Canonical Forms (SOP & POS)#

A canonical form is a systematic, mechanical way to write down a function straight from its truth table — no cleverness required. There are two, and they are duals of each other: sum of products, built from the rows where the output is 1, and product of sums, built from the rows where it is 0.

Sum of Products (from the 1s)#

For each row where the output is 1, write a minterm: the AND of every input variable, taken true where the variable is 1 and complemented where it is 0. OR all those minterms together and the result is the sum-of-products (SOP) form — it is 1 for exactly the input combinations that should give 1, and nothing else. For a function that is 1 on rows 3 and 5 of three variables, F = A′BC + AB′C, often written compactly as Σm(3, 5).

SOP is the more common of the two because it maps directly onto two levels of logic — a bank of AND gates feeding one OR gate — which is exactly the structure of a decoder (all minterms) with an OR, and of the programmable AND-OR array inside a PAL or CPLD.

Product of Sums (from the 0s)#

The dual approach uses the rows where the output is 0. For each, write a maxterm — the OR of every variable, complemented where it is 1 — and AND all the maxterms together. This product-of-sums (POS) form is 0 for exactly the rows that should be 0. It maps to the mirror structure, OR gates feeding an AND, and is the natural choice when a function is mostly 1s with a few 0s.

Canonical Versus Minimal#

Both forms are canonical: every term names every variable, so they are complete and unique but usually far from the smallest expression. They are the systematic starting point, not the finished design. Reducing a canonical form to a minimal one is the job of Boolean algebra and Karnaugh maps — the canonical form guarantees a correct expression, and minimization makes it a cheap one.