Cybernetics - Ashby

xholonWorkbook.xml

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<Notes><![CDATA[
Xholon
------
Title: Cybernetics - Ashby
Description: 
Url: http://www.primordion.com/Xholon/gwt/
InternalName: 5cfe135c89af9d503cb7053960b47c7d
Keywords: 

My Notes
--------
April 20, 2018
Ashby provides numerous examples in his book "An Introducttion to Cybernetics".
Some of these involve finite sets, where A × A → A
This is instead of infinite sets such as the set of positive integers ℤ, where ℤ × ℤ → ℤ
The problem with infinite sets, is that I can't represent their elements in Xholon as a small number of specific nodes.
But I can do this with some of Ashby's examples.

Chapter 1 [1]
---------
1/1. Cybernetics was defined by Wiener as “the science of control
and communication, in the animal and the machine”

1/2. The peculiarities of cybernetics. Many a book has borne the
title “Theory of Machines”, but it usually contains information
about mechanical things, about levers and cogs. Cybernetics, too,
is a “theory of machines”, but it treats, not things but ways of
behaving. It does not ask “what is this thing?” but “what does it
do?”
The truths of cybernetics are not conditional on their being derived from some other branch of science.

1/3. Cybernetics stands to the real machine—electronic, mechanical,
neural, or economic—much as geometry stands to a real object
in our terrestrial space.
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.

1/5. In keeping with this method, which works primarily with the
comprehensive and general, cybernetics typically treats any
given, particular, machine by asking not “what individual act will
it produce here and now?” but “what are all the possible behaviours
that it can produce?”

Even whether the 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 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 but closed to information and control—systems that are “information-tight”.

1/6. The uses of cybernetics.
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
suitable for representing the most diverse types of system.
It has been found repeatedly in science that the discovery that
two branches are related leads to each branch helping in the development
of the other.
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.

1/7. The complex system. The second peculiar virtue of cybernetics
is that it offers a method for the scientific treatment of the system
in which complexity is outstanding and too important to be
ignored Such systems are, as we well know, only too common in
the biological world!
Prominent among the methods for dealing with complexity is
cybernetics. 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 subject.
Cybernetics offers the hope of providing effective methods for
the study, and control, of systems that are intrinsically extremely complex.

Chapter 2
---------
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.

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
difficulties, so we shall avoid their consideration entirely. Instead,
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.

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.

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 then easy to see the form that the changes will take if the points
were to become infinitely numerous and close together.
KSW - see my Brusselator examples

Ashby Terminology (page 10)
-----------------
operand - that which is acted on
transform - what the operand is changed to
operator - the factor that acts on the operand
transition - the change that occurs
transformation - a set of transitions, on a set of operands

page 10 - sunshine
-------
2/2. 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
    pale skin → dark skin
is the transition.
The transition is specified by the two states and the indication
of which changed to which.

TRANSFORMATION

2/3. 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,
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.
KSW:
 - each of these 4 transitions is a change in state on an object, where the operand object is the same as the transform object
 - sunshine is a Xholon active object
 - skin, soil, plate, pigment are passive objects, with a state variable that may be discrete (2 possible values) or continuous

page 11 - letter transposition code
-------
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 alphabet,
Z being turned to A; so CAT would become DBU. The transformation is defined by the table:
A → B
B → C
…
Y → Z
Z → A

For convenience of printing, such a transformation can also be
expressed thus:
↓ A B … Y Z
  B C … Z A

2/4. Closure.

2/5. A transformation may have an infinite number of discrete
operands

2/6. Notation. Many transformations become inconveniently
lengthy if written out in extenso. Already, in S.2/3, we have been
forced to use dots ... to represent operands that were not given
individually. For merely practical reasons we shall have to
develop a more compact method for writing down our transformations.
Such as any of these three:
    Operand → operand plus three
    Op. → Op. + 3
    n → n + 3 (n = 1, 2, 3, 4)

2/7. The transformations mentioned so far have all been characterised
by being “single-valued”. A transformation is single-valued
if it converts each operand to only one transform.

2/8. Of the single-valued transformations, a type of some importance
in special cases is that which is one-one. In this case the
transforms are all different from one another.

A transformation that is single-valued but not one-one will be referred to as many-one.

2/9. 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 its operand.

2/10. Representation by matrix. All these transformations can be
represented in a single schema, which shows clearly their mutual
relations.
Thus every transformation can be shown as a matrix.

REPEATED CHANGE

2/11. Power.
such a transformation may be applied more than once, generating a series of changes analogous to the
series of changes that a dynamic system goes through when active
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”

2/12. The trial in the previous exercise will make clear the importance
of closure. An unclosed transformation such as W cannot be
applied twice; for although it changes h to k, its effect on k is
undefined, so it can go no farther. The unclosed transformation is
thus like a machine that takes one step and then jams.

2/13. Elimination. When a transformation is given in abbreviated
arm, such as n' = n + 1, the result of its double application must be found

2/14. Higher powers. Higher powers are found simply by adding
symbols for higher transforms, n"', etc., and eliminating the symbols
for the intermediate transforms.

