Ashby’s definition of a transducer (or machine with input, depending on the context) requires a set of closed single-valued transformations that are the canonical representation of the machine’s “laws” of operation, and a variable, the parameter. According to Ashby the behaviour of the (controlled) machine can only be selected from the outside by a controlling machine changing the value of the parameter.
In the following example taken from Ashby’s book pp. 49., two “transducers” (P and R) are joined together in a very simple (unidirectional) manner. When coupled together P controls (dominates) the behaviour of R without receiving any feedback from R. The canonical and block representation of R is given as follows:
P is defined with a singular closed transformation on three states. In order to couple the two transducers one has to decide which state of R <1, 2, 3> shall be defined by which state of P <i, j, k>. Ashby introduces here a new (coupling) transformation Z which may be “arbitrary and completely under the control of whoever arranges the coupling“. The only requirement is “that the two machines P and R work on a common time-scale, so that their changes keep in step“.
The complete integrated block diagram of Ashby’s transducer can be then depicted like this:
From the table of transformations below, it is evident that the coupled system for the initial conditions (b, j) briefly “selects” behaviour R3 in the first step. Immediately after that changes the behaviour to R2 and enters in an oscillatory equilibrium between the states c↔d and i↔k.
It is important to note at this point the fact that the behaviour selected by the transformation Z is not applied immediately (in the same step). If we want to keep the whole machine in sync, the behaviour selected in step tn must be applied only after at least a unit delay (in the next step tn+1). This fact is true and shall be more clear when we see Shannon’s description of a transducer in the next paragraph.
Shannon published his legendary Mathematical Theory of Communication in The Bell System Technical Journal back in 1948, eight years before the release of Asby’s Introduction to Cybernetics. Shannon’s description of the transducer (with memory) is much more simpler than Ashby’s, composed of just two functions with two variables. The full description is given here as follows:
Note that both functions f and g can be understood and described as complex multi-valued transformations similar to Ashby’s description from before.
The block diagram for Shannon’s transducer can be then depicted like this:
If we want to complicate things a little bit further, we can add to the simple unit delay another transformation (F) to get a proper “memory function”:
We can now define three functional transformations in a canonical notation with completely arbitrary elements as follows:
Note that this selection of variables is just an example to prove a point. As Ashby teach us, the selection can be practically anything as long as it follows the few rules described above.
We can now explore two timelines describing two different behaviours of the same transducer:
Note that even if the input in the first timeline is constant (a) the output of the system changes because the state changes. We can interpret this as if the system is “learning” and refining its “knowledge” by revisiting the same set of input data with an “upgraded” knowledge state.
In the second timeline the input sequence is repeating itself but we can see that, because the system is “state defined” and thus dynamical in nature, the state and output do not follow this regularity of the input. Note also that there is nothing “chaotic” in this system. The transformations are completely deterministic. However, even with a such a limited number of variables the system can exhibit some very complicated behaviour.
I’m not sure why Ashby never, as far as I know, adopted or even discussed Shannon’s description of the transducer in his enormous contribution to cybernetic theory and instead opted for a complex and cumbersome description like the one from above, requiring a parameter and an external controller system.
In my opinion, this elegant framework based on Shannon’s definition of a transducer with memory is far reaching and introduces multiple fundamental definitions for systems thinking and cybernetics:
- If the system is open to the flow of matter and energy but closed to the flow of information (as Ashby teaches us) then the control of the system must be internal, thus the distinction between control and controlled system from classical cybernetics is wrong and should be discarded from the discussion about complex dynamical systems.
- As a consequence of 1., strictly speaking, there is no transfer of information between systems either. What is transferred between systems (transducers) are messages, signals, etc. in the form of mater or energy (wave) structures. Information used for encoding a message by the source transducer (transmitter) is then re-created by the receiving transducer (receiver) at the destination and it this is obviously not the same information.
- As a consequence of 2., information and knowledge are also internal to the system. Information, once committed to memory (function F ), is used to build the knowledge (state α) of the system which (knowledge) is in turn used to extract new information (function g ) from external data as well as to formulate (control) the output (visible behaviour) of the system (function f ).
And one last (but not least) remark: This framework is so simple that can be very easily scaled and formalized in algorithm form (model) and used for simulating complex behaviour of basically any type of dynamical system with memory.