Type any boolean expression and see its standard product-of-sums form instantly here. Choose variable order, review maxterms, then download clean reports anytime you need.
Standard POS (canonical Product of Sums) is formed by multiplying maxterms for every truth-table row where the function equals 0. For a row with variable values, each maxterm uses the variable uncomplemented when the value is 0, and complemented when the value is 1.
| A | B | C | Expression | F | Maxterm when F=0 |
|---|---|---|---|---|---|
| 0 | 0 | 0 | A*(B + C') | 0 | (A + B + C) |
| 1 | 0 | 1 | A*(B + C') | 0 | (A' + B + C') |
| 1 | 1 | 0 | A*(B + C') | 1 | — |
| 1 | 1 | 1 | A*(B + C') | 1 | — |
Standard Product-of-Sums (POS) is the canonical way to express a boolean function as an AND of sum-clauses. Each clause is a maxterm, and the complete POS is uniquely determined once variable order is fixed. Canonical POS is especially useful when you need a reproducible, table-driven form rather than an algebraically simplified one.
The conversion is data-driven: evaluate the expression for every input combination, then focus on the rows where the output is 0. For each 0-row, build one maxterm that becomes 0 only for that same assignment. Multiplying all such maxterms produces a function that matches the original truth table exactly.
Π-notation compactly lists the indices of maxterms: F = Π M(i). Indices are binary numbers created from your chosen variable order (most-significant to least-significant). Changing the order does not change the logic, but it does change row indexing and the displayed maxterm list, so consistent ordering matters in reports.
Maxterms follow a consistent rule: if a variable is 0 in the 0-row, it appears uncomplemented in the sum; if it is 1, it appears complemented (for example, A'). This is the opposite of canonical SOP minterms, and mixing the rules is a common source of mistakes during manual conversion.
Engineers write logic in many styles: + or OR for addition-like OR, adjacency for AND, and different NOT symbols. This calculator accepts multiple conventions and can insert implicit AND to interpret expressions like A(B+C). Clean syntax reduces parsing ambiguity and improves consistency across teams.
XOR is not naturally “sum” or “product” in the canonical sense, but it is still a valid boolean operator. The safest approach is truth-table evaluation: compute outputs for all combinations, then generate maxterms from the 0-rows. This guarantees correctness even when the input uses XOR-heavy constructions.
Canonical conversion grows exponentially with variables: 2^n rows. That is why the tool enforces a practical variable limit; it keeps the page responsive and the export files readable. For larger circuits, consider decomposing expressions into smaller blocks, or use simplification before canonical expansion.
Canonical POS is valuable in education, documentation, and verification because it is unambiguous. CSV exports fit spreadsheets and automated checks, while PDF exports are convenient for sign-offs and lab reports. Keeping the expression, variable order, and Π-notation together creates a clean audit trail for design reviews.
It is the canonical product of maxterms formed from every truth-table row where the output equals 0. It is unique for a fixed variable order and matches the function exactly.
The logic result does not change, but row indices and the Π M(i) list depend on the chosen order. Use a consistent order to compare results across documents and tools.
If a variable is 0 in that row, include it uncomplemented in the sum. If it is 1, include its complement. The resulting sum evaluates to 0 only for that row.
Yes. Enable “implicit AND” to interpret adjacency such as AB or A(B+C) as multiplication. If you prefer explicit operators, write A*B or A AND B.
No. It generates the canonical POS by evaluating the full truth table. This prioritizes correctness and reproducibility over algebraic minimization.
Canonical conversion requires 2^n evaluations, so very large n becomes slow and produces huge outputs. Split the logic into subexpressions or simplify first, then convert.
Check that the truth-table output column matches your expected behavior, confirm the Π M(i) indices align with your variable order, and spot-check a few maxterms by substituting input values.
Accurate POS conversion helps verify digital logic designs fast.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.