So the triple application causes the same changes as would be
caused by a single application of m' = 8m – 21. If the original was T, this is T3.

2/15. Notation. 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 n,
the symbol n' does not show which one has acted. For this reason,
another symbol is sometimes used: if n is the operand, and transformation
T is applied, the transform is represented by T(n).

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 T2(n). In
exactly the same way T(n) may perhaps become operand to a
transformation U, which will give a transform U(T(n)).

V is said to be the product or composition of T and U

2/17. Kinematic graph.
KSW - this is about combining individual edges into a graph

Chapter 3
---------
THE DETERMINATE MACHINE

3/1. Having now established a clear set of ideas about transforma-
tions, we can turn to their first application: the establishment of an
exact parallelism between the properties of transformations, as
developed here, and the properties of machines and dynamic sys-
tems, as found in the real world.

determinate machine - that which behaves in the same way as does a closed single-valued transformation.

It should be noticed that the definition refers to a way of behav-
ing, not to a material thing. We are concerned in this book with
those aspects of systems that are determinate—that follow regular
and reproducible courses. It is the determinateness that we shall
study, not the material substance.

this Chapter will deal only with the machine in isolation—the machine to which nothing actively is being done.

As a simple and typical example of a determinate machine, con-
sider a heavy iron frame that contains a number of heavy beads
joined to each other and to the frame by springs.

state of a system - any well-defined condition or
property that can be recognised if it occurs again. Every system
will naturally have many possible states.

trajectory or line of behaviour

each transform is single-valued

examples:
- A bacteriological culture that has just been inoculated will
increase in “number of organisms present” from hour to hour.
- Next consider a clock, in good order and wound, whose hands,
pointing now to a certain place on the dial, will point to some
determinate place after the lapse of a given time.
- A series of changes as regular as those of the clock are not
readily found in the biological world, but the regular courses of
some diseases show something of the same features.
- Next consider an iron casting that has been heated so that its
various parts are at various but determinate temperatures.
- certain forms of reflex animal behaviour.
 - male and female threespined stickleback

By relating machine and transformation we enter the discipline
that relates the behaviours of real physical systems to the proper-
ties of symbolic expressions, written with pen on paper

To summarise: Every machine or dynamic system has many
distinguishable states. If it is a determinate machine, fixing its cir-
cumstances and the state it is at will determine, i.e. make unique
the state it next moves to. These transitions of state correspond to
those of a transformation on operands, each state corresponding to
a particular operand. Each state that the machine next moves to
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 record-
ing the next state. And since a determinate machine cannot go to
two states at once, the corresponding transformation must be sin-
gle-valued.

3/2. Closure.

3/3. The discrete machine.

3/4. Machine and transformation.
The parallelism between
machine and transformation is shown most obviously when we
compare the machine’s behaviour, as state succeeds state, with the
kinematic graph (S.2/17), as the arrows lead from element to ele-
ment.

VECTORS
3/5.
“compound” states
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 conve-
niently written thus: ( a 1 , a 2 , . . ., a n ), which means that the first
component has the particular value a 1 , the second the value a 2 ,
and so on.

Here are some well-known examples.
(1) A ship’s “position” at any moment cannot be described by a
simple number; two numbers are necessary: its latitude and its
longitude. “Position” is thus a vector with two components.
(2) “The weather at Kew” cannot be specified by a single num-
ber, but it can be specified to any desired completeness by our tak-
ing sufficient components. An approximation would be the
vector: height of barometer, temperature, cloudiness, humidity),
and a particular state might be (998 mbars, 56.2°F, 8, 72%).
(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:

Two vectors are considered equal only if each component of
the one is equal to the corresponding component of the other.

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
part of the transformation’s definition). Suppose, for instance, the
“system” consists of two coins, each of which may show either
Head or Tail.

Finally, nothing said so far excludes the possibility that some or
all of the components may themselves be vectors!

3/6. Notation.

3/7. (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 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. If he has done this, he will be better equipped to appre-
ciate the meaning of this section, which summarises the method.

3/8. After this discussion of differential equations,

3/9. “Unsolvable” equations.

3/10. Phase space.

3/11. What is a “system”?
every material object contains no less than
an infinity of variables and therefore of possible systems.

KSW - my ideas about modeling discrete systems using unit values (=1)

Chapter 4
---------
THE MACHINE WITH INPUT
4/1. In the previous chapter we studied the relation between trans-
formation and machine, regarding the latter simply as a unit. We
now proceed to find, in the world of transformations, what corre-
sponds 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”.


References
----------
(1) http://pespmc1.vub.ac.be/books/IntroCyb.pdf
    AN INTRODUCTION TO CYBERNETICS, by W. ROSS ASHBY

(2) https://en.wikipedia.org/wiki/Algebraic_structure
    see my Magma Xholon Workbook
    
    One set with operations
    =======================
     Simple structures: no binary operation:
     -----------------
      Set: a degenerate algebraic structure S having no operations.
      Pointed set: S has one or more distinguished elements, often 0, 1, or both.
      Unary system: S and a single unary operation over S.
      Pointed unary system: a unary system with S a pointed set.
    
    example from Ashby page 11, is a Unary system, but it has two unary operations over S (the set of Letters)

(3) https://en.wikipedia.org/wiki/William_Ross_Ashby
    W. Ross Ashby (6 September 1903 in London – 15 November 1972) was an English psychiatrist and a pioneer in cybernetics,
    the study of the science of communications and automatic control systems in both machines and living things.
    His first name was not used: he was known as Ross Ashby.
    His two books, Design for a Brain and An Introduction to Cybernetics, were landmark works.
    They introduced exact and logical thinking into the brand new discipline of cybernetics and were highly influential.
    
