8/5 AN INTRODUCTION TO CYBERNETICS 



ing" to be attached to the message and shall consider simply what 

 will happen in these determinate systems. 



For simplicity in the example, suppose that M can take any one 

 of four states: A, B, C, and D; that the parameters provide three 

 states Q, R, and S. These suppositions can be shown in tabular 

 form, which shows the essentials of the "transducer" (as in S.4/1): 



Given its initial state and the sequence of values given to the para- 

 meter, its output can be found without difficulty, as in S.4/1. 

 Thus, suppose it starts at B and that the input is at /?; it will change 

 to C. If the input goes next to Q, it will go from C to ^. The 

 results so far can be shown in tabular form : 



Input-state : R Q 



Transducer-state: B C A 



It can now easily be verified that if the initial state is B and the input 

 follows the sequence RQRSSQRRQSR, the output will follow 

 the sequence BCAABDBCBCCB. 



There is thus no difficulty, given the transducer, its initial state, 

 and the input sequence, in deducing its trajectory. Though the 

 example may seem unnatural with its arbitrary jumps, it is in fact 

 quite representative, and requires only more states, and perhaps the 

 passage to the limit of continuity to become a completely natural 

 representation. In the form given, however, various quantitative 

 properties are made obvious and easily calculable, where in the 

 continuous form the difficult technique of measure theory must be 

 used. 



Ex. 1 : Pass the same message (RQRSSQRRQSR) through the same 

 transducer, this time starting at A. 



£'.v. 2: Pass the message "Ri, Ra, R3, Ri, R2, R3" through the transducer of 

 S.4/1, starting it at a. 



Ex. 3: (Continued.) Encode the same message through the same transducer, 



starting it at b. 

 Ex. 4: (Continued.) Does a transducer's output depend, for given input, on 



its initial state? 



Ex. 5: If the transducer is «' = n — a, where o is a parameter, what will its 

 trajectory be if it starts at // = 10 and receives the input sequence 2, 1, 

 -3,-1,2,1? 



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