Communications Question

For this paper, you will consider the epistemological and ontological changes wrought by cybernetics in one of the following areas:

Human-machine distinctions and relationships

Human and animal behaviors and motivations

The fate and value of the individual

Structures and representations of gender

Modes of perceiving and interacting with “reality”

Channels and functions of communication

Economic and administrative management

Your argument should not be comprehensive in scope.  Instead, focus on a specific impact of a cybernetic idea or concept.  Here is a list of cybernetic and related ideas we have encountered, which you are free but not required to draw upon:

the science of control, control systems, information flow, and information transfer

prediction, probability, statistics, modeling, simulations, abstraction

feedback, negative feedback, self-adapting machines, homeostasis, entropy, order/disorder

the human-machine connection (or distinction), human and machine as analogies for each other, anthropomorphic machines

boundaries, transgressions, blurring, cyborgs, the Turing test

liberal humanism, selfhood, autonomy, human values

logic, binary code, digitization, digital representation, telephone switching circuitry

cybernetic control systems, military cybernetics, biocybernetics, state systems of control, servomechanisms, robot wars

time series, seriality, transmission rates

hyphenation, splicing, cyborgs, prosthetics, truncation, splitting, bodily or mental permeability, boundary problems

schizophrenia (and capitalism), the schizoid android, alcoholism, psychotropics, delusions, psychosis, immanent mind

teleological behavior, black boxes, causality, storage, recording, recall, operationalization of memory, technicization of perception,

conceptions of community, personal transformation, transcendental philosophy, counterculture

