Canonical normal form

In Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form, minterm canonical form, or Sum of Products as a disjunction of minterms. The De Morgan dual is the canonical conjunctive normal form, maxterm canonical form, or Product of Sums which is a conjunction of maxterms. These forms can be useful for the simplification of Boolean functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular.
Other canonical forms include the complete sum of prime implicants or Blake canonical form, and the algebraic normal form .

Minterms

For a boolean function of variables, a minterm is a product term in which each of the variables appears exactly once. Thus, a minterm is a logical expression of n variables that employs only the complement operator and the conjunction operator. A minterm gives a true value for just one combination of the input variables, the minimum nontrivial amount. For example, a ''b' c'', is true only when a and c both are true and b is false—the input arrangement where a = 1, b = 0, c = 1 results in 1.

Indexing minterms

There are 2ⁿ minterms of n variables, since a variable in the minterm expression can be in either its direct or its complemented form—two choices per variable. Minterms are often numbered by a binary encoding of the complementation pattern of the variables, where the variables are written in a standard order, usually alphabetical. This convention assigns the value 1 to the direct form and 0 to the complemented form ; the minterm is then. For example, minterm is numbered 110₂ = 6₁₀ and denoted.

Minterm canonical form

Given the truth table of a logical function, it is possible to write the function as a "sum of products" or "sum of minterms". This is a special form of disjunctive normal form. For example, if given the truth table for the arithmetic sum bit u of one bit position's logic of an adder circuit, as a function of x and y from the addends and the carry in, ci:

ci	x	y	u
0	0	0	0
0	0	1	1
0	1	0	1
0	1	1	0
1	0	0	1
1	0	1	0
1	1	0	0
1	1	1	1

Observing that the rows that have an output of 1 are the 2nd, 3rd, 5th, and 8th, we can write u as a sum of minterms and. If we wish to verify this: evaluated for all 8 combinations of the three variables will match the table.

Maxterms

For a boolean function of variables, a maxterm is a sum term in which each of the variables appears exactly once. Thus, a maxterm is a logical expression of variables that employs only the complement operator and the disjunction operator. Maxterms are a dual of the minterm idea, following the complementary symmetry of De Morgan's laws. Instead of using ANDs and complements, we use ORs and complements and proceed similarly. It is apparent that a maxterm gives a false value for just one combination of the input variables, i.e. it is true at the maximal number of possibilities. For example, the maxterm a′ + b + c′ is false only when a and c both are true and b is false—the input arrangement where a = 1, b = 0, c = 1 results in 0.

Indexing maxterms

There are again 2ⁿ maxterms of variables, since a variable in the maxterm expression can also be in either its direct or its complemented form—two choices per variable. The numbering is chosen so that the complement of a minterm is the respective maxterm. That is, each maxterm is assigned an index based on the opposite conventional binary encoding used for minterms. The maxterm convention assigns the value 0 to the direct form and 1 to the complemented form. For example, we assign the index 6 to the maxterm and denote that maxterm as M₆. The complement is the minterm, using de Morgan's law.

Maxterm canonical form

If one is given a truth table of a logical function, it is possible to write the function as a "product of sums" or "product of maxterms". This is a special form of conjunctive normal form. For example, if given the truth table for the carry-out bit co of one bit position's logic of an adder circuit, as a function of x and y from the addends and the carry in, ci:

ci	x	y	co
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	1

Observing that the rows that have an output of 0 are the 1st, 2nd, 3rd, and 5th, we can write co as a product of maxterms and. If we wish to verify this:
evaluated for all 8 combinations of the three variables will match the table.

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:

a	b	c	f
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, but it has an equivalent SoP form. In this trivial example, it is obvious that, and the smaller form has both fewer product terms and fewer variables within each term. The 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. 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 , 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.

Application example

The sample truth tables for minterms and maxterms above are sufficient to establish the canonical form for a single bit position in the addition of binary numbers, but are not sufficient to design the digital logic unless your inventory of gates includes AND and OR. Where performance is an issue, the available parts are more likely to be NAND and NOR because of the complementing action inherent in transistor logic. The values are defined as voltage states, one near ground and one near the DC supply voltage V_cc, e.g. +5 VDC. If the higher voltage is defined as the 1 "true" value, a NOR gate is the simplest possible useful logical element.
Specifically, a 3-input NOR gate may consist of 3 bipolar junction transistors with their emitters all grounded, their collectors tied together and linked to V_cc through a load impedance. Each base is connected to an input signal, and the common collector point presents the output signal. Any input that is a 1 to its base shorts its transistor's emitter to its collector, causing current to flow through the load impedance, which brings the collector voltage very near to ground. That result is independent of the other inputs. Only when all 3 input signals are 0 do the emitter-collector impedances of all 3 transistors remain very high. Then very little current flows, and the voltage-divider effect with the load impedance imposes on the collector point a high voltage very near to V_cc.
The complementing property of these gate circuits may seem like a drawback when trying to implement a function in canonical form, but there is a compensating bonus: such a gate with only one input implements the complementing function, which is required frequently in digital logic.
This example assumes the Apollo parts inventory: 3-input NOR gates only, but the discussion is simplified by supposing that 4-input NOR gates are also available.

Canonical and non-canonical consequences of NOR gates

A set of 8 NOR gates, if their inputs are all combinations of the direct and complement forms of the 3 input variables ci, x, and y, always produce minterms, never maxterms—that is, of the 8 gates required to process all combinations of 3 input variables, only one has the output value 1. That's because a NOR gate, despite its name, could better be viewed as the AND of the complements of its input signals.
The reason this is not a problem is the duality of minterms and maxterms, i.e. each maxterm is the complement of the like-indexed minterm, and vice versa.
In the minterm example above, we wrote but to perform this with a 4-input NOR gate we need to restate it as a product of sums, where the sums are the opposite maxterms. That is,

In the maxterm example above, we wrote but to perform this with a 4-input NOR gate we need to notice the equality to the NOR of the same minterms. That is,