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[\<- Boolean Algebra and DeMorgan's Theorem](3.md)
---
# Karnaugh Maps, Prime Implicants
## K-map structure, implicants, prime implicants
### Karnaugh Maps (K-Maps for short)
- Expresses same info as truth table but in different form
### Implicants
- Any product term that evaluates to true
- Two implicants that differ by just one variable (true and false) can be combined
- `X*Y + X*!Y` = `X`
- `X*Y*Z + X*Y*!Z` = `X*Y`]
- A "larger" implicant even though less variables
- An implicant that can't be combined with another is called a "prime" implicant (PI)
- Identifying the PI's allows us to synthesize simpler logic than canonical SOP
---
## Basics of using a K-map, with a 2-input function
### Minimizing with K Maps
- Circling adjacent cells represents combining two implicants into one
- A circle that "covers both values of a variable means it's not needed in the term
- It's OK for implicants to overlap
### K-map for the OR function
- We could define OR as F(A,B) = ∑m(1,2,3)
- If we synthesized using minterms we would get F = !A\*B + A\*!B + A\*B
- But we know OR is F=A+B
- Using a K-map identifies overlap
---
## Structure of a 3-input K-map
### 3-input functions
- One side must represent two variables
- Four values
- We want to keep the ability to circle adjacent cells to represent a larger implicant
- Enumerate as 00, 01, 11, 10
- Opportunity to "cover" four cells
- Edge cells on opposite sides are still "adjacent"
- 00 and 10 differ by one value
### Example 3-input K-map
- Can be drawn veritcally or horizontally (the example below shows both options)
- Also, order of variables doesn't matter
- As long as you correctly apply the concepts
---
## Discerning product terms
- Every implicant, prime or not, can be specified as a product term
- Individual cells are minterms
- A grouping of two means one of the variables is not needed because both the true and complement case are "covered"
- For the remaining two variables, llok to see if they are 0 or 1 for the two cells
- If 0, then the variable needs to be inverted before including in the product term
- A grouping of four reduces to one variable
---
## A few more 3-input K-map examples
- First map is "missing" a prime implicant
---
## 4-variable K-maps
### Going to four variables
- The vertical axis now covers two variables
- Same ordering considerations: 00 01 11 10
- Now we can have prime implicants that cover 8 cells
- Can't do 6
- Every time you're combining you're accounting for the true and false version of one of the variables
- Necessarily a power of 2
- Wrap-arounds in both directions
### 4-Input Examples
- Another missing prime implicant, in the upper left map
- Can you find it?
- The missing prime implicant is the grouping of the 01-10 with the 11-10
---
## Essential prime implicants
- Once **ALL** prime implicants (PI's) are identified, there is typically some amount of overlap
- Including all PI's in the solution equation might be unnecessary
- We'd like to find the simplest solution
- Some of the PI's will \*have\* to be in the solution equation -> they are essential
- Visually, they are the PI's that contain cells (minterms) not covered by any other PI
### Visualizing essential PI's
- K-map below has all PI's circled
- Only `!X3*!X4` is essential
- Minimal solution only needs to add greens
- Complete coverage with just 2 more terms
### K-map with no essentials
- To fully illustrate the point, note this K-map has no essentials
- Solution equation will have four terms
---
## Summary of the K-map process
### Synthesizing with K-maps
- Identify \*all\* PI's
- Start with largest, to avoid non-primes
- Don't forget to look for wrap-arounds
- Identify which PI's are "essential"
- An essential PI contains cells that are not part of any other PI
- Write the SOP equation, starting with the essential PI's
- Add the minimum number of non-essential PI's needed to "cover" all cases where the function should evaluate to true
---
[Additional K-map concepts: solving for 0's and use of don't cares ->](5.md)
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