AN INTRODUCTION TO
CYBERNETICS
by
W. ROSS ASHBY
M.A., M.D.(Cantab.), D.P.M.
Director of Research
Barnwood House, Gloucester
NEW YORK
JOHN WILEY & SONS INC.
440
FOURTH AVENUE
1956
AN INTRODUCTION TO
CYBERNETICS
By the same author
DESIGN FOR A BRAIN
First published 1956
PRINTED IN GREAT BRITAIN BY
WILLIAM CLOWES AND SONS, LIMITED, LONDON AND BECCLES
5r-
—
PREFACE
Many workers in the biological sciences—physiologists, psycholo-
—
are interested in cybernetics and would like to
and
methods
techniques to their own speciality. Many
apply
have, however, been prevented from taking up the subject by an
impression that its use must be preceded by a long study of electronics and advanced pure mathematics; for they have formed the
impression that cybernetics and these subjects are inseparable.
gists, sociologists
its
The author is convinced, however, that this impression is false.
The basic ideas of cybernetics can be treated without reference to
electronics, and they are fundamentally simple; so although advanced
techniques may be necessary for advanced applications, a great deal
can be done, especially in the biological sciences, by the use of quite
simple techniques, provided they are used with a clear and deep
understanding of the principles involved. It is the author’s belief
that if the subject is founded in the common-place and well understood, and is then built up carefully, step by step, there is no reason
why the worker with only elementary mathematical knowledge
should not achieve a complete understanding of its basic principles.
With such an understanding he will then be able to see exactly what
further techniques he will have to learn if he is to proceed further;
and, what is particularly useful, he will be able to see what techniques
he can safely ignore as being irrelevant to his purpose.
The book is intended to provide such an introduction. It starts
from common-place and well-understood concepts, and proceeds,
step by step, to show how these concepts can be made exact, and
how they can be developed until they lead into such subjects as
feedback, stability, regulation, ultrastability, information, coding,
and other cybernetic topics. Throughout the book no
noise,
knowledge of mathematics is required beyond elementary algebra;
in particular, the arguments nowhere depend on the calculus (the
few references to it can be ignored without harm, for they are
intended only to show how the calculus joins on to the subjects
The illustrations and examples are
discussed, if it should be used).
taken
from
the
mostly
biological, rather than the physical, sciences.
Its overlap with Design for a Brain is small, so that the two books are
They are, however, intimately related, and
are best treated as complementary; each will help to illuminate
the other.
almost independent.
PREFACE
It is
divided into three parts.
I deals with the principles of Mechanism, treating such
matters as its representation by a transformation, what is meant by
Part
“stability”, what is meant by “feedback”, the various forms of
independence that can exist within a mechanism, and how mechanisms can be coupled. It introduces the principles that must be
followed
when the system is so large and complex (e.g. brain or
It introduces also
society) that it can be treated only statistically.
the case when the system is such that not all of it is accessible to
direct observation—the so-called Black Box theory.
Part II uses the methods developed in Part I to study what is
meant by “information”, and how it is coded when it passes through
a mechanism. It applies these methods to various problems in
biology and tries to show something of the wealth of possible
It leads into Shannon’s theory; so after reading this
applications.
Part the reader will be able to proceed without difficulty to the study
of Shannon’s own work.
Part III deals with mechanism and information as they are used in
biological systems for regulation and control, both in the inborn
systems studied in physiology and in the acquired systems studied in
psychology. It shows how hierarchies of such regulators and
controllers can be built, and how an amplification of regulation is
thereby made possible. It gives a new and altogether simpler
account of the principle of ultrastability. It lays the foundation
for a general theory of complex regulating systems, developing
further the ideas of Design for a Brain.
Thus, on the one hand it
an
of
the
explanation
outstanding powers of regulation
provides
possessed by the brain, and on the other hand it provides the
principles by which a designer may build machines of like power.
Though the book is intended to be an easy introduction, it is not
—
intended to be merely a chat about cybernetics it is written for those
who want to work themselves into it, for those who want to achieve
an actual working mastery of the subject. It therefore contains
abundant easy exercises, carefully graded, with hints and explanatory
answers, so that the reader, as he progresses, can test his grasp of
what he has read, and can exercise his new intellectual muscles.
A
few exercises that need a special technique have been marked thus:
*Ex. Their omission will not affect the reader’s progress.
For convenience of reference, the matter has been divided into
sections; all references are to the section, and as these numbers are
shown at the top of every page, finding a section is as simple and
direct as finding a page.
The section is shown thus: S.9/14
Figures, Tables, and
indicating the fourteenth section in Chapter 9.
—
vi
PREFACE
Exercises have been numbered within their own sections; thus
simple reference, e.g.
Fig. 9/14/2 is the second figure in S.9/14.
Ex. 4, is used for reference within the same section. Whenever a
A
word is formally defined it is printed in bold-faced type.
I would like to express my indebtedness to Michael B, Sporn, who
checked all the Answers. I would also like to take this opportunity
to express my deep gratitude to the Governors of Barnwood House
and to Dr. G. W. T. H. Fleming for the generous support that made
these researches possible. Though the book covers many topics,
these are but means; the end has been throughout to make clear
what principles must be followed when one attempts to restore
normal function to a sick organism that is, as a human patient, of
fearful complexity.
It is
my faith that the new understanding may
lead to new and effective treatments, for the need is great.
W. Ross Ashby
Barnwood House
Gloucester
vu
CONTENTS
Page
V
Preface
Chapter
1 :
What is New
1
.
The peculiarities of cybernetics
The uses of cybernetics
1
4
.
PART ONE: MECHANISM
2:
3
:
Change
9
10
Repeated change
16
….
The Determinate Machine
Vectors
4:
….
Transformation
The Machine with Input
Coupling systems
Feedback
24
30
42
48
53
55
.
.
Independence within a whole
The very large system
5:
Stability
Disturbance
.
Equilibrium in part and whole
6:
61
.
The Black Box.
Isomorphic machines
Homomorphic machines
73
77
82
86
94
.
The very large Box
The incompletely observable Box
.
102
109
113
PART TWO: VARIETY
Quantity of Variety
121
.
127
130
Constraint
Importance of constraint
Variety in machines
134
.
vm
CONTENTS
8
:
….
Transmission of Variety
Inversion
140
.
145
Transmission from system to system
Transmission through a channel
9
:
151
154
Incessant Transmission
Entropy
Noise
161
…..
…..
The Markov chain
165
174
186
.
PART THREE: REGULATION AND CONTROL
10:
Regulation in Biological Systems
Survival
1 1 :
Requisite Variety
The law
.
.
.195
.
…….
Control
Some variations
12:
.
.
.
.
.
.
Markovian regulation
.
.
.
.
.
.
.
.
.219
Determinate regulation
Regulating THE Very Large System
Repetitive disturbance
Designing the regulator
.
.
.
.
.
Selection and machinery
Amplifying Regulation
What is an amplifier?
Amplification in the brain
Amplifying intelligence
References
Answers to Exercises
.
Index
.
.
.
.
235
238
240
244
247
.251
Quantity of selection
14:
225
.231
The power amplifier
Games and strategies
13:
202
206
213
.216
……
……
……
……
….
……
……
……
…..
…….
The Error-controlled Regulator
The Markovian machine
197
.
255
259
265
265
270
271
273
274
289
IX
72325
Chapter
WHAT IS NEW
Cybernetics was defined by Wiener as “the science of control
1/1.
and conununication, in the animal and the machine” in a
word, as the art of steermanship, and it is to this aspect that the
book will be addressed. Co-ordination, regulation and control
will be its themes, for these are of the greatest biological and practical
—
interest.
We must, therefore, make a study of mechanism; but some
introduction is advisable, for cybernetics treats the subject from a
new, and therefore unusual, angle. Without introduction. Chapter
2 might well seem to be seriously at fault.
The new point of view
should be clearly understood, for any unconscious vacillation between the old and the new is apt to lead to confusion.
The peculiarities of cybernetics. Many a book has borne the
“Theory of Machines”, but it usually contains information
about mechanical things, about levers and cogs. Cybernetics, too,
1/2.
title
a “theory of machines”, but it treats, not things but ways of
It does not ask “what is this thing?” but ”what does it
do?” Thus it is very interested in such a statement as “this variable
is undergoing a simple harmonic oscillation”, and is much less
concerned with whether the variable is the position of a point on a
wheel, or a potential in an electric circuit. It is thus essentially
is
behaving.
functional and behaviouristic.
Cybernetics started by being closely associated in many ways with
physics, but it depends in no essential way on the laws of physics or
on the properties of matter. Cybernetics deals with all forms of
behaviour in so far as they are regular, or determinate, or reproducible.
The materiahty is irrelevant, and so is the holding or not
of the ordinary laws of physics. (The example given in S.4/15 will
make this statement clear.) The truths of cybernetics are not
conditional on their being derived from some other branch of science.
Cybernetics has its own foundations. It is partly the aim of this
book to display them clearly.
1
1
1/3
AN INTRODUCTION TO CYBERNETICS
—
electronic, mechaniCybernetics stands to the real machine
or
economic
stands
much
as
to a real object
cal, neural,
geometry
—
1/3.
in our terrestrial space.
There was a time when “geometry”
meant such relationships as could be demonstrated on threedimensional objects or in two-dimensional diagrams. The forms
provided by the earth animal, vegetable, and mineral were larger
in number and richer in properties than could be provided by elementary geometry. In those days a form which was suggested by
geometry but which could not be demonstrated in ordinary space
was suspect or inacceptable. Ordinary space dominated geometry.
Today the position is quite different. Geometry exists in its own
It can now treat accurately and
right, and by its own strength.
a
and
of
forms
coherently
range
spaces that far exceeds anything
—
that terrestrial space can provide.
—
Today it is geometry that con-
tains the terrestrial forms, and not vice versa, for the terrestrial
forms are merely special cases in an all-embracing geometry.
The gain achieved by geometry’s development hardly needs to be
pointed out. Geometry now acts as a framework on which all
terrestrial forms can find their natural place, with the relations
between the various forms readily appreciable.
With this increased
understanding goes a correspondingly increased power of control.
Cybernetics is similar in its relation to the actual machine. It
takes as its subject-matter the domain of “all possible machines”,
and is only secondarily interested if informed that some of them have
not yet been made, either by Man or by Nature. What cybernetics
offers is the framework on which all individual machines may be
ordered, related and understood.
Cybernetics, then, is indifferent to the criticism that some of
1/4.
the machines it considers are not represented among the machines
found among us. In this it follows the path already followed with
obvious success by mathematical physics. This science has long
given prominence to the study of systems that are well known to be
non-existent
springs without mass, particles that have mass but no
volume, gases that behave perfectly, and so on. To say that these
entities do not exist is true; but their non-existence does not mean
that mathematical physics is mere fantasy; nor does it make the
physicist throw away his treatise on the Theory of the Massless
Spring, for this theory is invaluable to him in his practical work.
The fact is that the massless spring, though it has no physical
representation, has certain properties that make it of the highest
importance to him if he is to understand a system even as simple
—
as a watch.
