Non-canonical PoS and SoP Forms
It is often the case that the canonical minterm form can be simplified to an equivalent SoP form. This simplified form would still consist of a sum of product terms. However, in the simplified form it is possible to have fewer product terms and/or product terms that contain fewer variables. For example, the following 3-variable function:
| a | b | c | f(a,b,c) |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |
has the canonical minterm representation:, but it has an equivalent simplified form: . In this trivial example it is obvious that but the simplified form has both fewer product terms, and the term has fewer variables. The most simplified SoP representation of a function is referred to as a minimal SoP form.
In a similar manner, a canonical maxterm form can have a simplified PoS form.
While this example was easily simplified by applying normal algebraic methods, in less obvious cases a convenient method for finding the minimal PoS/SoP form of a function with up to four variables is using a Karnaugh map.
The minimal PoS and SoP forms are very important for finding optimal implementations of boolean functions and minimizing logic circuits.
Read more about this topic: Canonical Form (Boolean Algebra)
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