SYNTHESIS OF SWITCHING CIRCUITS 83 



First let us make the following definition: The symbol (ci , oa , • • • , On) 

 represents any set of numbers bi , bi , ■ ■ • , bn that may be obtained from 

 the set oi , a2 , • • • , «« by the following operations: 



1. Interchange of letters. 



2. A flow from a larger number to a smaller one, no flow, however, being 

 allowed to the number 1. Thus we would write 



1, 2, 4, 8 = (1, 2, 4, 8) 



4, 4, 1, 6 = (1, 2, 4, 8) 



1, 3, 10, 3, 10 = (1, 2, 4, 8, 12) 



but 2, 2 5^ (1, 3). It is possible to put the conditions that 



bi , bt , ■ ■ ■ , bn = {ai , a2 , • ■ ■ , an) (6) 



into a more mathematical form. Let the a, and the bi be arranged as non- 

 decreasing sequences. Then a necessary and sufficient condition for the 



relation (6) is that 



» t 



(1) llbi>T.ai s = 1,2, ••• ,n, 



i-l i 



n n 



(2) ^ bi ^ J2 di , and 



(3) There are the same number of I's among the Oi as among the bi . 

 The necessity of (2) and (3) is obvious. (1) follows from the fact that if 

 a,- is non-decreasing, flow can only occur toward the left in the sequence 



ai , a2 , as , • • ■ , Gn 



a 



and the sum ^ a,- can only increase. Also it is easy to see the suflSciency of 

 1 



the condition, for if 6i , 62 , • • • , bn satisfies (1), (2), and (3) we can get the 



bi by first bringing Ci up to bi by a flow from the c,- as close as possible to 



ci (keeping the "entropy" low by a flow between elements of nearly the 



same value), then bringing 02 up to 62 (if necessary) etc. The details 



are fairly obvious. 



Additive number theory, or the problem of decomposing a number into the 

 sum of numbers satisfying certain conditions, (in our case this definition is 

 generalized to "sets of numbers") enters through the following Lemma: 



Lemma: If ci , 02 , • • • , fln = (2, 4, 8, • • • , 2") 

 then we can decompose the Oi into the sum of two sets 



Ci = bi + Ci 



such that 



6i , 6a , • • • , 6n = (1, 2, 4, • • • , 2"-0 