WHAT IS NEW
1/5
The biologist knows and uses the same principle when he gives
to Amphioxus, or to some extinct form, a detailed study quite out
of proportion lo its present-day ecological or economic importance.
In the same way, cybernetics marks out certain types of mechanism (S.3/3) as being of particular importance in the general theory;
and ii does this with no regard for whether terrestrial machines
to make this form common.
Only after the study has
surveyed adequately the possible relations between machine and
machine does it turn to consider the forms actually found in some
particular branch of science.
happen
In keeping with this method, which works primarily with the
1/5.
comprehensive and general, cybernetics typically treats any given,
particular, machine by asking not “whai individual act will it
produce here and now?” but “what are all the possible behaviours
that it can produce ?”
It is in tliis way that information theory comes to play an essential
part in the subject
;
for information theory is characterised essentially
by its dealing always with a set of possibilities; both its primary
data and its final statements are almost always about the set as
such, and not about some individual element in the set.
This new point of view leads to the consideration of new types of
problem. The older point of view saw, say, an ovum grow into a
rabbit and asked “why does it do this ?
why does it not just stay
an ovum?” The attempts to answer this question led to the study
—
of energetics and to the discovery of many reasons why the ovum
should change it can oxidise its fat, and fat provides free energy;
it has phosphorylating
enzymes, and can pass its metabolites around
a Krebs’ cycle; and so on. In these studies the concept of energy
was fundamental.
—
Quite different, though equally valid, is the point of view of
It takes for granted that the ovum has abundant free
energy, and that it is so delicately poised metabolically as to be, in a
Growth of some form there will be; cybernetics
sense, explosive.
asks “why should the changes be to the rabbit-form, and not to a
dog-form, a fish-form, or even to a teratoma-form ?” Cybernetics
envisages a set of possibilities much wider than the actual, and then
asks why the particular case should conform to its usual particular
restriction.
In this discussion, questions of energy play almost no
the energy is simply taken for granted.
Even whether the
part
cybernetics.
—
system is closed to energy or open is often irrelevant; what is
important is the extent to which the system is subject to determining
and controlling factors. So no information or signal or determining
3
AN INTRODUCTION TO CYBERNETICS
1/6
factor may pass from part to part without its being recorded as a
significant
event.
Cybernetics might, in
fact,
be defined as the
study of systems that are open to energy hut closed to information and
control systems that are “information-tight” (S.9/19.).
—
The uses of cybernetics. After this bird’s-eye view of cyber1/6.
netics we can turn to consider some of the ways in which it promises
to be of assistance.
I shall confine
my attention to the applications
The review can only
be brief and very general. Many applications have aheady been
made and are too well known to need description here; more will
doubtless be developed in the future. There are, however, two
peculiar scientific virtues of cybernetics that are worth explicit
mention.
One is that it offers a single vocabulary and a single set of concepts
Until
suitable for representing the most diverse types of system.
that promise most in the biological sciences.
any attempt to relate the many facts known about, say,
servo-mechanisms to what was known about the cerebellum was
made unnecessarily difficult by the fact that the properties of servomechanisms were described in words redolent of the automatic
pilot, or the radio set, or the hydraulic brake, while those of the
cerebellum were described in words redolent of the dissecting room
recently,
—
and the bedside
aspects that are irrelevant
to
the similarities
between a servo-mechanism and a cerebellar reflex. Cybernetics
offers one set of concepts that, by having exact correspondences
with each branch of science, can thereby bring them into exact
relation with one other.
It has been found repeatedly in science that the discovery that
two branches are related leads to each branch helping in the develop-
(Compare S. 6/8.) The result is often a markedly
The infinitesimal calculus and astrothe
virus
and
the
protein molecule, the chromosomes and
nomy,
ment of the other.
accelerated growth of both.
heredity are examples that come to mind. Neither, of course, can
give proofs about the laws of the other, but each can give suggestions
that may be of the greatest assistance and fruitfulness.
The subject
Here I need only mention the fact that
cybernetics is likely to reveal a great number of interesting and
suggestive parallelisms between machine and brain and society.
And it can provide the common language by which discoveries in
one branch can readily be made use of in the others.
is
returned to in S.6/8.
The complex system. The second peculiar virtue of cyber1/7.
netics is that it offers a method for the scientific treatment of the
WHAT IS NEW
1/7
system in which complexity is outstanding and too important to be
Such systems are, as we well know, only too common in
ignored.
the biological world
In the simpler systems, the methods of cybernetics sometimes
show no obvious advantage over those that have long been known.
!
It is chiefly
when the systems become complex that the new methods
reveal their power.
For two centuries
Science stands today on something of a divide.
has been exploring systems that are either intrinsically simple
or that are capable of being analysed into simple components. The
fact that such a dogma as “vary the factors one at a time” could be
it
accepted for a century, shows that scientists were largely concerned
in investigating such systems as allowed this method; for this method
Not
is often fundamentally impossible in the complex systems.
until Sir Ronald Fisher’s work in the ’20s, with experiments con-
ducted on agricukural soils, did it become clearly recognised that
there are complex systems that just do not allow the varying of only
one factor at a time they are so dynamic and interconnected that
the alteration of one factor immediately acts as cause to evoke
—
Until recently,
science tended to evade the study of such systems, focusing its
attention on those that were simple and, especially, reducible (8.4/ 14).
alterations in others, perhaps in a great many others.
In the study of some systems, however, the complexity could not
be wholly evaded. The cerebral cortex of the free-living organism,
the ant-hill as a functioning society, and the human economic system
were outstanding both in their practical importance and in their
So today we see psychoses
intractability by the older methods.
untreated, societies dechning, and economic systems fahering, the
scientist being able to do little more than to appreciate the full
complexity of the subject he is studying. But science today is also
taking the first steps towards studying “complexity” as a subject
in its own right.
Prominent among the methods for dealing with complexity is
It rejects the vaguely intuitive ideas that we pick up
from handling such simple machines as the alarm clock and the
bicycle, and sets to work to build up a rigorous discipline of the
For a time (as the first few chapters of this book will show)
subject.
it seems rather to deal whh truisms and platitudes, but this is merely
because the foundations are built to be broad and strong. They
are built so that cybernetics can be developed vigorously, without
the primary vagueness that has infected most past attempts to
cybernetics.
grapple with, in particular, the complexities of the brain in action.
Cybernetics offers the hope of providing effective methods for the
5
AN INTRODUCTION TO CYBERNETICS
1/7
Study,
and control, of systems that are intrinsically extremely
It will do this by first marking out what is achievable (for
complex.
probably
many
of the investigations of the past attempted the
impossible), and then providing generalised strategies, of demonstrable value, that can be used uniformly in a variety of special cases.
In this way it offers the hope of providing the essential methods by
which to attack the ills
—
—
which at
psychological, social, economic
Part III of
present are defeating us by their intrinsic complexity.
this book does not pretend to offer such methods perfected, but it
attempts to offer a foundation on which such methods can be
constructed, and a start in the right direction.
PART ONE
MECHANISM
The properties commonly ascribed to any object
are, in last analysis,
names for its behavior.
(Herrick)
Chapter
CHANGE
2/1.
The most fundamental concept
in
cybernetics
is
that
of
“difference”, either that two things are recognisably different or that
one thing has changed with time. Its range of application need not
be described now, for the subsequent chapters will illustrate the
range abundantly. All the changes that may occur with time are
naturally included, for when plants grow and planets age and
machines move some change from one state to another is implicit.
So our first task will be to develop this concept of “change”, not
only making it more precise but making it richer, converting it to a
form that experience has shown to be necessary if significant developments are to be made.
Often a change occurs continuously, that is, by infinitesimal steps,
as when the earth moves through space, or a sunbather’s skin
darkens under exposure. The consideration of steps that are
infinitesimal, however, raises a number of purely mathematical
Instead,
difficulties, so we shall avoid their consideration entirely.
we shall assume in all cases that the changes occur by finite steps in
time and that any difference is also finite. We shall assume that the
change occurs by a measurable jump, as the money in a bank account
changes by at least a penny. Though this supposition may seem
artificial in a world in which continuity is common, it has great
advantages in an Introduction and is not as artificial as it seems.
When the differences are finite, all the important questions, as we
shall see later, can be decided by simple counting, so that it is easy to
be quite sure whether we are right or not. Were we to consider
continuous changes we would often have to compare infinitesimal
against infinitesimal, or to consider what we would have after adding
questions by nc
together an infinite number of infinitesimals
—
means easy to answer.
As a simple trick, the discrete can often be carried over into the
continuous, in a way suitable for practical purposes, by making a
graph of the discrete, with the values shown as separate points. It is
9
2/2
AN INTRODUCTION TO CYBERNETICS
then easy to see the form that the changes will take if the points
were to become infinitely numerous and close together.
In fact, however, by keeping the discussion to the case of the finite
we lose nothing. For having established with certainty
what happens when the differences have a particular size we can
difference
consider the case when they are rather smaller.
When this case is
known with certainty we can consider what happens when they are
We
smaller still.
can progress in this way, each step being well
established, until we perceive the trend then we can say what is the
limit as the difference tends to zero.
This, in fact, is the method
;
that the mathematician always does use if he wants to be really sure
of what happens when the changes are continuous.
Thus, consideration of the case in which all differences are finite
it gives a clear and
simple foundation; and it can
always be converted to the continuous form if that is desired.
The subject is taken up again in S.3/3.
loses nothing;
Next, a few words that will have to be used repeatedly.
Consider the simple example in which, under the influence of sunshine, pale skin changes to dark skin.
Something, the pale skin,
is acted on by a factor, the sunshine, and is
changed to dark skin.
That which is acted on, the pale skin, will be called the operand,
the factor will be called the operator, and what the operand is
changed to will be called the transform. The change that occurs,
which we can represent unambiguously by
2/2.
pale skin -^ dark skin
is
the transition.
The transition is specified by the two states and the indication of
which changed to which.
TRANSFORMATION
The single transition is, however, too simple. Experience has
shown that if the concept of “change” is to be useful it must be
enlarged to the case in which the operator can act on more than one
