Reed–Muller expansion

In Boolean logic, a Reed–Muller (or Davio) expansion is a decomposition of a boolean function.

For a boolean function $f(x_1,\ldots,x_n)$ we set with respect to $x_i$:

$\begin{align} f_{{x_i}}(x) & = f(x_1,\ldots,x_{i-1},1,x_{i+1},\ldots,x_n) \\[3pt] f_{\overline{x}_i}(x)& = f(x_1,\ldots,x_{i-1},0,x_{i+1},\ldots,x_n) \\[3pt] \frac{\partial f}{\partial x_i} & = f_{x_i}(x) \oplus f_{\overline{x}_i}(x)\, \\ \end{align}$

as the positive and negative cofactors of $f$, and the boolean derivation of $f$.

Then we have for the Reed–Muller or positive Davio expansion:

$f = f_{\overline{x}_i} \oplus x_i \frac{\partial f}{\partial x_i}.$

Similar to binary decision diagrams (BDDs), where nodes represent Shannon expansion with respect to the according variable, we can define a decision diagram based on the Reed–Muller expansion. These decision diagrams are called functional BDDs (FBDDs).

References

• Kebschull, U. and Rosenstiel, W., Efficient graph-based computation and manipulation of functional decision diagrams, Proceedings 4th European Conference on Design Automation, 1993, pp. 278–282