Simplification of Logic Expressions:Derivation of Minimal Sums for a Single Function by Other Means.

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Derivation of Minimal Sums for a Single Function by Other Means

Derivation of a minimal sum for a single function by Karnaugh maps is convenient for manual processing because designers can know the nature of logic networks better than automated minimization, to be described in Chapter 30, but its usefulness is limited to functions of four or five variables.

There are several methods for derivation of a minimal sum for a single function. A minimal sum can be found by forming a so-called a covering table where the minterms of a given function f are listed on the horizontal coordinate and all the prime implicants of f are listed on the vertical coordinate. A minimal sum can be described as a solution, that is, a minimal set of prime implicants that covers all the minterms [2]. The feasibility of this approach based on the covering table is limited by the number of minterms and prime implicants rather than the number of variables, which is the limiting factor for the feasibility of the derivation of minimal sums based on Karnaugh maps. This approach based on the table can be converted to an algebraic method, with prime implicants derived by the Tison method, as described by Procedure 4.6.1 in Ref. 3. This is much faster than the approach based on the table. Another approach is generation of irredundant disjunctive forms and then derive a minimal sum among them [4,5]. The feasibility of this approach is limited by the number of consensuses rather than the number of variables, minterms, or prime implicants.

As the number of variables, minterms, or prime implicants increases, the derivation of absolute minimal sums or even prime sums becomes too time-consuming, although the enhancement of the feasibility has been explored [1]. When too time-consuming, we need to resort to heuristic minimization, as will be described in Chapter 30 and Chapter 45, Section 45.3.

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