# charles consult blog

experience necessary

## sketches of cybernetics, part i.

After being invited to tweet about the cybernetition W. Ross Ashby, I tweeted a few ideas from cybernetics, and got no response – it occurred to me that no one knew what I was talking about. So, I thought I’d write an explanation of my first tweets on the subject to provide a background for the reader. I am going to be a brief as possible, without getting into philosophical matters such as epistemology, causation, and existence.

Ashby was a pioneer in the field of cybernetics. He made operational, important concepts such as: the determinate machine, markov machines, feedback, regulation, and requisite variety. His concepts such as the “black box” and stability are likely still part of most modern electrical engineering curricula.

These concepts are useful in situations where information is incomplete and or hidden, information is confusing, or when in complex and or competitive situations (think – “fog of war“). The subjects of stability and regulation are still relevant in light of current financial, economic, and political discussions. The subject matter is deep, long, and wide - I can only provide here, a brief glimpse into it.

This article was motivated by someone tweeting the well known quote:

“#Logic will get you from A to B. #Imagination will take you everywhere.” ~ Albert Einstein #systemsthinking #intuition

While I believe this to be true, I also believe that there needs to be something tethering imagination to reality, so I tweeted this:

Without logic, what’s the difference between imagination and hallucination? #systemsthinking #logic

I remembered that Ashby, in his book, An Introduction to Cybernetics1, started the subject matter using logic, developing the concept of change, and the procedure for making a truth table. I will introduce the subject by defining systems and describing their general behavior, then move to the logic of tabular methods/truth tables. We will then look at matrix, and graphical representations of change or transformation. Later articles will cover future states, various system categories and theories, information, stability, requisite variety, and control.

#### Part I: Introduction

Systems are easy. They have an “inertia”. They stay in one state, until energy is added, then they go to another. #Ashby #systemthinking

What are systems and how do they behave?

We shall define a system simply as anything (material or immaterial) that is predictable and describable.

How do systems behave? Large scale abstraction of the behavior a simple system – first, they exist in one state, then after a change in energy, they change to another state. The simplest example of state change is ice transforming to water via addition of heat (as opposed to the simplest system – that of existence or non-existence). A more active example is, a pendulum, swinging from one side (state) to the other. The pendulum example is also similar (depending on the precise behavior and mechanism) to other system metaphors such as: duality, flip flop, yin-yang, “hunting” in feedback systems, simple harmonic motion, etc. The pendulum is the one simplest intuitive system metaphors and can reflect a number of different underlying mechanisms.2

System behaviors can be described directly in terms of the observed changes in that system, but they are commonly given names such as: result, behavior, outcome, move, etc. This is because systems can take an endless list of metaphorical names such as: life, sport, game, relationship, interaction, dilemma, conversation, communication, institution, economy, society, machine, idea, spirit, thought, etc. These metaphors are consistent by our definition – anything that is predictable or describable.

#### Part II: The Observation and Representation of Change

Toffler tells us in Future Shock, that “change” is the thing to deal with. In I.T.C. Ashby shows us how. #systemsthinking #transdisciplinary

The first step is learning to observe and record meaningful information. The idea is to observe change because it is change that can indicate how things behave, and can point the way to reality.

Ashby begins with a simple example. If you have pale skin, and you stand in the sun, your skin will transform into dark skin. Written thusly:

pale skin —–> dark skin

The important concept is that of transformation. In this example, pale skin is transformed to dark skin by use of sunlight. Other transformation involving sunlight cold be:

cold soil —–> warm soil

closed flower —–> opened flower

brown leaves ——> green leaves

etc.

The set of transitions on a set of operands (the thing being transformed) is called a transformation, and can be written as:

A —–> B

B —–> C

. . .

Y —–> Z

Z —–> A

Note that the transformation is sufficiently defined merely by giving a set of operands and stating what happens to them, and not necessarily why. We can shelve the idea of causality for right now because we are dealing with the first step – observation.3

Truth table – tabular methods
Any set of transformations can describe the behavior of a system (in this context, being anything with a predictable and describable behavior). Physical systems tend to be deterministic systems, unless you are observing large numbers of objects or are getting quantum mechanical – then it becomes statistical. Systems can be written by various tabular forms (vertical, horizontal), and can be represented as we shall see, as a matrix.

