Two-Level Boolean Minimization

The Quine-McCluskey method is best illustrated with an example. Consider the completely specified Boolean function shown in (1A). It has been represented as a list of 0-terms, which are fully specified product terms with no don't-care entries. Next, each 0-term has an associated decimal value obtained by converting the binary number represented by the 0-term into a decimal number—for instance the value of 0000 is 0 and that of 1110 is 14. Each pair of 0-terms is checked to see if they can be merged into a single 1-term. Two 0-terms can be merged if they differ in exactly one position. The terms generated in our example by merging the pairs of 0-terms are shown in (1B). These terms are called 1-terms because they have exactly one “—“ entry. Next, these l-terms are examined in pairs to see if they can be merged into 2-terms. This can only be done if the 1-terms differ in exactly one position and they have their “—“ literal in the same position. Two 2-terms can be formed as shown in (1C). If a k-term is formed by merging two (kl)-terms, then the two (k-1) l-terms are not primes and are marked so that they can be discarded later. The process ends when no merging is possible in the final set of L-terms. The unmarked kterms, where O < k < L, are the complete set of prime terms of the function. The five prime terms for the function in this example are in bold type and labelled [A] through [E]. In the case of incompletely specified functions the initial list of 0-terms includes those 0-terms in the ON-set and the DC-set (don't-care set).