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Last Updated: Apr 23, 2024
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Canonical Forms

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Prerita Agarwal
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23 Jul, 2024 @ 01:30 PM

Introduction

In mathematics and computer science, the concept of canonical forms holds a profound significance, serving as a fundamental tool for simplification, standardization, and deeper comprehension of complex structures. From algebraic expressions to logic circuits and beyond, canonical forms provide a systematic representation that unveils hidden patterns, facilitates analysis, and enables efficient manipulation.

Canonical Forms

For example:

F(A,B,C) = A’+ABC. (In the first term B and C are missing).
F(A,B,C,D)=AB+BC+A’B’CD’.

What is Canonical Form?

In mathematics and computer science, a canonical form refers to a standard representation of an object or entity that possesses certain desirable properties. It serves as a unique and simplified form that aids in analysis, comparison, and manipulation. Canonical forms are often defined based on specific criteria or transformations, enabling consistent interpretation and facilitating efficient algorithms and computations.

Example

F(A,B,C)=A’B’C+ABC.
F(A,B,C,D)=A’BCD+AB’C’D+ABCD.

 

In this blog, we will be discussing only canonical form in detail. 

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Canonical Sum of Products

  • A product that contains each of n variables as factors either in complemented or uncomplemented form is called a minterm.
  • A minterm is given the value 1 for exactly one combination of the variables.
  • The sum of all minterms of ‘f’ for which ‘f’ assumes ‘1’ is called Canonical sum of product or disjunctive normal form.

 

Say: if a+bc+ac, this is the sum of product.

If abc+a’bc, this is the canonical sum of product as this contain each of n variables.

Example

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f=a’b’c+a’bc’+ab’c’+abc’

Short form: Σ(1,2,4,6)

First, we figured out the row where the value of ‘f’ is 1, then we start writing SOP, by denoting 0 as the complement form of the variable and 1 as the variable is.

Canonical Product of Sums

  • A sum term that contains each of ‘n’ variables as factors either in complemented or uncomplemented form is called a max term.
  • A max term gives the value ‘0’ for exactly one combination of the variables.
  • The product of all the maxterms of ‘f’ for which ‘f’ assumes 0 is called Canonical product of sum or conjunctive normal form. 

 

Example

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f=(a+b+c)(a+b’+c’)(a’+b+c’)(a’+b’+c)

Short form: Π(0,3,5,7)

First, we figured out the row where the value of ‘f’ is 0, then we start writing POS, by denoting 1 as the complement form of the variable and 0 as the variable is.

Converting Standard Form to Canonical Form

Convert the following function to canonical SOP.

f(x,y,z) = x’y + z’ + xyz

Ans = x’y.1 + z’.1.1 + xyz

= x’y(z+z’) + (x+x’)(y+y’)z + xyz

= x’yz+x’yz’+xyz’+xy’z’+x’yz’+x’y’z+xyz

Advantages of Canonical Form

  • Standardization: Canonical forms provide a standardized representation of objects or entities, promoting uniformity and consistency across various contexts.
  • Simplification: By reducing complex structures to their canonical forms, it becomes easier to understand, analyze, and manipulate them.
  • Efficient Algorithms: Canonical forms often lend themselves to the development of efficient algorithms and computational techniques.
  • Uniqueness: Canonical forms ensure a unique representation for each object or entity, eliminating ambiguity and ensuring consistent interpretation.

Disadvantages of Canonical Form

  • Loss of Information: Converting complex structures to canonical forms may result in the loss of information, potentially leading to a loss of fidelity or accuracy.
  • Computational Complexity: The process of deriving or transforming structures into canonical forms may incur computational overhead, especially for large-scale or intricate systems.
  • Context Dependency: The suitability of a canonical form may depend on the specific context or requirements of a problem, limiting its universality and applicability.

Frequently Asked Questions

What are the functional properties of canonical forms?

The Canonical sum of products or POS form of switching function is unique. Two switching functions f1(x1… xn) and f2(x1...xn) are said to be logically equivalent if and only if both functions have the same value for each and every combination of (x1,x2,..,xn). Two switching functions are equal if their Canonical Product of sum or Sum of product are identical.

What is the formula for canonical form?

In Boolean algebra, the formula for canonical form varies depending on whether it's in sum-of-products (SOP) form or product-of-sums (POS) form. In SOP form, it's expressed as a logical OR of logical AND terms, while in POS form, it's a logical AND of logical OR terms.

What is canonical form in logic gates?

In logic gates, canonical form refers to a standard representation of Boolean expressions using either sum-of-products (SOP) or product-of-sums (POS) format. It represents all possible combinations of inputs and outputs, aiding in simplification and analysis of logic circuits.

What is an example of canonical?

An example of a canonical representation is the sum-of-products (SOP) form in Boolean algebra, where expressions are written as a logical OR of logical AND terms. For instance, the canonical form of 𝐹(𝐴,𝐵,𝐶)=(𝐴⋅𝐵)+(𝐴‾⋅𝐵⋅𝐶)F(A,B,C)=(AB)+(ABC).

What is canonical and non-canonical form?

Canonical form refers to a standard and unique representation of an object or entity, while non-canonical form deviates from this standard. In logic, canonical forms adhere to specific rules and formats, facilitating systematic analysis and manipulation, whereas non-canonical forms may lack uniformity or standardization.

Conclusion

This article taught us about canonical forms. We individually discussed both sum of products and product of sums and even the steps to convert standard form to canonical form.

We hope you could easily take away all critical and conceptual techniques by walking over the given examples. 

Now, we strongly recommend you to understand the other related concepts in boolean algebra and enhance your learning. You can get a wide range of topics similar to this on booleanalgebra.

Recommended Reading - Canonical Cover In DBMS.

It's not the end. Learn and explore more.

Topics covered
1.
Introduction
2.
What is Canonical Form?
2.1.
Example
3.
Canonical Sum of Products
3.1.
Example
4.
Canonical Product of Sums
4.1.
Example
5.
Converting Standard Form to Canonical Form
6.
Advantages of Canonical Form
7.
Disadvantages of Canonical Form
8.
Frequently Asked Questions
8.1.
What are the functional properties of canonical forms?
8.2.
What is the formula for canonical form?
8.3.
What is canonical form in logic gates?
8.4.
What is an example of canonical?
8.5.
What is canonical and non-canonical form?
9.
Conclusion