394 Information Storage and Neural Control 



have had a set of past values specified by given functions 0,. In 

 this case the classes a, of the last theorem reduced to unit classes; 

 and the condition may be transformed into 



{Em, n) {v)n{i, <];) {Ej) : . {x)m : '\>{x) = . w , '^(x) = I : 

 ^i€j(({;, nt + p) : -^ : (:w)m(x)t — 1 . (f)i{n{t + 1) 

 + p, nx -\- 'p,w) = 4>j{nt + p, nx + p, w) : . 

 (:u, v) (w)m . 4>iix>'{u + 1) -\- p, nn + p, w) 

 = (t)i{n(v + 1) -]r p,nv -{- p,w). 



On account of limitations of space, we have presented the above 

 argument very sketchily; we propose to expand it and certain of 

 its implications in a further publication. 



The condition of the last theorem is fairly simple in principle, 

 though not in detail; its application to practical cases would, 

 however, require the exploration of some 2-" classes of functions, 

 namely the members of fjv(|ai, ••• , «..j). Since each of these is 

 a possible ^ of Theorem IX, this result cannot be sharpened. But 

 we may obtain a sufficient condition for the realizability of an S 

 which is very easily applicable and probably covers most practical 

 purposes. This is given by 



Theorem X 



Let us define a set of X" of S by the following recursion: 



1. Any TPE and any TPE whose arguments have been re- 

 placed by members of K belong to K; 



2. If Pri{zi) is a member of K, then (22)21 • Pri{zo), (£22)2:1 . 

 Pviiz-i), and C,„rXzi) • * belong to it, where C,„„ denotes the property 

 of being congruent to m modulo n, m < n. 



3. The set K has no further members. 

 Then every member of K is realizable. 



For, if Pr\{zi) is realizable, nervous nets for which 



A^,(2i) . = . Pry{z,) . SN,{zi) 

 iV,(zi) . ^ . Pn(z,)vSN,fzr) 



are the expressions of equation (4), realize (22)21 • Priiz-z) and 



