Simplifying Boolean Equations with Karnaugh Maps — Part: 2 Implicants, Prime Implicants and Essential Prime Implicants.

While forming a boolean equation from kmap there are few concepts which are important. So let us consider each concept one by one:

1] Implicants: Implicants in kmap refer to any product/minterm in the Sum of Product (SOP) or sum/maxterm in the Product of Sum (POS) of a Boolean equation.
Consider a Boolean equation, Y = ABC + BC + AC. Here, the implicants are ABC, BC and AC.

2] Prime Implicants (PI): In prime implicants we have groups which cover all ones. It can also be defined as all the possible groups. To understand prime implicants let us consider an example:

Suppose we have a 4-variable kmap with ones at positions 5, 11, 13, 14, and 15.

By following the rules of grouping kmaps, we can form 4 groups. Let us denote these 4 groups of 2 ones denoted by colors:

• violet = group 1
• red = group 2
• yellow = group 3
• brown = group 4

As the possible groups where all ones will be included are 4, the SOPs of these 4 groups are the prime implicants. Therefore, the equation for the above kmap will be represented using these prime implicants.

3] Essential Prime Implicant (EPI): Essential prime implicants are groups that have at least one minterm (or one) that is not covered by any other group or prime implicant. These are the groups or prime implicants considered in the final solution. To understand it more clearly, let’s refer back to the previous example (figure 1).

In this example, there are 4 prime implicants i.e., 4 groups.

• Group 1 covers minterms at 5th and 13th position.
• Group 2 covers minterms at 13th and 15th position.
• Group 3 covers minterms at 14th and 15th position.
• Group 4 covers a minterm at the 11th position.

From these, we identify the essential prime implicants:

• Group 1 is an essential prime implicant because it has a minterm (5th) that is unique and not present in any other group.
• Group 3 is an essential prime implicant because it has a minterm (14th) that is not present in any other group.
• Group 4 is an essential prime implicant because it has a minterm (11th) that is not present in any other group.
• Group 2 cannot be considered an essential prime implicant as both of its minterms (13th and 15th) are present in other groups.

Therefore, the final solution will only contain these 3 essential prime implicants.

4] Redundant prime implicants (RPI): The prime implicants for which each of its minterm is covered by some essential prime implicant are redundant prime implicants. Here in our above example group 2 is the redundant prime implicant as each of its minterm is covered by at least one essential prime implicant.

5] Selective prime implicant (SPI): These are neither Redundant prime implicants nor essential prime implicants. They may or may not appear in certain solutions.

Consider, the below example where we have 3 variable k map and minterms at 0, 2, 3, 5, 7 position. Here, we can have 4 prime implicants and let's denote them as follows:

• violet = group 1
• red = group 2
• yellow = group 3
• brown = group 4

Here,

• Group 1 and group 4 are essential prime implicants as they have at least 1 minterm i.e., 1 which is unique and is not present in any other prime implicant.
• Group 2 and group 3 here cannot be redundant prime implicants as redundant prime implicants have all minterms present in essential prime implicant both here minterm present at position 3 is not present in any essential prime implicant. Hence, Group 2 and Group 3 are selective prime implicants.

So, the above example has 2 selective implicants.
Hence, selective prime implicants are groups which are neither Redundant prime implicants nor essential prime implicant.

In the next blog, let us see how to form a boolean equation from a Kmap. So, Stay Tuned! :)

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