just those events which are considered to be relevant, and whose occurrence requires simultaneous participation of an environment. In describing the internal behaviour of a mechanism, we often need to consider events representing internal transitions of that
mechanism. Such events may denote the interactions and communications
between concurrently acting components from which the mechanism has been

3.5

Concealment

91

constructed, e.g., CHAIN2 (2.6 X4) and 2.6.2 X3. After construction of the mechanism, we conceal the structure of its components; and we also wish to conceal
all occurrences of actions internal to the mechanism. In fact, we want these actions to occur automatically and instantaneously as soon as they can, without
being observed or controlled by the environment of the process. If C is a finite
set of events to be concealed in this way, then
P\C
is a process which behaves like P , except that each occurrence of any event in
C is concealed. Clearly it is our intention that
α(P \ C) = (αP ) − C
Examples
X1 A noisy vending machine (2.3 X1) can be placed in a soundproof box
NOISYVM \ {clink, clunk}
Its unexercised capability of dispensing toffee can also be removed from its
alphabet, without affecting its actual behaviour. The resulting process is equal
to the simple vending machine
VMS = NOISYVM \ {clink, clunk, toffee}

When two processes have been combined to run concurrently, their mutual
interactions are usually regarded as internal workings of the resulting systems;
they are intended to occur autonomously and as quickly as possible, without
the knowledge or intervention of the system’s outer environment. Thus it is
the symbols in the intersection of the alphabets of the two components that
need to be concealed.
X2 Let
αP = {a, c}

P = (a → c → P )

αQ = {b, c}

Q = (c → b → Q )

as in (2.3.1 X1).
The action c in the alphabet of both P and Q is now regarded as an internal
action, to be concealed
(P || Q ) \ {c} = (a → c → µ X • (a → b → c → X
| b → a → c → X )) \ {c}
= a → µ X • (a → b → X
| b → a → X)



92

3

3.5.1

Nondeterminism

Laws

The first laws state that concealing no symbols has no effect, and that it makes
no difference in what order the symbols of a set are concealed. The remaining
laws of this group show how concealment distributes through other operators.
Concealment of nothing leaves everything revealed
L1 P \ {} = P
To conceal one set of symbols and then some more is the same as concealing
them all simultaneously.
L2 (P \ B) \ C = P \ (B ∪ C)
Concealment distributes in the familiar way through nondeterministic choice
L3 (P u Q ) \ C = (P \ C) u (Q \ C)
Concealment does not affect the behaviour of a stopped process, only its alphabet
L4 STOPA \ C = STOPA−C
The purpose of concealment is to allow any of the concealed events to occur
automatically and instantaneously, but make such occurrences totally invisible. Unconcealed events remain unchanged
L5 (x → P ) \ C = x → (P \ C)
=P \C

if x ∉ C
if x ∈ C

If C contains only events in which P and Q participate independently, concealment of C distributes through their concurrent composition
L6 If αP ∩ αQ ∩ C = {}, then
(P || Q ) \ C = (P \ C) || (Q \ C)
This is not a commonly useful law, because what we usually wish to conceal
are the interactions between concurrent processes, i.e., the events of αP ∩ αQ ,
in which they participate jointly.
Concealment distributes in the obvious way through symbol change by a
one-one function
L7 f (P \ C) = f (P ) \ f (C)
If none of the possible initial events of a choice is concealed, then the
initial choice remains the same as it was before concealment
L8 If B ∩ C = {}, then
(x : B → P (x)) \ C = (x : B → (P (x) \ C))

3.5

Concealment

93

Like the choice operator , the concealment of events can introduce nondeterminism. When several different concealed events can happen, it is not
determined which of them will occur; but whichever does occur is concealed
L9 If B ⊆ C, and B is finite and not empty, then
(x : B → P (x)) \ C = ux∈B (P (x) \ C)
In the intermediate case, when some of the initial events are concealed and
some are not, the situation is rather more complicated. Consider the process
(c → P | d → Q ) \ C
where
c ∈ C, d ∉ C
The concealed event c may happen immediately. In this case the total behaviour will be defined by (P \ C), and the possibility of occurrence of the event
d will be withdrawn. But we cannot reliably assume that d will not happen.
If the environment is ready for it, d may very well happen before the hidden
event, after which the hidden event c can no longer occur. But even if d occurs,
it might have been performed by (P \ C) after the hidden occurrence of c. In
this case, the total behaviour is as defined by
(P \ C)  (d → (Q \ C))
The choice between this and (P \ C) is nondeterministic. This is a rather
convoluted justification for the rather complex law
(c → P | d → Q ) \ C =
(P \ C) u ((P \ C)  (d → (Q \ C)))
Similar reasoning justifies the more general law
L10

If C ∩ B is finite and non-empty, then
(x : B → P (x)) \ C =
Q u (Q  (x : (B − C) → P (x)))

where
Q = ux∈B∩C P (x) \ C
A pictorial illustration of these laws is given in Section 3.5.4.
Note that \ C does not distribute backwards through . A counterexample
is
(c → STOP  d → STOP ) \ {c}

94

3

= STOP u (STOP  (d → STOP ))
= STOP u (d → STOP )

Nondeterminism

[L10]
[3.3.1 L4]

≠ d → STOP
= STOP  (d → STOP )
= ((c → STOP ) \ {c})  ((d → STOP ) \ {c})
Concealment reduces the alphabet of a process. We can also define an
operation which extends the alphabet of a process P by inclusion of symbols
of a set B
α(P+B ) = αP ∪ B
P+B = (P || STOPB )

provided B ∩ αP = {}

None of the new events of B will ever actually occur, so the behaviour of P+B
is effectively the same as that of P
L11

traces(PB+ ) = traces(P )

Consequently, concealment of B reverses the extension of the alphabet by B
L12

(P+B ) \ B = P

It is appropriate here to raise a problem that will be solved later, in Section 3.8. In simple cases, concealment distributes through recursion
(µ X : A • (c → X )) \ {c}
= µ X : (A − {c}) • ((c → X+{c} ) \ {c})
= µ X : (A − {c}) • X

[by L12, L5]

Thus the attempt to conceal an infinite sequence of consecutive events leads
to the same unfortunate result as an infinite loop or unguarded recursion. The
general name for this phenomenon is divergence.
The same problem arises even if the divergent process is infinitely often
capable of some unconcealed event, for example
(µ X • (c → X  d → P )) \ {c}
= µ X • ((c → X  d → P ) \ {c})
= µ X • (X \ {c}) u ((X \ {c})  d → (P \ {c}))

[by L10]

Here again, the recursion is unguarded, and leads to divergence. Even though
it seems that the environment is infinitely often offered the choice of selecting
d, there is no way of preventing the process from infinitely often choosing to
perform the hidden event instead. This possibility seems to aid in achieving the
highest efficiency of implementation. It also seems to be related to our decision
not to insist on fairness of nondeterminism, as discussed in Section 3.2.1. A
more rigorous discussion of divergence is given in Section 3.8.

3.5

Concealment

95

There is a sense, an important one, in which hiding is in fact fair. let
d ∈ αR, and consider the process
((c → a → P | d → STOP ) \ {c}) || (a → R)
= ((a → P \ {c}) u (a → P \ {c}  d → STOP )) || (a → R)

[L10]

= (a → P \ {c}) || (a → R) u (a → P \ {c}  d → STOP ) || (a → R)
= a → ((P \ {c}) || R)
This shows that a process which offers the choice between a hidden action c
and a nonhidden one d cannot insist that the nonhidden action shall occur.
If the environment (in this example, a → R) is not prepared for d, then the
hidden event must occur, so that the environment has the chance to interact
with the process (e.g. (a → P \ {c})) which results.
3.5.2

Implementation

For simplicity, we shall implement an operation which hides a single symbol
at a time
hide(P , c) = P \ {c}
A set of two or more symbols may be hidden by hiding one after the other,
since
P \ {c1, c2, . . . , cn} = (. . . ((P \ {c1}) \ {c2}) \ . . .) \ {cn}
The simplest implementation is one that always makes the hidden event occur
invisibly, whenever it can and as soon as it can
hide(P , c) = if P (c) = "BLEEP then
(λ x • if P (x) = "BLEEP then
"BLEEP
else
hide(P (x), c))
else
hide(P (c), c)
Let us explore what happens when the hide function is applied to a process which is capable of engaging in an infinite sequence of hidden events, for
example
hide(µ X • (c → X  d → P ), c)
In this case, the test (P (c) = "BLEEP ) will always yield FALSE , so the hide
function will always select its else clause, thereby immediately calling itself

96

3

Nondeterminism

recursively. There is no exit from this recursion, so no further communication
with the outside world will ever occur. This is the penalty for attempting to
implement a divergent process.
This implementation of concealment does not obey L2; indeed, the order
in which it hides the symbols is significant, as shown by the example
P = (c → STOP | d → a → STOP )
Then
hide(hide(P , c), d)
= hide(hide(STOP , c), d)
= STOP
and
hide(hide(P , d), c)
= hide(hide((a → STOP ), d), c)
= (a → STOP )
But as explained in Section 3.2.2, a particular implementation of a nondeterministic operator does not have to obey the laws. It is sufficient that both the
results shown above are permitted implementations of the same process
P \ {c, d} = (STOP u (a → STOP ))

3.5.3

Traces

If t is a trace of P , the corresponding trace of P \ C is obtained from t simply
by removing all occurrences of any of the symbols in C. Conversely each trace
of P \ C must have been obtained from some such trace of P . We can therefore
state
L1 traces(P \ C) = { t u (αP − C) | t ∈ traces(P ) }
provided that ∀ s : traces(P ) • ¬ diverges(P / s, C)
The condition diverges(P , C) means that P diverges immediately on concealment of C, i.e. that it can engage in an unbounded sequence of hidden
events. Thus we define
diverges(P , C) = ∀ n • ∃ s : traces(P ) ∩ C ∗ • #s > n

3.5

Concealment

97

Corresponding to a single trace s of P \ C, there can be several traces t of
the possible behaviour in which P has engaged which cannot be distinguished
after the concealment, i.e., t u (αP − C) = s. The next law states that after s it is
not determined which of the possible subsequent behaviours of P will define
the subsequent behaviour of (P \ C).
L2 (P \ C) / s = (ut ∈T P / t ) \ C
where T = traces(P ) ∩ { t | t u (αP − C) = s }
provided that T is finite and s ∈ traces(P \ C)
These laws are restricted to the case when the process does not diverge.
The restrictions are not serious, because divergences is never the intended result of the attempted definition of a process. For a fuller treatment of divergence
see Section 3.8.
3.5.4

Pictures

Nondeterministic choice can be represented in a picture by a node from which
emerge two or more unlabelled arrows; on reaching the node, a process passes
imperceptibly along one of the emergent arrows, the choice being nondeterministic.

P

Q

Figure 3.1
Thus P u Q is pictured as in Figure 3.1. The algebraic laws governing
nondeterminism assert identities between such processes, e.g., associativity
of u is illustrated in Figure 3.2.

P

=
Q

R

Figure 3.2

P

Q

R

98

3

Nondeterminism

Concealment of symbols may be regarded as an operation which simply
removes concealed symbols from all arrows which they label, so that these arcs
turn into unlabelled arrows. The resulting nondeterminism emerges naturally,
as shown in Figure 3.3.
\ { c, d }
c

d

P

Q

=

P \ { c, d }

Q \ { c, d }

Figure 3.3
But what is the meaning of a node if some of its arcs are labelled and
some are not? The answer is given by the law 3.5.1 L10. Such a node can be
eliminated by redrawing as shown in Figure 3.4.

a
R
b
P

=

c
Q

b
P

b

c
P

Q

c
Q

a
R

Figure 3.4
It is fairly obvious that such eliminations are always possible for finite
trees. They are also possible for infinite graphs, provided that the graph contains no infinite path of consecutive unlabelled arrows, as for example in Figure 3.5. Such a picture can arise only in the case of divergence, which we have
already decided to regard as an error.

a
P

Figure 3.5

3.6

Interleaving

99

As a result of applying the transformation L10, it is possible that the node
may acquire two emergent lines with the same label. Such nodes can be eliminated by the law given at the end of Section 3.3 (Figure 3.6).

a

b

P

a

=
Q

P

b

b
Q

R

R

Figure 3.6
The pictorial representation of processes and the laws which govern them
are included here as an aid to memory and understanding; they are not intended to be used for practical transformation or manipulation of large-scale
processes.

3.6

Interleaving

The || operator was defined in Chapter 2 in such a way that actions in the alphabet of both operands require simultaneous participation of them both, whereas
the remaining actions of the system occur in an arbitrary interleaving. Using
this operator, it is possible to combine interacting processes with differing alphabets into systems exhibiting concurrent activity, but without introducing
nondeterminism.
However, it is sometimes useful to join processes with the same alphabet
to operate concurrently without directly interacting or synchronising with each
other. In this case, each action of the system is an action of exactly one of the
processes If one of the processes cannot engage in the action, then it must
have been the other one; but if both processes could have engaged in the same
action, the choice between them is nondeterministic. This form of combination
is denoted
P ||| Q

P interleave Q

and its alphabet is defined by the usual stipulation
α(P ||| Q ) = αP = αQ
Examples
X1 A vending machine that will accept up to two coins before dispensing up
to two chocolates (1.1.3 X6)
(VMS ||| VMS ) = VMS2


100

3

Nondeterminism

X2 A footman made from four lackeys, each serving only one philosopher at
a time (see Section 2.5.4).

3.6.1

Laws

L1–L3 ||| is associative, symmetric, and distributes through u.
L4 P ||| STOP = P
L5 P ||| RUN = RUN

provided P does not diverge

L6 (x → P ) ||| (y → Q ) = (x → (P ||| (y → Q ))  y → ((x → P ) ||| Q ))
L7

If
and
then

P = (x : A → P (x))
Q = (y : B → Q (y))
P ||| Q = (x : A → (P (x) ||| Q )  y : B → (P ||| Q (y)))

Note that ||| does not distribute through . This is shown by the counterexample (where b ≠ c)
((a → STOP ) ||| (b → Q  c → R)) / hai
= (b → Q  c → R)
≠ ((b → Q ) u (c → R))
= ((a → STOP  b → Q ) ||| (a → STOP  c → R)) / hai
On the left-hand side of this chain, the occurrence of a can involve progress
only of the left operand of |||, so no nondeterminism is introduced. The left
operand stops, and the choice between b and c is left open to the environment.
On the right-hand side of the chain, the event a may be an event of either
operand of |||, the choice being nondeterministic. Thus the environment can
no longer choose whether the next event will be b or c.
L6 and L7 state that it is the environment which chooses between the
initial events offered by the operands of |||. Nondeterminism arises only when
the chosen event is possible for both operands.
Example
X1 Let R = (a → b → R), then
(R ||| R)
= (a → ((b → R) ||| R)  a → (R ||| (b → R)))

[L6]

= a → ((b → R) ||| R) u (R ||| (b → R))
= a → ((b → R) ||| R)
Also
(b → R) ||| R

[L2]

3.7

Specifications

101

= (a → ((b → R) ||| (b → R))  b → (R ||| R))
= (a → (b → ((b → R) ||| R))

[L6]
[as shown above]

 b → (a → ((b → R) ||| R)))
= µ X • (a → b → X

[since the recursion is guarded.]

 b → a → X)
Thus (R ||| R) is identical to the example 3.5 X2. A similar proof shows that
(VMS ||| VMS ) = VMS2.

3.6.2

Traces and refusals

A trace of (P ||| Q ) is an arbitrary interleaving of a trace from P with a trace
from Q . For a definition of interleaving, see Section 1.9.3.
L1 traces(P ||| Q ) =
{ s | ∃ t : traces(P ) • ∃ u : traces(Q ) • s interleaves (t , u) }
(P ||| Q ) can engage in any initial action possible for either P or Q ; and it can
therefore refuse only those sets which are refused by both P and Q
L2 refusals(P ||| Q ) = refusals(P  Q )
The behaviour of (P ||| Q ) after engaging in the events of the trace s is
defined by the rather elaborate formula
L3 (P ||| Q ) / s = ut ,u∈T (P / t ) ||| (Q / u)
where
T = { (t , u) | t ∈ traces(P ) ∧ u ∈ traces(Q ) ∧ s interleaves (t , u) }
This law reflects the fact that there is no way of knowing in which way a trace
s of (P ||| Q ) has been constructed as an interleaving of a trace from P and a
trace from Q ; thus after s, the future behaviour of (P ||| Q ) may reflect any one
of the possible interleavings. The choice between them is not known and not
determined.

3.7

Specifications

In Section 3.4 we saw the need to introduce refusal sets as one of the important
indirectly observable aspects of the behaviour. In specifying a process, we
therefore need to describe the desired properties of its refusal sets as well as
its traces. Let us use the variable ref to denote an arbitrary refusal set of a
process, in the same way as we have used tr to denote an arbitrary trace. As a
result, when P is a nondeterministic process the meaning of
P sat S (tr , ref )

102

3

Nondeterminism

is revised to
∀ tr , ref • tr ∈ traces(P ) ∧ ref ∈ refusals(P / tr ) ⇒ S (tr , ref )
Examples
X1 When a vending machine has ingested more coins than it has dispensed
chocolates, the customer specifies that it must not refuse to dispense a chocolate
FAIR = (tr ↓ choc < tr ↓ coin ⇒ choc ∉ ref )
It is implicitly understood that every trace tr and every refusal ref of the
specified process at all times should satisfy this specification.

X2 When a vending machine has given out as many chocolates as have been
paid for, the owner specifies that it must not refuse a further coin
PROFIT1 = (tr ↓ choc = tr ↓ coin ⇒ coin ∉ ref )

X3 A simple vending machine should satisfy the combined specification
NEWVMSPEC = FAIR ∧ PROFIT ∧ (tr ↓ choc ≤ tr ↓ coin)
This specification is satisfied by VMS . It is also satisfied by a vending machine
like VMS2 (1.1.3 X6) which will accept several coins in a row, and then give out
several chocolates.

X4 If desired, one may place a limit on the balance of coins which may be
accepted in a row
ATMOST2 = (tr ↓ coin − tr ↓ choc ≤ 2)

X5 If desired, one can insist that the machine accept at least two coins in a
row whenever the customer offers them
ATLEAST2 = (tr ↓ coin − tr ↓ choc < 2 ⇒ coin ∉ ref )

X6 The process STOP refuses every event in its alphabet. The following predicate specifies that a process with alphabet A will never stop
NONSTOP = (ref ≠ A)

3.7

Specifications

103

If P sat NOTSTOP , and if an environment allows all events in A, P must perform
one of them. Since (see X3 above)
NEWVMSPEC ⇒ ref ≠ {coin, choc}
it follows that any process which satisfies NEWVMSPEC will never stop.



These examples show how the introduction of ref into the specification of
a process permits the expression of a number of subtle but important properties; perhaps the most important of all is the property that the process must
not stop (X6). These advantages are obtained at the cost of slightly increased
complexity in proof rules and in proofs.
It is also desirable to prove that a process does not diverge. Section 3.8
describes how a divergent process is one that can do anything and refuse anything. So if there is a set which cannot be refused, then the process is not divergent. This justifies formulation of a sufficient condition for non-divergence
NONDIV = (ref ≠ A)
Fortunately
NONSTOP ≡ NONDIV
so proof of absence of divergence does not entail any more work than proof
of absence of deadlock.
3.7.1

Proofs

In the following proof rules, a specification will be written in any of the forms S ,
S (tr ), S (tr , ref ), according to convenience. In all cases, it should be understood
that the specification may contain tr and ref among its free variables.
By the definition of nondeterminism, (P u Q ) behaves either like P or like
Q . Therefore every observation of its behaviour will be an observation possible
for P or for Q or for both. This observation will therefore be described by the
specification of P or by the specification of Q or by both. Consequently, the
proof rule for nondeterminism has an exceptionally simple form
L1

If
and
then

P sat S
Q sat T
(P u Q ) sat (S ∨ T )

The proof rule for STOP states that it does nothing and refuses anything
L2A

STOPA sat (tr = hi ∧ ref ⊆ A)

Since refusals are always contained in the alphabet (3.4.1 L8) the clause ref ⊆ A
can be omitted. So if we omit alphabets altogether (which we shall do in future),

104

3

Nondeterminism

the law L2A is identical to that for deterministic processes (1.10.2 L4A)
STOP sat tr = hi
The previous law for prefixing (1.10.2 L4B) is also still valid, but it is not
quite strong enough to prove that the process cannot stop before its initial
action. The rule needs to be strengthened by mention of the fact that in the
initial state, when tr = hi, the initial action cannot be refused
L2B

If
then

P sat S (tr )
(c → P ) sat ((tr = hi ∧ c ∉ ref ) ∨ (tr0 = c ∧ S (tr 0 )))

The law for general choice (1.10.2 L4) needs to be similarly strengthened
L2

If
then

∀ x : B • P (x) sat S (tr , x)
(x : B → P (x)) sat
((tr = hi ∧ (B ∩ ref = {}) ∨ (tr0 ∈ B ∧ S (tr 0 , tr0 ))))

The law for parallel composition given in 2.7 L1 is still valid, provided that
the specifications make no mention of refusal sets. In order to deal correctly
with refusals, a slightly more complicated law is required
L3

If

P sat S (tr , ref )

and

Q sat T (tr , ref )

and

neither P nor Q diverges

then

(P || Q ) sat
(∃ X , Y , ref • ref = (X ∪ Y ) ∧ S (tr u αP , X ) ∧ T (tr u αQ , Y ))

The law for change of symbol needs a similar adaptation
L4

If
then

P sat (tr , ref )
f (P ) sat S (f −1∗ (tr ), f −1 (ref ))

provided f is one-one.

The law for  is surprisingly simple
L5

If

P sat S (tr , ref )

and

Q sat T (tr , ref )

and

neither P nor Q diverges

then

(P  Q ) sat (if tr = hi then (S ∧ T ) else (S ∨ T ))

Initially, when tr = hi, a set is refused by (P  Q ) only if it is refused by both
P and Q . This set must therefore be described by both their specifications.
Subsequently, when tr ≠ hi, each observation of (P  Q ) must be an observation either of P or of Q , and must therefore be described by one of their
specifications (or both).

3.8

Divergence

105

The law for interleaving does not need to mention refusal sets
L6

If

P sat S (tr )

and

Q sat T (tr )

and

neither P nor Q diverges

then

(P ||| Q ) sat (∃ s, t • (tr interleaves (s, t ) ∧ S (s) ∧ T (t )))

The law for concealment is complicated by the need to guard against divergence
L7

If

P sat (NODIV ∧ S (tr , ref ))

then (P \ C) sat ∃ s • tr = s u (αP − C) ∧ S (tr , ref ∪ C)
where NODIV states that the number of hidden symbols that can occur is
bounded by some function of the non-hidden symbols that have occurred
NODIV = #(tr u C) ≤ f (tr u (αP − C))
where f is some total function from traces to natural numbers.
The clause ref ∪ C in the consequent of law L7 requires some explanation.
It is due to the fact that P \ C can refuse a set X only when P can refuse the
whole set X ∪ C, i.e., X together with all the hidden events. P \ C cannot
refuse to interact with its external environment until it has reached a state
in which it cannot engage in any further concealed internal activities. This
kind of fairness is a most important feature of any reasonable definition of
concealment, as described in Section 3.5.1.
The proof method for recursion (1.10.2 L6) also needs to be strengthened.
Let S (n) be a predicate containing the variable n, which ranges over the natural
numbers 0, 1, 2, . . .
L8

If
and
then

S (0)
(X sat S (n)) ⇒ F (X ) sat S (n + 1))
(µ X • F (X )) sat (∀ n • S (n))

This law is valid even for an unguarded recursion, though the strongest specification which can be proved of such a process is the vacuous specification
true.

3.8

Divergence

In previous chapters, we have observed the restriction that the equations which
define a process must be guarded (Section 1.1.2). This restriction has ensured
that the equations have only a single solution (1.3 L2). It has also released us
from the obligation of giving a meaning to the infinite recursion
µX • X

106

3

Nondeterminism

Unfortunately, the introduction of concealment (Section 3.5) means that
an apparently guarded recursion is not constructive. For example, consider
the equation
X = c → (X \ {c})+{c}
This has as solutions both (c → STOP ) and (c → a → STOP ), a fact which may
be checked by substitution.
Consequently, any recursion equation which involves recursion under the
hiding operator is potentially unguarded, and liable to have more than one
solution. Which solution should be taken as the right one? We stipulate that
the right solution is the least deterministic, because this allows a nondeterministic choice between all the other solutions. With this understanding, we can
altogether remove the restriction that recursions must be guarded, and we can
give a (possibly nondeterministic) meaning to every expression of the form
µ X • F (X ), where F is defined in terms of any of the operators introduced in
this book (except /), and observing all alphabet constraints.
3.8.1

Laws

Since CHAOS is the most nondeterministic process it cannot be changed by
adding yet further nondeterministic choices; it is therefore a zero of u
L1 P u CHAOS = CHAOS
A function of processes that yields CHAOS if any of its arguments is CHAOS
is said to be strict. The above law (plus symmetry) states that u is a strict
function. CHAOS is such an awful process that almost any process which is
defined in terms of CHAOS is itself equal to CHAOS .
L2 The following operations are strict
/ s, ||, f , , \ C, |||, and µ X
However prefixing is not strict.
L3 CHAOS ≠ (a → CHAOS )
because the right-hand side can be relied upon to do a before becoming completely unreliable.
As mentioned before, CHAOS is the most unpredictable and most uncontrollable of processes. There is nothing that it might not do; furthermore, there
is nothing that it might not refuse to do!
L4 traces(CHAOSA ) = A∗
L5 refusals(CHAOSA ) = all subsets of A.

3.8

Divergence

3.8.2

107

Divergences

A divergence of a process is defined as any trace of the process after which the
process behaves chaotically. The set of all divergences is defined
divergences(P ) = { s | s ∈ traces(P ) ∧ (P / s) = CHAOSαP }
It follows immediately that
L1 divergences(P ) ⊆ traces(P )
Because / t is strict,
CHAOS / t = CHAOS
and it follows that the divergences of a process are extension-closed, in the
sense that
L2 s ∈ divergences(P ) ∧ t ∈ (αP )∗ ⇒ (s _ t ) ∈ divergences(P )
Since CHAOSA may refuse any subset of its alphabet A
L3 s ∈ divergences(P ) ∧ X ⊆ αP ⇒ X ∈ refusal(P / s)
The three laws given above state general properties of divergences of any
process. The following laws show how the divergences of compound processes
are determined by the divergences and traces of their components. Firstly, the
process STOP never diverges
L4 divergences(STOP ) = {}
At the other extreme, every trace of CHAOS leads to CHAOS
L5 divergences(CHAOSA ) = A∗
A process defined by choice does not diverge on its first step. Consequently,
its divergences are determined by what happens after the first step
L6 divergences(x : B → P (x)) = { hxi _ s | x ∈ B ∧ s ∈ divergences(P (x)) }
Any divergence of P is also a divergence of (P u Q ) and of (P  Q )
L7 divergences(P u Q ) = divergences(P  Q )
= divergences(P ) ∪ divergences(Q )
Since || is strict, a divergence of (P || Q ) starts with a trace of the nondivergent
activity of both P and Q , which leads to divergence of either P or of Q (or of
both)

108

3

Nondeterminism

L8 divergences(P || Q ) =
{ s _ t | t ∈ (αP ∪ αQ )∗ ∧
((s u αP ∈ divergences(P ) ∧ s u αQ ∈ traces(Q )) ∨
s u αP ∈ traces(P ) ∧ s u αQ ∈ divergences(Q ))
A similar explanation applies to |||
L9 divergences(P ||| Q ) =
{ u | ∃ s, t • u interleaves (s, t ) ∧
((s ∈ divergences(P ) ∧ t ∈ traces(Q )) ∨
(s ∈ traces(P ) ∧ t ∈ divergences(Q ))) }
Divergences of a process resulting from concealment include traces derived
from the original divergences, plus those resulting from the attempt to conceal
an infinite sequence of symbols
L10

divergences(P \ C) =
{ (s u (αP − C)) _ t |
t ∈ (αP − C)∗ ∧
(s ∈ divergences(P ) ∨
(∀ n • ∃ u ∈ C ∗ • #u > n ∧ (s _ u) ∈ traces(P ))) }

A process defined by symbol change diverges only when its argument diverges
L11

divergences(f (P )) = { f ∗ (s) | s ∈ divergences(P ) }

provided f is one-one.
It is a shame to devote so much attention to divergence, when divergence is
always something we do not want. Unfortunately, it seems to be an inevitable
consequence of any efficient of even computable method of implementation. It
can arise from either concealment or unguarded recursion; and it is part of the
task of a system designer to prove that for his particular design the problem
will not occur. In order to prove that something can’t happen, we need to use
a mathematical theory in which it can!

