Minimal Disjunctive Normal Form

The truth table of a -order Boolean function has rows. Its disjunctive normal form extracts all rows that return True. The prime implicants for these cases consolidate rules by introducing so-called "don't care" inputs that specify cases that do not depend on the value of a particular input. The minimal disjunctive form of a Boolean function is found by taking the smallest number of prime implicants.

The left plot presents a randomly generated truth table. The red vertical line separates the input bits on the left from the output on the right. The second plot tabulates the disjunctive normal form that includes only those cases which return True. The third plot shows all prime implicants and marks "don't care" inputs by cyan-colored squares. The right plot shows the minimal disjunctive normal form.