Product-of-sums

A product-of-sums (POS) expression is an AND of OR terms — sums are multiplied. It maps to a two-level circuit: a layer of OR gates feeding a single AND gate.

The canonical POS form lists every maxterm where $f=0$:

$f=\prod M(j_1,j_2,\dots )$

For example, if $f(x_1,x_2)$ is $0$ only on row $2$, then

$f=M_2=\overline{x_1}+x_2$

POS is the dual of SOP. The same function can be written either way — pick the form that yields fewer gates after simplification. A truth table with mostly $1$s has fewer maxterms than minterms, so POS will usually be the cheaper starting point; mostly $0$s favors SOP.

To minimize a POS expression with a Karnaugh Map, group $0$ cells (instead of $1$ cells), then complement each group’s literals when reading the term out. A group of $0$s where $C=0,D=1,B=1$ becomes the maxterm $C+\overline{D}+\overline{B}$.

In a circuit, a POS form gives two levels of gates after the input NOTs: ORs first, then ANDs. Same propagation behavior as SOP, just inverted gate roles. As with SOP, going to more levels via Multilevel synthesis can reduce fan-in at the cost of latency.