Essential prime implicant

Also called: EPI, essential PI, essential implicant

A prime implicant that is the only one covering some 1 of the function. It must appear in every minimal sum of products, so you choose these first.

A prime implicant is essential when it covers at least one 1 that no other prime implicant covers. That 1 is called a distinguished minterm.

Since that 1 has to be covered and only one group can do it, an essential prime implicant appears in every minimal sum of products. That makes them the natural first step:

List all prime implicants: every group that can't grow without taking in a 0.
For each 1, count how many prime implicants contain it.
Any 1 with a count of exactly one marks its group as essential. Circle it.
Cover the remaining 1s as cheaply as possible from the non-essential ones.

Don't-cares never make a group essential. A cell marked X doesn't need covering, so only real 1s count in step 2.

Some functions have no essential prime implicants at all. Every 1 sits in two or more groups, and you must choose. These are cyclic maps, and they can have more than one minimal answer.

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

Worked examples

Example

Essentials in a 3-variable map

F = Σm(0, 1, 3, 7), shown above.

1.
Prime implicants: m0 m1 → , m1 m3 → , m3 m7 → .
2.
m0 is only in , so that one is essential.
3.
m7 is only in , so that one is essential.
4.
Those two already cover m1 and m3, so is not needed.
5.
F = .

Example

Essentials in a 4-variable map

F = Σm(0, 4, 5, 7, 13, 15).

AB\CD	00	01	11	10
00	1m0	0m1	0m3	0m2
01	1m4	1m5	1m7	0m6
11	0m12	1m13	1m15	0m14
10	0m8	0m9	0m11	0m10

1.
Prime implicants: m0 m4 → , m4 m5 → , and the square m5 m7 m13 m15 → .
2.
m0 is only in : essential.
3.
m7, m13 and m15 are only in : essential.
4.
m4 and m5 are already covered, so is left out.
5.
F = .

Common mistakes

Assuming the largest group is always essential. A big group can be fully covered by smaller essential ones.
Counting a don't-care as the 1 that makes a group essential. Only real 1s need covering.
Thinking the essentials alone always cover the function. Often some 1s are left for a second step.

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Essential prime implicant is taught in Karnaugh Maps and Boolean Simplification.

Karnaugh Maps →Boolean Simplification →

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