    Despite being widely influential within cybernetics, systems theory and, more recently, complex systems,
    Ashby is not as well known as many of the notable scientists his work influenced, including Herbert A. Simon, Norbert Wiener, Ludwig von Bertalanffy, Stafford Beer, Stanley Milgram, Kevin Warwick, and Stuart Kauffman.
    
    Ashby kept a journal for over 44 years in which he recorded his ideas about new theories.
    He started May 1928, when he was medical student at St. Bartholomew's Hospital in London.
    Over the years, he wrote down a series of 25 volumes totalling 7,189 pages.
    In 2003, these journals were given to The British Library, London, and in 2008, they were made available online as The W. Ross Ashby Digital Archive.

(4) http://ashby.de/rossashby.info/index.html
    Ashby's journals

(5) https://archive.org/details/designforbrainor00ashb
    Design For a Brain

(6) amazon.ca
    An Introduction to Cybernetics, $16.48 CDN, paperback reprint
    Design for a Brain: The Origin of Adaptive Behavior, $17.76

]]></Notes> 

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  <!-- example from page 10 -->
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  <!-- example from page 11 -->
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  <Letters/>
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  <EncodedMessage superClass="Attribute_String"/>
  <DecodedMessage superClass="Attribute_String"/>
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  <!-- example from page 21-23 -->
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<PhysicalSystem>
  
  <!-- example from page 10, Exposure to Sunshine -->
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      <PhotographicPlate state="unexposed"/>
      <Pigment state="coloured"/>
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  <!-- example from page 11
  A single instance of a Set (the letters),
  with two unary operators (encode, decode).
  encode: L → L
  decode: L → L
  -->
  <Example roleName="Letter transposition code (page 11)">
    <!--<UncodedMessage>CAT</UncodedMessage>-->
    <UncodedMessage>THE CAT IN THE HAT</UncodedMessage>
    <EncodedMessage/> <!-- should become DBU -->
    <DecodedMessage/> <!-- should become CAT -->
    <Letters>
      <Letter roleName="A"/>
      <Letter roleName="B"/>
      <Letter roleName="C"/>
      <Letter roleName="D"/>
      <Letter roleName="E"/>
      <Letter roleName="F"/>
      <Letter roleName="G"/>
      <Letter roleName="H"/>
      <Letter roleName="I"/>
      <Letter roleName="J"/>
      <Letter roleName="K"/>
      <Letter roleName="L"/>
      <Letter roleName="M"/>
      <Letter roleName="N"/>
      <Letter roleName="O"/>
      <Letter roleName="P"/>
      <Letter roleName="Q"/>
      <Letter roleName="R"/>
      <Letter roleName="S"/>
      <Letter roleName="T"/>
      <Letter roleName="U"/>
      <Letter roleName="V"/>
      <Letter roleName="W"/>
      <Letter roleName="X"/>
      <Letter roleName="Y"/>
      <Letter roleName="Z"/>
    </Letters>
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  <!-- example from page 21-23, Kinematic graph -->
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T: ↓
A B C D E F G H I J K L M N P Q
D H D I Q G Q H A E E N B A N E
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}
xmlStr += "</KinematicGraph>\n";
me.append(xmlStr);
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  me["state"] = "bleached";
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  for (var i = 0; i < emsg.length; i++) {
    var eltr = emsg[i];
    var dltrNode = ltrsNode.xpath("Letter[@roleName='" + eltr + "']");
    //dmsg += dltrNode.transformD.role();
    if (dltrNode) {
      dmsg += dltrNode.transformD.role();
    }
    else {
      dmsg += eltr;
    }
  }
  dmsgNode.text(dmsg);
}
}
//# sourceURL=Decoderbehavior.js
]]></Decoderbehavior>

<SvgClient><Attribute_String roleName="svgUri"><![CDATA[data:image/svg+xml,
<svg width="100" height="50" xmlns="http://www.w3.org/2000/svg">
  <g>
    <title>Letters</title>
    <rect id="PhysicalSystem/Example[2]/Letters" fill="#98FB98" height="50" width="50" x="25" y="0"/>
    <g>
      <title>Sunshine</title>
      <rect id="PhysicalSystem/Example[1]/Operators/Sunshine" fill="#6AB06A" height="50" width="10" x="80" y="0"/>
    </g>
  </g>
</svg>
]]></Attribute_String><Attribute_String roleName="setup">${MODELNAME_DEFAULT},${SVGURI_DEFAULT}</Attribute_String></SvgClient>

</XholonWorkbook>