6/12 



AN INTRODUCTION TO CYBERNETICS 



HOMOMORPHIC MACHINES 



6/12. The definition given for isomorphism defines "equahty" 

 in the strictest sense— it allows that two machines (or two Black 

 Boxes) are "equal" only when they are so alike that an accidental 

 interchange of them would be subsequently indetectable, at least 

 by any test applied to their behaviours. 



There are, however, lesser degrees of resemblance. Thus two 

 pendulums, one beating seconds and the other half-seconds, are 

 obviously similar, yet they are not isomorphic in the strict sense. 

 There is, however, some similarity, which is shown by the fact that 

 they become isomorphic if they are measured on separate time- 

 scales, the one having half the values of the other. 



Two machines may also be related by a "homomorphism." This 

 occurs when a many-one transformation, apphed to the more complex, 

 can reduce it to a form that is isomorphic with the simpler. Thus 

 the two machines M and N 



g h 



N: a 



g h 

 h h 



may seem at first sight to have little resemblance. There is, however, 

 a deep similarity. (The reader will gain much if he reads no further 

 until he has discovered, if only vaguely, where the similarity lies; 

 notice the peculiarity of A^'s table, with three elements alike and 

 one different — can anything like that be seen in the table of A/?— if 

 cut into quadrants ?) 



Transform M by the many-one transformation T: 



T:\ 



a b c d e i j k I 

 hhhgg^^aa 



(which is single-valued but not one-one as in S.6/9) and we get 



\ h h h g g 



