Editing Canonical normal form (section)

==Minimal PoS and SoP forms==
It is often the case that the canonical minterm form is equivalent to a smaller SoP form. This smaller form would still consist of a sum of product terms, but have fewer product terms and/or product terms that contain fewer variables. For example, the following 3-variable function:

{| class="wikitable" style="margin: 1em auto 1em auto"
!width="50"|a
!width="50"|b
!width="50"|c
!width="50"|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 <math>f = a'bc + abc</math>, but it has an equivalent SoP form <math>f = bc</math>. In this trivial example, it is obvious that <math>bc = a'bc + abc</math>, and the smaller form has both fewer product terms and fewer variables within each term. The [[minimal element|minimal]] SoP representations of a function according to this notion of "smallest" are referred to as ''minimal SoP forms''. In general, there may be multiple minimal SoP forms, none clearly smaller or larger than another.<ref>{{cite book |last1=Lala |first1=Parag K. |title=Principles of Modern Digital Design |date=16 July 2007 |publisher=John Wiley & Sons |isbn=978-0-470-07296-7 |page=78 |url=https://books.google.com/books?id=olQ3EAAAQBAJ&dq=minimal%20sum%20of%20products%20form%20unique&pg=PA78 |language=en}}</ref> In a similar manner, a canonical maxterm form can be reduced to various minimal PoS forms.

While this example was simplified by applying normal algebraic methods [<math>f = (a' + a) b c</math>], in less obvious cases a convenient method for finding minimal PoS/SoP forms of a function with up to four variables is using a [[Karnaugh map]]. The [[Quine–McCluskey algorithm]] can solve slightly larger problems. The field of [[logic optimization]] developed from the problem of finding optimal implementations of Boolean functions, such as minimal PoS and SoP forms.