Minimization of Normal Forms

While it is easy (if time consuming) to generate a DNF formula from a truth table, this transformation is often expensive. The construction for DNFs uses a sub-term for every assignment with output true. This can be very inefficient - a function with $n$ inputs which simply outputs the first input would have $2^{n-1}$ sub-terms when a single variable suffices for the formula. For example,

$$\begin{eqnarray*} (A \land B \land C) \lor (A \land B \land \lnot C) \lor (A \la... ...lnot B \land C) \lor (A \land \lnot B \land \lnot C) & = & A \\ \end{eqnarray*}$$

Minimization of CNF formulas will not be discussed, but the same ideas are applied in essentially the same manner.