operand, inducing a characteristic transition in each. Thus the
operator “exposure to sunshine” will induce a number of transitions,
2/3.
among which are:
cold soil -> warm soil
unexposed photographic plate -^ exposed plate
coloured pigment -^ bleached pigment
Such a set of transitions, on a set of operands, is a transformation.
10
CHANGE
2/4
Another example of a transformation is given by the simple coding
that turns each letter of a message to the one that follows it in the
Z being turned to ^; so CAT would become DBU.
transformation is defined by the table:
alphabet,
The
Y->Z
Z^A
Notice that the transformation is defined, not by any reference to
what it “really” is, nor by reference to any physical cause of the
change, but by the giving of a set of operands and a statement of
what each is changed to. The transformation is concerned with
what happens, not with why it happens. Similarly, though we may
sometimes know something of the operator as a thing in itself (as
we know something of sunlight), this knowledge is often not essenwe
tial; what we must know is how it acts on the operands; that is,
must know the transformation that it effects.
For convenience of printing, such a transformation can also be
expressed thus:
,
^
A
B
B
…
C
…
Y Z
Z A
We shall use this form as standard.
2/4.
Closure.
When an operator acts on a set of operands it may
happen that the set of transforms obtained contains no element that
is
not already present in the set of operands, i.e. the transformation
Thus, in the transformation
creates no new element.
I
^
A
B
B
C
…
…
Y Z
Z A
When this
every element in the lower line occurs also in the upper.
The
the
transformation.
under
closed
is
the
set
of
occurs,
operands
and
a
a
transformation
between
a
relation
is
of
“closure”
property
particular set of operands; if either is ahered the closure may
It will be noticed that the test for closure is made, not by reference
to whatever may be the cause of the transformation but by reference
alter.
It can therefore be applied
to the details of the transformation itself.
even when we know nothing of the cause responsible for the changes.
11
AN INTRODUCTION TO CYBERNETICS
2/5
Ex. 1
If the operands are the positive integers
:
and 4, and the operator
1, 2, 3,
“add three to it”, the transformation is:
is
1
2
4
3
^4567
I
Is it closed?
Ex. 2: The operands are those EngUsh letters that have Greek equivalents (i.e.
excluding J, q, etc.), and the operator is “turn each EngUsh letter to its
Greek equivalent”. Is the transformation closed ?
^A
3
.
Are the following transformations closed or not
:
^:
j
^
”
a
*
^
^
a
a
a
:
B:\^^P’I
g J q P
^’•^
^’- ^
g f q
g f
Ex. 4: Write down, in the form of Ex. 3, a transformation that has only one
operand and is closed.
Mr. C, of the Eccentrics’ Chess Club, has a system of play that rigidly
prescribes, for every possible position, both for White and Black (except
for those positions in which the player is already mated) what is the player’s
Ex. 5
:
best next move.
to position.
The theory thus defines a transformation from position
On being assured that the transformation was a closed one,
and that C always plays by this system, Mr. D. at once offered to play C
for a large stake.
Was wise?
D
A transformation may have an infinite number of discrete
2/5.
operands; such would be the transformation
1
2
3
4
…
4
5
6
7
…
I
^
where the dots simply mean that the list goes on similarly without
Infinite sets can lead to difficulties, but in this book we shall
consider only the simple and clear. Whether such a transformation
is closed or not is determined by whether one cannot, or can
end.
some particular, namable, transform that does
not occur among the operands. In the example given above, each
particular transform, 142857 for instance, will obviously be found
among the operands. So that particular infinite transformation is
(respectively) find
closed.
Ex.
I
In ^ the operands are the even numbers from 2 onwards, and the transforms are their squares
4
6…
A-i^
^
4 16 36…
:
:
•
Is
A closed?
Ex. 2: In transformation B the operands are all the positive integers 1, 2, 3,
and each one’s transform is its right-hand digit, so that, for instance,
127
Is B closed?
7, and 6493 -> 3.
.
.
.
^
12
CHANGE
Notation.
2/6.
Many
2/6
transformations
become
inconveniently
lengthy if written out in extenso. Already, in S.2/3, we have been
forced to use dots … to represent operands ihat were not given
For merely practical reasons we shall have to develop
individually.
a more compact method for writing down our transformations,
though it is to be understood that, whatever abbreviation is used,
Several
the transformation is basically specified as in S.2/3.
abbreviations will now be described. It is to be understood that
they are a mere shorthand, and that they imply nothing more than
has already been stated expHcitly in the last few sections.
Often the specification of a transformation is made simple by
some simple relation that links all the operands to their respective
transforms. Thus the transformation of Ex. 2/4/1 can be replaced by
the single line
—
Operand > operand plus three.
The whole transformation can thus be specified by the general rule,
written more compactly,
Op.-^Op. + 3,
together with a statement that the operands are the numbers 1, 2,
And commonly the representation can be made even
3 and 4.
briefer, the two letters being reduced to one:
«^rt + 3
= 1,2, 3, 4)
(«
The word “operand” above, or the letter n (which means exactly
If we are thinking
of how, say, 2 is transformed, then “«” means the number 2 and
nothing else, and the expression tells us that it will change to 5. The
same expression, however, can also be used with n not given any
the same thing), may seem somewhat ambiguous.
It
It then represents the whole transformation.
particular value.
will be found that this ambiguity leads to no confusion in practice,
for the context will always indicate which meaning is intended.
Ex. 1
:
Condense into one line the transformation
1
2
3
II
12
13
I
^
E.X. 2
:
Condense similarly the transformations
rl->
7
a:io
2-^9
.3-^ 8
e:
fi-^i
2->l
l3->l
^
13
:
f’-^l
c: 2
l3->3
AN INTRODUCTION TO CYBERNETICS
2/7
We shall often require a symbol to represent the transform of such
a symbol as
n.
It can be obtained conveniently
by adding a
prime to the operand, so that, whatever n may be, n -> n
Thus,
if the operands of Ex. 1 are «, then the transformation can be written
as«’ = « + 10 (« = 1,2,3).
.
Ex. 3
:
Write out in full the transformation in which the operands are the three
numbers 5, 6 and 7, and in which //’ = « — 3.
Is it closed?
Ex. 4: Write out in full the transformations in which:
= 5n
= 2«2
Ci)n’
in)n’
{n
(«
= 5,6,7);
= – 1,0, 1).
Ex. 5: If the operands are all the numbers (fractional included) between and 1,
and «’ = ^ti, is the transformation closed? (Hint: try some representative
values for /?: |, |, i, 001, 0-99; try till you become sure of the answer.)
Ex. 6: (Continued) With the same operands,
n’
is
= l/(« + D?
the transformation closed
if
The transformations mentioned so far have all been characterby being “single-valued”. A transformation is single-valued
if it converts each operand to only one transform.
(Other types
are also possible and important, as will be seen in S.9/2 and 12/8.)
2/7.
ised
Thus the transformation
.
A
C D
B
^B A A D
is
single-valued; but the transformation
A
B
Box D
A
,
^
is
CD
D
Box C
not single-valued.
J
2/8.
Of the single-valued transformations, a type of some import-
ance in special cases is that which is one-one. In this case the transforms are all different from one another. Thus not only does each
operand give a unique transform (from the single-valuedness) but
each transform indicates (inversely) a unique operand. Such a
transformation is
.ABCDEFGH
^
F
H K L
G J E
M
This example is one-one but not closed.
On the other hand, the transformation of Ex. 2/6/2(e) is not one-one,
for the transform
“1” does not indicate a unique operand.
14
A
.
CHANGE
2/10
transformation that is single-valued but not one-one will be referred
to as many-one.
Ex. 1: The operands are the ten digits 0, 1, … 9; the transform is the third
decimal digit of logio («+4).
(For instance, if the operand is 3, we find
Is the transformation
in succession, 7, logio?, 0-8451, and 5; so 3-^5.)
one-one or many-one? (Hint: find the transforms of 0, 1, and so on in
succession ; use four-figure tables.)
The identity. An important transformation, apt to be
dismissed by the beginner as a nullity, is the identical transformation,
in which no change occurs, in which each transform is the same as
If the operands are all different it is necessarily oneits operand.
2/9.
An example is/ in Ex. 2/6/2.
one.
In condensed notation n’=n.
At the opening of a shop’s cash register, the transformation to be made
1
on its contained money is, in some machines, shown by a flag. What flag
shows at the identical transformation ?
Ex. 2 In cricket, the runs made during an over transform the side’s score from
one value to another. Each distinct number of runs defines a distinct
Ex.
:
:
transformation
is
:
thus if eight runs are scored in the over, the transformation
What is the cricketer’s name for the identical
n
S.
specified by ii’
=
+
transformation ?
Representation by matrix. All these transformations can
2/10.
be represented in a single schema, which shows clearly their mutual
relations.
(The method will become particularly useful in Chapter
9 and subsequently.)
Write the operands in a horizontal row, and the possible transforms
in a column below and to the left, so that they form two sides of a
Given a particular transformation, put a “-)-” at the
rectangle.
intersection of a row and column if the operand at the head of the
column is transformed to the element at the left-hand side; otherwise
insert a zero.
Thus the transformation
ABC
A
C
C
would be shown as
i
AN INTRODUCTION TO CYBERNETICS
2/11
If the transformation is large, dots can be used in the matrix if
their meaning is unambiguous.
Thus the matrix of the transforma-
= n -\- 2, and in which the operands are the positive
from
1
integers
onwards, could be shown as
tion in which n’
(The symbols in the main diagonal, from the top left-hand corner,
have been given in bold type to make clear the pos tional
relations.)
How are the +’s distributed in the matrix of an identical transformation?
Ex. 2: Of the three transformations, which is (a) one-one,
(b) single-valued but
not one-one, (c) not single-valued ?
Ex. 1
:
(i)
i
(ii)
(iii)
CHANGE
The generation and
properties
2/11
of such a
series
must now be
considered.
Suppose the second transformation of S.2/3 (call it Alpha) has
been used to turn an English message into code. Suppose the coded
message to be again so encoded by Alpha what effect will this have ?
The effect can be traced letter by letter. Thus at the first coding A
became B, which, at the second coding, becomes C; so over the
double procedure A has become C, or in the usual notation A-^ C.
—
Y-^A
Z^B.
and
Thus the
Similarly B-^ D; and so on to
double application o^ Alpha causes changes that are exactly the same
as those produced by a single application of the transformation
B
…
C D
…
,A
^
Y Z
A B
Thus, from each closed transformation we can obtain another
closed transformation whose effect, if applied once, is identical with
the first one’s effect if applied twice.
The second is said to be the
“square” of the first, and to be one of its “powers” (S.2/14). If the
first one was represented by T, the second will be represented by T^;
which is to be regarded for the moment as simply a clear and
convenient label for the new transformation.
Ex.\:\fA:\’^
^
c
^
c
^whatis/42?
a
Ex. 2: Write down some identity transformation; what is its square?
Ex. 3
:
What is A^l
(See Ex. 2/4/3.)
Ex. 4: What transformation is obtained when the transformation n’ = n + \
is appUed twice to the positive integers?
Write the answer in abbreviated
form, as «’ = …
(Hint: try writing the transformation out in full as
.
in S.2/4.)
Ex. 5: What transformation is obtained when the transformation
is applied twice to the positive integers ?
Ex. 6
:
If A^ is the transformation
;
«’
= In
AN INTRODUCTION TO CYBERNETICS
2/12
2/12.
The trial in the previous exercise will make clear the import-
ance of closure.
An unclosed transformation such as W cannot be
for although it changes h to k, its effect on k is
it
can go no further. The unclosed transformation is
so
undefined,
apphed twice;
thus like a machine that takes one step and then jams.
When a transformation is given in abbreviated
Elimination.
// + 1, the result of its double
form, such as «’
application must
be found, if only the methods described so far are used, by re-writing
2/13.
=
show every operand, performing the double
the transformation to
application, and then re-abbreviating.
