DESIGN FOR A BRAIN 21/11 



(The argument is simple and clear if it is supposed that each 

 part has a finite number of states possible, and if the number of 

 input states is also finite. The result for the infinite case, being 

 the limit of the finite case, is the same as that stated, but would 

 need a special technique for its discussion.) 



Suppose the system consists of p parts, each capable of being 

 in any one of s states, with p and s assumed finite. Then, whether 

 joined or not, the set of all the parts has s v states possible. (Put 

 s v = k, for convenience.) 



If the whole is richly joined, each of these k states may go, in a 

 transition, to any of the k states; for the transition of each part 

 is not restricted (since it is allowed to be conditional on, and to 

 vary with, the states of the other parts). The number of trans- 

 formations is thus k k . 



If, however, the parts are not joined, the transformations of 

 each part cannot vary with the states of the others ; so the trans- 

 formation of the whole must be built up by taking a single trans- 

 formation from each part. Each part, with s states, has s s 

 transformations; so the whole will have (s s ) r transformations 

 possible. This equals k s . 



As s is less than k, k s is less than k k ; whence the theorem. 



21/11. If X v . . . , X n is a state of equilibrium in a system 



dxi/dt =ft(x v . . . , x n \ 14, . . .) {i = 1 n) 



for certain a- values, and the system is then joined to some ?/'s by 

 making the a's functions of the y's, then X v . . . , X n will still 

 be a state of equilibrium (of the a>system) when the y's make the 

 a's take their original values. Thus the zeros of the/'s, and the 

 states of equilibrium of the ^-system, are not altered by the 

 operation of joining. 



21/12. On the other hand, the stabilities may be altered grossly. 



In the general case, when the/'s are unrestricted, this proposi- 

 tion is not easily given a meaning. But in the linear case (to 

 which all continuous systems approximate, S. 20/5) the meaning 

 is clear. Three examples will be given. 



Example 1: Two systems may give a stable whole if joined one 

 way, but an unstable whole if joined another way. Consider the 

 1 -variable systems dx/dt = x + ^V\ + 2h an ^ dy/dt = — 2r — 3y. 



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