Simplification of Logic Expressions:Derivation of Minimal Products by Karnaugh Map

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Derivation of Minimal Products by Karnaugh Map

Minimal products can be derived by the following procedure, based on a Karnaugh map.

Procedure 28.4: Derivation of Minimal Products on a Karnaugh Map Consider the case of a map for four or fewer variables (cases for five or more variables are similar, using more than one four-variable map).

1. Encircle 0-cells, instead of 1-cells, with rectangular loops, each of which consists of 2i 0-cells, choosing i as large as possible. These loops are called prime implicate loops because they represent prime implicates (not prime implicants). Examples of prime implicate loops are shown in Figure 28.12 (including the dotted-line loop). The prime implicate corresponding to a loop is formed by making a disjunction of literals, instead of a conjunction, using a noncomplemented variable corresponding to 0 of a coordinate of the map and a complemented variable corresponding to 1. (Recall that, in forming a prime implicant, variables corresponding to 0’s were complemented and those corresponding to 1’s were noncomplemented.) Thus, corresponding to the loops of Figure 28.12, we get the prime implicates, (x1 ∨ x4 ),(x1 ∨ x2 ∨ x3) and (x2 ∨ x3 ∨ x4 )

2. Each irredundant conjunctive form is derived by the conjunction of prime implicates correspond- ing to a set of loops, so that removal of any loop leaves some 0-cells uncovered by loops. An example is the set of two solid-line loops in Figure 28.12, from which the irredundant conjunctive

Simplification of Logic Expressions-0359

3. Among sets of a minimum number of prime implicate loops, the sets that contain as many of the largest loops as possible yield minimal products.

The don’t-care conditions are dealt with in the same manner as in case of minimal sums. In other words, whenever possible, we can form a larger prime implicate loop by interpreting some d ’s as 0-cells. Any prime-implicate loops consisting of only d-cells need not be formed. D Procedure 28.4 can be extended to five or more variables in the same manner as Procedures 28.2 and 28.3, although the procedure will be increasingly complex.

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