3.9

Mathematical theory of non-deterministic processes

The laws given in this chapter are distinctly more complicated than the laws
given in the two earlier chapters; and the informal justifications and examples
carry correspondingly less conviction. It is therefore even more important to
construct a proper mathematical definition of the concept of a nondeterministic process, and prove the correctness of the laws from the definitions of the
operators.

3.9

Mathematical theory of non-deterministic processes

109

As in Section 2.8.1, a mathematical model is based on the relevant directly
or indirectly observable properties of a process. These certainly include its
alphabet and its traces; but for a nondeterministic process there are also its
refusals (Section 3.4) and divergences (Section 3.8). In addition to refusals at
the first step of a process P , it is necessary also to take into account what P
may refuse after engaging in an arbitrary trace s of its behaviour. We therefore
define the failures of a process as a relation (set of pairs)
failures(P ) = { (s, X ) | s ∈ traces(P ) ∧ X ∈ refusals(P / s) }
If (s, X ) is a failure of P , this means that P can engage in the sequence of events
recorded by s, and then refuse to do anything more, in spite of the fact that
its environment is prepared to engage in any of the events of X . The failures
of a process are more informative about the behaviour of that process that its
traces or refusals, which can both be defined in terms of failures
traces(P ) = { s | ∃ X • (s, X ) ∈ failures(P ) }
= domain(failures(P ))
refusals(P ) = { X | (hi, X ) ∈ failures(P ) }
The various properties of traces (1.8.1 L6, L7, L8) and refusals (3.4.1 L8, L9,
L10, L11) can easily be reformulated in terms of failures (see conditions C1,
C2, C3, C4 under the definition D0 below).
We are now ready for the bold decision that a process is uniquely defined
by the three sets specifying its alphabet, its failures, and its divergences; and
conversely, any three sets which satisfy the relevant conditions uniquely define
a process. We will first define the powerset of A as the set of all its subsets
PA = {X | X ⊆ A}
D0 A process is a triple (A, F , D), where
A is any set of symbols (for simplicity finite)
F is a relation between A∗ and P A
D is a subset of A∗
provided that they satisfy the following conditions
C1 (hi, {}) ∈ F
C2 (s _ t , X ) ∈ F ⇒ (s, {}) ∈ F
C3 (s, Y ) ∈ F ∧ X ⊆ Y ⇒ (s, X ) ∈ F
C4 (s, X ) ∈ F ∧ x ∈ A ⇒ (s, X ∪ {x}) ∈ F ∨ (s _ hxi, {}) ∈ F
C5 D ⊆ domain(F )
C6 s ∈ D ∧ t ∈ A∗ ⇒ s _ t ∈ D
C7 s ∈ D ∧ X ⊆ A ⇒ (s, X ) ∈ F
(the last three conditions reflect the laws 3.8.2 L1, L2, L3).

110

3

Nondeterminism

The simplest process which satisfies this definition is also the worst
D1 CHAOSA = (A, (A∗ × P A), A∗ )
where
A∗ × P A is the Cartesian product { (s, X ) | s ∈ A∗ ∧ X ∈ P A }
This is the largest process with alphabet A, since every member of A∗ is both
a trace and a divergence, and every subset of A is a refusal after all traces.
Another simple process is defined
D2 STOPA = (A, {hi} × P A, {})
This process never does anything, can refuse everything, and has no divergences.
An operator is defined on processes by showing how the three sets of the
result can be derived from those of their operands. Of course it is necessary
to show that the result of the operation satisfies the six conditions of D0; this
proof is usually based on the assumption that its operands do so to start with.
The simplest operation to define is the nondeterministic or (u). Like many
other operations, it is defined only for operands with the same alphabet
D3 (A, F1, D1) u (A, F2, D2) = (A, F1 ∪ F2, D1 ∪ D2)
The resulting process can fail or diverge in all cases that either of its two operands can do so. The laws 3.2 L1, L2, L3 are direct consequences of this
definition.
The definitions of all the other operators can be given similarly; but it
seems slightly more elegant to write separate definitions for the alphabets,
failures and divergences. The definitions of the divergences have been given
in Section 3.8.2, so it remains only to define the alphabets and the failures.
D4

If
and
then

αP (x) = A for all x
B⊆A
α(x : B → P (x)) = A.

D5 α(P || Q ) = (αP ∪ αQ )
D6 α(f (P )) = f (α(P ))
D7 α(P  Q ) = α(P ||| Q ) = αP

provided αP = αQ

D8 α(P \ C) = αP − C
D9 failures(x : B → P (x)) =
{ hi, X | X ⊆ (αP − B) } ∪
{ hxi _ s, X | x ∈ B ∧ (s, X ) ∈ failures(P (x)) }

3.9

D10

Mathematical theory of non-deterministic processes

111

failures(P || Q ) =
{ s, (X ∪ Y ) | s ∈ (αP ∪ αQ )∗ ∧ (s u αP , X ) ∈ failures(P ) ∧
(s u αQ , Y ) ∈ failures(Q ) } ∪
{s, X | s ∈ divergences(P || Q ) }

D11

failures(f (P )) = { f ∗ (s), f (X ) | (s, X ) ∈ failures(P ) }

D12

failures(P  Q ) =
{ s, X | (s, X ) ∈ failures(P ) ∩ failures(Q ) ∨
(s ≠ hi ∧ (s, X ) ∈ failures(P ) ∪ failures(Q )) } ∪
{ s, X | s ∈ divergences(P  Q ) }

D13

failures(P ||| Q ) =
{ s, X | ∃ t , u • (t , X ) ∈ failures(P ) ∧ (u, X ) ∈ failures(Q ) } ∪
{ s, X | s ∈ divergences(P ||| Q ) }

D14

failures(P \ C) =
{ s u (αP − C), X | (s, X ∪ C) ∈ failures(P ) } ∪
{ s, X | s ∈ divergences(P \ C) }

The explanations of these laws may be derived from the explanations of the
corresponding traces and refusals, together with the laws for \.
It remains to give a definition for processes defined recursively by means
of µ. The treatment is based on the same fixed point theory as Section 2.8.2,
except that the definition of the ordering v is different
D15

(A, F1, D1) v (A, F2, D2) ≡ (F2 ⊆ F1 ∧ D2 ⊆ D1)

P v Q now means that Q is equal to P or better in the sense that it is less likely
to diverge and less likely to fail. Q is more predictable and more controllable
than P , because if Q can do something undesirable P can do it too; and if Q can
refuse to do something desirable, P can also refuse. CHAOS can do anything
at any time, and can refuse to do anything at any time. True to its name, it is
the least predictable and controllable of all processes; or in short the worst
L1 CHAOS v P
This ordering is clearly a partial order. In fact it is a complete partial order, with
a limit operation defined in terms of the intersections of descending chains of
failures and divergences
D16

F

n≥0 (A, Fn , Dn )

= (A,

T

n≥0

Fn ,

T

n≥0

Dn )

provided (∀ n ≥ 0 • Fn+1 ⊆ Fn ∧ Dn+1 ⊆ Dn )

112

3

Nondeterminism

The µ operation is defined in the same way as for deterministic processes (2.8.2
L7), except for the difference in the definition of the ordering, which requires
that CHAOS be used in place of STOP
D17

µ X : A • F (X ) =

F

i≥0

F i (CHAOSA )

The proof that this is a solution (in fact the most nondeterministic solution)
of the relevant equation is the same as that given in Section 2.8.2.
As before, the validity of the proof depends critically on the fact that all the
operators used on the right-hand side of the recursion should be continuous
in the appropriate ordering. Fortunately, all the operators defined in this book
(except /) are continuous, and so is every formula constructed from them. In
the case of the concealment operator, the requirement of continuity was one
of the main motivations for the rather complicated treatment of divergence.

Communication

4.1

4

Introduction

In previous chapters we have introduced and illustrated a general concept of
an event as an action without duration, whose occurrence may require simultaneous participation by more than one independently described process. In
this chapter we shall concentrate on a special class of event known as a communication. A communication is an event that is described by a pair c.v where
c is the name of the channel on which the communication takes place and v
is the value of the message which passes. Examples of this convention have
already been given in COPYBIT (1.1.3 X7) and CHAIN2 (2.6 X4).
The set of all messages which P can communicate on channel c is defined
αc(P ) = { v | c.v ∈ αP }
We also define functions which extract channel and message components of a
communication
channel(c.v) = c,

message(c.v) = v

All the operations introduced in this chapter can be defined in terms of
the more primitive concepts introduced in earlier chapters, and most of the
laws are just special cases of previously familiar laws. The reasons for introducing special notations is that they are suggestive of useful applications and
implementation methods; and because in some cases imposition of notational
restrictions permits the use of more powerful reasoning methods.

4.2

Input and output

Let v be any member of αc(P ). A process which first outputs v on the channel
c and then behaves like P is defined
(c!v → P ) = (c.v → P )
The only event in which this process is initially prepared to engage is the communication event c.v.

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4 Communication

A process which is initially prepared to input any value x communicable
on the channel c, and then behave like P (x), is defined
(c?x → P (x)) = (y : { y | channel(y) = c } → P (message(y)))
Example
X1 Using the new definitions of input and output we can rewrite 1.1.3 X7
COPYBIT = µ X • (in?x → (out !x → X ))
where αin(COPYBIT ) = αout (COPYBIT ) = {0, 1}



We shall observe the convention that channels are used for communication
in only one direction and between only two processes. A channel which is used
only for output by a process will be called an output channel of that process;
and one used only for input will be called an input channel. In both cases,
we shall say loosely that the channel name is a member of the alphabet of the
process.
When drawing a connection diagram (Section 2.4) of a process, the channels are drawn as arrows in the appropriate direction, and labelled with the
name of the channel (Figure 4.1).

left

right

P

down

Figure 4.1
Let P and Q be processes, and let c be an output channel of P and an input
channel of Q . When P and Q are composed concurrently in the system (P || Q ),
communication will occur on channel c on each occasion that P outputs a
message and Q simultaneously inputs that message. An outputting process
specifies a unique value for the message, whereas the inputting process is
prepared to accept any communicable value. Thus the event that will actually
occur is the communication c.v, where v is the value specified by the outputting
process. This requires the obvious constraint that the channel c must have the
same alphabet at both ends, i.e.,
αc(P ) = αc(Q )
In future, we will assume satisfaction of this constraint; and where no confusion can arise we will write αc for αc(P ). An example of the working of
this model for communication has been given in CHAIN2 (2.6 X4); and more
interesting examples will be given in Section 4.3 and subsequent sections.

4.2

Input and output

115

In general, the value to be output by a process is specified by means of an
expression containing variables to which a value has been assigned by some
previous input, as illustrated in the following examples.
Examples
X1 A process which immediately copies every message it has input from the
left by outputting it to the right
αleft (COPY ) = αright (COPY )
COPY = µ X • (left ?x → right !x → X )
If αleft = {0, 1}, COPY is almost identical to COPYBIT (1.1.3 X7).



X2 A process like COPY , except that every number input is doubled before it
is output
αleft = αright = N
DOUBLE = µ X • (left ?x → right !(x + x) → X )

X3 The value of a punched card is a sequence of eighty characters, which may
be read as a single value along the left channel. A process which reads cards
and outputs their characters one at a time
αleft = { s | s ∈ αright ∗ ∧ #s = 80 }
UNPACK = Phi
where

Phi = left ?s → Ps
Phxi = right !x → Phi
Phxi_s = right !x → Ps


X4 A process which inputs characters one at a time from the left, and assembles them into lines of 125 characters’ length. Each completed line is output on the right as a single array-valued message
αright = { s | s ∈ αleft ∗ ∧ #s = 125 }
PACK = Phi
where

Ps = right !s → Ph i

if #s = 125

Ps = left ?x → Ps _hxi

if #s < 125

Here, Ps describes the behaviour of the process when it has input and packed
the characters in the sequence s; they are waiting to be output when the line
is long enough.


116

4 Communication

X5 A process which copies from left to right, except that each pair of consecutive asterisks is replaced by a single “↑”
αleft = αright − {“↑”}
SQUASH = µ X • left ?x →
if x ≠ “*” then (right !x → X )
else left ?y → if y = “*” then (right !“↑” → X )
else (right !“*” → right !y → X ))



A process may be prepared initially to communicate on any one of a set of
channels, leaving the choice between them to the other processes with which
it is connected. For this purpose we adapt the choice notation introduced in
Chapter1. If c and d are distinct channel names
(c?x → P (x) | d?y → Q (y))
denotes a process which initially inputs x on c and then behaves like P (x),
or initially inputs y on channel d and then behaves like Q (y). The choice
is determined by whichever of the corresponding outputs is ready first, as
explained below.
Since we have decided to abstract from the timing of events and the speed
of processes which engage in them, the last sentence of the previous paragraph
may require explanation. Consider the case when the channels c and d are
output channels of two other separate processes, which are independent in
the sense that they do not directly or indirectly communicate with each other.
The actions of these two processes are therefore arbitrarily interleaved. Thus
if one processes is making progress towards an output on c, and the other is
making progress towards an output on d, it is not determined which of them
reaches its output first. An implementor will be expected, but not compelled,
to resolve this nondeterminism in favour of the first output to become ready.
This policy also protects against the deadlock that will result if the second
output is never going to occur, or if it can occur only after the first output,
as in the case when both the channels c and d are connected to the same
concurrent process, which outputs on one and then the other
(c!2 → d!4 → P )
Thus the presentation of a choice of inputs not only protects against deadlock but also achieves greater efficiency and reduces response times to proferred communications. A traveller who is waiting for a number 127 bus will
in general have to wait longer than one who is prepared to travel in either a
number 19 or a number 127, whichever arrives first at the bus stop. On the
assumption of random arrivals, the traveller who offers a choice will wait only
half as long—paradoxically, it is as though he is waiting twice as fast! To wait
for the first of many possible events is the only way of achieving this: purchase
of faster computers is useless.

4.2

Input and output

117

Examples
X6 A process which accepts input on either of the two channels left1 or left2,
and immediately outputs the message to the right
αleft1 = αleft2 = αright
MERGE = (left1?x → right !x → MERGE
| left2?x → right !x → MERGE )
The output of this process is an interleaving of the messages input from left1
and left2.

X7 A process that is always prepared to input a value on the left, or to output
to the right a value which it has most recently input
αleft = αright
VAR = left ?x → VARx
where
VARx = (left ?y → VARy
| right !x → VARx )
Here VARx behaves like a program variable with current value x. New values
are assigned to it by communication on the left channel, and its current value
is obtained by communication on the right channel. If αleft = {0, 1} the behaviour of VAR is almost identical to that of BOOL (2.6 X5).

X8 A process which inputs from up and left , outputs to down a function of
what it has input, before repeating
NODE (v) = µ X • (up?sum → left ?prod →
down!(sum + v × prod) → X )

X9 A process which is at all times ready to input a message on the left, and
to output on its right the first message which it has input but not yet output
BUFFER = Phi
where
Phi = left ?x → Phxi
Phxi_s = (left ?y → Phxi_s _hyi
| right !x → Ps )

118

4 Communication

BUFFER behaves like a queue; messages join the right-hand end of the queue
and leave it from the left end, in the same order as they joined, but after a
possible delay, during which later messages may join the queue.

X10 A process which behaves like a stack of messages. When empty, it responds to the signal empty. At all times it is ready to input a new message from
the left and put it on top of the stack; and whenever nonempty, it is prepared
to output and remove the top element of the stack
STACK = Phi
where
Phi = (empty → Phi | left ?x → Phxi )
Phxi_s = (right !x → Ps | left ?y → Phyi_hxi_s )
This process is very similar to the previous example, except that when empty it
participates in the empty event, and that it puts newly arrived messages on the
same end of the stored sequence as it takes them off. Thus if y is the newly
arrived input message, and x is the message currently ready for output, the
STACK stores hyi _ hxi _ s but the BUFFER stores hxi _ s _ hyi.

4.2.1

Implementation

In a LISP implementation of communicating processes, the event c.v is naturally
represented by the dotted pair c.v, which is constructed by
cons("c, v)
Input and output commands are conveniently implemented as functions which
first take a channel name as argument. If the process is not prepared to communicate on the channel, it delivers the answer "BLEEP . The actual value communicated is treated separately in the next stage, as described below.
If Q is the input command
c?x → Q (x))
then Q ("c) ≠ "BLEEP ; instead, its result is a function which expects the input
value x as its argument, and delivers as its result the process Q (x). Thus Q is
implemented by calling the LISP function
input ("c, λ x • Q (x))
which is defined
input (c, F ) = λ y • if y ≠ c then "BLEEP else F
It follows that Q / hc.vi is represented in LISP by Q ("c)(v), provided that hc.vi
is a trace of Q .

4.2

Input and output

119

If P is the output command
(c!v → P 0 )
then P ("c) ≠ "BLEEP ; instead, its result is the pair cons(v, P 0 ). Thus P is implemented by calling the LISP function
output ("c, v, P 0 )
which is defined
output (c, v, P ) = λ y •if y ≠ c then "BLEEP else cons(v, P )
It follows that v = car (P ("c)), and that P / hc.vi is represented in LISP by
cdr (P ("c)), provided that hc.vi is a trace of P .
In theory, if αc is finite, it would be possible to treat c.v as a single event,
passed as a parameter to the input and output commands. But this would be
grotesquely inefficient, since the only way of finding out what value is output
would be to test whether P (c.v) ≠ "BLEEP for all values v in αc, until the right
one is found. One of the justifications for introducing specialised notation
for input and output is to encourage and permit methods of implementation
which are significantly more efficient. The disadvantage is that the implementation of nearly all the other operators needs to be recoded in the light of this
optimisations.
Examples
X1 COPY = LABEL X • input ("left , λ x • output ("right , x, X ))



X2 PACK = P (NIL)
where
P = LABEL X •
(λ s • if length(s) = 125 then
output ("right , s, X (NIL))
else
input ("left ,
λ x • X (append(s, cons(x, NIL)))))

4.2.2

Specifications

In specifying the behaviour of a communicating process, it is convenient to
describe separately the sequences of messages that pass along each of the
channels. If c is a channel name, we define (see Section 1.9.6)
tr ↓ c = message∗ (tr u αc)

120

4 Communication

It is convenient just to omit the tr ↓, and write right ≤ left instead of tr ↓
right ≤ tr ↓ left .
Another useful definition places a lower bound on the length of a prefix
s ≤n t = (s ≤ t ∧ #t ≤ #s + N )
This means that s is a prefix of t , with not more than n items removed. The
following laws are obvious and useful
s ≤0 t ≡ (s = t )
s ≤n t ∧ t ≤m u ⇒ s ≤n+m u
s ≤ t ≡ ∃ n • s ≤n t
Examples
X1 COPY sat right ≤1 left



X2 DOUBLE sat right ≤1 double∗ (left )



X3 UNPACK sat right ≤ _/ left
where _/hs0 , s1 , . . . , sn−1 i = s0 _ s1 _ . . . _ sn−1

(see 1.9.2)

The specification here states that the output on the right is obtained by flattening the sequence of sequences input on the left.

X4 PACK sat ((_/ right ≤125 left ) ∧ (# ∗ right ∈ {125}∗ ))
This specification states that each element output on the right is itself a sequence of length 125, and the catenation of all these sequences is an initial
subsequence of what has been input on the left.

If ⊕ is a binary operator, it is convenient to apply it distributively to the corresponding elements of two sequences. The length of the resulting sequence
is equal to that of the shorter operand
s ⊕ t = hi
= hs0 ⊕ t0 i _ (s 0 ⊕ t 0 )

if s = hi or t = hi
otherwise

Clearly
(s ⊕ t )[i] = s[i] ⊕ t [i]

for i ≤ min(#s, #t ).

and
s ≤n t ⇒ (s ⊕ u ≤n t ⊕ u) ∧ (u ⊕ s ≤n u ⊕ t )

4.2

Input and output

121

Examples
X5 The Fibonacci sequence
h1, 1, 2, 3, 5, 8, . . .i
is defined by the recurrence relation
fib[0] = fib[1] = 1
fib[i + 2] = fib[i + 1] + fib[i]
The second line can be rewritten using the 0 operator to left-shift the sequence
by one place
fib 00 = fib 0 + fib
The original definition of the Fibonacci sequence may be recovered from this
more cryptic form by subscripting both sides of the equation
fib 00 [i] = (fib 0 + fib)[i]
⇒ fib 0 [i + 1] = fib 0 [i] + fib[i]

[1.9.4 L1]

⇒ fib[i + 2] = fib[i + 1] + fib[i]
Another explanation of the meaning of the equation is as a description of the
infinite sum, where the left shift is clearly displayed
1 , 1 , 2 ,
.

.

.

1 , 2 , 3 ,
.

.

3

,

5 , ...

fib

.
5

, ...

+ fib 0

.

2 , 3 , 5 , ...

= fib 00

In the above discussion, fib is regarded as an infinite sequence. If s is a finite
initial subsequence of fib (with #s ≥ 2) then instead of the equation we get the
inequality
s 00 ≤ s 0 + s
This formulation can be used to specify a process FIB which outputs the Fibonacci sequence to the right.
FIB sat (right ≤ h1, 1i ∨ (h1, 1i ≤ right ∧ right 00 ≤ right 0 + right ))

X6 A variable with value x outputs on the right the value most recently input
on the left, or x, if there is no such input. More formally, if the most recent

122

4 Communication

action was an output, then the value which was output is equal to the last item
in the sequence hxi _ left
VARx sat (channel(tr 0 ) = right ⇒ right 0 = (hxi _ left )0 )
where s 0 is the last element of s (Section 1.9.5).
This is an example of a process that cannot be adequately specified solely
in terms of the sequence of messages on its separate channels. It is also necessary to know the order in which the communications on separate channels
are interleaved, for example that the latest communication is on the right. In
general, this extra complexity will be necessary for processes which use the
choice operator.

X7 The MERGE process produces an interleaving (Section 1.9.3) of the two
sequences input on left1 and left2, buffering up to one message
MERGE sat ∃ r • right ≤1 r ∧ r interleaves (left1, left2)
X8 BUFFER sat right ≤ left




A process which satisfies the specification right ≤ left describes the behaviour of a transparent communications protocol, which is guaranteed to deliver
on the right only those messages which have been submitted on the left, and in
the same order. A protocol achieves this in spite of the fact that the place where
the messages are submitted is widely separated from the place where they are
received, and the fact that the communications medium which connects the
two places is somewhat unreliable. Examples will be given in Section 4.4.5.

4.3

Communications

Let P and Q be processes, and let c be a channel used for output by P and for
input by Q . Thus the set containing all communication events of the form c.v
is within the intersection of the alphabet of P with the alphabet of Q . When
these processes are composed concurrently in the system (P || Q ), a communication c.v can occur only when both processes engage simultaneously in
that event, i.e., whenever P outputs a value v on the channel c, and Q simultaneously inputs the same value. An inputting process is prepared to accept
any communicable value, so it is the outputting process that determines which
actual message value is transmitted on each occasion, as in 2.6 X4.
Thus output may be regarded as a specialised case of the prefix operator,
and input a special case of choice; and this leads to the law
L1 (c!v → P ) || (c?x → Q (x)) = c!v → (P || Q (v))
Note that c!v remains on the right-hand side of this equation as an observable
action in the behaviour of the system. This represents the physical possibility
of tapping the wires connecting the components of a system, and of thereby

4.3

Communications

123

keeping a log of their internal communications. It is also a help in reasoning
about the system.
If desired, such internal communications can be concealed by applying the
concealment operator described in Section 3.5 outside the parallel composition
of the two processes which communicate on the same channel, as shown by
the law
L2 ((c!v → P ) || (c?x → Q (x))) \ C = (P || Q (v)) \ C
where C = { c.v | v ∈ αc }
Examples will be given in Sections 4.4 and 4.5.
The specification of the parallel composition of communicating processes
takes a particularly simple form when channel names are used to denote the
sequences of messages passing on them. Let c be the name of a channel along
which P and Q communicate. In the specification of P , c stands for the sequence of messages communicated by P on c. Similarly, in the specification of
Q , c stands for the sequence of messages communicated by Q .
Fortunately, by the very nature of communication, when P and Q communicate on c, the sequences of messages sent and received must at all times be
identical. Consequently this sequence must satisfy both the specification of P
and the specification of Q . The same is true for all channels in the intersection
of their alphabets.
Consider now a channel d in the alphabet of P but not of Q . This channel
cannot be mentioned in the specification of Q , so the values communicated on
it are constrained only by the specification of P . Similarly, it is Q that determines the properties of the communications on its own channels. Consequently
a specification of the behaviour of (P || Q ) can be simply formed as the logical
conjunction of the specification of P with that of Q . However, this simplification is valid only when the specifications of P and Q are expressed wholly in
terms of the channel names, which is not always possible, as shown by 4.2.2
X6.
Example
X1 Let
P = (left ?x → mid!(x × x) → P )
Q = (mid?y → right !(173 × y) → Q )
Clearly
P sat (mid ≤1 square∗ (left ))
Q sat (right ≤1 173 × mid)
where (173 × mid) multiples each message of mid by 173. It follows that
(P || Q ) sat (right ≤1 173 × mid) ∧ (mid ≤1 square∗ (left ))

124

4 Communication

The specification here implies
right ≤ 173 × square∗ (left )
which was presumably the original intention.



When communicating processes are connected by the concurrency operator ||, the resulting formulae are highly suggestive of a physical implementation method in which electronic components are connected by wires along
which they communicate. The purpose of such an implementation is to increase the speed with which useful results can be produced.
The technique is particularly effective when the same calculation must be
performed on each member of a stream of input data, and the results must be
output at the same rate as the input, but possibly after an initial delay. Such
systems are called data flow networks.
A picture of a system of communicating processes closely represents their
physical realisation. An output channel of one processes is connected to a
like-named input channel of the other process, but channels in the alphabet of
only one process are left free. Thus the example X1 can be drawn, as shown in
Figure 4.2.

left

P

mid

Q

right

Figure 4.2

Examples
X2 Two streams of numbers are to be input from left1 and left2. For each
x read from left1 and each y from left2, the number (a × x + b × y) is to be
output on the right. The speed requirement dictates that the multiplications
must proceed concurrently. We therefore define two processes, and compose
them
X21 = (left1?x → mid!(a × x) → X21)
X22 = (left2?y → mid?z → right !(z + b × y) → X22)
X2 = (X21 || X22)
Clearly,
X2 sat (mid ≤1 a × left1 ∧ right ≤1 mid + b × left2)
⇒ (right ≤ a × left1 + b × left2)


4.3

Communications

125

X3 A stream of numbers is to be input on the left, and on the right is output
a weighted sum of consecutive pairs of input numbers, with weights a and b.
More precisely, we require that
right ≤ a × left + b × left 0
The solution can be constructed by adding a new process X23 to the solution
of X2
X3 = (X2 || X23)
where
X23 sat (left1 ≤1 left ∧ left2 ≤1 left 0 )
X23 can be defined
X23 = (left ?x → left1!x → (µ X • left ?x → left2!x → left1!x → X ))
It copies from left to both left1 and left2, but omits the first element in the
case of left2.
A picture of the network of X3 is shown in Figure 4.3.


left1

left

X23

X21

mid
right
X22
left2
Figure 4.3

When two concurrent processes communicate with each other by output
and input only on a single channel, they cannot deadlock (compare 2.7 L2). As
a result, any network of nonstopping processes which is free of cycles cannot
deadlock, since an acyclic graph can be decomposed into subgraphs connected
only by a single arrow.
However, the network of X3 contains an undirected cycle, and cyclic networks cannot be decomposed into subnetworks except with connections on
two or more channels; so in this case absence of deadlock cannot so easily be
assured. For example, if the two outputs left2!x → left1!x → in the loop of X3
were reversed, deadlock would occur rapidly.
In proving the absence of deadlock it is often possible to ignore the content
of the messages, and regard each communication on channel c as a single event

126

4 Communication

named c. Communications on unconnected channels can be ignored. Thus X3
can be written in terms of these events
(µ X • left1 → mid → X )
|| (µ Y • left2 → mid → Y )
|| (left1 → (µ Z • left2 → left1 → Z ))

= µ X3 • (left1 → left2 → mid → X3)
This proves that X3 cannot deadlock, using algebraic methods as in 2.3 X1.
These examples show how data flow networks can be set up to compute
one or more streams of results from one or more streams of input data. The
shape of the network corresponds closely to the structure of the operands and
operators appearing in the expressions to be computed. When these patterns
are large but regular, it is convenient to use subscripted names for channels,
and to introduce an iterated notation for concurrent combination

||i<n P (i) = (P (0) || P (1) || . . . || P (n − 1))
A regular network of this kind is known as an iterative array. If the connection
diagram has no directed cycles, the term systolic array is often used, since data
passes through the system much like blood through the chambers of the heart.
Examples
X4 The channels { leftj | j < n } are used to input the coordinates of successive
points in n-dimensional space. Each coordinate set is to be multiplied by a fixed
vector V of length n, and the resulting scalar product is to be output to the
right; or more formally
right ≤ Σn−1
j=0 Vj × leftj
It is specified that in each microsecond the n coordinates of one point are
to be input and one scalar product is to be output. The speed of each individual processor is such that it takes nearly one microsecond to do an input,
a multiplication, an addition and an output. It is therefore clear that at least n
processors will be required to operate concurrently. The solution to the problem should therefore be designed as an iterative array with at least n elements.
Let us replace the Σ in the specification by its usual inductive definition
mid0 = 0 ∗
midj+1 = Vj × leftj + midj

for j < n

right = midn
Thus we have split the specification into a conjunction of n + 1 component
equations, each containing at most one multiplication. All that is required is

4.3

Communications

127

to write a process for each equation: for j < n, we write
MULT0 = (µ X • mid0 !0 → X )
MULTj+1 = (µ X • leftj ?x → midj ?y → midj+1 !(Vj × x + y) → X )
MULTn+1 = (µ X • midn ?x → right !x → X )
NETWORK = ||j<n+2 MULTj
The connection diagram is shown in Figure 4.4.

