THE ERROR-CONTROLLED REGULATOR 



12/9 



*Ex. 4: (Continued.) What general rule, using matrix multiplication, allows 



the answer to be written down algebraically? (Hint: Ex. 9/6/8.) 

 ♦^.v. 5: Couple the Markovian machine (with states a, b, c and input-states 



«, iS) 



to the Markovian machine (with states e,/and input-states §, e, d) 



€: 



/ 



by the transformations 



b 

 8 



I 



/ 



What is the Markovian machine (without input) that results? (Hint: 

 Try changing the probabilities to and 1, so as to make the systems 

 determinate, and follow S.4/8; then make the probabilities fractional and 

 follow the same basic method.) 



*Ex. 6: (Continued.) Must the new matrix still be Markovian? 



*Ex. 7 : If M is a Markovian machine which dominates a determinate machine 

 A', show that A^ 's output becomes a Markov chain only after M has arrived 

 at statistical equilibrium (in the sense of S.9/6). 



12/9. Whether a given real machine appears Markovian or 

 determinate will sometimes depend on how much of the machine 

 is observable (S.3/11); and sometimes a real machine may be such 

 that an apparently small change of the range of observation may be 

 sufficient to change the appearances from that of one class to the 

 other. 



Thus, suppose a digital computing machine has attached to it a 

 long tape carrying random numbers, which are used in some process 

 it is working through. To an observer who cannot inspect the 

 tape, the machine's output is indeterminate, but to an observer 

 who has a copy of the tape it is determinate. Thus the question 

 "Is this machine really determinate?" is meaningless and inappro- 

 priate unless the observer's range of observation is given exactly. 

 In other words, sometimes the distinction between Markovian and 

 determinate can be made only after the system has been defined 

 accurately. (We thus have yet another example of how inadequate 

 is the defining of "the system" by identifying it with a real object. 



227 



