8/4 AN INTRODUCTION TO CYBERNETICS 



Often the method uses a "key-word" or some other factor that is 

 capable of changing the code from one form to another. Such a 

 factor corresponds, of course, to a parameter, giving as many 

 particular codings (or transformations) t/j, U2, ■ ■ ■ as there are 

 values to the factor. 



"Decoding" means applying such a transformation to the trans- 

 form C,- as will restore the original message M,: 



V- i ^^ ^' ^^ '" 



■ ^ Ml M2 M3 ... 



Such a transformation V is said to be the inverse of U; it may then 

 be written as t/"i. In general, only one-one transformations have 

 single-valued inverses. 



If the original message M^ is to be recoverable from the coded 

 form C,-, whatever value / may have, then both U and t/^ must be 

 one-one; for if both M, and M, were to be transformed to one form 

 Q, then the receiver of Q could not tell which of the M's had been 

 sent originally, and Q cannot be decoded with certainty. 



Next suppose that a set of messages, having variety v, is sent 

 coded by a one-one transformation U. The variety in the set of 

 coded forms will also be v. Variety is not altered after coding by a 

 one-one transformation. 



It follows that if messages of variety v are to pass through several 

 codes in succession, and are to be uniquely restorable to their 

 original forms, then the process must be one that preserves the variety 

 in the set at every stage. 



Ex. 1 : Is the transformation x' = logio x, applied to positive numbers, a one- 

 one coding? What is "decoding" it usually called? 

 £.v. 2: Is the transformation .v' = sin a-, applied to the positive numbers, a 



one-one coding? 

 Ex. 3: What transformation results from the application of, first, a one-one 



transformation and then its inverse ? 

 Ex. 4 : What transformation is the inverse of a?' = « + 7 ? 

 Ex. 5: What transformation is the inverse of x' = 2x + y, y' = x + v? 

 Ex. 6: If the coded form consists of three English letters, e.g. JNB, what is the 



variety of the possible coded forms (measured logarithmically)? 

 Ex. 1 : (Continued.) How many distinct messages can be sent through such a 



code, used once? 

 Ex. 8: Eight horses are running in a race, and a telegram will tell Mr. A. which 



came first and which second. What variety is there in the set of possible 



messages ? 

 Ex. 9: (Continued.) Could the set be coded into a single letter, printed either 



as capital or as lower case (small letters)? 



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