8/6 AN INTRODUCTION TO CYBERNETICS 



and these correspond to Mi, M2, . . ., Mg of U. Suppose the trans- 

 ducer is always started atA;ii is easy to verify that the corresponding 

 nine outputs will be (if we ignore the initial and invariable A) : 



CA, CB, CC, AC, AA, AB, BC, BC, BB. 



These are the Q, C2 . . .,Cgof U. Now the coding performed by the 

 transducer is not one-one, and there has been some loss of variety, 

 for there are now only eight distinguishable elements, BC being 

 duplicated. This transducer therefore fails to provide the possibility 

 for complete and exact decoding; for if BC arrives, there is no way 

 of telling whether the original message was SQ or SR. 



In this connexion it must be appreciated that an inability to 

 decode may be due to one of two very diiferent reasons. It may 

 be due simply to the fact that the decoder, which exists, is not at 

 hand. This occurs when a military message finds a signaller without 

 the code-book, or when a listener has a gramophone record (as a 

 coded form of the voice) but no gramophone to play it on. Quite 

 different is the inability when it is due to the fact that two distinct 

 messages may result in the same output, as when the output BC 

 comes from the transducer above. All that it indicates is that the 

 original message might have been SQ or SR, and the decoder that 

 might distinguish between them does not exist. 



It is easy to see that if, in each column of the table, every state had 

 been different then every transition would have indicated a unique 

 value of the parameter; so we would thus have been able to decode 

 any sequence of states emitted by the transducer. The converse 

 is also true; for if we can decode any sequence of states, each 

 transition must determine a unique value of the parameter, and thus 

 the states in a column must be all different. We have thus identified 

 the characteristic in the transducer that corresponds to its being a 

 perfect coder. 



Ex. 1 : In a certain transducer, which has 100 states, the parameters can take 

 108 combinations of values; can its output always be decoded? (Hint: 

 Try simple examples in which the number of transformations exceeds that 

 of the states.) 



Ex. 2: (To emphasise the distinction between the two transformations.) If 

 a transducer's input has 5 states, its output 7, and the message consists of 

 some sequence of 12, (i) how many operands has the transducer's trans- 

 formation, and (ii) how many has the coding transformation (/? 



Ex. 3: If a machine is continuous, what does "observing a transition" corres- 

 pond to in terms of actual instrumentation ? 



*Ex. 4: If the transducer has the transformation dx/dt = ax, where a is the 

 input, can its output always be decoded? (Hint: Solve for a.) 



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