method.
There is, however, a quicker
To demonstrate and explain it, let us write out in full
—
+
n
1, on the positive integers, showing
the results of its double application and, underneath, the general
symbol for what lies above
the transformation T: n’
:
1
2
3
……
2
3
4
…
n’
h
…
4
5
…
n”
.
r: j
^’-
.
.
.
n” is used as a natural symbol for the transform of n’, just as n’ is
the transform of n.
=
Now we are given that n’ n + 1. As we apply the same
transformation again it follows that n” must be 1 more than /;’.
Thus«”
n
1.
=
+
To specify the single transformation T^ we want an equation that
will show directly what the transform n” is in terms of the operand
n.
Finding the equation is simply a matter of algebraic elimination:
from the two equations n” = n’ + 1 and n’ = n -\-
\,
eliminate n’.
Substituting for n’ in the first equation we get (with brackets to show
the derivation) n”
n -\- 2.
(n
1)
1, i.e. n”
=
+
=
+
This equation gives correctly the relation between operand {n)
and transform {n”) when T- is applied, and in that way T^ is specified.
For uniformity of notation the equation should now be re-written
as m’ = m + 2.
This is the transformation, in standard notation,
whose single application (hence the single prime on m) causes the
same change as the double application of T. (The change from
n to w is a mere change of name, made to avoid confusion.)
The rule is quite general. Thus, if the transformation is
= 2n — 3, then a second application will give second transforms
n” that are related to the first by n” = 2n’ — 3. Substitute for n\
n’
using brackets freely:
//”
= 2(2/2 – 3) – 3
= 4« – 9.
18
CHANGE
2/14
So the double application causes the same change as a
application of the transformation m’
= 4ni — 9.
single
Higher powers. Higher powers are found simply by adding
symbols for higher transforms, n'”, etc., and eliminating the symbols
for the intermediate transforms.
Thus, find the transformation
caused by three applications of «’ = 2// — 3. Set up the equations
2/14.
relating step to step:
-3
= 2n’ — 3
;,'” = 2n” – 3
=2/2
n’
n”
Take the last equation and substitute for n”, getting
Now substitute for «’
n'”
= 2(2// – 3) – 3
= 4//’ – 9.
n'”
= 4(2/7 – 3) – 9
= 8/; – 21.
:
So the triple application causes the same changes as would be
caused by a single application of in’ = Sm — 21. If the original
was T, this is T^.
Eliminate n’ from n” = 3n’ and //’ = 3//. Form the transformation
corresponding to the result and verify that two applications of n’ = 3n
Ex. 1
:
gives the same result.
Ex. 2: Eliminate a’ from a” = a’ + S and a’ = a + S.
Ex. 3: Eliminate a” and a’ from a'” = la”, a”
Ex. 4: Eliminate k’ from k” ^
Ex. 1.
–
W+
Ex. 5: Eliminate tti’ from m” = log /;;’, m’
Ex. 6
:
2,
= la’, and a’ = la.
= – 3k + 2. Verify as in
k’
= log ?n,
Eliminate p’ from p” = (p’)’^, p’=p^
Ex. 7: Find the transformations that are equivalent to double applications, on
all the positive numbers greater than 1, of:
= 2« + 3;
= + n;
= + 2 log n.
(Hi) n’
(i)/;’
(ii) n’
III
Ex. 8: Find the transformation that
//’
= — 3a/ —
as in Ex.
1
to the positive
is
equivalent to a triple application of
integers and zero.
Verify
and negative
1 .
Ex. 9: Find the transformations equivalent to the second, third, and further
applications of the transformation //’ = 1/(1 + n).
(Note: the series
discovered by Fibonacci in the 12th century, 1, 1, 2, 3, 5, 8, 13, … is
extended by taking as next term the sum of the previous two; thus, 3 + 5
= 8, 5 + 8 = 13, 8 + 13 =
.
.
.,
etc.)
19
AN INTRODUCTION TO CYBERNETICS
2/15
Ex. 10: What
the result of applying the transformation n’
is
= 1/n twice,
when the operands are all the positive rational numbers (i.e. all the fractions) ?
Here is a geometrical transformation. Draw a straight line on paper
and mark its ends A and B. This line, in its length and position, is the
operand. Obtain its transform, with ends A’ and B’, by the transformationrule R: A’ is midway between A and B; B’ is found by rotating the line
Ex. 1 1
:
A’B about A’ through a right angle anticlockwise.
Draw such a line,
apply R repeatedly, and satisfy yourself about how the system behaves.
*Ex. 12: (Continued).
If familiar
with analytical geometry,
let
A
start
at
and B at (0,1), and find the limiting position. (Hint: Build up A’s
final AT-co-ordinate as a series, and sum; similarly for A’s j-co-ordinate.)
(0,0)
Notation.
2/15.
The notation that indicates the transform by the
addition of a prime (‘) is convenient if only one transformation is
under consideration; but if several transformations might act on
show which one has acted. For this
is
sometimes
used: if « is the operand, and
another
reason,
symbol
transformation T is applied, the transform is represented by T{n).
The four pieces of type, two letters and two parentheses, represent
one quantity, a fact that is apt to be confusing until one is used to it.
T{n), really n’ in disguise, can be transformed again, and would be
written T{T(n)) if the notation were consistent; actually the outer
«,
the symbol n’ does not
brackets are usually ehminated and the T’s combined, so that n”
The exercises are intended to make this notation
is written as T-(n).
the
for
familiar,
change is only one of notation.
Ex. 1
:
If/: i
1
2
3
T
1
2
whatis/(3)?/(l)?/2(3)?
Ex. 2: Write out in full the transformation g on the operands, 6, 7, 8, if ^(6) = 8,
^(7)
Ex. 3
:
= 7, ^(8) = 8.
Write out in full the transformation h on the operands a, p, y, S, if h(a)
= y, fi2(a) =
Ex. 4: If A{n) is n
j3,
/j3(a)
= §, /i4(a) = „.
+ 2, what is /i(I5)?
Ex. 5: If/(/0 is -//2 + 4, what is/(2)?
Ex. 6: If Tin) is 3n, what is T’^Ui)’]
(Hint: if uncertain, write out T in extenso.)
Ex. 7: If / is an identity transformation, and / one of its operands, what is /(O?
2/16.
Product.
We have just seen that after a transformation T
has been applied to an operand n, the transform T(n) can be treated
as an operand by T again, getting T(T(n)), which is written T^in).
In exactly the same way Tin) may perhaps become operand to a
20
CHANGE
U, which
transformation
2/17
give a transform
will
U{T{n)).
Thus,
if they are
_
,
a
b
c
d
b
d
a
b
,
,
and
T: i
a
b
c
d
d
c
d
b
U: i
Tand U appHed in
then T{b) is d, and V{T{b)) is U{d), which is b.
that order, thus define a new transformation, V, which is easily
found to be
^
c
b
d
c
V is said to be the product or composition of T and U. It gives
that
simply the result of T and U being applied in succession, in
order, one step each.
If
T{c)
U is apphed first, then U{b) is, in the example above, c, and
a; so T{U(b))
is
is a,
not the same as U(T(b)).
The product,
when U and T are applied in the other order is
‘^
a
b
b
d
For convenience, V can be written as UT, and IV as TU. It must
always be remembered that a change of the order in the product may
change the transformation.
not exist, if some
(It will be noticed that Kmay be impossible, i.e.
of r’s transforms are not operands for U.)
Ex. 1
:
Write out in full the transformation U^T.
Ex. 2: Write out in full: UTU.
*Ex. 3
:
Represent T and
U by matrices and then multiply these two matrices
in the usual way (rows into columns), letting the product and sum of +’s
be +
;
call the resulting matrix Mi.
Represent K by a matrix; call it M2.
Compare Mi and M2.
Kinematic graph. So far we have studied each transforma2/17.
tion chiefly by observing its effect, in a single action, on all its
Another method (applicable only
possible operands (e.g. S.2/3).
when the transformation is closed) is to study its effect on a single
operand over many, repeated, applications. The method corresponds, in the study of a dynamic system, to setting it at some initial
state and then allowing it to go on, without further interference,
through such a series of changes as its inner nature determines.
Thus, in an automatic telephone system we might observe all the
changes that follow the dialHng of a number, or in an ants’ colony
21
AN INTRODUCTION TO CYBERNETICS
2/17
we might observe all the changes that follow the placing of a piece
of meat near-by.
Suppose, for definiteness, we have the transformation
•”
^ ” ^
v.l^
^
D A E D D
U is applied to C, then to U{C), then to U\C), then to U\C)
and so on,- there results the series: C, E, D, D, D,
and so on,
If
.
with
D continuing for ever.
If
.
.
U is applied similarly to A there
with D continuing again.
D, D, D,
These results can be shown graphically, thereby displaying to the
glance results that otherwise can be apprehended only after detailed
results the series A,
study.
.
.
.
To form the kinematic graph of a transformation, the set of
operands is written down, each in any convenient place, and the
elements joined by arrows with the rule that an arrow goes from A
to B if and only if A is transformed in one step to B.
Thus U gives
the kinematic graph
C^E^D^A 5*1 -^ 5’2 —>
., will correspond to the operation of a
.
transformation,
TKSo), THSo),
.
.
converting
.
.,
operand
Sq
successively
to
T{S^,
etc.
A more complex example, emphasising that transformations do
not have to be numerical to be well defined, is given by certain forms
of reflex animal behaviour. Thus the male and female threespined stickleback form, with certain parts of their environment, a
determinate dynamic system. Tinbergen (in his Study of Instinct)
describes the system’s successive states as follows: “Each reaction
of either male or female is released by the preceding reaction of the
Each arrow (in the diagram below) represents a causal
partner.
relation that by means of dummy tests has actually been proved to
The male’s first reaction, the zigzag dance, is dependent on a
visual stimulus from the female, in which the sign stimuli “swollen
exist.
abdomen” and the special movements play a part.
The female
reacts to the red colour of the male and to his zigzag dance by swim-
ming right towards him. This movement induces the male to turn
round and to swim rapidly to the nest. This, in turn, entices the
female to follow him, thereby stimulating the male to point its head
into the entrance.
His behaviour now releases the female’s next
reaction: she enters the nest.
This again releases the quivering
The presence of
reaction in the male which induces spawning.
fresh eggs in the nest makes the male fertilise them.”
Tinbergen
summarises the succession of states as follows:
.
,
.
26
THE DETERMINATE MACHINE
3/1
Appearsx^
/ Zigzag
Courts/”
dance
Leads
Follows
Female < Enters \ Shows nest entrance Male nest\^ Trembles Spawns Fertilises He thus describes a typical trajectory. Further examples are hardly necessary, for the various branches of science to which cybernetics is applied will provide an abundance, and each reader should supply examples to suit his own speciality. By relating machine and transformation we enter the discipline that relates the behaviours of real physical systems to the properties of symbolic expressions, written with pen on paper. The whole The subject of "mathematical physics" is a part of this discipline. methods used in this book are however somewhat broader in scope, for mathematical physics tends to treat chiefly systems that are continuous and linear (S.3/7). The restriction makes its methods hardly applicable to biological subjects, for in biology the systems are almost always non-Hnear, often non-continuous, and in many The exercises i.e. expressible in number. below (S.3/4) are arranged as a sequence, to show the gradation from the very general methods used in this book to those commonly used in mathematical physics. The exercises are also important as illustrations of the correspondences between transformations and cases not even metrical, real systems. To summarise: Every machine distinguishable states. If it is or dynamic system has many a determinate machine, fixing its circumstances and the state it is at will determine, i.e. make unique, These transitions of state correspond the state it next moves to. to those of a transformation on operands, each state corresponding Each state that the machine next moves to to a particular operand. corresponds to that operand's transform. The successive powers of the transformation correspond, in the machine, to allowing double, treble, etc., the unit time-interval to elapse before recording And since a determinate machine cannot go to two the next state. states at once, the corresponding transformation must be singlevalued. 