MULT0
midn
left0

MULT1
mid1

leftn-1

MULTn
midn
MULTn+1

right

Figure 4.4

X5 This is similar to X4, except that m different scalar products of the same
coordinate sets are required almost simultaneously. Effectively, the channel
leftj (for j < n) is to be used to input the jth column of an infinite array; this
is to be multiplied by the (n × m) matrix M , and the ith column of the result
is to be output on righti , for i < m. In formulae
righti = Σj<n Mij × leftj
The coordinates of the result are required as rapidly as before, so at least m ×n
processes are required.

128

4 Communication

The solution might find practical application in a graphics display device
which automatically transforms or even rotates a two-dimensional representation of a three-dimensional object. The shape is defined by a series of points in
absolute space; the iterative array applies linear transformations to compute
the deflection on the x and y plates of the cathode ray tube; a third output
coordinate could perhaps control the intensity of the beam.

left

n

right

m

Figure 4.5
The solution is based on Figure 4.5. Each column of this array (except
the last) is modelled on the solution to X4; but it copies each value input on
its horizontal input channel to its neighbour on its horizontal output channel.
The processes on the right margin merely discard the values they input. It
would be possible to economise by absorbing the functions of these marginal
processors into their neighbours. The details of the solution are left as an
exercise.


4.3

Communications

129

X6 The input on channel c is to be interpreted as the successive digits of a
natural number C, starting from the least significant digit, and expressed with
number base b. We define the value of the input number as
C = Σi≥0 c[i] × b i
where c[i] < b for all i.
Given a fixed multiplier M , the output on channel d is to be the successive
digits of the product M × C. The digits are to be output after minimal delay.
Let us specify the problem more precisely. The desired output d is
d = Σi≥0 M × c[i] × b i
The jth element of d must be the jth digit, which can be computed by the
formula
d[j] = ((Σi≥0 M × c[i] × b i ) div b j ) mod b
= (M × c[j] + zj ) mod b
where zj = (Σi<j M × c[i] × b i ) div b j and div denotes integer division.
zj is the carry term, and can readily be proved to satisfy the inductive
definition
z0 = 0
zj+1 = ((M × c[j] + zj ) div b)
We therefore define a process MULT1(z), which keeps the carry z as a
parameter
MULT1(z) = c?x → d!(M × x + z) mod b → MULTI1((M × x + z) div b)
The initial value of z is zero, so the required solution is
MULT = MULT1(0)

X7 The problem is the same as X6, except M is a multi-digit number
M = Σi<n Mi × b i
A single processor can multiply only single-digit numbers. However, output
is to be produced at a rate which allows only one multiplication per digit.
Consequently, at least n processors are required. We will get each NODEi to
look after one digit Mi of the multiplier.

130

4 Communication

The basis of a solution is the traditional manual algorithm for multi-digit
multiplication, except that the partial sums are added immediately to the next
row of the table
. . . 153091

C

the incoming number

253

M

the multiplier

. . . 306182
. . . 765455
. . . 827275
. . . 459273
. . . 732023

M2 × C

computed by NODE2


M1 × C 
computed by NODE1

25 × C 


M0 × C 
computed by NODE0

M ×C 

cn-1

c1

c0
NODE0

NODEn-1
dn-1

d1

d0

Figure 4.6
The nodes are connected as shown in Figure 4.6. The original input comes
in on c0 and is propagated leftward on the c channels. The partial answers are
propagated rightward on the d channels, and the desired answer is output on
d0 . Fortunately each node can give one digit of its result before communicating with its left neighbour. Furthermore, the leftmost node can be defined to
behave like the answer to X6
NODEn−1 (z) = cn−1 ?x → dn−1 !(Mn−1 × x + z) mod b →
NODEn−1 ((Mn−1 × x + z) div b)
The remaining nodes are similar, except that each of them passes the input
digit to its left neighbour, and adds the result from its left neighbour to its
own carry. For k < n − 1
NODEk (z) = ck ?x → dk !(Mk × x + z) mod b →
ck+1 !x → dk+1 ?y →
NODEk (y + (Mk × x + z) div b)
The whole network is defined

||i<n NODEi (0)


4.4

Pipes

131

X7 is a simple example from a class of ingenious network algorithms, in
which there is an essential cycle in the directed graph of communication channels. But the statement of the problem has been much simplified by the assumption that the multiplier is known in advance and fixed for all time. In a
practical application, it is much more likely that such parameters would have
to be input along the same channel as the subsequent data, and would have
to be reinput whenever it is required to change them. The implementation of
this requires great care, but little ingenuity.
A simple implementation method is to introduce a special symbol, say
reload, to indicate that the next number or numbers are to be treated as a
change of parameter; and if the number of parameters is variable, an endreload
symbol may also be introduced.
Example
X8 Same as X4, except that the parameters Vj are to be reloaded by the number
immediately following a reload symbol. The definition of MULTj+1 needs to be
changed to include the multiplier as parameter
MULTj+1 (v) = leftj ?x →
if x = reload then (leftj ?y → MULTj+1 (y))
else (midj ?y → midj+1 !(v × x + y) → MULTj+1 (v))


4.4

Pipes

In this section we shall confine attention to processes with only two channels
in their alphabet, namely an input channel left and an output channel right .
Such processes are called pipes, and they may be pictured as in Figure 4.7.

left

left

P

Q

right

right

Figure 4.7
The processes P and Q may be joined together so that the right channel of
P is connected to the left channel of Q , and the sequence of messages output
by P and input by Q on this internal channel is concealed from their common

132

4 Communication

environment. The result of the connection is denoted
P >>Q
and may be pictured as the series shown in Figure 4.8.

left

P

Q

right

Figure 4.8
This connection diagram represents the concealment of the connecting
channel by not giving it a name. It also shows that all messages input on the
left channel of (P >>Q ) are input by P , and all messages output on the right
channel of (P >>Q ) are output by Q . Finally (P >>Q ) is itself a pipe, and may
again be placed in series with other pipes
(P >>Q )>>R,

(P >>Q )>>(R>>S ),

etc.

By 4.4.1 L1 >> is associative, so in future we shall omit brackets in such a series.
The validity of chaining processes by >> depends on the obvious alphabet
constraints
α(P >>Q ) = αleft (P ) ∪ αright (Q )
and a further constraint states that the connected channels are capable of
transmitting the same kind of message
αright (P ) = αleft (Q )
Examples
X1 A pipe which outputs each input value multiplied by four (4.2 X2)
QUADRUPLE = DOUBLE >>DOUBLE

X2 A process which inputs cards of eighty characters and outputs their text,
tightly packed into lines of 125 characters each (4.2 X3, X4)
UNPACK >>PACK
This process is quite difficult to write using conventional structured programming techniques, because it is not clear whether the major loop should iterate
once per input card, or once per output line. The problem is known by Michael
Jackson as structure clash. The solution given above contains a separate loop
in each of the two processes, which nicely matches the structure of the original
problem.


4.4

Pipes

133

X3 Same as X2, except that each pair of consecutive asterisks is replaced by
“↑” (4.2 X5)
UNPACK >>SQUASH >>PACK
In a conventional sequential program, this minor change in specification could
cause severe problems. It is nice to avoid such problems by the simple expedient of inserting an additional process. This kind of modularity has been
introduced and exploited by the designers of operating systems.

X4 Same as X2, except that the reading of cards may continue when the printer
is held up, and the printing can continue when the card reader is held up (4.2
X9)
UNPACK >>BUFFER>>PACK
The buffer holds characters which have been produced by the UNPACK process, but not yet consumed by the PACK process. They will be available for
input by the PACK process during times when the UNPACK process is temporarily delayed.
The buffer thus smoothes out temporary variations in the rate of production and consumption. However it can never solve the problem of long-term
mismatch between the rates of production and consumption. If the card reader
is on average slower than the printer, the buffer will be nearly always empty,
and no smoothing effect will be achieved. If the reader is faster, the buffer will
expand indefinitely, until it consumes all available storage space.

X5 In order to avoid undesirable expansion of buffers, it is usual to limit the
number of messages buffered. Even the single buffer provided by the COPY
process (4.2 X1) may be adequate. Here is a version of X4 which reads one card
ahead on input and buffers one line on output
COPY >>UNPACK >>PACK >>COPY
Note the alphabets of the two instances of COPY are different, a fact which
should be understood from the context in which they are placed.

X6 A double buffer, which accepts up to two messages before requiring output
of the first
COPY >>COPY
Its behaviour is similar to that of CHAIN2 (2.6 X4) and VMS2 (1.1.3 X6).
4.4.1

Laws

The most useful algebraic property of chaining is associativity
L1 P >>(Q >>R) = (P >>Q )>>R



134

4 Communication

The remaining laws show how input and output can be implemented in a
pipe; they enable process descriptions to be simplified by a form of symbolic
execution. For example, if the process on the left of >> starts with output of a
message v to the right, and the process on the right of >> starts with input from
the left, the message v is transmitted from the former process to the latter;
however the actual communication is concealed, as shown in the following law
L2 (right ! → P )>>(left ?y → Q (y)) = P >>Q (v)
If one of the processes is determined to communicate with the other, but the
other is prepared to communicate externally, it is the external communication that takes place first, and the internal communication is saved up for a
subsequent occasion
L3 (right !x → P )>>(right !w → Q ) =
right !w → ((right !v → P )>>Q )
L4 (left ?x → P (x))>>(left ?y → Q (y)) =
left ?x → (P (x)>>(left ?y → Q (y)))
If both processes are prepared for external communication, then either may
happen first
L5 (left ?x → P (x))>>(right !w → Q ) =
(left ?x → (P (x)>>(right !w → Q ))
| right !w → ((left ?x → P (x))>>Q ))
The law L5 is equally valid when the operator >> is replaced by >> R >>, since
pipes in the middle of a chain cannot communicate directly with the environment
L6 (left ?x → P (x))>>R>>(right !w → Q ) =
(left ?x → (P (x)>>R>>(right !w → Q ))
| right !w → ((left ?x → P (x))>>R>>Q ))
Similar generalisations may be made to the other laws
L7 If R is a chain of processes all starting with output to the right,
R>>(right !w → Q ) = right !w → (R>>Q )
L8 If R is a chain of processes all starting with input from the left,
(left ?x → P (x))>>R = left ?x → (P (x)>>R)

4.4

Pipes

135

Examples
X1 Let us define
R(y) = (right !y → COPY )>>COPY
So
R(y)
= (right !y → COPY )>>(left ?x → right !x → COPY )
= COPY >>(right !y → COPY )

[def COPY ]
[L2]


X2 COPY >>COPY
= (left ?x → right !x → COPY )>>COPY
= left ?x → ((right !x → COPY )>>COPY )
= left ?x → R(x)

[def COPY ]
[L4]
[def R(x)]


X3 From the last line of X1 we deduce
R(y)
= (left ?x → right !x → COPY )>>(right !y → COPY )
= (left ?x → ((right !x → COPY )>>(right !y → COPY ))

[L5]

| right !y → (COPY >>COPY ))
= (left ?x → right !y → R(x)

[L3, X2]

| right !y → left ?x → R(x))
This shows that a double buffer, after input of its first message, is prepared
either to output that message or to input a second message before doing so.
The reasoning of the above proofs is very similar to that of 2.3.1 X1.

4.4.2

Implementation

In the implementation of (P >>Q ) three cases are distinguished
1. If communication can take place on the internal connecting channel, it
does so immediately, without consideration of the external environment.
If an infinite sequence of such communications is possible, the process
diverges (Section 3.5.2).
2. Otherwise, if the environment is interested in communication on the left
channel, this is dealt with by P .

136

4 Communication

3. Or if the environment is interested in the right channel, this is dealt with
by Q .
For an explanation of the input and output operations, see Section 4.2.1.
chain(P , Q ) =
if P ("right ) ≠ "BLEEP and Q ("left ) ≠ "BLEEP then
chain(cdr (P ("right )), Q ("left )(car (P ("right ))))
else
λx •
if x = "right then
if Q ("right ) = "BLEEP then
"BLEEP
else
cons(car (Q ("right )), chain(P , cdr (Q ("right ))))

[Case 1]

[Case 2]
else
if x = "left then
if P (x) = "BLEEP then
"BLEEP
else
λ y • chain(P ("left )(y), Q )
else
"BLEEP

4.4.3

[Case 3]

Livelock

The chaining operator connects two processes by just one channel; and so it
introduces no risk of deadlock. If both P and Q are nonstopping, then (P >>Q )
will not stop either. Unfortunately, there is a new danger that the processes
P and Q will spend the whole time communicating with each other, so that
(P >>Q ) never again communicates with the external world. This case of divergences (Sections 3.5.1, 3.8) is illustrated by the trivial example
P = (right !1 → P )
Q = (left ?x → Q )
(P >>Q ) is obviously a useless process; it is even worse than STOP , in that
like an endless loop it may consume unbounded computing resources without
achieving anything. A less trivial example is (P >>Q ), where
P = (right !1 → P | left ?x → P1(x))
Q = (left ?x → Q | right !1 → Q1)

4.4

Pipes

137

In this example, divergence derives from the mere possibility of infinite internal
communication; it exists even though the choice of external communication on
the left and on the right is offered on every possible occasion, and even though
after such an external communication the subsequent behaviour of (P >>Q )
would not diverge.
A simple method to prove (P >>Q ) is free of livelock is to show that P is
left-guarded in the sense that it can never output an infinite series of messages
to the right without interspersing inputs from the left. To ensure this, we must
prove that the length of the sequence output to the right is at all times bounded
above by some well-defined function f of the sequence of values input from
the left; or more formally, we define
P is left-guarded ≡ ∃ f • P sat (#right ≤ f (left ))
Left-guardedness is often obvious from the text of P .
L1 If every recursion used in the definition of P is guarded by an input from
the left, then P is left-guarded.
L2 If P is left-guarded then (P >>Q ) is free of livelock.
Exactly the same reasoning applies to right-guardedness of the second operand
of >>
L3 If Q is right-guarded then (P >>Q ) is free of livelock.
Examples
X1 The following are left-guarded by L1 (4.1 X1, X2, X5, X9)
COPY , DOUBLE , SQUASH , BUFFER

X2 The following are left-guarded in accordance with the original definition,
because
UNPACK sat #right ≤ #(_/ left )
PACK sat #right ≤ #left

X3 BUFFER is not right-guarded, since it can input arbitrarily many messages
from the left without ever inputting from the right.

4.4.4

Specifications

A specification of a pipe can often be expressed as a relation S (left , right )
between the sequences of messages input on the left channel and the sequence
of messages output on the right. When two pipes are connected in series, the

138

4 Communication

sequence right produced by the left operand is equated with the sequence left
consumed by the right operand; and this common sequence is then concealed.
All that is known of the concealed sequence is that it exists. But we also need
to avert the risk of livelock. Thus we explain the rule
L1

If

P sat S (left , right )

and

Q sat T (left , right )

and

P is left-guarded or Q is right-guarded

then

P >>Q sat ∃ s • S (left , s) ∧ T (s, right )

This states that the relation between left and right which is maintained by
(P >>Q ) is the normal relational composition of the relation for P with the
relation for Q . Since the >> operator cannot introduce deadlock in pipes, we
can afford to omit reasoning about refusals.
Examples
X1 DOUBLE sat right ≤1 double∗ (left )
DOUBLE is left-guarded and right-guarded, so
(DOUBLE >>DOUBLE )
sat ∃ s • (s ≤1 double∗ (left ) ∧ right ≤1 double∗ (s))
≡ right ≤2 double∗ (double∗ (left ))
≡ right ≤2 quadruple∗ (left )

X2 Let us use recursion together with >> to give an alternative definition of a
buffer
BUFF = µ X • (left ?x → (X >>(right !x → COPY )))
We wish to prove that
BUFF sat (right ≤ left )
Assume that
X sat #left ≥ n ∨ right ≤ left
We know that
COPY sat right ≤ left
Therefore
(right !x → COPY )

4.4

Pipes

139

sat (right = left = hi ∨ (right ≤ hxi ∧ right 0 ≤ left ))
⇒ right ≤ hxi _ left
Since the right operand is right-guarded, by L1 and the assumption
(X >>(right !x → COPY ))
sat (∃ s • (#left ≥ n ∨ s ≤ left ) ∧ right ≤ hxi _ s)
⇒ (#left ≥ n ∨ right ≤ hxi _ left )
Therefore
left ?x → (. . .)
sat right = left = hi ∨
(left > hi ∧ (#left 0 ≥ n ∨ right ≤ hleft0 i _ left 0 ))
⇒ #left ≥ n + 1 ∨ right ≤ left
The desired conclusion follows by the proof rule for recursive processes (3.7.1
L8). The simpler law (1.10.2 L6) cannot be used, because the recursion is not
obviously guarded.

4.4.5

Buffers and protocols

A buffer is a process which outputs on the right exactly the same sequence
of messages as it has input from the left, though possibly after some delay;
furthermore, when non-empty, it is always ready to output on the right. More
formally, we define a buffer to be a process P which never stops, which is free
of livelock, and which meets the specification
P sat (right ≤ left ) ∧ (if right = left then left ∉ ref else right ∉ ref )
Here c ∉ ref means that the process cannot refuse to communicate on channel
c (Sections 3.7, 3.4). It follows that all buffers are left-guarded.
Example
X1 The following processes are buffers
COPY , (COPY >>COPY ), BUFF , BUFFER

Buffers are clearly useful for storing information which is waiting to be
processed. But they are even more useful as specifications of the desired behaviour of a communications protocol, which is intended to deliver messages
in the same order in which they have been submitted. Such a protocol consists
of two processes, a transmitter T and a receiver R, which are connected in
series (T >>R). If the protocol is correct, clearly (T >>R) must be a buffer.

140

4 Communication

In practice, the wire that connects the transmitter to the receiver is quite
long, and the messages which are sent along it are subject to corruption or
loss. Thus the behaviour of the wire itself can be modelled by a process WIRE ,
which may behave not quite like a buffer. It is the task of the protocol designer
to ensure that in spite of the bad behaviour of the wire, the system as a whole
acts as a buffer; i.e,
(T >>WIRE >>R)

is a buffer

A protocol is usually built in a number of layers, (T1 , R1 ) , (T2 , R2 ) , …
(Tn , Rn ) , each one using the previous layer as its communication medium
Tn >> . . . >>(T2 >>(T1 >>WIRE >>R1 )>>R2 )>> . . . >>Rn
Of course, when the protocol is implemented in practice, all the transmitters
are collected into a single transmitter at one end and all the receivers at the
other, in accordance with the changed bracketing
(Tn >> . . . >>T2 >>T1 )>>WIRE >>(R1 >>R2 >> . . . >>Rn )
The law of associativity of >> guarantees that this regrouping does not change
the behaviour of the system.
In practice, protocols must be more complicated than this, since singledirectional flow of messages is not adequate to achieve reliable communication
on an unreliable wire: it is necessary to add channels in the reverse direction,
to enable the receiver to send back acknowledgement signals for successfully
transmitted messages, so that unacknowledged messages can be retransmitted.
The following laws are useful in proving the correctness of protocols. They
are due to A. W. Roscoe.
L1 If P and Q are buffers, so are (P >>Q ) and (left ?x → (P >>(right !x → Q )))
L2 If T >>R = (left ?x → (T >>(right !x → R))) then (T >>R) is a buffer.
The following is a generalisation of L2
L3 If for some function f and for all z
(T (z)>>R(z)) = (left ?x → (T (f (x, z))>>(right !x → R(f (x, z)))))
then T (z)>>R(z) is a buffer for all z.

Examples
X2 The following are buffers by L1
COPY >>COPY , BUFFER>>COPY , COPY >>BUFFER, BUFFER>>BUFFER


4.4

Pipes

141

X3 It has been shown in 4.4.1 X1 and X2 that
(COPY >>COPY ) = (left ?x → (COPY >>(right !y → COPY )))
By L2 it is therefore a buffer.



X4 (Phase encoding) A phase encoder is a process T which inputs a stream
of bits, and outputs h0, 1i for each 0 input and h1, 0i for each 1 input. A
decoder R reverses this translation
T = left ?x → right !x → right !(1 − x) → T
R = left ?x → left ?y →if y = x then FAIL else (right !x → R)
where the process FAIL is left undefined.
We wish to prove by L2 that (T >>R) is a buffer
(T >>R)
= left ?x → ((right !x → right !(1 − x) → T )>>
(left ?x → left ?y →
if y = x then FAIL else (right !x → R)))
= left ?x → (T >> if (1 − x) = x then FAIL else (right !x → R))
= left ?x → (T >>(right !x → R))
Therefore (T >>R) is a buffer, by L2.



X5 (Bit stuffing) The transmitter T faithfully reproduces the input bits from
left to right, except that after three consecutive 1-bits which have been output,
it inserts a single extra 0. Thus the input 01011110 is output as 010111010.
The receiver R removes these extra zeroes. Thus (T >>R) must be proved to be
a buffer. The construction of T and R, and the proof of their correctness, are
left as an exercise.

X6 (Line sharing) It is desired to copy data from a channel left1 to right1 and
from left2 to right2. This can most easily be achieved by two disjoint protocols,
each using a different wire. Unfortunately, only a single wire mid is available,
and this must be used for both streams of data, as shown by Figure 4.9.

left1
left2

right1
T

mid

R

right2

Figure 4.9
Messages input by T must be tagged before transmission along mid, and

142

4 Communication

R must untag them and output them on the corresponding right channel
T = (left1?x → mid!tag1(x) → T
| left2?y → mid!tag2(y) → T )
R = mid?z →
if tag(z) = 1 then
(right1!untag(z) → R)
else
(right2!untag(z) → R)
This solution is quite unsatisfactory. If two messages are input on left1, but
the recipient is not yet ready for them, the whole system will have to wait, and
transmission between left2 and right2 may be seriously delayed. To insert
buffers on the channels will only postpone the problem for a short while. The
correct solution is to introduce another channel in the reverse direction, and
for R to send signals back to T to stop sending messages on the stream for
which there seems to be little demand. This is known as flow control.


4.5

Subordination

Let P and Q be processes with
αP ⊆ αQ
In the combination (P || Q ), each action of P can occur only when Q permits
it to occur; whereas Q can engage independently in the actions of (αQ − αP ),
without the permission and without the knowledge of its partner P . Thus
P serves Q as a slave or subordinate process, while Q acts as a master of
main process. When communications between a subordinate process and a
main process are to be concealed from their common environment, we use the
asymmetric notation
P // Q
Using the concealment operator, this can be defined
P // Q = (P || Q ) \ αP
This notation is used only when αP ⊆ αQ ; and then
α(P // Q ) = (αQ − αP )
It is usually convenient to give the subordinate process a name, say m,
which is used in the main process for all interactions with its subordinate. The
process naming technique described in Section 2.6.2 can be readily extended

4.5

Subordination

143

to communicating processes, by introducing compound channel names. These
take the form m.c, where m is a process name and c is the name of one of its
channels. Each communication on this channel is a triple
m.c.v
where αm.c(m : P ) = αc(P ) and v ∈ αc(P ).
In the construction (m : P // Q ), Q communicates with P along channels
with compound names of the form m.c and m.d; whereas P uses the corresponding simple channels c and d for the same communications. Thus for
example
(m : (c!v → P ) // (m.c?x → Q (x))) = (m : P // Q (v))
Since all these communications are concealed from the environment, the name
m can never be detected from the outside; it therefore serves as a local name
for the subordinate process.
Subordination can be nested, for example
(n : (m : P // Q ) // R)
In this case, all occurrences of events involving the name m are concealed
before the name n is attached to the remaining events, all of which are in the
alphabet of Q , and not of P . There is no way that R can communicate directly
with P , or even know of the existence of P or its name m.
Examples
X1 (for DOUBLE see 4.2 X2)
doub : DOUBLE // Q
The subordinate process acts as a simple subroutine called from within the
main process Q . Inside Q , the value of 2 × e may be obtained by a successive
output of the argument e on the left channel of doub, and input of the result
on the right channel
doub.left !e → (doub.right ?x → . . . )

X2 One subroutine may use another as a subroutine, and do so several times
QUADRUPLE =
(doub : DOUBLE // (µ X • left ?x → doub.left !x →
doub.right ?y → doub.left !y →
doub.right ?z → right !z → X ))

144

4 Communication

This is designed itself to be used as a subroutine
quad : QUADRUPLE // Q
This version of QUADRUPLE is similar to that of 4.4 X1, but does not have the
same double-buffering effect.

X3 A conventional program variable named m may be modelled as a subordinate process
m : VAR // Q
Inside the main process Q , the value of m can be assigned, read, and updated
by input and output, as described in 2.6.2 X2
m := 3; P

is implemented by

(m.left !3 → P )

x := m; P

is implemented by

(m.right ?x → P )

m := m + 3; P

is implemented by

(m.right ?y → m.left !(y + 3) → P )


A subordinate process may be used to implement a data structure with a
more elaborate behaviour than just a simple variable.
X4 (see 4.2 X9)
(q : BUFFER // Q )
The subordinate process serves as an unbounded queue named q. Within Q ,
the output q.left !v adds v to one end of the queue, and q.right ?y removes an
element from the other end, and gives its value to y. If the queue is empty, the
queue will not respond, and the system may deadlock.

X5 (see 4.2 X10)

A stack with name st is declared

st : STACK // Q
Inside the main process Q , st .left !v can be used to push the value v onto the
stack, and st .right ?x will pop the top value. To deal with the possibility that
the stack is empty, a choice construction can be used
(st .right ?x → Q1(x) | st .empty → Q2)
If the stack is non-empty, the first alternative is selected; if empty, deadlock is
avoided and the second alternative is selected.

A subordinate process with several channels may be used by several concurrent processes, provided that they do not use the same channel.

4.5

Subordination

145

X6 A process Q is intended to communicate a stream of values to R; these
values are to be buffered by a subordinate buffer process named b, so that
output from Q will not be delayed when R is not ready for input. Q uses
channel b.left for its output and R uses b.right for its input
(b : BUFFER // (Q || R))
Note that if R attempts to input from an empty buffer, the system will not
necessarily deadlock; R will simply be delayed until Q next outputs a value to
the buffer. (If Q and R communicate with the buffer on the same channel, then
that channel must be in the alphabet of both of them; and the definition of ||
would require them always to communicate simultaneously the same value—
which would be quite wrong.)

The subordination operator may be used to define subroutines by recursion. Each level of recursion (except the last) declares a new local subroutine
to deal with the recursive call(s).
X7 (Factorial)
FAC = µ X • left ?n →
(if n = 0 then
(right !1 → X )
else
(f : X // (f .left !(n − 1) →
f .right ?y → right !(n × y) → X )))
The subroutine FAC uses channels left and right to communicate parameters and results to its calling process; and it uses channels f .left and f .right to
communicate with its subordinate process named f . In these respects it is
similar to the QUADRUPLE subroutine (X2). The only difference is that the
subordinate process is isomorphic to FAC itself.

