Values of a, chosen anew at each step, will make n follow the series 10, 11,
15, 16, 20, 21, 25, 26, …, in which the differences are, alternately 1 and 4?
Iv) What values of a will make n advance by unit steps to 100 and then jump
Directly to 200?
Ex. 4: If a transducer has n operands and also a parameter that can take n values,
The set shows a triunique correspondence between the values of operand,
Transform, and parameter if (1) for given parameter value the transformation
Is one-one, and (2) for given operand the correspondence between parame-
Ter-value and transform is one-one. Such a set is
↓ a b c d
R1 c d a b
R2 b a c d
R3 d c b a
R4 a b d c
Show that the transforms must form a Latin square, i.e. one in which each
Row (and each column) contains each transform once and once only.
Ex. 5: A certain system of one variable V behaves as
190
--V' = ----- V + -----
10 P
Where P is a parameter. Set P at some value P1, e.g. 10, and find the limit
That V tends to as the transformation is repeated indefinitely often, call this
Limit V1. Then set P at another value P2, e.g. 3, and find the corresponding
Limit V2. After several such pairs of values (of P and limit-V) have been
Found, examine them to see if any law holds between them. Does V behave
Like the volume of a gas when subjected to a pressure P?
Ex. 6: What transformation, with a parameter a will give the three series of val-
Ues to n?:
a = 1: 0, → 1, → 2, → 3, → 4, …
a = 2: 0, → 4, → 8, → 12, → 16, …
a = 3: 0, → 9, → 18, → 27, → 36, …
(Hint: try some plausible expressions such as n' – n + a, n' = a2n, etc.)
Ex. 7: If n' = n + 3a, does the value given to a determine how large is n’s jump
At each step?
We can now consider the algebraic way of representing a
Transducer.
The three transformations
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R2: n' = n + 2R3: n' = n + 3R1: n' = n + 1
Can obviously be written more compactly as
Ra : n' = n + a,
And this shows us how to proceed. In this expression it must be
Noticed that the relations of n and a to the transducer are quite dif-
Ferent, and the distinction must on no account be lost sight of. n is
Operand and is changed by the transformation; the fact that it is an
operand is shown by the occurrence of n'. a is parameter and
Determines which transformation shall be applied to n. a must
Therefore be specified in value before n’s change can be found.
When the expressions in the canonical representation become
More complex, the distinction between variable and parameter
Can be made by remembering that the symbols representing the
operands will appear, in some form, on the left, as x' or dx/dt; for
The transformation must tell what they are to be changed to. So all
44
45
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 MA C HI N E WI TH I N PUT
When the expression for a transducer contains more than one
Meter, the number of distinct transformations may be as large e
Number of combinations of values possible to the parameters each
Combination may define a distinct transformation), but never
Exceed it.
Ex. 1: Find all the transformations in the transducer Uab when a can take the val-
Ues 0, 1, or 2, and b the values 0 or 1.
s' = (1 – a)s + abt
Uab : t' = (1 + b)t + (b – 1)a
How many transformations does the set contain?
Ex. 2: (continued.) if the vector (a,b) could take only the values (0,1), (1n1), and
How many transformations would the transducer contain?
Ex. 3: The transducer Tab,with variables p and q: f p = ap + bq
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Tab : p' = ap + bq
q' = bp + aq
Is started at (3,5). What values should be given to the parameters a and if
P,q) is to move, at one step, to (4,6)? (Hint: the expression for Tab can be
Regarded as a simultaneous equation.)
Ex. 4: (Continued.) Next find a value for (a,b) that will make the system move,
In one step, back from (4,6) to (3,5).
Ex. 5: The transducer n' = abn has parameters a and b, each of which can take
Any of the values o, 1, and 2. How many distinct transformations are there?
Such indistinguishable cases are said to be “degenerate”; the rule given at
The beginning of this section refers to the maximal number o transformations
That are possible; the maximal number need not always be achieved).
The parameters thus include the conditions in which the organism
Lives. In the chapters that follow, the reader must therefore be pre-
Pared to interpret the word “input” to mean either the few parame-
Ters appropriate to a simple mechanism or the many parameters
Appropriate to the free-living organism in a complex environment.
(The increase in the number of parameters does not necessarily
Imply any diminution in the rigour of the argument, for all the
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