94 BELL SYSTEM TECHNICAL JOURNAL 



Xi the value 0. This fixes some element in B namely, Xai where ai = 1. 

 There are two cases: 



(1) If this element is the first term, ai = 1, then we have 



0X2,--- ,X, 



1 Xa^ , ' • ■ , Xa^ 



Letting X2 , • • • , Xr range through their 2^~ possible sets of values gives 

 2"^"^ equalities between different functions of the set fi since these are 

 really 



f{X\ , X2 , • • • , Xr , Xr+1 , • ■ ■ , Xn) 



with Xi , X2 , • • • , Xr fixed at a definite set of values. 



(2) If the element in question is another term, say Xaj , we then give X2 

 in line A the opposite value, X2 = {X^^ = {X2 )'. Now proceeding as 

 before with the remaining r — 2 variables we establish 2"^^ equalities between 

 the fi . 



Now there are not more* than 2"w! relations 



NiSjf = f 



of the group invariant type that a function could satisfy, so that the number 

 of functions satisfying any non-trivial relation 



< 2"w!2*'". 



Since 



2"w! 2^^72^" -^0 sisn-^ 00 



we have: 



Theorem 13: Almost all functions have no non-trivial group invariance. 



It appears from Theorems 12 and 13 and from other results that almost all 

 functions are of an extremely chaotic nature, exhibiting no symmetries or 

 functional relations of any kind. This result might be anticipated from the 

 fact that such relations generally lead to a considerable reduction in the 

 number of elements required, and we have seen that ahnost all functions are 

 fairly high in "complexity". 



If we are synthesizing a function by the disjunctive tree method and the 

 function has a group invariance involving the variables 



-^1 , X2 , • • • , Xr 



at least T ^ of the terminals in the corresponding tree can be connected to 



* Ourfactorisreally less than this because, first, we must exclude iV, 5, = /; and second, 

 except for self inverse elements, one relation of this type implies others, viz. the powers 

 {NiSM = f. 



