Detection of totally symmetric Boolean functions with the aid of decimal combination tables

A method of detecting and identifying the total symmetry of an n-variable Boolean function F with the use of decimal combinational tables is presented. Instead of using maps, charts or truth tables, with permutations and complementations of variables, this method uses a table which gives for each minterm mi (that makes F=1) the number of adjacencies w1 ranging from 0 to n. The detection of the variables of symmetry is based on finding out from the table a pair of minterms m* and its is complement m* (or pairs of similar minterms) that among all minterms appearing in F (or in F¿) are distinguished by having w1 equalling n or 0. The parameter w1 is the total number of other minterms appearing in F (or in F¿) that differ from any function minterms, e.g. m by one complementation of literal, and is easily determined from the table of combinations. The minterms m and m* are likely contenders to define the literals of symmetry and merit further investigation. The identification of F as a totally symmetric function is completed if and when a set of a-numbers is determined from the combinational table. The paper describes the construction of the combinational table and illustrates the method with example.