This is a boringly familiar example of recursion, expressed in an unfamiliar but rather cumbersome notational framework. A less familiar idea is that
of using recursion together with subordination to implement an unbounded
data structure. Each level of the recursion stores a single component of the
structure, and declares a new local subordinate data structure to deal with the
rest.
X8 (Unbounded finite set) A process which implements a set inputs its members on its left channel. After each input, it outputs a YES if it has already input
the same value, and NO otherwise. It is very similar to the set of 2.6.2 X4, except that it will store messages of any kind
SET = left ?x → right !NO → (rest : SET // LOOP (x))

146

4 Communication

where
LOOP (x) = µ X • left ?y →
(if y = x then
right !YES → X
else
(rest .left !y →
rest .right ?z →
right !z → X ))
The set starts empty; therefore on input of its first member x is immediately outputs NO. It then declares a subordinate process called rest , which is
going to store all members of the set except x. The LOOP is designed to input
subsequent members of the set. If the newly input member is equal to x, the
answer YES is sent back immediately on the right channel. Otherwise, the new
member is passed on for storage by rest . Whatever answer (YES or NO) is sent
back by rest is passed on again, and the LOOP repeats.

X9 (Binary tree) A more efficient representation of a set is as a binary tree,
which relies on some given total ordering ≤ over its elements. Each node stores
its earliest inserted element, and declares two subordinate trees, one to store
elements smaller than the earliest, and one to store the bigger elements. The
external specification of the tree is the same as X8
TREE = left ?x →
right !NO →
(smaller : TREE //
(bigger : TREE //
LOOP ))
The design of the LOOP is left as an exercise.
4.5.1



Laws

The following obvious laws govern communications between a process and
its subordinates. The first law describes concealed communication in each
direction between the main and subordinate processes
L1A

(m : (c?x → P (x))) // (m.c!v → Q ) = (m : P (v)) // Q

L1B

(m : (d!v → P )) // (m.d?x → Q (x)) = (m : P ) // Q (v)

4.5

Subordination

147

If b is a channel not named by m, the main process can communicate on
b without affecting the subordinate
L2 (m : P // (b!e → Q )) = (b!e → (m : P // Q ))
The only process capable of making a choice for a subordinate process is its
main process
L3 (m : (c?x → P1(x) | d?y → P2(y))) // (m.c!v → Q ) = (m : P1(v) // Q )
If two subordinate processes have the same name, one of them is inaccessible
L4 m : P // (m : Q // R) = (m : Q // R)
Usually, the order in which subordinate processes are written does not matter
L5 m : P // (n : Q // R) = n : Q // (m : P // R)
provided that m and n are distinct names
The use of recursion in defining subordinate processes is sufficiently surprising to raise doubts as to whether it actually works. These doubts may be
slightly alleviated by showing how the combination evolves. The example below uses a particular trace of behaviour of the process, and shows how that
trace is produced. More important, it shows how other slightly differing traces
cannot be produced.
Example
X1 A typical trace of SET is
s = hleft .1, right .NO, left .2, right .NOi
The value of SET / s can be calculated using laws L1A, L1B, and L2:
SET / hleft .1i
= right !NO → (rest : SET // LOOP (1))
SET / hleft .1, right .NOi
= (rest : SET // LOOP (1))
SET / hleft .1, right .NO, left .2i
= (rest : SET //
(rest .left !2 → rest .right ?z → right !z → LOOP (1)))
= (rest : (right !NO → (rest : SET // LOOP (2)))) //
(rest .right ?z → right !z → LOOP (1)))
= rest : (rest : SET // LOOP (2)) // (right !NO → LOOP (1))
SET / s =

148

4 Communication

rest : (rest : SET // LOOP (2)) // LOOP (1)
It is obvious from this that hleft .1, right .NO, left .2, right .YES i is not a trace of
SET . The reader may check that
SET / s _ hleft .2, right .YES i = SET / s
and
SET / s _ hleft .5, right .NOi =
rest : (rest : (rest : SET // LOOP (5)) // LOOP (2)) // LOOP (1)

4.5.2

Connection diagrams

A subordinate process may be drawn inside the box representing the process
that uses it, as shown for 4.5 X1 in Figure 4.10. For nested subordinate processes, the boxes nest more deeply, as shown for 4.5 X2 in Figure 4.11.
A recursive process is one that is nested inside itself, like the picture of the
artist’s studio, in which there stands on an easel the completed painting itself,
which shows on the easel a completed painting…Such a picture in practice can
never be completed. Fortunately, for a process it is not necessary to complete
the picture—it evolves automatically as need during its activity. Thus (see 4.5.1
X1) we may picture successive stages in the early history of a set as shown in
Figure 4.12.
If we ignore the nesting of the boxes, this can be drawn as a linear structure
as in Figure 4.13. Similarly, the example TREE (4.5 X9) could be drawn as in
Figure 4.14.
The connection diagrams suggest how a corresponding network might be
constructed from hardware components, with boxes representing integrated
circuits and arrows representing wires between them. Of course in any practical realisation, the recursion must be unfolded to some finite limit before the
network can start its normal operation; and if this limit is exceeded during
operation, the network cannot successfully continue.
Dynamic reallocation and reconfiguration of hardware networks is a lot
more difficult than the stack-based storage allocation which makes recursion
in conventional sequential programs so efficient. Nevertheless, recursion is
surely justified by the aid that it gives in the invention and design of algorithms;
and if not by that, then at least by the intellectual joy which it gives to those
who understand it and use it.

4.5

Subordination

149

Q
doub.
left

doub.
right
doub:
DOUBLE

Figure 4.10

Q
quad.
left

quad.
right

quad:

LOOP

doub.
left

doub.
right
doub:
DOUBLE

Figure 4.11

150

4 Communication

SET / hleft .1, right .NOi =

right

left
LOOP(1)

rest.
left

rest.
right

rest: SET

SET / s =

right

left
LOOP(1)

rest.
left

rest.
right

LOOP(2)

...

Figure 4.12

4.5

Subordination

151

right

left
LOOP(x)

LOOP(y)

LOOP(z)

Figure 4.13

right

left

Figure 4.14

Sequential Processes

5.1

5

Introduction

The process STOP is defined as one that never engages in any action. It is
not a useful process, and probably results from a deadlock or other design
error, rather than a deliberate choice of the designer. However, there is one
good reason why a process should do nothing more, namely that it has already
accomplished everything that it was designed to do. Such a process is said
to terminate successfully. In order to distinguish between this and STOP , it
is convenient to regard successful termination as a special event, denoted by
the symbol ✓ (pronounced “success”). A sequential process is defined as one
which has ✓ in its alphabet; and naturally this can only be the last event in
which it engages. We stipulate that ✓ cannot be an alternative in the choice
construct
(x : B → P (x))

is invalid if ✓ ∈ B

SKIPA is defined as a process which does nothing but terminate successfully
αSKIPA = A ∪ {✓}
As usual, we shall frequently omit the subscript alphabet.
Examples
X1 A vending machine that is intended to serve only one customer with chocolate or toffee and then terminate successfully
VMONE = (coin → (choc → SKIP | toffee → SKIP ))

In designing a process to solve a complex task, it is frequently useful to
split the task into two subtasks, one of which must be completed successfully
before the other begins. If P and Q are sequential processes with the same
alphabet, their sequential composition
P ;Q

154

5

Sequential Processes

is a process which first behaves like P ; but when P terminates successfully,
(P ; Q ) continues by behaving as Q . If P never terminates successfully, neither
does (P ; Q ).
X2 A vending machine designed to serve exactly two customers, one after the
other
VMTWO = VMONE ; VMONE

A process which repeats similar actions as often as required is known as
a loop; it can be defined as a special case of recursion
∗

P = µ X • (P ; X )
= P ; P ; P ; ...

∗

α( P ) = αP − {✓}
Clearly such a loop will never terminate successfully; that is why it is convenient to remove ✓ from its alphabet.
X3 A vending machine designed to serve any number of customers
VMCT = ∗ VMONE
This is identical to VMCT (1.1.3 X3).



A sequence of symbols is said to be a sentence of a process P if P terminates successfully after engaging in the corresponding sequence of actions. The
set of all such sentences is called the language accepted by P . Thus the notations introduced for describing sequential processes may also be used to define
the grammar of a simple language, such as might be used for communication
between a human being and a computer.
X4 A sentence of Pidgingol consists of a noun clause followed by a predicate. A predicate is a verb followed by a noun clause. A verb is either bites or
scratches. The definition of a noun clause is given more formally below
αPIDGINGOL = {a, the, cat , dog, bites, scratches}
PIDGINGOL = NOUNCLAUSE ; PREDICATE
PREDICATE = VERB ; NOUNCLAUSE
VERB = (bites → SKIP | scratches → SKIP )
NOUNCLAUSE = ARTICLE ; NOUN
ARTICLE = (a → SKIP | the → SKIP )
NOUN = (cat → SKIP | dog → SKIP )
Example sentences of Pidgingol are
the cat scratches a dog
a dog bites the cat

5.1

Introduction

155


To describe languages with an unbounded number of sentences, it is necessary to use some kind of iteration or recursion.
X5 A noun clause which may contain any number of adjectives furry or prize
NOUNCLAUSE = ARTICLE ; µ X • (furry → X | prize → X
| cat → SKIP | dog → SKIP )
Examples of a noun clause are
the furry furry prize dog
a dog

X6 A process which accepts any number of as followed by a b and then the
same number of cs, after which it terminates successfully
An BC n = µ X • (b → SKIP
| a → (X ; (c → SKIP )))
If a b is accepted first, the process terminates; no as and no cs are accepted,
so their numbers are the same. If the second branch is taken, the accepted
sentence starts with a and ends with c, and between these is the sentence
accepted by the recursive call on the process X . If we assume that the recursive
call accepts an equal number of as and cs, then so will the non-recursive call
on An BC n , since it accepts just one more a at the beginning and one more c at
the end.
This example shows how sequential composition, used in conjunction with
recursion, can define a machine with an infinite number of states.

X7 A process which first behaves like An BC n , but the accepts a d followed by
the same number of es
An BC n DE n = ((An BC n ) ; d → SKIP ) || C n DE n
where C n DE n = f (An BC n ) for f which maps a to c, b to d, and c to e.
In this example, the process on the left of the || is responsible for ensuring
an equal number of as and cs (separated by a b). It will not allow a d until the
proper number of cs have arrived; but the es (which are not in its alphabet)
are ignored. The process on the right of || is responsible for ensuring an equal
number of es as cs. It ignores the as and the b, which are not in its alphabet.
The pair of processes terminate together when they have both completed their
allotted tasks.


156

5

Sequential Processes

The notations for defining a language by means of an accepting process
are as powerful as those of regular expressions. The use of recursion introduces some of the power of context-free grammars, but not all. A process can
only define those languages that can be parsed from left to right without backtracking or look-ahead. This is because the use of the choice operator requires
that the first event of each alternative is different from all its other first events.
Consequently, it is not possible to use the construction of X5 to define a noun
clause in which the word prize can be either a noun of an adjective or both,
e.g., the prize dog, the furry prize.
The use of  (Section 3.3) would not help, because it introduces nondeterminism, and allows an arbitrary choice of the clause which will analyse
the rest of the input. If the choice is wrong, the process will deadlock before
reaching the end of the input text. What is required to solve this problem is a
new kind of choice operator which provides angelic nondeterminism like or3
(Section 3.2.2). This new operator requires that the two alternatives run concurrently until the environment makes the choice; its definition is left as an
exercise.
Without angelic nondeterminism the language-defining method described
above is not as powerful as context-free grammars, because it requires left-toright parsability without back-tracking. However, the introduction of || permits
definition of languages which are not context-free, for example X7.
X8 A process which accepts any interleaving of downs and ups, except that it
terminates successfully on the first occasion that the number of downs exceeds
the number of ups
POS = (down → SKIP | up → (POS ; POS ))
If the first symbol is down, the task of POS is accomplished. But if the first
symbol is up, it is necessary to accept two more downs than ups. The only way
of achieving this is first to accept one more down than up; and then again to
accept one more down than up. Thus two successive recursive calls on POS
are needed, one after the other.

X9 The process C0 behaves like CT0 (1.1.4 X2)
C0 = (around → C0 | up → C1 )
Cn+1 = POS ; Cn
= POS ; . . . POS ; POS ; POS ; C0
{z
}
|
n times

for all n ≥ 0


We can now solve the problem mentioned in 2.6.2 X3, and encountered
again in 4.5 X4, that each operation on a subordinate process explicitly mentions the rest of the user process which follows it. The required effect can now
be more conveniently achieved by means of SKIP and sequential composition.

5.2

Laws

157

X10
A USER process manipulates two count variables named l and m (see
2.6.2 X3)
l : CT0 || m : CT3 || USER
The following subprocess (inside the USER) adds the current value of l to m
ADD = (l.around → SKIP
| l.down → (ADD ; (m.up → l.up → SKIP )))
If the value of l is initially zero, nothing needs to be done. Otherwise, l is
decremented, and its reduced value is added to m (by the recursive call to ADD).
Then m is incremented once more, and l is also incremented, to compensate
for the initial decrementation and bring it back to its initial value.


5.2

Laws

The laws for sequential composition are similar to those for catenation (Section
1.6.1), with SKIP playing the role of the unit
L1 SKIP ; P = P ; SKIP = P
L2 (P ; Q ) ; R = P ; (Q ; R)
L3 (x : B → P (x)) ; Q = (x : B → (P (x) ; Q ))
The law for the choice operator has corollaries
L4 (a → P ) ; Q = a → (P ; Q )
L5 STOP ; Q = STOP
When sequential operators are composed in parallel, the combination terminates successfully just when both components do so
L6 SKIPA || SKIPB = SKIPA∪B
A successfully terminating process participates in no further event offered by
a concurrent partner
L7 ((x : B → P (x)) || SKIPA ) = (x : (B − A) → (P (x) || SKIPA ))
In a concurrent combination of a sequential with a nonsequential processes, when does the combination terminate successfully? If the alphabet of
the sequential process wholly contains that of its partner, termination of the
partnership is determined by that of the sequential process, since the other
process can do nothing when its partner is finished.
L8 STOPA || SKIPB = SKIPB

if ✓ ∉ A ∧ A ⊆ B.

158

5

Sequential Processes

The condition for the validity of this law is very reasonable one, which should
always be observed when ✓ is in the alphabet of only one of a pair of processes
running concurrently. In this way, we avoid the problem of a process which
continues after engaging in ✓.
The laws L1 to L3 may be used to prove the claim made in 5.1 X9 that C0
behaves like CT0 (1.1.4 X2). This is done by showing that C satisfies the set of
guarded recursive equations used to define CT . The equation for CT0 is the
same as that for C0
C0 = (around → C0 | up → C1 )

[def C0 ]

For n > 0, we need to prove
Cn = (up → Cn+1 | down → Cn−1
Proof
LHS
= POS ; Cn−1
= (down → SKIP | up → POS ; POS ) ; Cn−1
= (down → (SKIP ; Cn−1 ) | up → (POS ; POS ) ; Cn−1 )

[def Cn ]
[def POS ]
[L3]

= (down → Cn−1 | up → POS ; (POS ; Cn−1 ))

[L1, L2]

= (down → Cn−1 | up → POS ; Cn )

[def Cn ]

= RHS

[def Cn ]

Since Cn obeys the same set of guarded recursive equations as CTn , they are
the same.
This proof has been written out in full, in order to illustrate the use of
the laws, and also in order to allay suspicion of circularity. What seems most
suspicious is that the proof does not use induction on n. In fact, any attempt
to use induction on n will fail, because the very definition of CTn contains the
process CTn+1 . Fortunately, an appeal to the law of unique solutions is both
simple and successful.

5.3

Mathematical treatment

The mathematical definition of sequential composition must be formulated in
such a way as to ensure the truth of the laws quoted in the previous section.
Special care needs to be exercised on
P ; SKIP = P
As usual, the treatment of deterministic processes is much simpler, and will
be completed first.

5.3

Mathematical treatment

5.3.1

159

Deterministic processes

Operations on deterministic processes are defined in terms of the traces of
their results. The first and only action of the process SKIP is successful termination, so it has only two traces
L0 traces(SKIP ) = {hi, h✓i}
To define sequential composition of processes, it is convenient first to define
sequential composition of their individual traces. If s and t are traces and s
does not contain ✓
(s ; t ) = s
_
(s h✓i) ; t = s _ t
(see Section 1.9.7 for a fuller treatment). A trace of (P ; Q ) consists of a trace
of P ; and if this trace ends in ✓, the ✓ is replaced by a trace of Q
L1 traces(P ; Q ) = { s ; t | s ∈ traces(P ) ∧ t ∈ traces(Q ) }
An equivalent definition is
L1A

traces(P ; Q ) = { s | s ∈ traces(P ) ∧ ¬ h✓i in s } ∪
{ s _ t | s _ h✓i ∈ traces(P ) ∧ t ∈ traces(Q ) }

This definition may be simpler to understand but it is more complicated to
use.
The whole intention of the ✓ symbol is that it should terminate the process
which engages in it. We therefore need the law
L2 P / s = SKIP

if s _ h✓i ∈ traces(P )

This law is essential in the proof of
P ; SKIP = P
Unfortunately, it is not in general true. For example, if
P = (SKIP{} || c → STOP{c} )
then traces(P ) = {hi, h✓i, hci, hc, ✓i, h✓, ci} and P / hi ≠ SKIP , even though
h✓i ∈ traces(P ). We therefore need to impose alphabet constraints on parallel
composition. (P || Q ) must be regarded as invalid unless
αP ⊆ αQ ∨ αQ ⊆ αP ∨ ✓ ∈ (αP ∩ αQ ∪ αP ∩ αQ )
For similar reasons, alphabet change must be guaranteed to leave ✓ unchanged,
so f (P ) is invalid unless
f (✓) = ✓

160

5

Sequential Processes

Furthermore, if m is a process name, we must adopt the convention that
m.✓ = ✓
Finally, we must never use ✓ in the choice construct
(✓ → P | c → Q )
This restriction also rules out RUNA when ✓ ∈ A.
5.3.2

Non-deterministic processes

Sequential composition of nondeterministic processes presents a number of
problems. The first of them is that a nondeterministic process like SKIP u
(c → SKIP ) does not satisfy the law L2 of the previous section. A solution of
this is to weaken 5.3.1 L2 to
L2A

s _ h✓i ∈ traces(P ) ⇒ (P / s) v SKIP

This means that whenever P can terminate, it can do so without offering any
alternative event to the environment. To maintain the truth of L2A, all restrictions of the previous section must be observed, and also
SKIP must never appear unguarded in an operand of 
✓ must not appear in the alphabet of either operand of |||
(It is possible that a slight change to the definitions of  and ||| might permit
relaxation of these restrictions.)
In addition to the laws given earlier in this chapter, sequential composition
of nondeterministic processes satisfies the following laws. Firstly, a divergent
process remains divergent, no matter what is specified to happen after its
successful termination
L1 CHAOS ; P = CHAOS
Sequential composition distributes through nondeterministic choice
L2A

P u Q ) ; R = (P ; R) u (Q ; R)

L2B

R ; (P u Q ) = (R ; P ) u (R ; Q )

To define (P ; Q ) in the mathematical model of nondeterministic processes (Section 3.9) requires treatment of its failures and divergences. But first
we describe its refusals (Section 3.4). If P can refuse X , and cannot terminate
successfully, it follows that X ∪ {✓} is also a refusal of P (3.4.1 L11). In this
case X is a refusal of (P ; Q ). But if P offers the option of successful termination, then in (P ; Q ) this transition may occur autonomously; its occurrence
is concealed, and any refusal of Q is also a refusal of (P ; Q ). The case where
successful termination of P is nondeterministic is also treated in the definition

5.4

Interrupts

161

D1 refusals(P ; Q ) = { X | (X ∪ {✓}) ∈ refusals(P ) } ∪
{ X | h✓i ∈ traces(P ) ∧ X ∈ refusals(Q ) }
The traces of (P ; Q ) are defined in exactly the same way as for deterministic processes. The divergences of (P ; Q ) are defined by the remark that
it diverges whenever P diverges; or when P has terminated successfully and
then Q diverges
D2 divergences(P ; Q ) = { s | s ∈ divergences(P ) ∧ ¬ h✓i in s } ∪
{ s _ t | s _ h✓i ∈ traces(P ) ∧ ¬ h✓i in s ∧
t ∈ divergences(Q ) }
Any failure of (P ; Q ) is either a failure of P before P can terminate, or it is a
failure of Q after P has terminated successfully
D3 failures(P ; Q ) = { (s, X ) | (s, X ∪ {✓}) ∈ failures(P ) } ∪
{ (s _ t , X ) | s _ h✓i ∈ traces(P ) ∧
(t , X ) ∈ failures(Q ) } ∪
{ (s, X ) | s ∈ divergences(P ; Q ) }

5.3.3

Implementation

SKIP is implemented as the process which accepts only the symbol "SUCCESS .
It does not matter what it does afterwards
SKIP = λ x • if x = "SUCCESS then STOP else "BLEEP
A sequential composition behaves like the second operand if the first operand
terminates; otherwise, the first operand participates in the first event, and the
rest of it is composed with the second operand
sequence(P , Q ) = if P ("SUCCESS ) ≠ "BLEEP then Q
else λ x •if P (x) = "BLEEP then "BLEEP
else sequence(P (x), Q )

5.4

Interrupts

In this section we define a kind of sequential composition (P 4 Q ) which does
not depend on successful termination of P . Instead, the progress of P is just
interrupted on occurrence of the first event of Q ; and P is never resumed. It
follows that a trace of (P 4 Q ) is just a trace of P up to an arbitrary point when

162

5

Sequential Processes

the interrupt occurs, followed by any trace of Q .
α(P 4 Q ) = αP ∪ αQ
traces(P 4 Q ) = { s _ t | s ∈ traces(P ) ∧ t ∈ traces(Q ) }
To avoid problems, we specify that ✓ must not be in αP .
The next law states that it is the environment which determines when Q
shall start, by selecting an event which is initially offered by Q but not by P
L1 (x : B → P (x)) 4 Q = Q  (x : B → (P (x) 4 Q ))
If (P 4 Q ) can be interrupted by R, this is the same as P interruptible by
(Q 4 R)
L2 (P 4 Q ) 4 R = P 4 (Q 4 R)
Since STOP offers no first event, it can never be triggered by the environment.
Similarly, if STOP is interruptible, only the interrupt can actually occur. Thus
STOP is a unit of
L3 P 4 STOP = P = STOP 4 P
The interrupt operator executes both of its operands at most once, so it distributes through nondeterministic choice
L4A

P 4 (Q u R) = (P 4 Q ) u (P 4 R)

L4B

(Q u R) 4 P = (Q 4 P ) u (R 4 P )

Finally, one cannot cure a divergent process by interrupting it; nor is it safe to
specify a divergent process after the interrupt
L5 CHAOS 4 P = CHAOS = P 4 CHAOS
In the remainder of this section, we shall insist that the possible initial
events of the interrupting process are outside the alphabet of the interrupted
process. Since the occurrence of interrupt is visible and controllable by the
environment, this restriction preserves determinism, and reasoning about the
operators is simplified. To emphasise the preservation of determinism, we
extend the definition of the choice operator. Provided that c ∉ B
(x : B → P (x) | c → Q ) ≡ (x : (B ∪ {c}) → (if x = c then Q else P (x)))
and similarly for more operands.
5.4.1

Catastrophe

Let “ be a symbol standing for a catastrophic interrupt event, which it is reasonable to suppose would not be caused by P ; more formally
“ ∉ αP

5.4

Interrupts

163

Then a process which behaves like P up to catastrophe and thereafter like Q
is defined
ˆ Q = P 4 (“ → Q )
P“
Here Q is perhaps a process which is intended to effect a recovery after cataˆ is distinguished from the event “ by
strophe. Note that the infix operator “
the circumflex.
The first law is just an obvious formulation of the informal description of
the operator
ˆ Q ) / (s _ h“i) = Q
L1 (P “

for s ∈ traces(P )

In the deterministic model, this single law uniquely identifies the meaning of
the operator. In a nondeterministic universe, uniqueness would require additional laws stating strictness and distributivity in both arguments.
The second law gives a more explicit description of the first and subˆ distributes back through →
sequent steps of the process. It shows how “
ˆ Q = (x : B → (P (x) “
ˆ Q) | “ → Q)
L2 (x : B → P (x)) “
This law too uniquely defines the operator on deterministic processes.
5.4.2

Restart

One possible response to catastrophe is to restart the original process again.
Let P be a process such that “ ∉ αP . We specify P̂ as a process which behaves
as P until “ occurs, and after each “ behaves like P from the start again. Such
a process is called restartable and is defined by the simple recursion
αP̂ = αP ∪ {“}
ˆ X)
P̂ = µ X • (P “
ˆ (P “
ˆ (P “
ˆ . . .))
=P“
This is a guarded recursion, since the occurrence of X is guarded by “. P̂ is
certainly a cyclic process (Section 1.8.3), even if P is not.
Catastrophe is not the only reason for a restart. Consider a process designed to play a game, interacting with its human opponent by means of a
selection of keys on a keyboard (see the description of the interact function of
Section 1.4). Humans sometimes get dissatisfied with the progress of a game,
and wish to start a new game again. For this purpose, a new and special key
(“) is provided on the keyboard; depression of this key at any point in the progress of the game will restart the game. It is convenient to define a game P
independently of the restart facility and then transform it into a restartable
game P̂ by using the operator defined above. This idea is due to Alex Teruel.
The informal definition of P̂ is expressed by the law
L1 P̂ / s _ h“i = P̂

for s ∈ traces(P )

164

5

Sequential Processes

But this law does not uniquely define P̂ , since it is equally well satisfied by
RUN . However, P̂ is the smallest deterministic process that satisfies L1.

5.4.3

Alternation

Suppose P and Q are processes which play games in the manner described in
Section 5.4.2; and a human player wishes to play both games simultaneously,
alternating between them in the same way as a chess master plays a simultaneous match by cycling round several weaker opponents. We therefore provide
a new key x , which causes alternation between the two games P and Q . This
is rather like an interrupt, in that the current game is interrupted at an arbitrary point; but it differs from the interrupt in that the current state of the
current game is preserved, so that it can be resumed when the other game is
later interrupted. The process which plays the games P and Q simultaneously
is denoted (P x Q ), and it is most clearly specified by the laws
L1

x ∈ (α(P x Q )) − αP − αQ )

L2 (P x Q ) / s = (P / s) x Q

if s ∈ traces(P )

L3 (P x Q ) / h x i = (Q x P )
We want the smallest operator that satisfies L2 and L3. A more constructive
description of the operator can be derived from these laws; it shows how x
distributes backward through →
L4 (x : B → P (x)) x Q = (x : B → (P (x) x Q ) | x → (Q x P ))
The alternation operator is useful not only for playing games. A similar
facility should be provided in a “friendly” operating system for alternating
between system utilities. For example, you do not wish to lose your place in
the editor on switching to a “help” program, nor vice versa.

