W. Ross Ashby's
An Introduction to Cybernetics
Notes

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Chapter 3: The Determinate Machine

This chapter establishes and exact parallelism between the properties of transformations and the properties of machines and dynamic systems, as found in the real world.

What is a "determinate machine"?

• A determinate machine is defined as that which behaves in the same way as does a closed single-valued transformation.
• The justification for this definition is that it works in practice.
• This chapter will deal only with the determinate machine, and only with the machine in isolation -- the machine to which nothing actively is being done.
• State of a system: Any well-defined condition or property that can be recognized if it occurs again. Every system will naturally have many possible states.
• The operands of a transformation correspond to the states of a system.
• The series of positions taken by the system in time corresponds to the series of elements generated by the successive powers of the transformation.
• Such a sequence of states defines a trajectory or line of behavior.
• The fact that a determinate machine, from one state, cannot proceed to both of two different states corresponds, in the transformation, to the restriction that each transform is single-valued.

The term "operator" is often poorly defined and somewhat arbitrary -- a concept of little scientific use. The transformation, however, is perfectly well defined, for it refers only to the facts of the changes, not to more or less hypothetical reasons for them. As an example of a well-defined but non-numerical transformation Ashby offers Tinbergen's series of behaviors during the courtship of the three-spined stickleback, which defines a typical trajectory.

By relating machine and transform we enter the discipline that relates the behaviors of real physical systems to the properties of symbolic expression, written with pen and paper. This discipline includes all of mathematical physics but is more inclusive; physics tends to treat cheifly systems that are continuous and linear, restrictions that make its methods harldy applicable to biological subjects, where systems are almost always nonlinear and non-continuous, and in many cases not even metrical.

Closure: The transformation that represents a machine must be closed. The typical machine can always be allowed to go in in time for a little longer, simply by the experimenter doing nothing. This means that no particular limit exists to the power that the transformationcan be raised to. Only the closed transformations allow this raising to any power.

The discrete machine: Most machines are smooth-working, while the transformations discussed thus far change by discrete jumps. However, these discrete transformations are the best introduction to the subject, and if needed the discrete can be changed to the continuous by making the time-slices small enough to approximate the continuous. Further real systems are observed at discrete points in time, just as in the transformation.

Machine and transformation:

• Each possible state of the machine corresponds uniquely to a particular element in the graph, and vice versa.
• Each succession of states that the machine passes through because of its inner dynamic nature corresponds to an unbroken chain of arrows through the corresponding elements.
• If the machine goes to a state and remains there, the element that corresponds to the state will have no arrow leaving it (or a re-entrant one).
• If the machine passes through a regularly recurring cycle of states, the graph will show a circuit of arrows passing through the corresponding elements.
• The stopping of a machine by the experimenter, and its restarting from some new, arbitrarily selected, state corresponds, in the graph, to a movement of the representative point from one element to another when the movement is due to the arbitrary action of the mathematician and not to an arrow.
• When a real machine and a transformation are so related, the transformation is the canonical representation of the machine, and the machine is said to embody the transformation.

Vectors

About vectors:

• Systems often have states whose specification demands further analysis. For example, the "state" of a marathon race at a certain hour would give the positions (states) of each runner at that hour.
• These simultaneously taken positions, as a set, specify the "state" of the race.
• Such a quantity is a vector, which is defined as a compound entity, having a definite number of components.
• A vector is conveniently written thus: (a₁, a₂, . . ., a_n).
• Two vectors are considered equal only if each component of one is equal to the corresponding component of the other. [Here, the components of the two vectors are the same and have the same values.]
• If two vectors have different components, then they are not comparable.
• When a vector is transformed, the operation is in no way different from any other transformation, provided we remember that the operand is the vector as a whole, not the individual components (though how they are to change is an essential part of the transformation). Example of a transformation on vector operands:

  (H,H) (H,T) (T,H) (T,T)
  (T,H) (T,T) (H,H) (H,T)

Notation: Transformations involving operands that are vectors are made up of sub-transformations that are applied simultaneously, i.e., always in step. The transformation F, operating on the vector (a, b, c), is equivalent to the simultandous action of the three sub-transformations, each acting on one component only:

	a'	=	b
F:	b'	=	c
	c'	=	b + c

[Note: Caution, Ashby's examples all depict a transform having only a single (vector) operand. It thus resembles a transformation having multiple single-variable operands.]

