Canonical normal form

In a Boolean algebra, any Boolean function can be expressed in a canonical form using the dual concepts of minterms and maxterms. All logical functions are expressible in canonical form, both as a "sum of minterms" and as a "product of maxterms". This allows for greater analysis into the simplification of these functions, which is of great importance in the minimization of digital circuits.

A Boolean function expressed as a disjunction (OR) of minterms is commonly known as the "sum of products" (SoP), and its De Morgan dual is the "product of sums" (PoS), which is a function expressed as a conjunction (AND) of maxterms.

Minterms

For a boolean function of $n$ variables ${x_{1},...x_{n}}$, a product term in which each of the $n$ variables appears once (either complemented, or uncomplemented) is called a minterm. Thus, a minterm is a logical expression of n variables consisting of only the logical conjunction operator and the complement operator.

For example, $abc$, $ab'c$ and $abc'$ are examples of minterms for a boolean function of the three variables $a$, $b$ and $c$.

There are 2ⁿ minterms of n variables - this is true since a variable in the minterm expression can either be in the form of itself or its complement - two choices per n variables.

Indexing minterms

In general, one assigns each minterm (ensuring the variables are written in the same order, usually alphabetic), an index based on the binary value of the minterm. A complemented term, like a' is considered a binary 0 and a noncomplemented term like a is considered a binary 1. For example, one would associate the number 6 with a b c'(110₂), and write the minterm expression as m₆. So m₀ of three variables is a'b'c'(000₂) and m₇ would be a b c(111₂).

Functional equivalence

It is apparent that minterm n gives a true value for the n+1 th unique function input for that logical function. For example, minterm 5, a b' c, is true only when a and c both are true and b is false - the input where a = 1, b = 0, c = 1 results in 1.

If one is given a truth table of a logical function, it is possible to write the function as a "sum of products". This is a special form of disjunctive normal form, qv. For example, if given the truth table

a b  f(a, b)
0 0  1
0 1  0
1 0  1
1 1  0

observing that the rows that have an output of 1 are the first and third, so we can write f as a sum of minterms m₀ and m₂.

If we wish to verify this:

f(a,b) = m₀ + m₂ = (a'b')+(ab')

then the truth table for this function, by direct computation, will be the same.

Maxterms

A maxterm is a logical expression of n variables consisting of only the logical disjunction operator and the complement operator. Maxterms are a dual of the minterm idea. Instead of using ANDs and complements, we use ORs and complements, and proceed similarly.

For example, the following are maxterms:

a+b'+c

a'+b+c

There are again 2ⁿ maxterms of n variables - this is true since a variable in the maxterm expression can also be in the form of itself or its complement - two choices per n variables.

Dualization

The complement of a minterm is the respective maxterm. This can be easily verified by using de Morgan's law. For example

m₁' = M₁

(a'b)' = a+b'

Indexing maxterms

Indexing maxterms is done in the opposite way as with minterms. One assigns each maxterm an index based on the order of its complements (again, ensuring the variables are written in the same order, usually alphabetic). For example, one might assign M₆ (Maxterm 6) to the maxterm a'+b'+c. Similarly M₀ of these three variables could be a+b+c and M₇ could be a'+b'+c'.

Functional equivalence

It is apparent that maxterm n now gives a false value for the n+1 th unique function input for that logical function. For example, maxterm 5, a'+b+c', is false only when a and c both are true and b is false - the input where a = 1, b = 0, c = 1 results in 0.

If one is given a truth table of a logical function, it is possible to write the function as a "product of sums". This a special form of conjunctive normal form, q.v. For example, if given the truth table

a b  f(a, b)
0 0  1
0 1  0
1 0  1
1 1  0

observing that the rows that have an output of 0 are the second and fourth, so we can write f as a product of maxterms M₁ and M₃.

If we wish to verify this:

f(a,b) = M₁ M₃ = (a+b')(a'+b')

then the truth table for this function, by direct computation, will be the same.

Non cannonical PoS and SoP forms

It is often the case that the cannonical minterm form can be simplifyed to an equivelent SoP form. This simplified form would still consist of a sum of product terms. However, in the simplifyed form it is posible to have either less product terms, and/or product terms that contain less variables (=that are shorter). 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 cannonical minterm representation: $f=a'bc+abc$ But it has an equivelent simplified form: $f=bc$ In this trivial example it is obvious that $bc=a'bc+abc$. 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 the similar manner, a cannonical maxterm form can have a simplified PoS form.

A convinient method for finding the minimal Pos/Sop form of a functions 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.

See also

• Canonical form
• List of Boolean algebra topics
• Logical graph