Regulation / Control / Guidance

Transformations

A fundamental notion in Ashby’s cybernetics theory, as described in his book “Introduction to Cybernetics” is that of a transformation. The term is used to describe any particular behaviour of a system. A transformation consists of a number of transitions (changes) on an ordered list of operands with associated results (the transforms). The elements of the transformation can be anything (even other transformations), and in general, there are no explicit requirements for the operands and transforms to be elements of the same domain. In addition, it is important to stress the fact that cybernetics in general is not interested in the underlying physical or other causes for the transformation.

If all the elements from the lower line (transforms) are also listed as operands on the upper line, the transformation is closed. Closure is of particular importance if a transformation is recursively applied when dealing with machine states, because for a non-closed transformation the machine will eventually stop (jam) when the resulting transform is not an operand.

Fig 1. Examples of transformations in Ashby’s notation

If each of the operands on the top is converted to only one transform on the bottom, the transformation is called single‑valued. If all transformations are single-valued and different they are said to be one-one. In the examples in Fig 1, transformation W is closed (all transforms are also operands), it is single valued (each operand has only one transition) but it is not one-one because two operands (p, s) will result with the same transform (q). Transformation R is not closed and can be understood as an “input-output” type of transformation, while for any given operand of U its appropriate transform represents the transformation that will be applied in some (other) coupled system.

Closed, single-valued transformations are interesting because they allow for the recursive application of the transformation on the results of the previous one. In the above examples, WxW on p will result with the transform r, or in a shorter notation W2(p)=r. Obviously this property also permits the coupling of two or more different transformations. If from the transformations above we define the coupling RxWxWxU with input 3 the result is W3, or RW2U(3)=W3. Note that all transformations are applied from left to right (e.g. R(3)=s W(s)=q W(q)=r U(r)=W3) and commutation is not allowed.

In Fig 2, an attempt is made to depict a block diagram of an”Ashby machine” going trough automatic recursive transformations which notation will, I believe, be useful in the discussion that follows. The block diagram on the left is the representation of the closed transformation W from above. The input vector for the transformation W is <p, q, r, s> while the output is <q, r, s>. Block z-1 is a “unit delay” ensuring the output symbol from the previous transformation is applied as an input in the next step.

Fig 2. Block diagram of the closed transformation W (left) and a generic recursive transformation (right)

State Change

Complex multi-valued transformations are described by Ashby in a matrix notation. In the following example the machine has three different “organizational” states (ways of behaving): A, B and C, that are associated with three closed transformations on a common set of operands (a, b, c and d). If the machine is in state A, the transformation WA will transform a→c but if the machine is in state C, the same machine will apply transformation WC which will now transform a→d.

Fig 3. A matrix and a block representation of a complex multi-valued transformation

Ashby identifies the vector <A, B, C> as the “parameter”, a “special input” the only purpose of which is to change the state of this “controlled” system as selected b an outside “controller”.

A natural question in this moment would be: “Are there other mechanisms that can change the behaviour (state) of a system?“. To answer this question we have to give a closer look to two descriptions for a transducer, one given by Ashby  in Chapter 4 of his book and another from Shannon in Part 1.8 of his seminal paper.