5.4.4

Checkpoints

Let P be a process which describes the behaviour of a long-lasting data base
system. When lightning (“) strikes, one of the worst responses would be to
restart P in its initial state, losing all the laboriously accumulated data of the
system. It would be much better to return to some recent state of the system
which is known to be satisfactory. Such a state is known as a checkpoint. We
therefore provide a new key c , which should be pressed only when the current
state of the system is known to be satisfactory. When “ occurs, the most recent
checkpoint is restored; or if there is no checkpoint the initial state is restored.
We suppose c and “ are not in the alphabet of P , and define Ch(P ) as the
process that behaves as P , but responds in the appropriate fashion to these
two events.
The informal definition of Ch(P ) is most succinctly formalised in the laws

5.4

Interrupts

165

L1 Ch(P ) / (s _ h“i) = Ch(P )

for s ∈ traces(P )

L2 Ch(P ) / (s _ h c i) = Ch(P /afters)

for s ∈ traces(P )

Ch(P ) can be defined more explicitly in terms of the operator Ch2(P , Q ),
where P is the current process and Q is the most recent checkpoint waiting
to be reinstated. If catastrophe occurs before the first checkpoint, the system
restarts, as described by the laws
L3 Ch(P ) = Ch2(P , P )
If P = (x : B → P (x))

L4

then Ch2(P , Q ) = (x : B → Ch2(P (x), Q )
| “ → Ch2(Q , Q )
| c → Ch2(P , P ))
The law L4 is suggestive of a practical implementation method, in which
the checkpointed state is stored on some cheap but durable medium such as
magnetic disc or tape. When c occurs, the current state is copied as the new
checkpoint; when “ occurs, the checkpoint is copied back as the new current
state. For reasons of economy, a system implementor ensures that as much
data as possible is shared between the current and the checkpoint states. Such
optimisation is highly machine and application dependent; it is pleasing that
the mathematics is so simple.
The checkpointing operator is useful not only for large-scale data base
systems. When playing a difficult game, a human player may wish to explore
a possible line of play without committing himself to it. So he presses the c
key to store the current position, and if his explorations are unsuccessful, use
of the “ key will restore the status quo.
These ideas of checkpointing have been explored by Ian Hayes.
5.4.5

Multiple checkpoints

In using a checkpointable system Ch(P ) it may happen that a checkpoint is
declared in error. In such cases, it may be desirable to cancel the most recent
checkpoint, and go back to the one before. For this we require a system which
retains two or more of the most recently checkpointed states. In principle,
there is no reason why we should not define a system Mch(P ) which retains
all checkpoints back to the beginning of time. Each occurrence of “ returns to
the state just before the most recent c , rather than the state just after it. As
always we insist
αMch(P ) = αP = { c , “}
A “ before a c goes back to the beginning
L1 Mch(P ) / s _ h“i = Mch(P )

for s ∈ traces(P )

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5

Sequential Processes

A “ after a c cancels the effect of everything that has happened back to and
including the most recent c
L2 Mch(P ) / s _ h c i _ t _ h“i = Mch(P ) / s

for (s u αP ) _ t ∈ traces(P )

A much more explicit description of Mch(P ) can be given in terms of a
binary operator Mch2(P , Q ), where P is the current process and Q is the stack
of checkpoints waiting to be resumed if necessary. The initial content of the
stack is an infinite sequence of copies of P
L3 Mch(P ) = µ X • Mch2(P , X )
= Mch2(P , Mch(P ))
= Mch2(P , Mch2(P , Mch2(P , . . .)))
On occurrence of c the current state is pushed down; on occurrence of “ the
whole stack is reinstated
L4 If P = (x : B → P (x)) then
Mch2(P , Q ) =
(x : B → Mch2(P (x), Q ) | c → Mch2(P , Mch2(P , Q )) | “ → Q )
The pattern of recursions which appear in L4 is quite ingenious, but the multiple checkpoint facility could be very expensive to implement in practice when
the number of checkpoints gets large.

5.4.6

Implementation

The implementation of the various versions of interrupt are based on laws
which show how the operators distribute through →. Consider for example
the alternation operator (5.4.3 L4)
alternation(P , Q ) = λ x • if x = x then
alternation(Q , P )
else if P (x) = "BLEEP then
"BLEEP
else
alternation(P (x), Q )
A more surprising implementation is that of MCh (5.4.5 L3, L4)
Mch(P ) = Mch2(P , Q )

5.5

Assignment

167

where
Mch2(P , Q ) = λ x • if x = “ then
Mch2(P , Mch2(P , Q ))
else if x = c then
Mch2(P , Mch2(P , Q ))
else if P (x) = "BLEEP then
"BLEEP
else
Mch2(P (x), Q )
When this function is executed, the amount of store used grows in proportion
to the number of checkpoints; and available storage is very rapidly exhausted.
Of course, the storage can be reclaimed by the garbage collector on each occurrence of “, but that is not really much consolation. As in the case of other
recursions, constraints if practical implementation enforce a finite bound on
the depth. In this case, the designer should impose a limit on the number of
checkpoints retained, and discard the earlier ones. But such a design is not so
elegantly expressible by recursion.

5.5

Assignment

In this section we shall introduce the most important aspects of conventional
sequential programming, namely assignments, conditionals, and loops. To
simplify the formulation of useful laws, some unusual notations will be defined.
The essential feature of conventional computer programming is assignment. If x is a program variable and e is an expression and P a process
(x := e ; P )
is a process which behaves like P , except that the initial value of x is defined
to be the initial value of the expression e. Initial values of all other variables
are unchanged. Assignment by itself can be given a meaning by the definition
(x := e) = (x := e ; SKIP )
Single assignment generalises easily to multiple assignment. Let x stand
for a list of distinct variables
x = x0 , x1 , . . . xn−1
Let e stand for a list of expressions
e = e0 , e1 , . . . en−1

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5

Sequential Processes

Provided that the lengths of the two lists are the same
x := e
assigns the initial value of ei to xi , for all i. Note that all the ei are evaluated
before any of the assignments are made, so that if y occurs in g
y := f ; z := g
is quite different from
y, z := f , g
Let b be an expression that evaluates to a Boolean truth value (either true or
false). If P and Q are processes
P <| b >
| Q

(P if b else Q )

is a process which behaves like P if the initial value of b is true, or like Q if the
initial value of b is false. The notation is novel, but less cumbersome than the
traditional
if b then P else Q
For similar reasons, the traditional loop
while b do Q
will be written
b∗Q
This may be defined by recursion
D1 b ∗ Q = µ X • ((Q ; X ) <| b >
| SKIP )
Examples
X1 A process that behaves like CTn (1.1.4 X2)
X1 = µ X • (around → X | up → (n := 1 ; X ))
<n
| = 0>
|
(up → (n := n + 1 ; X ) | down → (n := n − 1 ; X ))
The current value of the count is recorded in the variable n



X2 A process that behaves like CT0
n := 0 ; X1
The initial value of the count is set to zero.



5.5

Assignment

169

X3 A process that behaves like POS (5.1 X8)
n := 1 ; (n > 0) ∗ (up → n := n + 1 | down → n := n − 1)
Recursion has been replaced by a conventional loop.



X4 A process which divides a natural number x by a positive number y assigning the quotient to q and the remainder to r
QUOT = (q := x + y ; r := x − q × y)

X5 A process with the same effect as , which computes the quotient by the
slow method of repeated subtraction
LONGQUOT = (q := 0 ; r := x ; ((r ≥ y) ∗ (q := q + 1 ; r := r − y)))

In a previous example (4.5 X3) we have shown how the behaviour of a variable can be modelled by a subordinate process which communicates its value
with the process which uses it. In this chapter, we have deliberately rejected
that technique, because it does not have the properties which we would like.
For example, we want
(m := 1 ; m := 1) = (m := 1)
but unfortunately
(m.left !1 → m.left !1 → SKIP ) ≠ (m.left !1 → SKIP )
5.5.1

Laws

In the laws for assignment, x and y stand for lists of distinct variables; e, f (x),
f (e) stand for lists of expressions, possible containing occurrences of variables
in x or y; and f (e) contains ei whenever f (x) contains xi for all indices i. For
simplicity, in the following laws we shall assume that all expressions always
give a result, for any values of the variables they contain.
L1 (x := x) = SKIP
L2 (x := e ; x := f (x)) = (x := f (e))
L3 If x, y is a list of distinct variables (x := e) = (x, y := e, y)
L4 If x, y, z are of the same length as e, f , g respectively
(x, y, z := e, f , g) = (x, z, y := e, g, f )

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5

Sequential Processes

Using these laws, it is possible to transform every sequence of assignments
into a single assignment to a list of all the variables involved.
When <b
| >
| is considered as a binary infix operator, it possesses several
familiar algebraic properties
L5–L6 <b
| >
| is idempotent, associative, and distributes through <c
| >
|
L7 P <| true >
| Q =P
L8 P <| false >
| Q =Q
L9 P <| ¬ b >
| Q = Q <| b >
| P
L10

P <| b >
| (Q <| b >
| R) = P <| b >
| R

L11

P <| (a <| b >
| c) >
| Q = (P <| a >
| Q ) <| b >
| (P <| c >
| Q)

L12

x := e ; (P <| b(x) >
| Q ) = (x := e ; P ) <| b(e) >
| (x := e ; Q )

L13

(P <| b >
| Q ) ; R = (P ; R) <| b >
| (Q ; R)

To deal effectively with assignment in concurrent processes, it is necessary
to impose a restriction that no variable assigned in one concurrent process
can ever be used in another. To enforce this restriction, we introduce two new
categories of symbol into the alphabets of sequential processes
var (P ) the set of variables that may be assigned within P
acc(P ) the set of variables that may be accessed in expressions within P .
All variables which may be changed may also be accessed
var (P ) ⊆ acc(P ) ⊆ αP
Similarly, we define acc(e) as the set of variables appearing in e. Now if P and
Q are to be joined by ||, we stipulate that
var (P ) ∩ acc(Q ) = var (Q ) ∩ acc(P ) = {}
Under this condition, it does not matter whether an assignment takes place
before a parallel split, or within one of its components after they are running
concurrently
L14

((x := e ; P ) || Q ) = (x := e ; (P || Q ))
provided that x ⊆ var (P ) − acc(Q ) and acc(e) ∩ var (Q ) = {}

An immediate consequence of this is
(x := e ; P ) || (y := f ; Q ) = (x, y := e, f ; (P || Q ))
provided that x ⊆ var (P ) − acc(Q ) − acc(f )
and y ⊆ var (Q ) − acc(P ) − acc(e)

5.5

Assignment

171

This shows how the alphabet restriction ensures that assignments within
one component process of a concurrent pair cannot interfere with assignments
within the other. In an implementation, sequences of assignments may be carried out either together or in any interleaving, without making any difference
to the externally observable actions of the process.
Finally, concurrent combination distributes through the conditional
P || (Q <| b >
| R) = (P || Q ) <| b >
| (P || R)

L15

provided that acc(b) ∩ var (P ) = {}.
This law again states that it does not matter whether b is evaluated before
or after the parallel split.
We now deal with the problem which arises when expressions are undefined for certain values of the variables they contain. If e is a list of expressions, we define D e as a Boolean expression which is true just when all
the operands of e are within the domains of their operators. For example, in
natural number arithmetic,
D(x ÷ y) = (y > 0)
D(y + 1, z + y) = true
D(e + f ) = D e ∧ D f
D(r − y) = y ≤ r
It is reasonable to insist that D e is always defined, i.e.,
D(D e) = true
We deliberately leave completely unspecified the result of an attempt to
evaluate an undefined expression—anything whatsoever may happen. This is
reflected by the use of CHAOS in the following laws.
L16’

(x := e) = (x := e <| D e >
| CHAOS )

L17’

P <| b >
| Q = ((P <| b >
| Q ) <| D b >
| CHAOS )

Furthermore, the laws L2, L5, and L12 need slight modification
L2’

(x := e; x := f (x)) = (x := f (e) <| D e >
| CHAOS )

L5’

(P <| b >
| P ) = (P <| D b >
| CHAOS )

5.5.2

Specifications

A specification of a sequential process describes not only the traces of the
events which occur, but also the relationship between these traces, the initial
values of the program variables, and their final values. To denote the initial
value of a program variable x, we simply use the variable name x by itself. To

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5

Sequential Processes

denote the final value, we decorate the name with a superscript ✓, as in x ✓ . The
value of x ✓ is not observable until the process is terminated, i.e., the last event
of the trace is ✓. This fact is represented by not specifying anything about x ✓
unless tr 0 = ✓.
Examples
X1 A process which performs no action, but adds one to the value of x, and
terminates successfully with the value of y unchanged
tr = hi ∨ (tr = h✓i ∧ x ✓ = x + 1 ∧ y ✓ = y)

X2 A process which performs an event whose symbol is the initial value of
the variable x, and then terminates successfully, leaving the final values of x
and y equal to their initial values
tr = hi ∨ tr = hxi ∨ (tr = hx, ✓i ∧ x ✓ = x ∧ y ✓ = y)

X3 A process which stores the identity of its first event as the final value of x
#tr ≤ 2 ∧ (#tr = 2 ⇒ (tr = hx ✓ , ✓i ∧ y ✓ = y))

X4 A process which divides a nonnegative x by a positive y, and assigns the
quotient to q and the remainder to r
DIV = (y > 0 ⇒
tr = hi ∨ (tr = h✓i ∧ q ✓ = (x ÷ y) ∧
r ✓ = x − (q ✓ × y) ∧ y ✓ = y ∧ x ✓ = x))
Without the precondition, this specification would be impossible to meet in its
full generality.

X5 Here are some more complex specifications which will be used later
DIVLOOP =
(tr = hi ∨ (tr = h✓i ∧ r = (q ✓ − q) × y + r ✓ ∧
r ✓ < y ∧ x ✓ = x ∧ y ✓ = y))

T (n) = r < n × y


5.5

Assignment

173

All variables in these and subsequent specifications are intended to denote
natural numbers, so subtraction is undefined if the second operand is greater
than the first.
We shall now formulate the laws which underlie proofs that a process
satisfies its specification. Let s(x, tr , x ✓ ) be a specification. In order to prove
that SKIP satisfies this specification, clearly the specification must be true when
the trace is empty; furthermore, it must be true when the trace is h✓i and
the final values of all variables x ✓ are equal to their initial values. These two
conditions are also sufficient, as stated in the following law
If S (x, hi, x ✓ )

L1

and S (x, h✓i, x ✓ )
then SKIP sat S (x, tr , hx ✓ )
X6 The strongest specification satisfied by SKIP is
SKIPA sat (tr = hi ∨ (tr = h✓i ∧ x ✓ = x))
where x is a list of all variables in A and x ✓ is a list of their ticked variants. X6
is an immediate consequence of L1 and vice versa.

X7 We can prove that
SKIP sat (r < y ⇒ (T (n + 1) ⇒ DIVLOOP ))
Proof :
(1) Replacing tr by hi in the specification gives
r < y ∧ T (n + 1) ⇒ hi = hi ∨ . . .
which is a tautology.
(2) Replacing tr by h✓i and final values by initial values gives
r < y ∧ T (n + 1) ⇒
(h✓i = hi ∨ (h✓i = h✓i ∧ x = x ∧
y = y ∧ r = ((q − q) × y + r ∧ r < y)))
which is also a trivial theorem. This result will be used in X10.



It is a precondition of successful assignment x := e that the expressions e
on the right-hand side should be defined. In this case, if P satisfies a specification S (x), (x := e ; P ) satisfies the same specification, after modification to
reflect the fact that the initial value of x is e.

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5

Sequential Processes

L2 If P sat S (x) then
(x := e; P ) sat (D e ⇒ S (e))
The law for simple assignment can be derived from L2 on replacing P by SKIP ,
and using X6 and 5.2 L1
L2A

x0 := e sat (D e ∧ tr ≠ hi ⇒ tr = h✓i ∧ x0✓ = e ∧ x1✓ = x1 ∧ . . .)

A consequence of L2 is that for any P , the strongest fact one can prove about
(x := 1/0 ; P ) is
(x := 1/0 ; P ) sat true
Whatever non-vacuous goal you may wish to achieve, it cannot be achieved by
starting with an illegal assignment.
Examples
X8
SKIP sat (tr ≠ hi ⇒ tr = h✓i ∧ q ✓ = q ∧ r ✓ = r ∧ y ✓ = y ∧ x ✓ = x)
therefore
(r := x − q × y ; SKIP ) sat (x ≥ q × x ∧ tr ≠ hi ⇒
tr = h✓i ∧ q ✓ = q ∧
r ✓ = (x − q × y) ∧ y ✓ = y ∧ x ✓ = x)
therefore
(q := x ÷ y ; r := x − q × y) sat (y > 0 ∧ x ≥ (x ÷ y) × y ∧ tr ≠ hi ⇒
tr = h✓i ∧ q ✓ = (x ÷ y) ∧
r ✓ = (x − (x ÷ y) × y) ∧
y ✓ = y ∧ x ✓ = x)
The specification on the last line is equivalent to DIV , defined in X4.
X9 Assume
X sat (T (n) ⇒ DIVLOOP )
therefore
(r := r − y ; X ) sat (y ≤ r ⇒
(r − y < n × y ⇒
(tr = hi ∨ tr = h✓i ∧ (r − y) = . . .)))



5.5

Assignment

175

therefore
(q := q + 1 ; r := r − y ; X ) sat (y ≤ r ⇒ (r < (n + 1) × y ⇒ DIVLOOP 0 ))
where
DIVLOOP 0 =
(tr = hi ∨ (tr = h✓i ∧ (r − y) = (q ✓ − (q + 1)) × y + r ✓ ∧
r ✓ < y ∧ x ✓ = x ∧ y ✓ = y))
By elementary algebra of natural numbers
y ≤ r ⇒ (DIVLOOP 0 ≡ DIVLOOP )
therefore
(q := q + 1 ; r := r − y ; X ) sat (y ≤ r ⇒ (T (n + 1) ⇒ DIVLOOP ))
This result will be used in X10.



For general sequential composition, a much more complicated law is required, in which the traces of the components are sequentially composed, and
the initial state of the second component is identical to the final state of the
first component. However, the values of the variables in this intermediate state
are not observable; only the existence of such values is assured
L3

If P sat S (x, tr , x ✓ )
and Q sat T (x, tr , x ✓ )
and P does not diverge
then (P ; Q ) sat (∃ y, s, t • tr = (s ; t ) ∧ S (x, s, y) ∧ T (y, t , x ✓ ))

In this law, x is a list of all variables in the alphabet of P and Q , x ✓ is a list of
their subscripted variants, and y a list of the same number of fresh variables.
The specification of a conditional is the same as that of the first component
if the condition is true, and the same as that of the second component if false.
L4

If P sat S and Q sat T
then (P <| b >
| Q ) sat ((b ∧ S ) ∨ (¬ b ∧ T ))

An alternative form of this law is sometimes more convenient
L4A

If P sat (b ⇒ S ) and Q sat (¬ b ⇒ S )
then (P <| b >
| Q ) sat S

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5

Sequential Processes

Example
X10

Let
COND = (q := q + 1 ; r := r − y ; X ) <| r ≥ y >
| SKIP

and
X sat (T (n) ⇒ DIVLOOP )
then
COND sat (T (n + 1) ⇒ DIVLOOP )
The two sufficient conditions for this conclusion have been proved in X7 and
X9; the result follows by L4A.

The proof of a loop uses the recursive definition given in 5.5 D1, and the
law for unguarded recursion (3.7.1 L8). If R is the intended specification of the
loop, we must find a specification S (n) such that S (0) is always true, and also
(∀ n • S (n)) ⇒ R
A general method to construct S (n) is to find a predicate T (n, x), which describes the conditions on the initial state x such that the loop is certain to
terminate in less than n repetitions. Then define
S (n) = (T (n, x) ⇒ R)
Clearly, no loop can terminate in less than no repetitions, so if T (n, x) has been
correctly defined T (0, x) will be false, and consequently S (0) will be true. The
result of the proof of the loop will be ∀ n • S (n), i.e.,
∀ n • (T (n, x) ⇒ R)
Since n has been chosen as a variable which does not occur in R, this is equivalent to
(∃ n • T (n, x)) ⇒ R
No stronger specification can possibly be met, since
∃ n • T (n, x)
is the precondition under which the loop terminates in some finite number of
iterations.
Finally, we must prove that the body of the loop meets its specification.
Since the recursive equation for a loop involves a conditional, this task splits
into two. Thus we derive the general law

5.5

L5

Assignment

177

If ¬ T (0, x) and T (n, x) ⇒ D b
and SKIP sat (¬ b ⇒ (T (n, x) ⇒ R))
and (X sat T (n, x) ⇒ R)) ⇒ ((Q ; X ) sat (b ⇒ (T (n + 1, x) ⇒ R)))
then (b ∗ Q ) sat ((∃ n • T (n, x)) ⇒ R)
Example

X11 We wish to prove that the program for long division by repeated subtraction (5.5 X5) meets its specification DIV . The task splits naturally in two. The
second and more difficult part is to prove that the loop meets some suitably
formulated specification, namely
(r ≥ y) ∗ (q := q + 1 ; r := r − y) sat (y > 0 ⇒ DIVLOOP )
First we need to formulate the condition under which the loop terminates in
less than n iterations
T (n) = r < n × y
Here T (0) is obviously false; the clause ∃ n • T (n) is equivalent to y > 0, which
is the precondition under which the loop terminates. The remaining steps of
the proof of the loop have already been taken in X7 and X5. The rest of the
proof is a simple exercise.

The laws given in this section are designed as a calculus of total correctness
for purely sequential programs, which contain no input or output. If Q is such
a program, then a proof that
Q sat (P (x) ∧ tr ≠ hi ⇒ tr = h✓i ∧ R(x, x ✓ )

(1)

established that if P (x) is true of the initial values of the variables when Q
is started, then Q will terminate and R(x, x ✓ ) will describe the relationship
between the initial values x and the final values x ✓ . Thus (P (x), R(x, x ✓ )) form
a precondition/postcondition pair in the sense of Cliff Jones. If R(x ✓ ) does not
mention the initial values x, the assertion (1) is equivalent to
P (x) ⇒ wp(Q , R(x))
where wp is Dijkstra’s weakest precondition.
Thus in the special case of noncommunicating programs, the proof methods are mathematically equivalent to ones that are already familiar, though
the explicit mention of “tr = hi” and “tr = h✓i” makes them notationally more
clumsy. This extra burden is of course necessary, and therefore more acceptable, when the methods are extended to deal with communicating sequential
processes.

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5

5.5.3

Sequential Processes

Implementation

The initial and final states of a sequential process can be represented as a
function which maps each variable name onto its value. A sequential process
is defined as a function which maps its initial states onto its subsequent behaviour. Successful termination (✓) is represented by the atom "SUCCESS . A
process which is ready to terminate will accept this symbol, which it maps, not
onto another process, but onto the final state of its variables.
The process SKIP takes an initial state as a parameter, accepts "SUCCESS
as its only action, and delivers its initial state as its final state
SKIP = λ s • λ y • if y ≠ "SUCCESS then "BLEEP else s
An assignment is similar, except that its final state is slightly changed
assign(x, e) = λ s • λ y • if y ≠ "SUCCESS then
"BLEEP
else
update(s, x, e)
where update(s, x, e) is the function λ y • if y = x then eval(e, s) else s(y) and
eval(s, e) is the result of evaluating the expression e in state s.
If e is undefined in state s, we do not care what happens. Here, for simplicity, we have implemented only the single assignment. Multiple assignment is
a little more complicated.
To implement sequential composition, it is necessary first to test whether
the first operand has successfully terminated. If so, its final state is passed on
to the second operand. If not, the first action is that of the first operand
sequence(P , Q ) = λ s • if P (s)("SUCCESS ) ≠ "BLEEP then
Q (P (s)("SUCCESS ))
else
λ y • if P (s)(y) = "BLEEP then
"BLEEP
else
sequence(P (s)(y), Q )
The implementation of the conditional is as a conditional
condition(P , b, Q ) = λ s • if eval(b, s) then P (s) else Q (s)
The implementation of the loop (b ∗ Q ) is left as an exercise.

5.5

Assignment

179

Note that the definition of sequence given above is more complicated than
that given in Section 5.3.3, because it takes a state as its first argument, and it
has to supply the state as the first argument of its operands. Unfortunately,
a similar complexity has to be introduced into the definitions of all the other
operators given in earlier chapters. A simpler alternative would be to model
variables as subordinate processes; but this would probably be a great deal less
efficient than the use of conventional random access storage. When considerations of efficiency are added to those of mathematical convenience, there are
adequate grounds for introducing the assignable program variable as a new
primitive concept, rather than defining it in terms of previously introduced
concepts.

Shared Resources

6.1

6

Introduction

In Section 4.5 we introduced the concept of a named subordinate process
(m : R), whose sole task is to meet the needs of a single main process S ; and
for this we have used the notation
(m : R // S )
Suppose now that S contains or consists of two concurrent processes (P || Q ),
and that both P and Q require the services of the same subordinate process
(m : R). Unfortunately, it is not possible for P and Q both to communicate
with (m : R) along the same channels, because these channels would have to
be in the alphabet of both P and Q ; and then the definition of || would require
that communications with (m : R) take place only when both P and Q communicate the same message simultaneously—which (as explained in 4.5 X6) is
far from the required effect. What is needed is some way of interleaving the
communications between P and (m : R) with those between Q and (m : R). In
this way, (m : R) serves as a resource shared between P and Q ; each of them
uses it independently, and their interactions with it are interleaved.
When the identity of all the sharing processes is known in advance, it is
possible to arrange that each sharing process uses a different set of channels
to communicate with the shared resource. This technique was used in the
story of the dining philosophers (Section 2.5); each fork was shared between
two neighbouring philosophers, and the footman was shared among all five.
Another example was 4.5 X6, in which a buffer was shared between two processes, one of which used only the left channel and the other used only the
right channel.
A general method of sharing is provided by multiple labelling (Section 2.6.4),
which effectively creates enough separate channels for independent communication with each sharing process. Individual communications along these
channels are arbitrarily interleaved. But this method requires that the names
of all the sharing processes are known in advance; and so it is not adequate
for a subordinate process intended to serve the needs of a main process which

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Shared Resources

splits into an arbitrary number of concurrent subprocesses. This chapter introduces techniques for sharing a resource among many processes, even when
their number and identities are not known in advance. It is illustrated by examples drawn from the design of an operating system.

6.2

Sharing by interleaving

The problem described in Section 6.1 arises from the use of the combinator || to
describe the concurrent behaviour of processes; and this problem can often be
avoided by using instead the interleaving form of concurrency (P ||| Q ). Here, P
and Q have the same alphabet and their communications with external (shared)
processes are arbitrarily interleaved. Of course, this prohibits direct communication between P and Q ; but indirect communication can be re-established
through the services of a shared subordinate process of appropriate design, as
shown in 4.5 X6 and in X2 below.
Example
X1 (Shared subroutine)
doub : DOUBLE // (P ||| Q )
Here, both P and Q may contain calls on the subordinate process
(doub.left !v → doub.right ?x → SKIP )
Even though these pairs of communications from P and Q are arbitrarily interleaved, there is no danger than one of the processes will accidentally obtain
an answer which should have been received by the other. To ensure this, all
subprocesses of the main process must observe a strict alternation of communications on the left channel with communications on the right channel of the
shared subordinate. For this reason, it seems worthwhile to introduce a specialised notation, whose exclusive use will guarantee observance of the required
discipline. The suggested notation is reminiscent of a traditional procedure
call in a high-level language, except that the value parameters are preceded by
! and the result parameters by ?, this
doub!x?y = (doub.left !x → doub.right ?y → SKIP )

The intended effect of sharing by interleaving is illustrated by the following series of algebraic transformations. When two sharing processes both
simultaneously attempt to use the shared subroutine, matched pairs of communications are taken in arbitrary order, but the components of a pair of communications with one process are never separated by a communication with

6.2

Sharing by interleaving

183

another. For convenience, we use the following abbreviations


d!v for d.left !v 
within a main process

d?x for d.right ?x 


!v for right !v 
within a subordinate process

?x for left ?x 
Let
D = ?x →!(x + x) → D
P = d!3 → d?y → P (y)
Q = d!4 → d?z → Q (z)
R = (d : D // (P ||| Q ))
(as in X1 above), then
P ||| Q
= d!3 → ((d?y → P (y)) ||| Q )

d!4 → (P ||| (d?z → Q (z)))
[by 3.6.1 L7]
The sharing processes each start with an output to the shared process. It is the
shared process that is offered the choice between them. But the shared process
is willing to accept either, so after hiding the choice becomes nondeterministic
(d : D // (P ||| Q ))
= ((d : (!3 + 3 → D)) // ((d?y → P (y)) ||| Q ))
u
((d : (!4 + 4 → D)) // (P ||| (d?z → Q (z))))
= (d : D // (P (6) ||| Q )) u (d : D // (P ||| Q (8)))
[4.5.1 L1, 3.5.1 L5, etc.]
The shared process offers its result to whichever of the sharing processes is
ready to take it. Since one of these processes is still waiting for output, it is the
process which provided the argument that gets the result. That is why strict
alternation of output and input is so important in calling a shared subroutine.
Example
X2 (Shared data structure) In an airline flight reservation system, bookings are
made by many reservation clerks, whose actions are interleaved. Each reserva-

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Shared Resources

tion adds a passenger to the flight list, and returns an indication whether that
passenger was already booked or not. For this oversimplified example, the set
implemented in 4.5 X8 will serve as a shared subordinate process, named by
the flight number
AG109 : SET // (. . . (CLERK ||| CLERK ||| . . .) . . .)
Each CLERK books a passenger by the call
AG109!pass no?x
which stands for
(AG109.left !pass no → AG109.right ?x → SKIP )

In these two examples, each occasion of use of the shared resource involves exactly two communications, one to send the parameters and the other
to receive the results; after each pair of communications, the subordinate process returns to a state in which it is ready to serve another process, or the same
one again. But frequently we wish to ensure that a whole series of communications takes place between two processes, without danger of interference by a
third process. For example, a single expensive output device may have to be
shared among many concurrent processes. On each occasion of use, a number
of lines constituting a file must be output consecutively, without any danger
of interleaving of lines sent by another process. For this purpose, the output
of a file must be preceded by an acquire which obtains exclusive use of the
resource; and on completion, the resource must be made available again by a
release.
Examples
X3 (Shared line printer)
LP = acquire → µ X • (left ?x → h!s → X | release → LP )
Here, h is the channel which connects LP to the hardware of the line printer.
After acquisition, the process LP copies successive lines from its left channel
to its hardware, until a release signal returns it to its original state, in which it
is available for use by any other processes. This process is used as a shared
resource
lp.acquire → . . . lp.left !“A. JONES” → . . .
lp.left !nextline → . . . lp.release →


6.2

Sharing by interleaving

185

X4 (An improvement on X3) When a line printer is shared between many
users, the length of paper containing each file must be manually detached after
output from the previous and the following files. For this purpose, the printing
paper is usually divided into pages, which are separated by perforations; and
the hardware of the printer allows an operation throw, which moves the paper
rapidly to the end of the current page—or better, to the next outward-facing
fold in the paper stack. To assist in separation of output, files should begin
and end on page boundaries, and a complete row of asterisks should be printed
at the end of the last page of the file, and at the beginning of the first page. To
prevent confusion, no complete line of asterisks is permitted to be printed in
the middle of a file
LP = (h!throw → h!asterisks →
acquire → h!asterisks →
µ X • (left ?s → if s ≠ asterisks then
h!s → X
else
X
| release → LP ))
This version of LP is used in exactly the same way as the previous one.



