FORMAL REALIZABILITY THEORY — I 223 



search alono will toll wholhor or not one obtains in this way the kinds 

 of device which are si<;iiilicaiit. For example, one would like f>;eiiei-al 

 i-e;i]izaliiiil\' theorems for si nictures containiiifi; xacuum lubes wilh 

 fr(Mluency-in(le))en(l(Mit li-anscouductances, \a('uum luhes with non- 

 \anishiiiu; transit times, unilateral devices with si)ecified parasitics, etc. 



2.7") Actually, the i)ostulates as we have fi;i\en them are certainly not 

 adeiiuate for such an ambitious program. Exigencies of the presentation 

 ha\e dictated a number of condensations and compromises. It is hoped 

 that the basic ideas are still e\'ident even if not isolated indix'idually in 

 separate and entirely independent postulates. In any event, it is the 

 author's firm belief that the presentation as given is at least illustrative 

 of the kind of approach, and the level of mathematical detail, which 

 will be needed if one is ever to provide a truly adequate realizability 

 theory: a theory which will cover, for example, the broad range of active 

 linear systems which present-day technology allows us to consider. 



2.8 Apart from the network theoretic concepts, which must be evalu- 

 ated by their effectiveness in solving problems — an assessment which is 

 by no means yet complete — this paper is strongly marked b}^ an idio- 

 syncracy of its author: a consistent and insistent use of geometric ideas 

 and terminology. This is based on the personal experience that linear 

 algebra achieves logical unity and a freedom from encumbering notation 

 when viewed in this way. A general reference covering most of the linear 

 algebra (geometrj^) rec^uired here is P. R. Halmos' elegant monograph^. 



2.9 For a proof solely of 1.1, which has already been three times proved 

 in the literature, ' ' this paper provides an apparatus which is too 

 cimibersome. There is even a sense in which 1.1 alone provides a charac- 

 terization of all finite passive devices, for it seems to be generally ac- 

 cepted that, by the use of ideal transformers, any finite passive network 

 can be represented as a network which has an impedance matrix to 

 which is adjoined suitable ideal transformers. Therefore we cannot claim 

 that, in using this cumbrous apparatus to characterize all finite passive 

 2/;-poles (including the degenerate ones), we have offered anything not 

 already provided by a simpler proof of 1.1. 



Three things may be said in rebuttal. First, we have already empha- 

 sized that the apparatus here exhibited was designed for more problems 

 than that to which it is here ai)plied. It is presented in the belief that 

 it will prove of further use. 



Second, even in the study of passive networks, it has schemed to the 

 author helpful to look at the manifold things which are not passive net- 



