W. Ross Ashby's
An Introduction to Cybernetics
Notes

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Chapter 2: Change

The most fundamental concept in cybernetics is that of "difference," either that two things are recognisably different or that one thing has changed over time.

Often a change occurs continuously, that is, by infinitesimal steps.

• However, continuous change raises a number of purely mathematical difficulties, so we shall avoid their consideration entirely.
• We shall assume that all changes occur by finite steps in time and that any differences are also finite -- that change occurs by a measurable jump. This has the advantage of allowing us to decide any question by simple counting, making it easy to determine whether we are right or not.
• The discrete can often be carried over to the continuous, for practical purposes, by graphing the discrete values, which will make it easy to see the form that the changes will take if the points become infinitely numerous and close together.
• However, we lose nothing by keeping discussion to the case of finite differences (by taking finer and finer time-slices and observing the convergence to the continuous case). The discrete can always be converted to the continous form if that is desired.

At this point, Ashby introduces some basic terms and their definitions:

• Operand: That which is acted upon.
• Operator: The factor acting upon the operand.
• Transform: What the operand is changed to.
• Transition: The change that occurs.
• Transformation: A set of transitions, on a set of operands. (Here the operator is acting on more than one operand.)

We can represent a transormation by a set of operands arranged in a row and, in the next row, the set of transforms corresponding to the operands:

a b c d e f
b c d e f a

Here, a is transformed to b, b to c, c to d, and so on. An important property of a transformation is:

• Closure: The set of transforms obtained contains no element that is not already present in the set of operands, i.e., the transformation introduces no new element. The set of operands is said to be closed under the transformation. The above example is closed. So is:

  a b c d e f
  b a d d a f

  Note that in this case not all the operands reappear in the transform. However, no new element is introduced.

Notation:

• Many transformations become inconveniently long when written out completely. However, in many cases some simple relationn can be found that links all the operants to their respective transforms. The transformation can then be written more compactly:

  n --> n + 3  (n = 1, 2, 3, 4)

• In some cases we need a symbol for the transform of such a symbol as n. We will simply add a prime to the operator:

  n' = n + 3  (n = 1, 2, 3, 4)

  In this example, the transform of n = n'.

Some other terms (in addition to closed) that may describe a given transformation:

• Single-valued: A transformation is single-valued if it converts each operand to only one transform. Thus, A --> B is single-valued but A --> B or C is not.
• One-one: A type of single-valued transformation in which the transforms are all different from one another. Not only does each operand give a unique transform, but each transform gives (inversely) a unique operand.
• Many-one: A single-valued transformation that is not one-one. Here, different operands may yeild the same transform so that, inversely, the same transform may yield several operands.
• Identity: The identical transform produces no change: the transform is identical to the operand in each case.

Representation by Matrix: Every transformation can be represented by a matrix: The columns give the all operands and the rows give all the possible transforms. The intersections of row and column are marked with a "+" where the transform does occur and with "0" otherwise. Example:

A B C
A C C

If the transformation is large (e.g., all positive integers), dots may be used in the matrix, if the meaning is unambiguous.

Repeated Change

With respect to these repeated applications:

• The double application of a single transformation produces changes equivalent to those of the single application of another single transformation.
• The second is said to be the square of the first and to be one of its powers.
• If the original transformation is symbolized as T, then its square is symbolized as T².

The importance of closure is that an unclosed transformation cannot be applied twice.

Elimination:

• When a transformation is written in the appreviated form, such as n' = n + 1, the double-application can be written as n'', and higher powers in like fashion.
• We can then use substitution and elimination to obtain each power. The second power of the above is n'' = n' + 1 = (n + 1) + 1 = n + 2.
• Higher powers can be found by like substitution and eliminating the symbols for intermediate powers.

Notation: If more than one transformation can act on the same operand, then we need some way to indicate which transformation was used:

• If transformation T is used on operand n, then this will be symbolized as T(n).
• The double-application of T is T²(n), etc.
• If transformation T is applied to operand n and transformation U is applied to the transform of operand n, the result is the product, symbolized U(T(n)). [The order in which the two transformations are applied matters! T(U(n)) gives a different result.]
• Application of transformation T then U defines a new single transformation. Call it V. V is said to be the product or composition of T and U.
• V may not exist if some of T's transforms are not operands for U.

Kinematic graph:

• When a transformation is closed, one can study its effect on a single operand over many, repeated, applications. The method corresponds, in the study of a dynamic system, to setting it to some initial state and then allowing it to go on, without further interference, through such a series of changes as its inner nature determines.
• The series of transitions can be represented by writing down each element and then joining them such that each arrow goes to another element only if the first element is transformed in one step to the second element. The resulting graph is called a kinematic graph of the transformation.
• The graph has a button-and-string appearance, but the positions of the buttons (and thus the shape of the graph) are not important, depending as they do on where one happened to write the element.
• If you start with a given element and follow the arrows through the successive connected elements, you have a moving point termed the representative point.
• By starting at any state and following the chain of arrows we can verify that, under repeated transformation, the representative point always moves either to some state at which it stops, or to some cycle around which it moves indefinitely. These regions are the graph's basins. [Note: Ashby's two possibilities have since been augmented by a third: the "strange attractor" of chaos theory. It is not treated here.]