In the last two examples, the use of the signals acquire and release prevent
arbitrary interleaving of lines from distinct files, and they do so without introducing the danger of deadlock. But if more than one resource is to be shared
in this fashion, the risk of deadlock cannot be ignored.
Example
X5 (Deadlock) Ann and Mary are good friends and good cooks; they share a
pot and a pan, which they acquire, use and release as they need them
UTENSIL = (acquire → use → use → . . . → release → UTENSIL)
pot : UTENSIL // pan : UTENSIL // (ANN ||| MARY )
Ann cooks in accordance with a recipe which requires a pot first and then a
pan, whereas Mary needs a pan first, then a pot
ANN = . . . pot .acquire → . . . pan.acquire → . . .
MARY = . . . pan.acquire → . . . pot .acquire → . . .
Unfortunately, they decide to prepare a meal at about the same time. Each of
them acquires her first utensil; but when she needs her second utensil, she
finds that she cannot have it, because it is being used by the other.

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6

Shared Resources

eat
serve
pan.release
pan.use
pot.release
pan.use
pan.acquire
pot.use
pot.acquire
mix
peel
ANN

pan.
acquire
MARY

pot.
acquire

pot.
release

eat
pan.
release

Figure 6.1
The story of Ann and Mary can be visualised on a two-dimensional plot
(Figure 6.1), where the life of Ann is displayed along the vertical axis and Mary’s
life on the horizontal. The system starts in the bottom left hand corner, at the
beginning of both their lives. Each time Ann performs an action, the system
moves one step upward. Each time Mary performs an action, the system moves
one step right. The trajectory shown on the graph shows a typical interleaving
of Ann’s and Mary’s actions. Fortunately, this trajectory reaches the top right
hand corner of the graph where both cooks are enjoying their meal.
But this happy outcome is not certain. Because they cannot simultaneously
use a shared utensil, there are certain rectangular regions in the state space
through which the trajectory cannot pass. For example in the region hatched
both cooks would be using the pan, and this is not possible. Similarly, exclusion on the use of the pot prohibits entry into the region hatched . Thus if the
trajectory reaches the edge of one of these forbidden regions, it can only follow the edge upward (for a vertical edge) or rightward (for a horizontal edge).
During this period, one of the cooks is waiting for release of a utensil by the
other.
Now consider the zone marked with dots . If ever the trajectory enters
this zone, it will inevitably end in deadlock at the top right hand corner of the
zone. The purpose of the picture is to show that the danger of deadlock arises
solely as a result of a concavity in the forbidden region which faces towards
the origin: other concavities are quite safe. The picture also shows that the
only sure way of preventing deadlock is to extend the forbidden region to

6.3

Shared storage

187

cover the danger zone, and so remove the concavity. One technique would be
to introduce an additional artificial resource which must be acquired before
either utensil,and must not be released until both utensils have been released.
This solution is similar to the one imposed by the footman in the story of the
dining philosophers (Section 2.5.3) where permission to sit down is a kind of
resource, of which only four instances are shared among five philosophers.
An easier solution is to insist that any cook who is going to want both utensils
must acquire the pan first. This example is due to E. W. Dijkstra.

The easier solution suggested for the previous example generalises to any
number of users, and any number of resources. Provided that there is a fixed
order in which all users acquire the resources they want, there is no risk of
deadlock. Users should release the resources as soon as they have finished with
them; the order of release does not matter. Users may even acquire resources
out of order, provided that at the time of acquisition they have already released
all resources which are later in the standard ordering. Observance of this
discipline of resource acquisition and release can often be checked by a visual
scan of the text of the user processes.

6.3

Shared storage

The purpose of this section is to argue against the use of shared storage; the
section may be omitted by those who are already convinced.
The behaviour of systems of concurrent processes can readily be implemented on a single conventional stored program computer, by a technique
known as timesharing, in which a single processor executes each of the processes in alternation, with process change on occurrence of interrupt from an
external device or from a regular timer. In this implementation, it is very easy
to allow the concurrent processes to share locations of common storage, which
are accessed and assigned simply by means of the usual machine instructions
within the code for each of the processes.
A location of shared storage can be modelled in our theory as a shared
variable (4.2 X7) with the appropriate symbolic name, for example
(count : VAR //
(count .left !0 → (P ||| Q )))
Shared storage must be clearly distinguished from the local storage described
in 5.5. The simplicity of the laws for reasoning about sequential processes derives solely from the fact that each variable is updated by at most one process;
and these laws do not deal with the many dangers that arise from arbitrary
interleaving of assignments from different processes.
These dangers are most clearly illustrated by the following example.

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Shared Resources

Example
X1 (Interference) The shared variable count is used to keep a count of the total
number of occurrences of some important event. On each occurrence of the
event, the relevant process P or Q attempts to update the count by the pair of
communications
count .right ?x; count .left !(x + 1)
Unfortunately, these two communications may be interleaved by a similar pair
of communications from the other process, resulting in the sequence
count .right ?x →
count .right ?y →
count .left !(y + 1) →
count .left !(x + 1) → . . .
As a consequence, the value of the count is incremented only by one instead
of two. This kind of error is known as interference, and it is an easy mistake in
the design of processes which share common storage. Further, the actual occurrence of the fault is highly nondeterministic; it is not reliably reproducible,
and so it is almost impossible to diagnose the error by conventional testing
techniques. As a result, I suspect that there are several operating systems in
current use which regularly produce slightly inaccurate summaries, statistics,
and accounts.

A possible solution to this problem is to make sure that no change of
process takes place during a sequence of actions which must be protected
from interleaving. Such a sequence is known as a critical region. On an implementation by a single processor, the required exclusion is often achieved by
inhibiting all interrupts for the duration of the critical region. This solution
has an undesirable effect in delaying response to interrupts; and worse, it fails
completely as soon as a second processing unit is added to the computer.
A better solution was suggested by E. W. Dijkstra in his introduction of
the binary exclusion semaphore. A semaphore may be described as a process
which engages alternatively in actions named P and V
SEM = (P → V → SEM )
This is declared as a shared resource
(mutex : SEM // . . .)
Each process, on entry into a critical region, must send the signal
mutex.P

6.4

Multiple resources

189

and on exit from the critical region must engage in the event
mutex.V
Thus the critical region in which the count is incremented should appear
mutex.P →
count .right ?x → count .left !(x + 1) →
mutex.V → . . .
Provided that all processes observe this discipline, it is impossible for two processes to interfere with each other’s updating of the count. But if any process
omits a P or a V , or gets them in the wrong order, the effect will be chaotic,
and will risk a disastrous or (perhaps worse) a subtle error.
A much more robust way to prevent interference is to build the required
protection into the very design of the shared storage, taking advantage of
knowledge of its intended pattern of usage. For example, if a variable is to
be used only for counting, then the operation which increments it should be a
single atomic operation
count .up
and the shared resource should be designed like CT0 (1.1.4 X2)
count : CT0 // (. . . P ||| Q . . .)
In fact there are good reasons for recommending that each shared resource
be specially designed for its purpose, and that pure storage should not be
shared in the design of a system using concurrency. This not only avoids the
grave dangers of accidental interference; it also produces a design that can
be implemented efficiently on networks of distributed processing elements as
well as single-processor and multiprocessor computers with physically shared
store.

6.4

Multiple resources

In Section 6.2, we described how a number of concurrent processes with different behaviour could share a single subordinate process. Each sharing process
observes a discipline of alternating input and output, or alternating acquire
and release signals, to ensure that at any given time the resource is used by at
most one of the potentially sharing processes. Such resources are known as
serially reusable. In this section we introduce arrays of processes to represent
multiple resources with identical behaviour; and indices in the array ensure
that each element communicates safely with the process that has acquired it.

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6

Shared Resources

We shall therefore make substantial use of indices and indexed operators,
with obvious meaning. For example

||i<12 Pi = (P0 || P1 || . . . || P11 )
|||i<4 P = (P ||| P ||| P ||| P )
||i≥0 Pi = (P0 || P1 || . . .)
i≥0 (f (i) → Pi ) = (f (0) → P0 | f (1) → P1 | . . .)
In the last example, we insist that f is a one-one function, so that the choice
between the alternatives is made solely by the environment.
Examples
X1 (Re-entrant subroutine) A shared subroutine that is serially reusable can
be used by only one calling process at a time. If the execution of the subroutine
requires a considerable calculation, there could be corresponding delays to
the calling processes. If several processors are available to perform the calculations, there is good reason to allow several instances of the subroutine to
proceed concurrently on different processors. A subroutine capable of several
concurrent instances is known as re-entrant, and it is defined as an array of
concurrent processes
doub : (||i<27 (i : DOUBLE )) // . . .
A typical call of this subroutine could be
(doub.3.left !30 → doub.3.right ?y → SKIP )
The use of the index 3 ensures that the result of the call is obtained from the
same instance of doub to which the arguments were sent, even though some
other concurrent process may at the same time call another instance of the
array, resulting in an interleaving of the messages
doub.3.left .30, . . . doub.2.left .20,
. . . doub.3.right .60, . . . doub.2.right .40, . . .
When a process calls a re-entrant subroutine, it really does not matter
which element of the array responds to the call; any one that happens to be
free will be equally good. So rather than specifying a particular index 2 or 3, a
calling process should leave the selection arbitrary, by using the construct

i≥0 (doub.i.left !30 → doub.i.right ?y → SKIP )
This still observes the essential discipline that the same index is used for sending the arguments and (immediately after) for receiving the result.


6.4

Multiple resources

191

In the example shown above, there is an arbitrary limit of 27 simultaneous
activations of the subroutine. Since it is fairly easy to arrange that single processor divides its attention among a much larger number of processes, such
arbitrary limits can be avoided by introducing an infinite array of concurrent
processes
doub : (||i≥0 i : D)
where D can now be designed to serve only a single call and then stop
D = left ?x → right !(x + x) → STOP
A subroutine with no bound on its re-entrancy is known as a procedure.
The intention in using a procedure is that the effect of each call

i≥0 (doub.i.left !x → doub.i.right ?y → SKIP )
should be identical to the call of a subordinate process D declared immediately
adjacent to the call
(doub : D // (doub.left !x → doub.right ?y → SKIP ))
This latter is known as a local procedure call, since it suggests execution of
the procedure on the same processor as the calling process; whereas the call
of a shared procedure is known as a remote call, since it suggests execution
on a separate possibly distant processor. Since the effect of remote and local
calls is intended to be the same, the reasons for using the remote call can only
be political or economic—e.g., to keep the code of the procedure secret, or to
run it on a machine with special facilities which are too expensive to provide
on the machine on which the using processes run.
A typical example of an expensive facility is a high-volume backing store,
such as a disk or bubble memory.
X2 (Shared backing storage) A storage medium is split into B sectors which
can be read and written independently. Each sector can store one block of
information, which it inputs on the left and outputs on the right. Unfortunately
the storage medium is implemented in a technology with destructive read-out,
so that each block written can be read only once. Thus each sector behaves
like COPY (4.2 X1) rather than VAR (4.2 X7). The whole backing store is an
array of such sectors, indexed by numbers less than B
BSTORE = ||i<B i : COPY

This store is intended for use as a subordinate process
(back : BSTORE // . . .)

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6

Shared Resources

Within its main process, the store may be used by the communications
back.i.left ?bl → . . . back.i.right ?y → . . .
The backing store may also be shared by concurrent processes. In this case,
the action

i<B (back.i.left !bl → . . .)
will simultaneously acquire an arbitrary free sector with number i, and write
the value of bl into it. Similarly, back.i.right ?x will in a single action both
read the content of sector i into x and release this sector for use on another
occasion, very possibly by another process. It is this simplification that is the
real motive for using COPY to model the behaviour of each sector: the story
of destructive read-out is just a story.

Of course, successful sharing of this backing store requires the utmost
discipline on the part of the sharing processes. A process may input from a
sector only if the same process has most recently output to that very sector;
and each output must eventually be followed by such an input. Failure to observe such disciplines will lead to deadlock, or even worse confusion. Methods
of enforcing this discipline painlessly will be introduced after the next example,
and will be extensively illustrated in the subsequent design of modules of an
operating system (Section 6.5).
X3 (Two line printers) Two identical line printers are available to serve the
demands of a collection of using processes. They both need the kind of protection from interleaving that was provided by LP (6.2 X4). We therefore declare
an array of two instances of LP , each of which is indexed by a natural number
indicating its position in the array
LP2 = (0 : LP || 1 : LP )
This array may itself be given a name for use as a shared resource
(lp2 : LP2 // . . .)
Each instance of LP is now prefixed twice, once by a name and once by an index;
thus communications with the using process have three or four components,
e.g.,
lp.0.acquire, lp.1.left .“A.JONES”, . . .
As in the case of a re-entrant procedure, when a process needs to acquire
one of an array of identical resources, it really cannot matter which element
of the array is selected on a given occasion. Any element which is ready to
respond to the acquire signal will be acceptable. A general choice construction
will make the required arbitrary choice

i≥0 (lp.i.acquire → . . . lp.i.left !x → . . . lp.i.release → SKIP )

6.4

Multiple resources

193

Here, the initial lp.i.acquire will acquire whichever of the two LP processes
is ready for this event. If neither is ready, the acquiring process will wait; if
both are ready, the choice between them is nondeterministic. After the initial
acquisition, the bound variable i takes as its value the index of the selected
resource, and all subsequent communications will be correctly directed to that
same resource.

When a shared resource has been acquired for temporary use within another process, the resource is intended to behave exactly like a locally declared
subordinate process, communicating only with its using subprocess. Let us
therefore adapt the familiar notation for subordination, and write
(myfile :: lp // . . . myfile.left !x . . .)
instead of the much more cumbersome construction

i≥0 (lp.i.acquire → . . . lp.i.left !x . . . ; lp.i.release → SKIP )
Here, the local name myfile has been introduced to stand for the indexed name
lp.i, and the technicalities of acquisition and release have been conveniently
suppressed. The new “::” notation is called remote subordination; it is distinguished from the familiar “:” notation in that it takes on its right, not a
complete process, but the name of a remotely positioned array of processes.
X4 (Two output files) A using process requires simultaneous use of two line
printers to output two files, f1 and f2
(f1 :: lp // (f2 :: lp // . . . f1.left !s1 → f2.left !s2 → . . .))
Here, the using process interleaves output of lines to the two different files;
but each line is printed on the appropriate printer. Of course, deadlock will
be the certain result of any attempt to declare three printers simultaneously;
it is also a likely result of declaring two printers simultaneously in each of two
concurrent processes, as shown in the history of Ann and Mary (6.2 X5).

X5 (Scratch file) A scratch file is used for output of a sequence of blocks.
When the output is complete, the file is rewound, and the entire sequence of
blocks is read back from the beginning. When all the blocks have been read, the
scratch file will then give only empty signals; no further reading or writing is
possible. Thus a scratch file behaves like a file output to magnetic tape, which
must be rewound before being read. The empty signal serves as an end-of-file
marker
SCRATCH = WRITEhi
WRITEs = (left ?x → WRITEs _hxi
| rewind → READs )
READhxi_s = (right !x → READs )
READhi = (empty → READhi )

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6

Shared Resources

This may conveniently be used as a simple unshared subordinate process
(myfile : SCRATCH // . . . mfile.left !v . . . myfile.rewind . . .
. . . (myfile.right ?x → . . .
| myfile.empty → . . .) . . .)
It will serve later as a model for a shared process.



X6 (Scratch files on backing store) The scratch file described in X5 can be
readily implemented by holding the stored sequence of blocks in the main
store of a computer. But if the blocks are large and the sequence is long, this
could be an uneconomic use of main store, and it would be better to store the
blocks on a backing store.
Since each block in a scratch file is read and written only once, a backing
store (X2) with destructive read-out will suffice. An ordinary scratch file (held
in main store) is used to hold the sequence of indices of the sectors of backing
store on which the corresponding actual blocks of information are held; this
ensures that the correct blocks are read back, and in the correct sequence
BSCRATCH = (pagetable : SCRATCH //
µ X • (left ?x → (i<B back.i.left !x → pagetable.left !i → X )
| rewind →
pagetable.rewind →
µ Y • (pagetable.right ?i →
back.i.right ?x → right !x → Y
| pagetable.empty → empty → Y )))
BSCRATCH uses the name back to address a backing store (X2) as a subordinate
process. This should be supplied
SCRATCHB = (back : BSTORE // BSCRATCH )
SCRATCHB can be used as a simple unshared subordinate process in exactly
the same way as the scratch file of X5
(myfile : SCRATCHB // . . . myfile.left !v . . .)
The effect is exactly the same as use of SCRATCH , except that the maximum
length of the scratch file is limited to B blocks.

X7 (Serially reused scratch files) Suppose we want to share the scratch file
on backing store among a number of interleaved users, who will acquire, use,
and release it one at a time in the manner of a shared line printer (6.2 X3). For
this purpose, we must adapt BSCRATCH to accept acquire and release signals.

6.4

Multiple resources

195

If a user releases his scratch file before reading to the end, there is a danger
that the unread blocks on backing store will never be reclaimed. This danger
can be averted by a loop that reads back these blocks and discards them
SCAN = µ X • (pagetable.right ?i →
back.i.right ?x →
X
| pagetable.empty → SKIP )
A shared scratch file acquires its user, and then behaves as BSCRATCH .
The release signal causes an interrupt (Section 5.4) to the SCAN process
SHBSCR = acquire →
(BSCRATCH 4 (release → SCAN ))
The serially reusable scratchfile is provided by the simple loop
∗

SHBSCR

which uses BSTORE as a subordinate process
back : BSTORE // ∗ SHBSCR

X8 (Multiplexed scratch files) In the previous two examples only one scratch
file is in use at a time. A backing store is usually sufficiently large to allow
many scratch files to exist simultaneously, each occupying a disjoint subset
of the available sectors. The backing store can therefore be shared among
an unbounded array of scratch files. Each scratch file acquires a sector when
needed by outputting to it, and releases it automatically on inputting that block
again. The backing store is shared by the technique of multiple labelling (Section 2.6.4), using as labels the same indices (natural numbers) which are used
in constructing the array of sharing processes
FILESYS = N : (back : BSTORE ) // (||i≥0 i : SHBSCR)
where N = { i | i ≥ 0 }
This filing system is intended to be used as a subordinate process, shared
by interleaving among any number of users
filesys : FILESYS // . . . (USER1 ||| USER2 ||| . . .)
Inside each user, a fresh scratch file can be acquired, used, and released by
remote subordination
myfile : filesys // (. . . myfile.left !v . . . myfle.rewid . . . myfile.right ?x . . .)

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6

Shared Resources

which is intended (apart from resource limitations) to have exactly the same
effect as the simple subordination of a private scratch file (X5)
(myfile : SCRATCH //
. . . myfile.left !v . . . myfile.rewind . . . myfile.right ?x . . .)



The structure of the filing system (X8) and its mode of use is a paradigm
solution of the problem of sharing a limited number of actual resources (sectors on backing store) among an unknown number of users. The users do not
communicate directly with the resources; there is an intermediary virtual resource (the SHBSCR) which they declare and use as though it were a private
subordinate process. The function of the virtual resource is twofold
(1) it provides a nice clean interface to the user; in this example, SHBSCR glues
together into a single contiguous scratch file a set of sectors scattered on
backing store.
(2) It guarantees a proper, disciplined access to the actual resources; for example, the process SHBSCR ensures that each user reads only from sectors
allocated to that user, and cannot forget to release sectors on finishing with
a scratch file.
Point (1) ensures that the discipline of Point (2) is painless.
The paradigm of actual and virtual resources is very important in the
design of resource-sharing systems. The mathematical definition of the paradigm
is quite complicated, since it uses an unbounded set of natural numbers to implement the necessary dynamic creation of new virtual processes, and of new
channels through which to communicate with them. In a practical implementation on a computer, these would be represented by control blocks, pointers to
activation records, etc. To use the paradigm effectively, it is certainly advisable
to forget the implementation method. But for those who wish to understand
it more fully before forgetting it, the following further explanation of X8 may
be helpful.
Inside a user processor a scratch file is created by remote subordination
myfile :: filesys // (. . . myfile.left !v . . . myfile.rewind . . . myfile.right ?x . . .)
By definition of remote subordination this is equivalent to
(i≥0 filesys.i.acquire →
filesys.i.left !v . . . filesys.i.rewind . . . filesys.i.right ?x . . .
filesys.i.release → SKIP )
Thus all communications between filesys and its users begin with filesys.i . . .
where i is the index of the particular instance of SHBSCR which has been acquired by a particular user on a particular occasion. Furthermore, each occa-

6.4

Multiple resources

197

sion of its use is surrounded by a matching pair of signals filesys.i.acquire and
filesys.i.release.
On the side of the subordinate process, each virtual scratchfile begins by
acquiring its user, and then continues according to the pattern of X5 and X6
(acquire → . . . left ?x . . . rewind . . . right !v . . . release . . .)
All other communications of the virtual scratch file are with the subordinate
BSTORE process, and are concealed from the user. Each instance of the virtual
scratch file is indexed by a different index i, and then named by the name
filesys. So the externally visible behaviour of each instance is
(filesys.i.acquire →
filesys.i.left ?x . . . filesys.i.rewind . . . filesys.i.right !v . . .
filesys.i.release)
This exactly matches the user’s pattern of communication as described above.
The matching pairs of acquire and release signals ensure that no user can
interfere with a scratch file that has been acquired by another user.
We turn now to communications within FILESYS between the virtual scratch
files and the backing store. These are concealed from the user, and do not even
have the name filesys attached to them. The relevant events are:
i.back.j.left .v denotes communication of block v from the ith element of
the array of scratch files to the jth sector of backing store
i.back.j.right .v denotes a communication in the reverse direction.
Each sector of the backing store behaves like COPY . After indexing with a
sector number j and naming by back, the jth sector behaves like
µ X • (back.j.left ?x → back.j.right !x → X )
After multiple labelling by natural numbers it behaves like
µ X • (i≥0 i.back.j.left ?x → (k≥0 k.back.j.right !x → X ))
This is now ready to communicate on any occasion with any element of the
array of virtual scratch files. Each individual scratch file observes the discipline of reading only from those sectors which the scratch file itself has most
recently written to.
In the above description the role of the natural numbers i and j is merely
to permit any scratch file to communicate with any sector on disc, and to communicate safely with the user that has acquired it. The indices therefore serve
as a mathematical description of a kind of crossbar switch which is used in
a telephone exchange to allow any subscriber to communicate with any other
subscriber. A crude picture of this can be drawn as in Fig 6.2.

198

6

COPY

Shared Resources

sectors of the backing store

crossbar

SHBSCR

scratch files

crossbar

users
USER

Figure 6.2
If the number of sectors in the backing store is infinite, FILESYS behaves
exactly like a similarly constructed array of simple scratch files

||i≥0 i : (acquire → (SCRATCH 4 (release → STOP )))
With a backing store of finite size, there is a danger of deadlock if the
backing store gets full at a time when all users are still writing to their scratch
files. In practice, this risk is usually reduced to insignificance by delaying
acquisition of new files when the backing store is nearly full.

6.5

Operating systems

The human users of a single large computer submit programs on decks of cards
for execution. The data for each program immediately follows it. The task of
a batch processing operating system is to share the resources of the computer
efficiently among these jobs. For this we postulate that each user’s program
is executed by a process called JOB, which inputs the cards of the program on
channel cr .right , runs the program on the data which immediately follows it
in the card reader, and outputs the results of execution on the channel lp.left .
We do not need to know about the internal structure of JOB—in early days it

6.5

Operating systems

199

used to be a FORTRAN monitor system. However, we need to rely on the fact
that it will terminate successfully within a reasonable internal after starting.
The alphabet of JOB is therefore defined
αJOB = {cr .right , lp.left , ✓}
If LPH represents the hardware of the line printer and CRH represents the
hardware of the card reader, a single job for a single user will be executed by
JOB1 = (cr : CRH // lp : LPH // JOB)
An operating system that runs just one job and then terminates is not
much use. The simplest method of sharing a single computer among many
users is to run their jobs serially, one after the other
BATCH0 = (cr : CRH // lp : LPH // JOB)
But this design ignores some important administrative details, such as separation of files output by each job, and separation of card decks containing each
job from the previous job, so that one job cannot read the cards containing its
successor. To solve these problems we use the LP process defined in 6.2 X4,
and a CR process defined below (X1)
JOBS = ∗ ((cr .acquire → lp.acquire → JOB) ;
(cr .release → lp.release → SKIP ))
BATCH1 = (cr : CR // lp : LP // JOBS )
BATCH1 is an abstract description of the simplest viable operating system,
sharing a computer among many users whose jobs are executed one at a time in
succession. The operating system expedites the transition between successive
jobs, and protects each job from possible interference by its predecessors.
Examples
X1 (A shared card reader) A special separator card is inserted at the front
of each jobfile loaded into the card reader. The card reader is acquired to read
all cards of a single jobfile and is then released. If the user attempts to read
beyond a separator card, further copies of the separator card are supplied,
without further input from the card reader. If the user fails to read up to the
separator card, the left-over cards are flushed out. Superfluous separators are
ignored. Input from the hardware is achieved by h?x.
The shared card reader needs to read one card ahead, so the value of the
buffered card is used as an index
CR = h?x →if x = separator then CR else (acquire → CRx )

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6

Shared Resources

CRx = (right !x → h?y →
if y ≠ separator then
CRy
else
µ X • (right !separator → X | release → CR)
| release →
µ X • (h?y →if y = separator then CR else X ))
After ignoring an initial subsequence of separators, this process acquires
its user and copies on its right channel the sequence of nonseparator cards
which it reads from the hardware. On detecting a separator card, its value is
replicated as necessary until the user releases the resource. But if the user
releases the reader before the separator card is reached, the remaining cards
of the deck up to the next separator card must be flushed out by reading and
ignoring them.

The BATCH1 operating system is logically complete. However, as the hardware of the central processor gets faster, it outstrips the capacity of readers
and printers to supply input and transmit output. In order to establish a match
between input, output, and processing speeds, it is necessary to use two or
more readers and printers. Since only one job at a time is being processed,
the extra readers should be occupied in reading ahead the card file for the following job or jobs, and the extra printers should be occupied in printing the
output file for the previous job or jobs. Each input file must therefore be held
temporarily in a scratch file during the period between actual input on a card
reader and its consumption by JOB; and each output file must be similarly buffered during the period between production of the lines by JOB and its actual
output on the line printer. This technique is known as spooling.
The overall structure of a spooled operating system is
OPSYS1 =
insys : INSPOOL //
outsys : OUTSPOOL //
BATCH
Here BATCH is like BATCH1, except that it uses remote subordination to acquire any one of the waiting input files, and also to acquire an output file which
is destined for subsequent printing
BATCH =
∗

(cr :: insys // lp :: outsys // JOB)

The spoolers are defined in the next two examples.

