SOP and POS Canonical Forms

The words are used loosely and the marks depend on which is meant, so here they are side by side. Every row below is a statement about form, not about the function: the same function has all of them at once.

mᵢ is the product term that is 1 at exactly index i; Mᵢ is the sum term that is 0 at exactly index i. A minterm is written by complementing each variable whose bit is 0; a maxterm by complementing each variable whose bit is 1 — the opposite way round, which is the whole difficulty.

That is why the minterm and maxterm table above prints both columns side by side rather than describing them, and why mᵢ′ = Mᵢ sits underneath it: complementing a minterm by De Morgan turns the product into a sum and every literal into its complement, which is exactly the maxterm at the same index.

The rule is a set complement over {0 … 2ⁿ−1}, minus the don't-cares, and that is all. On the page's default Σm(1, 4, 5, 6, 7) over three variables is ΠM(0, 2, 3). Two facts people expect to be harder: the index sets do not change when you convert — the same numbers move from one list to the other — and the variable order matters, because the indices are positional.

For SOP, multiply each term by (x + x′) for every variable x it is missing [Complement, then Identity], distribute [Distributive] and drop duplicates [Idempotent]. On the default, A is missing B and C, so it splits into four minterms; B′·C is missing A, so it splits into two; the duplicate A·B′·C is removed by idempotence, leaving five — Σm(1, 4, 5, 6, 7).

For POS the dual works: add x·x′ to each sum term that is missing x, then factor. Both walkthroughs are printed in full under Show the expansion working above, generated by the engine rather than written out here. The count check that catches an error: after expanding, the number of distinct minterms must equal the number of 1-rows in the truth table.

It is the unique fingerprint of a function. Two expressions are equal exactly when their canonical forms match, which is how this toolkit's equivalence check works underneath — it compares truth masks, the same statement one step further compressed. Canonical SOP is also where Quine–McCluskey starts: it is the first row of the combining table.

The default is A + B'C. Its canonical SOP is A′ · B′ · C + A · B′ · C′ + A · B′ · C + A · B · C′ + A · B · C — 5 terms · 15 literals. Its minimal SOP is B′ · C + A — 2 terms · 3 literals. Same function; five times the literals. Its canonical POS is (A + B + C) · (A + B′ + C) · (A + B′ + C′) and its minimal POS (A + C) · (A + B′). Minimal SOP is the cheapest of the four here — 2 terms · 3 literals, against 5 terms · 15 literals for the canonical SOP.

Nobody builds the canonical form. It exists to be compared and to be minimised — by named laws, on a map or in a table. All three reach the same answer.

Don't-cares belong to neither list, so |Σm| + |ΠM| + |d| = 2ⁿ. The consequence is stronger than it looks: a function with kdon't-cares has no single canonical form at all, but a family of 2ᵏof them, one per assignment of the don't-cares — all equally consistent with the specification. This page shows the one that reads every don't-care as 0, labels it as that, and states the alternative rather than implying uniqueness.

Canonical expansion is O(2ⁿ) and needs no minimiser at all, so it stays exact to 12 variables and the truth mask itself reaches 24. The row-by-row minterm and maxterm table stops at 8 variables, and says so where it stops rather than truncating quietly — the two lists and both canonical forms stay exact past it. Minimal forms are exact to 12 variables and near-minimal, badged as a heuristic, to 20. Equally minimal covers are enumerated to 200, and the badge says so when the cap is reached rather than printing a count nothing proved.