So he examines the record. Usually his first concern is to see
Whether the Box is absolute if the input state is given. He does this
By collecting:
(i) all the transitions that followed the input state α, sorting
Them into what g went to, what h went to, and so on through
All the output states;
(ii) the same for input β;
Iii) and so on through all the observed input states.
What he tries, in other words, is to fill in a set of transformations
Like those of S.4/1, and he examines what he gets to see if they are
Single- valued.
Thus, if the given protocol is tested, and if every one of the 16
Transforms is recorded, there results:
↓
α
β
F
Fff
Hhh
g
j
.
H
Jjj
Hh
J
Ff
Ff
One way is to alter the set of inputs and outputs— to take more
Variables into account— and then to see if the new system (equiv-
Alent to a new Box, S.3/11) is determinate. Thus a chemist may
Find that a system’s behaviour is at first not determinate, but that
When the presence of traces of chloride is taken into account it
Becomes determinate. A great deal of research consists of such
Searches for a suitable set of variables.
A second way is to abandon the attempt to find strict determi-
Nacy and to look for statistical determinacy, i.e. determinacy in
Averages etc. The experimenter, with extensive records available,
Then studies them in long sections, to see whether, if the details are
Not predictable from step to step, the averages (or similar statis-
Tics) are predictable from section to section. He may find that the
Records show the statistical determinateness of the Markov chain;
(but discussion of this will be left to Chapter 9, for until then we
Shall be concerned only with machines that are determinate from
Step to step).
To summarise: once the protocol has been obtained, the sys-
Tem’s determinateness can be tested, and (if found determinate) its
Canonical representation can be deduced.
Ex. 1: Deduce the kinematic graph for input at α directly from the protocol of the
System of S.6/3.
Ex. 2: (Continued.) and for input at f.
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Ex. 3: A system with only one input state gave the following sequence of states
As output:
D G A H C L H C L H C F C…
Is it absolute?
Ex. 4: A system has two variables, x and y, each of which can take the values 0,
1 or 2. The input can take two values, α or β. The protocol gave:
Time:
Input:
X:
Y:
Time:
Input:
X:
Y:
1 2 3 4 5
α α α α α
1 0 0 0 0
1 0 1 0 1
6 7 8 9 10 11 12 13
β α α α α α α α
0 1 2 2 1 0 0 0
0 2 1 0 1 0 1 0
(No transition was observed from g with input at β.) Within each
Cell the letters are all equal, so the table can be simplified to:
↓ f g h j
α f j j f
β h . h f
With a statement that throughout the protocol this closed single-
Valued transformation was observed.
Thus by direct re-coding of the protocol the experimenter can
Demonstrate that the behaviour is machine-like, and he can
Deduce its canonical representation.
It should be noticed that he has deduced it from direct observa-
Tion of the Box’s actual behaviour. He has relied on no “bor-
Rowed” knowledge. Whatever he may have expected, and
Regardless of the confidence of his expectation, the final deduc-
Tion depends only on what actually happened. Thus, in any con-
Flict between what he, or others, expected and what was found,
These empirical results are final as a statement of the Box’s nature.
Should the system not be determinate, i.e. the transformation
Not single-valued, he can proceed in either of two ways.
90
14 15 16 17 18 19 20 21 22 23 24 25
β α α β β β α β β β α β
0 0 1 0 1 1 1 1 1 2 2 1
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1 2 1 0 2 1 0 1 0 0 2 1
Is it a machine with input ?
Ex. 5: (Continued.) What is its transformation if the input is held at α?
Ex. 6: If a machine has m input-states and n output-states, what is the least
Number of steps of observation sufficient for its complete study?
91
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 BL AC K B O X
Ex. 7: Two Black Boxes are of identical external appearance, and each has a sin-
gle input α and a single output x, each a numerical variable. They were
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