6.5

Operating systems

201

X2 (Spooled output) A single virtual printer uses a temporary scratch file
(6.4 X5) to store blocks that have been output by its using process. When the
using process signals release of the virtual printer, then an actual printer (6.4
X3) is acquired to output the content of the temporary file
VLP = (temp : SCRATCH //
µ X • left ?x → temp.left !x → X
| release → temp.rewind →
(actual :: lp //
µ Y • (temp.right ?y → actual.left !y → Y
| temp.empty → SKIP )))
The requisite unbounded array of virtual line printers is defined
VLPS = ||i≥0 i : (acquire → VLP )
Since we want the actual line printers (6.4 X3) to be used only in spooling mode,
we can declare them local to the spooling system using multiple labelling to
share them among all elements of the array VLPS as in 6.4 X8
OUTSPOOL = (N : (lp : LP2) // VLPS )

X3 (Spooled input) Input spooling is very similar to output spooling, except
that a real card reader is acquired first, and is released at the end of the input
for a single job; a user process is then acquired to execute the job, and the
contents of the cards are output to it
VCR = temp : SCRATCH //
(actual :: cr //
(µ X • actual.right ?x → if x = separator then
SKIP
else
temp.left !x → X ));
(temp.rewind → acquire →
(µ Y • (temp.right ?x → right !x → Y
| temp.empty → right !separator → Y )) 4
(release → SKIP ))

INSPOOL = (N : cr : (0 : cr || 1 : cr )) // (||i≥0 i : VCR)


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6

Shared Resources

The input and output spoolers now supply an unbounded number of virtual card readers and virtual line printers for use of the JOB process. As a
result, it is possible for two or more JOB processes to run concurrently, sharing these virtual resources. Since no communication is required between these
jobs, simple interleaving is the appropriate sharing method. This technique is
known as multiprogramming; or if more than one actual hardware processor
is used, it is known as multiprocessing. However, the logical effects of multiprogramming and multiprocessing are the same. Indeed, the operating system
defined below has the same logical specification as OPSYS1 defined above
OPSYS = insys : INSPOOL // outsys : OUTSPOOL // BATCH4
where
BATCH4 = (|||i<4 BATCH )
In mathematics, the change to multiprogramming has been remarkably simple:
historically it caused much greater agony.
In the design of the VLP process within OUTSPOOL (X2) the subordinate
process SCRATCH was used to store the lines produced by each JOB until they
are output on an actual printer. In general the output files are too large to be
held in the main store of a computer, and should be held on a backing store
as illustrated in 6.4 X8. All the temporary files should share the same backing
store, so we need to replace the subordinate process
temp : SCRATCH // . . .
within VLP by a declaration of a remote subordinate process
temp :: filesys // . . .
and then declare the filing system (6.4 X8) as a subordinate process of the
output spooler
(filesys : FILESYS // OUTSPOOL)
If the volume of card input is significant, a similar change must be made to
INSPOOL. If a separate backing store is available for this purpose, the change is
easy. If not, we will need to share a single backing store between the temporary
files of both the input and the output spoolers. This means that FILESYS must
be declared as a subordinate process, shared by multiple labelling between
both spoolers; and this involves a change in the structure of the system. We
will do this redesign in a top-down fashion, trying to re-use as many of the
previously defined modules as possible.

6.5

Operating systems

203

The operating system is composed from a batched multiprogramming system BATCH4, and an input-output system, serving as a subordinate process
OP = IOSYSTEM // BATCH4
The input-output system shares a filing system between an input spooler and
an output spooler
IOSYSTEM = SH : (filesys : FILESYS ) //
(lp : OUTSPOOL 0 || cr : INSPOOL 0 )
and SH = { lp.i | i ≥ 0 } ∪ { cr .i | i ≥ 0 }, and OUTSPOOL 0 and INSPOOL 0 are
the same as X2 and X3, except that
temp : SCRATCH
is replaced by the equivalent remote subordination
temp :: filesys
In the design of the four operating systems described in this chapter
(BATCH1, OPSYS1, OPSYS , and OP ) we have emphasised above all else the
virtue of modularity. This means that we have been able to re-use large parts
of the earlier systems within the later systems. Even more important, every
decision of detail is isolated within one or two modules of the system. Consequently, if a detail must be changed, it is very easy to identify which module must be altered, and the alteration can be confined to that single module.
Among the easy changes are
• the number of line printers
• the number of card readers
• the number of concurrent batches
But not all changes will be so easy: a change to the value of the separator card
will affect three modules, CR (X1), INSPOOL (X3) and JOB.
There are also a number of valuable improvements to the system which
would require very significant changes in its structure
1. The user jobs should also have access to the filing system, and to multiple
virtual input and output devices.
2. Users’ files should be stored permanently between the jobs they submit.
3. A method of checkpointing for rapid recovery from breakdown may be
needed.
4. If there is a backlog of jobs input but not yet executed, some method is
needed to control the order in which waiting jobs are started. This point
is taken up more fully in the next section.

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6

Shared Resources

One of the problems encountered in making these improvements is the
impossibility of sharing resources between a subordinate and its main process,
in those cases where the technique of multiple labelling is not appropriate. It
seems that a new definition of subordination is required, in which the alphabet
of the subordinate is not a subset of the alphabet of the main process. But this
is a topic which is left for future research.

6.6

Scheduling

When a limited number of resources is shared among a greater number of potential users, there will always be the possibility that some aspiring users will
have to wait to acquire a resource until some other process has released it. If
at the time of release there are two or more processes waiting to acquire it,
the choice of which waiting process will acquire the resource has been nondeterministic in all examples given so far. In itself, this is of little concern;
but suppose, by the time a resource is released again, yet another process
has joined the set of waiting processes. Since the choice between waiting processes is again nondeterministic, the newly joined process may be the lucky
one chosen. If the resource is heavily loaded, this may happen again and again.
As a result, some of the processes may happen to be delayed forever, or at least
for a wholly unpredictable and unacceptable period of time. This is known as
the problem of infinite overtaking (Section 2.5.5).
One solution to the problem is to ensure that all resources are lightly used.
This may be achieved either by providing more resources, or by rationing their
use, or by charging a heavy price for the services provided. In fact, these are
the only solutions in the case of a resource which is consistently under heavy
load. Unfortunately, even a resource which is on average lightly loaded will
quite often be heavily used for long periods (rush hours or peaks).
The problem can sometimes be mitigated by differential charging to try to
smooth the demand, but this is not always successful or even possible. During
the peaks, it is inevitable that, on the average, using processes will be subject to
delay. It is important to ensure that these delays are reasonably consistent and
predictable—you would much prefer to know that you will be served within the
hour than to wonder whether you will have to wait one minute or one day.
The task of deciding how to allocate a resource among waiting users is
known as scheduling. In order to schedule successfully, it is necessary to know
which processes are currently waiting for allocation of the resource. For this
reason, the acquisition of a resource cannot any longer be regarded as a single
atomic event. It must be split into two events
please, which requests the allocation
thankyou, which accompanies the actual allocation of the resource.
For each process, the period between the please and the thankyou is the period
during which the process has to wait for the resource. In order to identify

6.6

Scheduling

205

the requesting process, we will index each occurrence of please, thankyou
and release by a different natural number. The requesting process acquires
its number on each occasion by the same construction as remote subordination (6.4, X3)

i≥0 (res.i.please;
res.i.thankyou;
...;
res.i.release → SKIP )
A simple but effective method of scheduling a resource is to allocate it to
the process which has been waiting longest for it. This policy is known as first
come first served (FCFS) or first in first out (FIFO). It is the queueing discipline
observed by passengers who form themselves into a line at a bus stop.
In such a place as a bakery, where customers are unable or unwilling to
form a line, there is an alternative mechanism to achieve the same effect. A
machine is installed which issues tickets with strictly ascending serial numbers.
On entry to the bakery, a customer takes a ticket. When the server is ready,
he calls out the lowest ticket number of a customer who has taken a ticket
but has not yet been served. This is known as the bakery algorithm, and is
described more formally below. We assume that up to R customers can be
served simultaneously.
Example
X1 (The bakery algorithm) We need to keep the following counts
p customers who have said please
t customers who have said thankyou
r customers who have released their resources
Clearly, at all times r ≤ t ≤ p. Also, at all times, p is the number that will
be given to the next customer who enters the bakery, and t is the number of
the next customer to be served; furthermore, p − t is the number of waiting
customers, and R + r − t is the number of waiting servers. All counts are
initially zero, and can revert to zero again whenever they are all equal—say at
night after the last customer has left.
One of the main tasks of the algorithm is to ensure that there is never simultaneously a free resource and a waiting customer; whenever such a situation
arises, the very next event must be the thankyou of a customer obtaining the
resource
BAKERY = B0,0,0

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6

Shared Resources

Bp,t ,r = if 0 < r = t = p then
BAKERY
else if R + r − t > 0 ∧ p − t > 0 then
t .thankyou → Bp,t +1,r
else
(p.please → Bp+1,t ,r
| (i<t i.release → Bp,t ,r +1 ))
The bakery algorithm is due to Leslie Lamport.



Discussion

7.1

7

Introduction

The main objective of my research into communicating processes has been to
find the simplest possible mathematical theory with the following desirable
properties
1. It should describe a wide range of interesting computer applications, from
vending machines, through process control and discrete event simulation,
to shared-resource operating systems.
2. It should be capable of efficient implementation on a variety of conventional and novel computer architectures, from time-sharing computers
through microprocessors to networks of communicating microprocessors.
3. It should provide clear assistance to the programmer in his tasks of specification, design, implementation, verification and validation of complex
computer systems.
It is not possible to claim that all these objectives have been achieved in
an optimal fashion. There is always hope that a radically different approach,
or some significant change to the detailed definitions, would lead to greater
success in one or more of the objectives listed above. This chapter initiates a
discussion of some of the alternatives which I and others have explored, and
an explanation of why I have not adopted them. It also gives me an opportunity
to acknowledge the influence of the original research of other workers in the
field. Finally, I hope to encourage further research both into the foundations
of the subject and into its wider practical application.

7.2

Shared storage

The earliest proposals in the 1960s for the programming of concurrent operations within a single computer arose naturally from contemporaneous developments in computer architecture and operating systems. At that time processing power was scarce and expensive, and it was considered very wasteful

208

7

Discussion

that a processor should have to wait while communicating with slow peripheral
equipment, or even slower human beings. Consequently, cheaper specialpurpose processors (channels) were provided to engage independently in input–
output, thereby freeing the central processor to engage in other tasks. To keep
the valuable central processor busy, a timesharing operating system would ensure that there were several complete programs in the main store of the computer, and at any time several of the programs could be using the input–output
processors, while another program used the central processor. At the termination of an input–output operation, an interrupt would enable the operating
system to reconsider which program should be receiving the attention of the
central processor.
The scheme described above suggests that the central processor and all
the channels should be connected to all the main storage of the computer; and
accesses from each processor to each word of store were interleaved with those
from the other processors. Nevertheless, each program under execution was
usually a complete job submitted by a different user, and wholly independent
of all the other jobs.
For this reason, great care was expended in the design of hardware and
software to divide the store into disjoint segments, one for each program, and
to ensure that no program could interfere with the store of another. When it
became possible to attach several independent central processors to the same
computer, the effect was to increase the throughput of jobs; and if the original
operating system were well structured this could be achieved with little effect
on the code of the operating system, and even less on the programs for the
jobs which it executed.
The disadvantages of sharing a computer among several distinct jobs were
1. The amount of storage required goes up linearly with the number of jobs
executed.
2. The amount of time that each user has to wait for the result of his jobs is
also increased, except for the highest priority jobs.
It therefore seems tempting to allow a single job to take advantage of the
parallelism provided by the hardware of the computer, by initiating several
concurrent processes within the same area of storage allocated to a single
program.
7.2.1

Multithreading

The first proposal of this kind was based on a jump (go to command). If L is
a label of some place in the program, the command fork L transfers control to
the label L, and also allows control to pass to the next command in sequence.
From then on, the effect is that two processors execute the same program at the
same time; each maintains its own locus of control threading its way through
the commands. Since each locus of control can fork again, this programming
technique is known as multithreading.

7.2

Shared storage

209

Having provided a method for a process to split in two, some method is
also required for two processes to merge into one. A very simple proposal is
to provide a command join which can be executed only when two processes
execute it simultaneously. The first process to reach the command must wait
until another one also reaches it. Then only one process goes ahead to execute
the following command.
In its full generality, multithreading is an incredibly complex and errorprone technique, not to be recommended in any but the smallest programs.
In excuse, we may plead that it was invented before the days of structured
programming, when even FORTRAN was still considered to be a high-level programming language!
A variation of the fork command is still used in the UNIXTM operating
system. The fork does not mention a label. Its effect is to take a completely
fresh copy of the whole of storage allocated to the program, and to allocate the
copy to a new process. Both the original and the new process resume execution
at the command following the fork. A facility is provided for each process to
discover whether it is the parent or the offspring. The allocation of disjoint
storage areas to the processes removes the main difficulties and dangers of
multithreading, but it can be inefficient both in time and in space. This means
that concurrency can be afforded only at the outermost (most global) level of
a job, and its use on a small scale is discouraged.
7.2.2

cobegin . . . coend

A solution to the problems of multithreading was proposed by E. W. Dijkstra:
make sure that after a fork the two processors execute completely different
blocks of program, with no possibility of jumping between them. If P and Q
are such blocks, the compound command
cobegin P ; Q coend
causes P and Q to start simultaneously, and proceed concurrently until they
have both ended. After that, only a single processor goes on to execute the
following commands. This structured command can be implemented by the
unstructured fork and join commands, using labels L and J
fork L; P ; go to J ; L:Q ; J : join
The generalisation to more than two component processes is immediate and
obvious
cobegin P ; Q ; . . . ; R coend
One great advantage of this structured notation is that it is much easier
to understand what is likely to happen, especially if the variables used in each
of the blocks are distinct from the variables used in the others (a restriction
that can be checked or enforced by a compiler for a high-level language). In

210

7

Discussion

this case, the processes are said to be disjoint, and (in the absence of communication) the concurrent execution of P and Q has exactly the same effect as
their sequential execution in either order
begin P ; Q end

=

begin Q ; P end

=

cobegin P ; Q coend

Furthermore, the proof methods for establishing correctness of the parallel
composition can be even simpler than the sequential case. That is why Dijkstra’s proposal forms the basis for the parallel construct in this book. The main
change is notational; to avoid confusion with sequential composition, I have
introduced the || operator to separate the processes; and this permits the use
of simple brackets to surround the command instead of the more cumbersome
cobegin . . . coend .
7.2.3

Conditional critical regions

The restriction that concurrent processes should not share variables has the
consequence that they cannot communicate or interact with each other in any
way, a restriction which seriously reduces the potential value of concurrency.
After reading this book, the introduction of (simulated) input and output channels may seem the obvious solution; but in earlier days, an obvious
technique (suggested by the hardware of the computers) was to communicate by sharing main storage among concurrent processes. Dijkstra showed
how this could be safely achieved by critical regions (Section 6.3) protected by
mutual exclusion semaphores. I later proposed that this method should be
formalised in the notations of a high-level programming language. A group
of variables which is to be updated in critical regions within several sharing
processes should be declared as a shared resource, for example
shared n : integer
shared position : record x, y : real end
Each critical region which updates this variable is preceded by a with clause,
quoting the variable name
with n do n := n + 1;
with position do begin x := x + δx; y := y + δy end
The advantage of this notation is that a compiler automatically introduces the
necessary semaphores, and surrounds each critical region by the necessary P
and V operations. Furthermore, it can check at compile time that no shared
variable is ever accessed or updated except from within a critical region protected by the relevant semaphore.
Cooperation between processes which share store sometimes requires another form of synchronisation. For example, suppose one process updates a
variable with the objective that other processes should read the new value.

7.2

Shared storage

211

The other processes must not read the variable until the updating has taken
place. Similarly, the first process must not update the variable again until all
the other processes have read the earlier updating.
To solve this problem, a convenient facility is offered by the conditional
critical region. This takes the form
with sharedvar when condition do critical region
On entry to the critical region, the value of the condition is tested. If it is true,
the critical region is executed normally. But if the condition is false, this entry
into the critical region is postponed, so that other processes are permitted to
enter their critical regions and update the shared variable. On completion of
each such update, the condition is retested. If it has become true, the delayed
process is permitted to proceed with its critical region; otherwise that process
is suspended again. If more than one delayed process can proceed, the choice
between them is arbitrary.
To solve the problem of updating and reading a message by many processes, declare as part of the resource an integer variable to count the number
of processes that must read the message before it is updated again
shared message : record count : integer ; content : . . . end
message.count := 0;
The updating process contains a critical region
with message when count = 0 do
begin content := . . . ;
...;
count := number of readers
end
Each reading process contains a critical region
with message when count > 0 do
begin my copy := content ; count = count − 1 end
Conditional critical regions may be implemented by means of semaphores.
Compared with direct use of synchronisation semaphores by the programmer,
the overhead of conditional critical regions may be quite high, since the conditions of all processes waiting to enter the region must be retested on every exit
from the region. Fortunately, the conditions do not have to be retested more
frequently than that, because restrictions on access to shared variables ensure
that the condition tested by a waiting process can change value only when the
shared variable itself changes value. All other variables in the condition must

212

7

Discussion

be private to the waiting process, which obviously cannot change them while
it is waiting.

7.2.4

Monitors

The development of monitors was inspired by the class of SIMULA 67, which
was itself a generalisation of the procedure concept of ALGOL 60. The basic
insight is that all meaningful operations on data (including its initialisation and
perhaps also its finalisation) should be collected together with the declaration
of the structure and type of the data itself; and these operations should be
invoked by procedure calls whenever required by the processes which share
the data. The important characteristic of a monitor is that only one of its
procedure bodies can be active at one time; even when two processes call a
procedure simultaneously (either the same procedure or two different ones),
one of the calls is delayed until the other is completed. Thus the procedure
bodies act like critical regions protected by the same semaphore.
For example a very simple monitor acts like a count variable. In notations
based on PASCAL PLUS it takes the form
1

monitor count ;

2

var n : integer ;

3

procedure∗ up; begin n := n + 1 end ;

4

procedure∗ down; when n > 0 do begin n := n − 1 end ;

5

function∗ grounded : Boolean; begin grounded := (n = 0) end ;

6

begin n := 0;

7

· · ·;

8

if n ≠ 0 then print (n)

9

end

Line 1
2

declares the monitor and gives it the name count .
declares the shared variable n local to the monitor; it
is inaccessible except within the monitor itself.





3 





4 





5 

declare three procedures within their bodies; the
asterisks ensure they can be called from the program
which uses the monitor;

6

the monitor starts execution here;

7

the three fat dots are an inner statement, which stands
for the block that is going to use this monitor;

8

the final value of n (if non-zero) is printed on exit from
the user block.

7.2

Shared storage

213

A new instance of this monitor can be declared local to a block P
instance rocket : count ; P
Within the block P , the starred procedures may be called by commands
rocket .up; . . . rocket .down; . . . ; if rocket .grounded then . . .
However an unstarred procedure or variable such as n cannot be accessed
from within P and observance of this restriction is enforced by a compiler.
The mutual exclusion inherent in the monitor ensures that the procedure of
the monitor can be safely called by any number of processes within P , and
there is no danger of interference in updating n. Note that an attempt to call
rocket .down when n = 0 will be delayed until some other process within P
calls rocket .up. This ensures that the value of n can never go negative.
The effect of the declaration of an instance of a monitor is explained by
a variation of the copy rule for procedure calls in ALGOL 60. Firstly, a copy
is taken of the text of the monitor; the using block P is copied in place of the
three dots inside the monitor, and all local names of the monitor are prefixed
by the name of the instance, as shown below
rocket .n : integer ;
procedure rocket .up; begin rocket .n := rocket .n + 1 end ;
procedure rocket .down;
when rocket .n > 0 do begin rocket .n := rocket .n − 1 end ;
function rocket .grounded : Boolean;
begin rocket .grounded := (rocket .n = 0) end ;
begin rocket .n := 0;
· · ·;
if rocket .n ≠ 0 then print (rocket .n)
end
Note how the copy rule has made it impossible for the user process to forget
to initialise the value of n, or to forget to print its final value of n, or to forget
to print its final value when necessary.
The inefficiency of repeated testing of entry conditions has led to the
design of monitors with a more elaborate scheme of explicit waiting and explicit signalling for resumption of waiting processes. These schemes even allow
a procedure call to suspend itself in the middle of its execution, after automatically releasing exclusion of the suspended process. In this way, a number
of ingenious scheduling techniques can be efficiently implemented; but I now
think that the extra complexity is hardly worthwhile.

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7

7.2.5

Discussion

Nested monitors

A monitor instance can be used like a semaphore to protect a single resource
such as a line printer which must not be used by more than one process at a
time. Such a monitor could be declared
monitor singleresource;
var free : Boolean;
procedure ∗ acquire; when free do free := false;
procedure ∗ release; begin free := true end ;
begin free := true; · · · end
However, the protection afforded by this monitor can be evaded by a process
which uses the resource without acquiring it, or frustrated by one which forgets
to release it afterwards. Both of these dangers can be averted by a construction
similar to that of the virtual resource (6.4 X4). This takes the form of a monitor
declared locally within the actual resource monitor shown above. The name
of the virtual resource is starred to make it accessible for declaration by user
processes. However, the stars are removed from ∗ acquire and ∗ release, so
that these can be used only within the virtual resource monitor, and cannot be
misused by other processes.
monitor singleresource;
free : Boolean;
procedure acquire;
when free do free := false;
procedure release;
begin
free := true;
end
monitor ∗ virtual;
procedure ∗ use(l : line); begin . . . end ;
begin
acquire; · · ·; release
end
begin
free := true; · · ·;
end

7.2

Shared storage

215

An instance of this monitor is declared
instance lpsystem : singleresource; P
A block within P which requires to output a file to a line printer is written
instance mine : lpsystem.virtual;
begin . . . mine.use(l1); . . . mine.use(l2); . . . end
The necessary acquisition and release of the line printer are automatically inserted by the virtual monitor before and after this user block, in a manner
which prevents antisocial use of the line printer. In principle, it would be possible for the using block to split into parallel processes, all of which use the
instance mine of the virtual monitor, but this is probably not the intention here.
A monitor which is to be used only by a single process is known in PASCAL
PLUS as an envelope, and it can be implemented more efficiently without exclusion or synchronisation; and the compiler checks that it is not inadvertently
shared.
The meaning of these instance declarations can be calculated by repeated
application of the copy rule, with the result shown below
var lpsystem.free : Boolean;
procedure lpsystem.acquire;
when lpsystem.free do lpsystem.free := false;
procedure lpsystem.release; begin lpsystem.free := true end ;
begin lpsystem.free := true
..
.
begin
procedure mine.lpsystem.use(l : line); begin . . . end ;
lpsystem.acquire;
. . . mine.lpsystem.use(l1);
. . . mine.lpsystem.use(l2);
lpsystem.release;
end ;
..
.
end
The explicit copying shown here is only for initial explanation; a more experienced programmer would never wish to see the expanded version, or even
think about it.

216

7

Discussion

These notations were used in 1975 for the description of an operating system similar to that of Section 6.5; and they were later implemented in PASCAL
PLUS. Extremely ingenious effects can be obtained by a mixture of starring and
nesting; but the PASCAL- and SIMULA-based notations seem rather clumsy, and
explanations in terms of substitution and renaming are rather difficult to follow. It was Edsger W. Dijkstra’s criticisms of these aspects that first impelled
me towards the design of communicating sequential processes.
However, it is now clear from the constructions of Section 6.5 that the
control of sharing requires complication, whether it is expressed within the
conceptual framework of communicating processes or within the copy-rule and
procedure-call semantics of PASCAL PLUS. The choice between the languages
seems partly a matter of taste, or perhaps efficiency. For implementation of
an operating system on a computer with shared main storage, PASCAL PLUS
probably has the advantage.
7.2.6

AdaTM

Facilities offered for concurrent programming in Ada are an amalgam of the
remote procedure call of PASCAL PLUS, with the less structured form of communication by input and output. Processes are called tasks, and they communicate by call statements (which are like procedure calls with output and input
parameters), and accept statements (which are like procedure declarations in
their syntactic form and in their effect). A typical accept statement might be
accept put (V : in integer ; PREV : out integer ) do
PREV := K ; K := V end
A corresponding call might be
put (37 , X )
The identifier put is known as an entry name.
An accept and a call statement with the same name in different tasks as
executed when both processes are ready to execute them together. The effect
is as follows
1. Input parameters are copied from the call to the accepted process.
2. The body of the accept statement is executed.
3. The values of the output parameters are copied back to the call.
4. Then both tasks continue execution at their next statements.
The execution of the body of an accept is known as a rendezvous, since the
calling and accepting task may be thought to be executing it together. The
rendezvous is an attractive feature of Ada, since it simplifies the very common
practice of alternating output and input, without much complicating the case
when only input or only output is required.

7.2

Shared storage

217

The Ada analogue of  is the select statement which takes the form
select
accept get (v : out integer ) do v := B[i] end ; i := i + 1; . . .
or accept put (v : in integer ) do B[j] := v end ; j := j + 1; . . .
or . . .
end select
Exactly one of the alternatives separated by or will be selected for execution,
depending on the choice made by the calling task(s). The remaining statements of the selected alternative after the end of the accept are executed on
completion of the rendezvous, concurrently with continuation of the calling
task. Selection of an alternative can be inhibited by falsity of a preceding when
condition, for example
when not full ⇒ accept . . .
This achieves the effect of a conditional critical region.
One of the alternatives in a select statement may begin with a delay instead
of an accept. This alternative may be selected if no other alternative is selected
during elapse of a period greater than the specified number of seconds’ delay.
The purpose of this is to guard against the danger that hardware or software
error might cause the select statement to wait forever. Since our mathematical
model deliberately abstracts from time, a delay cannot be faithfully represented, except by allowing wholly nondeterministic selection of the alternative
beginning with the delay.
One of the alternatives in a select statement may be the statement terminate.
This alternative is selected when all tasks which might call the given task have
terminated; and then the given task terminates too. This is not as convenient
as the inner statement of PASCAL PLUS, which allows the monitor to tidy up
on termination.
A select statement may have an else clause, which is selected if none of
the other alternatives can be selected immediately, either because all the when
conditions are false, or because there is no corresponding call already waiting
in some other task. This would seem to be equivalent to an alternative with
zero delay.
A call statement may also be protected against arbitrary delay by a delay
statement or an else clause. This may lead to some inefficiencies in implementation on a distributed network of processors.
Tasks in Ada are declared in much the same way as subordinate processes
(Section 4.5); but like monitors in PASCAL PLUS, each one may serve any number of calling processes. Furthermore, the programmer must arrange for the
task to terminate properly. The definition of a task is split into two parts,
its specification and its body. The specification gives the task name and the
names and parameter types of all entries through which the task may be called.

218

7

Discussion

This is the information required by the writer of the program which uses the
task, and by the compiler of that program. The body of the task defines its
behaviour, and may be compiled separately from the using program.
Each task in Ada may be given a fixed priority. If several tasks are capable of making progress, but only a lesser number of processors are available,
the tasks with lower priority will be neglected. The priority of execution of a
rendezvous is the higher of the priorities of the calling and of the accepting
tasks. The indication of priority is called a pragma; it is intended to improve
critical response times compared with non-critical ones, and it is not intended
to affect the logical behaviour of the program. This is an excellent idea, since it
separates concern for abstract logical correctness from problems of real time
response, which can often be more easily solved by experiment or by judicious
selection of hardware.
Ada offers a number of additional facilities. It is possible to test how many
calls are waiting on an entry. One task may abruptly terminate (abort ) another
task, and all tasks dependent upon it. Tasks may access and update shared
variables. The effect of this is made even more unpredictable by the fact that
compilers are allowed to delay, reorder, or amalgamate such updating just as
if the variable were not shared. There are some additional complexities and
interaction effects with other features which I have not mentioned.
Apart from the complexities listed in the preceding paragraph, tasking in
Ada seems to be quite well designed for implementation and use on a multiprocessor with shared storage.

