Reed-Muller expansion

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In logic synthesis Reed-Muller (or Davio) expansion is a decomposition of a boolean function.

For a boolean function $f (x 1,..., x n)$ we set with respect to $x i$ :

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

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 the 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).

[edit] 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