In the same way you assigned arrows between letters, you can put the same information in a tabular array. A, B, and C go on the top and side – where the transform goes to is denoted by an “+”.

Here, A transforms to A, and B and C transform to C. Visible in both vertical and horizontal tables. For both columns and rows, they are labeled A, B, C, vertically and horizontally.

Note that if you replace the “+” by “1″, you have the idea of a digital (0, 1) representation. Although it’s more complicated than I’ve suggested, the digital form is important because, Claude Shannon proved in his masters thesis4, that any analog assemblage of switching and routing equipment had an equivalent digital representation, thus, digital representations are “complete” representations for “real” systems.

Thus, a table of transformations can be equivalent to a described system, and has an equivalent transformation matrix. Matrices, like the systems they represent, can be classified by looking at the relationship within the elements. In the following examples, first is a 3 x 3 matrix to show it is the same as a table. Then we have a 4 x 4 matrix with {A, B, C, D,} as a set of states that the system goes to (state A to state B, etc).

Matrix representations

$T = \left[ {\begin{array}{ccc} + & 0 & 0 \\ 0 & 0 & 0\\ 0 & + & + \\ \end{array} } \right]$

The matrix T is the equivalent matrix for the table given above, with the states {A, B, C}. The first row is the transformation of A to itself, the second row is no transformation, the third row is both B and C transforming to C. The columns also correspond to {A, B, C}. The matrices are square, that is the number of rows, columns, and states, are equal.

$I = \left[ {\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\\0 & 0 & 0 & 1 \\ \end{array} } \right]$

The identity matrix here, represents a transformation of the states {A, B, C, D} in which no change occurs (that is, A -> A, B -> B, etc.) and is represented by a matrix with “1″s down the main diagonal and “0″s elsewhere. This is also called an “identity transformation”. There are other kinds of matrices representing other kinds of relationships.

$M = \left[ {\begin{array}{cccc} + & 0 & 0 & +\\ 0 & 0 & + & 0 \\ + & 0 & 0 & 0 \\ 0 & + & 0 & +\\ \end{array} } \right]$

This is a matrix that is not single valued (notice that the first and last columns have repeating elements). This is equivalent to A -> A and A -> D, and or D -> B and D -> D, thus two elements each going to more than one state.

$M = \left[ {\begin{array}{cccc} 0 & + & 0 & 0\\ 0 & 0 & 0 & + \\ + & 0 & 0 & 0 \\ 0 & 0 & + & 0\\ \end{array} } \right]$

This is a matrix that is single valued. Ashby writes, “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. Thus not only does each operand give a unique transform (from the single-valued-ness) but each transform indicates (inversely) a unique operand”.5

This is the matrix form of a function. Note: The algebraic expression y = f(x) is a function if it passes the vertical line test (a relation is a function if there are no vertical lines that intersect the graph at more than one point). It is a one to one function if it passes both the vertical line test and the horizontal line test (If a horizontal line  intersects a function’s graph more than once, then the function is not one-to-one.).  For the sake of clarity, these are also called injunctive functions.

$M = \left[ {\begin{array}{cccc} 0 & 0 & 0 & 0\\ + & 0 & 0 & + \\ 0 & + & 0 & 0 \\ 0 & 0 & + & 0\\ \end{array} } \right]$

This is a matrix that is single valued but not one to one. Called many to one. Not by definition a mathematical function. The efficacy of the matrix or tabular method is that it works even if there is no clear functional or mathematical relationship (because you are concerned here with behavioral consistency and not necessarily logical consistency).

Ashby goes into enough math concerning concepts like closure, and identity elements, to convince the reader that the cybernetics is mathematically correct, while logically and physically consistent.6 Closure comes up when you are dealing with repeated transformations, and identity transformations occur when there is a transformation with no change. These are important, but not important to develop the main idea, that all systems that can be observed can be written as a matrix (or truth table) representation (these days a spreadsheet would be used – or a programming language).