27 AN INTRODUCTION TO CYBERNETICS 3/2 Ex. Name two states that are related as operand and transform, with time as the operator, taking the dynamic system from : : (a) Cooking ; (d) Embryo- (/) Endocrinology (g) EconMeteorology (h) Animal behaviour (/) Cosmology. (Meticulous accuracy is logical development ; (e) ; omics not required.) ; 3/2. (c) The petrol engine ; (b) Lighting a fire ; Closure. now be seen. ; ; Another reason for the importance of closure can The typical machine can always be allowed to go on in lime for a little longer, simply by the experimenter doing nothing! This means that no particular limit exists to the power that the transformation can be raised to. Only the closed transformations Thus the transformaallow, in general, this raising to any power. tion T d a b c e b m f f g g c f e T\a) is c and T\a) is /;/. But T{m) is not defined, With a as initial state, this transformation does not define what happens after five steps. Thus the transformaThe full significance tion that represents a machine must be closed. is not closed. so T\a) is not defined. of this fact will appear in S.10/4. The discrete machine. At this point it may be objected that most machines, whether man-made or natural, are smooth-working, while the transformations that have been discussed so far change by These discrete transformations are, however, the discrete jumps. 3/3. Their great advantage is their absolute freedom from subtlety and vagueness, for every one of their This simproperties is unambiguously either present or absent. best introduction to the subject. makes possible a security of deduction that is essential if The subject was touched further developments are to be reliable. plicity on in S.2/1. In any case the discrepancy is of no real importance. The discrete change has only to become small enough in its jump to approximate as closely as is desired to the continuous change. It must further be remembered that in natural phenomena the observations are almost invariably made at discrete intervals; the "continuity" ascribed to natural events has often been put there by the observer's imagination, not by actual observation at each of an infinite number of points. Thus the real truth is that the natural system is observed at discrete points, and our transformation represents it at discrete There can, therefore, be no real incompatibility. points. 28 THE DETERMINATE MACHINE 3/4 Machine and transformation. The parallelism between machine and transformation is shown most obviously when we 3/4. compare the machine's behaviour, as state succeeds state, with the kinematic graph (S.2/17), as the arrows lead from element to element. If a particular machine and a particular graph show full correspondence it will be found that: (1) Each possible state of the machine corresponds uniquely to a The correspondence particular element in the graph, and vice versa. is one-one. (2) Each succession of states that the machine passes through because of its inner dynamic nature corresponds to an unbroken chain of arrows through the corresponding elements. (3) If the machine goes to a state and remains there (a state of equiUbrium, S.5/3) the element that corresponds to the state will have no arrow leaving it (or a re-entrant one, S.2/17). (4) If the machine passes into a regularly recurring cycle of states, the graph will show a circuit of arrows passing through the corres- ponding elements. (5) The stopping of a machine by the experimenter, and its refrom some new, arbitrarily selected, state corresponds, in the graph, to a movement of the representative point from one element to another when the movement is due to the arbitrary action of the mathematician and not to an arrow. When a real machine and a transformation are so related, the transformation is the canonical representation of the machine, and the machine is said to embody the transformation. starting Ex. 1 A culture medium is inoculated with a thousand bacteria; their number doubles in each half-hour. Write down the corresponding transformation. : Ex. 2: (Continued.) Find n after the 1st, 2nd, 3rd, . . ., 6th steps. Draw the ordinary graph, with two axes, showing the culture's changes in number with time, (ii) Draw the kinematic graph of Ex. 3 : (Continued.) (i) the system's changes of state. Ex. 4: A culture medium contains 10^ bacteria and a disinfectant that, in each minute, kills 20 per cent of the survivors. of survivors as a transformation. Ex. 5 : (Continued.) (i) Express the change in the number Find the numbers of survivors after 1 , 2, 3, 4, 5 minutes. To what limit does the number tend as time goes on indefinitely? Ex. 6: Draw the kinematic graph of the transformation in which n' is, in a table (ii) of four-figure logarithms, the rounded-off right-hand digit of logio(/J What would be the behaviour of a corresponding machine ? Ex. 7 : + 70). (Continued, but with 70 changed to 90.) Ex. 8: (Continued, but with 70 changed to graph ? 29 10.) How many basins has this AN INTRODUCTION TO CYBERNETICS 3/5 Ex. 9: In each decade a country's population diminishes by 10 per cent, but in the same interval a million immigrants are added. Express the change from decade to decade as a transformation, assuming that the changes occur in finite steps. Ex. 10: (Continued.) If the country at one moment has twenty million in- habitants, find what the population will be at the next three decades. Ex. Find, in any way you can, at what number the population remain stationary. (Hint: when the population is "stationary" what relation exists between the numbers at the beginning and at the end of the decade? what relation between operand and transform?) 1 1 (Continued.) : will — Ex. 12: A growing tadpole increases in length each day by 1-2 mm. this as a transformation. Express Ex. 13: Bacteria are growing in a culture by an assumed simple conversion of food to bacterium; so if there was initially enough food for 10* bacteria and the bacteria now number n, then the remaining food is proportional to If the law of mass action holds, the bacteria will increase in each 108 _ „. (number of bacteria) x (amount of remaining food). In this particular culture the bacteria are _ n). Express the changes from increasing, in each hour, by 10 8/j (108 hour to hour by a transformation. Ex. 14: (Continued.) If the culture now has 10,000,000 bacteria, find what the interval by a number proportional to the product number will be after 1, 2, . . ., : 5 hours. (Continued.) Draw an ordinary graph with two axes showing how the number of bacteria will change with time. Ex. 1 5 : VECTORS 3/5. In the previous sections a machine's "state" has been regarded as something that is known as a whole, not requiring more detailed States of this type are particularly common in specification. biological systems where, for instance, characteristic postures or expressions or patterns can be recognised with confidence though no analysis of their components has been made. The states desSo are the types of cribed by Tinbergen in S.3/1 are of this type. cloud recognised by the meteorologist. The earher sections of this chapter will have made clear that a theory of such unanalysed states can be rigorous. Nevertheless, systems often have states whose specification demands (for whatever reason) further analysis. Thus suppose a news item over the radio were to give us the "state", at a certain hour, of a Marathon race now being run; it would proceed to give, These for each runner, his position on the road at that hour. So the "state" of positions, as a set, specify the "state" of the race. the race as a whole is given by the various states (positions) of the Such "compound" states various runners, taken simultaneously. are extremely common, and the rest of the book will be much 30 THE DETERMINATE MACHINE 3/5 It should be noticed that we are now beginning to consider the relation, most important in machinery, that exists between the whole and the parts. Thus, it often happens that the state of the whole is given by a list of the states taken, at that moment, by each of the parts. Such a quantity is a vector, which is defined as a compound entity, having a definite number of components. It is conveniently written concerned with them. thus: (fli, a^, . . ., a„), which means that the first component has the particular value a^, the second the value 02, and so on. vector is essentially a sort of variable, but more complex than A the ordinary numerical variable met with in elementary mathematics. It is a natural generalisation of "variable", and is of extreme importance, especially in the subjects considered in this book. The reader is advised to make himself as familiar as possible with it, applying it incessantly in his everyday life, until it has become as ordinary and well understood as the idea of a variable. It is not too much to say that his famiharity with vectors will largely determine his success with the rest of the book. Here are some well-known examples. (1) single A ship's "position" at any moment cannot be described by a number; two numbers are necessary: its latitude and its a vector with two components. One ship's position might, for instance, be given by the vector (58°N, 17'W). In 24 hours, this position might undergo the longitude. "Position" transition (58"N, (2) is thus 17°W)^ (59°N, 20°W). "The weather at Kew" cannot be specified by a single number, but can be specified to any desired completeness by our taking sufficient components. An approximation would be the vector: (height of barometer, temperature, cloudiness, humidity), and a A weather particular state might be (998 mbars, 56-2°F, 8, 72%). prophet is accurate if he can predict correctly what state this present state will change to. (3) Most of the administrative "forms" that have to be filled in are really intended to define some vector. Thus the form that the motorist has to fill in: Age of car : . Horse-power: Colour: is , , merely a vector written vertically. Two vectors are considered equal only if each component of the 31 AN INTRODUCTION TO CYBERNETICS 3/5 one is equal to the corresponding component of the other. Thus if there is a vector (w,x,y,z), in which each component is some number, and if two particular vectors are (4,3,8,2) and (4,3,8,1), then these two particular vectors are unequal; for, in the fourth component, 2 is not equal to 1. (If they have different components, e.g. (4,3,8,2) and {H,T), then they are simply not comparable.) When such a vector is transformed, the operation is in no way different from any other transformation, provided we remember that the operand is the vector as a whole, not the individual com- ponents (though how they are to change is, of course, an essential Suppose, for instance, the part of the transformation's definition). "system" consists of two coins, each of which may show either Head or Tail. The system has four states, which are (//,//) {H,T) {T,H) and (r,T). Suppose now my small niece does not like seeing two heads up, but always alters that to {T,H), and has various other preferences. It might be found that she always acted as the transformation , ^. ^ {H,H) {H,T) {T,H) {T,T) (T,H) iT,T) {T,H) (H,H) ' As a transformation on four elements, A'^ differs in no way from those considered in the earlier sections. There is no reason why a transformation on a set of vectors should not be wholly arbitrary, but often in natural science the transformation has some simplicity. Often the components change in some way that is describable by a more or less simple rule. if M were Thus : ^. ^ iH,H) (H,T) (T,H) {T,H) {T,T) iH,H) {H,T) , {T,T) could be described by saying that the first component always changes while the second always remains unchanged. Finally, nothing said so far excludes the possibility that some or all of the components may themselves be vectors But (E.g. S.6/3.) we shall avoid such complications if possible. it ! Ex. I : Using /IBCas first operand, find the transformation generated by repeated application of the operator "move the left-hand letter to the right" (e.g. ABC-^BCA). Ex. 2: (Continued.) Express the transformation as a kinematic graph. Ex. 3: Using (1,-1) as first operand, find the other elements generated by repeated application of the operator "interchange the two numbers and then multiply the new left-hand number by minus one ". 