This optional section summarizes the method of mathematical physics. The physicist starts by naming his variables -- x₁, x₂, . . ., x_n. The basic equations of the transformation can then always be obtained by the following fundamental method:

1. Take the first variable, x₁, and consider what state it will change to next. If it changes by finite steps it will change to x₁'; if continuously the next state will be x₁ + dx₁. (In the latter case he may equivalently consider the value of dx₁/dt.)
2. Use what is known about the system, and the laws of physics, to express the value of x₁', or dx₁/dt (i.e., what x₁ will be) in terms of the values that x₁, ..., x_n (and any other necessary factors) have now. In this way some ewuation such as

   x₁' = 2ax₁ - x₃   or   dx₁/dt = 4k sin x₃

   is obtained.

3. Repeat the process for each variable in turn until the whole transformation is written down.

The set of equations so obtained -- giving, for each variable in the system, what it will be as a function of the present values of the variables and of any other necessary factores -- is the canonical representation of the system. It is the standard form to which all descriptions of a determinate dynamic system may be brought.

• If the functions of the canonical representation are all linear, the system is said to be linear.
• Given an initial state, the trajectory or line of behavior may now be computed by finding the powers of the transformation.

Unsolvable equations: If a closed and single-valued transformation is given, and also an initial state, the the trajectory from that state is both determined (i.e. single-valued) and can be found by computation (by taking the powers of the transformation). This process of deducing a trajectory when given a transformation and an initial state is mathematically called "integrating" the transformation. (When the equations are differential, the process is called "solving" the equations.) One cannot always obtain a solution analytically, by solving differential equations, for some such equations are "nonintegral" or "unsolvable." However, one can always obtain the trajectory by computation, to get what is wanted in practice.

Phase space:

• When the components of a vector are numerical variables, we can represent the variables on a cartesian graph whose axes represent the variables. (For a two-variable vector this would take the form of an x, y graph in two-dimensional space.) Individual vector values would then be plotted as points on the graph. If we start with a particular vector, we can use this as an operand and determine its transform, then repeat the process to determine the trajectory. We can then plot all the points of this trajectory. Doing this for a whole series of initial states yields the phase-space of the system. (This is just the kinematic graph of the system; however, because the numerical quantities are represented by the graph's axes, the button-and-string freedom of previous kinematic graphs is lost.)
• The phase-space shows at a glance the whole range of trajectories. In this way some property may be displayed, or thesis proved, with the greatest of ease, where the algebraic form would have been obscure.
• Beyond three dimensions this representation is no longer possible, but a sketch representing the higher-dimensional structure may still be useful, especially when what is significant are the general topological, rather than the detailed, properties.
• The words phase space are sometimes used to describe the empty space of the graph before the arrows are drawn in. The context will make clear which meaning is intended.

What is a "system"?:

Every real determinate machine or dynamic system corresponds to a closed, single-valued transformation. However, it does not follow that the correspondence is always obvious. In fact, it may prove difficult to identify the set of variables that will yield such a correspondence. In scientific investigation the object is often to find the right set of variables that will do so (e.g., for a pendulum, the vector angular deviation, angular velocity yields the desired transformation, while angular deviation alone does not).

What is "the system"? It is not simply the material object, for every material object contains no less than an infinity of variables and therefore of possible systems. However, in the world around us only certain sets of facts are capable of yielding transformations that are closed and single. The experimenter, attempting to account for some particular behavior, generates lists of variables and attempts to find a list that will generate a closed, single-valued transformation. Those variables then define the system.

(Sometimes what is wanted is that certain probabilities shall be single-valued; what is important is the probability, not the variable giving the probability. A system with such probabilities may be predictable in the sense that the probabilities are constant.)

We can state that every determinate dynamic system corresponds to a single-valued transformation simply because science refuses to study the other types, such as the one-variable pendulum, dismissing them as "chaotic" or "non-sensical." It is we who ultimately decide what we will accept as "machine-like" and what we will reject.