Understanding Canonical Standard Forms

In digital logic design, a Boolean expression can be manipulated into many algebraic variations while still representing the exact same underlying logic. To accurately map expressions to truth tables, compare functions, or prepare them for Karnaugh Map (K-Map) simplification, we must convert them into a Canonical Form. An expression is considered "canonical" when every single variable in the given domain is explicitly present (either complemented or uncomplemented) in every term.

Sum of Products (SOP) & Minterms

A Canonical SOP expression highlights the conditions where the circuit's output evaluates to logic 1 (TRUE). Each individual product term within a Canonical SOP expression is known as a Minterm.

• In a Minterm, a variable is written uncomplemented (e.g., $A$) if its truth table value is 1, and complemented (e.g., $A'$) if its value is 0.
• The entire expression is constructed by logically ORing (+) the individual ANDed ($\cdot$) product terms together.
• In standard mathematical notation, canonical SOP is abbreviated using the Greek summation symbol alongside its decimal equivalents: $\Sigma(m)$.

Product of Sums (POS) & Maxterms

A Canonical POS expression is the dual of SOP. It highlights the conditions where the circuit's output evaluates to logic 0 (FALSE). Each individual sum term within a Canonical POS expression is known as a Maxterm.

• In a Maxterm, the mapping rule is inverted: a variable is written uncomplemented (e.g., $A$) if its truth table value is 0, and complemented (e.g., $A'$) if its value is 1.
• The entire expression is constructed by logically ANDing ($\cdot$) the individual ORed (+) sum terms together.
• In standard mathematical notation, canonical POS is abbreviated using the Greek product symbol alongside its decimal equivalents: $\Pi(M)$.