A minimal SOP is the cheapest sum of products for a function. The usual order of priorities:
- the fewest product terms (each multi-literal term needs its own AND gate),
- then the fewest literals (each is one gate input).
Key facts:
- It uses only prime implicants. If a term is not a prime implicant, you could drop a literal and still cover only 1s, making it cheaper. So every term of a minimal SOP is prime.
- It does not always use every prime implicant. Some primes are redundant, often because they are a consensus of two others. See consensus theorem.
- It may not be unique. Some functions have two equally cheap answers.
- It is not always cheaper than the minimal POS. Check both when cost matters; see minimal pos.
How to find one: algebra (boolean simplification) works but gives no guarantee. Combining minterms systematically, then choosing a minimal cover of prime implicants, does. A karnaugh map makes that choice visual for up to four variables.
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 |