Implicant


In Boolean logic, the term implicant has either a generic or a particular use. In the generic use, it refers to the hypothesis of an implication. In the particular use, a product term P is an implicant of a Boolean function F, denoted , if P implies F.
For instance, implicants of the function
include the terms,,,,
as well as some others.

Prime implicant

A prime implicant of a function is an implicant that cannot be covered by a more general implicant. W. V. Quine defined a prime implicant to be an implicant that is minimal—that is, the removal of any literal from P results in a non-implicant for F. An essential prime implicant is a prime implicant that covers an input combination, for which the function is true, that no combination of other prime implicants is able to cover.
Using the example above, one can easily see that while is a prime implicant, and are not. From the latter, multiple literals can be removed to make it prime:
  • , and can be removed, yielding.
  • Alternatively, and can be removed, yielding.
  • Finally, and can be removed, yielding.
The process of removing literals from a Boolean term is called expanding the term. Expanding by one literal doubles the number of input combinations for which the term is true. Using the example function above, we may expand to or to without changing the cover of.
The sum of all prime implicants of a Boolean function is called its complete sum, minimal covering sum, or Blake canonical form.