4/3 AN INTRODUCTION TO CYBERNETICS 



4/3. When the expression for a transducer contains more than one 

 parameter, the number of distinct transformations may be as large 

 as the number of combinations of values possible to the parameters 

 (for each combination may define a distinct transformation), but 

 can never exceed it. 



Ex. 1 : Find all the transformations in the transducer f/,,^ when a can take the 

 values 0, 1, or 2, and b the values or 1. 



J J j s' = {\ — a)s + abt 

 ^'^b- [/' = (] +b)t + {b- \)a. 



How many transformations does the set contain ? 



Ex. 2: (Continued.) If the vector ia,b) could take only the values (0,1), (1,1), 



and (2,0), how many transformations would the transducer contain? 

 Ex. 3 : The transducer T^b, with variables p and q : 



J, . ( p' = ap + bq 



^"f'- \q' = bp + aq 



is started at (3,5). What values should be given to the parameters a and 

 b if {p,q) is to move, at one step, to (4,6) ? (Hint : the expression for 7^;, 

 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 0, 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 of 

 transformations that are possible; the maximal number need not always be 

 achieved). 



4/4. Input and output. The word "transducer" is used by the 

 physicist, and especially by the electrical engineer, to describe any 

 determinate physical system that has certain defined places of input, 

 at which the experimenter may enforce changes that affect its 

 behaviour, and certain defined places of output, at which he observes 

 the changes of certain variables, either directly or through suitable 

 instruments. It will now be clear that the mathematical system 

 described in S.4/1 is the natural representation of such a material 

 system. It will also be clear that the machine's "input" corresponds 

 to the set of states provided by its parameters; for as the parameters 

 or input are altered so is the machine's or transducer's behaviour 

 affected. 



With an electrical system, the input is usually obvious and 

 restricted to a few terminals. In biological systems, however, the 

 number of parameters is commonly very large and the whole set of 

 them is by no means obvious. It is, in fact, co-extensive with the set 

 of "all variables whose change directly affects the organism". The 



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