32 THE DETERMINATE MACHINE 3/6 Express the transformation as a kinematic graph. Ex. 4: (Continued.) Ex. 5: The first operand, .v, is the vector (0,1,1); the operator Fis defined thus: (i) the left-hand number of the transform is the same as the middle number of the operand ; the middle number of the transform is the same as the right-hand number of the operand; (iii) the right-hand number of the transform is the sum of the operand's (ii) middle and right-hand numbers. Thus, F{x) is ( 1 1 ,2), and F\x) is (1 ,2,3). compare Ex. 2/14/9.) Find F\x), F\x), F\x). , (Hint : The last exercise will have shown the clumsiness Notation. of trying to persist in verbal descriptions. The transformation F is in fact made up of three sub-transformations that are applied simultaneously, i.e. always in step. Thus one sub-transformation 3/6. on the left-hand number, changing acts 0^1^1-^2-^3-^5, etc. If we call successively through the three components it and c, then F, operating on the vector {a, b, c), is equivalent to the simultaneous action of the three sub-transformations, each a, b, acting on one component only: =^ =c F:^b' fa' Ic' =b + c Thus, a' = b says that the new value of a, the left-hand number in the transform, is the same as the middle number in the operand; and so on. Let us try some illustrations of this new method; no new idea is involved, only a new manipulation of symbols. (The reader is advised to work through all the exercises, since many important features appear, and they are not referred to elsewhere.) Ex. 1 : If the operands are of the form (a,b), and one of them is (i,2), find the vectors produced by repeated application to it of the transformation T: a' b' =b = -a (Hint: find r(i2),r2(i,2), etc.) Ex. 2: If the operands are vectors of the form {v,w,x,y,z) and U is =w =v \w' uUx' = X f v' y =z find Via), where a Ex. 3 (Continued.) : a, Uia), 3 = (2,1,0,2,2). Draw the kinematic graph of U if its only operands are U\a\ etc. 33 AN INTRODUCTION TO CYBERNETICS 3/6 Ex. 4: (Continued.) How would the graph alter if further operands were added? Ex. 5: Find the transform of (3,-2,1) by A if the general form is (^,A,y) and the transformation is = 2g-h [g' A:^h'= h-j ir = g + h Ex. 6: Arthur and Bill agree to have a gamble. Each is to divide his money into two equal parts, and at the umpire's signal each is to pass one part over to the other player. Each is then again to divide his new wealth into two equal parts and at a signal to pass a half to the other; and so on. Arthur started with 8/- and Bill with 4/-. by the vector (8,4). Ex. 7: (Continued.) above. Represent the initial operand Find, in any way you can, all its subsequent transforms. Express the transformation by equations as in Ex. 5 Ex. 8: (Continued.) Charles and David decide to play a similar game except that each will hand over a sum equal to a half of what the other possesses. If they start with 30/- and 34/- respectively, what will happen to these quantities? Ex. 9: (Continued.) Express the transformation by equations as in Ex. 5. Ex. 10: If, in Ex. 8, other sums of money had been started with, who in general would be the winner? Ex. 11 In an aquarium two species of animalcule are prey and predator. In each day, each predator destroys one prey, and also divides to become two If today the aquarium has m prey and // predators, express predators. : their changes as a transformation. Ex. 12: (Continued.) What is the operand of this transformation? Ex. 13: (Continued.) If the state was initially (150,10), find how it changed over the first four days. Ex. 14: A certain pendulum swings approximately in accordance with the transformation x' — \(x — y), y' — \{x where x is its angular ;'), + deviation from the vertical and y is its angular velocity; x' and y' are It starts from the state (10,10); find how its their values one second later. angular deviation changes from second to second over the first eight seconds. (Hint: find x', x", x'", etc.; can they be found without cal- culating y', y", etc?) (Continued.) Draw an ordinary graph (with axes for x and /) showing how .v's value changed with time. Is the pendulum frictionless ? Ex. 1 5 : Ex. 16: In a certain economic system a new law enacts that at each yearly readjustment the wages shall be raised by as many shillings as the price index exceeds 100 in points. The economic effect of wages on the price index is such that at the end of any year the price index has become equal to the wage rate at the beginning of the year. Express the changes of wage-level and price-index over the year as a transformation. Ex. 17: (Continued.) If this year starts with the wages at 110 and the price index at 1 10, find what their values will be over the next ten years. Ex. 18: (Continued.) will change. Draw an ordinary graph to show how prices and wages Is the law satisfactory? 34 THE DETERMINATE MACHINE 3/7 The system is next changed so that its transformation becomes .v' = j(x + y), y' — j(x — y) + 100. It starts with wages and Calculate what will happen over the next ten years. prices both at 110. Ex. 20: (Continued.) Draw an ordinary graph to show how prices and wages Ex. 19: (Continued.) will change. Ex. 21: Compare the graphs of Exs. 18 and 20. be described in the vocabulary of economics? How would the distinction Ex. 22: If the system of Ex. 19 were suddenly disturbed so that wages fell to 80 and prices rose to 120, and then left undisturbed, what would happen over the next ten years? (Hint: use (80,120) as operand.) Ex. 23 : (Continued.) Draw an ordinary graph to show how wages and prices would change after the disturbance. Ex. 24: Is transformation Tone-one between the vectors (xi, ^2) and the vectors (xi',X2V J. = 2.V1 + X2 + X2 \X2 = fx\' A"! (Hint: If versa ?) (.vi,a-2) is given, is (.vi',.Y2') And vice uniquely determined? *Ex. 25: Draw the kinematic graph of the 9-state system whose components are residues: x' =X+y How many basins has it? (This section may be omitted.) The previous section is of fundamental importance, for it is an introduction to the methods of mathematical physics, as they are applied to dynamic systems. The 3/7. reader is therefore strongly advised to work through all the exercises, for only in this way can a real grasp of the principles be obtained. done this, he will be better equipped to appreciate the If he has meaning of this section, which summarises the method. The physicist starts by naming his variables Xj, X2, — . . . x„. The basic equations of the transformation can then always be obtained by the following fundamental method: — (l)Take the first variable, x^, and consider what state it will change to next. If it changes by finite steps the next state will be Xi', if continuously the next state will be .Vj + ^Vj. (In the latter case he may, equivalently, consider the value of dxjdt.) (2) Use what is known about the system, and the laws of physics, what x^ will be) in terms to express the value of .Yj', or clxjdt (i.e. of the values that .Yj, ., x„ (and any other necessary factors) have now. In this way some equation such as . X,' is . = 2a.Yj — .Y3 or obtained. 35 dxjch = 4k sin X3 AN INTRODUCTION TO CYBERNETICS 3/8 (3) Repeat the process for each variable in turn until the whole transformation is written down. The set of equations so obtained giving, for each variable in the system, what it will be as a function of the present values of the variables and of any other necessary factors is the canonical // is a standard form to which all representation of the system. descriptions of a determinate dynamic system may be brought. If the functions in the canonical representation are all linear, the system is said to be linear. Given an initial state, the trajectory or line of behaviour may now be computed by finding the powers of the transformation, as in — — S.3/9. *Ex. 1 Convert the transformation (now in canonical form) : =y =z — x"^ dzjdt = z + 2xy dxidt dyjdt to a differential equation of the third order in one variable, x. Eliminate y and z and their derivatives.) (Hint: *Ex. 2: The equation of the simple harmonic oscillator is often written d2x dr2+-Convert this to canonical form Invert the process used in Ex. *Ex. 3 : in =^ two independent variables. (Hint: 1.) Convert the equation d2x dx 2 to canonical form in two variables. After this discussion of differential equations, the reader who used to them may feel that he has now arrived at the "proper" way of representing the effects of time, the arbitrary and discrete He tabular form of S.2/3 looking somewhat improper at first sight. should notice, however, that the algebraic way is a restricted way, applicable only when the phenomena show the special property of The tabular form, on the other hand, can be continuity (S.7/20). used always; for the tabular form includes the algebraic. This is of some importance to the biologist, who often has to deal with phenomena that will not fit naturally into the algebraic form. When this happens, he should remember that the tabular form can always provide the generality, and the rigour, that he needs. The rest of this book will illustrate in many ways how naturally and easily the tabular form can be used to represent biological systems. 3/8. is 36 THE DETERMINATE MACHINE 3/10 ''Unsolvable'" equations. The exercises to S.3/6 will have shown beyond question that if a closed and single-valued transformation is given, and also an initial state, then the trajectory from that state is both determined (i.e. single-valued) and can be found by computation. For if the initial state is x and the transformation T, then the successive values (the trajectory) of x is the series 3/9. X, T{x), T2(x), T^ix), T\x), and so on. This process, of deducing a trajectory when given a transforma- tion and an initial state, is, mathematically, called "integrating" the (The word is used especially when the transformation is a set of differential equations, as in S.3/7; the process is then transformation. also called "solving" the equations.) If the reader has worked all through S.3/6, he is probably already given a transformation and an initial state, he can always obtain the trajectory. He will not therefore be discouraged if he hears certain differential equations referred to as "nonThese words have a purely technical integrable" or "unsolvable". meaning, and mean only that the trajectory cannot be obtained if one is restricted to certain defined mathematical operations. satisfied that, Mechanism of Economic Systems shows clearly how the economist may want to study systems and equations that are of the type called "unsolvable"; and he shows how the economist can, in practice, get what he wants. Tustin's 3/10. Phase space. When the components of a vector are numerical variables, the transformation can be shown in geometric form; and this form sometimes shows certain properties far more clearly and obviously than the algebraic forms that have been considered so far. As example of the method, consider the transformation y' of Ex. 3/6/7. If = ^x + iy we take axes x and y, we can represent each by the point whose x-co-ordinate and whose j-co-ordinate is 4. The state of the system is thus particular vector, such as (8,4), is 8 represented initially by the point P in Fig. 3/10/1 (I). The transformation changes the vector to (6,6), and thus changes The movement is, of course, none other than the change drawn in the kinematic graph of S.2/17, now drawn in a plane with rectangular axes which contain numerical scales. This two-dimensional space, in which the operands and transforms can be represented by points, is called the phase-space of the system. (The "button and string" freedom of S.2/17 is no longer possible.) the system's state to P'. 37 AN INTRODUCTION TO CYBERNETICS 3/10 In II of the same figure are shown enough arrows to specify Here the generally what happens when any point is transformed. arrows show the other changes that would have occurred had other states been taken as the operands. It is easy to see, and to prove geometrically, that all the arrows in this case are given by one rule: with any given point as operand, run the arrow at 45° up and to the left (or down and to the right) till it meets the diagonal represented by the line y = x. Fig. 3/10/1 The usefulness of the phase-space (II) can now be seen, for the whole range of trajectories in the system can be seen at a glance, In this way it often happens frozen, as it were, into a single display. that some property may be displayed, or some thesis proved, with the greatest ease, where the algebraic form would have been obscure. Such a representation in a plane is possible only when the vector has two components. When it has three, a representation by a three-dimensional model, or a perspective drawing, is often still When the number of components exceeds three, actual useful. representation is no longer possible, but the principle remains, and a sketch representing such a higher-dimensional structure may still be most useful, especially when what is significant are the general topological, rather than the detailed, properties. (The words "phase space" are sometimes used to refer to the empty space before the arrows have been inserted, i.e. the space into which any set of arrows may be inserted, or the diagram, such as II above, containing the set of arrows appropriate to a particular transformation. The context usually intended.) 