Canonical form

Also called: canonical forms, canonical expression, Boolean canonical form

A standard way of writing a Boolean function so that each function has exactly one such expression: the sum of minterms or the product of maxterms.

The same function can be written as endless different expressions. A canonical form fixes one standard way to write it, so that each function has exactly one canonical expression. There are two:

Canonical SOP (sum of minterms, Σm): one minterm for each row where F = 1, ORed together.
Canonical POS (product of maxterms, ΠM): one maxterm for each row where F = 0, ANDed together.

Why they are useful:

They come straight from a truth table, with no cleverness needed.
They make comparison easy. Two expressions are equivalent exactly when they have the same minterm list.
They are the starting point for systematic simplification.

What they are not: cheap. Every term contains all n variables. For F(A, B, C) = Σm(4, 5, 6, 7) the canonical SOP has 12 literals, while the simplest form is just A.

Which to write? Use whichever list is shorter. A function with 13 ones out of 16 rows has a 13-term canonical SOP but only a 3-term canonical POS.

Non-canonical SOP and POS forms, where terms may leave out variables, are sometimes called standard forms.

Worked example

Example

Both canonical forms of XOR

F(A, B) = 1 when A and B differ. Write both canonical forms.

1.
F = 1 on rows 01 and 10 (rows 1 and 2): F = Σm(1, 2) = .
2.
F = 0 on rows 00 and 11 (rows 0 and 3): F = ΠM(0, 3).
3.
M0 = and M3 = , so F = .
4.
Multiplying the POS out gives = , the same function.

Common mistakes

Calling any SOP canonical. In a canonical SOP every term contains every variable.
Thinking a canonical form is the simplest form. It is the most complete, and usually the most expensive.

Practice Canonical form

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Canonical form is taught in Boolean Simplification and Combinational Logic.

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