Canonical and Standard Form

Boolean algebra is the foundation of digital electronics, enabling the design and analysis of logic circuits. Canonical and standard forms are key concepts in Boolean algebra, used to simplify and optimize these circuits for better performance.

• Canonical forms provide a unique representation of Boolean functions using minterms or maxterms.
• Standard forms simplify logic expressions into sum of products or product of sums for easier implementation.
• These forms help reduce hardware complexity and improve efficiency in digital design.
• Mastering canonical and standard forms is key for troubleshooting and enhancing circuit reliability.

Canonical Forms in Boolean Algebra

Canonical forms in Boolean algebra provide a standard way to represent logic functions using minterms or maxterms.

• Minterms: A minterm is the product of different Boolean variables, each occurring exactly once, where the output is 1. If the value of the Boolean variable is 1, we take the variable without complementing it and if the value is 0, we take the variable with complement.
• Maxterms: A maxterm is the sum of different Boolean variables, each occurring exactly once, where the output is 0. If the value of the Boolean variable is 0, we take the variable without complementing it and if the value is 1, we take the variable with complement.

Truth table representing minterm and maxterm:

There are two types of canonical forms that are commonly used:

1. Canonical Sum of Products (SOP)

In the Canonical Sum of Products form, every term in the Boolean expression is minterm. The sum (OR) of these minterms represents the final expression. For this sum we only consider minterms where output of Boolean function is 1. Each minterm contains all input variables either in true or in complemented form.

2. Canonical Product of Sums (POS)

In the Canonical Product of Sums form, every term in the Boolean expression is maxterm. The product (AND) of these maxterms represents the final expression. For this product we only consider maxterms where output of Boolean function is 0. Each maxterm contains all input variables either in true or in complemented form.

Example

Consider two Boolean variable A, B and a Boolean function F. The Boolean function F is equal to A XOR B.

F(A, B) = A ⊕ B

Truth Table for F(A, B):

Canonical Sum of Products (SOP) Form for the Boolean Function 'F':

F(A, B) = m₁ + m₂ = A'.B + A.B'

Canonical Product of Sums (POS) Form for the Boolean Function 'F':

F(A, B) = M₀ + M₃ = (A + B).(A' + B')

These forms ensure every Boolean expression has a unique representation, making analysis and simplification easier. Every Boolean function has only one canonical SOP and one canonical POS form, ensuring consistency in digital circuit design. It is used in Karnaugh maps, Quine-McCluskey method and logic minimization to optimize circuits for speed and cost.

Standard Forms in Boolean Algebra

Standard forms provide simplified ways to represent Boolean functions while maintaining clarity and efficiency in digital circuit design. Unlike canonical forms, they don't require every term to contain all variables.

There are two common standard forms used to express Boolean functions:

1. Standard Sum of Products (SOP)

The Standard Sum of Products is a simplified version of canonical SOP where product terms don't need to contain all variables. In this form the Boolean expression is written as a sum (OR operation) of multiple product terms (AND operations). Each product term contains input variables either in true or in complemented form.

For example,

F(A, B, C) = A'B'C + A'B + AB

Some product terms in this expression does not contain all of the Boolean variables such as, A'B or AB.

2. Standard Product of Sums (POS)

In this standard form, the Boolean expression is written as a product (AND operation) of sum terms (OR operations). It represents functions as AND operations on OR terms, offering a practical alternative to canonical POS. This optimized form reduces circuit complexity while maintaining logical equivalence.

For example,

F = (A+C)(B+C)(A'+B'+C)

Some of the sum terms in this expression does not contain all of the Boolean variables such as, (A+B) or (B+C).

Standard forms strike a balance between completeness and simplicity, making them practical for real-world digital design while maintaining clear logical relationships. These forms serve as the working language for digital engineers, bridging the gap between theoretical Boolean algebra and physical circuit implementation.

Difference between Canonical Form and Standard Form

In Canonical Form, every Boolean function is expressed in its most complete and precise form, based directly on its truth table. It contains every variable either in true or complemented form in its terms. The canonical forms are unique for each Boolean function because they are derived directly from the truth table without any simplification or omission of terms.

On the other hand, Standard Form offers a more flexible and simplified representation of the Boolean function. While it still involves the basic Sum of Products (SOP) or Product of Sums (POS) structure, it does not require all minterms or maxterms to be included. The Standard Forms are not unique because there can be multiple ways to simplify the function using Boolean identities, leading to different but equivalent expressions.

Conversion Between Canonical and Standard Forms

Boolean functions can be represented in canonical forms (complete minterm/maxterm expansions) or standard forms (simplified expressions). Converting between these forms is essential for circuit optimization

Converting Canonical Form into Standard Form

1. Start with canonical form (SOP or POS)
2. Apply Boolean algebra rules to simplify
3. Eliminate redundant terms when possible

Example:

Canonical SOP: F(A, B, C) = A'B'C + A'BC + AB'C + ABC

Step-by-Step Simplification:

Group terms with common variables:

(A′B′C + A′BC) + (AB′C + ABC)

Factor out common terms:

A′C(B′+B) + AC(B′ + B)

Apply B′+B=1:

A′C(1)+AC(1)

Final Standard SOP:

F=A′C+AC

Further simplification (optional):

F = C(A′ + A) = C (Fully minimized)

Standard Form: F(A, B, C) = C

Converting Standard Form into Canonical Form

• Identify missing variables in each product term.
• Expand using X = X(Y + Y′) (for SOP) or X = X + YY' (for POS) to include all variables.
• Remove duplicates (if any).

Example:

Standard SOP: F = A + BC

Step-by-Step Expansion:

In first term i.e. A, B and C are missing.

A = A(B + B′)(C + C′) = ABC + AB′C + ABC′ + AB′C′

In second term i.e. BC, A is missing.

BC = (A + A′)BC = ABC + A′BC

Combining all minterms:

F = ABC + AB′C + ABC′ + AB′C′ + ABC + A′BC

Remove duplicates (ABC appears twice):

F = A′BC + AB′C′ + AB′C + ABC′ + ABC

Canonical Form: F(A, B, C) = A′BC + AB′C′ + AB′C + ABC′ + ABC