THE BLACK BOX 6/9 



It is important because most systems have both difficult and easy 

 patches in their properties. When an experimenter comes to a 

 difficult patch in the particular system he is investigating he may, 

 if an isomorphic form exists, find that the corresponding patch in 

 the other form is much easier to understand or control or investigate. 

 And experience has shown that the ability to change to an isomorphic 

 form, though it does not give absolutely trustworthy evidence (for 

 an isomorphism may hold only over a certain range), is nevertheless 

 a most useful and practical help to the experimenter. In science it 

 is used ubiquitously. 



6/9. It must now be shown that this concept of isomorphism, vast 

 though its range of applicability, is capable of exact and objective 



oc 



p 



i 



d^^C 



a==bb 



cl=^c 



I J 



i\ 



k^ 



Fig. 6/9/1 



definition. The most fundamental definition has been given by 

 Bourbaki; here we need only the form suitable for dynamic systems. 

 It applies quite straightforwardly once two machines have been 

 reduced to their canonical representations. 



Consider, for instance, the two simple machines M and TV, with 

 canonical representations 



M: 



N 



They show no obvious relation. If, however, their kinematic 

 graphs are drawn, they are found to be as in Fig. 6/9/1. Inspection 

 shows that there is a deep resemblance. In fact, by merely 

 rearranging the points in A'^ without disrupting any arrow (S.2/17) 

 we can get the form shown in Fig. 6/9/2. These graphs are identical 

 with M's graphs, apart from the labelling. 



More precisely: the canonical representations of two machines are 

 isomorphic if a one-one transformation of the states (input and 



7 97 



