Detection of Group Invariance or Total 

 Symmetry of a Boolean Function* 



By E. J. McCLUSKEY, Jr. 



(Manuscript received June 26, 1956) 



A method is presented for determining whether a Boolean function pos- 

 sesses any group invariance; that is, whether there are any permutations or 

 primings of the independent variables which leave the function unchanged. 

 This method is then extended to the detection of functions which are totally 

 symmetric. 



1 GROUP INVARIANCE 



For some Boolean transmission functions (transmissions, for short) it 

 is possible to permute the variables, or prime some of the variables, or 

 both permute and prime variables without changing the transmission. 

 The following material presents a method for determining, for any given 

 transmission, which of these operations (if any) can be carried out with- 

 out changing the transmission. 



The permutation operations will be represented symbolically as fol- 

 lows: 



Si2z...nT will represent the transmission T with no variables permuted 

 8213.. -nT will represent the transmission T with the xi and X2 variables 



interchanged, etc. 

 Thus *Si432T(.x-i , X2 , xs , X4) = T(xi , Xi , Xs , X2') 

 The symbolic notation for the priming operation will be as follows: 

 Noooo-.-oT will represent the transmission T with no variables primed 

 A^ono. --oT will represent the transmission T with the .r2 and ;i;3 variables 



primed, etc. 

 Thus NiowT(xi , :r2 , X3 , Xa) = T(xi, X2 , Xs, Xi). 



The notation for the priming operator can be shortened by replacing 

 the binary subscript on N by its decimal equivalent. Thus N9T is equiv- 



* This paper is derived from a thesis submitted to the Massachusetts Institute 

 of Technology in partial fulfillment of the requirements for the degree of Doctor 

 of Science on April 30, 1956. 



1445 . 



