Grouping table for the minimisation of n-variable boolean functions

Abstract

The construction of a decimal grouping table and its use to determine essential and nonessential prime implicants for the minimisation of an n-variable Boolean function are presented in the paper. In existing tabular methods, e.g. the Quine-McCluskey technique, each fundamental product is represented by a row of binary 1s and 0s and the finding of a set of prime implicants necessitates the formation of successive tables of binary characters, and only after an exhaustive search in the tables can one discover any prime implicants. Dealing with binary characters is rather tedious, and searching through several tables to establish a prime implicant is time consuming. The proposed grouping table offers the convenience of using decimal minterm numbers and the advantage of using one table in the search for prime implicants. In the grouping table the decimal equivalent of function terms appear in a column and the entries to a row corresponding to a function term N are the decimal equivalent of product terms related to N by one change of variable. From these entries, only those terms which appear in the function under investigation are selected and only these need to be considered for the minimisation of the problem. Thus, unlike other tabular methods, the grouping table provides all possible combinational terms for each fundamental product term as its row terms and also the facility of at-a-glance comparison of all function terms by referring to the same table. In the paper, a method of minimising Boolean functions with the aid of grouping table is illustrated with examples.