MINIMIZATION OF BOOLEAN FUNCTIONS 1443 



l)i-ocodure for determining the i)rinie iniplicants is completely systematic 

 it is capable of being programmed on a digital computer. The arrange- 

 ment of terms introduced here then results in a considerable saving in 

 both time and storage space over previous methods, making it possible 

 to solve larger problems on a given computer. It should be pointed out 

 that this procedure can be programmed on a decimal machine by using 

 the decimal labels instead of the binary characters introduced. 



A method was presented for choosing the minimum sum terms from 

 the list of prime iniplicants by means of a table of prime implicants. 

 This is again similar to a method presented l:)y Quine. Howe\'er, Quine 

 did not give any systematic procedure for handling cyclic prime impli- 

 cant tables; that is, tables with more than one cross in each column. In 

 this paper a procedure is given for obtaining a minimum sum from a 

 cyclic prime implicant table. In general, this procedure requires enumera- 

 tion of several possible minimum sums. If a transmission has any non- 

 trivial group invariances it may be possible to avoid enumeration or to 

 reduce considerably^ the amount of enumeration necessary. A method 

 for doing this is given. 



The process of enumeration used for selecting the terms of the mini- 

 mum sum from a cyclic prime implicant table is not completely satis- 

 factory since it can be quite lengthy. In seeking a procedure which does 

 not require enumeration, the method involving the group invariances of 

 a transmission was discovered. This method is an improvement over 

 complete enumeration, but still has two shortcomings. There are trans- 

 missions which have no nontrivial group invariances but which give 

 rise to cyclic prime implicant tables. For such transmissions it is still 

 necessary to resort to enumeration. Other transmissions which do possess 

 nontrivial group invariances still reciuire enumeration after the in- 

 variances have been used to simplify the process of selecting minimum 

 sum terms. More research is necessary to determine some procedure 

 which will not require any enumeration for cyclic prime implicant 

 tables. Perhaps the concept of group invariance can be generalized so 

 as to apply to all transmissions which result in cyclic prime implicant 

 tables. 



13 ACKNOW'LEDGEMENTS 



The author wishes to acknowledge his indebtedness to Professor S. H. 

 Caldwell, Professor D. A. Huffman, Professor W. K. Linvill, and S. H. 

 Unger with whom the author had many stimulating discussions. Thanks 

 are due also to W. J. Cadden, C. Y. Lee, and G. H. Mealy for their 

 helpful comments on the preparation of this paper. 



