Editing Implicant

{{Use dmy dates|date=May 2019|cs1-dates=y}}
In [[Boolean logic]], the term '''implicant''' has either a generic or a particular meaning.  In the generic use, it refers to the hypothesis of an implication ([[wiktionary:implicant|implicant]]).  In the particular use, a [[product term]] (i.e., a conjunction of literals) ''P'' is an '''implicant''' of a Boolean function ''F'', denoted  <math> P \le F</math>, if ''P'' implies ''F'' (i.e., whenever ''P'' takes the value 1 so does ''F'').
For instance, implicants of the function
:<math>f(x,y,z,w)=xy+yz+w</math>
include the terms <math>xy</math>, <math>xyz</math>, <math>xyzw</math>, <math>w</math>, 
as well as some others.

== {{anchor|Essential prime implicant}}Prime implicant ==<!-- Section header "Prime implicant" used by redirects -->
A '''prime implicant''' of a function is an implicant (in the above particular sense) that cannot be covered by a more general, (more reduced, meaning with fewer [[literal (computer programming)|literal]]s) implicant. [[W.&nbsp;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''.  '''Essential prime implicants''' (also known as '''core prime implicants''') are prime implicants that cover an output of the function that no combination of other prime implicants is able to cover.<ref>{{cite web | url=https://philosophy-question.com/library/lecture/read/207320-what-are-the-essential-prime-implicants | title=What are the essential prime implicants? }}</ref>

Using the example above, one can easily see that while <math>xy</math> (and others) is a prime implicant, <math>xyz</math> and <math>xyzw</math> are not. From the latter, multiple literals can be removed to make it prime:

*<math>x</math>, <math>y</math> and <math>z</math> can be removed, yielding <math>w</math>.
*Alternatively, <math>z</math> and <math>w</math> can be removed, yielding <math>xy</math>.
*Finally, <math>x</math> and <math>w</math> can be removed, yielding <math>yz</math>.

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 (in binary Boolean algebra). Using the example function above, we may expand <math>xyz</math> to <math>xy</math> or to <math>yz</math> without changing the cover of <math>f</math>.<ref>De Micheli, Giovanni. ''Synthesis and Optimization of Digital Circuits''. McGraw-Hill, Inc.,  1994</ref>

The sum of all prime implicants of a Boolean function is called its '''complete sum''', '''minimal covering sum''', or  [[Blake canonical form]].

==See also==
* [[Quine&ndash;McCluskey algorithm]]
* [[Karnaugh map]]
* [[Petrick's method]]

==References==
{{reflist}}

==External links==
* [http://www.cs.ualberta.ca/~amaral/courses/329/webslides/Topic4-KarnaughMaps/sld021.htm Slides explaining implicants, prime implicants and essential prime implicants]
* [http://web.cecs.pdx.edu/~mcnames/ECE171/Lectures/Lecture10.html Examples of finding essential prime implicants using K-map]

[[Category:Boolean algebra]]