TH E ERR O R- CO N TR O LLED REG U LA TO R
Real objects may provide a variety of equally plausible “sys-
Tems”, which may differ from one another grossly in those prop-
Erties we are interested in here, and the answer to a particular
Question may depend grossly on which system it happens to be
Applied to.) (Compare S.6/22.)
The close relation between the Markovian machine and the
Determinate can also be shown by the existence of mixed forms.
Thus, suppose a rat has partly learned the maze, of nine cells, shown
In Fig. 12/11/1,
Single-valued, more than one arrow can go from each state. Thus
The Markovian machine
↓
a
b
c
A
0.2
0.8
.
B
0.3
0.7
.
C
0.1
0.5
0.4
Has the graph of Fig. 12/11/1, in which each arrow has a fraction
Indicating the probability that that arrow will be traversed by the
Representative point.
Fig. 12/10/1
In which G is the goal. For reasons that need not be detailed here,
The rat can get no sensory clues in cells 1, 2, 3 and 6 (lightly shaded),
So when in one of these cells it moves at random to such other cells
As the maze permits. Thus, if we put it repeatedly in cell 3 it goes
With equal probability to 2 or to 6. (I assume equal probability
Merely for convenience.) In cells 4, 5, 7, 8 and G, however, clues
Are available, and it moves directly from cell to cell towards G.
Thus, if we put it repeatedly in cell 5 it goes always to 8 and then to
G. Such behaviour is not grossly atypical in biological work.
The matrix of its transitions can be found readily enough. Thus,
From 1 it can go only to 2 (by the maze’s construction). From 2 it
Goes to 1, 3, or 5 with equal probability. From 4 it goes only to 5.
From G, the only transition is to G itself. So the matrix can be
Built up.
Ex.: Construct a possible matrix of its transition probabilities.
Fig. 12/11/1
Stability. The Markovian machine will be found on exami-
Nation to have properties corresponding to those described in Part I,
Though often modified in an obvious way. Thus, the machine’s kin-
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Ematic graph is constructible; though, as the transformation is not
228
In this particular example it can be seen that systems at c will all
Sooner or later leave it, never to return.
A Markovian machine has various forms of stability, which
Correspond to those mentioned in Chapter 5. The stable region is
A set of states such that once the representative point has entered
A state in the set it can never leave the set. Thus a and b above form
A stable region.
A state of equilibrium is simply the region shrunk to a single
State. Just as, in the determinate system, all machines started in a
Basin will come to a state of equilibrium, if one exists, so too do
The Markovian; and the state of equilibrium is sometimes called
An absorbing state. The example of S.9/4 had no state of equilib-
Rium. It would have acquired one had we added the fourth position
“on a fly-paper”, whence the name.
Around a state of equilibrium, the behaviour of a Markovian
Machine differs clearly from that of a determinate. If the system
Has a finite number of states, then if it is on a trajectory leading to
A state of equilibrium, any individual determinate system must
Arrive at the state of equilibrium after traversing a particular tra-
Jectory and therefore after an exact number of steps. Thus, in the
229
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
TH E ERR O R- CO N TR O LLED REG U LA TO R
First graph of S.2/17, a system at C will arrive at D in exactly two
Steps. If the system is Markovian, however, it does not take a
Unique number of steps; and the duration of the trajectory can be
Predicted only on the average. Thus suppose the Markovian
Machine is
↓ ab
A1 1/2
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B0 1/2
With a a state of equilibrium. Start a great number of such systems
All at b. After the first step, half of them will have gone to a and half
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