K2 Choosing Prime Implicants

Of all of the possible ways to choose from among the

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Of all of the possible ways to choose from among the ten remaining prime implicants (we have chosen three and eliminated two of the original fifteen), there are ten different covers that require five prime implicants. (I leave this as an exercise for the reader.) Of those ten covers, four of them require three groups of four and two groups of two. The other six all require two groups of three and three groups of two. Since smaller groups represent gates having higher numbers of inputs, the latter six covers each has a higher classical cost than do the former four. Herein lies the point: if we follow one pathway’s worth of choices to derive a cover, there is no guarantee that we have followed the pathway to the most minimal cover. Further, complex functions may end up requiring us to make a second choice part of the way down a path that proceeds from having made an initial choice. With that in mind, we proceed best by following one pathway’s worth of choices all the way to a cover, and then returning to all points where we made a choice, and determining the cover that results from having made the other choice. At that point, we can choose the cover (or covers) that satisfies the requirement of having the smallest classical cost.
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