What is the difference between canonical form and standard form?

The main difference between canonical and standard form is that canonical form is a way of representing boolean outputs of digital circuits using Boolean Algebra whereas standard form is a simplified version of canonical form that represents boolean outputs of digital circuits using Algebra.

Digital circuits operate using digital signals. These signals have discrete binary values: they are zero and one. Zero means the false state while one means the true state. Boolean algebra is a type of algebra that helps to represent binary numbers and binary variables. Canonical form is a method of representing Boolean functions of Boolean algebra, while standard form is a simplified version of canonical form.

Key Areas Covered

1. What are minimum terms and maximum terms?
     – Definition, Functionality
2. What is canonical form?
     – Definition, Functionality
3. What is the standard form?
     – Definition, Functionality
4. What is the difference between canonical form and standard form?
     – Comparison of key differences

Key terms

Canonical Form, Maximum Terms, Minimum Terms, Standard Form

What are minterms and maxterms?

For example, suppose we have two variables such as a and b. After performing the AND operation, we call the Boolean product terms of these two variables as minterms or standard product terms. These minterms are a’b’, a’b, ba’, and ab.

After performing the OR operation, we call the Boolean product terms of these two variables as max terms or standard sum terms. These peak terms are a + b, a + b’, a’ + b and a’ + b’.

The min and max terms of a and b are as follows.

What is the canonical form?

A truth table represents inputs and outputs. If there are ‘n’ number of input variables, then there are 2 n ^Number of outputs or combinations or units and zeros. There are two methods in canonical form to represent an output variable. They are canonical SoP form and canonical PoS form.

Canonical SoP stands for Canonical Sum of Products. This form considers them in terms It is called the form of least terms of min. First, it is necessary to recognize the minterms that have 1 as the output variable. After identifying them in terms, the logical OR is used to find the Boolean expression equivalent to the output variable.

Also, Canonical PoS stands for Canonical Product of Sums. This form refers to maximum terms. It’s called the Max Product Terms Form. First, it is necessary to recognize the maximal terms that have 0 as the output variable. After identifying the max terms, the logical AND is used to find the Boolean expression equivalent to the output variable.

What is the standard form?

Another method of representing Boolean outputs is by using the standard form. It is divided into two forms as SoP standard form and PoS standard form.

Standard SoP stands for standard sum-of-products form. It is not necessary for each product term to contain all literals. Therefore, the terms of the product may or may not be the minimum terms. There are two steps involved in getting the standard SoP form of the output variable. The first step is to obtain the canonical SoP form of the output variable. The second step is to simplify the boolean function in the Canon SoP form. If there is more than one output variable, it is possible to follow the same method for other variables. There may be situations where it is impossible to simplify the canonical form of SoP. So the standard and canonical SoP forms are similar.

The standard PoS form means the standard product of sums. It is not necessary for each product term to contain all literals. Therefore, the sum terms may or may not be the max terms. There are two steps involved in getting the standard POS form of the output variable. The first step is to obtain the canonical benchmark form of the output variable. The second step is to simplify the boolean function in the Canon PoS form. If there is more than one output variable, it is possible to follow the same method for other variables. There may also be situations where it is impossible to simplify the PoS canonical form. In such cases, the standard and canonical PoS forms are similar.

Difference Between Canonical and Standard Form

Definition

The canonical form is a representation that helps describe Boolean outputs from digital circuits that use Boolean functions. In contrast, the standard form is a simplified version of the canonical form. These definitions contain the main difference between canonical form and standard form. 

Division

Another difference between canonical form and standard form is that canonical form is divided into canonical SoP form and canonical PoS form, while standard form is divided into standard SoP form and standard PoS form. 

Complexity

Also, the canonical form is more complex while the standard form is simple.

conclusion

In short, canonical form and standard form are a method of representing Boolean functions of Boolean algebra. The difference between canonical and standard form is their complexity as standard form is a simplified version of canonical form.

Reference:

1. “Canonical Normal Form.” Wikipedia , Wikimedia Foundation, October 23, 2018, Available here.

Courtesy image:

1. “1539113” (CC0) via Maxpixel