THE BLACK BOX 6/13 



It will be found that the repetitions do not contradict one another, 

 and that the table can equally well be given as 



which is isomorphic with A^. 



Examination of M shows now where the resemblance to A'^ lies. 

 Within M the transitions occur in blocks; thus a, b and c always go 

 to some one of a, b or c. And the blocks in M undergo transitions 

 in the same way as the states in N. N is thus equivalent to a sim- 

 plified version of M. 



The relation can be displayed in another way. Suppose first the 

 two machines are viewed by some one who can distinguish all the 

 five states of M; he will report simply that M is different from N 

 (i.e. not isomorphic) and more complex. Suppose next that they 

 are viewed by some observer with less power of discrimination, one 

 who cannot discriminate between a, b, and c, but lumps them all 

 together as, say, A ; and who also lumps d and e together as B, i and 

 j as r, and k and / as zl. This new observer, seeing this simplified 

 version of M, will report that it is isomorphic with A'^. Thus two 

 machines are homomorphic when they become alike if one is merely 

 simplified, i.e. observed with less than full discrimination. 



Formally, if two machines are so related that a many-one trans- 

 formation can be found that, applied to one of the machines, gives 

 a machine that is isomorphic with the other, then the other (the 

 simpler of the two) is a homomorphism of the first. 



Ex. : Is isomorphism simply an extreme case of homomorphism? 



Problem : What other types of homomorphism are there between machine 

 and machine? 



6/13. If the methods of this book are to be applied to biological 

 systems, not only must the methods become sufficiently complex to 

 match the systems but the systems must be considerably simplified 

 if their study is ever to be practical. No biological system has yet 

 been studied in its full complexity, nor is hkely to be for a very long 

 time. In practice the biologist always imposes a tremendous 

 simplification before he starts work: if he watches a bird building 

 its nest he does not see all the intricate pattern of detailed neuronic 

 activities in the bird's brain; if he studies how a lizard escapes from 

 its enemies he does not observe the particular molecular and ionic 

 changes in its muscles; if he studies a tribe at its council meeting he 



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