Product of sums maxterm calculator

Enter the number of input variables and those zero-row indices, and this calculator converts every index into its binary assignment, writes the corresponding maxterm, and multiplies the sum terms into one canonical POS expression. The result also includes compact product-of-maxterms notation, ordered indices, variable names, and a term-by-term breakdown that is ready to check or copy into combinational logic work.

Turn zero rows into canonical maxterms

Canonical POS starts with the truth-table rows for which the Boolean function equals zero. Number the rows in ordinary binary order, with the first variable as the most significant bit, and submit those row numbers as the maxterm list. For each selected row, the calculator pads the binary representation to the declared variable count and constructs one sum containing every variable exactly once. A zero bit produces the variable without complementation; a one bit produces the complemented variable, shown with an apostrophe. This polarity is chosen because the resulting sum must evaluate to zero on that specific assignment. For example, binary 101 for variables A, B, and C becomes (A' + B + C'). The calculator sorts the submitted indices before building the result, so an unordered input still produces the conventional ascending maxterm order. Duplicate indices are rejected because a canonical maxterm list represents a set of truth-table rows, and repeating a factor adds no logical information. The term breakdown keeps the source index, padded binary row, literals, and completed parenthesized sum together for straightforward verification.

Read and verify the product of sums result

The main expression places all generated maxterms next to one another, which denotes Boolean multiplication: each parenthesized OR term is combined by AND. The accompanying notation uses ΠM followed by the sorted zero-row indices, the compact form commonly used in logic courses, specifications, Karnaugh-map exercises, and digital design documentation. Verification is mechanical. Select any listed index, assign its binary bits to the variables, and inspect the matching sum term. Every literal in that term becomes zero, making the term zero and therefore forcing the full product to zero. For a row not in the list, no maxterm is designed to vanish on that complete assignment, so every included sum remains one and the product evaluates to one. This confirms that the expression represents exactly the supplied zero set. The output retains variable ordering because changing that order changes the numeric interpretation of every truth-table row. When custom names are supplied, they must be unique, valid identifiers, and equal in count to the declared variables. Otherwise, predictable default names are generated automatically for a copy-ready result.

Use canonical POS in a logic design workflow

Canonical product of sums is an exact representation, not necessarily the smallest circuit. It is especially useful as a dependable bridge between a truth table and later optimization. Start by confirming the zero rows, generate the POS expression, and use the detailed terms to catch index, bit-order, or complement mistakes before simplifying. You can then group zeros on a Karnaugh map, apply Boolean algebra, or feed the expression into a separate minimization process. Keeping the canonical form beside the minimized form gives reviewers a traceable reference for proving that optimization did not alter the function. The method also helps when implementing active-low behavior, checking decoder-based designs, or documenting test vectors, because each maxterm points directly to one failing or disabled input assignment. Inputs are bounded to sixteen variables and indices must fit the corresponding truth-table range, preventing ambiguous truncation or silently ignored high bits. Computation is deterministic and local to the request: it uses no network, stored state, random values, or clock. Browser use is free, while automated API execution costs $0.002 per request.