THE BLACK BOX 6/13 



This is the machine as seen by the first observer (call him One). 

 Suppose now that another observer (call him Two) was unable to 

 distinguish states a and d, and also unable to distinguish h and e. 

 Let us give the states new names for clarity : 



a d c h e 



K L M 



The second observer, seeing states K,Lov M would find the machine's 

 behaviour determinate. Thus when at K (really a or d) it would 

 always go to M (either b or e), and so on. He would say that it 

 behaved according to the closed transformation 



K L M 

 ^ M K M 



and that this was single-valued, and thus determinate. 



The new system has been formed simply by grouping together 

 certain states that were previously distinct, but it does not follow 

 that any arbitrary grouping will give a homomorphism. Thus suppose 

 yet another observer Three could distinguish only two states: 



a b c d e 



P Q 



He would find that P changed sometimes to Q (when P was really 

 at a) and sometimes to P (when P was really at b or c). The change 

 from P is thus not single-valued, and Three would say that the 

 machine (with states P and Q) was not determinate. He would 

 be dissatisfied with the measurements that led to the distinction 

 between P and Q and would try to become more discriminating, 

 so as to remove the unpredictability. 



A machine can thus be simplified to a new form when its states are 

 compounded suitably. Scientific treatment of a complex system 

 does not demand that every possible distinction be made. 



Ex. 1 : What homomorphism combines Odd and Even by the operation of 



addition? 

 Ex. 2: Find all possible simplifications of the four-state system 



.abed 

 ^ b b d c 



which leaves the result still a determinate machine. 



Ex. 3 : What simplification is possible in 



lx'=-y 

 \y =x^ + y, 



if the result is still to be a determinate machine? 



105 



