Implicant
From Wikipedia, the free encyclopedia
In Boolean logic, an implicant is a "covering" (sum term or product term) of one or more minterms in a sum of products (or maxterms in a product of sums) of a boolean function. Formally, a product term P in a sum of products is an implicant of the Boolean function F if P implies F. More precisely:
- P implies F (and thus is an implicant of F) if F also takes the value 1 whenever P equals 1.
where
- F is a Boolean function of n variables.
- P is a product term.
This means that P < = F with respect to the natural ordering of the Boolean space. For instance, the function
- f(x,y,z,w) = xy + yz + w
is implied by xy, by xyz, by xyzw, by w and many others; these are the implicants of f.
A prime implicant of a function is an implicant that cannot be covered by a more general (more reduced - meaning with fewer literals) implicant. W.V. Quine defined a prime implicant of F to be an implicant that is minimal - that is, if the removal of any literal from P results in a non-implicant for F. Essential prime implicants are prime implicants that cover an output of the function that no other prime implicant (or sum thereof) is able to cover.
Using the example above, one can easily see that while xy (and others) is a prime implicant, xyz and xyzw are not. From the latter, multiple literals can be removed to make it prime:
- x, y and z can be removed, yielding w.
- Alternatively, z and w can be removed, yielding xy.
- Finally, x and w can be removed, yielding yz.
The process of adding literals to a Boolean term is called expanding the term. Expanding by one literal halves the number of input combinations for which the term is true (in binary Boolean algebra).
The sum of all prime implicants of a Boolean function is called the complete sum of that function.
