Non-essential prime implicant

Also called: nonessential prime implicant, non-essential PI, selective prime implicant, optional prime implicant

A prime implicant that is not the only cover for any 1. It may be needed to cover leftover 1s, or it may be redundant and left out.

Every prime implicant is either essential or non-essential. A non-essential one covers only 1s that some other prime implicant also covers.

After you choose the essential prime implicants, each non-essential one falls into one of two groups:

Redundant: all its 1s are already covered by the essentials. Leave it out. See redundant group.
Selective: it covers some 1 that the essentials missed, but so does at least one other prime implicant. You must pick from among these, choosing the cheapest combination that covers everything.

When there is a choice, compare total cost: fewer terms first, then fewer literals. If two options tie, both are minimal, and the function has more than one correct answer.

A map where every prime implicant is non-essential is a cyclic k map.

A\BC	00	01	11	10
0	1m0	1m1	1m3	0m2
1	0m4	0m5	1m7	1m6

Worked example

Example

Choosing between two selective PIs

F = Σm(0, 1, 3, 6, 7), shown above.

1.
Prime implicants: (m0 m1), (m1 m3), (m3 m7), (m6 m7).
2.
m0 is only in , and m6 is only in . Both are essential.
3.
They cover m0, m1, m6, m7. Only m3 is left.
4.
m3 is in both and . Either one works, and both cost 2 literals.
5.
Two minimal answers: (shown) or .

Common mistakes

Assuming every non-essential prime implicant can be dropped. Selective ones may be needed to finish the cover.
Including all the selective ones "to be safe". Pick only as many as the leftover 1s require.
Thinking a non-essential group isn't prime. It's still as large as it can be; it just isn't the only cover for any 1.