38 makes obvious which is THE DETERMINATE MACHINE 3/11 Ex.: Sketch the phase-spaces, with detail merely sufficient to show the main features, of some of the systems in S.3/4 and 6. What is a ''system'"? In S.3/1 it was stated that every real determinate machine or dynamic system corresponds to a closed, single-valued transformation; and the intervening sections have It does not, however, illustrated the thesis with many examples. follow that the correspondence is always obvious; on the contrary, any attempt to apply the thesis generally will soon encounter certain difficulties, which must now be considered. Suppose we have before us a particular real dynamic system a swinging pendulum, or a growing culture of bacteria, or an automatic pilot, or a native village, or a heart-lung preparation and we want to discover the corresponding transformation, starting 3/11. — — from the beginning and working from first principles. Suppose it is We actually a simple pendulum, 40 cm long. provide a suitable recorder, draw the pendulum through 30° to one side, let it go, and its position every quarter-second. We find the successive deviations to be 30° (initially), 10°, and —24° (on the other side). record So our first estimate of the transformation, under the given conditions, is Y 30° 10° 10° -24° I Next, as good scientists, we check that transition from 10°: we draw the pendulum aside to 10°, let it go, and find that, a quarter-second later, it is at +3°! Evidently the change from 10° is not single- — the system is contradicting itself. What are we to do now ? Our difficulty is typical in scientific investigation and is fundamental: we want the transformation to be single-valued but it will not come so. We cannot give up the demand for singleness, for to do so would be to give up the hope of making single-valued predictions. Fortunately, experience has long since shown what valued to be done: the system must be re-defined. At this point we must be clear about how a "system" is to be defined. Our first impulse is to point at the pendulum and to is say "the system is that thing there". This method, however, has a fundamental disadvantage: every material object contains no less than an infinity of variables and therefore of possible systems. The real pendulum, for instance, has not only length and position; it has also mass, temperature, electric conductivity, crystalline structure, chemical impurities, some radio-activity, velocity, reflecting power, tensile strength, a surface film of moisture, bacterial contamination, 39 3/11 AN INTRODUCTION TO CYBERNETICS an optical absorption, elasticity, shape, specific gravity, and so on and on. Any suggestion that we should study "all" the facts is What is unrealistic, and actually the attempt is never made. necessary is that we should pick out and study the facts that are relevant to some main interest that is already given. The truth is that in the world around us only certain sets of facts are capable of yielding transformations that are closed and singlevalued. The discovery of these sets is sometimes easy, sometimes difficult. The history of science, and even of any single investigation, abounds in examples. Usually the discovery involves the other method for the defining of a system, that of listing the variables that are to be taken into account. The system now means, not a This list can be varied, and the thing, but a list of variables. experimenter's commonest task is that of varying the list ("taking other variables into account") until he finds a set of variables that Thus we first considered the pendulum gives the required singleness. as if it consisted solely of the variable "angular deviation from the vertical"; we found that the system so defined did not give singleness. If we were to go on we would next try other definitions, for instance the vector: (angular deviation, mass of bob), which would also be found to fail. Eventually we would try the vector: (angular deviation, angular velocity) and then we would find that these states, defined in this way, would give the desired singleness (cf. Ex. 3/6/14). Some of these discoveries, of the missing variables, have been of major scientific importance, as when Newton discovered the importance of momentum, or when Gowland Hopkins discovered the importance of vitamins (the behaviour of rats on diets was not Sometimes the discovery single-valued until they were identified). is scientifically trivial, as when single-valued results are obtained only after an impurity has been removed from the water-supply, or a loose screw tightened; but the singleness is always essential. (Sometimes what is wanted is that certain probabilities shall be This more subtle aim is referred to in S.7/4 and 9/2. single- valued. It is not incompatible with what has just been said: it merely means that it is the probability that is the important variable, not the Thus, if I study a roulettewheel scientifically I may be interested in the variable ''probability of the next throw being Red", which is a variable that has numerical values in the range between and 1, rather than in the variable variable that is giving the probability. 40 THE DETERMINATE MACHINE 3/11 throw", which is a variable that has only two A system that includes the latter variable is almost certainly not predictable, whereas one that includes the former (the probability) may well be predictable, for the probability has a constant value, of about a half.) The "absolute" system described and used in Design for a Brain is just such a set of variables. It is now clear why it can be said that every determinate dynamic system corresponds to a single-valued transformation (in spite of the fact that we dare not dogmatise about what the real world We can make the statement contains, for it is full of surprises). simply because science refuses to study the other types, such as the one-variable pendulum above, dismissing them as "chaotic" or ''colour of the next values: Red and Black. It is we who decide, ultimately, what we will accept "non-sensical". as "machine-like" and what we will reject. (The subject is resumed in S.6/3.) 41 4 Chapter THE MACHINE WITH INPUT In the previous chapter we studied the relation between transformation and machine, regarding the latter simply as a unit. We now proceed to find, in the world of transformations, what 4/1. corresponds to the fact that every ordinary machine can be acted on by various conditions, and thereby made to change its behaviour, as a crane can be controlled by a driver or a muscle controlled by a nerve. For this study to be made, a proper understanding must be had of what is meant by a "parameter". So far, each transformation has been considered by itself; we must now extend our view so as to consider the relation between one transformation and another. Experience has shown that just the same methods (as S.2/3) applied again will suffice; for the change from transformation A to transformation B is nothing but the transition A-> B.
(In S.2/3 it was implied that the elements of a
transformation may be anything that can be clearly defined: there
therefore no reason why the elements should not themselves be
is
transformations.) Thus, if Tj, T2, and Ti, are three transformations,
there is no reason why we should not define the transformation U:
U: i
^
All that
is
^’
^^
^’
T2
T2
Ti
necessary for the avoidance of confusion
is
that the
changes induced by the transformation T^ should not be allowed to
become confused with those induced by U\ by whatever method is
appropriate in the particular case the two sets of changes must
be kept conceptually distinct.
An actual example of a transformation such as U occurs when a
boy has a toy-machine T^, built of interchangeable parts, and then
dismantles it to form a new toy-machine T2. (In this case the
changes that occur when Ti goes from one of its states to the next
when Ti “works”) are clearly distinguishable from the change
that occurs when Tj changes to T2.)
(i.e.
Changes from transformation to transformation may, in general,
be wholly arbitrary.
We shall, however, be more concerned with
42
THE MACHINE WITH INPUT
the special case in
same
set
4/1
which the several transformations act on the
of operands.
Thus,
if
the four
common operands are
a, b, c, and d, there might be three transformations, R^, R2, and R^:
.abed
e
.abed
bade
d d b
These can be written more compactly as
^’
.abed
d c d b
AN INTRODUCTION TO CYBERNETICS
4/2
A real machine whose behaviour can be represented by such a set
of closed single-valued transformations will be called a transducer
or a machine with input (according to the convenience of the context).
The set of transformations is its canonical representation. The
parameter, as something that can vary, is its input.
Ex. I:
US
*
is
I
I
b
a,
how many other closed and single-valued transformations can be formed
on the same two operands ?
Ex. 2: Draw the three kinematic graphs of the transformations Ry, Ri, Ri
above. Does change of parameter- value change the graph ?
Ex. 3: With R (above) at /?i, the representative point is started at c and allowed
to move two steps (to RiHc)); then, with the representative point at this
new state, the transformation is changed to Rz and the point allowed to
move two more steps.
Where is it now ?
Ex. 4: Find a sequence of R’s that will take the representative point (i) from d
to a, (ii) from c to a.
Ex. 5: What change in the transformation corresponds to a machine having
one of its variables fixed? What transformation would be obtained if the
system
x’= — X + 2y
X – y
y’=
were to have its variable x fixed at the value 4?
Form a table of transformations affected by a parameter, to show that a
parameter, though present, may in fact have no actual eff’ect.
Ex. 6
4/2.
:
We can now consider the algebraic way of representing a
transducer.
The three transformations
Ri:n’ =
n+\
R2: n
=n+2
Ry. n’
=n+3
can obviously be written more compactly as
R^: n
= n + a,
and this shows us how to proceed. In this expression it must be
noticed that the relations of n and a to the transducer are quite
different, and the distinction must on no account be lost sight of.
n is operand and is changed by the transformation; the fact that it is
an operand is shown by the occurrence of n’
a is parameter and
determines which transformation shall be applied to n. a must
.
therefore be specified in value before w’s change can be found.
When the expressions in the canonical representation become
more complex, the distinction between variable and parameter can
be made by remembering that the symbols representing the operands
will appear, in some form,
on the left, as x’ or dxjdt; for the trans44
THE MACHINE WITH INPUT
4/2
formation must tell what they are to be changed to. So all quantities that appear on the right, but not on the left, must be parameters.
The examples below will clarify the facts.
Ex. 1
:
What are the three transformations obtained by giving parameter a the
— 1, 0, or +1 in 7″^:
values
jg’
= {\ -a)g + {a- \)h
lah.
2^+
«•[/?’=
Ex. 2: What are the two transformations given when the parameter a takes the
or I in 5?:
value
= (\ – a)J + log (1 + a + sin ah)
= (1 + sin ay>(°-‘)/(.
Ex. 3: The transducer n’ = n + a^, in which a and n can take only positive
(i) At what value should a be kept
integral values, is started at « = 10.
h’
j’
if,
n is to remain at 10? (ii) At what
value should a be kept if n is to advance in steps of 4 at a time (i.e. 10, 14,
18,
.)?
(iii) What values of a, chosen anew at each step, will make n
follow the series 10, 11, 15, 16, 20, 21, 25, 26,
., in which the differences
are alternately 1 and 4? (iv) What values of a will make n advance by unit
steps to 100 and then jump directly to 200?
in spite of repeated transformations,
.
.
.
.
Ex. 4: If a transducer has n operands and also a parameter that can take n
values, the set shows a triunique correspondence between the values of operand, transform, and parameter if (1) for given parameter value the transformation is one-one, and (2) for given operand the correspondence between
parameter-value and transform is one-one. Such a set is
1
AN INTRODUCTION TO CYBERNETICS
4/3
When the expression for a transducer contains more than one
4/3.
parameter, the number of distinct transformations may be as large
as the number of combinations of values possible to the parameters
(for each combination may define a distinct transformation), but
can never exceed it.
Ex. 1
:
Find all the transformations in the transducer f/,,^ when a can take the
or 1.
1, or 2, and b the values
values 0,
JJ
^’^b-
j s’
[/’
= {\ — a)s + abt
= (] +b)t + {b- \)a.
How many transformations does the set contain ?
Ex. 2: (Continued.)
If the vector ia,b)
could take only the values (0,1), (1,1),
and (2,0), how many transformations would the transducer contain?
Ex. 3 The transducer T^b, with variables p and q
:
:
= ap + bq
J,
^”f’\q’ = bp + aq
.
(
p’
is started at (3,5).
What values should be given to the parameters a and
b if {p,q) is to move, at one step, to (4,6) ? (Hint the expression for 7^;,
ca…