Graphical representations – Kinematic graphs:

There is another useful representation, that of a kinematic graph.

These are two representations of the same 5 state system {A, B, C, D, E} in tabular and linear graphic form. This graphical form can be re-arranged to provide a more convenient and revealing shape, where you can more easily see that state D is an endpoint, coming from two different states {A, E}, as seen below.

Kinematic graphs are just a way of representing a system with the connections shown, and are equivalent to both matrices and linked lists (a computer data structure). This is important because there are important behaviors that can be deduced from these maps. Important properties include: endpoints, cycles, and basins.7

The endpoint a state that has been reached where the transformations stop at one particular place or behavior, such as, when a piece of paper is burnt – it’s starting state is unburnt paper, its final transformation is into “ash.” A cycle is a behavior that goes from one state to another in a cycle – such as a stop light – it goes from green, to yellow, to red, then back to green. Finally, there is the basin, that is like an endpoint, where it goes to a state, and unless there is an addition of energy, the system will stay at that state – like a car that has run out of gas.

#### Part III Representation of possible states

The important thing about a system is that is should be predictable – that is, if it is in one state, there should be a predictable change of state (e.g. if it is summer, you know that autumn is next). There are ways of representing possible future states, Ashby calls this representation a “phase space” of possible behaviors, moves or outcomes.8

Phase space

All the possible behaviors of a system are called its “phase space”. Often – momentum vs position #systemthinking #transdisciplinary #Ashby

To be continued. . . .

Footnotes

1. W. Ross Ashby,  An Introduction to Cybernetics, Chapman & Hall Ltd., 1957. Although this book has been a part of my personal library since the mid-seventies, I owe a debt of gratitude to Mr. Ashby for writing the book, and his family and estate for releasing the copyrights of this book into the public domain. The book is available in pdf form here.
2. Although a pendulum can approximate many large scale system behaviors (depending on how the pendulum is driven or damped), trigonometric functions (i.e. sin(x) and cos(x) – the x and y components of circles and waves) do it better. In fact, first and second order differential equations (being themselves, system representations) can be solved by breaking their solution equations down into their trigonometric components.
3. Causality – the question of what “why” means and about “why” something behaves the way that it does, is subject to interpretation. Physics is considered to be a reasonably determinate machine.  Mechanical causality is sufficient for physical aspects of existence and can be handled with mathematics (e.g. differential equations). In subject matters where large numbers are important (economics, sociology, ecology, etc.) the use of probability and statistical inference become more useful. Historical and cultural factors, power structures, and politics shape our ideas of causality as well – the larger, more important the issue, the more complex causation becomes. Finally, there are subject matters such as art, philosophy, religion, number theory, computability, etc., which point out that which is indeterminate. Mathematical and reductionist approaches are thus insufficient for all subject matters. That said, there are still “indeterminate approaches” to non-structured information and situations.
4. Claude Shannon, “A Symbolic Analysis of Relay and Switching Circuits,” unpublished MS Thesis, Massachusetts Institute of Technology, Aug. 10, 1937 – direct in applicability and broad in scope, it is widely considered the “best masters thesis ever”.  Soon after, he wrote A Mathematical Theory of Communication, for Bell Labs. This is the basis thesis for the field of Information Theory, and is used from communication systems, to business models, to secrecy systems.
5. Ashby, An Introduction to Cybernetics, 2/8.
6. Id. Identity matrix, 2/9, 2/10.
7. These kinematic expressions can be drawn as “digraphs” or “directed graphs”. Digraphs have properties such as; sources, sinks, adjacency, grouping, that make them useful for planning and decision-making. Also useful for activities that have linked constraints, such as critical path analysis. See NIC class, Introduction to Graphs.
8.  Ashby, An Introduction to Cybernetics, Phase space, Sections: 3/10, 3/11, and 6/10.

Category: knowledge

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