Regulation / Control / Guidance

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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 behavior 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, or even systems science in general, is not interested in the underlying physical substance that causes the transformation, just its form.

If all the elements from the lower line (the transforms) are also listed as operands on the upper line, the transformation is said to be closed. A closure is of particular importance if the transformation is recursively applied to itself as when dealing with machine states because if the transformation is not closed 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 in 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 the transform represents another transformation that may 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 in 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, I made an attempt to depict a block diagram of an”Ashby machine” going through 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 Fig 1. above. The input vector for the transformation W is <p, q, r, s> while the output is <q, r, s>. The block z-1 represents 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 behavior (state) of a system?”. To answer this question we have to give a closer look at 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. But that’s a different discussion.

Structural hierarchies and domains

In Maturana 2011 answer to T Froese and J Stewart 2010 the biologist reiterates one of the fundamental notions in autopoiesis, the duality of the autopoietic system:

All systems are composite entities that exist in two not intersecting operational-relational domains, the domain of the operation of their components, and the domain of their operation as totalities. Due to this the totality does not operate as an argument in what happens with its components, and the components do not operate as arguments in what happens with the totality.

In other words, the function(s) of the components even if they are the direct cause for it, do not define the behavior of the totality in its environment.

There is a whole hierarchy of non-intersecting phenomenal domains (structures) that can be identified in any system:

  • Physical (atoms, forces …)
  • Chemical (molecules, reactions …)
  • Biological (living organisms, reproduction, evolution)
  • Social

See how HA Simon explains this hierarchy of multiple non-intersecting domains in “biological and physical systems”:

The hierarchical structure of biological systems is a familiar fact. Taking the cell as the building block, we find cells organized into tissues, tissues into organs, organs into systems. Moving downward from the cell, well-defined subsystems-for example, nucleus, cell membrane, microsomes, mitochondria, and so on-have been identified in animal cells.

The hierarchic structure of many physical systems is equally clear-cut. I have already mentioned the two main series. At the microscopic level we have elementary particles, atoms, molecules, macromolecules. At the macroscopic level we have satellite systems, planetary systems, galaxies. Matter is distributed throughout space in a strikingly non-uniform fashion. The most nearly random distributions we find, gases, are not random distributions of elementary particles but random distributions of complex systems, i.e. molecules.

It is rather naive of Simon to select a cell as the building block if the system under inquiry is an animal’s organism as a totality in its environment. The structure of the resulting “system” spreads through multiple non-intersecting domains and it is too big and too complex to analyze. All building blocks (elements) of a system must be selected from the same (next lower) phenomenal domain (no mix and match of elements from multiple domains is allowed).

Also, Simon’s “physical systems” – even if they may have an identifiable hierarchy – are not systems at all, in the sense that they are just structures without a clearly identifiable intrinsic purpose or behavior, that is, they are a “system off” something (energy, matter, information), not a “system for” something (a need, purpose, function) with an input/response relation. To paraphrase physicist Howard H. Pattee, structures are passive and completely under the influence of inexorable natural laws, while systems are free to “invent” their local arbitrary rules and behave accordingly.

Hierarchies can be observed as “top-down” (elements on the lower level “obey” rules from the higher level) or “bottom-up” (properties on a higher level are generated or emerge from the workings of lower levels).

The former is the traditional cybernetic controller vs. controlled view of the system while the latter is an emergent (kihbernetic) approach to systems as totalities. The emergence of novel properties in the next (higher) non-intersecting domain of interaction is supported (generated) by the interaction of elements in the adjacent (lower level) domain.

E.g. in order for a chemical reaction to occur, all necessary conditions at the next lower physical (atomic) level must be satisfied. The “wetness” property of H2O does not exist on the atomic level (domain), it “emerges” only at the molecular level.

Hierarchy as a Tool

In the same paper, HA Simon is using the famous parable of the two watchmakers, Hora and Tempus, to promote the idea that hierarchical (modular) structures are somehow superior to other types of structures:

The watches the men made consisted of about 1,000 parts each. Tempus had so constructed his that if he had one partly assembled and had to put it down-to answer the phone say-it immediately fell to pieces and had to be reassembled from the elements. The better the customers liked his watches, the more they phoned him, the more difficult it became for him to find enough uninterrupted time to finish a watch.

The watches that Hora made were no less complex than those of Tempus. But he had designed them so that he could put together subassemblies of about ten elements each. Ten of these subassemblies, again, could be put together into a larger subassembly; and a system of ten of the latter subassemblies constituted the whole watch. Hence, when Hora had to put down a partly assembled watch in order to answer the phone, he lost only a small part of his work, and he assembled his watches in only a fraction of the man-hours it took Tempus.

Simon will also venture into a dubious discussion about the effects of a hierarchical and modular organization on the evolution of living systems as if living systems are constructed from readily available components the same way Hora and Tempus are building their watches. It is worth mentioning here that a hierarchy is not an inherent property of any machine or structure (living or not). Hierarchy is just a construct (tool) used by the observer to manage complexity while defining the system. A living machine is not “built of” sub-assemblies readily available in the environment as Simon would like you to believe. Cell division (growth) does not follow either Tempus or Hora’s “production” methods.

Living “production” is autopoietic (entities are building themselves), while Simon’s watchmakers are alopoietic entities (building something else than themselves).

The Illusion of Control

A hierarchy that is identified within the same domain (system) has to do with control. The elements of all dynamical systems are grouped in just three distinct layers (regulation, control, and guidance).

The structure of a system is defined within the boundaries it shares with the environment. The system boundary defines two non-intersecting phenomenal domains, the outside (behavioral) and the internal (functional) domain of interaction. The system’s control capability can not be extended outside this boundary with the environment. The system has full control only over its internal (functional) domain elements. Any change in its niche is purely incidental, caused by the system’s behavior and the results of its interaction (structural coupling) with this environment.

The distinction classical Cybernetics makes between control and controlled (sub)systems is not particularly useful for dynamical (state-defined) systems. Such systems are closed to information as well as control. Control is never accepted from another phenomenal domain. The “control system” is thus an integral part of and distributed through the system (not a subsystem).

As indicated before there are only three levels of control (functions) within every dynamical system:

Regulation  – monitoring outside disturbances and maintaining internal system states within the allowed (defined) limits that assure its integrity.

Control (proper) – monitoring the state of regulators and setting their regulation points and boundaries in such way as to provide a proper response to excessive disturbances threatening the integrity of the regulators, and for an orderly transition to new desired states

Guidance – monitoring trends and setting new desired states (goals) for the system to prepare for future challenges

Note that only the “regulatory envelope” of the system has to endure the full force of the variety in the environment. The most common strategy at this level is filtration of all unwanted influences and allowing just the needed ones to affect the system.

The other two levels (functions) have to deal only with the known internal variety of the system.

Communication

  • Within the system communication is fulfilled by signals, variable changes, and state transitions; for outside communication, the system has to produce messages in the form of observable behavior and/or artifacts.
  • Information and knowledge are internal to the system, not a commodity that is exchanged between systems through the environment.
  • The system uses knowledge to extract information and then incorporates (integrates) this (new) information into a new knowledge state

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