7.3

Communication

The exploration of the possibility of structuring a program as a network of
communicating processes was also motivated by spectacular progress in the
development of computer hardware. The advent of the microprocessor rapidly
reduced the cost of processing power by several orders of magnitude. However, the power of each individual microprocessor was still rather less than
that of a typical computer of the traditional and still expensive kind. So it
would seem to be most economical to obtain greater power by use of several
microprocessors cooperating on a single task. These microprocessors would
be cheaply connected by wires, along which they could communicate with each
other. Each microprocessor would have its own local main store, which it could
access at high speed, thus avoiding the expensive bottleneck that tends to result from allowing only one processor at a time to access a shared store.
7.3.1

Pipes

The simplest pattern of communication between processing elements is just
single-directional message passing between each process and its neighbour in
a linear pipe, as described in Section 4.4. The idea was first propounded by
Conway who illustrated it by examples similar to 4.4 X2 and X3, except that all

7.3

Communication

219

components of a pipe were expected to terminate successfully instead of running forever. He proposed that the pipe structure should be used for writing
a multiple-pass compiler for a programming language. On a computer with
adequate main storage, all the passes are simultaneously active, and control
transfers between the passes together with the messages, thus simulating parallel execution. On a computer with less main storage, only one pass at a time
is active, and sends its output to a file in backing store. On completion of each
pass, the next pass starts, taking its input from the file produced by its predecessor. However, the final result of the compiler is exactly the same, in spite
of the radical differences in the manner of execution. It is characteristic of a
successful abstraction in programming that it permits several different implementation methods which are efficient in differing circumstances. In this case,
Conway’s suggestion could have been very valuable for software implementation on a computer range offering widely varying options of store size.
The pipe is also the standard method of communication in the UNIXTM
operating system, where the notation ‘|’ is used instead of ‘>>’.

7.3.2

Multiple buffered channels

The pipe construction allows a linear chain of processes to communicate in a
single direction only, and it does not matter whether the message sequence is
buffered or not. The natural generalisation of the pipe is to permit any process
to communicate with any other process in either direction; and at first sight it
seems equally natural to provide buffering on all the communication channels.
In the design of the RC4000 operating system, a facility for buffered communication was implemented in the kernel; and was used for communication between
modules providing services at a higher level. On a grander scale, a store-andforward packet switching network, like ARPAnet in the United States, inevitably
interposes buffering between the source and destination of messages.
When the pattern of communication between processes is generalised from
a linear chain to a network that may be cyclic, the presence or absence of buffering can make a vital difference to the logical behaviour of the system. The
presence of buffering is not always favourable: for example, it is possible to
write a program that can deadlock if the length of the buffer is allowed to
grow greater than five, as well as a different program that will deadlock unless the buffer length is allowed to exceed five. To avoid such irregularities,
the length of all buffers should be unbounded. Unfortunately, this leads to
grave problems of implementation efficiency when main storage is filled with
buffered messages. The mathematical treatment is also complicated by the
fact that every network is an infinite-state machine, even when the component processes are finite. Finally, for rapid and controllable interaction between
humans and computers, buffers only stand in the way, since they can interpose delay between a stimulus and a response. If something goes wrong in
processing a buffered stimulus, it is much more difficult to trace the fault
and recover from it. Buffering is a batch-processing technique, and should be

220

7

Discussion

avoided wherever fast interactions are more important than heavy processor
utilisation.
7.3.3

Functional multiprocessing

A deterministic process may be defined in terms of a mathematical function
from its input channels to its output channels. Each channel is identified with
the indefinitely extensible sequence of messages which pass along it. Such
functions are defined in the usual way by recursion on the structure of the input
sequences, except that the case of an empty input sequence is not considered.
For example, a process which outputs the result of multiplying each number
input by n is defined
prodn (left ) = hn × left0 i _ prodn (left 0 )
A function which takes two sorted streams (without duplicates) as parameters
and outputs their sorted merge (suppressing duplicates) is defined
merge2(left1, left2) =
if left10 < left20 then
hleft1 i _ merge2(left10 , left2)
0

else if left20 < left10 then
hleft2 i _ merge2(left1, left2 0 )
0

else
hleft20 i _ merge2(left10 , left2 0 )
An acyclic network can be represented by a composition of such functions. For
example, a function which merges three sorted input streams can be defined
merge3(left1, left2, left3) = merge2(left1, merge2(left2, left3))
Figure 7.1 shows a connection diagram of this function.

merge3
left1
left2

merge2
merge2

left3

Figure 7.1

7.3

Communication

221

A cyclic network can be constructed by a set of mutually recursive equations. For example, consider the problem attributed by Dijkstra to Hamming,
namely to define a function which outputs in ascending sequence all numbers
which have only 2, 3, and 5 as non-trivial factors. The first such number is 1;
and if x is any such number, so are 2 × x, 3 × x, and 5 × x. We therefore use
three processes prod2 , prod3 , and prod5 to generate these products, and feed
them back into the process merge3, which ensures they are eventually output
in ascending order (Figure 7.2).

prod2
merge3
prod3

1

prod5

Figure 7.2

The function which outputs the desired result has no inputs; it is simply
defined
Hamming =
h1i _ merge3(prod2 (Hamming), prod3 (Hamming), prod5 (Hamming))
The functional approach to multiprocessor networks is very different from
that taken in this book in the following respects
1. A general implementation requires unbounded buffering on all channels.
2. Each value output into the buffer must be retained in the buffer until all the
inputting processes have taken it, which they may do at different times.
3. There is no possibility for a process to wait for one of two inputs, whichever
one arrives first.
4. The processes are all deterministic.
Recent research has been directed towards reducing the inefficiency of 1
and 2, and towards relaxing the restrictions 3 and 4.

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7

7.3.4

Discussion

Unbuffered communication

For many years now, I have chosen to take unbuffered (synchronised) communication as basic. My reasons have been
1. It matches closely a physical realisation on wires which connect processing
agents. Such wires cannot store messages.
2. It matches closely the effect of calling and returning from subroutines
within a single processor, copying the values of the parameters and the
results.
3. When buffering is wanted, it can be implemented simply as a process; and
the degree of buffering can be precisely controlled by the programmer.
4. Other disadvantages of buffers have been mentioned at the end of Section 7.3.2.
Of course, none of these arguments carry absolute conviction. For example, if buffered communication were taken as primitive, this would make
no logical difference in the common case of alternation of subroutine call and
return; and synchronisation can be achieved in all other cases by following
every output by input of an acknowledgement, and every input by output of
an acknowledgement.

7.3.5

Communicating sequential processes

This was the title of my first complete exposition of a programming language
based on concurrency and communication. That early proposal differs from
this book in two significant respects.
(1) Parallel composition
Channels are not named. Instead, the component processes of a parallel construction have unique names, prefixed to them by a pair of colons
[a :: P || b :: Q || . . . || c : R]
Within process P , the command b!v outputs the value v to the process named
b. This value is input by a command a?x occurring within the process Q . The
process names are local to the parallel command in which they are introduced,
and communications between the component processes are hidden.
The advantage of this scheme is that there is no need to introduce into
the language any concept of channel or channel declaration. Furthermore, it
is logically impossible to violate the restriction that a channel is between two
processes only, and one of them uses it for input and the other for output. But
there are some disadvantages, both in practice and in theory.

7.3

Communication

223

1. A serious practical disadvantage is that a subordinate process needs to
know the name of its using process; this complicates construction of libraries of subordinate processes.
2. A disadvantage in the underlying mathematics is that parallel composition is an operation with a variable number of parameters and cannot be
reduced to a binary associative operator like ||.
(2) Automatic termination
In the early version, all processes of a parallel command were expected to
terminate. The reason for this was the hope that the correctness of a process
could be specified in the same way as for a conventional program by a postcondition, i.e., a predicate intended to be true on successful termination. (That
hope was never fulfilled, and other proof methods (Section 1.10) now seem
more satisfactory). The obligation that a subordinate process should terminate
imposes an awkward obligation on the using process to signal its termination
to all subordinates. An ad hoc convention was therefore introduced. A loop
of the form
∗

[a?x → P  b?x → Q  . . .]

terminates automatically on termination of all of the processes a, b, …from
which input is requested. This enables the subordinate process to complete
any necessary finalisation code before termination—a feature which had proved
useful in SIMULA and PASCAL PLUS.
The trouble with this convention is that it is complicated to define and
implement; and methods of proving program correctness seem no simpler
with it than without. Now it seems to me better (as in Section 4.5) to relax the
restriction that simple subordinate processes must terminate; and take other
measures (Section 6.4) in the more complicated cases.
7.3.6

Occam

In contrast to Ada, occam is a very simple programming language, and very
closely follows the principles expounded in this book. The most immediately
striking differences are notational; occam syntax is designed to be composed
at a screen with the aid of a syntax checking editor; it uses prefix operators
instead of infix, and it uses indentation instead of brackets.
SEQ
P
Q
R

for (P ; Q ; R)

224

7

Discussion

for (P || Q || R)

PAR
P
Q
R

for (c?x → P  d?y → Q )

ALT
c?x
P
d?y
Q

for (P <| B >
| Q)

IF
B
P
NOT B
Q
WHILE B

for (B ∗ P )

P
The ALT construct corresponds to the Ada select statement, and offers a
similar range of options. Selection of an alternative may be inhibited by falsity
of a Boolean condition B
B & c?x
P
The input may be replaced by a SKIP , in which case the alternative may be
selected whenever the Boolean guard is true; or it may be replaced by a wait,
which allows the alternative to be selected after elapse of a specified interval.
The occam language does not have any distinct notations for pipes (Section 4.4), subordinate processes (Section 4.5), or shared processes (Section 6.4).
All the required patterns of communication must be achieved by explicit identity of channel names. To help in this, procedures may be declared with channels as parameters. For example, the simple copying process may be declared
PROC copy(CHAN left , right ) =
WHILE TRUE
VAR x :
SEQ
left ?x
right !x :

7.3

Communication

225

The double buffer COPY >>COPY can now be constructed
CHAN mid :
PAR
copy(left , mid)
copy(mid, right )
A chain of n buffers may be constructed using an array of n channels and an
iterative form of the parallel construct, which constructs n − 2 processes, one
for each value of i between 0 and n − 3 inclusive
CHAN mid[n − 1] :
PAR
copy(left , mid[0])
PAR i = [0 FOR n − 2]
copy(mid[i], mid[i + 1])
copy(mid[n − 2], right )
Because occam is intended to be implemented with static storage allocation
on a fixed number of processors, the value n in the above example must be a
constant. For the same reason, recursive procedures are not allowed.
A similar construction may be used to achieve the effect of subordinate
processes, for example
PROC double(left , right ) =
WHILE TRUE
VAR x :
SEQ
left ?x
right !(x + x) :
This may be declared subordinate to a single using process P
CHAN doub.left , doub.right :
PAR
double(doub.left , doub.right )
P
Inside P a number is doubled by
doub.left !4; doub.right ?y . . .
Processes may be shared using arrays of channels (with one element per
using process) and an iterative form of the ALT construction. For example,
take an integrator, which after each new input outputs the sum of all numbers

226

7

Discussion

it has input so far
CHAN add[n], integral[n] :
PAR
VAR sum, x :
SEQ
sum := 0
ALT i = [0 FOR n]
add[i]?x
SEQ
sum := sum + x
integral[i]!sum
PAR i = [0 FOR n]
. . . user processes . . .
Like Ada, occam allows a programmer to assign relative priorities to processes combined in parallel. This is done by using PRI PAR instead of PAR;
and the earlier processes in the list have higher priority. The screen-editing
facilities provided with the language facilitate reordering of processes when necessary. A similar option is offered for the ALT construction, namely PRI ALT .
This ensures that if more than one alternative is ready for immediate selection, the textually earliest will be chosen—otherwise the effect is the same as
the simple ALT . Of course, the programmer is urged to ensure that his programs are logically correct, independent of the assignment of priorities.
There are also facilities for distributing processes among distinct processors, and for specifying which physical pins on each processor are to be
used for each relevant channel of the occam program, and which pin is to be
used for loading the code of the program itself.

7.4

Mathematical models

Recognition of the idea that a programming language should have a precise
mathematical meaning or semantics dates from the early 1960s. The mathematics provides a secure, unambiguous, precise and stable specification of the
language to serve as an agreed interface between its users and its implementors. Furthermore, it gives the only reliable grounds for a claim that different
implementations are implementations of the same language. So mathematical semantics are as essential to the objective of language standardisation as
measurement and counting are to the standardisation of nuts and bolts.
In the later 1960s an even more important role for mathematical semantics
was recognised, that of assisting a programmer to discharge his obligation of
establishing correctness of his program. Indeed R. W. Floyd suggested that the
semantics be formulated as a set of valid proof rules, rather than as an explicit

7.4

Mathematical models

227

mathematical model. This suggestion has been adopted in the specification of
PASCAL and Euclid and Gypsy.
The early design of Communicating Sequential Processes (Section 7.3.5)
had no mathematical semantics, and it left open a number of important design
questions, for example
1. Is it permissible to nest one parallel command inside another?
2. If so, is it possible to write a recursive procedure which calls itself in
parallel?
3. Is it theoretically possible to use output commands in guards?
The mathematical model given in this book answers “yes” to all these questions.

7.4.1

A calculus of communicating systems

The major breakthrough in the mathematical modelling of concurrency was
made by Robin Milner. The objective of his investigation was to provide a
framework for constructing and comparing different models, at different levels
of abstraction. So he starts with the basic syntax of expressions intended to
denote processes, and he defines a series of equivalences between the expressions, of which the most important are
strong equivalence
observational equivalence
observational congruence
Each equivalence defines a different model of concurrency. The initials CCS
usually refer to the model obtained by adopting observational congruence as
the definition of equality between processes.
The basic notations of CCS are illustrated by the following correspondences
a.P

corresponds to a → P

(a.P ) + (b.Q )

corresponds to (a → P | b → Q )

NIL

corresponds to STOP

More important than these notational distinctions are differences in the treatment of hiding. In CCS, there is a special symbol τ which stands for the occurrence of a hidden event or an internal transition. The advantage of retaining
this vestigial record of a hidden event is that it can be freely used to guard recursive equations and so ensure that they have unique solutions, as described
in Section 2.8.3. A second (but perhaps less significant) advantage is that processes which can engage in an unbounded sequence of τs do not all reduce to

228

7

Discussion

CHAOS ; so possibly useful distinctions can be drawn between divergent processes. However, CCS fails to distinguish a possibly divergent process from
one that is similar in behaviour but nondivergent. I expect this would make
efficient implementation of the full CCS language impossible.
CCS does not include u as a primitive operator. However, nondeterminism
can be modelled by use of τ, for example
(τ.P ) + (τ.Q )

corresponds to P u Q

(τ.P ) + (a.Q )

corresponds to P u (P  (a → Q ))

But these correspondences are not exact, because in CCS nondeterminism
defined by τ would not be associative, as shown by the fact that the trees
in Figure 7.3 are distinct.

P

R

Q

R

P

P

Q

Q

R

Figure 7.3
Furthermore, prefixing does not distribute through nondeterminism, because the trees in Figure 7.4 are distinct when P ≠ Q

a
a

a
a

P

Q

P

Q

a

P

Q

Figure 7.4
These examples show that CCS makes many distinctions between processes which would be regarded as identical in this book. The reason for this
is that CCS is intended to serve as a framework for a family of models, each
of which may make more identifications than CCS but cannot make less. To
avoid restricting the range of models, CCS makes only those identifications
which seem absolutely essential. In the mathematical model of this book we
have pursued exactly the opposite goal—we have made as many identifications
as possible, preserving only the most essential distinctions. We have therefore
a far richer set of algebraic laws. It is hoped that these laws will be practic-

7.4

Mathematical models

229

ally useful in reasoning about designs and implementations; in particular, they
permit more transformations and optimisations than CCS.
The basic concurrent combinator of CCS is denoted by the single bar |. It
is rather more complicated than the || combinator, in that it includes aspects
of hiding, nondeterminism and interleaving as well as synchronisation.
Each event in CCS has two forms, either simple (a) or overbarred (a).
When two processes are put together to run concurrently, synchronisation occurs only when one process engages in a barred event and the other engages in
the corresponding simple event. Their joint participation in such an event is
hidden by immediate conversion to τ. However, synchronisation is not compulsory; each of the two events can also occur visibly and independently as an
interaction with the outer environment. Thus in CCS
(a.P ) | (b.Q ) = a.(P | (b.Q )) + b.((a.P ) | Q )
(a.P ) | (a.Q ) = a.(P | (a.Q )) + a.((a.P ) | Q )
(a.P ) | (a.Q ) = τ.(P | Q ) + a.(P | (a.Q )) + a.((a.P ) | Q )
Consequently, only two processes can engage in a synchronisation event; if
more than two processes are ready, the choice of which pair succeeds is nondeterministic
(a.P ) | (a.Q ) | (a.R) = τ.(P | (a.Q ) | R) +
τ.((a > P ) | Q | R) +
a.(P | (a.Q ) | (a.R)) +
a.((a.P ) | Q | (a.R)) +
a.((a.P ) | (a.Q ) | R)
Because of the extra complexity of the parallel operator, there is no need
for a concealment operator. Instead, there is a restriction operator \, which
simply prevents all occurrence of the relevant events, and removes them from
the alphabet of the process, together with their overbarred variant. The effect
is illustrated by the following laws of CCS
(a.P ) \ {a} = (a.P ) \ {a} = NIL
(P + Q ) \ {a} = (P \ {a}) + (Q \ {a})
((a.P ) | (a.Q )) \ {a} = τ.((P | Q ) \ {a})
((a.P ) | (a.Q ) | (a.R)) \ {a} = τ.((P | (a.Q ) | R) \ {a}) +
τ.(((a.P ) | Q | R) \ {a})
The last law above illustrates the power of the CCS parallel combinator in
achieving the effect of sharing the process (a.R) among two using processes
(a.P ) and (a.Q ). It was an objective of CCS to achieve the maximum expressive
power with as few distinct primitive operators as possible. This is the source
of the elegance and power of CCS, and greatly simplifies the investigation of
families of models defined by different equivalence relations.

230

7

Discussion

In this book, I have taken a complementary approach. Simplicity is sought
through design of a single simple model, in terms of which it is easy to define as
many operators as seem appropriate to investigate a range of distinct concepts.
For example, the nondeterministic choice u introduces nondeterminism in its
purest form, and is quite independent of environmental choice represented by
(x : B → P (x)). Similarly, || introduces concurrency and synchronisation, quite
independent of nondeterminism or hiding, each of which is represented by a
distinct operator. The fact that the corresponding concepts are distinct is perhaps indicated by the simplicity of the algebraic laws. A reasonably wide range
of operators seems to be needed in the practical application of useful mathematical theories. Minimisation of operator sets is also useful, more especially
in theoretical investigations.
Milner has introduced a form of modal logic to specify the observable
behaviour of a process. The modality
a S

describes a process which may do a and then behave as described by S , and
the dual
a S

describes a process that if it starts with a must behave like S afterwards. A
calculus of correctness is defined which permits a proof that a process P meets
specification S , a fact which is expressed in the traditional logical notation
P îS
The calculus is very different from that governing the sat notation, because it
is based on the structure of the specification rather than the structure of the
programs. For example, the rule for negation is
If it is not true that P î F , then P î ¬ F
This means that the whole process P must be written before the proof of its
correctness starts. In contrast, the use of sat permits proof of the correctness
of a compound process to be constructed from a proof of correctness of its
parts. Furthermore, there is never a need to prove that a process does not
satisfy its specification. Modal logic is a subject of great theoretical interest,
but in the context of communicating processes it does not yet show much
promise for useful application.
In general, equality in CCS is a strong relation, since equal processes must
resemble each other both in their observable behaviour and in the structure of
their hidden behaviour. CCS is therefore a good model for formulating and exploring various weaker definitions of equivalence, which ignore more aspects
of the hidden behaviour. Milner accomplishes this by introducing the concept
of observational equivalence. This involves definition of the set of observations or experiments that may be made on a process; then two processes are

7.4

Mathematical models

231

equivalent if there is no observation that can be made of one of them but not
the other—a nice application of the philosophical principle of the identity of
indiscernibles. The principle was taken as the basis of the mathematical theory in this book, which equates a process with the set of observations that can
be made of its behaviour. A sign of the success of the principle is that two
processes P and Q are equivalent if and only if they satisfy the same specifications
∀S • P î S ≡ Q î S
Unfortunately, it doesn’t always work as simply as this. If two processes
are to be regarded as equal, the result of transforming them by the same function should also be equal, i.e.,
(P ≡ Q ) ⇒ (F (P ) ≡ F (Q ))
Since τ is supposed to be hidden, a natural definition of an observation would
lead to the equivalence
(τ.P ) ≡ P
However, (τ.P + τ.NIL) should not be equivalent to (P + NIL), which equals P ,
since the former can make a nondeterministic choice to deadlock instead of
behaving like P .
Milner’s solution to this problem is to use the concept of congruence in
place of equivalence. Among the experiments which can be performed on the
process P is to place it in an environment F (P ) (where F is composed of other
processes by means of operators in the language), and then to observe the
behaviour of this assembly. Processes P and Q are (observationally) congruent
if for every F expressible in the language the process F (P ) is observationally
equivalent to F (Q ). According to this definition τ.P is not congruent to P .
The discovery of a full set of laws of congruence is a significant mathematical
achievement.
The need for the extra complexity of observational congruence is due to
the inability to make sufficiently penetrating observations of the behaviour of
P , without placing it in an environment F (P ). That is why we have had to introduce the concept of a refusal set, rather than just a refusal of a single event.
The refusal set seems to be the weakest kind of observation that efficiently
represents the possibility of nondeterministic deadlock; and it therefore leads
to a much weaker equivalence, and to a more powerful set of algebraic laws
than CCS.
The description given above has overemphasised the differences of CCS,
and has overstated the case for practical application of the approach taken
in this book. The two approaches share their most important characteristic,
namely a sound mathematical basis for reasoning about specifications, designs
and implementations; and either of them can be used for both theoretical investigations and for practical applications.

Select Bibliography

Conway, M. E. ‘Design of a Separable Transition-Diagram Compiler,’ Comm.
ACM 6 (7), 8–15 (1963)
The classical paper on coroutines.
Hoare, C. A. R. ‘Monitors: An Operating System Structuring Concept,’ Comm.
ACM 17 (10), 549–557 (1974)
A programming language feature to aid in construction of operating
systems.
Hoare, C. A. R. ‘Communicating Sequential Processes,’ Comm. ACM 21 (8),
666-677 (1978)
A programming language design—an early version of the design
propounded in this book.
Milner, R. A Calculus of Communicating Systems, Lecture Notes in Computer
Science 92, Springer Verlag, New York (1980)
A clear mathematical treatment of the general theory of concurrency and
synchronisation.
Kahn, G. ‘The Semantics of a simple language for Parallel Programming,’ in
Information Processing, 74, North Holland, Amsterdam pp. 471–475
(1984)
An elegant treatment of functional multiprogramming.
Welsh, J. and McKeag, R. M. Structured System Programming, Prentice–Hall,
London, pp. 324 (1980)
An account of PASCAL PLUS and its use in structuring an operating
system and compiler.
Filman, R. E. and Friedman, D. P. Coordinated Computing, Tools and
Techniques for Distributed Software, McGraw–Hill pp. 370 (1984)
A useful survey and further bibliography.

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Dahl, O–J ‘Hierarchical Program Structures,’ in Structured Programming,
Academic Press, London pp. 175–220 (1982)
An introduction to the influential ideas of SIMULA 67.
(INMOS Ltd.) occamTM Programming Manual Prentice–Hall International, pp.
100 (1984).
(ANSI/MIL–STD 1815A) AdaTM Reference Manual
Chapter 9 describes the tasking feature.
Brookes, S. D., Hoare, C. A. R., and Roscoe, A. W. ‘A Theory of
Communicating Sequential Processes,’ Journal ACM 31 (7), 560–599
(1984)
An account of the mathematics of nondeterminism processes, but
excluding divergences.
Brookes, S. D. and Roscoe, A. W. ‘An Improved Failures Model for
Communicating Sequential Processes,’ in Proceedings NSF–SERC Seminar
on Concurrency, Springer Verlag, New York, Lecture Notes in Computer
Science (1985)
The improvement is the addition of divergences.

Index

acc, 179
accept, 226
accepting process, 163
acknowledgement, 150
acquire, 192
activation records, 205
actual resource, 224
Ada, 226
after, 33
algebraic laws, 15, 91
ALGOL 60, 222
alphabet, 1, 3
alternation, 173
alternative, 7
angelic nondeterminism, 89
arcs, 12
ARPAnet, 229
assignment, 176
backing storage, 200
backtracking, 164
batch processing, 208
binary choice, 10, 90
binary exclusion semaphore, 197
binary tree, 155
bit stuffing, 151
BOOL, 67
BUFFER, 125
catastrophe, 171
catenation, 22
CCS, 237
chain, 79

CHAIN2, 66
chaining, 141
change-giving, 7
channel, 121
CHAOS , 113
checkpoints, 173
choice, 7
choice2, 18
class, 222
CLOCK , 5
cobegin, 219
coend , 219
COLLEGE , 60
communication, 121
complete partial order (c. p. o.), 79
composition, 38
concealment, 95
concurrency, 48
conditional critical regions, 220
conditionals, 177
congruence, 237
connection diagrams, 55
constructive, 81
context-free grammar, 164
continuous, 79
control block, 205
COPY , 123
copy rule, 223
copybit, 9
critical region, 197
crossbar switch, 206
CT , 11
cyclic, 34

236

data base system, 173
data flow, 134
data structure, 153
deadlock, 46
deterministic processes, 76
dining philosophers, 56
disjoint, 218
distributed processing, 198
distributive, 23
divergence, 99
diverges, 101
domain, 115
DOUBLE , 123
double buffer, 142
end-of-file marker, 202
entry name, 226
Euclid, 237
event, 1
factorial, 154
failures, 115
FCFS, 214
FIB, 129
FIFO, 214
finite set, 28
fixed point, 78
flight reservation, 192
flow control, 151
footman, 62
fork, 218
FORTRAN monitor system, 208
free variables, 39
functional multiprocessing, 230
guarded, 6
Gypsy, 237
Hamming, 230
head, 24
hide, 100
implementation, 17
independent choice, 87
infinite overtaking, 63
inner statement, 227
input channel, 122

Index

instance, 225
interaction, 45
interference, 196
interleaving, 36
interruptible, 171
interrupts, 170
inverse, 34
ismember , 28
isprefix, 29
istrace, 33
join, 219
label, 69
LABEL, 18
LACKEY , 73
language, 162
laws, 15
layers, 149
lazy evaluation, 20
least upper bound, 79
left shift, 129
left-guarded, 146
length, 26
limit, 79
LISP, 17
LISPkit, 20
livelock, 146
local procedure call, 200
LONGQUOT , 178
loop, 162
main process, 152
master, 152
mathematical semantics, 237
menu, 10
menu, 19
MERGE , 125
message, 121
modularity, 141
monitor, 222
monotonic, 26
multiple checkpoints, 174
multiple choice, 88
multiple labelling, 72
multiprocessor, 198

Index

multithreading, 218
mutual recursion, 11
nested monitors, 224
NEWCOLLEGE , 62
NIL, 19
node, 102
NOISYVM , 49
nondestructive, 83
NONSTOP , 108
observational equivalence, 237
occam, 234
operating system, 208
or1, 89
or2, 89
or3, 89
output channel, 122
PACK , 123
packet switching, 229
partial order, 25, 79
PASCAL PLUS, 222, 225
path, 32
phase encoding, 150
PHIL, 59
picture, 12
PIDGINGOL, 162
pipes, 140
pointers, 205
post-condition, 233
pragma, 228
precondition, 182
prefix, 3
priority, 228
procedure, 200
process, 2
process labelling, 69
protection, 224
protocol, 148
QUOT , 178
RC4000, 229
re-entrant subroutine, 199
ready sets, 94
receiver, 149

237

recursion, 4
refusals, 93
regular expressions, 164
release, 192
remote call, 200
rendezvous, 226
scalar product, 134
scheduling, 213
scratch file, 202
sector, 200
selection, 38
semaphore, 197
sentence, 162
separate compilation, 229
separator card, 209
sequential composition, 166
sequential programming, 176
serially reusable, 198
SET , 155
shared card reader, 209
shared data structure, 192
SHARED LACKEY , 73
shared storage, 196
SIMULA 67, 222
singleton, 24
SKIP , 161
slave, 152
specification, 39
spooled input, 211
spooled output, 210
spooling, 210
stack, 126
star, 25
state, 11
static binding, 20
STOP , 3
strict, 23
structure clash, 141
subordination, 151
subroutine, 153
subscription, 37
success, 161
successful termination, 38
symbolic execution, 142
synchronisation, 46

238

systolic array, 134
tail, 24
then, 3
timesharing, 196
tr , 39
trace, 21, 48, 54
transition, 12
transmitter, 149
transparent, 130
unfolding, 5
unguarded recursion, 99
unique solutions, 81
UNIX, 219
UNPACK , 123
VAR, 125
var , 179
virtual resource, x, 205
VMC, 8
VMCRED, 9
VMCT , 8
VMS , 6
VMS2, 9
weighted sum, 133
WIRE , 149

Index